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# Spectroscopy of diagnostically-important magnetic-dipole lines in highly-
charged 3dn ions of tungsten
Yu. Ralchenko yuri.ralchenko@nist.gov I.N. Draganić Current address: Oak
Ridge National Laboratory, Oak Ridge TN 37831-6372 D. Osin J.D. Gillaspy J.
Reader National Institute of Standards and Technology, Gaithersburg, Maryland
20899-8422
###### Abstract
An electron beam ion trap (EBIT) is used to measure extreme ultraviolet
spectra between 10 nm and 25 nm from highly-charged ions of tungsten with an
open $3d$ shell (W XLVIII through W LVI). We found that almost all strong
lines are due to the forbidden magnetic-dipole (M1) transitions within
$3d^{n}$ ground configurations. A total of 37 spectral lines are identified
for the first time using detailed collisional-radiative (CR) modeling of the
EBIT spectra. A new level-merging scheme for compactification of rate
equations is described. The CR simulations for Maxwellian plasmas show that
several line ratios involving these M1 lines can be used to reliably diagnose
temperature and density in hot fusion devices.
###### pacs:
## I Introduction
Spectroscopy of highly-charged ions of high-Z elements is currently the
subject of extensive research. From a theoretical viewpoint, the accurately
measured wavelengths, energy levels and transition probabilities provide
crucial tests for advanced theories of atomic structure in a regime where
relativistic and quantum-electrodynamic effects become very strong. As for
applications, since tungsten is currently considered to be a primary candidate
for the plasma-facing material in the ITER divertor region Hawryluk09 , the
spectra of its ions in a wide range of wavelengths are being studied under
various conditions. It is not surprising, therefore, that a large number of
research papers on the spectra of tungsten ions measured with electron beam
ion traps (EBIT) 11850EL ; 9743EL ; 11159EL ; 9064EL ; 8472EL ; 9000EL ;
12286EL ; 12366EL ; 8219TP ; 14870EL ; PhysRevA.81.012505 ; 15007EL , tokamaks
12101EL ; 14670EL ; 15280EL , stellarators 15343EL and other high-
temperature-plasma devices were published over the last decade. A detailed
compilation of the recent results on spectral lines and spectra of W can be
found in Refs. 12177EL ; 14671EL .
In the ITER plasma, the tungsten ions will be transported from the relatively
cold divertor region to the plasma core with temperatures on the order of 20
keV. Although considerable efforts are to be spent to minimize radiative power
losses due to emission from highly-charged ions of W, very useful information
for plasma diagnostics can be derived from isolated spectral lines. For
instance, the electron temperature $T_{e}$ can be easily found from the ratios
of strong lines from different ions through the dependence of the ionization
balance on $T_{e}$, and the ion temperature can be derived from the line
shapes. Determination of the electron density $n_{e}$ from spectral lines,
however, is not as straightfoward. Most often it involves a comparison of
allowed and forbidden lines, and thus this technique relies upon knowledge of
wavelengths and transition probabilities of the involved spectral lines. At
high densities, when level populations approach the local thermodynamic
equilibrium (LTE), or Boltzmann, limit, forbidden lines with transition
probabilities many orders of magnitude smaller than those for allowed
electric-dipole (E1) transitions are too weak to be observed in the spectrum.
In low density plasmas, however, the populations of the excited levels which
decay only via forbidden transitions can be relatively high and therefore
result in strong intensities. For each forbidden line there exists a
transition range of electron densities where electron collisions are
comparable to the radiative decay rate. It is in this range of densities that
one may hope to use a particular forbidden line for density diagnostics.
Currently, more than 80 forbidden lines in tungsten ions, from W28+ to W57+,
are known from experimental measurements ASD . The high-multipole lines from W
ions were observed, for instance, in x-rays 9823EL ; 10172EL ; 11848EL ;
9000EL ; PhysRevA.81.012505 , extreme ultraviolet (EUV) 12101EL ; 8219TP ;
12366EL , vacuum ultraviolet 12340EL and ultraviolet (UV) 11850EL ; 9743EL
ranges of spectra in tokamaks and EBITs. The probabilities of forbidden
transitions show very strong increase with the ion charge while the
collisional damping of spectral lines becomes less effective due to a decrease
of cross sections. As a result, the forbidden lines are more prominent in the
spectra of multiply-charged ions.
Forbidden transitions in highly-charged ions are also a subject of active
theoretical research with emphasis on their use in plasma diagnostics. The
visible/UV magnetic-dipole (M1) $J=2-3$ line in Ti-like ions was analyzed for
density diagnostics in hot plasmas since the pioneering work of Feldman et al.
10637EL . Continuing this work, Feldman et al. 7416TP ; 11893EL performed a
systematic study of density-sensitive M1 lines in Ti-like ions and in various
N-shell ions. Recently Jonauskas et al. 8699TP calculated wavelengths and
transition probabilities of M1 lines in $4d^{n}$ configurations of W ions
using large-scale configuration-interaction methods. Also, Quinet et al.
8768TP performed Hartree-Fock calculations of allowed and forbidden
transitions in W I–III, addressing in particular the diagnostics of fusion
plasmas. An extensive calculation of atomic characteristics of eight
isoelectronic sequences of tungsten ions in a broad range of wavelengths and
transitions was recently performed by Safronova and Safronova using the
relativistic many-body perturbation theory (RMBPT) 8725TP . The number of
theoretical works on spectral lines and transition probabilities in ions of
tungsten is too large to cite here, so we refer the reader to the
bibliographic databases at the National Institute of Standards and Technology
(NIST) for an extensive list of publications on tungsten NISTbib .
An example of a density-sensitive line ratio in highly-charged tungsten ions
can be provided by the ratio of electric-quadrupole (E2) and magnetic-octupole
(M3) lines in Ni-like W46+ 12263EL . These two close lines at about 0.793 nm
are due to two $3d^{10}$–$3d^{9}4s$ parity-conserving transitions. They were
experimentally resolved only recently PhysRevA.81.012505 , although the
unresolved spectral feature was known for several years from tokamak 9823EL
and EBIT 9000EL measurements. A large difference in transition probabilities
for these lines, on the order of $10^{6}$, results in a different response to
collisional destruction of level populations. As was shown in Ref. 12263EL ,
the E2/M3 line ratio in W46+ can be used for density diagnostics in the range
of typical values of $n_{e}$ in tokamaks.
The goal of the present work is to study the magnetic-dipole transitions
within the ground configurations of the $3d^{n}$ ions of tungsten using the
NIST EBIT. Previously we reported several EUV lines in Co-, Ca- and K-like
ions 12286EL ; 12366EL within configurations $3d^{9}$, $3d^{2}$ and $3d$,
respectively. Here we extend our measurements to include the remaining ions of
tungsten with open $3d$ shell. Using detailed collisional-radiative (CR)
modeling, we identify the measured spectral lines in the EUV range of spectra
between 10 nm and 25 nm. In addition, we perform CR simulations to explore
potential use of the newly identified lines for diagnostics of hot fusion
plasmas.
The paper is organized as follows. Section II describes the experiment and
measurement of the spectra. Details of the CR modeling are presented in
Section III. We then discuss the identification of the new M1 lines. Section V
presents the analysis of the line ratios that can be used for density
diagnostics in fusion plasmas. Finally, the last Section summarizes our
conclusions.
## II Experiment
The measurements of spectra from the $3d^{n}$ ions of tungsten were performed
at the NIST EBIT facility Gillaspy_1997 using a grazing-incidence EUV
spectrometer Blagojevic_2005 . The photons were collected by a spherical gold-
coated mirror. A 1:1 image of the EBIT plasma column was focused onto the
spectrometer entrance slit. The mirror center was at an equal distance of 480
mm from the EBIT axis and the spectrometer entrance slits. The photons were
dispersed with a reflection flat-field grating with 1200 lines/mm. The grazing
incidence angle for both the mirror and the grating was 3∘. The slit width was
kept at 500 $\mu\text{m}$ resulting a constant resolving power of about 350.
The EUV spectra were directly recorded by a liquid nitrogen cooled back-
illuminated charge-coupled device (CCD) that was placed in the focal plane of
the grating at a distance of 235 mm. The CCD detector has an array of
1340$\times$400 pixels (20$\mu\text{m}\times\text{20}\mu\text{m}$ each). A
detailed description of our EUV spectroscopic system can be found in
Blagojevic_2005 .
The spectra were measured in two separate runs, one in 2008 (run A) and
another in 2010 (run B). The nominal electron beam energies in run A were 4500
eV, 4750 eV, 5000 eV, 5250 eV, 5500 eV, 6000 eV, and 7000 eV, and the observed
spectra were in the range between 4.5 nm and 19.5 nm. A theoretical analysis
of the spectra indicated that some additional lines may have longer
wavelengths and therefore the second run of measurements was initiated. The
beam energies for run B were selected to be complementary to those for run A,
namely, 4665 eV, 4840 eV, 5155 eV, 5355 eV, 5755 eV, and 6500 eV, and the
observed spectral window was shifted to 8 nm to 26 nm by translating the
detector in the focal plane. The electron beam current for both runs was 150
mA and the trap depth was approximately 220 V. The trap was emptied and
reloaded every 11 s.
The measured spectra of tungsten were calibrated with lines from lighter
elements. Reference spectra of Ne, Ar, O, and Fe were measured at several
energies between 2 keV and 9 keV for run A. The calibration of spectra for run
B was performed with lines from N, O, Fe and Kr at beam energies between 1 keV
and 16 keV. Both gas injection gas_injection and metal vapour vacuum arc ion
source (MEVVA) MEVVA systems were utilized in the calibration runs. The
observed calibration lines were fitted with the statistically-weighted
Gaussian line profiles. The calibration curve was a fourth-order polynomial
fit of the line centers (CCD pixel number) to the known wavelengths. The
weighting in the fit to the calibration curve was based on the quadrature sum
of the statistical uncertainty of our observation of the calibration line
center, the accuracy of the calibration line wavelength, and estimated
systematic measurement uncertainty. When a wavelength was measured at various
beam energies, the final wavelength was taken to be the weighted average of
the corresponding values (with exceptions noted below), while the total error
in the final wavelength was taken to be the quadrature sum of the total
uncertainty from the calibration curve and the reduced statistical uncertainty
from the average at various energies. The statistical uncertainties in the
line positions were typically less than 0.001 nm. The final accuracy of the W
spectral lines was 0.003 nm.
The measured spectra for tungsten are shown in Figs. 1 (beam energies $E_{B}$
= 4500 eV to 5250 eV) and 2 (beam energies $E_{B}$ = 5355 eV to 7000 eV). The
run-A spectra are shown in black and the run-B spectra are presented in red
(color online only). The spectral region in the figures is limited to
$\lambda$= 10 nm to 20 nm since almost all M1 lines from the $3d^{n}$ ions are
within this range. Only one line, $\lambda\approx$ 21.203 nm in the V-like
ion, was found above 20 nm, and therefore we do not show the run B spectra at
longer wavelengths. The identified transitions in various ions of tungsten are
indicated by vertical dashed lines in the plots. The measured spectra also
contain a few impurity lines from oxygen (e.g., at 15 nm) and xenon, which are
marked by asterisks. The highest-energy spectrum of $E_{B}$ = 7000 eV also
shows a few lines from Ar- and Cl-like ions which have already been identified
in our previous work 12366EL .
Figure 1: Tungsten spectra between 10 nm and 20 nm for beam energies between
4500 eV and 5250 eV. The identified transitions are indicated by vertical
dashed lines. The spectra from run A are shown in black and the spectra from
run B are shown in red. Asterisks show the strongest impurity lines. Figure 2:
Tungsten spectra between 10 nm and 20 nm for beam energies between 5355 eV and
7000 eV. The identified transitions are indicated by vertical dashed lines.
The spectra from run A are shown in black and the spectra from run B are shown
in red. Asterisks show the strongest impurity lines.
## III Collisional-radiative modeling of EBIT spectra
Generally, identification of measured spectral lines greatly benefits from
applying methods that include comparisons of different physical parameters,
such as wavelengths and intensities. While atomic structure methods for simple
ions can calculate wavelengths with the accuracy at the level of 0.01% or even
better, the simulations for multi-electron ions with open shells may not be as
precise as needed for unambiguous line identification. Another often used
technique in EBIT studies is the analysis of the variation of line intensity
with beam energy. This method, however, would only be of marginal value when
neighboring ions do not differ much in ionization energy, and therefore it is
difficult to uniquely associate a line with a specific ionization stage.
The most reliable analysis of spectral lines can be accomplished with a
collisional-radiative modeling of EBIT plasmas. For any given set of plasma
parameters, such as beam energy and density, the CR simulations can produce a
detailed synthetic spectrum containing lines from a number of ions. A
comparison of calculated line positions and line intensities with the spectra
measured at several energies provides practically unambiguous identification
of spectral lines. This method was successfully used in our previous
publications 9000EL ; 8219TP ; 12286EL ; 14870EL ; 12366EL in order to
analyze and identify dozens of spectral lines from highly-charged heavy ions
in x-ray and EUV regions.
In this work we implement the non-Maxwellian collisional-radiative code NOMAD
NOMAD for the calculation of spectra from tungsten ions in the EBIT. The
solution of the steady-state rate equation
$\hat{A}\cdot\hat{N}=0$ (1)
provides populations of all relevant atomic states and, consequently,
intensities of spectral lines. Here $\hat{N}$ is the vector of populations of
atomic states included in simulations and $\hat{A}$ is the rate matrix
describing physical processes that affect state populations. The detailed
representation of Eq. (1) is:
$\displaystyle\sum_{j>i}{N_{z,j}\cdot\left(A_{z,ij}^{rad}+n_{e}R_{z,ij}^{dx}\right)}+\sum_{j<i}{N_{z,j}n_{e}R_{z,ij}^{ex}}+\sum_{k}{n_{e}R_{z-1,ki}^{ion}}+\sum_{k}{n_{e}R_{z+1,ki}^{rr}}+\delta_{i1}n_{0}R_{z+1}^{cx}$
$\displaystyle-
N_{z,i}\left(\sum_{j<i}{\left(A_{z,ji}^{rad}+n_{e}R_{z,ji}^{dx}\right)}+\sum_{j>i}{n_{e}R_{z,ji}^{ex}}+\sum_{k}{n_{e}R_{z,ki}^{ion}}+\sum_{m}{n_{e}R_{z,mi}^{rr}}+\delta_{i1}n_{0}R_{z}^{cx}\right)=0$
(2)
where $N_{z,i}$ is the population of atomic state $j$ in an ion $z$,
$A_{z,ij}^{rad}$ is the radiative transition probability, $n_{e}$ is the
electron density, $R_{z,ij}^{ex}$, $R_{z,ij}^{dx}$, and $R_{z-1,ki}^{ion}$ are
the rate coefficients for electron-impact excitation, deexcitation and
ionization, respectively, $R_{z,mi}^{rr}$ is the rate coefficient for
radiative recombination, $n_{0}$ is the density of the neutrals in the trap,
and $R_{z}^{cx}$ is the rate coefficient for the charge exchange (CX) between
neutrals and W ions. Unlike Maxwellian plasmas, dielectronic capture (DC) is
normally neglected in EBIT collisional-radiative simulations since this
resonant process requires an accurate match of the beam energy with the DC
energy.
Charge exchange between the W ions and neutrals in the ion trap can affect the
ionization distribution. The Kronecker factor $\delta_{i1}$ in Eq. (III)
indicates that in our model the CX connects only the ground states of adjacent
ions. It is worth noting that since the ion charge is so high ($z\approx$ 50),
the contribution of double CX may be comparable to the single CX. We are
unaware of any calculations or measurements of single or multiple CX cross
sections between highly-charged tungsten and neutral atoms or molecules, and
therefore the Classical Trajectory Monte Carlo cross section scaling CTMC
$\sigma_{cx}=z\cdot 10^{-15}~{}cm^{2}$ (3)
was used in calculations. In fact, the precise value of this parameter is not
very important as it enters the rate equations as a factor in the product
$n_{0}v_{0}\sigma_{cx}$, where $v_{0}$ is the relative velocity between
neutrals and tungsten ions. Neither $n_{0}$ nor $v_{0}$ are accurately known
for our experimental conditions, so that the product $n_{0}v_{0}$ was used as
the only free parameter in CR simulations.
The NOMAD code solves the rate equations (1) using externally calculated basic
atomic data. For the present work the energy levels, radiative transition
probabilities (up to electric and magnetic octupoles) and electron-impact
collisional cross sections were calculated with the relativistic Flexible
Atomic Code (FAC), which is described in detail in Ref. FAC . The relativistic
atomic structure (including quantum-electrodynamics corrections) and collision
methods implemented in FAC are well suited for highly-charged ions of heavy
elements. Our CR model contains 15 ions, from Zn-like W44+ to Si-like W60+.
Since the ions with ionization potentials $I_{z}$ larger than the beam energy
$E_{b}$ have very small populations, only 6 to 8 ionization stages were kept
for each specific simulation. The atomic states for each ion contained single
and double ($\Delta n=0$ only) excitations from the ground configuration.
Single excitations from the $3l$ subshells were included up to $n$ = 5 for
most of the open-shell ions, and up to $n=7$ or 8 for closed-shell (Ni-like)
ions or ions with one or two electrons above closed shells. The double
excitations are included only for $\Delta n=0$ within n=3 shell.
In our previous works on high-Z ions with open $s$ and $p$ shells 9000EL ;
8219TP ; 12286EL ; 14870EL ; 12366EL all singly- and doubly-excited states
included in the CR modeling were atomic levels, i.e., the fine-structure
components. Since open $3d^{n}$ shells allow many more permitted combinations
of angular momenta, the total number of atomic levels due to single and double
excitations from $3s^{2}3p^{6}3d^{n}$ increases drastically and becomes
untractable with available computational facilities. While a typical number of
levels in CR modeling of $4s^{2}4p^{n}$ and $3s^{2}3p^{n}$ ions was on the
order of 1000 per ionization stage, the excitations from $3s^{2}3p^{6}3d^{n}$
can generate 10,000 levels or more, and thus the total number of levels
becomes prohibitively large.
In order to reduce the size of the rate equations to an acceptable level, the
atomic states within each $3d^{n}$ ion were divided into two groups. The first
group was composed of the ground configuration levels $3s^{2}3p^{6}3d^{n}$ and
singly- and doubly-excited levels within the same n=3 shell, i.e.,
$3s^{2}3p^{5}3d^{n+1}$, $3s3p^{6}3d^{n+1}$, $3s^{2}3p^{4}3d^{n+2}$,
$3s3p^{5}3d^{n+2}$, and $3p^{6}3d^{n+2}$. The levels in this first group were
retained without modification as the fine-structure components. The levels in
the second group, i.e., $\Delta n\geq 1$ excitations $3s^{2}3p^{6}3d^{n-1}kl$,
$3s^{2}3p^{5}3d^{n}kl$ and $3s3p^{6}3d^{n}kl$ with $k\geq 4$, were joined into
generalized atomic states, which are referred to as the “superterms” below.
The procedure of level grouping can be exemplified for the $3d^{n-1}kl$
configuration. Each of the atomic levels within this configuration can be
described by the following set of quantum numbers in jj-coupling (FAC level
notations are given in this coupling scheme):
$(((3d_{-}^{a})_{j_{-}},(3d_{+}^{b})_{j_{+}})_{j_{c}},(kl)_{j_{k}})_{J}$ or
simply $((j_{-},j_{+})_{j_{c}},j_{k})_{J}$ where $a+b=n-1$, $j_{-}$ and
$j_{+}$ are the momenta of the $3d_{-}$ and $3d_{+}$ sub-shells, $j_{c}$ is
the total angular momentum of the core $3d^{n-1}$, $j_{k}=l\pm 1/2$ is the
momentum of the optical electron $kl$, and $J$ is the total angular momentum.
Here and below we use notation $l_{\pm}$ for an $l$ electron with $j=l\pm
1/2$. The simplest procedure in level grouping would be to join them according
to the atomic jj-terms $((j_{-},j_{+})_{j_{c}},j_{k})$. However, the reduction
in the total number of states is rather small. Even the next level of
grouping, based on the core momentum $j_{c}$, results in several thousands of
states per ion. Therefore the excited levels in the present work were joined
according to the $(j_{-},j_{+})$ pairs. For the high excited configurations
with a hole in the $3s$ or $3p$ subshell, the levels were combined by the
three momenta $(j_{h},j_{-},j_{+})$ where $j_{h}$ is the hole momentum. We
found that while such grouping significantly reduces the total number of
states per ion, the resulting set of superterms provides a sufficiently dense
representation of atomic structure for each of the $3d^{n}$ ions in our CR
model. The actual reduction in the number of states in an ion can reach an
order of magnitude: for instance, for the V-like ion with $3d^{5}$ ground
configuration the originally generated 10801 levels are reduced to 791 fine-
structure levels and 465 superterms only. This method of including lowest
atomic levels and highly-excited generalized states is similar to the recently
proposed hybrid CR models HYBRID for high-density plasma kinetics, where the
highly-excited levels are combined into even more general groups of states,
namely, configurations or superconfigurations.
Another feature of our calculations is the additional correction of calculated
energies for the $3d^{n}$ levels. For each of these ions we performed another
calculation with the FAC code including, in addition to configurations
mentioned above, all possible excitations within the n=3 complex. (Obviously,
the total complex was already included for $3d^{9}$ and $3d^{8}$ ions.) The
energies of the ground state configurations in the CR model were then replaced
by the newly calculated energies which, although different by a fraction of a
percent only, still improved the agreement with the experimental energies and
wavelengths. A similar procedure was applied in our recent work on EUV spectra
from highly-charged ions of Hf, Ta and Au DragJPB .
The transition probabilities and cross sections between the levels and the
superterms or between the superterms were derived from the FAC results for
transitions between atomic levels using statistical averaging. The collisional
cross sections were then convolved with a 45-eV-wide Gaussian electron-energy
distribution function of the EBIT beam in order to generate the rate
coefficients. The final set of rate equations (1) was solved in the steady-
state approximation for a grid of beam energies between 4300 eV and 7000 eV.
For each energy the charge-exchange parameter $n_{0}v_{0}$ varied between
$10^{13}$ and $3\cdot 10^{14}$ cm-2s-1. The ionization distributions, level
populations, and spectral line intensities were calculated for each
combination of $E_{b}$ and $n_{0}v_{0}$, and the spectral patterns were
compared with the measured spectra to find the best agreement. The best value
of $n_{0}v_{0}$ was found to be about 1014 cm-2s-1. This agrees with our
order-of-magnitude estimates of 107 cm-3 for the density of neutrals and 107
cm/s for the relative velocity.
An example of comparison between the experimental and calculated spectra is
presented in Fig. 3. The simulated spectrum for the beam energy of 5150 eV
with $n_{0}v_{0}=$ $10^{14}$ cm-2s-1 and CX cross section from Eq. (3)
(bottom) agrees very well with the measured spectral pattern at the nominal
beam energy of 5250 eV (top); this energy difference is attributed to the
space charge effects in the trap. The three strong lines at 12.4 nm, 13.0 nm,
and 15.0 nm marked by asterisks are due to xenon and oxygen impurities. Also,
Fig. 3 (top) shows the second order spectrum shifted along the vertical axis
in order to indicate a few relatively weak second-order lines. One can see
that both line positions and line intensities are reproduced in our
simulations very accurately so that most of the lines can be identified from
the visual comparison. Such a good agreement was observed for all cases
considered in the present work.
Figure 3: Comparison of experimental spectrum at the nominal beam energy of
5250 eV (top) and calculated spectrum at 5150 eV and $n_{0}v_{0}$=1014
cm-2s-1. The second order spectrum is shown by the shifted line and the
strongest impurity lines from Xe (12.4 nm and 13.0 nm) and O (15.0 nm) are
indicated by asterisks.
## IV Line identification
Table LABEL:Tab1 presents the strongest identified lines between 10 nm and 25
nm in the experimental spectra of runs A and B. Almost all lines in this table
are the forbidden magnetic-dipole transitions within the ground configurations
$3d^{n}$ of tungsten ions from Co-like W47+ to K-like W55+. The only exception
is the 18.468-nm M1 line within the lowest excited configuration
$3p^{5}3d^{2}$ of the K-like ion. Four of the observed lines, namely, 18.567
nm in the Co-like ion, 17.080 nm and 14.959 nm in the Ca-like ion and 15.962
nm in the K-like ion, are already known from our previous measurements 12366EL
. The other lines in Table LABEL:Tab1 are reported here for the first time. As
discussed above, the uncertainty of the measured wavelengths is $\pm$0.003 nm.
Table 1: Identified magnetic dipole lines in the experimental spectra between 10 nm and 25 nm. The previously known lines are marked by asterisks. The FAC level numbers within ions are given in square brackets. Other theoretical works: a–11741EL , b–7416TP , c–8725TP , d–7367TP . Ion | Sequence | $\lambda_{exp}$ | $\lambda_{th}$ | | | A
---|---|---|---|---|---|---
charge | | (nm) | (nm) | Lower level | Upper level | (s-1)
47 | Co | 18.567* | 18.640,18.6229a | $3d^{9}$ [1] ($d_{+}^{5}$)5/2 | $3d^{9}$ [2] ($d_{-}^{3}$)3/2 | 2.47(6)
48 | Fe | 15.511 | 15.525 | $3d^{8}$ [1] ($d_{+}^{4}$)4 | $3d^{8}$ [6] (($d_{-}^{3}$)3/2,($d_{+}^{5}$)5/2)4 | 1.01(6)
48 | Fe | 17.502 | 17.489 | $3d^{8}$ [2] ($d_{+}^{4}$)2 | $3d^{8}$ [7] (($d_{-}^{3}$)3/2,($d_{+}^{5}$)5/2)1 | 1.71(6)
48 | Fe | 18.878 | 18.956 | $3d^{8}$ [2] ($d_{+}^{4}$)2 | $3d^{8}$ [5] (($d_{-}^{3}$)3/2,($d_{+}^{5}$)5/2)2 | 1.93(6)
48 | Fe | 18.988 | 19.075 | $3d^{8}$ [1] ($d_{+}^{4}$)4 | $3d^{8}$ [4] (($d_{-}^{3}$)3/2,($d_{+}^{5}$)5/2)3 | 3.22(6)
49 | Mn | 14.166 | 14.139 | $3d^{7}$ [1] ($d_{+}^{3}$)9/2 | $3d^{7}$ [10] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)2)7/2 | 1.76(5)
49 | Mn | 15.368 | 15.354 | $3d^{7}$ [1] ($d_{+}^{3}$)9/2 | $3d^{7}$ [9] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)4)11/2 | 1.89(5)
49 | Mn | 17.106 | 17.137 | $3d^{7}$ [1] ($d_{+}^{3}$)9/2 | $3d^{7}$ [5] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)2)9/2 | 2.23(6)
49 | Mn | 18.276 | 18.303 | $3d^{7}$ [3] ($d_{+}^{3}$)5/2 | $3d^{7}$ [10] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)2)7/2 | 2.72(5)
49 | Mn | 18.670 | 18.741 | $3d^{7}$ [2] ($d_{+}^{3}$)3/2 | $3d^{7}$ [8] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)2)1/2 | 2.56(6)
49 | Mn | 18.880 | 18.972 | $3d^{7}$ [1] ($d_{+}^{3}$)9/2 | $3d^{7}$ [4] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)2)7/2 | 3.57(6)
49 | Mn | 19.047 | 19.127 | $3d^{7}$ [2] ($d_{+}^{3}$)3/2 | $3d^{7}$ [7] (($d_{-}^{3}$)3/2,($d_{+}^{4}$)4)5/2 | 1.03(6)
50 | Cr | 12.779 | 12.685 | $3d^{6}$ [1] ($d_{+}^{2}$)4 | $3d^{6}$ [15] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)5/2)3 | 4.05(5)
50 | Cr | 13.137 | 13.105 | $3d^{6}$ [1] ($d_{+}^{2}$)4 | $3d^{6}$ [12] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)5/2)4 | 3.68(4)
50 | Cr | 13.886 | 13.848 | $3d^{6}$ [2] ($d_{+}^{2}$)2 | $3d^{6}$ [15] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)5/2)3 | 4.92(5)
50 | Cr | 14.193 | 14.176 | $3d^{6}$ [2] ($d_{+}^{2}$)2 | $3d^{6}$ [13] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)5/2)2 | 3.20(5)
50 | Cr | 15.363 | 15.363 | $3d^{6}$ [1] ($d_{+}^{2}$)4 | $3d^{6}$ [10] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)3/2)3 | 1.01(6)
50 | Cr | 17.133 | 17.153 | $3d^{6}$ [1] ($d_{+}^{2}$)4 | $3d^{6}$ [8] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)9/2)5 | 6.57(5)
50 | Cr | 17.826 | 17.823 | $3d^{6}$ [3] ($d_{+}^{2}$)0 | $3d^{6}$ [14] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)5/2)1 | 1.32(6)
50 | Cr | 19.239 | 19.317 | $3d^{6}$ [1] ($d_{+}^{2}$)4 | $3d^{6}$ [5] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)9/2)4 | 3.02(6)
50 | Cr | 19.684 | 19.791 | $3d^{6}$ [1] ($d_{+}^{2}$)4 | $3d^{6}$ [4] (($d_{-}^{3}$)3/2,($d_{+}^{3}$)9/2)3 | 2.56(6)
51 | V | 14.531 | 14.511 | $3d^{5}$ [1] ($d_{+}$)5/2 | $3d^{5}$ [9] (($d_{-}^{3}$)3/2,($d_{+}^{2}$)2)7/2 | 1.21(5)
51 | V | 17.215 | 17.260 | $3d^{5}$ [1] ($d_{+}$)5/2 | $3d^{5}$ [5] (($d_{-}^{3}$)3/2,($d_{+}^{2}$)2)3/2 | 3.75(6)
51 | V | 17.660 | 17.709 | $3d^{5}$ [1] ($d_{+}$)5/2 | $3d^{5}$ [3] (($d_{-}^{3}$)3/2,($d_{+}^{2}$)4)7/2 | 1.59(6)
51 | V | 18.996 | 19.098 | $3d^{5}$ [4] (($d_{-}^{3}$)3/2,($d_{+}^{2}$)4)11/2 | $3d^{5}$ [13] (($d_{-}^{2}$)3,($d_{+}^{3}$)9/2)11/2 | 2.31(6)
51 | V | 21.203 | 21.370 | $3d^{5}$ [1] ($d_{+}$)5/2 | $3d^{5}$ [2] (($d_{-}^{3}$)3/2,($d_{+}^{2}$)4)5/2 | 3.40(6)
52 | Ti | 13.543 | 13.521 | $3d^{4}$ [5] (($d_{-}^{3}$)3/2,($d_{+}$)5/2)3 | $3d^{4}$ [17] (($d_{-}^{2}$)0,($d_{+}^{2}$)4)4 | 1.09(6)
52 | Ti | 16.890 | 16.922 | $3d^{4}$ [2] (($d_{-}^{3}$)3/2,($d_{+}$)5/2)1 | $3d^{4}$ [7] (($d_{-}^{2}$)2,($d_{+}^{2}$)4)2 | 4.70(6)
52 | Ti | 17.846 | 17.905 | $3d^{4}$ [3] (($d_{-}^{3}$)3/2,($d_{+}$)5/2)4 | $3d^{4}$ [10] (($d_{-}^{2}$)2,($d_{+}^{2}$)4)5 | 1.65(6)
52 | Ti | 19.319 | 19.427,19.6b | $3d^{4}$ [1] ($d_{-}^{4}$)0 | $3d^{4}$ [2] (($d_{-}^{3}$)3/2,($d_{+}$)5/2)1 | 3.31(6)
52 | Ti | 19.445 | 19.568 | $3d^{4}$ [3] (($d_{-}^{3}$)3/2,($d_{+}$)5/2)4 | $3d^{4}$ [8] (($d_{-}^{2}$)2,($d_{+}^{2}$)4)4 | 3.02(6)
53 | Sc | 12.312 | 12.291 | $3d^{3}$ [1] ($d_{-}^{3}$)3/2 | $3d^{3}$ [7] (($d_{-}^{2}$)0,($d_{+}$)5/2)5/2 | 2.75(5)
53 | Sc | 15.785 | 15.812 | $3d^{3}$ [4] (($d_{-}^{2}$)2,($d_{+}$)5/2)9/2 | $3d^{3}$ [12] (($d_{-}$)3/2,($d_{+}^{2}$)4)11/2 | 1.42(6)
53 | Sc | 16.027 | 16.056 | $3d^{3}$ [1] ($d_{-}^{3}$)3/2 | $3d^{3}$ [6] (($d_{-}^{2}$)2,($d_{+}$)5/2)1/2 | 1.02(6)
53 | Sc | 17.216 | 17.271 | $3d^{3}$ [1] ($d_{-}^{3}$)3/2 | $3d^{3}$ [3] (($d_{-}^{2}$)2,($d_{+}$)5/2)3/2 | 2.74(6)
53 | Sc | 18.867 | 18.971 | $3d^{3}$ [1] ($d_{-}^{3}$)3/2 | $3d^{3}$ [2] (($d_{-}^{2}$)2,($d_{+}$)5/2)5/2 | 3.41(6)
54 | Ca | 14.959* | 14.984,15.010c | $3d^{2}$ [1] ($d_{-}^{2}$)2 | $3d^{2}$ [4] (($d_{-}$)3/2,($d_{+}$)5/2)2 | 1.81(6),1.798(6)c
54 | Ca | 17.080* | 17.147,17.157c | $3d^{2}$ [1] ($d_{-}^{2}$)2 | $3d^{2}$ [3] (($d_{-}$)3/2,($d_{+}$)5/2)3 | 3.68(6),3.683(6)c
54 | Ca | 19.177 | 19.281,19.294c | $3d^{2}$ [2] ($d_{-}^{2}$)0 | $3d^{2}$ [6] (($d_{-}$)3/2,($d_{+}$)5/2)1 | 1.72(6),1.771(6)c
55 | K | 15.962* | 16.003 | $3d$ [1] ($d_{-}$)3/2 | $3d$ [2] ($d_{+}$)5/2 | 2.59(6),1.48(6)d
55 | K | 18.468 | 18.536 | $3p^{5}3d^{2}$ [6] (($p_{+}^{3}$)3/2,($d_{-}^{2}$)2)7/2 | $3p^{5}3d^{2}$ [9] ((($p_{+}^{3}$)3/2,$d_{-}$)3,$d_{+}$)9/2 | 2.99(6)
The atomic levels in Table LABEL:Tab1 are described in jj-coupling, as
calculated by the FAC code. The $l_{\pm}$ groups with total zero angular
momentum are not shown. For instance, the excited level of the $3d^{9}$
configuration of the Co-like ion has six $3d_{+}$ electrons with momentum
projections from $m_{j}$ = -5/2 to $m_{j}$ = +5/2 which are omitted in the
notation. The numbers in square brackets in the level notation columns show
the calculated level number within the corresponding ion (the ground level is
number 1 and so on).
There are several lines from neighboring ions that have very close
wavelengths, e.g., 18.878 nm in Fe-like and 18.880 nm in Mn-like ion, or
15.368 nm in Mn-like and 15.363 nm in Cr-like ion. For such cases the
wavelengths were determined from a spectrum where one of the lines was strong
while the other was weak due to the shifted ionization distribution. The
wavelengths for other lines were obtained by averaging over several measured
spectra.
Table LABEL:Tab1 also shows our calculated wavelengths and transition
probabilities as well as several other theoretical results 11741EL ; 7416TP ;
7367TP ; 8725TP . In most cases the present results agree quite well with the
measured wavelengths although for several lines the difference is rather
large, as much as 0.5 %. This probably reflects difficulties in atomic
structure calculations for such complex ions. Most of the calculated
transition probabilities are between $10^{5}$ s-1 and $5\cdot 10^{6}$ s-1. The
only line with a smaller probability of A = $3.68\cdot 10^{4}$ s-1 is the
13.137 nm J=4–J=4 transition in Cr-like W. Note also that the recent RMBPT
calculations for Ca-like W 8725TP agree with our transition probabilities to
within a few percent.
The energy structure of the $3d^{n}$ ions and population flux analysis explain
why only the forbidden M1 lines are visible between 10 nm ($\Delta E\approx$
124 eV) and 25 nm ($\Delta E\approx$ 50 eV) under EBIT conditions. Normally,
the strongest E1 lines in a collisionally-dominated spectrum are due to
transitions between the ground configuration and lowest excited configuration
of opposite parity. There is, however, a relatively large energy gap between
$3p^{6}3d^{n}$ and $3p^{5}3d^{n+1}$. Our calculations with FAC show that the
$3p-3d$ excitation energy in tungsten ions is about (300–400) eV. Figure 4
shows the calculated energy levels below 500 eV for Co-like through K-like
ions of W. The levels of the ground configuration are represented by
horizontal lines, and the vertical bars show the spread of the
$3p^{5}3d^{n+1}$ configuration. For W47+, W48+,W49+, and W55+, the energy gap
between these configurations is larger than 124 eV, so that the corresponding
E1 lines have wavelengths smaller than 10 nm. For the remaining ions, the
highest $3d^{n}$ levels are rather close to the lower edge of $3p^{5}3d^{n+1}$
manifold. However, only a few possible EUV transitions obey the $|\Delta
J|\leq 1$ selection rule for E1 lines and moreover, those transitions are
greatly suppressed by small branching ratios due to stronger soft x-ray decays
into the lowest levels of the ground configuration.
Figure 4: Calculated energy levels of the $3p^{6}3d^{n}$ configurations in
W47+ through W55+. The vertical bars at the top show the spread of the
$3p^{5}3d^{n+1}$ configurations below 500 eV.
Some transitions between higher-$n$ states, for instance, n=4–5 transitions in
Fe-like and Mn-like ions, also fall into the 10–25 nm range. In low-density
plasmas, the populations of the lowest excited levels of $3d^{n}$ are much
higher that those of the high-excited states and therefore only M1 lines are
strong. When the density is high, the populations approach the Boltzmann
values which are of the same order for all levels in an ion. Since E1 rates
between the higher states are much stronger than the M1 probabilities, only
the E1 lines will be present at high densities.
As mentioned above, four of the spectral lines in table LABEL:Tab1 have
already been observed in our previous experiments. While the new wavelengths
for the lines from Ca- and K-like ions agree with the known values within
experimental uncertainties, the 18.567$\pm$0.003 nm wavelength for the M1
transition in Co-like ion is shifted with respect to our previous value of
18.578$\pm$0.002 nm 8219TP . To address this problem, we reexamined the 4228
eV spectrum of Ref. 8219TP where this line was identified for the first time.
As was pointed out in the original publication, the line was strongly blended
by third-order lines; our current analysis shows that its wavelength should
therefore have been assigned a larger uncertainty. For the lowest beam
energies of the present experiment, the M1 line in the Co-like ion is the
strongest in the spectrum (Fig. 1) so that its wavelength was determined very
reliably. Therefore, the presently measured wavelength of 18.567$\pm$0.003 nm
replaces the previous value of Ref. 8219TP . Our new wavelength agrees better
with the semi-empirical wavelength of 18.541$\pm$0.032 nm 7017EL .
## V Diagnostics with the M1 lines
The diagnostic potential of the M1 lines is based on several features. First,
the intensity ratios for lines from different ions can conveniently serve as a
diagnostics of temperature and ionization balance. It is also helpful that the
spectral window for these lines is rather narrow, which reduces dependence of
spectrometer efficiency on wavelength. Finally, and most importantly, these
forbidden lines can be used to diagnose electron density in fusion plasmas.
Various methods have been developed for spectroscopic diagnostics of electron
density GriemBook . Such techniques make use, for instance, of line intensity
ratios or collisional widths of isolated lines. The line ratio diagnostics is
normally based on comparison of allowed and forbidden lines which are
populated by similar mechanisms (normally, by excitation from the ground
state) and whose radiative decay rates differ by orders of magnitude. For low
densities, when collisional depopulation rates are much smaller than either of
the radiative rates, the intensities of both allowed and forbidden lines vary
linearly with density and therefore their ratio is independent of $n_{e}$.
When collisional rates become comparable with the probabilities of forbidden
transitions, the ratio shows sensitivity to $n_{e}$, typically over one or two
orders in $n_{e}$. A well-known example is the resonance-to-intercombination-
line ratio in He-like ions which has been widely used in plasma diagnostics
Kunze .
While the measured low-density spectra from the $3d^{n}$ ions of tungsten
contain no strong E1 lines, the decay rates for the observed M1 lines vary by
as much as two orders of magnitude, and hence one may expect at least some
sensitivity of line ratios to density variations. In order to analyze the
$n_{e}$-dependence of the M1 line intensities under typical conditions of hot
fusion plasmas, we performed another set of calculations with NOMAD using a
Maxwellian electron energy distribution function and including dielectronic
recombination within the Burgess-Merts-Cowan-Magee approximation BMCM . The
steady-state solutions of the rate equations were determined for electron
densities in the range of $n_{e}$ = $10^{10}-10^{17}$ cm-3 at electron
temperatures $T_{e}^{z}\approx I_{z}$ where $I_{z}$ is the ionization
potential of the ion under study. It was recently shown Wiondist that unlike
the low-Z elements, the typical temperatures of the maximal abundance for
highly-charged W ions are on the order or even larger than the corresponding
ionization potentials, and therefore the condition $T_{e}^{z}\approx I_{z}$ is
well justified for hot steady-state plasmas. Also, since the excitation
energies of the levels within ground configurations are much smaller than
$T_{e}^{z}$, the intensity ratios for the EUV lines would only be marginally
sensitive to electron temperature. Finally, since the intensity ratios involve
only lines from the same ionization stage, the conclusions will not depend on
ionization distribution.
We have already mentioned above that the collisional depopulation of the upper
level is the primary physical process resulting in density sensitivity of a
spectral line. The effect of collisions on level population can be
approximately parameterized by their fraction in the total depopulation rate
(see also Kunze ):
$\alpha_{i}(n_{e})=\frac{\sum_{j}{n_{e}R_{ij}^{col}}}{\sum_{j<i}{A_{ij}^{rad}}+\sum_{j}{n_{e}R_{ij}^{col}}}$
(4)
where $\sum{n_{e}R_{ij}^{col}}$ includes all collisional processes
depopulating level $i$, such as excitation, deexcitation and ionization. The
low-density coronal limit corresponds to $\alpha_{i}\rightarrow 0$, and in the
high-density Boltzmann (LTE) limit $\alpha_{i}\rightarrow 1$. The transitional
region for an atomic state can be defined by the condition
$0.1<\alpha_{i}(n_{e})<0.9,$ (5)
which covers about two orders of magnitude in $n_{e}$, as follows from Eq.
(4). This region of $n_{e}$ determines the range of densities for a spectral
line that are most promising for diagnostics when compared with other lines
that are still in a coronal limit. Figure 5 shows the calculated transitional
regions of $n_{e}$ for the upper levels listed in table LABEL:Tab1 including,
for completeness, the levels of Co-like $3d^{9}$ and K-like $3d$ ions. It
follows from this plot that, for instance, level 9 in the Mn-like ion, from
which the 15.368 nm line originates, decays only radiatively ($\alpha<0.1$) at
densities smaller that $2\cdot 10^{13}$ cm-3, while electron collisions are
the dominant depletion mechanism for this level above $1.5\cdot 10^{15}$ cm-3.
One can see that for almost every ion there exist levels with rather different
ranges of $\alpha_{i}$ variations, and therefore one may expect to find
several pairs of lines whose intensity ratio may be used for density
diagnostics. Moreover, the transitional regions for a large number of levels
overlap with the typical range of $n_{e}$ in core fusion plasmas (marked by
the vertical dashed lines).
Figure 5: Transitional ($0.1<\alpha_{i}<0.9$) regions of densities for the
upper levels of M1 lines from table LABEL:Tab1. The numbers are the calculated
level numbers in a corresponding ion. The range of typical electron densities
of core fusion plasmas is shown by vertical dashed lines.
The best $n_{e}$-sensitive line intensity ratios for ions of tungsten from
W48+ to W53+ are presented in Fig. 6 and are discussed below. The line
intensities were defined as $I=N\cdot A\cdot\Delta E$ where $N$ is the upper
level population (in cm-3), $A$ is the transition probability (in s-1), and
$\Delta E$ is the photon energy (in J).
Figure 6: Density-sensitive line ratios for ions of tungsten from W48+ to
W53+.
### V.1 Fe-like W48+
Among the four identified lines in the Fe-like ion, the ratio of lines
originating from levels 4 and 6 offers the best sensitivity to density
variations. However, the transitional regions of $n_{e}$ for these levels, as
follows from Fig. 5, are not too different and therefore the line ratio would
not change significantly. Indeed, as shown in the top left panel of Fig. 6,
the intensity ratio for the spectral lines at 18.988 nm and 15.511 nm varies
only within a factor of 2.3 between $3\cdot 10^{14}$ cm-3 and 1016 cm-3.
Another complication arises from the overlap between the 18.988 nm lines and
the 19.047 nm line in the next Mn-like ion W49+.
### V.2 Mn-like W49+
Although the seven identified transitions in Mn-like W49+ offer several pairs
of lines of potential usage in density diagnostics (top right panel in Fig.
6), it would be rather challenging for experimentalists to reliably isolate
most of these lines from the other lines that originate from the neighboring
ions. It seems that the 18.670/18.276 ratio that increases by a factor of 5
between $10^{14}$ cm-3 and 1016 cm-3 may actually be the best choice since
these two lines are well isolated in the measured spectrum.
### V.3 Cr-like W50+
Among all $3d^{n}$ ions of tungsten, the Cr-like ion offers the largest number
of $n_{e}$-sensitive line pairs. This is due to both the largest number of
identified lines and the most separated regions of density sensitivity for
different lines. The population of level 12, which is responsible for the
strong and well isolated line at 13.137 nm, starts to deviate from coronal
behavior at electron densities as low as $9\cdot 10^{12}$ cm-3 (Fig. 5). The
strongest radiative transition from this level has probability of only
$3.68\cdot 10^{4}$ s-1 as compared with the typical M1 probabilities of 105
s-1 and larger (Table LABEL:Tab1) and therefore this level becomes
collisionally depleted at lower electron densities. The line ratios involving
the 13.137 nm line, presented in the middle left panel of Fig. 6, show the
strongest dependence on $n_{e}$: for instance, the 19.684/13.137 ratio of two
strong and isolated lines varies by a factor of 30 between 1012 cm-3 and 1016
cm-3. The ratios with other lines (middle right panel) vary within smaller
limits, from 3 to 6.
### V.4 V-like W51+
The transitional regions for the levels of the V-like ion are mostly between
$2\cdot 10^{14}$ cm-3 and $2\cdot 10^{16}$ cm-3. However, electron collisions
start to depopulate level 9 at much lower densities of about $2\cdot 10^{13}$
cm-3 and thus the line ratios involving a strong isolated line at 14.531 nm
can be very sensitive to electron density. For instance, the ratio
18.996/14.531 varies by two orders of magnitude, between 0.1 and 10, for the
electron density range of (1013–1017) cm-3.
### V.5 Ti-like W52+
Figure 5 shows that the transitional regions of $n_{e}$ for the $3d^{4}$
levels in Ti-like ion are very close, and therefore the intensity ratios for
the observed M1 lines do not exhibit significant $n_{e}$-dependence. We only
present a single line ratio 16.890/13.543 (dashed line in the bottom left
panel of Fig. 6) which only changes within a factor of 3 within the discussed
range of $n_{e}$.
### V.6 Sc-like W53+
The two line ratios for the Sc-like ion, 18.867/12.312 and 17.216/12.312, show
significant variation between 1014 cm-3 and 1016 cm-3. The former ratio is not
monotonic, as is seen from Fig. 6: the low-density limit of about 2 drops to
approximately 1.3 before climbing to the value of 10 at the high-density
limit. This dip results from the presence of other metastable levels of
$3d^{3}$ that interact with the upper levels of those lines.
## VI Conclusions
In this paper we presented measurements and identifications of forbidden
magnetic-dipole lines within ground configurations of all $3d^{n}$ ions of
tungsten, from Co-like W47+ to K-like W55+. Two sets of EUV spectra between 10
nm and 20 nm, independently measured two years apart, excellently agreed with
each other, thereby confirming a very good reproducibility of the results. A
total of 37 new spectral lines were identified in the spectra.
The identification of the observed M1 lines was based on extensive
collisional-radiative modeling of spectra from non-Maxwellian plasma of EBIT.
We introduced a new scheme for level grouping in order to reduce the total
number of states in our CR model to a tractable level. The calculated spectra
agree very well with the measured ones, thereby providing unambiguous
determination of line identifications.
The importance of the measured magnetic-dipole lines for spectroscopic
diagnostics of hot plasmas stems from several factors. First, the lines are
located within a rather narrow range of wavelengths, which facilitates their
measurements and reduces dependence on variation of spectrometer efficiency
with wavelength. Second, most of the lines are well isolated with only a few
overlapping to some degree. Third, the intensity ratios of spectral lines from
different ions can be used to infer electron temperature and ionization
balance over a large range of plasma density. Finally, as was shown here with
a detailed collisional-radiative modeling of Maxwellian plasmas, a large
number of line intensity ratios are sensitive to electron density in the range
of magnetic fusion devices. All these features make the M1 lines in $3d^{n}$
ions of tungsten especially useful for plasma diagnostics.
###### Acknowledgements.
We thank J.M. Pomeroy, J.N. Tan, and S.M. Brewer for assistance during the
first experimental phase of this work. This work is supported in part by the
Office of Fusion Energy Sciences of the U.S. Department of Energy and by the
Research Associate Program of the National Research Council.
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|
arxiv-papers
| 2011-02-03T19:16:00 |
2024-09-04T02:49:16.829543
|
{
"license": "Public Domain",
"authors": "Yu. Ralchenko, I.N. Dragani\\'c, D. Osin, J.D. Gillaspy, J. Reader",
"submitter": "Yuri Ralchenko",
"url": "https://arxiv.org/abs/1102.0752"
}
|
1102.1057
|
# Common Fermi Surface Topology and Nodeless Superconducting Gap in
K0.68Fe1.79Se2 and (Tl0.45K0.34)Fe1.84Se2 Superconductors Revealed from Angle-
Resolved Photoemission Spectroscopy
Lin Zhao1, Daixiang Mou1, Shanyu Liu1, Xiaowen Jia1, Junfeng He1, Yingying
Peng1, Li Yu1, Xu Liu1, Guodong Liu1, Shaolong He1, Xiaoli Dong1, Jun Zhang1,
J. B. He2, D. M. Wang2, G. F. Chen2, J. G. Guo1, X. L. Chen1, Xiaoyang Wang3,
Qinjun Peng3, Zhimin Wang3, Shenjin Zhang3, Feng Yang3, Zuyan Xu3, Chuangtian
Chen3 and X. J. Zhou1,∗
1Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China
2Department of Physics, Renmin University of China, Beijing 100872, China
3Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China
(February 5, 2011)
###### Abstract
We carried out high resolution angle-resolved photoemission measurements on
the electronic structure and superconducting gap of K0.68Fe1.79Se2 (Tc=32 K)
and (Tl0.45K0.34)Fe1.84Se2 (Tc=28 K) superconductors. In addition to the
electron-like Fermi surface near M($\pi$,$\pi$), two electron-like Fermi
pockets are revealed around the zone center $\Gamma$(0,0) in K0.68Fe1.79Se2.
This observation makes the Fermi surface topology of K0.68Fe1.79Se2 consistent
with that of (Tl,Rb)xFe2-ySe2 and (Tl,K)xFe2-ySe2 compounds. A nearly
isotropic superconducting gap ($\Delta$) is observed along the electron-like
Fermi pocket near the M point in K0.68Fe1.79Se2 ($\Delta$$\sim$ 9 meV) and
(Tl0.45K0.34)Fe1.84Se2 ($\Delta$$\sim$ 8 meV). The establishment of a
universal picture on the Fermi surface topology and superconducting gap in the
AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) superconductors will provide important
information in understanding the superconductivity mechanism of the iron-based
superconductors.
###### pacs:
74.70.-b, 74.25.Jb, 79.60.-i, 71.20.-b
The latest discovery of superconductivity with a Tc above 30 K in a new
AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) systemJGGuo ; Switzerland ; Mizuguchi ;
MHFang ; GFChen has triggered a new wave of broad interest in the iron-based
high temperature superconductorsKamihara ; ZARenSm ; RotterSC ; MKWu11 ;
CQJin111 . A couple of unique characteristics of the AxFe2-ySe2 system provide
new perspectives that ask for rethinking and re-examination of ideas which
have been proposed for other iron-based superconductors, such as the effect of
Fe vacancy and structural modulation on superconductivityMHFang ; GFChen ;
ZWang ; PZavalij , the nature of the underlying parent compoundMHFang ; GFChen
; QMSi ; YiZhou , the role of electron scattering across the bands between the
zone center $\Gamma$(0,0) and zone corner M($\pi$,$\pi$) on superconductivity,
and the pairing symmetry of this new system with a distinct Fermi surface
topologyYiZhou ; FWang . Band structure calculations of AxFe2-ySe2LJZhang ;
IRShein ; XWYan suggest that the large electron doping in this system leads
to the disappearance of the hole-like Fermi surface pockets around the
$\Gamma$ point that are commonly present in other Fe-based compounds. In this
case, the peculiar Fermi surface topology near $\Gamma$ in the AxFe2-ySe2
superconductors would make it unlikely to have electron scatterings from the
hole-like bands near $\Gamma$ to the electron-like bands near M that are
considered to play an important role in the electron pairing and
superconductivity in the Fe-based superconductors by some theoriesKuroki ;
FeSCMagnetic . Experimental investigations on the electronic structure and the
superconducting gap of AxFe2-ySe2 superconductors are thus crucial for
understanding the physical properties and the pairing mechanism in the iron-
based superconductors.
Figure 1: Fermi surface mapping of K0.68Fe1.79Se2 superconductor (Tc=32 K)(a)
and (Tl0.45K0.34)Fe1.84Se2 superconductor (Tc=28 K) (b) measured by using
h$\nu$=21.2 eV light source. Near the M($\pi$,$\pi$) point, one Fermi surface
sheet is clearly observed which is marked as $\gamma$ (for the sake of
clarity, we refer the four equivalent M points in the first BZ as M1, M2, M3
and M4). Near the $\Gamma$(0,0) point, in addition to a tiny Fermi pocket
observed which is marked as $\alpha$, a weak large Fermi surface sheet (marked
as $\beta$) is also discernable. Figure 2: Band structure and photoemission
spectra of K0.68Fe1.79Se2 measured along typical high symmetry cuts. (a). Band
structure along the Cut 1 crossing the $\Gamma$ point measured by using
h$\nu$=21.2 eV light source; the location of the cut is shown on the top of
Fig. 2a. (b). Corresponding EDC second derivative image of Fig. 2a. The
$\alpha$ band and two Fermi crossings of the $\beta$ band ($\beta_{L}$ and
$\beta_{R}$) are marked. Two inverse-parabolic GA and GB bands are also
marked. (c). Band structure along the Cut 2 crossing the $\Gamma$ point
measured by using h$\nu$=6.994 eV VUV laser. (d). Corresponding EDC second
derivative image of Fig. 2c. (e). Band structure along the Cut 3 crossing the
M2 point measured by using h$\nu$=21.2 eV. (f). Corresponding EDC second
derivative image of Fig. 2e. Two Fermi crossings of the $\gamma$ band
($\gamma_{L}$ and $\gamma_{R}$) are marked. The photoemission spectra (EDCs)
corresponding to the Cut1, Cut2 and Cut3 are shown in (g), (h) and (i),
respectively.
Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to
directly measure the electronic structure and superconducting gap of
superconductorsDamascelli . Some initial ARPES measurements on KxFe2-ySe2 did
not observe Fermi surface near $\Gamma$TQian or observed only a trace of a
tiny electron-like pocket near $\Gamma$YZhang . These seem to be in agreement
with the band structure calculationsLJZhang ; IRShein ; XWYan . However, in
the ARPES measurement on (Tl,Rb)xFe2-ySe2DXMou , two electron-like Fermi
surface sheets are observed near $\Gamma$, with the large one having a similar
size as the one near the electron-like pocket around M. The existence of two
electron-like pockets near $\Gamma$ is also reported in (Tl,K)xFe2-ySe2XPWang
. These results raise an obvious issue on whether the Fermi surface topology
of KxFe2-ySe2 is different from (Tl,Rb,K)xFe2-ySe2; the resolving of this
issue is important for sorting out general electronic structure features in
understanding the Fe-based superconductors.
In this paper, we report the observation of two electron-like Fermi surface
sheets around the zone center $\Gamma$(0,0) in K0.68Fe1.79Se2 superconductor
(Tc=32 K) revealed from our high resolution ARPES measurements. This is
different from the previous ARPES reports that no Fermi pocket or only one
tiny Fermi pocket is present near $\Gamma$ in KxFe2-ySe2TQian ; YZhang . The
observation of two electron-like Fermi pockets near $\Gamma$ makes the Fermi
surface topology of KxFe2-ySe2 consistent with that in (Tl,Rb)xFe2-ySe2DXMou
and (Tl,K)xFe2-ySe2XPWang , thus establishing a coherent picture of Fermi
surface topology in the AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) system. We
observe nearly isotropic superconducting gap ($\Delta$) around the Fermi
pocket near M in K0.68Fe1.79Se2 ($\Delta$$\sim$ 9 meV) and
(Tl0.45K0.34)Fe1.84Se2 ($\Delta$$\sim$ 8 meV). The general picture on the
Fermi surface topology and its associated superconducting gap in the
AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) superconductors will provide key
insights in understanding the iron-based superconductors.
High resolution angle-resolved photoemission (ARPES) measurements were carried
out by using our lab system equipped with a Scienta R4000 electron energy
analyzerGDLiu . We used Helium discharge lamp as the light source which
provides photons with an energy of h$\upsilon$= 21.218 eV (Helium I), as well
as vacuum ultraviolet (VUV) laser which provides h$\upsilon$= 6.994 eV
photons. The energy resolution was set at 10 meV for the Fermi surface mapping
(Fig. 1) and band structure measurements (Fig. 2) and at 4 meV for the
superconducting gap measurements (Figs. 3 and 4). The angular resolution is
$\sim$0.3 degree. The Fermi level is referenced by measuring on a clean
polycrystalline gold that is electrically connected to the sample. The
K0.68Fe1.79Se2 and (Tl0.45K0.34)Fe1.84Se2 single crystals were grown by the
Bridgeman methodGFChen . The composition of the crystals were analyzed by the
energy dispersive X-ray (EDX) spectroscopy. Electrical resistivity and DC
magnetic susceptibility measurements show that the crystals exhibit a sharp
superconducting transition at Tc$\sim$32 K (transition width of $\sim$1 K) for
K0.68Fe1.79Se2 and Tc$\sim$28 K (transition width of $\sim$1 K) for
(Tl0.45K0.34)Fe1.84Se2. The crystal was cleaved in situ and measured in vacuum
with a base pressure better than 5$\times$10-11 Torr.
Fig. 1 shows Fermi surface mapping of K0.68Fe1.79Se2 (Fig. 1a) and
(Tl0.45K0.34)Fe1.84Se2 (Fig. 1b) superconductors. The band structure of
K0.68Fe1.79Se2 along two typical high symmetry cuts are shown in Fig. 2. An
electron-like Fermi surface is clearly observed around M($\pi$,$\pi$), similar
to previous ARPES results on KxFe2-ySe2TQian ; YZhang , (Tl,Rb)xFe2-ySe2DXMou
and (Tl,K)xFe2-ySe2XPWang . Near the $\Gamma$ point, a tiny Fermi pocket
(denoted as $\alpha$) is obvious which is possibly formed by an electron-like
band with its bottom nearly touching the Fermi level. In addition, one can
observe a rather weak but discernable electron-like Fermi surface sheet
(denoted as $\beta$) near $\Gamma$ in both K0.68Fe1.79Se2 (Fig. 1a) and
(Tl0.45K0.34)Fe1.84Se2 (Fig. 1b), with its size being similar to that of the
electron-like pocket near M.
Figure 3: Temperature dependence of the superconducting gap of K0.68Fe1.79Se2
(Tc$\sim$32 K) along the $\gamma$ Fermi pocket near M. (a-e) show
photoemission images taken at different temperatures along a cut near M3; the
location of the cut is marked in the bottom-left inset of (h). (f).
Photoemission spectra measured at different temperatures at the Fermi crossing
kF of the $\gamma$ band, as marked in (a). (g). The corresponding symmetrized
EDCs of (f). (h). Temperature dependence of the measured superconducting gap
(empty red circles). The black dashed line is a curve following the BCS form.
The existence of the $\beta$ Fermi pocket near the $\Gamma$ point in
K0.68Fe1.79Se2 can also be identified from the measured band structure (Fig.
2a and Fig. 2b). We note that the feature of the $\beta$ band (Fig. 2a) and
its associated Fermi surface (Fig. 1a) near $\Gamma$ are rather weak in
K0.68Fe1.79Se2, much weaker than in (Tl,Rb)xFe2-ySe2DXMou ; this is probably
why it was not revealed beforeTQian ; YZhang . We also notice that the band
structure of K0.68Fe1.79Se2 near the $\Gamma$ point (Figs. 2a, 2b, 2c and 2d)
presents some new features that were not observed before. As shown in Fig. 2b,
in addition to the electron-like $\alpha$ band and the electron-like $\beta$
band, at least two more bands are clearly present within the measured energy
window. The observation of the hole-like GB band is consistent with other
measurements on KxFe2-ySe2TQian ; YZhang that is also commonly observed in
(Tl,Rb)xFe2-ySe2DXMou and (Tl,K)xFe2-ySe2XPWang . However, the presence of a
new GA band is very clear in our measurement (Figs. 2b and 2d) which was not
observed in the previous measurementsTQian ; YZhang . The revelation of this
GA band is important when comparing the experimental results with the band
structure calculations and considering electron scatterings between various
bands.
The observation of two electron-like Fermi pockets, $\alpha$ and $\beta$,
around $\Gamma$ in K0.68Fe1.79Se2 is interesting. It is distinct from other
Fe-based compounds where hole-like Fermi surface sheets are expected around
the $\Gamma$ pointDJSingh1111 ; Kuroki . It is also different from band
structure calculationsLJZhang ; IRShein ; XWYan ; YZhang ; TQian and previous
ARPES measurementsYZhang ; TQian on KxFe2-ySe2 that only suggest
disappearance of hole-like Fermi surface sheets near the $\Gamma$ point. It
becomes now consistent with the ARPES measurements on (Tl,Rb)xFe2-ySe2DXMou
and (Tl,K)xFe2-ySe2XPWang to provide a general picture on the Fermi surface
topology in the AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) superconductors.
Now we turn to investigate the superconducting gap in the K0.68Fe1.79Se2 and
(Tl0.45K0.34)Fe1.84Se2 superconductors. Since the $\beta$ feature near
$\Gamma$ is too weak to give reasonable information on the superconducting
gap, we will focus in this paper on the superconducting gap along the $\gamma$
Fermi surface near M. Figs. 3(a-e) show the photoemission images measured on
K0.68Fe1.79Se2 along a cut near M (its location shown in the bottom-left inset
of Fig. 3h) at different temperatures. The photoemission spectra (energy
distribution curves, EDCs) on the Fermi momentum at different temperatures are
shown in Fig. 3f. To visually inspect possible gap opening and remove the
effect of Fermi distribution function near the Fermi level, these original
EDCs are symmetrized to get spectra in Fig. 3g, following the procedure that
is commonly used in high temperature cuprate superconductorsMNorman . As seen
from Fig. 3g, there is a clear superconducting gap opening below Tc$\sim$ 32 K
which is closed above Tc. The superconducting gap size is extracted from the
peak position of the symmetrized EDCs in this paperMNorman (Fig. 3g); it is
$\sim$9 meV at 12 K and its temperature dependence roughly follows the BCS-
type form (Fig. 3h).
In order to measure the momentum-dependence of the superconducting gap, we
took high resolution Fermi surface mapping of the $\gamma$ Fermi pocket at M
for K0.68Fe1.79Se2 (Fig. 4a) and (Tl0.45K0.34)Fe1.84Se2 (Fig. 4e)
superconductors. Fig. 4b shows photoemission spectra around the $\gamma$ Fermi
pocket (Fig. 4a) measured in the superconducting state (T= 15 K); the
corresponding symmetrized photoemission spectra are shown in Fig. 4c. The
superconducting gap (Fig. 4d), extracted by picking up the peak position of
the symmetrized EDCs (Fig. 4c), is nearly isotropic with a size of (9$\pm$2)
meV. By the same procedure, the superconducting gap around the $\gamma$ Fermi
pocket near M for the (Tl0.45K0.34)Fe1.84Se2 superconductor (Fig. 4h) is also
nearly isotropic with a size of (8$\pm$2) meV.
Figure 4: Momentum dependent superconducting gap of K0.68Fe1.79Se2
superconductor (Tc=32 K) and (Tl0.45K0.34)Fe1.84Se2 superconductor (Tc=28 K)
measured along the $\gamma$ Fermi surface sheet near M at a temperature of 15
K. (a). High resolution Fermi surface mapping of K0.68Fe1.79Se2 near M3; the
corresponding Fermi crossings are marked by empty black circles. (b) and (c)
show several typical EDCs along the $\gamma$ Fermi surface and their
corresponding symmetrized EDCs, respectively. (d). Momentum dependence of the
superconducting gap along the $\gamma$ Fermi surface sheet (solid red
circles). (e), (f), (g) and (h) show, respectively, the high resolution Fermi
surface mapping near M2, EDCs along the Fermi surface, their corresponding
symmetrized EDCs, and the obtained momentum-dependent superconudtcing gap for
the (Tl0.45K0.34)Fe1.84Se2 superconductor.
In summary, we have identified two electron-like Fermi pockets near the
$\Gamma$ point in K0.68Fe1.79Se2 and (Tl0.45K0.34)Fe1.84Se2 superconductors.
This has established a consistent picture on the Fermi surface topology in the
AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) superconductors. The distinct Fermi
surface topology in the AxFe2-ySe2 superconductors definitely asks for re-
evaluation of the pairing mechanisms, based on electron scatterings between
the bands near $\Gamma$ and the bands near M, proposed before for other Fe-
based superconductors Kuroki ; FeSCMagnetic . We have observed nearly
isotropic superconducting gap around the $\gamma$ Fermi pocket near the M
point in K0.68Fe1.79Se2 ($\Delta$$\sim$ 9 meV) and (Tl0.45K0.34)Fe1.84Se2
($\Delta$$\sim$ 8 meV). These are consistent with other ARPES
measurementsYZhang ; DXMou ; XPWang to build a general picture on an
isotropic superconducting gap along the $\gamma$ Fermi surface near M. These
results, together with the observation of nearly isotropic superconducting gap
along the $\beta$ pocket near $\Gamma$DXMou ; XPWang , indicate that the
AxFe2-ySe2 superconductors are nodeless in its gap structure, a fact that
appears to favor an s-wave symmetry or a nodeless d-wave symmetryYiZhou ;
FWang . These rich information on the Fermi surface topology and the
associated superconducting gap will provide crucial information and
constraints on understanding the superconductivity mechanism in the Fe-based
superconductors.
XJZ thanks the funding support from NSFC (Grant No. 10734120) and the MOST of
China (973 program No: 2011CB921703).
∗Corresponding author: XJZhou@aphy.iphy.ac.cn
## References
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* (2) A. Krzton-Maziopa et al., arXiv:1012.3637.
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* (21) I. I. Mazin et al., Phy. Rev. Lett. 101, 057003(2008); F. Wang eta l., Phys. Rev. Lett. 102, 047005(2009); A. V. Chubukov et al., Phys. Rev. B 78, 134512(2008); V. Stanev et al., Phys. Rev. B 78, 184509(2008); F. Wang et al., Europhys. Lett. 85, 37005 (2009); I. I.Mazin and M. D. Johannes, Nature Phys. 5, 141(2009).
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* (25) D. X. Mou et al., arXiv:1101.4556.
* (26) X.-P. Wang et al., arXiv: 1101.4923.
* (27) G. D. Liu et al., Rev. Sci. Instruments 79, 023105 (2008).
* (28) M. R. Norman et al., Phys. Rev. B 57, R11093 (1998). The gap size can be obtained by either fitting the symmetrized EDCs using the phenonolological formula as proposed in the paper, or picking up the peak position. We found that, when the selected energy window is large to cover the overall peak, the gap value from the fitting procedure tends to be (2$\sim$3) meV larger than that obtained directly from the peak position. The gap size in this paper is obatined by picking the peak position while it was obtained by fitting the symetrized EDCs over a large energy window in DXMou .
|
arxiv-papers
| 2011-02-05T07:22:25 |
2024-09-04T02:49:16.843043
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lin Zhao, Daixiang Mou, Shanyu Liu, Xiaowen Jia, Junfeng He, Yingying\n Peng, Li Yu, Xu Liu, Guodong Liu, Shaolong He, Xiaoli Dong, Jun Zhang, J. B.\n He, D. M. Wang, G. F. Chen, J. G. Guo, X. L. Chen, Xiaoyang Wang, Qinjun\n Peng, Zhimin Wang, Shenjin Zhang, Feng Yang, Zuyan Xu, Chuangtian Chen and X.\n J. Zhou",
"submitter": "Xingjiang Zhou",
"url": "https://arxiv.org/abs/1102.1057"
}
|
1102.1351
|
# Relativistic Quantum Games in Noninertial Frames
Salman Khan, M. Khalid Khan
Department of Physics, Quaid-i-Azam University,
Islamabad 45320, Pakistan sksafi@phys.qau.edu.pk
###### Abstract
We study the influence of Unruh effect on quantum non-zero sum games. In
particular, we investigate the quantum Prisoners’ Dilemma both for entangled
and unentangled initial states and show that the acceleration of the
noninertial frames disturbs the symmetry of the game. It is shown that for
maximally entangled initial state, the classical strategy $\hat{C}$
(cooperation) becomes the dominant strategy. Our investigation shows that any
quantum strategy does no better for any player against the classical
strategies. The miracle move of Eisert et al [2] is no more a superior move.
We show that the dilemma like situation is resolved in favor of one player or
the other.
PACS: 02.50.Le, 03.67.Bg,03.67.Ac, 03.65.Aa.
Keywords: Quantum games; Unruh effect; Noninertial frames
## 1 Introduction
Quantum game theory, began from the seminal paper of Meyer [1]. It deals with
classical games in the domain of quantum mechanics. For the last few years
much valuable work has been done in this area. Various quantum protocols have
been developed and many classical games have been extended to the domain of
quantum mechanics. It has been shown that quantum superposition and prior
quantum entanglement between the players’ states ensure quantum players to
outperform the classical counterparts through quantum mechanical
strategies[2-9]. Quantum entanglement is one of the powerful tools of quantum
mechanics and plays the role of a kernel in quantum information and quantum
computation. A prior quantum entanglement between two spatially separated
parties increases the number of classical information communicated between
them to twice the number of classical bits communicated in the case of
unentangled state [10, 11]. Recently, the behavior of prior entanglement
shared between two spatially separated parties has been extended to the
relativistic setup in noninertial frames [12, 13, 14, 15, 16, 17] and
interesting results have been obtained. Alsing et al. [12] have shown that the
entanglement between the two modes of a free Dirac field is degraded by the
Unruh effect and asymptotically reaches a nonvanishing minimum value in the
limit of infinite acceleration.
In this paper, we study the influence of Unruh effect on the payoffs function
of the players in the quantum non-zero sum games. In particular, we
concentrate on the quantum Prisoners’ Dilemma [2]. We show that the payoffs
function of the players are strongly influenced by the acceleration of the
noninertial frame and the symmetry of the game is disturbed. It is shown that
under some particular situations, the classical strategy $\hat{C}$ becomes the
dominant strategy and the classical strategy profiles ($\hat{C},\hat{C}$) and
($\hat{D},\hat{D}$) are no more the Pareto optimal and the Nash equilibrium,
respectively. We show that the dominance of the quantum player ceases in the
presence of acceleration of the noninertial frame. In the infinite limit of
acceleration, new Nash equilibrium arises. Furthermore, the dilemma like
situation under every condition, we consider here, is resolved in the favor of
one player or the other or both.
## 2 The Prisoners’ Dilemma
The Prisoners’ Dilemma is a well known non-zero sum game, which has a
widespread applications in many areas of science. Each one of the two players
(Alice and Bob) has to choose one of the two pure strategies simultaneously.
The two pure strategies are called cooperation ($C$) and defection ($D$). The
reward to the action of a player depends not only on his own strategy but also
on the strategy of his opponent. The classical payoff matrix of the game has
the structure given in Table $1$. The first number in each pair of the matrix
corresponds to Alice’s payoff and the second number in a pair to Bob’s payoff.
This is a symmetric noncooperative game where each player tries to maximize
his/her own payoff. The catch of the dilemma is that $D$ is the dominant
strategy, that is, rational reasoning forces each player to defect, and
thereby doing substantially worse than if they would both decide to cooperate.
The quantum form of the Prisoners’ Dilemma was studied for the first time by
Eisert et al [2].
Table 1: Payoff matrix for the classical Prisoners’ Dilemma. The first entry
in a pair of numbers denotes the payoff of Alice and the second entry
represents Bob’s payoff.
Bob: $C$ Bob: $D$ Alice: $C$ $3,3$ $0,5$ Alice: $D$ $5,0$ $1,1$
## 3 Calculation
We consider that Alice and Bob share an entangled initial state
$|\psi_{i}\rangle=\hat{J}|00\rangle_{A,B}$ of two qubits (one for each player)
at a point in flat Minkowski spacetime. The subscripts $A,B$ of the ket stand,
respectively, for Alice and Bob, which means that the first entry in the ket
corresponds to Alice and the second entry corresponds to Bob. The unitary
operator $\hat{J}$ is an entangling operator and is given by
$\hat{J}=\mathrm{exp}[i\frac{\gamma}{2}\hat{D}_{1}\otimes\hat{D}_{1}],$ (1)
where $\gamma\in[0,\pi/2]$ and is a measure of the degree of entanglement in
the initial state. The initial state is maximally entangled when
$\gamma=\pi/2$. The operator $\hat{D}_{1}$ is given by
$\hat{D}_{1}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right),$ (2)
The entangling operator $\hat{J}$ must be symmetric with respect to the
interchange of the two players in order to execute a fair game and must be
known to both players for the knowledge of the degree of entanglement in the
initial state. The initial state, after the entangling operator is applied,
becomes
$|\psi_{i}\rangle=\cos\frac{\gamma}{2}|00\rangle_{A,B}+i\sin\frac{\gamma}{2}|11\rangle_{A,B}.$
(3)
\put(-320.0,220.0){} | |
---|---|---
Figure 1: (color online) Rindler spacetime diagram: A uniformly accelerated
observer Bob ($B$) moves on a hyperbola with constant acceleration $a$ in
region $I$ and a fictitious observer anti-Bob ($\bar{B}$) moves on a
corresponding hyperbola in causally diconnected region $II$. The coordinates
$\tau$ and $\zeta$ are the Rindler coordinates in Bob’s frame, which represent
constant proper time and constant position, respectively. Lines $H^{\pm}$ are
the horizons that represent Bob’s future and past and correspond to
$\tau=+\infty$ and $\tau=-\infty$. Alice and Bob share an entangled initial
state at point $P$ and $Q$ is the point where Alice crosses Bob’s future
horizon.
We consider that Bob then moves with a uniform acceleration and Alice stays
stationary. Each player is equipped with a device which is sensitive only to a
single mode in their respective regions. To cover Minkowski space, two
different sets of Rindler coordinates ($\tau,\xi$) (see Fig. ($1$)) that
differe from each other by an overall change in sign and define two Rindler
regions ($I,II$) are necessary (for detail see [12] and references therein). A
uniformly accelerated particle (observer) in one Rindler region is causally
disconnected from the other Rindler region at the opposite side. Thus an
observer in region $I$ has no access to the information that leaks into region
$II$. The opposite is true for an observer in region $II$. An observer in
region $II$ is called anti-observer (anti-particle) of the observer in region
$I$. The inaccessible information that leaks into the opposite region is as
the system is decohered. The decohrence effects in quantum games in inertial
frames are studied by a number of authors [18, 19, 20]. Particularly, in Ref.
[18] the decoherence effects on quantum Prisoners’ Dilemma has been studied
using various quantum channels. However, the results of our calculations in
the relativistic set up of the game in noninertial frames are different from
the one obtained in Refs. [18, 19]. The creation operator ($a_{k}$) of
particle and annihilation operator ($b_{k}$) of antiparticle in Minskowski
space are related to the creation operator $c_{k}^{I}$ in region $I$ and
annihilation operator $d_{k}^{II{\dagger}}$ in region $II$ by the following
Bogoliubov transformation
$\left(\begin{array}[]{c}a_{k}\\\
b_{k}^{{\dagger}}\end{array}\right)=\left(\begin{array}[]{cc}\cos
r&-e^{-i\phi}\sin r\\\ e^{i\phi}\sin r&\cos
r\end{array}\right)\left(\begin{array}[]{c}c_{k}^{I}\\\
d_{k}^{II{\dagger}}\end{array}\right),$ (4)
where $k$ represents a single mode in each region and $\phi$ is an unimportant
phase that can always be absorbed into the definition of the operators and $r$
is the dimensionless acceleration parameter given by $\cos
r=\left(e^{-2\pi\omega c/a}+1\right)^{-1/2}$. The constants $\omega$, $c$ and
$a$, in the exponential stand, respectively, for Dirac particle’s frequency,
speed of light in vacuum and Bob’s acceleration. The parameter $r=0$ when
acceleration $a=0$ and $r=\pi/4$ when $a=\infty$. We see that the
transformation in Eq. (4) mixes a particle in region $I$ and an antiparticle
in region $II$. A similar transformation exists for an antiparticle’s operator
in region $I$ and a particle’s operator in region $II$ [12]. In fact, a given
Minskowski mode of a particular frequency spreads over all positive Rindler
frequencies ($\omega/(a/c)$) that peaks about the Minskowski frequency [21,
22]. However, to simplify our problem we consider a single mode only in the
Rindler region $I$, an approximation that results into Eq. (4). This is valid
if the observers’ detectors are highly monochromatic that detects the
frequency $\omega_{A}\sim\omega_{B}=\omega$.
From Eq. (4) one can find that
$a_{k}=\cos rc_{k}^{I}-e^{-i\phi}\sin rd_{k}^{II{\dagger}}.$ (5)
From the accelerated Bob’s frame, with the help of Eq. (5), one can show that
the Minkowski vacuum state is found to be a two-mode squeezed state
$|0\rangle_{M}=\cos r|0\rangle_{I}|0\rangle_{II}+\sin
r|1\rangle_{I}|1\rangle_{II}.$ (6)
Note that in Eq. (6) we put $I$ and $II$ in the subscript of the kets to
represent the Rindler modes in region $I$ and region $II$, respectively. Eq.
(6) shows that the noninertial observer that moves with a constant
acceleration in region $I$ sees a thermal state instead of the vacuum state.
This effect is called the Unruh effect [23, 24]. Similarly, using the adjoint
of Eq. (5) one can easily show that the excited state in Minkowski spacetime
is related to Rindler modes as follow
$|1\rangle_{M}=|1\rangle_{I}|0\rangle_{II}.$ (7)
In terms of Minkowski mode for Alice and Rindler modes for Bob, the entangled
initial state of Eq. (3) by using Eqs. (6) and (7) becomes
$\displaystyle|\psi\rangle_{A,I,II}$ $\displaystyle=$
$\displaystyle\cos\frac{\gamma}{2}\cos
r|0\rangle_{A}|0\rangle_{I}|0\rangle_{II}$ (8)
$\displaystyle+\cos\frac{\gamma}{2}\sin
r|0\rangle_{A}|1\rangle_{I}|1\rangle_{II}+i\sin\frac{\gamma}{2}|1\rangle_{A}|1\rangle_{I}|0\rangle_{II}.$
Since Bob is causally disconnected from region $II$, we must take trace over
all the modes in region $II$. This leaves the following mixed density matrix
between the two players,
$\rho_{A,BI}=\left(\begin{array}[]{cccc}\cos^{2}r\cos^{2}\frac{\gamma}{2}&0&0&-i\cos
r\cos\frac{\gamma}{2}\sin\frac{\gamma}{2}\\\
0&\cos^{2}\frac{\gamma}{2}\sin^{2}r&0&0\\\ 0&0&0&0\\\ i\cos
r\cos\frac{\gamma}{2}\sin\frac{\gamma}{2}&0&0&\sin^{2}\frac{\gamma}{2}\end{array}\right).$
(9)
In the quantum Prisoners’ Dilemma, the strategic moves of Alice and Bob are
unitary operators which are given by [2]
$\hat{U}_{N}(\alpha,\theta)=\left(\begin{array}[]{cc}e^{i\alpha_{N}}\cos\frac{\theta_{N}}{2}&i\sin\frac{\theta_{N}}{2}\\\
i\sin\frac{\theta_{N}}{2}&e^{-i\alpha_{N}}\cos\frac{\theta_{N}}{2}\end{array}\right),$
(10)
where, the subscript $N=A,B$ represent Alice and Bob, $\theta\in[0,\pi]$ and
$\alpha\in[0,2\pi]$. If cooperation and defection are associated with the
state $|0\rangle$ and the state $|1\rangle$, respectively, then the quantum
strategy $\hat{C}$ corresponds to $\hat{U}_{N}(0,0)$ and the quantum strategy
$\hat{D}$ corresponds to $\hat{U}_{N}(0,\pi)$. To ensure that the classical
game be a subset of the quantum one, Eisert et al. [2] argued that the
operator $\hat{J}$ must commute with the tensor product of any pair of the
moves $\hat{C}$ and $\hat{D}$. Since fermionic system in noninertial frames is
a physically realizable system, we hope that the encoding of the game might be
practically possible. Once decisions are made, the final density matrix prior
to the measurement becomes [2]
$\rho=\hat{J}^{{\dagger}}\left(\hat{U}_{A}\otimes\hat{U}_{B}\right)\rho_{A,I}\left(\hat{U}_{A}^{{\dagger}}\otimes\hat{U}_{B}^{{\dagger}}\right)\hat{J},$
(11)
where $\hat{J}^{{\dagger}}$ is applied to disentangle the final density
matrix. The expected payoffs of the players are then found by using the
following equation
$P_{N}^{j_{1}j_{2}}=\sum_{i}\$_{N}^{j_{1(i)}j_{2(i)}}\rho_{ii},$ (12)
where $\rho_{ii}$ ($i\in[0,1]$) are the diagonal elements of the final density
matrix and $\$_{N}^{j_{1}(i)j_{2}(i)}$ ($j_{1},j_{2}\in[C,D]$) are the
classical payoffs of the players from Table $1$.
## 4 Results and discussion
The payoffs of the players for unentangled initial state ($\gamma=0$), when
each of them is allowed to play one of the two classical strategies, that is,
$\hat{C}=\hat{U}_{N}(0,0)$ or $\hat{D}=\hat{U}_{N}(0,\pi)$, are given in Table
$2$. The payoffs become the function of $r$.
Table 2: The payoff matrix of the players’ payoffs as a function of the
acceleration of Bob’s frame. The first entry in every pair corresponds to
Alice’s payoff and the second entry corresponds to Bob’s payoff. The initial
state of the game is unentangled and the players are allowed to select a move
from the two pure classical moves.
Bob: $\hat{C}$ Bob: $\hat{D}$ Alice: $\hat{C}$ $3\cos^{2}r,4-\cos 2r$
$3\sin^{2}r,4+\cos 2r$ Alice: $\hat{D}$ $3+2\cos 2r,\sin^{2}r$ $3-2\cos
2r,\cos^{2}r$
One can easily see that the results of Table $2$ reduce to the classical
results of Table $1$ when the acceleration $a=0$ ($r=0$). The presence of
acceleration in the payoff functions of the players disturbs the symmetry of
the game. Neither the strategy profile ($\hat{C},\hat{C}$) nor the strategy
profile ($\hat{D},\hat{D}$) is an equilibrium outcome of the game in the range
of acceleration $0<r\leq\pi/4$. In this range of acceleration, Alice always
wins by playing $\hat{D}$ and always loses by playing $\hat{C}$. The dilemma
like situation is resolved in the favor of Alice. At infinite acceleration
($r=\pi/4$), the strategy profiles ($\hat{C},\hat{C}$) $=$ ($\hat{C},\hat{D}$)
$=$ ($3/2,4$), which means that if Alice plays $\hat{C}$, Bob strategy becomes
irrelevant and he wins all the time. Similarly, the strategy profiles
($\hat{D},\hat{C}$) $=$ ($\hat{D},\hat{D}$) $=$ ($3,3/2$), Alice is victorous,
regardless of what strategy Bob executes. Non of the strategy profiles is
either Pareto optimal or Nash equilibrium.
However, for a maximal entangled state ($\gamma=\pi/2$), the situation is
entirely different. When both the players are restricted only to the classical
region of moves, the payoffs of the players for different strategy profiles
are given by
$\displaystyle P_{A,B}^{CC}$ $\displaystyle=$ $\displaystyle 1+\cos
r+\cos^{2}r+\frac{5}{4}\sin^{2}r,$ $\displaystyle P_{A,B}^{DD}$
$\displaystyle=$ $\displaystyle\frac{1}{8}(17-8\cos r-\cos 2r),$
$\displaystyle P_{A}^{CD}$ $\displaystyle=$ $\displaystyle
P_{B}^{DC}=\frac{1}{2}\cos^{2}\frac{r}{2}(9+\cos r),$ $\displaystyle
P_{A}^{DC}$ $\displaystyle=$ $\displaystyle P_{B}^{CD}=\frac{1}{2}(9-\cos
r)\sin^{2}\frac{r}{2}.$ (13)
\put(-220.0,220.0){} | |
---|---|---
Figure 2: (color online) The payoffs for the maximally entangled initial state
are plotted against the acceleration parameter $r$ of Bob’s frame. The players
are allowed to choose only the classical moves. The subscripts stand for the
players and the superscripts represent a strategy profile.
It can easily be seen from the payoffs function of Eq. (13) that the payoff
matirx is symmetric and that for $r=0$, the classical results are obtained.
Also, the strategy profiles ($\hat{C},\hat{C}$) and ($\hat{D},\hat{D}$) are
equilibrium points for the whole range of the acceleration of Bob’s frame.
However, unlike the classical form and unentangled initial state of the
quantum form in inertial frames of the game, the strategy $\hat{C}$ in this
case becomes the dominant strategy and it always results in payoff $>2.83$ for
all values of the acceleration of Bob’s frame. Moreover, the strategy profile
($\hat{C},\hat{C}$) becomes the Nash equilibrium and the strategy profile
($\hat{D},\hat{D}$) becomes the Pareto optimal of the game for all values of
acceleration $a$. The payoffs of Eq. (13), as function of $r$ for all the
possible strategy profiles, are plotted in Fig. $2$. It can be seen from the
figure that playing $\hat{C}$ is the best option for any player and hence
resolves the dilemma like situation.
Now we consider the case in which the players are allowed to choose any
strategy from the allowed quantum mechanical strategic space. We first
consider the quantum strategy $\hat{Q}$ of Eisert et al. [2], which is given
by
$\hat{Q}=\hat{U}\left(0,\pi/2\right)=\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right).$ (14)
The payoffs of the players when Alice chooses $\hat{Q}$ are given by
$P_{A,B}^{Q\theta_{B}}=\frac{1}{4}[9-\cos r((\cos r\mp 5)\cos\theta_{B}+2\cos
2\alpha_{B}(\cos\theta_{B}+1)\pm 5)],$ (15)
where $\theta_{B}=0$ or $\pi$ gives strategy $\hat{C}$ or strategy $\hat{D}$
respectively. Now, if Bob plays $\hat{C}$, then $P_{A}^{QC}=P_{B}^{QC}$ is an
equilibrium point of the game. If Bob plays $\hat{D}$ then
$P_{B}^{QD}=P_{A}^{CD}>P_{B}^{QC}>P_{A}^{QD}$ for all values of of the
acceleration of the Bob’s frame. This means that the quantum strategy
$\hat{Q}$ does no better for Alice against any of the two classical strategies
of Bob. In other words, $\hat{D}$ is the dominant strategy for Bob against
Alice strategy $\hat{Q}$. The same is true for Alice, if Bob plays the quantum
strategy $\hat{Q}$. In fact the strategy profile ($\hat{Q}$, $\hat{C}$) or
($\hat{C}$, $\hat{Q}$) is a Pareto optimal outcome. However, if both players
execute $\hat{Q}$, the payoffs $P_{A}^{QQ}=P_{B}^{QQ}=P_{A,B}^{CC}$ and hence
the strategy profile ($\hat{Q},\hat{Q}$) is the Nash equilibrium.
Finally we consider the unfair game and the effect of the miracle move of
Eisert et al. [2]. That is, if one player is restricted to the classical
strategic space, then, in the case of inertial frames, the quantum player
outsmarts the classical player all the time if he or she plays the miracle
move $\hat{M}$,
$\hat{M}=\hat{U}\left(\frac{\pi}{2},\frac{\pi}{2}\right)=\frac{i}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right).$ (16)
However, this is not true in the case of noninertial frames. Let Alice plays
$\hat{M}$ and Bob is restricted to the classical strategies, the payoffs of
the players become
$\displaystyle P_{A}^{M\theta_{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(-3\cos^{2}r\sin\theta_{B}+\cos
r(\sin\theta_{B}-7)+9),$ $\displaystyle P_{B}^{M\theta_{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(7\cos^{2}r\sin\theta_{B}+\cos
r(\sin\theta_{B}+3)+9).$ (17)
It can easily be checked that $P_{A}^{M\theta_{B}}<P_{B}^{M\theta_{B}}$
irrespective of what strategy Bob executes. This result is symmetric with
respect to the interchange of the players. That is, if Alice is restricted to
the classical strategies and Bob plays $\hat{M}$, then, the payoffs of the
players in Eq. (17) interchage and $\theta_{B}$ is replaced with $\theta_{A}$.
The quantum player should never go for playing the quantum miracle move of the
inertial frames. The dominance of quantum player over the classical one ceases
in the case of noninertial frames. However, the miracle move $\hat{M}$ always
results in a winning payoff against the quantum move $\hat{Q}$. Logically,
putting $r=0$ in Eq. (17) should give the results of quantum Prisoners’
Dilemma in the inertial frames but this is not so. Eq. (17) gives inverted
results when $r=0$, that is, Alice’s payoff becomes Bob’s payoff of the
inertial frame and vice versa. We have no explanation for this inconsistency.
## 5 Conclusion
We study the influence of Unruh effect on the payoffs function of the players
in the quantum Prisoners’ Dilemma. For unentangled initial state, the Unruh
effect gives rise to an asymmetric payoff matrix in contrast to the payoff
matrix for the classical form and quantum form in the inertial frames of the
game. It is shown that for unentangled initial state, Alice wins all the time
if she plays $\hat{D}$ and loses if she plays $\hat{C}$. As a result non of
the classical strategies profile is either Perato optimal or Nash equilibrium.
We have shown that the Unruh effect limits the dominance of the quantum
player. The classical moves $\hat{C}$ or $\hat{D}$ becomes dominant against
the quantum moves depending on the initial state entanglement. It is shown
that the miracle move $\hat{M}$ of the inertial frames becomes the worst move
that always results in loss against any classical move. Nevertheless, against
the quantum move $\hat{Q}$, it always gives a winning payoff. It is shown that
the dilemma like situation is resolved in favor of one or the other player or
for both players depending on the degree of entanglement in the initial state
of the game.
## 6 Acknowledgment
Salman Khan is thankful to World Federation of Scientists, Geneva,
Switzerland, for partially supporting this work under the National Scholarship
Program for Pakistan.
## References
* [1] Meyer D A 1999 Phys. Rev. Lett. 82 1052
* [2] Eisert J et al 1999 Phys. Rev. Lett. 83, 3077
* [3] Marinatto L and Weber T 2000 Phys. Lett. A 272, 291
* [4] Li H, Du J and Massar S 2002 Phys. Lett. A 306 73
* [5] Lo C F and Kiang D 2004 Phys. Lett. A 321 94
* [6] Flitney A P and Abbott D 2002 Phys. Rev. A 65, 062318
* [7] Iqbal A and Toor A H 2002 Phys. Rev. A 65, 052328
* [8] Flitney A P Ng J and Abbott D 2002 Physica A 314 35
* [9] Goldenberg L, Vaidman L and Wiesner S 1999 Phys. Rev. Lett. 82, 3356
* [10] Bennett C H and Wiesner S J 1992 Phys. Rev. Lett. 69(20), 2881
* [11] Brassard G 2003 Found. Phys. 33(11) 1593
* [12] Alsing P M, Fuentes-Schuller I, Mann R B and Tessier T E 2006 Phys. Rev. A 74 032326\.
* [13] Ling Y et al 2007 J. Phys. A: Math. Theor. 40 9025\.
* [14] Gingrich R M and Adami C 2002 Phys. Rev. Lett. 89 270402\.
* [15] Pan Q and Jing J 2008 Phys. Rev. A 77 024302
* [16] Fuentes-Schuller I and Mann R B 2005 Phys. Rev. Lett. 95 120404
* [17] Terashima H and Ueda M 2003 Int. J. Quantum Inf. 1 93\.
* [18] Chen L K, Ang H, Kiang D, Kwek L.C and Lo C F 2003 Phys. Lett. A 316 317
* [19] Flitney A P and Abbott D 2005 J. Phys. A: Math. Gen. 38 449
* [20] Salman Khan, Ramzan M and Khan M K 2010 Int. J. Theo. Phys. 49 31
* [21] Takagi S 1986 Prog. Theor. Phys. Suppl. 88 1
* [22] Alsing P M, McMahon D and Milburn G J 2004 J. Opt. B: Quantum Semiclass. Opt. 6 834
* [23] Davies P C W 1975 J. Phys. A 8 609
* [24] Unruh W G 1976 Phys. Rev. D 14 870
Figures Captions
Figure $1$. (color online) Rindler spacetime diagram: A uniformly accelerated
observer Bob ($B$) moves on a hyperbola with constant acceleration $a$ in
region $I$ and a fictitious observer anti-Bob ($\bar{B}$) moves on a
corresponding hyperbola in causally diconnected region $II$. The coordinates
$\tau$ and $\zeta$ are the Rindler coordinates in Bob’s frame, which represent
constant proper time and constant position, respectively. Lines $H^{\pm}$ are
the horizons that represent Bob’s future and past and correspond to
$\tau=+\infty$ and $\tau=-\infty$. Alice and Bob share an entangled initial
state at point $P$ and $Q$ is the point where Alice crosses Bob’s future
horizon.
Figure $2$. (color online) The payoffs for the maximally entangled initial
state are plotted against the acceleration parameter $r$ of Bob’s frame. The
players are allowed to choose only the classical moves. The subscripts stand
for the players and the superscripts represent a strategy profile.
|
arxiv-papers
| 2011-02-07T16:25:01 |
2024-09-04T02:49:16.850001
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Salman Khan, M. K. Khan",
"submitter": "Salman Khan",
"url": "https://arxiv.org/abs/1102.1351"
}
|
1102.1353
|
# Quantum Stackelberg duopoly in noninertial frame
Salman Khan, M. Khalid Khan
Department of Physics, Quaid-i-Azam University,
Islamabad 45320, Pakistan sksafi@phys.qau.edu.pk
###### Abstract
We study the influence of Unruh effect on quantum Stackelberg duopoly. We show
that the acceleration of noninertial frame strongly effects the payoffs of the
firms. The validation of the subgame perfect Nash equilibrium is limited to a
particular range of acceleration of the noninertial frame. The benefit of
initial state entanglement in the quantum form of the duopoly in inertial
frame is adversely affected by the acceleration. The duopoly can become as a
follower advantage only in a small region of the acceleration.
PACS: 02.50.Le, 03.67.Bg,03.67.Ac, 03.65.Aa
Keywords: Stackelberg duopoly; Unruh effect; Noninertial frames
Game theory is the mathematical study of interaction among independent, self
interested agents. It emerged from the work of Von Neumann [1], and is now
used in various disciplines like economics, biology, medical sciences, social
sciences and physics [2, 3]. Due to dramatic development in quantum
information theory [4], the game theorists [5-9] have made strenuous efforts
to extend the classical game theory into the quantum domain. The first attempt
in this direction was made by Meyer [10] by quantizing a simple coin tossing
game. Applications of quantum games are reviewed by several authors [11, 12].
A formulation of quantum game theory based on the Schmidt decomposition is
presented by Ichikawa et al. [13].
In quantum games, results different from the classical counterparts are
obtained by using the fascinating feature of quantum mechanics ”the
entanglement”. Recently, the study of quantum entanglement of various fields
has been extended to the relativistic setup [14, 15, 16, 17, 18, 19] and
interesting results about the behavior of entanglement have been obtained.
Alsing et al [14] have shown that the entanglement between two modes of a free
Dirac field is degraded by the Unruh effect and asymptotically reaches a
nonvanishing minimum value in the infinite acceleration.
In this letter, we study the influence of Unruh effect on the payoffs function
of the firms in the quantum Stackelberg duopoly. We show that the payoffs
function of the firms are strongly influenced by the acceleration of the
noninertial frame. It is shown that for small values of acceleration the
duopoly is leader advantage and it becomes the follower advantage in the range
of large values of acceleration. Unlike the quantum form of the duopoly in
inertial frames, the benefit of initial state entanglement is adversely
affected in the noninertial frames. We show that for a maximally entangled
initial state, the Unruh effect damps the payoffs considerably as compared to
the case of unentangled initial state. Furthermore, it is shown that the Unruh
effect limits the validation of the subgame perfect Nash equilibrium outcome
to a particular range of values of the acceleration of the frame. The payoffs
of the firms vanish, irrespective of the initial state entanglement, at a
particular value of the acceleration.
The Stackelberg duopoly is a market game, which is a modified form of the
Cournot duopoly. In the Cournot duopoly, two firms simultaneously put a
homogeneous product into a market and guess that what action the opponent will
take. The Stackelberg duopoly is a dynamic model of duopoly in which one firm,
say firm $A$, moves first and the other firm, say $B$, goes after. Before
making its decision, firm $B$ observes the move of firm $A$. This transforms
the static nature of the Cournot duopoly to a dynamic one. Firm $A$ is usually
called the leader and firm $B$ the follower, on this basis the game is also
called the leader-follower model [20]. In the classical Stackelberg duopoly,
it is assumed that firm $B$ will respond optimally to the strategic decision
of firm $A$. As firm $A$ can precisely predict firm $B$’s strategic decision,
firm $A$ chooses its move in such a way that maximizes its own payoff. This
informational asymmetry makes the Stackelberg duopoly as the first mover
advantage game. The quantum Stackelberg duopoly has been studied under various
circumstances and interesting results have been obtained [21, 22, 23, 24]
We consider two firms, $A$ and $B$, that share an entangled initial state of
two qubits at a point in flat Minkowski spacetime. Then firm $B$ moves with a
uniform acceleration and firm $A$ stays stationary. Let the two modes of
Minkowski spacetime that correspond to firm $A$ and firm $B$ are,
respectively, given by $|n\rangle_{A}$ and $|n\rangle_{B}$. We assume that the
firms share the following entangled initial state
$|\psi_{i}\rangle=\cos\theta|00\rangle_{A,B}+\sin\theta|11\rangle_{A,B}$ (1)
\put(-220.0,220.0){} | |
---|---|---
Figure 1: Rindler spacetime diagram: A uniformly accelerated observer B (firm
B) moves on a hyperbola with acceleration $a$ in region $I$ and is causally
disconnected from region $II$.
where $\theta$ is a measure of entanglement. The state is maximally entangled
at $\theta=\frac{\pi}{4}$. The first entry in each ket of Eq. (1) corresponds
to firm $A$ and the second entry corresponds to firm $B$. From the accelerated
firm $B$’s frame, the Minkowski vacuum state is found to be a two-mode
squeezed state [14]
$|0\rangle_{M}=\cos r|0\rangle_{I}|0\rangle_{II}+\sin
r|1\rangle_{I}|1\rangle_{II},$ (2)
where $\cos r=\left(e^{-2\pi\omega c/a}+1\right)^{-1/2}$. The constant
$\omega$, $c$ and $a$, in the exponential stand, respectively, for Dirac
particle’s frequency, light’s speed in vacuum and firm $B$’s acceleration. In
Eq. (2) the subscripts $I$ and $II$ of the kets represent the Rindler modes in
region $I$ and $II$, respectively, in the Rindler spacetime diagram (see Fig.
($1$)). The excited state in Minkowski spacetime is related to Rindler modes
as follow [14]
$|1\rangle_{M}=|1\rangle_{I}|0\rangle_{II}.$ (3)
In terms of Minkowski modes for firm $A$ and Rindler modes for firm $B$, the
entangled initial state of Eq. (1) by using Eqs. (2) and (3) becomes
$|\psi\rangle_{A,I,II}=\cos\theta\cos
r|0\rangle_{A}|0\rangle_{I}|0\rangle_{II}+\cos\theta\sin
r|0\rangle_{A}|1\rangle_{I}|1\rangle_{II}+\sin\theta|1\rangle_{A}|1\rangle_{I}|0\rangle_{II}.$
(4)
Since firm $B$ is causally disconnected from region $II$, we must take trace
over all the modes in region $II$. This leaves the following density matrix
between the two firms,
$\rho_{A,I}=\left(\begin{array}[]{cccc}\cos^{2}r\cos^{2}\theta&0&0&\cos
r\cos\theta\sin\theta\\\ 0&\cos^{2}\theta\sin^{2}r&0&0\\\ 0&0&0&0\\\ \cos
r\cos\theta\sin\theta&0&0&\sin^{2}\theta\end{array}\right).$ (5)
In the quantum Stackelberg duopoly, each firm has two possible strategies $I$,
the identity operator and $C$, the inversion operator or Pauli’s bit-flip
operator. Let $x$ and $1-x$ stand for the probabilities of $I$ and $C$ that
firm $A$ applies and $y$, $1-y$, are the probabilities that firm $B$ applies,
respectively. The final density matrix is given by [25]
$\displaystyle\rho_{f}$ $\displaystyle=$ $\displaystyle xyI_{A}\otimes I_{B}\
\rho_{A,I}\ I_{A}^{{\dagger}}\otimes
I_{B}^{{\dagger}}+x\left(1-y\right)I_{A}\otimes C_{B}\ \rho_{A,I}\
I_{A}^{{\dagger}}\otimes C_{B}^{{\dagger}}$ (6)
$\displaystyle+y\left(1-x\right)C_{A}\otimes I_{B}\ \rho_{A,I}\
C_{A}^{{\dagger}}\otimes I_{B}^{{\dagger}}$
$\displaystyle+\left(1-x\right)\left(1-y\right)C_{A}\otimes C_{B}\ \rho_{A,I}\
C_{A}^{{\dagger}}\otimes C_{B}^{{\dagger}},$
where $\rho_{A,I}$ is the density matrix given by Eq. (5).
Suppose that the players’ moves in the quantum Stackelberg duopoly are given
by probabilities lying in the range $[0,1]$. In the classical form of the
duopoly, the moves of firms $A$ and $B$ are given by quantities $q_{1}$ and
$q_{2}$, which have values in the range $[0,\infty)$. We assume that firms $A$
and $B$ agree on a function that uniquely defines a real positive number in
the range $(0,1]$ for every quantity $q_{1}$, $q_{2}$ in $[0,\infty)$. Such a
function is given by $1/(1+q_{i})$, so that firms $A$ and $B$ find $x$ and
$y$, respectively, as
$x=\frac{1}{1+q_{1}}\mathrm{,\qquad}y=\frac{1}{1+q_{2}}$ (7)
The payoffs of firms $A$ and $B$ are given by the following trace operations
$P_{A}\left(q_{1},q_{2}\right)=\mathrm{Tr}\left[\rho_{f}P_{A}^{\mathrm{op}}\left(q_{1},q_{2}\right)\right]\mathrm{,\qquad}P_{B}\left(q_{1},q_{2}\right)=\mathrm{Tr}\left[\rho_{f}P_{B}^{\mathrm{op}}\left(q_{1},q_{2}\right)\right],$
(8)
where $P_{A}^{\mathrm{op}}$, $P_{B}^{\mathrm{op}}$ are payoff operators of the
firms and are given by
$\displaystyle P_{A}^{\mathrm{op}}\left(q_{1},q_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{q_{1}}{q_{12}}\left(k\rho_{11}-\rho_{22}-\rho_{33}\right),$
$\displaystyle P_{B}^{\mathrm{op}}\left(q_{1},q_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{q_{2}}{q_{12}}\left(k\rho_{11}-\rho_{22}-\rho_{33}\right),$
(9)
where $\rho_{ii}$ are the diagonal elements of the final density matrix, $k$
is a constant as given in Ref. [20] and $q_{12}$ is given by
$q_{12}=\frac{1}{\left(1+q_{1}\right)\left(1+q_{2}\right)}.$ (10)
The backward-induction outcome in the Stackelberg duopoly is found by first
finding the reaction function $R_{2}\left(q_{1}\right)$ of firm $B$ to an
arbitrary quantity $q_{1}$ chosen by firm $A$. It is found by differentiating
firm $B$’s payoff with respect to $q_{2}$, and maximizing the result for
$q_{1}$ and can be written as
$R_{2}\left(q_{1}\right)=\max P_{B}\left(q_{1},q_{2}\right)$ (11)
Once firm $B$ chooses this quantity, firm $A$ can compute its optimization
problem by differentiating its own payoff with respect to $q_{1}$ and then
maximizing it to find the value $q_{1}=q_{1}^{\ast}$. Using the value of
$q_{1}^{\ast}$ in Eq. (11), we can get the value of $q_{2}^{\ast}$. These
quantities define the backward-induction outcome of the quantum Stackelberg
duopoly and represent the subgame perfect Nash equilibrium. The payoffs of the
firms at the subgame perfect Nash equilibrium can be found using Eq. (8).
\put(-220.0,220.0){} | |
---|---|---
Figure 2: (color online) The payoffs are plotted at the subgame perfect Nash
equilibrium against the acceleration $r$ for unentangled initial state. The
value of $k$ is set to $1$. The solid line represents the payoff of firm $A$
and the dotted line represents the Payoff of firm $B$.
The subgame perfect Nash equilibrium outcome of the duopoly becomes
$\displaystyle q_{1}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}\theta(k\cos^{2}r-\sin^{2}r)}{2(\cos^{2}r\cos^{2}\theta+\sin^{2}\theta)}$
$\displaystyle q_{2}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{4\cos^{2}\theta(k\cos^{2}r-\sin^{2}r)(\cos^{2}r\cos^{2}\theta+\sin^{2}\theta)}{\begin{array}[]{c}(3-k+12\cos
2r+(1+k)\cos 4r)\cos^{4}\theta\\\
-8\cos^{2}\theta((-4+k^{2})\cos^{2}r+k\sin^{2}r)\sin^{2}\theta+16\sin^{4}\theta\end{array}}$
(14)
It is important to note that the result of Eq. (14) for unentangled initial
state ($\theta=0$) reduces to the classical result when we put the
acceleration $r=0$. Similarly the results of Ref. [21] for the maximal
entangled initial state are retrieved for $\theta=\pi/4$ and $r=0$. In the
classical form of the duopoly the subgame perfect Nash equilibrium is a point,
whereas in this case, it is a function of both entanglement angle $\theta$ and
the acceleration $r$ of firm $B$’s frame. The payoffs of the firms at the
subgame perfect Nash equilibrium for unentangled initial state, when $k=1$,
are given as
$\displaystyle P_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\cos^{2}2r\sec^{2}r$ $\displaystyle P_{B}$
$\displaystyle=$ $\displaystyle\frac{\cos^{2}r\cos 2r}{4(3+\cos 2r)}$ (15)
The payoffs of the firms for a maximally entangled initial state, with $k=1$,
become
$\displaystyle P_{A}$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}2r}{8(3+\cos 2r)}$ $\displaystyle P_{B}$
$\displaystyle=$ $\displaystyle\frac{\cos^{2}2r(3+\cos 2r)\sec^{2}r}{32(6+\cos
2r)}$ (16)
\put(-220.0,220.0){} | |
---|---|---
Figure 3: (color online) The payoffs are plotted at the subgame perfect Nash
equilibrium against the acceleration $r$ for maximally entangled initial
state. The value of $k$ is set to $1$. The solid line represents the payoff of
firm $A$ and the dotted line represents the Payoff of firm $B$.
The existence of the Nash equilibrium requires that the firms’ moves
($q_{1}^{\ast}$ and $q_{2}^{\ast}$) should have positive values. It can easily
be checked from Eq. (14) that for both unentangled and maximally entangled
initial states the move of firm $A$ becomes negative for $r\geq\pi/4$. Hence
no Nash equilibrium exists for the values of $r$ at which $q_{1}^{\ast}$
becomes negative. Thus the range of the acceleration in which the acceleration
parameter $r$ is given by $\pi/4\leq r\leq\pi/2$ is not a physically
meaningful range for the Stackelberg duoply. To see how the payoffs are
influenced by the acceleration in its physically meaningful range, we plot it
against the acceleration parameter $r$. In Fig. $2$, we show the plot of the
firms’ payoffs against $r$ for the unentangled initial state. It can be seen
that for smaller values of the acceleration, the duopoly is leader advantage
and the payoffs decrease with the increasing value of the acceleration. At
$r=0.66$ there happens a critical point at which both firms are equally
benefitted. From this point onward, the payoff of firm $A$ rapidly decreases
and becomes zero at $r=0.76$. The duopoly becomes follower advantage in the
region $0.66<r<0.78$. The payoff of the follower firm reaches zero at
$r=0.78$. The payoffs of the firms for the maximally entangled initial state
are plotted in Fig. $3$. It can be seen that the payoffs of the firms are
highly damped as compared to the case of unentangled initial state and the
duopoly is follower advantage for the whole range of the acceleration in which
the Nash equilibrium exists. The payoffs of both firms becomes zero at
$r=0.75$. In Fig. $4$, we plot the payoffs of the firms against the
entanglement angle $\theta$. It is seen that the payoffs decrease with the
increasing degree of entanglement in the initial state. The duopoly is
follower advantage for smaller value of $\theta$ and becomes leader advantage
as the degree of the initial state entanglement increases.
\put(-220.0,220.0){} | |
---|---|---
Figure 4: (color online) The payoffs are plotted at the subgame perfect Nash
equilibrium against the entanglement angle $\theta$. The values other
parameters are chosen as $k=1$, $r=2\pi/9$. The solid line represents the
payoff of firm $A$ and the dotted line represents the Payoff of firm $B$.
In conclusion, we study the influence of Unruh effect on the payoffs function
of the quantum Stackelberg duopoly. We have shown that the Unruh effect limits
the validation of the subgame perfect Nash equilibrium outcome to certain
range of acceleration of firm $B$’s frame. The acceleration damps the payoffs
function both for unentangled and entangled initial states. However, the
damping is heavy when the initial state is maximally entangled and the duopoly
always benefit the firm that moves first. For an unentangled initial state, a
critical point that correspond to a particular value of the acceleration
exists at which both firms are equally benefitted. For larger values of
acceleration the duopoly becomes a follower advantage. We show that
irrespective of the degree of entanglement in the initial state, the payoffs
function vanish when the acceleration of firm $B$ frame reaches to $\pi/4$.
Acknowledgment
Salman Khan is thankful to World Federation of Scientists for partially
supporting this work under the National Scholarship Program for Pakistan.
## References
* [1] von Neumann J 1951 Appl. Math. Ser. 12 36
* [2] Piotrowski E W, Sladkowski J 2002 Physica A 312 208
* [3] Baaquie B E 2001 Phys. Rev. E 64
* [4] Nielson M A ,Chuang I L 2000 Quantum Computation and Quantum Information ( Cambridge: Cambridge University Press)
* [5] Eisert J, Wilkens M and Lewenstein M 1999 Phy. Rev. Lett 83 3077
* [6] Benjamin S C and Hayden P M 2001 Phys. Rev. Lett. 87 0689801
* [7] Marinatto L and Weber T 2001 Phys. lett. A 280 249
* [8] Flitney A P and Abbott D 1999 Phys. Rev. A 65 062318
* [9] Lo C F and Kiang D 2003 Phys. Lett. A 318 333
* [10] Meyer D A 1999 Phys. Rev. Lett. 82 1052
* [11] Cheon T and Tsutsui I 2006 Phys. Lett. A 348 147
* [12] Ichikawa T and Tsutsui I 2007 Ann. Phys. 322 531
* [13] Ichikawa T, Tsutsui I and Cheon T 2008 J. Phys. A: Math. Theor. 41 135303
* [14] Alsing P M, Fuentes-Schuller I, Mann R B and Tessier T E 2006 Phys. Rev. A 74 032326\.
* [15] Ling Y et al 2007 J. Phys. A: Math. Theor. 40 9025\.
* [16] Gingrich R M and Adami C 2002 Phys. Rev. Lett. 89 270402\.
* [17] Pan Q and Jing J 2008 Phys. Rev. A 77 024302\.
* [18] Fuentes-Schuller I and Mann R B 2005 Phys. Rev. Lett. 95 120404\.
* [19] Terashima H and Ueda M 2003 Int. J. Quantum Inf. 1 93\.
* [20] Gibbons R 1992 Game theory for Applied Economists (Princeton University Press, Princeton, NJ)
* [21] Iqbal A and Toor A H 2002 Phys. Rev. A 65 052328
* [22] Zhu X and Kuang L M 2007 J. Phys. A: Math. Theor. 40 7729
* [23] Zhu X and Kuang L M 2008 Commun. Theor. Phys. 49 111
* [24] Khan S, Ramzan M and Khan M K 2010 J. Phys. A: Math. Theor. 43 375301
* [25] Marinatto L and Weber T 2000 Phys. Lett. A 272 291
Figures Captions
Figure $1$. Rindler spacetime diagram: A uniformly accelerated observer firm
$B$ ($B$) moves on a hyperbola with acceleration $a$ in region $I$ and is
causally disconnected from region $II$.
Figure $2$. (color online) The payoffs are plotted at the subgame perfect Nash
equilibrium against the acceleration $r$ for unentangled initial state. The
value of $k$ is set to $1$. The solid line represents the payoff of firm $A$
and the dotted line represents the Payoff of frim $B$.
Figure $3$. (color online) The payoffs are plotted at the subgame perfect Nash
equilibrium against the acceleration $r$ for maximally entangled initial
state. The value of $k$ is set to $1$. The solid line represents the payoff of
firm $A$ and the dotted line represents the Payoff of frim $B$.
Figure $4$. (color online) The payoffs are plotted at the subgame perfect Nash
equilibrium against the entanglement angle $\theta$. The values other
parameters are chosen as $k=1$,$r=2\pi/9$. The solid line represents the
payoff of firm $A$ and the dotted line represents the Payoff of frim $B$.
|
arxiv-papers
| 2011-02-07T16:36:27 |
2024-09-04T02:49:16.855037
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Salman Khan, M.K. Khan",
"submitter": "Salman Khan",
"url": "https://arxiv.org/abs/1102.1353"
}
|
1102.1696
|
# Spinor atom-molecule conversion via laser-induced three-body recombination
H. Jing1,2, Y. Deng1, and P. Meystre2 1Department of Physics, Henan Normal
University, Xinxiang 453007, China
2B2 Institute, Department of Physics and College of Optical Sciences, The
University of Arizona, Tucson, Arizona 85721
###### Abstract
We study theoretically several aspects of the dynamics of coherent atom-
molecule conversion in spin-1 Bose-Einstein condensates. Specifically, we
discuss how for a suitable dark-state condition the interplay of spin-exchange
collisions and photoassociation leads to the stable creation of an atom-
molecule pairs from three initial spin-zero atoms. This process involves $two$
two-body interactions and can be intuitively viewed as an effective three-body
recombination. We investigate the relative roles of photoassociation and of
the initial magnetization in the “resonant” case where the dark state
condition is perfectly satisfied. We also consider the ”non-resonant” regime,
where that condition is satisfied either approximately – the so-called
adiabatic case – or not at all. In the adiabatic case, we derive an effective
non-rigid pendulum model that allows one to conveniently discuss the onset of
an antiferromagnetic instability of an “atom-molecule pendulum,” as well as
large-amplitude pair oscillations and atom-molecule entanglement.
###### pacs:
42.50.-p, 03.75.Pp, 03.70.+k
## I Introduction
Recent years have witnessed rapid advances in the manipulation of the spin
degrees of freedom of ultracold atoms Meystre ; spin ; spin 1 ; spin domains ;
spin-2-2 ; Cr . By magnetically steering two-body collisions, a broad range of
effects has been observed, including atomic magnetism Ho98 ; Ohmi98 ; Law98 ;
Pu99 , coherent spin mixing spin mixing ; spin-2-2 , topological excitations
votex , and an atomic analog of the Einstein-de Haas effect de Haas . The
optical control of atomic spin dynamics has also attracted much experimental
interest Dum ; Chapman ; APB . For example, Dumke et al. Dum and Hamley et
al. Chapman have investigated the photoassociation (PA) diagnosis Dum and PA
spectroscopy Chapman of spin-1 atoms, opening the way to studies of PA-
controlled regular HJ or chaotic J. C. spin dynamics.
In a very recent experiment, the ro-vibrational ground-state molecules were
successfully prepared via the all-optical association of laser-cooled atoms
Inouye , which has triggered the investigation of coherent PA of a wide
variety of ultracold atomic and molecular systems Carr . A result of
particular relevance for the present study is an experiment by Kobayashi et
al., who used a coherent two-color PA technique to create spinor molecules in
a spin-1 atomic Bose condensate APB . In particular, these authors found that
for strong PA couplings the atomic spin oscillations are significantly
suppressed and the dominant process is scalar-like atom-molecule conversion.
That is, only the populations of the spin components that are associated into
molecules are observed to decrease, while the other spin component remains
almost unchanged on the experimentally relevant timescale APB .
In this paper we show that under appropriate two-photon resonance conditions
quantum interferences between optical PA and atomic spin mixing can lead to
the existence of a dark state of the spin-down atoms, which can in turn be
exploited in the stable formation of a spinor atom-molecule pair from three
initial spin-zero atoms. This process, which involves $two$ two-body
interactions, can be thought of as an effective three-body spin-exchange
effect. The important role of the initial magnetization in creating the atom-
molecule pairs is also analyzed. We also analyze dynamical features that occur
in the “non-resonant” regime where no dark state is formed, including large-
amplitude coherent oscillations of the atom-molecule pairs population and an
antiferromagnetic instability. As such, these manifestations of the interplay
between two-color PA and spin-exchange collisions sheds significant new
insight into the study of quantum spin gases and ultracold chemistry Carr .
The article is organized as follows. Section II discusses the ”resonant”
situation where the dynamics of the system is characterized by the existence
of a dark state. We first introduce our model, which we then apply to the
description of scalar-like photoassociation APB . We then derive a dark-state
condition for the spin-down atoms and show that when satisfied, it results in
the stable resonant creation of atom-molecule pairs. The role of the initial
atomic magnetization is also discussed. Section III then turns to the non-
resonant regime. We show that in that case the system can be described in
terms of a nonrigid pendulum model. Two important dynamical manifestations of
this regime, large-amplitude atom-molecule oscillations, and a regime of
antiferromagnetic instability are explicitly discussed. Finally Section IV is
a summary and conclusion.
Figure 1: (Color online). (a) Schematic of coherent two-color PA in a spin-1
atomic condensate. Here $\delta$ and $\Delta$ are the one- and two-photon
detunings of the laser fields with Rabi frequencies $\Omega_{1,2}(t)$, and
$\gamma$ accounts for the spontaneous decay of the excited state $|m\rangle$.
(b) Scalar-like atom-molecule conversion as observed in a recent experiment of
Kobayashi et al. APB . (c) Effective three-body recombination resulting from
the interplay of $two$ two-body interactions (see text).
## II The model
This section introduces our model and exploit it to describe the main features
of scalar-like PA APB . We also discuss a regime of stable atom-molecule pair
formation, and analyze the role of initial magnetization in the system
dynamics.
### II.1 Theoretical model
The system that we consider is illustrated in Fig. 1. It consists of a spin-1
atomic condensate undergoing spin-changing two-body collisions and coupled via
2-photon coherent PA to a ground-state diatomic molecular condensate.
Denoting by $\hat{\psi}_{i,j=0,\pm 1}$ and $\hat{\psi}_{m,g}$ the annihilation
operators of the three atomic components and of the excited or ground-state
molecules, respectively, the Hamiltonian of the binary atomic and molecular
condensate is $(\hbar=1)$
$\hat{{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{c\rm
oll}+\hat{\mathcal{H}}_{\rm PA},$ (1)
where
$\displaystyle\hat{\mathcal{H}}_{0}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\left[\sum_{i=-1,0,1}\hat{\psi}_{i}^{{\dagger}}\left(V+E_{i}\right)\hat{\psi}_{i}\right.$
(2) $\displaystyle+$
$\displaystyle\left.\left(\delta-\frac{1}{2}i\gamma\right)\hat{\psi}_{m}^{{\dagger}}\hat{\psi}_{m}+(\Delta+\delta)\hat{\psi}_{g}^{{\dagger}}\hat{\psi}_{g}\right],$
$\displaystyle\hat{\mathcal{H}}_{\rm coll}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d\bf{r}\left[c_{0}^{\prime}\hat{\psi}_{i}^{\dagger}\hat{\psi}_{j}^{\dagger}\hat{\psi}_{j}\hat{\psi}_{i}\right.$
(3) $\displaystyle+$
$\displaystyle\left.c_{2}^{\prime}\hat{\psi}_{i}^{\dagger}(F_{\kappa})_{ij}\hat{\psi}_{j}\hat{\psi}_{k}^{\dagger}(F_{\kappa})_{kl}\hat{\psi}_{l}\right],$
$\displaystyle\hat{\mathcal{H}}_{\rm PA}$ $\displaystyle=$ $\displaystyle\int
d{\bf
r}\left[-\Omega_{2}\hat{\psi}_{g}^{\dagger}\hat{\psi}_{m}+\Omega_{1}\hat{\psi}_{m}^{\dagger}\hat{\psi}_{0}\hat{\psi}_{-1}+H.c.\right].$
(4)
Here $V$ is the trap potential, $E_{i}$ is the energy of the spin state $i$
with a static magnetic field lifting their degeneracy, $F_{\kappa=x,y,z}$ are
spin-1 matrices, and
$c_{0}^{\prime}=4\pi(a_{0}+2a_{2})/3M$
and
$c_{2}^{\prime}=4\pi(a_{2}-a_{0})/3M$
where $a_{0,2}$ are $s$-wave scattering lengths Ho98 . Finally
$\Omega_{i},i=\\{1,2\\}$ are the Rabi frequencies of the PA fields, and
$\gamma$ is a phenomenological decay factor. The detunings $\delta$ and
$\Delta$ between the PA fields and the atomic and molecular levels are defined
in Fig. 1. We have ignored the kinetic energy of the particles by assuming a
dilute and homogeneous ensemble. Note also that this model ignores collisions
between the molecules since there is currently no knowledge of their strength.
To extract the main aspects of the system dynamics we invoke a single-mode
approximation, a simplification that has proven successful in describing key
aspects of related systems in the past spin ; Law98 ; Pu99 . It amounts to
approximating the fields operators of the three spin components of the atomic
condensate as
$\hat{\psi}_{i}(\vec{r},t)=\sqrt{N}\hat{a}_{i}(t)\phi(\vec{r})\exp(-i\mu
t/\hbar),$
where $N$ is the initial atomic number, $\mu$ the chemical potential,
$\phi(\vec{r})$ is the normalized condensate wave function for each spin
component, satisfying $\hat{\mathcal{H}}_{S}\phi(\vec{r})=\mu\phi(\vec{r})$
with $\int d\vec{r}|\phi(\vec{r})|^{2}=1$, and $\hat{a}_{i}(t)$ are bosonic
annihilation operators. The molecular condensate is described likewise in a
single-mode approximation, with the annihilation operators $\hat{m}$ and
$\hat{g}$ describing excited and ground-sate molecules.
For large enough detunings $\delta$ the intermediate molecular state
$|m\rangle$ can be adiabatically eliminated adiabatic elimination ,
simplifying the Heisenberg equations of motion of the atom-molecule system to
$\displaystyle i\frac{d\hat{a}_{+}}{d\tau}$ $\displaystyle=$
$\displaystyle\chi_{2}(\rho_{+}+\rho_{0}-\rho_{-})\hat{a}_{+}+\chi_{2}\hat{a}_{0}^{2}\hat{a}_{-}^{\dagger},$
$\displaystyle i\frac{d\hat{a}_{0}}{d\tau}$ $\displaystyle=$
$\displaystyle\chi_{2}(\rho_{+}+\rho_{-})\hat{a}_{0}-\omega\rho_{-}\hat{a}_{0}+2\chi_{2}\hat{a}_{+}\hat{a}_{-}\hat{a}_{0}^{\dagger}+\Omega\hat{g}\hat{a}_{-}^{\dagger},$
$\displaystyle i\frac{d\hat{a}_{-}}{d\tau}$ $\displaystyle=$
$\displaystyle-\Gamma\hat{a}_{-}+\chi_{2}\hat{a}_{0}^{2}\hat{a}_{+}^{\dagger}+\Omega\hat{g}\hat{a}_{0}^{\dagger},$
$\displaystyle i\frac{d\hat{g}}{d\tau}$ $\displaystyle=$
$\displaystyle\Omega\hat{a}_{0}\hat{a}_{-}+(\Delta+\delta-\delta^{\prime})\hat{g},$
(5)
where
$\displaystyle c_{0,2}$ $\displaystyle=$ $\displaystyle c_{0,2}^{\prime}\int
d\mathbf{r}|\phi(\mathbf{r)|^{4}},$ (6) $\displaystyle\delta^{\prime}$
$\displaystyle=$
$\displaystyle\frac{\Omega_{2}^{2}}{c_{0}N\delta}\left(1+\frac{i\gamma}{2\delta}\right)$
(7)
and we have introduced the dimensionless variables $\tau=c_{0}Nt$,
$\chi_{2}=c_{2}/c_{0}$, $\omega=\Omega_{1}^{2}/(c_{0}N\delta)$,
$\Gamma=\omega\rho_{0}-\chi_{2}(\rho_{-}+\rho_{0}-\rho_{+})$, and
$\Omega=\frac{\Omega_{1}\Omega_{2}}{c_{0}N\delta}.$
### II.2 Scalar-like photoassociation
In their recent experiment on two-color PA of the spinor atoms 87Rb APB ,
Kobayashi et al. observed the spin-selective formation of the molecular state
$|2,-1\rangle$ from reactant atoms in the state $|1,-1\rangle$ and
$|1,0\rangle$. One important feature of their experimental results is that
while the populations of the reactant atoms decreased, the population of the
state $|1,1\rangle$ remained almost unchanged. This is the situation
illustrated in Fig. 1(b) APB .
To test our model against that experiment we assume that the energy degeneracy
of the atomic magnetic sublevels is lifted by a static magnetic field and that
the atomic condensate is initially prepared in the state
$f=[\sqrt{0.2},\sqrt{0.6},\sqrt{0.2}]$ APB . The experiment used two lasers of
maximum powers $I_{1}=I_{2}/2=10W$, detuning $\delta=2\pi\times 300$MHz and
$\Omega/\sqrt{I}=7{\rm MHz(Wcm}^{-2})^{-\frac{1}{2}}$, which yields in our
case $\Omega_{1}=139$ MHz and $\Omega_{2}=$197 MHz. As we will see in the
following these values are well beyond the regime of atom-molecule pair
formation, and as illustrated in Fig. 2 our model does confirm that the two-
color PA of atoms into molecules is scalar-like in this case.
Figure 2: (Color online) Scalar-like atom-molecule conversion of 87Rb atoms,
with essentially unchanged population of the spin-up state APB . The initial
condition is $f=[\sqrt{0.2},\sqrt{0.6},\sqrt{0.2}]$, and
$\Omega=\Omega_{m}{\rm sech}(t/4)$ with $|{\Omega_{m}}/{\chi_{2}}|=1.44\times
10^{4}$ APB . The other parameters are $\chi_{2}=-0.01$,
$\delta=-100\chi_{2}$, $\gamma=10|\chi_{2}|$, and $c_{0}N=10^{5}s^{-1}$.
### II.3 Stable atom-molecule pair formation
The scalar-like photoassociation sketched in the previous subsection results
from the binding of a pair of Rb atoms of spin-$0$ and spin-down. We now
consider the case of PA from spin-0, but in the presence of spin-changing
collisions, the situation sketched in Fig. 1(c). Specifically, we assume that
the atomic condensate is initially prepared in the spin-$0$ state
$|1,0\rangle$. Spin-exchange collisions couple then a pair of spin-0 atoms to
a pair of atoms with opposite spins, $2A_{0}\rightarrow
A_{\downarrow}+A_{\uparrow}$ Chapman , while PA fields of appropriate
wavelengths selectively combine a spin-down atom and a spin-0 atom into the
molecular ground state $|g\rangle$ via a virtual transition to an excited
molecular $|m\rangle$, $A_{0}+A_{\downarrow}\rightarrow A_{0}A_{\downarrow}$
APB . The outcome of these combined mechanisms is the creation of an atom-
molecule pair from three spin-0 atoms, $3A_{0}\rightarrow
A_{0}A_{\downarrow}+A_{\uparrow}$, a process that can be intuitively thought
of as an effective, spin-dependent three-body recombination. As such, this
process is quite different from both the scalar-like PA of the previous
subsection APB and the purely atomic laser-catalyzed spin mixing HJ .
We found numerically that in this case the stable atom-molecule pair formation
is possible, provided that the dark-state condition
$\Omega(t)=-\chi_{2}\sqrt{\frac{\rho_{0}\rho_{+}}{\rho_{g}}},$ (8)
for the spin-down atomic state is satisfied Ling ; dark state . This result is
easily confirmed from Eqs. (II.1), which show that when condition (8) is
satisfied the spin-down atomic state remains essentially unoccupied. That
situation is illustrated in Fig. 3, which shows the efficient stable creation
of atom-molecule pairs in this case.
Figure 3: (Color online) Atom-molecule pairs formation as a function of time
for 87Rb atoms, under the dark state condition for spin-down atoms. The
initial state is $f=[0,1,0]$ and the other parameters are the same as Fig. 2.
### II.4 Role of magnetization
The initial magnetization
$\mathcal{M}=\rho_{+}-\rho_{-}-\rho_{g}$ (9)
of a spin gas prepared in the state $f=[0,1,0]$ is $\mathcal{M}=0$. In this
subsection we consider the role of that initial magnetization in the creation
of atom-molecule pairs. We find that in contrast to the case of scalar-like
molecule formation, the initial magnetization now plays a significant role, as
illustrated in Figs. 4 and 5.
Figure 4 shows the evolution of the population of ground-state molecules for
several values of the initial magnetization, under the generalized dark-state
condition (8). For $\mathcal{M}\geq 0$ the ground-state molecules are produced
efficiently and reach a steady-state population $\rho_{g}=(1-\mathcal{M})/3$;
for $\mathcal{M}<0$, in contrast, this population exhibits large oscillations
– see also Fig. 5, which shows more details of the oscillations of $\rho_{g}$
for negative magnetizations – and do not appear to reach a steady state. This
is due to the simple fact that for $\mathcal{M}<0$, the populations of spin-
down atomic state are not zero and thus that state is a ”bright” state that
does not remain uncoupled to the other atomic states during association.
Figure 4: (Color online) Spinor molecules population for several values of the
initial magnetization $\mathcal{M}$ under the dark-state condition, with
$\chi_{2}=-0.01$ and $\delta=100|\chi_{2}|$. The stable formation of spinor
molecules is possible only for $\mathcal{M}\geq 0$. Figure 5: (Color online)
Large-amplitude oscillations of the spinor molecules population for negative
values of the magnetization $\mathcal{M}<0$. All other parameters are as in
Fig. 4.
## III Non-resonant regime
The dynamics of atom-molecule pair formation in the case of negative
magnetization indicates that the presence or absence of an atomic dark state
plays a key role in that process. In this section we further investigate the
“non-resonant” situation where no dark state exists. We consider specifically
two examples: The first one is an ‘adiabatic’ case characterized by an
approximate dark-state condition. In this case the system dynamics can be
understood in terms of an effective nonrigid pendulum model that permits to
discuss an antiferromagnetic instability of the atom-molecule pendulum. In a
second example, we briefly discuss a situation where the dark-state condition
is strongly violated.
### III.1 Adiabatic case
Figure 6 shows an example of atom-molecule-pair oscillations for a non-
resonant situation and starting from spin-0 87Rb atoms. (Note that pair
formation implies that $\rho_{+}\simeq\rho_{g}$.) As would be intuitively
expected, the numerical integration of Eqs. (II.1) confirms that the creation
of atom-molecule pairs is only possible for PA field strengths that allow for
the simultaneous occurrence of spin-exchange collisions and atom-molecule
conversion. For the initial atomic state $f=[0,1,0]$, we find that the Rabi
frequencies of the PA fields should be such that
$\Omega=-\chi_{2}\sqrt{\rho_{0}}\leq|\chi_{2}|$
or equivalently
$\Omega_{1}\Omega_{2}\leq|N\delta c_{2}|,$
which gives $\Omega_{1}\leq 0.3\pi$MHz, and $\Omega_{2}\leq 0.6\pi$MHz for the
case $\Omega_{2}/\Omega_{1}$ = 2 considered here.
Figure 6: (Color online) Coherent atom-molecule oscillations as a function of
time for 87Rb atoms. The dashed line is the population of the spin-up atoms.
The initial atomic state is $f=[0,1,0]$, $\Omega=0.75|\chi_{2}|$, and the
other parameters are as in Fig. 2.
We remark that for an atomic condensate initially prepared in the spin-0
state, and assuming that the dark-state condition (8) is approximately
satisfied, the first derivatives of the slowly-varying amplitudes for spin-
down atoms can be neglected, $i\dot{\hat{a}}_{-}\approx 0$ Pu2000 ; adiabatic
elimination 1 . It is then possible to describe the system by the approximate
effective three-state Hamiltonian
$\displaystyle\hat{\mathcal{H}}_{\rm eff}$ $\displaystyle=$
$\displaystyle\chi_{3}(\hat{a}_{0}^{\dagger
3}\hat{a}_{+}\hat{g}+\hat{a}_{0}^{3}\hat{a}_{+}^{\dagger}\hat{g}^{\dagger}$
(10) $\displaystyle+$
$\displaystyle\frac{1}{\Gamma}(\Omega^{2}\hat{\rho}_{0}\hat{\rho}_{g}+\chi_{2}^{2}\hat{\rho}_{0}^{2}\hat{\rho}_{+})+\chi_{2}\hat{\rho}_{0}\hat{\rho}_{+},$
where $\chi_{3}={\Omega\chi_{2}}/{\Gamma}$. The first term in this Hamiltonian
describes the creation of atom-molecule pairs from three spin-0 atoms through
a laser-induced effective three-body recombination three body .
For short enough times, it is possible to neglect the depletion of the spin-0
population and to treat $\hat{a}_{0}$ as a c-number, $\hat{a}_{0}\rightarrow
N^{1/2}$. Linearizing the Hamiltonian (10), the second line reduces then to a
simple self-interacting contribution, and the Heisenberg equations of motion
for the remaining operators $\hat{a}_{+}$ and $\hat{g}$ have the solution
$\displaystyle\hat{a}_{+}(t)$ $\displaystyle=$
$\displaystyle\hat{a}_{+}(0)\cosh\chi^{\prime}_{3}t-i\hat{g}^{{\dagger}}(0)\sinh\chi^{\prime}_{3}t,$
$\displaystyle\hat{g}(t)$ $\displaystyle=$
$\displaystyle\hat{g}(0)\cosh\chi^{\prime}_{3}t-i\hat{a}_{+}^{{\dagger}}(0)\sinh\chi^{\prime}_{3}t,$
(11)
with $\chi^{\prime}_{3}=N^{3/2}\chi_{3}$. These solutions are well-known to be
indicative of quantum entanglement of the created atom-molecule pairs. As such
this system is formally a matter-wave analog of optical parametric down
conversion in quantum optics Meystre ; Pu2000 .
### III.2 Antiferromagnetic instability
Within the mean-field approach, the spatial part of the atomic and molecular
wave functions can be written as $\sqrt{N}e^{-i\mu t/\hbar}\zeta$, where
$\zeta\sqrt{\rho_{i}}e^{i\theta_{i}}$ or $\sqrt{\rho_{g}}e^{i\theta_{g}}$ and
$\theta_{i}$ represents the phase of the $i$-th Zeeman state spin . Within
this description the dynamics of the system can be expressed in terms of the
coupled equations
$\displaystyle\dot{\rho}_{0}$ $\displaystyle=$
$\displaystyle{3\chi_{3}}{\rho_{0}^{3/2}}\sqrt{(1-\rho_{0})^{2}-\mathcal{M}^{2}}\sin\theta,$
$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle-{\Theta}+\chi_{2}(1+\mathcal{M}-2\rho_{0})+{\frac{1}{\Gamma}}[\chi_{2}^{2}\rho_{0}(3+3\mathcal{M}-4\rho_{0})$
(12) $\displaystyle+$
$\displaystyle\Omega^{2}({\frac{3}{2}}-{\frac{3m}{2}}-{\frac{5\rho_{0}}{2}})+\Omega^{2}+(\Delta+\delta-\delta^{\prime})\Gamma]$
$\displaystyle+$
$\displaystyle{\frac{\Omega\chi_{2}}{2\Gamma}}{\frac{\sqrt{\rho_{0}}[(1-\rho_{0})(9-13\rho_{0})-9\mathcal{M}^{2}]}{\sqrt{(1-\rho_{0})^{2}-\mathcal{M}^{2}}}}\cos\theta,$
where
$\theta=3\theta_{0}-(\theta_{+}+\theta_{g})$ (13)
and
$\Theta=E_{g}+E_{+}-3E_{0}.$ (14)
These nonlinear equations support the two phase-independent fixed-point
solutions $\rho_{0}=0$ and $\rho_{0}=1-|\mathcal{M}|$, as well as phase-
dependent solutions for $\theta=0$ or $\pi$.
Equations (12) describe a nonrigid pendulum with energy functional
$\mathcal{E}=\lambda_{1}\cos\theta+\lambda_{2},$ (15)
where
$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle{3\chi_{3}}{\rho_{0}^{3/2}}\sqrt{(1-\rho_{0})^{2}-\mathcal{M}^{2}},$
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}}{\Gamma}\left[\chi_{2}^{2}\rho_{0}\left(\frac{3}{2}+\frac{3}{2}\mathcal{M}-\frac{4}{3}\rho_{0}\right)\right.$
(16) $\displaystyle+$
$\displaystyle\left.\frac{\Omega^{2}}{2}\left(3-3\mathcal{M}-\frac{5}{2}\rho_{0}\right)\right]$
$\displaystyle-$
$\displaystyle\rho_{0}\left(\Theta+\frac{\Omega^{2}}{\Gamma}+\Delta+\delta-\delta^{\prime}\right)$
$\displaystyle+$ $\displaystyle\chi_{2}\rho_{0}(1+\mathcal{M}-\rho_{0}).$
This approach allows one to study simply the stability of the magnetic domain
structure of the system. Specifically, we follow the approach of Ref. Zhang
and consider instabilities associated with a change in the sign of
$d{\mathcal{E}}/{d{\cal M}}$. For example, $dE/d{\cal M}>0$ for ${\cal M}>0$
and $dE/d{\cal M}<0$ for ${\cal M}<0$ implies that the magnetization always
oscillates around zero, and no domain forms. Following this approach we find
that, in contrast to the situation for purely atomic gases Sadler ; Zhang , an
instability of the domain structure can occur for both ferromagnetic and anti-
ferromagnetic atoms. One finds readily from Eqs. (15) and (III.2),
$\displaystyle\frac{d\cal E}{d{\cal M}}$ $\displaystyle=$
$\displaystyle\frac{3\chi_{3}}{2}{\cal
M}\left[1-\frac{\rho_{0}^{3/2}cos\theta}{\sqrt{(1-\rho_{0})^{2}-{\cal
M}^{2}}}\right]+\chi_{2}\rho_{0}$ (17) $\displaystyle+$
$\displaystyle\frac{3\rho_{o}}{2\Gamma}(\chi_{2}^{2}\rho_{0}-\Omega^{2}).$
Figure 7: (Color online) Surfaces of $d\mathcal{E}$/$d\mathcal{M}=0$ (green
solid lines) for (a) ferromagnetic 87Rb atoms ($\theta=0$, $\chi_{2}=-0.01$);
and (b) anti-ferromagnetic 23Na atoms ($\theta=\pi$, $\chi_{2}=0.01$). The red
forbidden line is determined by the condition of conserved total atomic number
or $\rho_{0}+|\cal{M}|$ $\leq 1$ (see Ref. Zhang ).
Figure 7 shows the resulting surfaces of $d\mathcal{E}$/$d{\cal M}=0$ for the
ferromagnetic and anti-ferromagnetic cases. The plus or minus sign denotes
$d\mathcal{E}$/$d\mathcal{M}>0$ or $d\mathcal{E}$/$d\mathcal{M}<0$. Here the
condensate size is already assumed to be much larger than the healing length
$\mathcal{L}_{s}=2\pi/\sqrt{2M|c_{2}^{\prime}|n}$ at least in one direction so
that instability-induced domains can appear Zhang .
As already mentioned, for $d\mathcal{E}$/$d\mathcal{M}<0$ an increase in
$\mathcal{M}$ leads to lower energy while for $d\mathcal{E}$/$d\mathcal{M}>0$
it leads to a higher energy. Hence the (+, -) boundary delimitates the domain
of dynamic instability (see e.g. Ref. Zhang for more details). We observe
that in contrast to the case of a pure sample of 87Rb atoms, which is
characterized by a wide instability region Zhang , in the case at hand this
region can be significantly reduced by an appropriate tuning of the lasers. We
also note that in the case of anti-ferromagnetic atoms such as 23Na, where no
dynamical instability exists for a pure atomic sample, for our hybrid system,
an instability can now develop for a wide range of parameters, see Fig. 7b.
One point to emphasize is that the antiferromagnetic instability can be
experimentally observed without any laser fields, i.e. for $\Omega=0$ –
although these fields are of course required for the formation of molecules.
We also remark that the spin mixing of spin-2 molecules is slow enough in
comparison with the effective three-body recombination process that it can be
safely ignored here. However, thermalization and spontaneous decay of the
ground-state molecules are expected to be major challenges for the observation
of coherent oscillations of atom-molecule pairs spin-2-2 .
### III.3 Violation of the dark-state condition
As a final special case we now consider the situation when
$|\Omega/\chi_{2}|>1$, in which case the dark-state condition (8) is
completely violated. Figure 8 shows that for increasing values of
$\Omega/\chi_{2}$, the amplitude of the oscillations in molecular population
first increase, and then decreases until $|\Omega/\chi_{2}|=1$. Beyond that
critical value the molecular oscillations become strongly damped, and
eventually population transfer to the molecular ground state essentially
disappears, as illustrated in the figure for $|\Omega/\chi_{2}=1.5$. As
illustrated in Fig. 8(b) the population oscillations of spin-$0$ atoms is also
strongly suppressed in that regime of strong PA.
Finally Fig. 8(b) also illustrates how different choices of the initial atomic
state result in different dynamics of the spinor atom-molecule system. In
particular, an atomic sample initially in the spin-0 state remains completely
unperturbed by the strong PA fields (far from the dark-sate resonance
condition). Note that the scalar-like atom-molecule conversion illustrated in
Fig. 2 corresponds to fields that strongly violate the condition (8), with
$|\Omega/\chi_{2}|=1.44\times 10^{4}\gg 1$. In that case the only parameters
of practical relevance are the initial atomic state and the strengths of the
PA fields.
Figure 8: (Color online) (a) Molecular oscillations for several values of
$|\Omega/\chi_{2}|$, which label the curves, and the initial atomic state
$|0,1,0\rangle$. (b) Atomic spin populations for the initial atomic states
$|f_{1}\rangle=|0,1,0\rangle$ and
$|f_{2}\rangle=|\sqrt{0.25},\sqrt{0.5},\sqrt{0.25}\rangle$, and for
$|\Omega/\chi_{2}|=1.5$. Other parameters are as in Fig. 2.
## IV Summary and Conclusion
In conclusion, we have studied a number of aspects of coherent
photoassociation in a spinor Bose condensate, with emphasis on the creation of
atom-molecule pairs from the initial spin-zero atoms. This process, which
involves $two$ two-body interactions, can be conveniently described by an
effective three-body spin-dependent recombination mechanism – the term ”three-
body recombination” being used here to differentiate our proposal from the
recent two-color PA experiment (that involves the scalar-like association of
spinor atoms) APB . We have shown in particular that the spin-down atoms can
be kept in a dark state for appropriate conditions in both the initial states
of the atoms and PA fields, leading to the formation of atom-molecule pairs.
For comparison we also considered the regimes with PA fields strong enough to
violate the dark-state condition.
Although it shares the similar usage of PA fields and spin-dependent
collisions, the present work is different from previous results on laser-
catalyzed atomic spin oscillations HJ , which did not involve the formation of
molecules. In addition, the simulations of experimentally observed scalar-like
features in associating spinor atoms, the study of the roles of magnetization
and of the initial atomic state, and the antiferromagnetic instability of a
hybrid atom-molecule system are also the new results.
In view of the rapid experimental advances in all-optical association of
laser-cooled atoms Inouye , it can be expected that the coherent PA of quantum
spin gases, in particular, the atom-molecule pair formation in a spinor
sample, should become experimentally observable in the near future APB .
Laser-controlled spinor reactions can provide a new testing ground to address
a number of questions in many-body physics, cold chemistry, and quantum
information science. Future work will study the creation of heteronuclear
spinor molecules from a two-species atomic spin gas hetero , and the spinor
reactions in an optical lattice Daley , with and without the long-range
dipole-dipole interactions de Haas . We also plan to study the cavity-assisted
amplification of spinor molecules CPA , the bistability of a spinor atom-
molecule “pendulum” Ying , and the spinor trimer formation Carr ; trimer .
This work is supported by the U.S. Office of Naval Research, by the U.S.
National Science Foundation, by the U.S. Army Research Office, and by the
National Science Foundation of China under Grant Numbers 10874041 and
10974045.
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|
arxiv-papers
| 2011-02-08T19:30:34 |
2024-09-04T02:49:16.862467
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui Jing, Y. Deng, and P. Meystre",
"submitter": "H Jing",
"url": "https://arxiv.org/abs/1102.1696"
}
|
1102.1729
|
DAMTP-2010-123
KIAS-P11003
Framed BPS States, Moduli Dynamics, and Wall-Crossing
Sungjay Lee111s.lee@damtp.cam.ac.uk♢ and Piljin Yi222piljin@kias.re.kr♠
♢DAMTP, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, UK
♠School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
We formulate supersymmetric low energy dynamics for BPS dyons in strongly-
coupled $N=2$ Seiberg-Witten theories, and derive wall-crossing formulae
thereof. For BPS states made up of a heavy core state and $n$ probe (halo)
dyons around it, we derive a reliable supersymmetric moduli dynamics with $3n$
bosonic coordinates and $4n$ fermionic superpartners. Attractive interactions
are captured via a set of supersymmetric potential terms, whose detail depends
only on the charges and the special Kähler data of the underlying $N=2$
theories. The small parameters that control the approximation are not electric
couplings but the mass ratio between the core and the probe, as well as the
distance to the marginal stability wall where the central charges of the probe
and of the core align. Quantizing the dynamics, we construct BPS bound states
and derive the primitive and the semi-primitive wall-crossing formulae from
the first principle. We speculate on applications to line operators and
Darboux coordinates, and also about extension to supergravity setting.
###### Contents
1. 1 Introduction
2. 2 Classical Dynamics of Probe Dyons
1. 2.1 Semiclassical Core State
2. 2.2 Probe Dyons and Electromagnetic Forces
3. 2.3 Massive Moduli Dynamics of Probe Dyons
4. 2.4 Fermionic Partners
3. 3 Massive ${\cal N}=4$ Mechanics onto Moduli Space
1. 3.1 Toy Model: Flat $R^{3}$ Target
2. 3.2 Toy Model with ${\cal N}=1$ Superfields
3. 3.3 Massive ${\cal N}=4$ Theory onto Conformally Flat $R^{3}$
4. 4 Quantum BPS States near WMS
1. 4.1 ${\cal N}=4$ Low Energy Dynamics of Dyons near MSW
2. 4.2 Quantization and Supercharges
3. 4.3 BPS Bound States and Marginal Stability
5. 5 Wall-Crossing from Moduli Dynamics
1. 5.1 Primitive Wall-Crossing: $\gamma_{c}+\gamma_{h}$
2. 5.2 Semi-Primitive Wall Crossing: $\gamma_{c}+n\gamma_{h}$
6. 6 Conclusion and Discussion
7. A BPS Equation for the Semiclassical Core
8. B More on ${{\cal N}}=4$ Quantum Mechanics
1. B.1 Massless and curved
2. B.2 Massive and curved
3. B.3 Supercharges and Hamiltonian
9. C Review of KS Invariant and Line Operator
## 1 Introduction
The wall-crossing in supersymmetric theories refers to the phenomenon where
certain one-particle BPS states [1, 2] disappear from the spectrum as the
vacuum moduli or parameters are changed continuously. The naive stability
argument of BPS states relies on the short-multiplet structure due to
partially preserved supersymmetry, but this is really applicable only when we
consider dynamical processes in a given vacuum. When we change vacuum or
parameters, even continuously, the state itself can disappear from the
spectrum altogether at which point the supermultiplet structure of the state
becomes a moot issue.
Although the wall-crossing had an early precursor in the context of
supersymmetric kinks in two-dimensional ${N}=(2,2)$ theories [3, 4], it was in
the context of ${N}=2$ supersymmetric theories in four dimensions, such as
Seiberg-Witten theory [5, 6] and Calabi-Yau compactification of type II string
theories, that the phenomenon came under wide scrutiny. The co-dimension one
surface across which a BPS state disappear is called the marginal stability
wall (MSW), and their presence renders the problem of finding BPS spectrum
extremely complicated. For the simplest of Seiberg-Witten theories, monodromy
properties [5] alone can determine the spectrum [7] but this is more an
exception than a rule.
Despite such early discoveries, the space-time picture of exactly what happens
to the state upon the wall-crossing remained unclear until it was uncovered in
the context of 1/4 BPS dyons in $N=4$ super Yang-Mills theory, which preserve
four supersymmetries just as 1/2 BPS objects of $N=2$ theories do. It was
found in Ref. [8] that such BPS dyons must be, typically, thought of as a
loose bound states of more than one dyonic centers with mutually non-local
charges. The distances between such centers are not free but determined by the
vacuum moduli $u_{i}$’s
$R_{AB}=R_{AB}(u_{i};\\{q^{i}_{C},p^{i}_{C}\\})\ ,$ (1.1)
with $(q^{i}_{A},p^{i}_{A})$ being the charges of the $A$-th center. In
particular, some of $R_{AB}$ was shown to diverge as a MSW is approached; this
happens simply because scalar forces and electromagnetic forces do not cancel
each other between dyons of mutually nonlocal charges and the equilibrium
distance is determined by a detailed balance of classical forces: state by
state, the wall-crossing has a very mundane and classical explanation.
This finding is immediately applicable to weakly coupled $N=2$ theories as
well, because $N=2$ theory BPS solitons can be classically embedded to a $N=4$
theory. There will be differences at quantum level because the supermultiplet
structures (and flavor structures) are different, but the above space-time
picture of wall-crossing is essentially classical and quite robust.
For both $N=2$ and $N=4$ theories, this multi-center nature and the subsequent
wall-crossing were soon elevated to the semiclassical level [9, 10, 11]. The
quantum low energy dynamics of magnetic monopoles was derived rigorously from
the super Yang-Mills theories in question [12], and dyons were constructed as
quantum bound states of monopoles with certain conjugate momenta turned on
[13, 10]. What used to be the classical orbit size is now represented by the
quantum bound state size, and is still determined by the vacuum moduli and
charges as in (1.1). The size of the bound state is again divergent as a wall
of marginal stability is approached, across which the state no longer exists
as quantum and BPS one-particle state.
In such supersymmetric low energy dynamics of solitons, precise state counting
is a simple matter of finding bound state wavefunctions or computing index of
ceratin Dirac operators on the moduli space. For instance, the bound state of
a pair of dyons of charge $\gamma_{1}+\gamma_{2}$ has been constructed and
counted when the total magnetic charge is a dual root. The degeneracy on one
side of a wall [10]#1#1#1In this note, we take the convention that Schwinger
products take values in ${\mathbf{Z}}/2$. can be written as
$|\,\Omega(\gamma_{1}+\gamma_{2})|=2|\langle\gamma_{1},\gamma_{2}\rangle|\ ,$
(1.2)
where we introduced $\Omega(\gamma)$, the second helicity trace for the
supermultiplet of charge $\gamma$, as
$\displaystyle\Omega=-\frac{1}{2}\text{tr}\big{(}(2J_{3})^{2}(-1)^{2J_{3}}\big{)}\
.$ (1.3)
A simplest generalization of this is a chain of dyons with nearest neighbor
interactions, namely $\langle\gamma_{A},\gamma_{B}\rangle\neq 0$ if and only
if $|A-B|=1$. Whenever such a state exists as a quantum BPS state, the
degeneracy takes a simple form [14],
$|\,\Omega(\gamma_{1}+\gamma_{2}+\gamma_{3}+\cdots)|=\Big{|}\prod_{A}2\langle\gamma_{A},\gamma_{A+1}\rangle\Big{|}\
.$ (1.4)
They were shown to exist only when each and every one of
$\langle\gamma_{A},\gamma_{A+1}\rangle$ obeys certain inequalities defined by
the vacuum moduli, which amounts to being on the “right” side of several
MSW’s, basically one for each interacting pair. The formula clearly suggests
that such states can be constructed iteratively by attaching one kind of dyons
at a time, already hinting at a simple universal wall-crossing formula.
The next breakthrough came from ${N}=2$ supergravity analysis by Denef who
also found the multi-centered nature of BPS black holes and the subsequent
wall-crossing in the context of attractor flow solutions [15, 16]. The
approach gave a universal and explicit constraints for the relative positions
of charge centers, say, charge $\gamma_{A}$ at $\vec{x}_{A}$,
$\displaystyle\sum_{B\neq
A}\frac{\langle\gamma_{B},\gamma_{A}\rangle}{|\,\vec{x}_{B}-\vec{x}_{A}|}={\text{Im}\big{[}\zeta_{T}^{-1}Z(\gamma_{A})\big{]}}\
.$ (1.5)
where $Z(\gamma_{A})$ is the central charge of $\gamma_{A}$ and $\zeta_{T}$ is
the phase factor of the total central charge $Z_{T}=\sum_{A}Z(\gamma_{A})$. In
supergravity, this simplifies and supersedes the field theory results which we
abstractly noted as (1.1).
The wall-crossing for supergravity black hole solutions is again due to a
divergent distance between the charge centers, which is dictated by long
distance classical physics, just as as in the field theory soliton picture of
BPS dyons: the sign of the left hand side of (1.5) is independent of vacuum
moduli while that of the right hand side can flip the sign as we change
vacuum. Clearly at some point where the right hand side approaches zero from
the positive side, one of the distances has to diverge, beyond which the
solution can no longer exist. This is most useful since the sizes of the
states can be found without detailed construction. However, there is no
information on how a given charge state is split into what charge centers,
unlike the explicit constructions of multi-center solution/quantum states in
the field theory story.
Although supergravity solutions themselves were not amenable to explicit and
precise quantum counting, Denef further went on to conjecture general two-body
wall-crossing formula that extends the above field theory result to arbitrary
(magnetic) charges [17]. With spin content taken into account, the formula
reads,
$\Omega(\gamma_{1}+\gamma_{2})=-(-1)^{2|\langle\gamma_{1},\gamma_{2}\rangle|}\,2|\langle\gamma_{1},\gamma_{2}\rangle|\,\Omega(\gamma_{1})\,\Omega(\gamma_{2})\
,$ (1.6)
which was later extended by Denef and Moore to the semi-primitive cases [18],
captured in a generating function,
$\displaystyle\sum_{n=0}\Omega(\gamma_{1}+n\gamma_{2})\,q^{n}=\Omega(\gamma_{1})\prod_{k=1}\
\Big{[}1-(-1)^{2k\langle\gamma_{2},\gamma_{1}\rangle}q^{k}\Big{]}^{2k|\langle\gamma_{2},\gamma_{1}\rangle|\Omega(k\gamma_{2})}\
,$ (1.7)
counting the BPS states of charges $\gamma_{1}+n\gamma_{2}$ in terms of
degeneracies of states with charges $\gamma_{1}$ and $n\gamma_{2}$. These
spurred much activities toward general solutions to the wall-crossing problem,
and was integrated recently into more general Kontsevich-Soibelman’s wall-
crossing formalism [19].
Despite evidences that support the semi-primitive wall-crossing formulae of
Denef-Moore (which in turn support Kontsevich-Soibelman formalism), it has
been rigorously tested only in specific cases. The most systematic example of
this is the ${N}=2$ weak coupling analysis that preceded the conjecture, but
the limitation of weak coupling limit casts some shadows on its general
usefulness. It would be very useful if we can find a similarly systematic
method of constructing and counting BPS bound states, and apply to diverse BPS
objects, such as those dyons that appear in the generic strongly coupling
region of Seiberg-Witten theory. In this paper, we wish to initiate a new
framework that can count and construct BPS states, without referring either to
specific subset of charges or to weak electric coupling, but applicable to a
large class of ${N}=2$ theories and BPS states thereof.
One common lesson from earlier studies of multi-centered BPS states is that
non-Abelian completion of the state at charge centers is not essential for
understanding wall-crossing, since the latter is essentially a long distance
phenomenon from the spacetime viewpoint. The supergravity solutions are all
Abelian while, for solitons, it is the long range Coulomb-type interactions
that determined the multi-center nature of the state. Related is the notion of
the “framed” BPS state [20]. The main idea there was to treat one or more
component dyons as an external object, and the remainders as dynamical object
around such a background. This way, one can treat the former as the
background, in which the latter moves around and sometime becomes
supersymmetrically bound to the core state. This split of the state into two
parts can simplify the state construction and counting substantially.
Inspired by these ideas, we wish to consider dyons moving around purely
Abelian dyonic background. In effect, we will split the state in question into
the heavy “core state” of total charge $\gamma_{c}$ and light “halo” or
“probe” of charge $\gamma_{h}$. For our purpose, it is the ratio of the two
masses that matters, so this can be for instance achieved by approaching a
singular point where the probe dyon becomes massless. The low energy dynamics
of the probe dyon is quite natural thing to do there since, precisely at such
a singular point, the probe dyon would be the lightest particle among charged
states. However, there are other circumstances where one part become
relatively light compared to the other, and our framework will apply.
Another useful fact is that, as far as wall-crossing behavior goes, we only
need information near the relevant MSW’s, away from which the BPS spectrum is
continuous. This allows another small quantity to play with, by taking vacuum
very near a marginal stability wall. As we will see later, the distance to the
MSW plays a role very similar to the weak electric coupling in that it
controls the nonrelativistic approximation. In the end, we find that the
dynamics between the core and the probe reduces to massive supersymmetric
quantum mechanics with two kinds of potentials.
These two lead us to a new model of low energy dynamics for dyons in the
strongly coupled region of ${N}=2$ field theory. Although similar in spirit to
the old moduli dynamics of solitons, an essential difference here is that the
small electric coupling constant is no longer needed; this is what allows us
to apply the technique to much wider class of BPS states than previously
possible.#2#2#2 In fact, it should be possible to extend this framework to
include gravity and discuss quantum bound states of charged BPS black holes.
The quantum mechanics has four supercharges, as required by the BPS condition,
but comes with only $3n$ bosonic coordinates, three for each probe dyon, and
$4n$ fermionic coordinates. Compared to the conventional moduli dynamics of
weakly coupled regime, we are missing one angular collective coordinate for
each dyon. This has something to do with the fact that we start with dyons,
rather than monopoles, as basic building blocks.
With this new low energy dynamics in place, we can compute how many BPS bound
states of the core and the probe dyons can form, and under what condition. At
the end of day we derive, via a first-principle computation, the semi-
primitive wall-crossing formula with $\gamma_{1}=\gamma_{c}$ and
$\gamma_{2}=\gamma_{h}$. In this note there is in fact no restriction on
$\gamma_{c}$, as far as such a state actually exist and all of its component
dyon centers can be made heavy. Thus we in effect are computing
$\Omega(\gamma_{c}+n\gamma_{h})$ with the only restriction that the dyon
$\gamma_{h}$ is primitive and become massless somewhere in the vacuum moduli
space. We wish to emphasize that, alternatively, we may think of the theory as
a setup for finding framed BPS state with line operator of charge $\gamma_{c}$
and halos $\gamma_{h}$ [20].
This paper is organized as follows. In Section 2, we write down the long-
distance Abelian form of the core state in terms of the central charge
function, while the probe dyons are treated as quantized solitons in that
background. As a result we find a bosonic low energy Lagrangian of the probe
dyons purely in terms of quantities that can be constructed out of the central
charge functions. This reproduces some of general results, such as distances
between two charge-centers, obtained from supergravity attractor flow
analysis, even though we are dealing with field theory states.
Section 3 discusses how one can construct a ${\cal N}=4$ supersymmetric
Lagrangian with $3n$ bosonic coordinates and $4n$ fermionic coordinates, by
extending previous studies by Coles and Papadopoulos [21] and also by [22].
These previous works constructed massless supersymmetric theories of similar
kind, which is, however, missing the crucial elements of potentials. Without
the latter, the bound states we are interested in cannot form at all. We
construct in particular massive theories in which degrees of freedom are
cataloged by $SO(4)_{R}=SU(2)_{L}\times SU(2)_{R}$ algebra with bosons in
$({\bf 3},{\bf 1})$ (thus, the first $SU(2)_{L}$ also serves as a rotation
group) and fermions in $({\bf 2},{\bf 2})$ representations. The four
supercharges are also in $({\bf 2},{\bf 2})$. The Lagrangian has $SU(2)_{R}$
symmetry manifest while $SU(2)_{L}$ can be explicitly broken by the
background.
Section 4 shows how the general discussion of section 3 makes contact with the
probe dyon dynamics of section 2 under the assumption the vacuum moduli of the
underlying ${N}=2$ theory is very near the MSW. The latter assumption controls
the energy scale of the potential energy, and allows a nonrelativistic
approximation possible. We then quantize the resulting dynamics and derive the
bound states for $\gamma_{c}+\gamma_{h}$, and again shows how the bound state
size diverges as one approach MSW and how the bound state is impossible on the
other side of MSW.
Section 5 elevates this to a primitive wall-crossing formula, and extends
further to the cases of $\gamma_{c}+n\gamma_{h}$ by invoking spin-statistics
theorem. This derives, in particular, the semi-primitive wall-crossing formula
from a first principle computation.
We then conclude in Section 6 with summary and other comments especially on
how one can make use of this formalism to compute the line operator
expectation values and how one can extend the formalism to the supergravity
setting. Some computational details are summarized in Appendices.
## 2 Classical Dynamics of Probe Dyons
In this section, we construct the semiclassical form of the core state,
entirely in terms of the central charge function, and describe energetics and
dynamics of a probe dyon in the core state background. This leads us to a
bosonic Lagrangian of the probe dyon, which will be supersymmetrized and
quantized in the later section.
Although the exercise here applies to any core state one can imagine, as long
as there solve the relevant semi-classical BPS equation of the effective
Abelian theory, we are eventually interested in core states that actually
exist as quantum BPS states. It is known that the former does not always imply
the latter [23, 10]. Alternatively, for the framed BPS states, the core state
should correspond to a supersymmetric line operator. Either way, we are
interested in case where the supersymmetric lift of this probe bosonic
dynamics would make sense in the context of the underlying four-dimensional
theory.
### 2.1 Semiclassical Core State
We start by recalling semiclassical properties of $N=2$ dyons when expressed
in terms of the low energy theory of Seiberg and Witten. Traditionally the
smooth solitons are possible only when we include the entire non-Abelian
origin, but this is practical only in the weakly coupled limit.
To avoid such restrictions, a more convenient starting point is to write the
BPS equation in the Abelian low energy description of Seiberg and Witten. This
approach was investigated previously [24, 25] with emphasis on split flow
picture of the classical soliton and gave an interesting parallel to the
string web picture [26] of $N=4$ 1/4 BPS dyons. These solutions are invariably
singular at the charge centers, since there is no non-Abelain mechanism to
stop the Coulomb-like divergence at origin, which was controlled ad hoc by
introducing UV cutoffs.
For our purpose, however, this divergence is of little consequence,
essentially because we will be using this solution as background. As long as
we can ascertain existence of quantum state of such a charge and as long as we
put correct boundary condition at such singular points, forcing the probe dyon
wavefunction to vanish there fast enough, there would be no physical problem
associated with it. It is entirely analogous to the Hydrogen atom problem of
undergraduate quantum mechanics, where finite and trustworthy bound states are
obtained even though the Hamiltonian is naively singular at origin.
Using the SUSY transformation rule for gaugino along the particular direction
parameterized by a phase factor $\zeta$, one can obtain the BPS equations
$\displaystyle\vec{\cal F}_{i}-i\zeta^{-1}\vec{\nabla}\phi_{i}=0\ ,$ (2.1)
where $i$ labels the unbroken $U(1)$ gauge groups, and $\vec{\cal F}$ denotes
the complexified field strength 3-vector $\vec{B}+i\vec{E}$. See appendix A
for details. There is also an electric version of this equation
$\displaystyle\vec{\cal F}_{D}^{i}-i\zeta^{-1}\vec{\nabla}\phi^{i}_{D}=0\ ,$
(2.2)
with $\vec{\cal F}_{D}^{i}\equiv\tau^{ij}\vec{\cal F}_{j}$ and
$\tau^{ij}=\frac{\partial^{2}}{\partial\phi_{i}\partial\phi_{j}}F_{\text{SW}}(\phi)\
,$ (2.3)
where $F_{\text{SW}}(\phi)$ is the Seiberg-Witten prepotential of the given
theory. Since it is $\text{Re}{\cal F}_{D}$ that enters the Gauss constraint,
the field strengths are such that [24]
$\displaystyle\text{Re}\,\int_{S^{2}_{\infty}}{\cal F}_{i}=4\pi
P^{i},\qquad\text{Re}\,\int_{S^{2}_{\infty}}{\cal F}_{D}^{i}=-4\pi Q^{i},$
(2.4)
with the total magnetic charges $P^{i}$ and the total electric charges $Q^{i}$
In particular imagine a semiclassical core state of charges
$\gamma_{c}=(P^{i},Q^{i})=\sum_{A}\gamma_{c,A},$ with
$\gamma_{A}=(P^{i}_{A},Q^{i}_{A})$, distributed into several dyonic cores at
$\vec{x}^{A}$, and the field strength takes the following asymptotic forms,
$\displaystyle\text{Re}\,\vec{\cal
F}^{i}=\sum_{A}\frac{P_{A}^{i}(\vec{x}-\vec{x}_{A})}{|\vec{x}-\vec{x}_{A}|^{3}}=\vec{\nabla}\left(-\sum_{A}\frac{P_{A}^{i}}{|\vec{x}-\vec{x}_{A}|}\right)\
,$ $\displaystyle\text{Re}\,\vec{\cal
F}_{D}^{i}=-\sum_{A}\frac{Q_{A}^{i}(\vec{x}-\vec{x}_{A})}{|\vec{x}-\vec{x}_{A}|^{3}}=\vec{\nabla}\left(\sum_{A}\frac{Q_{A}^{i}}{|\vec{x}-\vec{x}_{A}|}\right)\
.$ (2.5)
One can show that $\zeta$ can be identified as the phase factor of central
charge $Z_{\text{core}}$ of this core state#3#3#3See Appendix A.
$\displaystyle
Z_{\text{core}}=|Z_{\text{core}}|\zeta=Q^{i}\phi_{i}(\infty)+P_{i}\phi_{D}^{i}(\infty)\
.$ (2.6)
This semiclassical description is, strictly speaking, valid away from
$\vec{x}=\vec{x}_{A}$’s.
Note that the positions, $\vec{x}_{A}$’s, of the centers would be restricted
by an analog of (1.5). Precise positions of these centers is, however,
immaterial for counting BPS bound states, as long as the relevant core state
actually exists as quantum and BPS bound state. This happens because one ends
up computing supersymmetric indices, which are robust under small deformations
of the supercharges. More important is how the core electromagnetic charge is
distributed into such centers. See section 4 for related discussions.
### 2.2 Probe Dyons and Electromagnetic Forces
Let us now introduce a probe particle of charge $\gamma_{h}=(p_{i},q_{i})$, in
a background created by such a core state. It will be considered as a probe
particle in the external electromagnetic field by the massive core state.
Using the equations (2.1,2.2), one obtains
$\displaystyle q\cdot\vec{\cal F}+p\cdot\vec{\cal
F}_{D}=i\zeta^{-1}\vec{\nabla}{\cal Z}_{h}\ ,$ (2.7)
where ${\cal Z}_{h}=q\cdot\phi+p\cdot\phi_{D}$ is now understood as position-
dependent. We introduced the notation ${\cal Z}$ to emphasize that this
quantity is position-dependent. The usual central charge ${Z}$ is related to
it as $Z={\cal Z}(\infty)$.
The real and imaginary part of the relation will give us hints how to
construct the low-energy Lagrangian of probe dyon in the background of core
particle. The real part can be succinctly written as
$\displaystyle\vec{\nabla}V_{\text{Coulomb}}=-\vec{\nabla}\text{Re}\Big{[}\zeta^{-1}{\cal
Z}_{h}\Big{]}\ ,$ (2.8)
where
$\displaystyle V_{\text{Coulomb}}$ $\displaystyle=$
$\displaystyle\text{Re}(\tau)_{ij}\sum_{A}\frac{p^{i}P^{j}_{A}}{|\vec{x}-\vec{x}_{A}|}$
(2.9)
$\displaystyle+\left(\text{Re}(\tau)\right)^{-1}_{ij}\sum_{A}\frac{\big{(}q_{i}+\text{Im}(\tau)_{ij}p^{j}\big{)}\big{(}Q_{j,A}+\text{Im}(\tau)_{ij}P^{j}_{A}\big{)}}{|\vec{x}-\vec{x}_{A}|}$
is nothing but the Coulomb potential energy felt by the probe dyon due to the
core state. The real part of this equation is even simpler
$\displaystyle\vec{\nabla}\left(\sum_{A}\frac{Q_{A}\cdot p-P_{A}\cdot
q}{|\vec{x}-\vec{x}_{A}|}\right)=-\vec{\nabla}\text{Im}\Big{[}\zeta^{-1}{\cal
Z}_{h}\Big{]}\ ,$ (2.10)
or equivalently
$\displaystyle\vec{\nabla}\left(\sum_{A}\frac{\langle\gamma_{c,A},\gamma_{h}\rangle}{|\vec{x}-\vec{x}_{A}|}\right)=-\vec{\nabla}\text{Im}\Big{[}\zeta^{-1}{\cal
Z}_{h}\Big{]}\ .$ (2.11)
which, as we will presently see, encodes the Lorentz force on the probe
dyon.#4#4#4The normalization of charges and the sign convention for Schwinger
product differs from that of Refs. [15, 18]
$\langle\gamma,\gamma^{\prime}\rangle=\frac{1}{2}\langle\gamma^{\prime},\gamma\rangle_{\text{Denef-
Moore}}$ Related is the fact that our $\zeta$ is $-\zeta_{\text{Denef-
Moore}}$.
We first discuss the invariant expression of the minimal coupling under the
Montonen-Olive duality. Recall that, from the BPS equations, one can conclude
that $(\vec{\cal F},\vec{\cal F}_{D})$ transform under the duality
transformation as vector representation like $(\phi,\phi_{D})$. For example,
let us consider the S-duality transformation of $(\vec{\cal F},\vec{\cal
F}_{D})$
$\displaystyle\vec{B}\ \to\ -\text{Im}(\tau)\vec{E}+\text{Re}(\tau)\vec{B}\
,\qquad\vec{E}\ \to\ \text{Im}(\tau)\vec{B}+\text{Re}(\tau)\vec{E}\ .$ (2.12)
Then, one can easily show that, under the S-duality transformation,
$\displaystyle q\ \to\ p\ ,\qquad p\ \to\ -q\ ,$ (2.13)
where we used, for the last transformation, the fact that $\tau\to-\tau^{-1}$.
When the probe dyon moves (slowly) under the electromagnetic field of core
particle, the minimal coupling terms therefore become [27, 28]
$\displaystyle{\cal
L}_{\text{int}}=q^{i}\vec{v}\cdot\vec{A}^{i}+p^{i}\vec{v}\cdot\vec{\tilde{A}}^{i}+q^{i}A_{0}^{i}+p^{i}\tilde{A}_{0}^{i}\
,$ (2.14)
which is the duality invariant expression. Here $A_{\mu}$ and
$\tilde{A}_{\mu}$ are defined as
$\displaystyle\text{Re}\vec{\cal F}=\vec{\nabla}\times\vec{A}\ ,$
$\displaystyle\text{Re}\vec{\cal F}_{D}=\vec{\nabla}\times\vec{\tilde{A}}\ ,$
$\displaystyle\text{Im}\vec{\cal F}=\vec{\nabla}\cdot A_{0}\ ,$
$\displaystyle\text{Im}\vec{\cal F}_{D}=\vec{\nabla}\cdot\tilde{A}_{0}\ .$
(2.15)
Using the BPS equation (2.7), the interaction terms can be managed into a
rather simpler form
$\displaystyle{\cal L}_{\text{int}}=-\vec{v}\cdot\vec{\cal
W}+\text{Re}\Big{[}\zeta^{-1}{\cal
Z}_{h}(x)\Big{]}-\text{Re}\Big{[}\zeta^{-1}{\cal Z}_{h}(\infty)\Big{]}\ ,$
(2.16)
where the vector $\vec{w}$ satisfies the relation below
$\displaystyle\vec{\nabla}\times\vec{\cal
W}=\vec{\nabla}\text{Im}\Big{[}\zeta^{-1}{\cal Z}_{h}(x)\Big{]}\ .$ (2.17)
Note that, in (2.16), $\text{Re}[\zeta^{-1}{\cal
Z}_{h}(\infty)]=\text{Re}[\zeta^{-1}Z_{h}]$ represents the lowest possible
energy the probe dyon can attain.
### 2.3 Massive Moduli Dynamics of Probe Dyons
Finally we come to the effect of the long range scalar field on the dyon. The
low-energy Lagrangian of probe dyon $\gamma_{h}$ moving in the background of
core particle $\gamma_{c}$ can take the following form
$\displaystyle{\cal L}^{\text{bosonic}}={\cal L}_{\text{kin}}+{\cal
L}_{\text{int}}\ ,$ (2.18)
where the kinetic term must be [27, 28]
$\displaystyle{\cal L}_{\text{kin}}=-\big{|}{\cal
Z}_{h}(x)\big{|}\sqrt{1-v^{2}}\simeq-|{\cal Z}_{h}(x)|+\frac{1}{2}\big{|}{\cal
Z}_{h}(x)\big{|}\vec{v}^{2}+{\cal O}(v^{4})$ (2.19)
with ${\cal Z}_{h}(x)=q\cdot\phi+p\cdot\phi_{D}$, since $|{\cal Z}_{h}(x)|$ is
the effective inertia of the probe dyon. Adding all these together, we find
the classical Lagrangian,
$\displaystyle{\cal L}^{\text{bosonic}}=\frac{1}{2}\big{|}{\cal
Z}_{h}(x)\big{|}\vec{v}^{2}-\big{|}{\cal
Z}_{h}(x)\big{|}+\text{Re}\left(\zeta^{-1}{\cal
Z}_{h}(x)\right)-\text{Re}\Big{[}\zeta^{-1}{\cal
Z}_{h}(\infty)\Big{]}-\vec{v}\cdot\vec{\cal W}$ (2.20)
with $\vec{\nabla}\times\vec{\cal
W}=\vec{\nabla}\text{Im}\left(\zeta^{-1}{\cal Z}_{h}(x)\right)$.
This Lagrangian has the classical ground state at $\vec{x}=\vec{x}_{*}$ where
$\big{|}{\cal Z}_{h}(x_{*})\big{|}=\text{Re}[\zeta^{-1}{\cal Z}_{h}(x_{*})]$,
with the ground state energy $\text{Re}[\zeta^{-1}{\cal Z}_{h}(\infty)]$. We
wish to elevate this, later, to ${\cal N}=4$ quantum mechanics, so it is more
convenient to separate out the ground state energy. Thus, our starting point
is the bosonic Lagrangian,
$\displaystyle{\cal L}_{\text{moduli}}^{\text{bosonic}}={\cal
L}^{\text{bosonic}}+\text{Re}\Big{[}\zeta^{-1}{\cal Z}_{h}(\infty)\Big{]}\ ,$
(2.21)
so that supersymmetric bound states would have zero energy. This also
reproduces an analog of Denef’s formula [15] for the probe dyons since,
$\displaystyle\sum_{A}\frac{\langle\gamma_{c,A},\gamma_{h}\rangle}{|\vec{x}_{A}-\vec{x}_{*}|}=\text{Im}\big{[}\zeta^{-1}{\cal
Z}_{h}(\infty)\big{]}\ .$ (2.22)
from Eq. (2.11) and $\text{Im}[\zeta^{-1}{\cal Z}_{h}(x_{*})]=0$. This is the
same as (1.5) once we realize that total central charge $Z_{T}=Z_{c}+Z_{h}$ is
dominated by $Z_{c}$ since $Z_{h}/Z_{c}$ is very small; $\zeta_{T}$ is
approximately equal to $\zeta$.
### 2.4 Fermionic Partners
We have derived a classical (thus purely bosonic) Lagrangian that describe the
dynamics of a probe dyon in the background of the core state, with 3 bosonic
collective coordinates per each probe dyon. Without much effort, we can
further deduce that each probe dyon will also come with 4 fermionic degrees of
freedom, giving $4n$ fermionic variables as opposed to $3n$ bosonic variables.
The simplest way to see those four fermionic variables is to recall that a BPS
particle, of a given charge, in $N=2$ theory are at least in the half-
hypermultiplet, with spin content
$[{1/2}]\oplus 2[{0}]\,.$ (2.23)
This spin content can be generated only if the dyon comes with a pair of
complex fermionic degrees of freedom in a spin 1/2 multiplet, which translates
to four real fermionic coordinates. They are, when we consider the dyon in
isolation, also Goldstino modes coming from the four supercharges broken by
the BPS state. More generally, the probe dyon could be in a BPS multiplet of
type,
$[s]\otimes\left([{1/2}]\oplus 2[{0}]\right)\,,$ (2.24)
with an angular momentum multiplet $[s]$ of spin $s$, in which case $[s]$
typically arises because the probe dyon is itself a composite or has,
otherwise, some internal light degrees of freedom. What matters for us is that
we still have the same four fermionic collective coordinates whose coupling to
the bosonic ones are tightly constrained by the ${\cal N}=4$ supersymmetries.
When we consider the special limit of solitonic dyons in weakly coupled
theories, this mismatch between the bosonic and the fermionic degrees of
freedom can be understood easily [29, 30, 37]. Solitonic dyons arise there
from excitation of a monopole soliton with particular $U(1)$ momenta turned on
[31]. While the initial monopole soliton comes with four bosonic and four
fermionic collective coordinates, one angular bosonic coordinate is traded
away in favor of its conjugate momentum (which is physically the electric
charge). This procedure, however, leaves the four fermionic coordinates
intact. It has to be so, since the dyon is still BPS and the necessary half-
hypermultiplet structure would be generated using all four of these fermionic
degrees of freedom. Nor does this reduce the ${\cal N}=4$ supersymmetry of the
remaining dynamics, although their embedding into the underlying field theory
is rotated in response to the new electromagnetic charges.
## 3 Massive ${\cal N}=4$ Mechanics onto Moduli Space
An odd fact, when we consider a supersymmetric lift of the above Lagrangian
for probe dyons, is that the low energy dynamics involves $3$ bosonic
collective coordinates for each probe dyon, yet, there should be 4 fermionic
counterparts. Supersymmetry with mismatching bosonic and fermionic degrees of
freedom is in principle possible for quantum mechanics because there is no
notion of spin, but still construction of such theories, especially with
extended supersymmetry, was not widely studied. The only known example is
certain (massless) class of supersymmetric nonlinear sigma models by Coles et.
al. [21], which were later specialized in the context of extremely charged
black holes of the same charges [22]. Neither of these studies considered
massive versions, as needed here, however.
Similar situation existed a dozen years ago when low energy dynamics of
solitonic monopoles were studied for weakly coupled ${N}=2,4$ Yang-Mills
theories. The conventional massless moduli dynamics [32, 33, 34] with
$4n$-dimensional target manifolds without potential were found to be
inadequate for dyons in generic Coulombic vacuum when the rank of the gauge
group is two or larger [8]. The problem was the lack of potential terms in
this older formulation. The low energy dynamics of monopoles had to be
reformulated so that both the potentials and ${\cal N}=4$ supersymmetry are
manifest. Later, such massive ${\cal N}=4$ quantum mechanics were found,
simply by twisting supercharged by triholomorphic Killing vector fields on the
moduli space [9, 11, 10, 12].#5#5#5Some related mathematical structures were
first studied in Refs. [35] while its potential connection to dyons was
previously hinted by Ref. [36]. This lead to a whole machinery whereby dyon
spectra in the weakly coupled limit of $N=2,4$ Yang-Mills theories were
constructed explicitly [14]. See Ref. [37] for a broad overview of this
development.
In this section, we wish to investigate how the new kind of classical low
energy dynamics of section 2 can be also elevated to one with ${\cal N}=4$
supersymmetry. We will find that massive ${\cal N}=4$ supersymmetric mechanics
with mismatching bosonic and fermionic degrees of freedom is possible and
will, specifically, build a massive (i.e. with potential) supersymmetric
Lagrangian with 3 bosonic coordinates and 4 fermionic coordinates. This
restriction to the lowest possible target dimension simplifies the
construction greatly, in part because the target manifold turned out to be
conformally flat $R^{3}$, and yet still good enough for deriving semi-
primitive wall-crossing formula.#6#6#6 For generalization that can address
many probe dyons with non-negligible mutual interactions, we need to consider
higher dimensional target manifolds, which is left for a future work.
### 3.1 Toy Model: Flat $R^{3}$ Target
As a toy model, let us pretend that the bosonic moduli space is flat $R^{3}$
and see how scalar and vector potentials on $R^{3}$ can be incorporated into
the quantum mechanics in a manner consistent with four supercharges.
${\cal N}=1$ supersymmetry is easy to incorporate. We start with the usual
transformation rule
$\displaystyle\delta x^{a}=-i\epsilon\psi^{a}\
,\qquad\delta\psi^{a}=\epsilon{\dot{x}}^{a}\ ,\qquad$ (3.1)
under which the following free Lagrangian that is invariant
$\displaystyle{\cal
L}^{(0)}=\frac{1}{2}{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}\psi^{a}{\dot{\psi}}^{a}\
.$ (3.2)
Since we are dealing with quantum mechanics, rather than a field theory, we
can add any number of fermions, as long as we let them be invariant under the
above supersymmetry transformation. As we will see shortly, however, extended
supersymmetry would not leave this extra fermion intact.
For our purpose, one extra fermion $\lambda$ is needed for each triplet of
$(x^{a},\psi^{a})$, so we may start with
$\displaystyle{\cal
L}^{(0)}=\frac{1}{2}{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}\psi^{a}{\dot{\psi}}^{a}+\frac{i}{2}\lambda{\dot{\lambda}}\
,$ (3.3)
where
$\displaystyle\delta\lambda=0\ .$ (3.4)
Incorporation of an external gauge field $w$ on $R^{3}$ is equally easy.
Adding a minimal coupling $-\dot{x}^{a}w_{a}$ to the Lagrangian and noting the
supersymmetry transformation property,
$\displaystyle\delta\big{(}-w_{a}{\dot{x}}^{a}\big{)}=$
$\displaystyle+i\epsilon\psi^{a}{\dot{x}}^{b}\big{(}\partial_{a}w_{b}-\partial_{b}w_{a}\big{)}+\text{total
derivative}\ $ $\displaystyle=$
$\displaystyle+i\partial_{a}w_{b}\big{(}\epsilon\psi^{a}{\dot{x}}^{b}-\epsilon\psi^{b}{\dot{x}}^{a}\big{)}\
,$ (3.5)
one finds a canceling term of type
$\displaystyle\delta\big{(}+i\partial_{a}w_{b}\psi^{a}\psi^{b}\big{)}=+i\partial_{a}w_{b}\big{(}{\dot{x}}^{a}\epsilon\psi^{b}-{\dot{x}}^{b}\epsilon\psi^{a}\big{)}\
.$ (3.6)
In summary, the following Lagrangian has ${\cal N}=1$ supersymmetry
$\displaystyle{\cal
L}=\frac{1}{2}{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}\psi^{a}{\dot{\psi}}^{a}+\frac{i}{2}\lambda{\dot{\lambda}}-w_{a}{\dot{x}}^{a}+i\partial_{a}w_{b}\psi^{a}\psi^{b}\
.$ (3.7)
In order to introduce the bosonic potential to the above model, we modify the
transformation rule of the auxiliary fermion $\lambda$ as
$\displaystyle\delta x^{a}=-i\epsilon\psi^{a}\
,\qquad\delta\psi^{a}=\epsilon{\dot{x}}^{a}\ ,\qquad\delta\lambda=\epsilon K\
,$ (3.8)
upon which we find
$\displaystyle\delta\left(\frac{1}{2}{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}\psi^{a}{\dot{\psi}}^{a}+\frac{i}{2}\lambda{\dot{\lambda}}-w_{a}{\dot{x}}^{a}+i\partial_{a}w_{b}\psi^{a}\psi^{b}\right)=-i\epsilon\lambda{\dot{x}}^{a}\partial_{a}K\
.$ (3.9)
The canceling term for this is
$\displaystyle\delta\big{(}+i\partial_{a}K\psi^{a}\lambda\big{)}=$
$\displaystyle
i\epsilon\lambda{\dot{x}}^{a}\partial_{a}K-i\epsilon\psi^{a}K\partial_{a}K$
$\displaystyle=$ $\displaystyle
i\epsilon\lambda{\dot{x}}^{a}\partial_{a}K+\delta x^{a}K\partial_{a}K\ ,$
(3.10)
while one must add one more to close the transformation algebra,
$\displaystyle\delta\big{(}-\frac{1}{2}K^{2}\big{)}=-\delta
x^{a}K\partial_{a}K\ .$ (3.11)
In summary, the Lagrangian
$\displaystyle{\cal
L}=\frac{1}{2}{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}\psi^{a}{\dot{\psi}}^{a}+\frac{i}{2}\lambda{\dot{\lambda}}-w_{a}{\dot{x}}^{a}-\frac{1}{2}K^{2}+i\partial_{a}w_{b}\psi^{a}\psi^{b}+i\partial_{a}K\psi^{a}\lambda\
$ (3.12)
has ${\cal N}=1$ supersymmetry, for any $K$ and $w$.
We eventually wish to formulate dyon dynamics with ${\cal N}=4$
supersymmetries. For conventional supersymmetric quantum mechanics, this
requires the target manifold to be $4n$ dimensional and hyperKähler, which is
clearly inappropriate for our $3n$ dimensional target. Nevertheless, the BPS
nature of the dyons and existence of BPS bound states implies that there
should exist such an ${\cal N}=4$ lift.
To find the relevant supersymmetries and the subsequent restrictions on the
potentials, note that, since the number of bosons and the number of fermions
mismatch by 3 to 4, we can organize the degrees of freedom using
$SO(4)=SU(2)_{L}\times SU(2)_{R}$ algebra. Let the bosons, $x^{a},a=1,2,3$,
transform as $({\bf 3,1})$ representation while the fermions,
$(\psi^{a},\lambda)$, are naturally in $({\bf 2,2})$ and better denoted as
$\psi^{m},m=1,2,3,4$ with $\psi^{4}=\lambda$. Thus, $a,b,\dots$ are the vector
indices of $SU(2)_{L}$ while $m,n,\dots$ are vector indices of $SO(4)$. The
${\cal N}=4$ supersymmetries are then naturally in $({\bf 2,2})$ under this
$SO(4)$, since it should relate bosons to fermions. Thus the four
supersymmetry transformation parameters will be denoted by $\epsilon_{m}$.
A useful method of relating $SU(2)_{L}$ objects to $SO(4)$ object is to employ
’t Hooft’s self-dual symbol $\eta^{a}_{mn}$. Based on previous experience of
embedding $SO(3)\simeq SU(2)_{L}$ into $SO(4)$, such as in Yang-Mills
instanton construction, one can guess the following ${\cal N}=4$ SUSY
transformation rules
$\displaystyle\delta x^{a}=i\eta^{a}_{mn}\epsilon^{m}\psi^{n}\
,\qquad\delta\psi_{m}=\eta^{a}_{mn}\epsilon^{n}{\dot{x}}^{a}+\epsilon_{m}K\ ,$
(3.13)
with the ’t Hooft self-dual symbol $\eta$ defined as [38]
$\eta^{a}_{bc}=\epsilon_{abc},\qquad\eta^{a}_{b4}=\delta^{a}_{b}=-\eta^{a}_{4b}\
.$ (3.14)
which, for $\epsilon^{4}\equiv\epsilon$, matches (3.8).
This suggests that the Lagrangian (3.12) can be extended to admit ${\cal N}=4$
supersymmetries, if we can organize the fermion bilinears in terms of $\eta$
symbol as
$\displaystyle{\cal
L}=\frac{1}{2}\sum_{a=1}^{3}{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}\sum_{m=1}^{4}\psi^{m}{\dot{\psi}}^{m}+\frac{i}{2}\eta^{a}_{mn}\partial_{a}K\psi^{m}\psi^{n}-w_{a}{\dot{x}}^{a}-\frac{1}{2}K^{2}\
,$ (3.15)
which matches (3.12) if we impose
$\displaystyle\epsilon^{abc}\partial_{a}K=\partial_{b}w_{c}-\partial_{c}w_{b}\
.$ (3.16)
One can indeed show that the above Lagrangian is invariant under the ${\cal
N}=4$ SUSY transformation rules (3.13). This Lagrangian is manifestly
invariant under $SU(2)_{R}$. The $SU(2)_{L}$ invariance is broken only to the
extent that $K$ breaks the rotational invariance. If $K$ is spherically
symmetric, for instance, the full $SO(4)$ symmetry would be restored.
Let us discuss the closure of the ${\cal N}=4$ algebra. For bosonic variables,
one can show
$\displaystyle\delta_{\zeta}\delta_{\epsilon}x^{a}=-i\eta^{a}_{mn}\eta^{b}_{pn}\epsilon^{m}\zeta^{p}{\dot{x}}^{b}+i\eta^{a}_{mn}\epsilon^{m}\zeta^{n}K\
,$ (3.17)
which implies
$\displaystyle\big{(}\delta_{\zeta}\delta_{\epsilon}-\delta_{\epsilon}\delta_{\zeta}\big{)}x^{a}=-2i\epsilon^{m}\zeta^{m}{\dot{x}}^{a}\
.$ (3.18)
Here we used the following identity
$\eta^{a}_{mn}\eta^{b}_{pn}=\delta^{ab}\delta_{mp}+\epsilon^{abc}\eta^{c}_{mp}$.
Let us now in turn consider the case of fermionic variables.
$\displaystyle\big{(}\delta_{\zeta}\delta_{\epsilon}-\delta_{\epsilon}\delta_{\zeta}\big{)}\psi_{m}=$
$\displaystyle-2i\epsilon^{n}\zeta^{n}{\dot{\psi}}_{m}+i\big{(}\epsilon^{n}\zeta^{m}+\epsilon^{m}\zeta^{n}\big{)}{\dot{\psi}}^{n}+i\eta^{a}_{pq}\partial_{a}K\big{(}\epsilon^{p}\zeta^{m}+\epsilon^{m}\zeta^{p}\big{)}\psi^{q}\
,$ $\displaystyle=$ $\displaystyle-2i\epsilon^{n}\zeta^{n}{\dot{\psi}}_{m}\ ,$
(3.19)
where for the last equality we used the equation of motion of $\psi^{m}$
$\displaystyle{\dot{\psi}}^{q}+\eta^{a}_{qn}\partial_{a}K\psi^{n}=0\ .$ (3.20)
We therefore conclude that the ${\cal N}=4$ SUSY algebra is given by
$\displaystyle\big{\\{}Q_{m},Q_{n}\big{\\}}=2\delta_{mn}H\ ,$ (3.21)
with the Hamiltonian $H$.
### 3.2 Toy Model with ${\cal N}=1$ Superfields
We can write the above Lagrangian by introducing ${\cal N}=1$ superspace with
an anti-commuting coordinate $\theta$. Following the notation in Ref. [22], we
define the bosonic and the fermionic superfields as
$\Phi^{a}=x^{a}-i\theta\psi^{a},\quad\Lambda=i\lambda+i\theta b\ .$ (3.22)
The supersymmetry generator and the supercovariant derivatives are then,
$Q=\partial_{\theta}+i\theta\partial_{t},\quad
D=\partial_{\theta}-i\theta\partial_{t}\ .$ (3.23)
Our toy model based on flat $R^{3}$, with scalar and vector potentials, can be
written in a superspace form as
${\cal L}=\int
d\theta\;\left(\frac{i}{2}D\Phi^{a}\partial_{t}\Phi^{a}-\frac{1}{2}\Lambda
D\Lambda+iK(\Phi)\Lambda-iw(\Phi)_{a}D\Phi^{a}\right)\ .$ (3.24)
Although only ${\cal N}=1$ supersymmetry is manifest, we saw that ${\cal N}=4$
supersymmetry will emerge if the condition $*dK=dw$ is imposed. This form of
the Lagrangian is useful because it could be generalized to the curved moduli
space almost immediately.
### 3.3 Massive ${\cal N}=4$ Theory onto Conformally Flat $R^{3}$
Recall that, for a single probe dyon, there are three quantities that appears
in the bosonic moduli dynamics. The scalar and the vector potentials, as we
already incorporated into ${\cal N}=4$ toy model above, and most crucially,
the position-dependent mass term $|{\cal Z}_{h}|$ for the coordinates $x^{a}$.
Thus, in addition to the above interaction terms, we wish to replace $R^{3}$
by a conformal flat $R^{3}$ whose metric is
$g_{ab}=f\delta_{ab}\ ,$ (3.25)
with $f$ later to be identified with $|{\cal Z}_{h}|$. In fact, as can be
inferred from the massless version in Refs. [21, 22], ${\cal N}=4$
supersymmetry restricts the three-dimensional metric to be conformally flat.
We defer detailed construction to appendix B, and simply state here that the
desired Lagrangian, now with potentials, has the superspace form
$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle\int
d\theta\;\biggl{(}\frac{i}{2}f(\Phi)D\Phi^{a}\partial_{t}\Phi^{a}-\frac{1}{2}f(\Phi)\Lambda
D\Lambda$ (3.26)
$\displaystyle+\frac{1}{4}\epsilon_{abc}\partial_{a}f(\Phi)D\Phi^{b}D\Phi^{c}\Lambda+i{\cal
K}(\Phi)\Lambda-i{\cal W}(\Phi)_{a}D\Phi^{a}\biggr{)}\ ,$
with the condition
$\partial_{a}{\cal K}=\epsilon_{abc}\,\partial_{b}{\cal W}_{c}$ (3.27)
imposed. In terms of component fields, this equals
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}f\left({\dot{x}}^{a}{\dot{x}}^{a}+i\psi^{m}\nabla_{t}\psi^{m}\right)$
(3.28) $\displaystyle-\frac{1}{4\cdot 4!}\left(2\partial^{2}f-f^{-1}(\partial
f)^{2}\right)\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}$
$\displaystyle-\frac{1}{2f}{\cal K}^{2}-{\cal
W}_{a}{\dot{x}}^{a}+\frac{i}{2}f^{1/2}\partial_{a}\big{(}f^{-1/2}{\cal
K}\big{)}\eta^{a}_{mn}\psi^{m}\psi^{n}\ ,$
where the covariant derivative for fermions is given by
$\displaystyle\nabla_{t}\psi^{m}={\dot{\psi}}^{m}+\frac{1}{2}\epsilon_{abc}{\dot{x}}^{a}f^{-1}\partial_{b}f\eta^{c}_{mn}\psi^{n}\
.$ (3.29)
As in the flat case, the degrees of freedom and the supercharges are cataloged
by $SO(4)=SU(2)_{L}\times SU(2)_{R}$ algebra, and the Lagrangian is manifestly
invariant under $SU(2)_{R}$. The $SU(2)_{L}$ keeps track of how $f$ and ${\cal
K}$ (and thus ${\cal W}$ also) transform under spatial rotations, and become a
symmetry whenever these quantities are rotationally invariant.
This $SO(4)$ structure and $SU(2)_{R}$ symmetry tells us an extended ${\cal
N}=4$ supersymmetry exists, as in the flat $R^{3}$ example. It is not
difficult to see that
$\displaystyle\delta_{\epsilon}x^{a}$ $\displaystyle=$ $\displaystyle
i\eta^{a}_{mn}\epsilon^{m}\psi^{n}\ ,\qquad$
$\displaystyle\delta_{\epsilon}\psi_{m}$ $\displaystyle=$
$\displaystyle\eta^{a}_{mn}\epsilon^{n}{\dot{x}}^{a}+\epsilon_{m}b\;,$ (3.30)
with four Grassman parameters $\epsilon^{m}$ leaves the Lagrangian invariant.
The only difference from the flat case, (3.17), is that $K$ is replaced by its
generalized form, namely on-shell value of the ${\cal N}=1$ auxiliary field
$b$,
$\displaystyle b=\frac{1}{f}\,\left({\cal
K}+\frac{i}{4}\eta^{a}_{pq}\partial_{a}f\psi^{p}\psi^{q}\right)\ .$ (3.31)
The superalgebra remains the same as the flat case,
$\displaystyle\big{\\{}Q_{m},Q_{n}\big{\\}}=2\delta_{mn}H\ ,$ (3.32)
where we denoted the four supercharges by $Q_{m}$ as before and the
Hamiltonian by $H$. For completeness, we also record the classical form of the
Hamiltonian,
$\displaystyle H_{classical}$ $\displaystyle=$
$\displaystyle\frac{1}{2f}\pi^{a}\pi^{a}+\frac{1}{4\cdot
4!}\left(2\partial^{2}f-f^{-1}(\partial
f)^{2}\right)\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}$ (3.33)
$\displaystyle+\frac{1}{2f}\,{\cal
K}^{2}-\frac{i}{2}f^{1/2}\partial_{a}\big{(}f^{-1/2}{\cal
K}\big{)}\eta^{a}_{mn}\psi^{m}\psi^{n}\ ,$
with the covariantized momenta
$\pi^{a}=p_{a}+{\cal
W}_{a}-\frac{i}{4}\epsilon_{abc}\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}\ .$
(3.34)
The quantum Hamiltonian differs from this by normal ordering issue, and can
also be found in appendix B.
## 4 Quantum BPS States near WMS
### 4.1 ${\cal N}=4$ Low Energy Dynamics of Dyons near MSW
Let us stop here and ask under what circumstances we actually expect to see a
sensible low energy dynamics of dyons to appear. The old setting based on
dyons as quantum bound states of excited magnetic solitons was possible by
resorting to weakly coupled regime. There, the basic requirements was that the
energy due to electric charges and also due to motion of the solitons are of
higher order. Thus, the reduction to quantum mechanics is controlled two small
parameters; typical speed of the magnetic soliton and the electric coupling
constant [32].
Here, however, we are here dealing with dyons of generic charges at generic
coupling, and must find different criteria to justify reduction to low energy
quantum mechanics. Note that the weak coupling requirement and the small speed
requirement of old moduli dynamics is in fact interrelated. That happens was
that the moduli dynamics of ${N}=2$ and ${N}=4$ monopoles usually acquire a
bosonic potential of order $e^{2}$, so for typical states the small electric
coupling is necessary to ensure small velocities.
In the present low energy dynamics of probe dyons around a core state, the
size of the potential is instead controlled by how far are the phases of
central charges of core and halo particles are aligned. Thus, by staying very
near MSW, we have a good control over the potentials. Furthermore, the massgap
between this sector and the rest is also substantial, and controls possible
interference from other charged particles.#7#7#7The latter is easiest to see
when the small mass ratio is achieved by being near a singular point of the
vacuum moduli space. The relevant coupling that governs the interaction of the
field theory would be a dualized coupling which becomes small as the singular
point is approached. So it is the proximity to the MSW and also the mass ratio
of the two parts that now control the reduction to the low energy quantum
mechanics.
With this mind, we compare (2.21) against the supersymmetric Lagrangian
(3.28). One can see the supersymmetric uplift may work only if
$\displaystyle f=|{\cal Z}_{h}|\ ,\qquad\frac{1}{2f}{\cal K}^{2}=|{\cal
Z}_{h}|-{\rm Re}[\zeta^{-1}{\cal Z}_{h}],\qquad\vec{\nabla}\times\vec{\cal
W}=\vec{\nabla}\left(\text{Im}[\zeta^{-1}{\cal Z}_{h}]\right)\ .$ (4.1)
but the requisite ${\cal N}=4$ relationship between ${\cal K}$ and ${\cal W}$,
$*d{\cal K}=d{\cal W}$, is not yet apparent. Thankfully, this condition is
satisfied precisely when the criteria for the low energy approximation are
met, as we described above.
To see the latter, write $\zeta^{-1}{\cal Z}_{h}=|\,{\cal
Z}_{h}|\,e^{i\beta}$. Near the wall of marginal stability, the angle $\beta$
at spatial infinity is very small whereas its value at classical vacuum is 0.
Recall that the bound states we wish to find and count are all peaked at the
classical vacuum manifold. This allows us to expand relevant quantities in
small $\beta$. As we move closer to charge centers, $\vec{x}_{A}$’s, $\beta$
can grow again but the precisely form of the background at such charge centers
are not to be trusted and also happily immaterial for our purpose of finding
BPS bound states. Therefore, we take the value of $\beta$ to be small
everywhere and find
${\cal K}^{2}=2|\,{\cal Z}_{h}|^{2}(1-\cos\beta)\simeq|\,{\cal
Z}_{h}|^{2}\beta^{2}\simeq|\,{\cal
Z}_{h}|^{2}(\sin\beta)^{2}=\left(\text{Im}[\zeta^{-1}{\cal Z}_{h}]\right)^{2}\
.$ (4.2)
Thus, for all practical purpose, we may identify ${\cal
K}=\text{Im}[\zeta^{-1}{\cal Z}_{h}]$ and the ${\cal N}=4$ requirement (3.27)
is obeyed automatically. This completes the derivation of ${\cal N}=4$ moduli
dynamics in (3.28) of a probe dyon in a given core state background.
The function ${\cal K}$ can be generally written, from (2.11), as
${\cal K}={\cal
K}_{0}-\sum_{A}\frac{\langle\gamma_{c,A},\gamma_{h}\rangle}{|\,\vec{x}-\vec{x}_{A}|}\
,$ (4.3)
with $\gamma_{c,A}$ centers of the core states at $\vec{x}_{A}$ and also
${\cal K}_{0}\equiv\text{Im}[\zeta^{-1}{\cal Z}_{h}(\infty)].$ Details of
$f=|{\cal Z}_{h}|$ won’t matter much for the purpose of constructing bound
states, it turns out, as long as we keep track of its singular behaviors at
charge centers.
Before we start the detailed analysis, let us note again that the
semiclassical core state here is not really a good representation very near
its charge center(s), where the non-Abelian nature of the states becomes
relevant. Naturally, the low energy dynamics of probe dyons is plagued by the
same issue. However, this hardly matters near MSW because the bound state (if
any) would be very large and determined entirely by Abelian part of the low
energy field theory: Whatever singularity at Coulombic centers cannot alter
such wavefunction significantly, as long as we impose the boundary condition
at centers intelligently enough. This should become more obvious when we
discuss actual bound state wavefunctions in section 4.3. For this reason, we
may as well take the above form of ${\cal K}$, $f$, etc literally, and
consider supersymmetric bound states thereof, with some care given to the
boundary condition of the wavefunctions at centers $\vec{x}_{A}$.
### 4.2 Quantization and Supercharges
Let us start with the canonical commutators. The conjugate momenta of bosons
are denoted as $p_{a}$,
$[p_{a},x^{b}]=-i\delta^{b}_{a}\ ,$ (4.4)
whereas the normalized fermions, $\hat{\psi}^{m}\equiv{f}^{1/2}\psi^{m}$, are
more convenient for writing out the remaining canonical commutators,
$\\{\hat{\psi}^{m},\hat{\psi}^{n}\\}=\delta^{mn},\qquad[p_{a},\hat{\psi}^{m}]=0=[x^{a},\hat{\psi}^{n}]\
.$ (4.5)
With this we can now write the four supercharges as
$\displaystyle Q_{m}=-\eta^{a}_{mn}\psi^{n}(p_{a}+{\cal
W}_{a})+\frac{i}{4}\eta^{a}_{mn}f^{-1}\partial_{a}f\psi^{n}+\frac{i}{4}\partial_{a}f\eta^{a}_{pq}\psi^{[p}\psi^{q}\psi^{m]}+{\cal
K}\psi^{m}\ .$ (4.6)
For the proof that these are right supercharges, see appendix B. In
particular, the supercharge associated with $\epsilon^{4}$ is
$Q=Q_{4}=\psi^{a}\left(p_{a}+{\cal
W}_{a}\right)-\frac{i}{4}f^{-1}\partial_{a}f\psi^{a}+\frac{i}{4}\partial_{a}f\epsilon_{abc}\psi^{b}\psi^{c}\lambda+\lambda{\cal
K}\ .$ (4.7)
Since the superalgebra implies $\\{Q_{m},Q_{n}\\}=2\delta_{mn}H$, the ground
state of the system can be found by demanding that it be annihilated by
$Q_{4}$.
### 4.3 BPS Bound States and Marginal Stability
The canonical commutator of the fermions
$\\{\hat{\psi}^{m},\hat{\psi}^{n}\\}=\delta^{mn}$ (4.8)
is a Clifford algebra which can be represented by Dirac matrices,
$\sqrt{2}\;\hat{\psi}^{a}=\gamma^{a}=\left(\begin{array}[]{ll}0&\sigma^{a}\\\
\sigma^{a}&0\end{array}\right),\qquad\sqrt{2}\hat{\lambda}=\sqrt{2}\;\hat{\psi}^{4}=\gamma^{4}=\left(\begin{array}[]{cc}0&i\\\
-i&0\end{array}\right)\ ,$ (4.9)
and wavefunctions can be regarded as 4-component spinors on $R^{3}$. Also
useful is the chirality operator
$\Gamma\equiv\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\ .$ (4.10)
Under the above representation, one of supercharge $Q_{4}$ now has a simple
form,
$\sqrt{2f}\;Q_{4}=\gamma^{a}(p_{a}+{\cal
W}_{a})-\frac{i}{2}(\partial_{a}\text{log}f)\gamma^{a}\,\frac{1-\Gamma}{2}+{\cal
K}\gamma^{4}\ ,$ (4.11)
or more explicitly,
$\sqrt{2f}\;Q_{4}=\left(\begin{array}[]{cc}0&\sigma\cdot(p+{\cal W})+i{\cal
K}\\\ \sigma\cdot(p+{\cal W})-i\sigma\cdot\partial(\log f^{1/2})-i{\cal
K}&0\end{array}\right)\ .$ (4.12)
We wish to find supersymmetric ground states, $Q_{4}\Psi=0$. Since ${\cal
H}\Psi=0$ then, such states would actually preserve all four supercharges.
Such states are then automatically BPS with respect to the $N=2$ field theory
with the energy $\text{Re}[\zeta^{-1}{Z}_{h}(\infty)]$, as can be seen from
(2.21), not counting the core state energy.
Write the four-component wavefunction $\Psi$ as
$\Psi=\left(\begin{array}[]{c}f^{-1/2}{\cal U}\\\ {\cal V}\end{array}\right)\
,$ (4.13)
upon which two component wavefunctions ${\cal U}$ and ${\cal V}$ obey
$\displaystyle\left(\sigma\cdot(p+{\cal W})-i{\cal K}\right){\cal U}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\left(\sigma\cdot(p+{\cal
W})+i{\cal K}\right){\cal V}$ $\displaystyle=$ $\displaystyle 0,$ (4.14)
With the supersymmetry condition $d{\cal K}=*d{\cal W}$, it is easy to see
that the first equation cannot have a normalizable solution while the second
may. Denoting the respective operators as ${\cal D}_{\pm}$,
${\cal D}_{\mp}{\cal D}_{\pm}=\left(p+{\cal W}\right)^{2}+{\cal
K}^{2}+\sigma^{a}\left(\partial_{a}{\cal K}\pm\partial_{a}{\cal K}\right)\ ,$
(4.15)
which shows that ${\cal D}_{+}{\cal D}_{-}$ is a positive definite operator
while ${\cal D}_{-}{\cal D}_{+}$ is not. Only the latter can have zero modes.
Thus, we arrived at the conclusion that the counting of BPS bound states
between the core dyon and the probe dyon becomes that of counting normalizable
two-component zero modes ${\cal V}$ of the operator ${\cal D}_{+}$, with the
final form of the BPS bound state
$\Psi=\left(\begin{array}[]{c}0\\\ {\cal V}\end{array}\right)\ ,$ (4.16)
with ${\cal D}_{+}{\cal V}=0$.
It is illuminating to solve this equation for the particular case of
spherically symmetry core state. The vector potential would be that of a Dirac
monopole, so we denote
${\cal W}=-gA_{\text{Dirac}},\qquad
g=-\langle\gamma_{c},\gamma_{h}\rangle,\quad A_{Dirac}=-\cos\theta d\phi\ ,$
(4.17)
from which follows the scalar potential
${\cal K}={\cal K}_{0}+\frac{g}{r}\ .$ (4.18)
In this case $SU(2)_{L}$ also becomes a symmetry, allowing an explicit
solution to the bound state problem. The number $g$ is half-integer quantized,
as dictated by the Dirac quantization of this quantum mechanics, and also from
its field theory origin as the Schwinger product of the two quantized charge
vectors. The angular and spin part of the wavefunction is classified by
spinorial monopole spherical harmonic tensor. The lowest possible angular
momentum would be than $j=|g|-1/2$, since the charge interacting with such a
Dirac monopole, ${\cal W}$, is endowed with a well-known angular momentum
$-g\hat{r}$. Tensoring with the intrinsic spin 1/2, the minimum possible value
$j=|g|-1/2$ follows.
Denoting the corresponding the lowest-lying two-component angular momentum
eigen-states $\eta_{j=|g|-1/2,m}$ of $SO(3)\simeq SU(2)_{L}$ we rely on Kazama
et.al. [39] for reduction of the above to the radial equation,
${\cal
V}=h(r)\eta_{j=|g|-1/2,m},\qquad\left(-i\frac{g}{|\,g|}\times\left[\frac{d}{dr}+\frac{1}{r}\right]+i{\cal
K}(r)\right)h(r)=0\ .$ (4.19)
Integrating the latter equation, we find
$h(r)=\frac{1}{r}\exp\left(\frac{g}{|g|}\int^{r}{\cal
K}(r)\right)=C\,r^{|\langle\gamma_{c},\gamma_{h}\rangle|-1}\exp\left(-[\text{sgn}(\langle\gamma_{c},\gamma_{h}\rangle)\cdot{\cal
K}_{0}]\cdot r\right)\ ,$ (4.20)
with the normalization constant $C$. Note that this gives a normalizable
ground state if and only if the half-integer-quantized
$\langle\gamma_{c},\gamma_{h}\rangle$ is not zero and has the same sign as
${\cal K}_{0}={\text{Im}[\zeta^{-1}{\cal Z}_{h}(\infty)]}$.#8#8#8$L^{2}$
normalizability requirement from $r=0$ region is satisfied as long as
$|\langle g_{c},\gamma_{h}\rangle|$ is not zero, so does not impose additional
restriction. The latter condition is also reflected on the fact that the
probability density of this wavefunction is peaked at radial size
$\frac{\langle\gamma_{c},\gamma_{h}\rangle}{\text{Im}[\zeta^{-1}{\cal
Z}_{h}(\infty)]}\ ,$ (4.21)
which, for a single-center core state, exactly mirrors the classical orbit
radius in (2.22).
The sign of $\langle\gamma_{c},\gamma_{h}\rangle$ is determined by the charges
of the core state and the probe state, and does not change as we move along
the vacuum moduli space. However, the ${\cal K}_{0}=\text{Im}[\zeta^{-1}{\cal
Z}_{h}(\infty)]$ does change its sign across the marginal stability wall
between the core state and the probe state. Classically, this happens because
$g{\cal K}_{0}<0$ would make the potential repulsive. The upshot is that the
BPS bound states of one side where
$\langle\gamma_{c},\gamma_{h}\rangle/\text{Im}[\zeta^{-1}{\cal
Z}_{h}(\infty)]>0$ disappear as we move to the other side where
$\langle\gamma_{c},\gamma_{h}\rangle/\text{Im}[\zeta^{-1}{\cal
Z}_{h}(\infty)]<0$, as was originally found in the supergravity setting.
With this exercise, we learned a few things:
1. $\bullet$
Normalizable bound state between the core state and the probe state is
realized only when the Schwinger product of the two charge is nonzero.
2. $\bullet$
Normalizable bound state between the core state and the probe state is
realized only when the Schwinger product of the two charge is of the same sign
relative to the value of $\text{Im}[\zeta^{-1}{\cal Z}_{h}]$ at spatial
infinity.
3. $\bullet$
When such normalizable states exist, the degeneracy is
$2j+1=2|\langle\gamma_{c},\gamma_{h}\rangle|$.
Much of the above statements are properties of a Dirac operator with ${\cal
D}_{\pm}$ as the chiral and the anti-chiral parts; there must be an index
theorem associated with them.
In fact, the structure of the operators are essentially that of an
electrically charged fermionic field around the magnetic monopole, except that
we do not see the non-Abelian structure that regulate the short-distance
behavior of the core state. Similar issues in the context of quantization in
the backgrounds of non-Abelian monopoles vs. Dirac monopoles (or more
precisely Wu-Yang monopoles [40]) have been studied in depth decades ago,
where it was found that with proper boundary condition at origins of the
latter, behaviors of the two are essentially the same [41]. The boundary
condition is constrained by the requirement that the Dirac operator
constructed out of ${\cal D}_{\pm}$ should be Hermitian, which is known in the
literature as the self-adjoint extension.
This is related to the fact that, even though the two potentials of the
quantum mechanics are singular at origin, the wavefunctions found are regular
everywhere and in particular suppressed strongly at origin. If we attempted to
solve for ${\cal D}_{-}{\cal U}=0$, the radial eigen-function of ${\cal U}$
would have the behavior $r^{-|\langle\gamma_{c},\gamma_{h}\rangle|-1}$ at
origin and is clearly unacceptable. This again shows that only ${\cal D}_{+}$
can have a solution. In particular, the supersymmetric bound state are
trustworthy even though the quantum mechanics itself would be corrected, at
small $r$, by non-Abelian nature of such objects.
Therefore, the index problem of the above operator is on par with that of zero
mode problems around non-Abelian monopoles; the Callias index theorem [42, 29,
30] should apply. We thus anticipate that the number of zero energy bound
states is additive; when the core state is composed of many centers of charges
$\gamma_{c,A}$ with $\langle\gamma_{c,A},\gamma_{h}\rangle{\cal K}_{0}>0$, the
number of the bound state of the probe dyon is the naive one,
$2|\langle\gamma_{c},\gamma_{h}\rangle|=\Big{|}\sum_{A}2\langle\gamma_{c,A},\gamma_{h}\rangle\Big{|}\
,$ (4.22)
since $\gamma_{c}=\sum_{A}\gamma_{c,A}$.
## 5 Wall-Crossing from Moduli Dynamics
### 5.1 Primitive Wall-Crossing: $\gamma_{c}+\gamma_{h}$
So far, we ignored the precise supermultiplet structures; Our approximation
allowed us to treat the supermultiplet structure of the core state as a
separate sector, while we extracted only partial sector of the probe dyons
which would have been responsible for building a half-hypermultiplet. More
generally, the probe dyon can come with higher spin states, such as $N=2$
vector multiplet or higher, so we may decompose the Hilbert space of the
combined core-probe system as
${\cal H}_{\text{core}}\otimes{\cal
H}_{\text{probe}}^{\text{reduced}}\otimes{\cal H}_{\text{moduli dynamics}}\ .$
(5.1)
The reduced Hilbert space denotes part of the free Hilbert space of a BPS
particle that multiplies the half-hypermultiplet,
${\cal H}={\cal H}^{\text{reduced}}\otimes\left([{1/2}]\oplus 2[{0}]\right)\
.$ (5.2)
When the probe dyon is in the half-hypermultiplet,#9#9#9 Recall that usual
hypermultiplet forms when the CTP conjugate states are taken into account.
${\cal H}^{\text{reduced}}_{\text{probe}}$ would have only one state, while in
the vector multiplet, it would be the angular momentum 1/2 Hilbert space, etc.
The decomposition (5.1) can be understood easily. The core part of the Hilbert
space is inert, so can be treated as non-dynamical. Of the probe, the half-
hypermultiplet part are generated by the universal would-be Goldstino modes
which become no longer free due to the presence of the core state. Instead
they participate in the moduli dynamics we constructed and thus belong to
${\cal H}_{\text{moduli dynamics}}$. Note that these four modes would become
free at $r=\infty$, regaining its nature as Goldstino. The remaining part
${\cal H}^{\text{reduced}}_{\text{probe}}$ accounts for extra degeneracies and
spin content of the probe supermultiplet, which should represent additional
structure on top of the low energy dynamics.
On the other hand, the second helicity trace (1.3), which is the relevant
index for $N=2$ theories, takes value
$\Omega\left([j]\otimes\left([{1/2}]\oplus
2[{0}]\right)\right)=(-1)^{2j}(2j+1)$ (5.3)
for the irreducible angular momentum multiplet $[j]$, and can also be
expressed as
$\Omega\left({\cal H}\right)=\text{tr}_{{\cal
H}^{\text{reduced}}}(-1)^{2j_{3}}\ .$ (5.4)
The degrees of freedom for the core state does not participate in the
dynamics, so we have the decomposition
$\displaystyle\Omega\left({\cal H}_{\text{core}}\otimes{\cal
H}_{\text{probe}}^{\text{reduced}}\otimes{\cal H}_{\text{moduli
dynamics}}\right)$ (5.5) $\displaystyle=$ $\displaystyle\Omega\left({\cal
H}_{\text{core}}\right)\times\text{tr}_{{\cal
H}_{\text{probe}}^{\text{reduced}}}(-1)^{2j_{3}}\times\text{tr}_{{\cal
H}_{\text{moduli dynamics}}}(-1)^{2J_{3}}$ $\displaystyle=$
$\displaystyle\Omega\left({\cal
H}_{\text{core}}\right)\times\Omega\left({{\cal
H}_{\text{probe}}}\right)\times\text{tr}_{{\cal H}_{\text{moduli
dynamics}}}(-1)^{2j_{3}}\ .$
Combining with the supersymmetric bound state we found above, this reproduces
the primitive wall-crossing formula of Denef,
$\Delta\Omega(\gamma_{c}+\gamma_{h})=-(-1)^{2|\langle\gamma_{c},\gamma_{h}\rangle|}\;2|\langle\gamma_{c},\gamma_{h}\rangle|\;\Omega(\gamma_{c})\,\Omega(\gamma_{h})\
.$ (5.6)
### 5.2 Semi-Primitive Wall Crossing: $\gamma_{c}+n\gamma_{h}$
The semi-primitive wall-crossing formula of Denef and Moore conjectures how
many BPS states of charge $\gamma_{c}+n\gamma_{h}$ appears across a MSW, for
positive integer $n$ In order to compute the degeneracies of such states we
must consider $n$ number of $\gamma_{h}$ charges in the core state background
of $\gamma_{c}$. The Lagrangian would be
${\cal L}=\sum_{i=1}^{n}{\cal L}_{(i)}+{\cal L}_{hh}\ ,$ (5.7)
where ${\cal L}_{(i)}$ denotes the one-particle Lagrangian for $i$-th probe
dyon, all of which are of the identical form. ${\cal L}_{hh}$ captures the
interaction among (identical) probe particles.
In our approximation, the latter can be ignored as long as the charges are
such that
$|\langle\gamma_{c},\gamma_{h}\rangle|\gg|\langle\gamma_{h},\gamma_{h}^{\prime}\rangle|\
.$ (5.8)
In particular this is the case if the probe charges are all mutually local,
e.g., the same or proportional to each other. Then, the latter term ${\cal
L}_{hh}$ represent the second order correction to the former’s first order
form and can be safely ignored. The only nontrivial remnant is the matter of
statistics, as in any quantum mechanics of many identical particles.
In addition, there is also a logical possibility that one-particle BPS states
of non-primitive charge $k\gamma_{h}$ exist. In supergravity, such states are
always there, since black holes can have any quantized charges. In field
theory setting, the situation is a little unclear. In five dimensions, multi-
instanton bound state do exist in the maximally supersymmetric Yang-Mills
theory as quantum one-particle states. However, they are tied to the UV
completion of this theory which is the mysterious $(2,0)$ theories. In the
more familiar four-dimensional Yang-Mills setting, we are yet to see such an
example. Nevertheless, we will include the possibility that the probe dyon of
our moduli dynamics is non-primitive. Then, counting the degeneracy of the
bound states $\gamma_{c}+n\gamma_{h}$ is basically identical to partition of
$n\gamma_{h}$ into identical halo particles of
$n\gamma_{h}=(\sum_{i}m_{i}k_{i})\gamma_{h}$ with some cares on the statistics
of each dyon of charge $k_{i}\gamma_{h}$. If it turns out that such non-
primitive states do not exist,#10#10#10The result of the previous section is
suggestive in this regard. The bound states exist only if the Schwinger
product of the two constituent charges are nonzero. Even if we take into
account the finite core mass, we expect that a single-particle bound state of
type $k\gamma+\gamma$ probably does not exist, which in induction suggests
absence of the state of charge $k\gamma$ for $k\geq 2$ altogether. An
interesting question is how this feature is modified in the realm of
supergravity, where black holes of large non-primitive charges appears. we
may simply set $\Omega(k\gamma_{h})=0$ for $k\geq 2$.
The question of statistics lead us to consider the intrinsic spin of the
individual probe particle in the moduli dynamics. While the quantum mechanics
by itself won’t tell us about statistics of the particle, we can invoke the
usual spin-statistics relation and instead ask about the spin. Recall that the
canonical commutators,
$\\{\hat{\psi}^{m},\hat{\psi}^{n}\\}=\delta^{mn}\ ,$ (5.9)
implies that the spatial rotation generators of $SU(2)_{L}$ acting on the
wavefunction are
$-\frac{i}{4}\,[\hat{\psi}^{a},\hat{\psi}^{b}]-\frac{i}{4}\,\epsilon_{abc}\,[\hat{\psi}^{c},\hat{\psi}^{4}]=\frac{1}{2}\,\epsilon_{abc}\left(\begin{array}[]{cc}0&0\\\
0&\sigma^{a}\end{array}\right)\ .$ (5.10)
This shows that the 4-component wavefunction, $\Psi$, consists of a single
spin doublet ${\cal V}$ in the lower half and a pair of spin singlet states
combined into the upper half part, ${\cal U}$. Recall that the bound states
can appear only in the ${\cal V}$ sector; the supercharge $Q_{4}$ is
effectively positively definite on ${\cal U}$ as we saw in section 4.3.
Therefore the BPS bound state of a (half-)hypermultiplet probe and the core
always involve of a spin 1/2 wavefunction.
More generally, the probe might be in a bigger multiplet, where ${\cal
H}^{\text{reduced}}_{\text{probe}}$ is also part of the data that enters the
probe dynamics although we simply factored it out. Taking into account the
latter, we can see that the probe particle can be seen as a particle of spin
content in the moduli quantum mechanics
${\cal H}^{\text{reduced}}_{\text{probe}}\otimes([1/2]\oplus 2[0])\ ,$ (5.11)
but the BPS bound state appears only in the sector ${\cal
H}^{\text{reduced}}_{\text{probe}}\otimes[1/2]$. For example, if ${\cal
H}^{\text{reduced}}_{\text{probe}}=[S]$, the total spin of the probe dyon that
is involved in the bound state formation is $S\pm 1/2$. Therefore, as far as
supersymmetric bound state formation goes, that the probe dyon can be treated
as if it is Boson or Fermion for $2S$ odd or even, respectively.
Such assignment of statistics is precisely what we expect on the field theory
ground: Note that $S=0$ correspond to the hypermultiplet while $S=1/2$ to the
vector multiplet. When one construct BPS dyons in the weakly coupled theory,
the simplest method is to excite massive electrically charged and
$L^{2}$-normalizable modes around magnetic soliton [8]. When the charged field
is in the hypermultiplet, the relevant excitations arise all from the Dirac
field and the Fermi statistics rule when we try to construct the dyons. For a
vector multiplet, additional modes arise both from the vector field, so the
Bosonic statistics become dominant. This naive construction works verbatim for
$N=4$ Yang-Mills theories, while for $N=2$ only slightly modified (i.e.,
degeneracy shift by unit) as seen from more rigorous index computation [14,
17]. When we phrase the $N=2$ result in terms of vector multiplet
contributions vs. hypermultiplet contributions, we see the above statistics
assignment emerging.
Interestingly, this statistics is correlated with the sign of index $\Omega$
of the probe dyon since
$\Omega\Big{[}[S]\otimes([{1/2}]\oplus 2[{0}])\Big{]}=(-1)^{2S}(2S+1)\ .$
(5.12)
Thus, in the context of our probe moduli dynamics, probe dyons with positive
$\Omega$ should behave as Fermions, while probe dyons with negative $\Omega$
should behave as Bosons. More generally, ${\cal
H}^{\text{reduced}}_{\text{probe}}$ can be a direct sum of more than one spin
sectors. We write
${\cal H}^{\text{reduced}}_{\text{probe}}=\oplus_{\sigma}[S_{\sigma}]={\bf
R}_{+}\oplus{\bf R}_{-}\ ,$ (5.13)
with ${\bf R}_{\pm}$ denoting the decomposition according to the sign
$(-1)^{2S_{\sigma}}$. Thus,
$\Omega_{\text{probe}}=\text{dim}{\bf R}_{+}-\text{dim}{\bf R}_{-}\ .$ (5.14)
For the purpose of the moduli quantum mechanics here, then, we effectively
have $\text{dim}{\bf R}_{+}$ Fermions and $\text{dim}{\bf R}_{-}$ Bosons of
the same probe charge.
Once this statistics issue is cleared, one can construct the generating
function for the index $\Omega(\gamma_{c}+n\gamma_{h})$ as follows
$\displaystyle\sum_{n=0}^{\infty}\Omega(\gamma_{c}+n\gamma_{h})q^{n}=\Omega(\gamma_{c})\cdot\text{Tr}\Big{[}\big{(}-1\big{)}^{2J_{3}}q^{N}\Big{]}\
.$ (5.15)
We used here the notation Tr to emphasize that it is performed also over the
dyons of various charges $k\gamma_{h}$ as well as over the individual Fock
space with the number operator $N$ that counts the multiple probe dyons of the
same charge. Let us split the number operator
$N=\sum_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}kN^{B}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}+\sum_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}kN^{F}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}$
with $N^{B}$ for bosons and $N^{F}$ for fermions. Here
$\big{|}j^{3}_{\text{ext}}\big{|}\leq\big{|}\langle\gamma_{c},k\gamma_{h}\rangle\big{|}-\frac{1}{2}$
and $\big{|}j^{3}_{\sigma}\big{|}\leq S_{\sigma}$. The relevant trace then
becomes
$\displaystyle\text{Tr}\Big{[}\big{(}-1\big{)}^{2J_{3}}q^{N}\Big{]}$
$\displaystyle=$
$\displaystyle\sum_{N^{B/F}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}}\
(-1)^{\sum_{k,j^{3}_{\text{ext}},j^{3}_{a}}(2j^{3}_{\text{ext}}+2j^{3}_{\sigma})\big{(}N^{B}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}+N^{F}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}\big{)}}q^{\sum_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}k\big{(}N^{B}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}+N^{F}_{k,j^{3}_{\text{ext}},j^{3}_{\sigma}}\big{)}}\
,$
which can be summed explicitly as
$\displaystyle\prod_{k}\prod_{j^{3}_{\text{ext}},j^{3}_{\sigma}}\bigg{(}\sum_{N^{B}=0}^{\infty}\Big{[}(-1)^{2k|\langle\gamma_{c},\gamma_{h}\rangle|}q^{k}\Big{]}^{N^{B}}\bigg{)}\cdot\prod_{k}\prod_{j^{3}_{\text{ext}},j^{3}_{\sigma}}\bigg{(}\sum_{N^{F}=0}^{1}\Big{[}-(-1)^{2k|\langle\gamma_{c},\gamma_{h}\rangle|}q^{k}\Big{]}^{N^{F}}\bigg{)}$
(5.16) $\displaystyle=$ $\displaystyle\prod_{k}\
\Big{[}1-(-1)^{2k|\langle\gamma_{c},\gamma_{h}\rangle|}q^{k}\Big{]}^{\text{dim}(j_{\text{ext}})\cdot\big{(}\text{dim}({\bf
R}_{+})-\text{dim}({\bf R}_{-})\big{)}}$ $\displaystyle=$
$\displaystyle\prod_{k}\
\Big{[}1-(-1)^{2k\langle\gamma_{c},\gamma_{h}\rangle}q^{k}\Big{]}^{2|\langle\gamma_{c},k\gamma_{h}\rangle|\Omega(k\gamma_{h})}\
.$
It shows that the generating function is
$\displaystyle\sum_{n=0}\Omega(\gamma_{c}+n\gamma_{h})q^{n}=\Omega(\gamma_{c})\prod_{k=1}\
\Big{[}1-(-1)^{2k\langle\gamma_{h},\gamma_{c}\rangle}q^{k}\Big{]}^{2k|\langle\gamma_{h},\gamma_{c}\rangle|\Omega(k\gamma_{h})}\
.$ (5.17)
This is precisely the semi-crossing wall-crossing formula conjectured by Denef
and Moore [18], provided that the one-particle states of charge
$\gamma_{c}+n\gamma_{h}$ are absent on the other side of the wall. Note that
the latter assumption is guaranteed by our moduli dynamics. Thus, by staying
near the walls of marginal stability and adjusting the probe dyon to be much
lighter than the core, we have derived the semi-primitive wall-crossing
formulae from the first principle.
## 6 Conclusion and Discussion
We have derived a ${\cal N}=4$ supersymmetry low energy dynamics that govern
probe dyons interacting with relatively heavy core states, in the long
distance approximation. The proximity of the Coulomb vacuum to the marginal
stability wall acts as a crucial control parameter that allows this non-
relativistic quantum mechanical description, and we were able to reproduce the
conjectured primitive and semi-primitive wall-crossing formulae for Seiberg-
Witten theory dyons.
An important technological step here was to incorporate the potential energy
of the probe particles, due to the core state, into the supersymmetric quantum
mechanics. Because the latter comes with different bosonic and fermionic
degrees of freedom, a nonconventional form of the supersymmetric low energy
theory emerged, but in a manner consistent with the BPS structure of the
underlying $N=2$ field theory in question.
As we mentioned early on, our approximation scheme was inspired by the notion
of framed BPS state in presence of a line operator. See Appendix C for a short
review on line operator in relation to the wall-crossing. In a sense the line
operator provides a setting where our computation would become an exact
description and can aid evaluation of the line operator expectation values.
The vacuum expectation of line operator is in effect a $(-1)^{F}$ weighted
trace over the Hilbert space with a particular charge object $\Gamma$ inserted
as an external object,
$\displaystyle\langle L_{\Gamma}\rangle=\text{Tr}_{{\cal
H}_{\Gamma}}\Big{[}(-1)^{F}e^{-2\pi R\hat{H}}\Big{]}\
,\qquad\hat{H}=\big{\\{}{\cal Q}_{\zeta}^{\dagger},{\cal Q}_{\zeta}\big{\\}}\
,$ (6.1)
where ${\cal Q}_{\zeta}$ denote the supercharges preserved by the line
operator. It was conjectured that this observable can be expanded into
$\displaystyle\langle
L_{\Gamma}\rangle_{\gamma_{h}}=\sum_{\gamma_{h}}\Omega(\Gamma+\gamma_{h}){\cal
X}_{\gamma_{h}}\ ,$ (6.2)
where ${\cal X}_{\gamma_{h}}$’s are the Darboux coordinates of [44]. The semi-
classical analysis on the conjectured form of $\langle L_{\Gamma}\rangle$
would be interesting and illuminating as in Ref. [43]. As noted by Gaiotto
et.al [44, 45], this asserts the much needed continuity property of ${\cal
X}$’s over the vacuum moduli space that plays a central role justifying KS
formalism in the context of $N=2$ Seiberg-Witten theory. Our low energy
quantum mechanics is consistent with this claim since
$\displaystyle\Omega(\Gamma+\gamma_{h},\zeta){\cal X}_{\gamma_{h}}(\zeta,R)$
(6.3) $\displaystyle=$ $\displaystyle
e^{-2\pi\text{Re}[\zeta^{-1}Z_{\gamma_{h}}]}\text{tr}_{{\Gamma+g_{h}}}\big{[}(-1)^{F}e^{-2\pi
RH_{\text{moduli}}-i\theta\cdot Q}\sigma(Q)\big{]}\times(\cdots)\ ,$
where the first two terms follows from discussions in section 2, while
$\sigma(Q)$ denotes the quadratic refinement, as argued in Ref. [20]. The
trace is over the quantum mechanical Hilbert space for the charge
$\Gamma+\gamma_{h}$, while the parenthesis denotes subleading loop
contribution in the given charge sector.
An important generalization of our analysis is to study the wall-crossing
phenomena in the $N=2$ supergravity. In fact, the formalism we developed is
more natural for the supergravity system, since the horizon provides natural
cut-off at short distance and renders the Abelian description of the core
state exact. That is, one can hide the any potential subtlety associated with
the Coulombic centers behind the horizon. Quantum mechanical description of
more than one extremally charged black hole has been studied previously, but
only in the context of same charge black holes, which is a particular limit of
our dynamics without potential terms. We are poised to consider many black
holes with mutually non-local and interacting center, and elevate Denef’s old
discussion black hole halos to fully quantum level.
In both field theory and the supergravity version of such a low energy quantum
mechanics, there is a simpler way to count bound states. As long as the true
moduli space defined by ${\cal K}=0$ is compact, the relevant supercharge
would be Fredholm, and one could compute the index by concentrating on the
true moduli space defined by ${\cal K}=0$. The quantum mechanics then would
reduce to a supersymmetric Landau level problem on a curved $2n$ dimensional
manifold, and can be presumably counted by computing the volume of this true
moduli space. A similar idea has been recently used in [46, 47], but our
approach provides a rigorous derivation of such a method and thus the precise
state counting. Details of this computation will be presented elsewhere.
Finally, though we have focused on the moduli space dynamics of framed BPS
particles in $D=4$ $N=2$ supersymmetric gauge theories, our analysis can be
potentially applied to study the wall-crossing phenomena of any supersymmetric
theories in presence of higher dimensional external objects. One potential
application is a study of the wall-crossing formulae of the four-dimensional
gauge theories in presence of a surface operator, which has been conjectured
in [48] as a hybrid of 2D Ceccoti-Vafa WCF [3] and 4D Kontsevich-Soibelman WCF
[19]. Our analysis also would be useful to study the wall-crossing formulae of
two-dimensional ${\cal N}=(2,2)$ massive $\mathbb{CP}^{n}$ models in relation
to that of four-dimensional ${\cal N}=2$ SQCD [49, 50, 51, 52].
Acknowledgement
We would like to thank Nick Dorey, Kazuo Hosomichi, Ki-Myeong Lee, and Andrew
Neitzke for valuable discussions. P.Y. is supported in part by the National
Research Foundation of Korea (NRF) funded by the Ministry of Education,
Science and Technology via the Center for Quantum Spacetime (grant number
2005-0049409) and also by Basic Science Research Program (grant number
2010-0013526).
Appendix
## Appendix A BPS Equation for the Semiclassical Core
This appendix reviews the BPS equation, of Seiberg-Witten low energy theory,
for long-range Abelian fields for any given core charges. One can easily read
off $N=2$ SUSY variation rules in four dimensions from $N=1$ SUSY variation
rules in six dimensions
$\displaystyle\delta\lambda_{A}=\frac{1}{2}F_{MN}\Gamma^{MN}\epsilon_{A}\ ,$
(A.1)
where $\lambda$ and $\epsilon$ are six-dimensional chiral spinors,
$\displaystyle\Gamma^{012345}\lambda_{A}=\lambda_{A}\
,\qquad\Gamma^{012345}\epsilon_{A}=\epsilon_{A}\ .$ (A.2)
Here $A=1,2$ are the R-symmetry indices. Let us decompose the six-dimensional
gamma matrices $\Gamma^{M}$ as
$\displaystyle\Gamma^{\mu}=$ $\displaystyle\gamma^{\mu}\otimes{\bf 1}_{2}\
,\qquad\Gamma^{4}=\gamma_{c}\otimes\tau^{2}\
,\qquad\Gamma^{5}=\gamma_{c}\otimes\tau^{1}\
,\qquad\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{pmatrix}\ ,$ (A.3)
where $i\gamma_{c}=\gamma^{0123}$. In the above representation, the gaugino
$\lambda_{A}$ can be decomposed into $\lambda_{A}=\lambda_{\alpha
A}\oplus\bar{\lambda}^{{\dot{\alpha}}}_{A}$. As usual, $\alpha,{\dot{\alpha}}$
denote the 4-D chiral/anti-chiral spinor indices. One can then rewrite (A.1)
as
$\displaystyle\delta\lambda_{\alpha
A}=\frac{1}{2}F_{\mu\nu}{\sigma^{\mu\nu}}_{\alpha}^{\ \beta}\epsilon_{\beta
A}+i{\sigma^{\mu}}_{\alpha{\dot{\alpha}}}\bar{\epsilon}^{\dot{\alpha}}_{A}D_{\mu}\phi\
,\qquad\phi=A_{4}+iA_{5}\ .$ (A.4)
With $Z_{c}=|Z_{c}|\,\zeta$, the core state configuration should satisfy the
following relation
$\displaystyle\Big{[}\big{(}Q^{A}+i\zeta^{-1}\bar{Q}^{A}\bar{\sigma}^{0}\big{)}\varepsilon_{A},\lambda_{B}\Big{]}=0$
(A.5)
or equivalently
$\displaystyle-i\vec{\tau}\varepsilon_{B}\cdot\big{(}\vec{B}+i\vec{E}-i\zeta^{-1}\vec{\nabla}\phi\big{)}-\zeta^{-1}\varepsilon_{B}\partial_{t}\phi=0\
,$ (A.6)
that is,
$\displaystyle\vec{\cal F}-i\zeta^{-1}\vec{\nabla}\phi=0\
,\qquad\partial_{t}\phi=0\ .$ (A.7)
One quick way to show that $\zeta$ represents the phase factor of $Z_{c}$ is
to look at the energy for the configuration (A.7), say, for rank one example:
performing the usual trick of completing the square with (A.7) in mind, one
obtain
$\displaystyle{\cal E}=\frac{1}{8\pi}\int d^{3}{\bf x}\
\text{Im}\tau\Big{[}\vec{B}^{2}+\vec{E}^{2}+|\vec{\nabla}\phi|^{2}\Big{]}=\text{Re}\Big{[}\zeta^{-1}Z_{c}\Big{]}\
,$ (A.8)
with $Z_{c}=P\phi_{D}(\infty)+Q\phi(\infty)$. This shows that
$\zeta^{-1}Z_{c}=|Z_{c}|$.
## Appendix B More on ${{\cal N}}=4$ Quantum Mechanics
Here we present more on ${\cal N}=4$ Lagrangian with conformal $R^{3}$ target
manifold. Here, we first derive the massless case with curved background and
then add potential terms, which provides an alternate path to (3.28). Then, we
spend some time on supercharge operators and quantum Hamiltonian.
### B.1 Massless and curved
First of all, we wish to fill the gap between sections 3.1 and 3.3 with a
derivation of massless ${\cal N}=4$ theory onto conformally flat $R^{3}$,
which turned out to be regarded as a special case of theories in Ref. [22]. In
next subsection, we demonstrate that how the massive Lagrangian of section 3.3
emerges by combining the result of section 3.1 with this massless case. Based
on the educated guess and group theoretical consideration, one possible
candidate for ${\cal N}=4$ SUSY transformation rules are following
$\displaystyle\delta x^{a}=i\eta^{a}_{mn}\epsilon^{m}\psi^{n}\
,\qquad\delta\psi^{m}=\eta^{a}_{mn}\epsilon^{n}{\dot{x}}^{a}+\alpha\epsilon_{m}\eta^{a}_{pq}f^{-1}\partial_{a}f\psi^{p}\psi^{q}\
,$ (B.1)
where $\alpha$ will be determined. Here $\eta^{a}_{mn}$ denotes the ’t Hooft
tensor with the convention $\eta^{3}_{12}=\eta^{3}_{34}=+1$.
To start, consider a standard kinetic term for flat target manifold,
$\displaystyle{\cal
L}^{(0)}=\frac{1}{2}f{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}f\psi^{m}{\dot{\psi}}^{m}\
,$ (B.2)
whose variation under the ${\cal N}=4$ SUSY transformations is
$\displaystyle\delta\big{(}\frac{1}{2}f{\dot{x}}^{a}{\dot{x}}^{a}\big{)}=$
$\displaystyle\frac{i}{2}\eta^{b}_{mn}\partial_{b}f\epsilon^{m}\psi^{n}{\dot{x}}^{a}{\dot{x}}^{a}+if\eta^{a}_{mn}\epsilon^{m}{\dot{\psi}}^{n}{\dot{x}}^{a}\
,$ $\displaystyle\delta\big{(}\frac{i}{2}f\psi^{m}{\dot{\psi}}^{m}\big{)}=$
$\displaystyle-\frac{1}{2}\eta^{a}_{pq}\partial_{a}f\epsilon^{p}\psi^{q}\psi^{m}{\dot{\psi}}^{m}-if\eta^{a}_{mn}\epsilon^{m}{\dot{\psi}}^{n}{\dot{x}}^{a}-\frac{i}{2}\eta^{a}_{mn}\partial_{b}f{\dot{x}}^{a}{\dot{x}}^{b}\epsilon^{m}\psi^{n}$
$\displaystyle+i\alpha\eta^{a}_{mn}\partial_{a}f\psi^{m}\psi^{n}\epsilon^{p}{\dot{\psi}}^{p}+\frac{i}{2}\alpha\eta^{a}_{mn}f^{-1}\partial_{a}f\partial_{l}f{\dot{x}}^{l}\psi^{m}\psi^{n}\epsilon^{p}\psi^{p}\
.$ (B.3)
1. $\bullet$
One can reorganize the velocity-square terms in (B.1) into
$\displaystyle\frac{i}{2}\partial_{b}f\epsilon^{m}\psi^{n}\Big{[}\eta^{b}_{mn}{\dot{x}}^{a}-$
$\displaystyle\eta^{a}_{mn}{\dot{x}}^{b}\Big{]}{\dot{x}}^{a}=\frac{i}{2}\epsilon_{eab}\epsilon_{ecd}{\dot{x}}^{a}{\dot{x}}^{c}\partial_{b}f\eta^{d}_{mn}\epsilon^{m}\psi^{n}$
$\displaystyle=$
$\displaystyle+\frac{i}{2}\eta^{c}_{pm}\eta^{e}_{np}\epsilon_{eab}{\dot{x}}^{a}{\dot{x}}^{c}\partial_{b}f\epsilon^{m}\psi^{n}$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\epsilon_{eab}\cdot{\dot{x}}^{a}\partial_{b}f\eta^{e}_{np}\delta\psi^{n}\psi^{p}-\frac{i}{2}\alpha{\dot{x}}^{a}f^{-1}\partial_{a}f\partial_{b}f\eta^{b}_{mn}\psi^{m}\psi^{n}\epsilon^{p}\psi^{p}$
$\displaystyle+\frac{i}{6}\alpha
f^{-1}\partial_{a}f\partial_{a}f\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\delta\psi^{q}\
.$ (B.4)
2. $\bullet$
The first term in the last equality of ($\bullet$ ‣ B.1) implies that we have
to add the following term
$\displaystyle\delta\big{(}+\frac{i}{4}\epsilon_{abc}{\dot{x}}^{a}$
$\displaystyle\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}\big{)}=+\frac{i}{2}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}f\eta^{c}_{mn}\delta\psi^{m}\psi^{n}$
$\displaystyle-\frac{1}{4}\epsilon_{abc}\eta^{a}_{pq}\eta^{c}_{mn}\partial_{b}f\epsilon^{p}{\dot{\psi}}^{q}\psi^{m}\psi^{n}-\frac{1}{4}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}\partial_{d}f\eta^{c}_{mn}\eta^{d}_{pq}\epsilon^{p}\psi^{q}\psi^{m}\psi^{n}\
.$ (B.5)
3. $\bullet$
Using the identities of ’t Hooft tensor
$\displaystyle\epsilon_{abc}\eta^{c}_{mn}\eta^{a}_{pq}=\delta_{mp}\eta^{b}_{nq}-\delta_{np}\eta^{b}_{mq}+\delta_{nq}\eta^{b}_{mp}-\delta_{mq}\eta^{b}_{np}\
,$
$\displaystyle\eta^{d}_{pq}\eta^{c}_{mn}+\eta^{d}_{pm}\eta^{c}_{nq}+\eta^{d}_{pn}\eta^{c}_{qm}+\eta^{d}_{ps}\eta^{c}_{rs}\epsilon_{qmnr}=0\
,$ (B.6)
one can massage the second and third terms in ($\bullet$ ‣ B.1) into
followings:
$\displaystyle-\frac{1}{4}\epsilon_{abc}\eta^{a}_{pq}\eta^{c}_{mn}\partial_{b}f\epsilon^{p}{\dot{\psi}}^{q}\psi^{m}\psi^{n}=\frac{1}{2}\eta^{a}_{mn}\partial_{a}f\epsilon^{m}\psi^{n}\cdot\psi^{p}{\dot{\psi}}^{p}-\frac{1}{2}\eta^{a}_{mn}\partial_{a}f\psi^{m}{\dot{\psi}}^{n}\cdot\epsilon^{p}\psi^{p}\
,$ (B.7)
and
$\displaystyle-\frac{1}{4}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}\partial_{d}f\eta^{c}_{mn}\eta^{d}_{pq}\epsilon^{p}\psi^{q}\psi^{m}\psi^{n}=$
$\displaystyle+\frac{1}{12}{\dot{x}}^{a}\partial_{b}\partial_{d}f\eta^{d}_{ps}\epsilon_{abc}\eta^{c}_{rs}\epsilon_{qmnr}\epsilon^{p}\psi^{q}\psi^{m}\psi^{n}$
$\displaystyle=$
$\displaystyle+\frac{1}{12}\epsilon_{mnpq}\partial^{2}f\psi^{m}\psi^{n}\psi^{p}\delta\psi^{q}$
$\displaystyle-\frac{1}{12}{\dot{x}}^{a}\partial_{a}\partial_{c}f\eta^{c}_{pl}\epsilon^{p}\psi^{q}\psi^{m}\psi^{n}\epsilon_{qmnl}\
.$ (B.8)
In summary, one can show that
$\displaystyle\delta\big{(}+\frac{i}{4}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}\big{)}=$
$\displaystyle+\frac{i}{2}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}f\eta^{c}_{mn}\delta\psi^{m}\psi^{n}+\frac{1}{2}\eta^{a}_{mn}\partial_{a}f\epsilon^{m}\psi^{n}\cdot\psi^{p}{\dot{\psi}}^{p}$
$\displaystyle+\frac{1}{4}\eta^{a}_{mn}\partial_{a}f\psi^{m}\psi^{n}\cdot\epsilon^{p}{\dot{\psi}}^{p}+\frac{1}{12}\epsilon_{mnpq}\partial^{2}f\psi^{m}\psi^{n}\psi^{p}\delta\psi^{q}\
.$ (B.9)
4. $\bullet$
Here one can determine, from the fourth term in second equality of (B.1) and
third term in ($\bullet$ ‣ B.1), the value of the coefficient $\alpha$ by
$\displaystyle\alpha=+\frac{i}{4}$ (B.10)
5. $\bullet$
Collecting all the results so far, one can have
$\displaystyle\delta\Big{(}\frac{1}{2}f{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}f\psi^{m}{\dot{\psi}}^{m}+\frac{i}{4}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}\Big{)}$
$\displaystyle=\frac{1}{12}\epsilon_{mnpq}\partial^{2}f\psi^{m}\psi^{n}\psi^{p}\delta\psi^{q}-\frac{1}{24}f^{-1}\partial_{a}f\partial_{a}f\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\delta\psi^{q}\
.$
At the end of the day, this gives the massless ${\cal N}=4$ non-linear sigma
model therefore takes the following form
$\displaystyle{\cal L}^{(0)}=$
$\displaystyle\frac{1}{2}f{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}f\psi^{m}{\dot{\psi}}^{m}+\frac{i}{4}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}$
$\displaystyle-\frac{1}{48}\partial_{a}^{2}f\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}+\frac{1}{96}f^{-1}(\partial_{a}f)^{2}\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}\
,$ (B.12)
where the covariant derivative for fermions is defined as
$\displaystyle\nabla_{t}\psi^{m}={\dot{\psi}}^{m}+\frac{1}{2}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}\text{log}f\eta^{c}_{mn}\psi^{n}\
.$ (B.13)
The above massless Lagrangian is invariant under the ${\cal N}=4$ SUSY
transformation
$\displaystyle\delta x^{a}=i\eta^{a}_{mn}\epsilon^{m}\psi^{n}\
,\qquad\delta\psi_{m}=\eta^{a}_{mn}\epsilon^{n}{\dot{x}}^{a}+\frac{i}{4}\epsilon_{m}\eta^{a}_{pq}f^{-1}\partial_{a}f\psi^{p}\psi^{q}\
.$ (B.14)
This is the curved space version of (3.2).
### B.2 Massive and curved
Now we wish to add potential terms to this by twisting the supersymmetry
transformation rules. From discussion of section 3.2, it is clear that the
right thing to do, at least in the context of ${\cal N}=1$ supersymmetry, is
to shift the fermion transformation rule as
$\displaystyle\delta x^{a}=i\eta^{a}_{mn}\epsilon^{m}\psi^{n}\
,\qquad\delta\psi_{m}=\eta^{a}_{mn}\epsilon^{n}{\dot{x}}^{a}+\epsilon_{m}\frac{1}{f}\left({\cal
K}+\frac{i}{4}\eta^{a}_{pq}\partial_{a}f\psi^{p}\psi^{q}\right)\ ,$ (B.15)
since the last piece multiplying $\epsilon_{m}$ is nothing but the on-shell
value of the auxiliary field $b$. The corresponding Lagrangian from (3.28)
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}f\Big{[}{\dot{x}}^{a}{\dot{x}}^{a}+i\psi^{m}\nabla_{t}\psi^{m}-\frac{1}{4!}\epsilon_{mnpq}\big{\\{}\nabla^{2}f-(\partial_{a}\text{log}f)^{2}\big{\\}}\psi^{m}\psi^{n}\psi^{p}\psi^{q}\Big{]}$
$\displaystyle-{\cal W}_{a}{\dot{x}}^{a}+i\partial_{b}{\cal
W}_{c}\psi^{b}\psi^{c}+if^{1/2}\partial_{a}(f^{-1/2}{\cal
K})\psi^{a}\lambda-\frac{i}{4}\epsilon_{abc}{\cal
K}f^{-1}\partial_{a}f\psi^{b}\psi^{c}-\frac{1}{2f}{\cal K}^{2}\ $
is indeed consistent with the above massless one in (B.1).
To show that this Lagrangian is invariant under this transformation, we split
it into three parts, ${\cal L}={\cal L}^{(0)}+{\cal L}^{(1)}+{\cal L}^{(2)}$,
as
$\displaystyle{\cal L}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}f{\dot{x}}^{a}{\dot{x}}^{a}+\frac{i}{2}f\psi^{m}{\dot{\psi}}^{m}+\frac{i}{4}\epsilon_{abc}{\dot{x}}^{a}\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}-{\cal
W}_{a}{\dot{x}}^{a}$
$\displaystyle-\frac{1}{48}\partial_{a}^{2}f\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}+\frac{1}{96}f^{-1}(\partial_{a}f)^{2}\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}\
,$ $\displaystyle{\cal L}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{i}{2}f^{1/2}\partial_{a}\big{(}f^{1/2}K\big{)}\eta^{a}_{mn}\psi^{m}\psi^{n}\
,$ $\displaystyle{\cal L}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}fK^{2}\ ,$ (B.17)
where we introduced $K\equiv f^{-1}{\cal K}$. ${\cal L}^{(0)}$ is already
invariant under (B.14), so we have only $K$-dependence pieces in $\delta{\cal
L}^{(0)}$, which is
$\displaystyle\delta{\cal L}^{(0)}$ $\displaystyle=$ $\displaystyle-
if^{1/2}\partial_{a}(f^{1/2}K){\dot{x}}^{a}\epsilon^{m}\psi_{m}-i\epsilon_{abc}{\dot{x}}^{a}f^{1/2}\partial_{b}(f^{1/2}K)\eta^{c}_{mn}e_{m}\psi_{n}$
(B.18)
$\displaystyle-\frac{1}{12}K\Big{[}\partial_{a}^{2}f-\frac{1}{2}f^{-1}(\partial_{a}f)^{2}\Big{]}\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\epsilon^{q}\
.$
After some tedious computation, we find
$\displaystyle\delta{\cal L}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\delta\Big{(}f^{1/2}\partial_{a}(f^{-1/2}\cdot
fK)\Big{)}\eta^{a}_{mn}\psi^{m}\psi^{n}+if^{1/2}\partial_{a}(f^{1/2}K)\delta\psi^{m}\psi^{n}$
(B.19) $\displaystyle=$
$\displaystyle\frac{1}{12}K\partial_{a}^{2}f\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\epsilon^{q}-\frac{1}{24}Kf^{-1}(\partial_{a}f)^{2}\epsilon_{mnpq}\psi^{m}\psi^{n}\psi^{p}\psi^{q}$
$\displaystyle+if^{1/2}\partial_{a}(f^{1/2}K){\dot{x}}^{a}\epsilon^{m}\psi_{m}+i\epsilon_{abc}{\dot{x}}^{a}f^{1/2}\partial_{b}(f^{1/2}K)\eta^{c}_{mn}e_{m}\psi_{n}$
$\displaystyle+\delta(\frac{1}{2}fK^{2})\ ,$
which, combined with $\delta{\cal L}^{(0)}$, give us
$\displaystyle\delta\big{(}{\cal L}^{(0)}+{\cal
L}^{(1)}\big{)}=\delta\big{(}\frac{1}{2}fK^{2}\big{)}=-\delta{\cal L}^{(2)}\
,$ (B.20)
as required.
### B.3 Supercharges and Hamiltonian
Using the ${\cal N}=4$ supersymmetric variation rules (3.3) , the Nöther
charges of the ${\cal N}=4$ supersymmetry therefore become
$\displaystyle Q_{m}=-\eta^{a}_{mn}\psi^{n}(p_{a}+{\cal
W}_{a})+\frac{i}{4}\eta^{a}_{mn}f^{-1}\partial_{a}f\psi^{n}+\frac{i}{4}\partial_{a}f\eta^{a}_{pq}\psi^{[p}\psi^{q}\psi^{m]}+{\cal
K}\psi^{m}\ .$ (B.21)
For completeness, let us check whether the above supercharges give the correct
supersymmetric transformation rules for bosons and fermions. One can read off
from the Lagrangian (3.28) the canonical quantization
$\displaystyle\big{[}x^{a},p_{b}\big{]}=i\delta^{a}_{b}\
,\qquad\big{\\{}\psi^{m},\psi^{n}\big{\\}}=f^{-1}\delta^{mn}\
,\qquad\big{[}p_{a},\psi^{m}\big{]}=\frac{i}{2}f^{-1}\partial_{a}f\psi^{m}\ .$
(B.22)
One can show
$\displaystyle\big{\\{}-\eta^{a}_{mp}\psi^{p}(p_{a}+{\cal
W}_{a}),\psi^{n}\big{\\}}$ (B.23) $\displaystyle=$
$\displaystyle-\eta^{a}_{mn}f^{-1}(p_{a}+{\cal
W}_{a})-\frac{i}{2}\eta^{a}_{mp}f^{-1}\partial_{a}f\psi^{p}\psi^{n}$
$\displaystyle=$
$\displaystyle-\eta^{a}_{mn}{\dot{x}}^{a}-\frac{i}{4}\eta^{a}_{mp}f^{-1}\partial_{a}f\big{\\{}\psi^{p},\psi^{n}\big{\\}}+\frac{i}{2}f^{-1}\partial_{a}f\eta^{a}_{np}\psi^{[m}\psi^{p]}\
,$ $\displaystyle=$
$\displaystyle-\eta^{a}_{mn}{\dot{x}}^{a}+\frac{i}{2}f^{-1}\partial_{a}f\eta^{a}_{np}\psi^{[m}\psi^{p]}-\frac{i}{4}f^{-2}\partial_{a}f\eta^{a}_{mn}\
,$
where we used for the second equality the definition of momentum operator
$p_{a}$
$\displaystyle p_{a}+{\cal
W}_{a}=f{\dot{x}}^{a}+\frac{i}{4}\epsilon_{abc}\partial_{b}f\eta^{c}_{mn}\psi^{m}\psi^{n}\
.$ (B.24)
One can also show that
$\displaystyle\big{\\{}\frac{i}{4}\partial_{a}f\eta^{a}_{pq}\psi^{[p}\psi^{q}\psi^{m]},\psi^{n}\big{\\}}$
(B.25) $\displaystyle=$
$\displaystyle\delta_{mn}\frac{i}{4}f^{-1}\partial_{a}f\eta^{a}_{pq}\psi^{p}\psi^{q}+\frac{i}{2}f^{-1}\partial_{a}f\eta^{a}_{np}\psi^{p}\psi^{m}+\frac{i}{4}f^{-2}\partial_{a}f\eta^{a}_{mn}\
,$
where we used an identity of ’t Hooft tensor
$\displaystyle\epsilon_{abc}\eta^{b}_{mn}\eta^{c}_{pq}=\eta^{a}_{mp}\delta_{nq}-\eta^{a}_{np}\delta_{mq}+\eta^{a}_{nq}\delta_{mp}-\eta^{a}_{mq}\delta_{np}\
.$ (B.26)
It implies that
$\displaystyle\big{\\{}Q_{m},\psi_{n}\big{\\}}=-\eta^{a}_{mn}{\dot{x}}^{a}+\delta_{mn}f^{-1}\left({\cal
K}+\frac{i}{4}f^{-1}\partial_{a}f\eta^{a}_{pq}\psi^{p}\psi^{q}\right)\ ,$
(B.27)
while the action of supercharges on the bosons follows immediately,
$\displaystyle\big{[}Q_{m},x^{a}\big{]}=i\eta^{a}_{mn}\psi^{n}\ .$ (B.28)
These are precisely the supersymmetry transformation rules in (3.3).
Finally, we wish to determine the quantum form of the Hamiltonian using
$Q_{4}^{2}=H\ .$
Let us first write
$Q_{4}=\psi^{a}(p+{\cal W})_{a}+\lambda({\cal K}+Z)\ ,$
where
$Z=\frac{i}{2}\,\partial_{a}f\psi^{a}\lambda+\frac{i}{4}\,\epsilon_{abc}\partial_{a}f\psi^{b}\psi^{c}\
.$
Using $\\{Q_{4},\lambda\\}=({\cal K}+Z)/f$ and
$\\{Q_{4},\psi^{a}\\}=\dot{x}^{a}=f^{-1}\pi_{a}$, with the supercovariant
momentum operator
$\pi_{a}=(p+{\cal
W})_{a}+\Gamma_{a},\qquad\Gamma_{a}\equiv\frac{i}{2}\,\partial_{b}f\psi^{[b}\psi^{a]}-\frac{i}{2}\,\epsilon_{abc}\partial_{b}f\psi^{c}\lambda\
,$
we find
$\displaystyle\\{Q_{4},Q_{4}\\}$ $\displaystyle=$
$\displaystyle\\{Q_{4},\psi^{a}(p+{\cal W})_{a}+\lambda({\cal K}+Z)\\}$ (B.29)
$\displaystyle=$ $\displaystyle\frac{1}{f}\pi^{a}(p+{\cal
W})_{a}+\frac{1}{f}({\cal K}+Z)^{2}$ $\displaystyle-\psi^{a}[Q_{4},(p+{\cal
W})_{a}]-\lambda[Q_{4},{\cal K}+Z]\ .$
Let us separate out terms involving either ${\cal W}$ or ${\cal K}$ from the
last two terms. Using $d{\cal K}=*d{\cal W}$, we find
$\displaystyle\\{Q_{4},Q_{4}\\}$ $\displaystyle=$
$\displaystyle\frac{1}{f}\pi^{a}(p+{\cal W})_{a}+\frac{1}{f}({\cal
K}+Z)^{2}-2i\partial_{a}{\cal
K}\psi^{a}\lambda-i\epsilon_{abc}\partial_{a}{\cal K}\psi^{b}\psi^{c}$ (B.30)
$\displaystyle+\left(\psi^{a}[(p+{\cal
W})_{a},\psi^{b}]+\lambda[Z,\psi^{b}]\right)(p+{\cal W})_{b}$
$\displaystyle+\left(\psi^{a}[(p+{\cal
W})_{a},\lambda]+\lambda[Z,\lambda]\right)Z$
$\displaystyle+2\psi^{a}\lambda[(p+{\cal W})_{a},Z]\ .$
By explicit computation one can see that
$\displaystyle\left(\psi^{a}[(p+{\cal
W})_{a},\psi^{b}]+\lambda[Z,\psi^{b}]\right)$ $\displaystyle=$
$\displaystyle\frac{1}{f}\Gamma_{b}$ $\displaystyle\left(\psi^{a}[(p+{\cal
W})_{a},\lambda]+\lambda[Z,\lambda]\right)$ $\displaystyle=$ $\displaystyle 0\
.$ (B.31)
Since $[(p+{\cal W})_{a},\Gamma_{a}]=0$ upon the summation over $a$,
$\displaystyle\frac{1}{f}\pi^{a}(p+{\cal
W})_{a}+\frac{1}{f}\,\Gamma_{b}(p+{\cal W})_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{f}\pi_{a}\pi_{a}-\frac{1}{f}\,\Gamma_{a}\Gamma_{a}\ .$
(B.32)
Finally expanding $({\cal K}+Z)^{2}$ out, we complete the potential terms
associated with ${\cal K}$ from ${\cal K}^{2}+2{\cal K}Z$, but have a leftover
piece $Z^{2}$. So combining them all, we have
$\displaystyle\\{Q_{4},Q_{4}\\}$ $\displaystyle=$
$\displaystyle\frac{1}{f}\pi^{a}\pi_{a}+\frac{1}{f}{\cal
K}^{2}-2if^{1/2}\partial_{a}(f^{-1/2}{\cal
K})\psi^{a}\lambda-i\epsilon_{abc}f^{1/2}\partial_{a}(f^{-1/2}{\cal
K})\psi^{b}\psi^{c}$ (B.33)
$\displaystyle+\frac{1}{f}Z^{2}-\frac{1}{f}\,\Gamma_{b}\Gamma_{b}+2\psi^{a}\lambda[(p+{\cal
W})_{a},Z]$
The last line can be organized in terms of the curvature of the fermion
bundle,
$\displaystyle[D_{a},D_{b}]=F_{abmn}\psi^{m}\psi^{n},\qquad
D_{a}\equiv\partial_{a}+i\Gamma_{a}\ ,$ (B.34)
and has the explicit form,
$\displaystyle-\frac{1}{2}F_{abmn}\psi^{a}\psi^{b}\psi^{m}\psi^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{48}\,\left(2(\partial^{2}f)-f^{-1}(\partial
f)^{2}\right)\epsilon_{mnkl}\psi^{m}\psi^{n}\psi^{k}\psi^{l}$ (B.35)
$\displaystyle-\frac{1}{4}\,f^{-2}(\partial^{2}f)+\frac{1}{8}f^{-3}(\partial
f)^{2}\ ,$
Thus, the Hamiltonian $H=\\{Q_{4},Q_{4}\\}/2$ is
$\displaystyle
H=\frac{1}{2f}\pi_{a}\pi_{a}-\frac{1}{4}F_{abmn}\psi^{a}\psi^{b}\psi^{m}\psi^{n}+\frac{1}{2f}{\cal
K}^{2}-\frac{i}{2}\,\eta^{a}_{mn}f^{1/2}\partial_{a}(f^{-1/2}{\cal
K})\psi^{m}\psi^{n}$ (B.36)
Although $SU(2)_{R}$ is not manifest in the curvature piece, it is actually
$SU(2)_{R}$ invariant as can be seen from (B.35). This coincides with the
classical Hamiltonian up to normal ordering; the curvature pieces generate
extra terms because quantum $\psi$’s obey not the Grassman algebra but the
Clifford algebra.
Note that the kinetic term is slightly unconventional in its choice of normal
ordering. Because of this, the inner product in the Hilbert space of this
quantum mechanics should be defined as
$||\,\Psi||^{2}=\int dx^{3}f\Psi^{\dagger}\Psi\ .$ (B.37)
More usual choice of kinetic term/inner product is related to our convention
by rescaling of the wavefunction by a factor of $f^{1/4}$.
## Appendix C Review of KS Invariant and Line Operator
The idea of the framed BPS state originally arises in study of four-
dimensional $N=2$ supersymmetric theories in presence of an external particle
of charge $\Gamma$, called line operator $L_{\Gamma}$. The line operator can
be characterized by the phase factor $\zeta$ of its central charge
$Z_{\Gamma}$. Compactifying the theory on a circle, it has been conjectured in
[20] that the vacuum expectation value of $L_{\Gamma}$ can be expanded in
terms of the Darboux coordinates ${\cal X}_{\gamma}$ with integer coefficients
$\displaystyle\langle
L_{\Gamma}\rangle=\sum_{\gamma}\Omega(\Gamma+\gamma){\cal X}_{\gamma}\ ,$
(C.1)
which provides us a direct physical interpretation of Darboux coordinates.
Each integer coefficient $\Omega(\Gamma+\gamma)$ here represents the
supersymmetric index of a framed BPS state of charge $\gamma$ bounded to
$L_{\Gamma}$. The Darboux coordinates are very useful to compute the
hyperKähler metric on the Coulomb branch of four-dimensional theories on a
circle.
The expectation value of the line operator depends on both $\zeta$ and the
Coulomb branch parameter $a$ in four-dimensional theories. Due to the fact
that the physical observable $\langle L_{\Gamma}\rangle$ should not have any
discontinuities as $\zeta$ and $a$ change, important consequences of (C.1) are
that one can understand how the Kontsevich-Soibelman invariant naturally
arises, and that provides the origin of the thermodynamic Bethe ansatz
equation the Darboux coordinates should satisfy.
Let us now review in this section the central importance of semi-primitive
wall-crossing formula to derive the Kontsevich-Soibelman BPS invariant in the
context of line operators. For more details, it is referred to [20].
As discussed in the main context, the Witten index
$\Omega(\Gamma+\gamma,\zeta)$ can jump once the phase of central charge for a
certain probe(halo) particle of $\gamma_{h}$ is parallel to that of the
external particle of $\Gamma$ denoted by $\text{arg}(\zeta)$. That is, when
$\zeta$ moves across the so-called BPS ray
$l_{h}=\big{\\{}\zeta~{}\big{|}~{}Z_{h}/\zeta\in R_{+}\big{\\}}$, the index
could have discontinuity. One advantage on computation of the index jump in
presence of line operator is that the wall-crossing phenomena is essentially
restricted to the semi-primitive ones.
Let us now consider the vacuum expectation value of the line operator
conjectured as in (C.1)
$\displaystyle\langle
L_{\Gamma}\rangle=\sum_{\gamma}\Omega(\Gamma+\gamma){\cal X}_{\gamma}\ ,$
where ${\cal X}_{\gamma}$ satisfy a multiplication rule below
$\displaystyle{\cal X}_{\gamma_{1}}{\cal
X}_{\gamma_{2}}=(-1)^{2\langle\gamma_{1},\gamma_{2}\rangle}{\cal
X}_{\gamma_{1}+\gamma_{2}}\ .$ (C.2)
Let us then increase the phase parameter $\text{arg}(\zeta)$ so that it moves
across the BPS ray $l_{h}$.
1. $\bullet$
Look at the relation (2.22). If $\langle\gamma_{c},\gamma_{h}\rangle>0$, we
have a stable bound state between core and halo particles before $\zeta$ cross
the BPS ray $l_{h}$. Then, one can reorganize (C) before across the ray into
the following form
$\displaystyle\langle L_{\Gamma}\rangle_{-}$ $\displaystyle=$
$\displaystyle\sum_{\gamma_{c}}{\cal
X}_{\gamma_{c}}\cdot\sum_{n=0}\Omega(\gamma_{c}+n\gamma_{h})(-1)^{2n\langle\gamma_{c},\gamma_{h}\rangle}{\cal
X}_{\gamma_{h}}^{n}\ ,$ (C.3) $\displaystyle=$
$\displaystyle\sum_{\gamma_{c}}\Omega(\gamma_{c}){\cal
X}_{\gamma_{c}}\prod_{n=1}\Big{[}1-{\cal
X}_{\gamma_{h}}^{n}\Big{]}^{2n\langle\gamma_{c},\gamma_{h}\rangle\Omega(n\gamma_{h})}\
.$
Note that we used the semi-primitive wall-crossing formula (5.17) for the last
equality. Since we loose the Fock space of halo particles after across the ray
$l_{h}$, one can say that
$\displaystyle\langle
L_{\Gamma}\rangle_{+}=\sum_{\gamma_{c}}\Omega_{\gamma_{c}}{\cal
X}_{\gamma_{c}}\ .$ (C.4)
One can therefore conclude that, since $\langle L_{\Gamma}\rangle$ should be
continuous across the ray, ${\cal X}_{\gamma_{c}}$ is required to jump across
the wall by the amount
$\displaystyle{\cal X}_{\gamma_{c}}\ \to\ {\cal
X}_{\gamma_{c}}\prod_{n=1}\Big{[}1-{\cal
X}_{\gamma_{h}}^{n}\Big{]}^{2n\langle\gamma_{h},\gamma_{c}\rangle\Omega(n\gamma_{h})}=\prod_{n=1}{\cal
K}_{n\gamma_{h}}^{\Omega(n\gamma_{h})}({\cal X}_{\gamma_{c}})\ ,$ (C.5)
where
$\displaystyle{\cal K}_{\gamma_{h}}({\cal X}_{\gamma_{c}})={\cal
X}_{\gamma_{c}}\Big{[}1-{\cal
X}_{\gamma_{h}}\Big{]}^{2\langle\gamma_{h},\gamma_{c}\rangle}\ .$ (C.6)
It is noteworthy here that this is the desired discontinuity how the Darboux
coordinate ${\cal X}_{\gamma}$ jumps across the BPS ray $l_{h}$.
2. $\bullet$
Let us now in turn consider the converse, i.e.,
$\langle\gamma_{c},\gamma_{h}\rangle<0$. According to (2.22), there is no
stable bound state between the core and halo particle before the $\zeta$
across the BPS ray $l_{h}$. Then, one can rewrite (C) before across the ray as
$\displaystyle\langle
L_{\Gamma}\rangle_{-}=\sum_{\gamma_{c}}\Omega_{\gamma_{c}}{\cal
X}_{\gamma_{c}}\ .$ (C.7)
Since we gain the Fock space of halo particles after across the ray $l_{h}$,
one can say that
$\displaystyle\langle L_{\Gamma}\rangle_{+}$ $\displaystyle=$
$\displaystyle\sum_{\gamma_{c}}\Omega(\gamma_{c}){\cal
X}_{\gamma_{c}}\prod_{n=1}\Big{[}1-{\cal
X}_{\gamma_{h}}^{n}\Big{]}^{-2n\langle\gamma_{c},\gamma_{h}\rangle\Omega(n\gamma_{h})}\
.$ (C.8)
${\cal X}_{\gamma_{c}}$ is again required to jump across the wall by the same
amount
$\displaystyle{\cal X}_{\gamma_{c}}\ \to\ {\cal
X}_{\gamma_{c}}\prod_{n=1}\Big{[}1-{\cal
X}_{\gamma_{h}}^{n}\Big{]}^{2n\langle\gamma_{h},\gamma_{c}\rangle\Omega(n\gamma_{h})}=\prod_{n=1}{\cal
K}_{n\gamma_{h}}^{\Omega(n\gamma_{h})}({\cal X}_{\gamma_{c}})\ ,$ (C.9)
which is the same to (C.6).
Let us now consider two chambers of ${\cal
M}_{\text{Coulomb}}\times\mathbb{C}^{*}$, the Coulomb branch and
$\zeta$-plane, separated by walls of marginal stability. The physical
observable $\langle L_{\Gamma}\rangle$ should not depend on choice of a path
connecting those two chambers. The different paths however in general cross
different set of walls of marginal stability. One can therefore conclude, from
the fact that there are infinitely many possible line operators, that a path-
ordered product of transformations below
$\displaystyle{\cal I}=\prod_{\gamma_{h}}^{\curvearrowleft}\prod_{n}\ {\cal
K}_{n\gamma_{h}}^{\Omega(n\gamma_{h})}$ (C.10)
defines an invariant over the Coulomb branch ${\cal M}_{\text{Coulomb}}$.
${\cal I}$ is indeed the so-called Kontsevich-Soibelman invariant.
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|
arxiv-papers
| 2011-02-08T21:04:54 |
2024-09-04T02:49:16.869320
|
{
"license": "Public Domain",
"authors": "Sungjay Lee and Piljin Yi",
"submitter": "Sungjay Lee",
"url": "https://arxiv.org/abs/1102.1729"
}
|
1102.1866
|
# A quantum group for the Einstein equations
Giuseppe Iurato111e-mail: iurato@dmi.unict.it
###### Abstract
In this paper, we expose the construction of a possible, simple matrix quantum
group structure (according to Woronowicz), related to elementary formal
aspects of the Einstein field equations of General Relativity, and its
possible symmetries.
Mainly, we present a simple application of the results achieved by M. Dubois-
Violette and G. Launer in [1], where is built up a first matrix quantum group
structure (in the sense of S.L. Woronowics $-$ see [8], 2.1) associated to an
arbitrary non-degenerate bilinear form.
Precisely, we apply, almost verbatim, these considerations to a generalization
of the Einstein field equations (1915), in purely covariant form given
by222According to the Robertson-Noonan sign convention (1968) (see [4]). (see
[6], 1.13.5, and [5], 4.0, 4.1)
$G_{ij}=R_{ij}-\frac{1}{2}Rg_{ij}=-8\pi GT_{ij},\qquad i,j=0,1,2,3,$ $None$
where $G_{ij}$ is the Einstein curvature tensor, $R_{ij}$ is the Ricci
curvature tensor, $g_{ij}$ is the Lorentz metric, $R$ is the Ricci scalar, $G$
is the gravitational constant, and $T_{ij}$ is the so called Hilbert tensor
(see [9], Chapter 7) with $T_{ij}=T_{ij}(g_{lh},\partial
g_{lh};\psi_{k},\partial\psi_{k})$ in the presence of a set of physical fields
$\psi_{k}\ \ k=1,...,p$.
In the geometrized units, it is $G=1$ (see [7]).
We recall that the Einstein field equations (1) may be deduced both by a
variational Palatini’s argument (see [9]) and, inductively, by the newtonian
Poisson’s equation $\Delta\phi=4\pi G\rho$.
Following the latter way, it is assumed that the field equations for the
gravitational field, that we may call generalized Einstein (field) equations,
should have a general form of the type (see [9], Chapter 4, [10], Cap. II, §
2.1, and [4], Chapter 17, § 17.1)
$G_{ij}(g_{lh},g_{lh,r},g_{lh,rt},...)=k\pi T_{ij},\qquad i,j=0,1,2,3,$ $None$
where $G_{ij}$ is a yet to be determined tensor function of the metric tensor
$g_{lh}$ and some of its derivatives, and $k$ is a real constant.
Many physical reasons (see [10], § 2.1, and [4], § 17.1) restricts the class
of the possible functions $G_{ij}$, satisfying (2), to a well-defined tensor,
namely the Einstein curvature tensor mentioned above, obtaining the known
equations (1).
On the other hand, by earlier Weyl’s and Cartan’s results culminated in
Lovelock’s statement (see [16]), if we seek a tensor equation of the form
$G_{ij}=T_{ij}$, where the components $A_{ij}$ involve the metric tensor
$g_{ij}$ and its first and second derivatives (hence, assuring second-order
partial differential equations generalizing the Poisson one), and if $A_{ij}$
have vanishing divergence $A_{ij;j}=0$, then the equation must be of the form
$aG_{ij}+bg_{ij}=-8\pi kT_{ij}$, where $a$ and $b$ are constants; Einstein’s
choice is then $a=1,b=0$ ($b$ is said to be the cosmological constant).
At this point, taking into account the geometrical meaning of the Einstein’s
equations333These arguments shall be the matter of another paper. according to
[17], it is possible to consider the following Einstein bilinear form
$\Omega_{ij}\doteq G_{ij}+8\pi T_{ij},\qquad i,j=0,1,2,3,$ $None$
that, for now, we suppose to be non-degenerate; its zero values are the
generalized Einstein equations (2). In $(2^{\prime})$, we suppose, a priori,
$G_{ij}$ to be an arbitrary bilinear form (of $\mathbb{R}^{4}$), while
$T_{ij}$ is the Hilbert tensor.
In [1] (see also [11], Example 4.62), it is considered a finite family
$\\{T({\alpha})\\}_{\alpha\in\Xi}$ of $(r_{\alpha},s_{\alpha})$-tensors on
$\mathbb{R}^{n}$ and the group $G$ of the automorphisms of $\mathbb{R}^{n}$
that preserve $T({\alpha})$ in the following sense
$u_{k_{1}}^{i_{1}}...u_{k_{r_{\alpha}}}^{i_{r_{\alpha}}}T(\alpha)_{j_{1}...j_{s_{\alpha}}}^{k_{1}...k_{r_{\alpha}}}=u_{j_{1}}^{k_{1}}...u_{j_{s_{\alpha}}}^{k_{s_{\alpha}}}T(\alpha)_{k_{1}...k_{s_{\alpha}}}^{i_{1}...i_{r_{\alpha}}}\qquad\forall\alpha\in\Xi,$
$None$
supposing invertible the generic matrix $u=\|u_{j}^{i}\|\in G$.
In matrix quantum group theory (see [2]), one can considers the elements
$u_{j}^{i}$ as linear coordinate functions on $G$, which assigns to each $g\in
G$ its matrix elements (respect to a given base), namely
$u_{j}^{i}(g)=g_{j}^{i}$, and that one can also interprets as generating the
unital associative algebra $Fun(G)$ of functions on $G$, under the relations
(3).
The latter is a commutative Hopf algebra, with usual comultiplication given by
$(\Delta(f))(g_{1},g_{2})=f(g_{1}g_{2})$, so that the cocommutativity, or not,
of this algebra, is related to the commutativity, or not, of the group $G$;
furthermore, the coproduct is induced by $\Delta u_{j}^{i}=u_{k}^{i}\otimes
u^{k}_{j}$, since $u_{j}^{i}(g)=g_{j}^{i}$.
Hence, following [1], we could say that (3) defines a first (matrix) quantum
group structure preserving each $T(\alpha)$; moreover, we restricts our study
to the case in which $T(\alpha)$ is a given non-degenerate bilinear form
$\Omega_{ij}$ on $\mathbb{R}^{4}$, with dual $\Omega^{ij}$ (given by the
inverse matrix), that is we suppose $r_{\alpha}=0,s_{\alpha}=2$, $card\ \Xi=1$
and444However, the following considerations holds true also for any $n\geq 2$.
$n=4$.
If $\Omega$ is a bilinear form on $\mathbb{R}^{4}$ with components (respect to
a given base) $\Omega_{ij}$, and $\tilde{\Omega}$ is a bilinear form on its
dual with components (respect to the dual base) $\tilde{\Omega}^{ij}$, then,
as known, $\tilde{\Omega}\otimes\Omega$ is identified with the endomorphisms
of $\mathbb{R}^{4}\otimes\mathbb{R}^{4}$ with components
$\Omega^{i_{1}i_{2}}\Omega_{j_{1}j_{2}}$; likewise, if $u$ and $v$ are
endomorphisms of $\mathbb{R}^{4}$ with components $u_{j}^{i}$ and $v_{j}^{i}$,
then $u\otimes v$ is identified with the endomorphism of
$\mathbb{R}^{4}\otimes\mathbb{R}^{4}$ with components
$u_{j_{1}}^{i_{1}}v_{j_{2}}^{i_{2}}$.
Let $\Omega$ be the non-degenerate bilinear form with components (in the
canonical base) $\Omega_{ij}$ given by (3); the matrix of its components
$\Omega_{ij}$, will be denoted again by $\Omega$. Associated to $\Omega$ is
its dual $\Omega^{-1}$ of $\mathbb{R}^{4}\otimes\mathbb{R}^{4}$, that is the
bilinear form on the dual of $\mathbb{R}^{4}$ ($\cong\mathbb{R}^{4}$), with
components $\Omega^{ij}$ defined by $\Omega^{ik}\Omega_{kj}=\delta_{j}^{i}$;
the matrix of the components $\Omega^{ij}$ will be again denoted by
$\Omega^{-1}$, the inverse of the matrix $\Omega$ (that there exists because
$\Omega$ is non-degenerate).
Let $\mathcal{A}_{\mathbb{R}}(\Omega)$ be the unital associative
$\mathbb{R}$-algebra generated by the scalars $t_{j}^{i}\in\mathbb{R}\ \
i,j=0,1,2,3$, with the relations
$\Omega_{ij}t_{k}^{i}t_{l}^{j}=\Omega_{kl},\qquad\Omega^{ij}t^{k}_{i}t_{j}^{l}=\Omega^{kl},\qquad
k,l=0,1,2,3,$
where $\Omega_{kl},\Omega^{kl}\in\mathbb{R}$ are identified, respectively,
with
$\Omega_{kl}1_{\mathcal{A}},\Omega^{kl}1_{\mathcal{A}}\in\mathcal{A}_{\mathbb{R}}(\Omega)$,
if $1_{\mathcal{A}}$ is the unit of $\mathcal{A}_{\mathbb{R}}(\Omega)$.
Hence, it is possible to prove (see [1]) that
1. 1.
there exists a unique homomorphism of algebras, say
$\Delta:\mathcal{A}_{\mathbb{R}}(\Omega)\rightarrow\mathcal{A}_{\mathbb{R}}(\Omega)\otimes\mathcal{A}_{\mathbb{R}}(\Omega)$,
such that $\Delta t_{j}^{i}=t_{k}^{i}\otimes t_{j}^{k}\ \ i,j=0,1,2,3$;
2. 2.
there exists a unique homomorphism of algebras, say
$\varepsilon:\mathcal{A}_{\mathbb{R}}(\Omega)\rightarrow\mathbb{R}$, such that
$\varepsilon(t_{j}^{i})=\delta_{j}^{i}\ \ i,j=0,1,2,3$;
3. 3.
there exists a unique linear antimultiplicative mapping, say
$S:\mathcal{A}_{\mathbb{R}}(\Omega)\rightarrow\mathcal{A}_{\mathbb{R}}(\Omega)$,
such that555Setting $t=\|t_{j}^{i}\|$, it is $S(t)=(\Omega^{-1})^{t}t\Omega$.
$S(t_{j}^{i})=\Omega^{ik}t_{k}^{l}\Omega_{lj}\ \ i,j=0,1,2,3,$ and
$S(1_{\mathcal{A}})=1_{\mathcal{A}}$.
Furthermore, $\Delta$ is a coproduct, $\varepsilon$ is a counit, and $S$ is an
antipode666In general, there is no antipode for a generic tensor $T(\alpha)$.
since $S(t_{k}^{i})t_{j}^{k}=t^{i}_{k}S(t_{j}^{k})$, so that, denoted by $m$
the product of the algebra $\mathcal{A}_{\mathbb{R}}(\Omega)$, we have that
$(\mathcal{A}_{\mathbb{R}}(\Omega),m,1_{\mathcal{A}},\Delta,\varepsilon,S)$ is
a Hopf algebra, called the Hopf algebra of the Einstein bilinear form
$\Omega$.
This Hopf algebra defines (in the terminology of [1]; see also [12], Appendix
2) the quantum group of the non-degenerate bilinear form $\Omega$, that we may
call the Einstein quantum group; hence, as usual, we may think
$\mathcal{A}_{\mathbb{R}}(\Omega)$ as a kind of algebra of ’representative
functions’ on this quantum group.
Besides, this quantum group extends the classical group of the linear
transformations of $\mathbb{R}^{4}$ which preserves $\Omega$, and, therefore,
such a quantum object may represents further generalized symmetries of the
Einstein bilinear form $\Omega$.
The matrix $t=\|t_{j}^{i}\|\in M^{(4,4)}(\mathcal{A}_{\mathbb{R}}(\Omega))$ is
a multiplicative matrix (see [3]) whose entries generates
$\mathcal{A}_{\mathbb{R}}(\Omega)$, obtaining an example of matrix quantum
group.
Note. All the above considerations about $\mathcal{A}_{\mathbb{R}}(\Omega)$,
holds for an arbitrary non-degenerate bilinear form $\Omega$ of
$\mathbb{R}^{n}$, with $n\geq 2$.
Given a non-degenerate bilinear form $\Omega$ on $\mathbb{R}^{n}$ ($n\geq 2$),
with components $\Omega_{ij}$ (respect to the canonical base), we may define
the quadratic homogeneous algebra777For brief recalls on homogeneous algebras,
see [13] or [12], Appendix 1, and references therein.
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ generated by the elements $x^{j}\ \
j=1,...,n$, with the relations $\Omega_{ij}x^{i}x^{j}=0$.
In [12], § 2. (see also [13]), it is proved as
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ be a Gorenstein and Koszul algebra of
global dimension 2. Conversely, it is possible to prove that any quadratic
algebra generated by $n$ elements $x^{j}$, finitely generated in degree 1 and
finitely presented with relations of degree $\geq 2$, which is Gorenstein and
Koszul of low global dimension 2, is an algebra of the type
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ for a certain non-degenerate bilinear form
$\Omega$.
Moreover, if $\Omega\stackrel{{\scriptstyle\chi}}{{\rightarrow}}\Omega\circ M$
is the action given by $(\Omega\circ M)(x,y)=\Omega(Mx,My)$ for each $M\in
GL_{n}(\mathbb{R})$ and $x,y\in\mathbb{R}^{n}$, then it follows that $\chi$
preserves the non-degeneracy of bilinear forms, and
$\mathcal{Q}_{\mathbb{R}}(\Omega)\cong\mathcal{Q}_{\mathbb{R}}(\Omega^{\prime})$
if and only if $\Omega$ and $\Omega^{\prime}$ belong to the same
$GL_{n}(\mathbb{R})$-orbit of $\chi$, that is, if and only if
$\Omega^{\prime}=\Omega\circ M$ for some $M\in GL_{n}(\mathbb{R})$.
Therefore, since the action of $\chi$ corresponds to a change of generators in
$\mathcal{A}_{\mathbb{R}}(\Omega)$, it follows that
$\mathcal{A}_{\mathbb{R}}(\Omega)$ only depends by the orbit of $\Omega$ under
$\chi$.
So, we may define the moduli space
$\mathcal{M}(\mathcal{Q}_{\mathbb{R}}(\Omega))$ of
$\mathcal{Q}_{\mathbb{R}}(\Omega)$, to be the space of all
$GL_{n}(\mathbb{R})$-orbits of $\chi$.
Furthermore, taking into account what has been said above about
$\mathcal{A}_{\mathbb{R}}(\Omega)$ in $\mathbb{R}^{n}$, by Proposition 20 of
[12], Appendix 2, follows that there is a unique algebra homomorphism
$\Delta_{t}:\mathcal{Q}_{\mathbb{R}}(\Omega)\rightarrow\mathcal{A}_{\mathbb{R}}(\Omega)\otimes\mathcal{Q}_{\mathbb{R}}(\Omega)$
such that $\Delta_{t}(x^{j})=t^{j}_{i}\otimes x^{i}$ for all $j=1,...,n$,
endowing $\mathcal{Q}_{\mathbb{R}}(\Omega)$ of a
$\mathcal{A}_{\mathbb{R}}(\Omega)$-comodule structure.
Hence, the quantum group of $\Omega$ coacts on the quantum space corresponding
to $\mathcal{Q}_{\mathbb{R}}(\Omega)$, that is
$\mathcal{Q}_{\mathbb{R}}(\Omega)$ corresponds to the natural quantum space
for the coaction of $\mathcal{A}_{\mathbb{R}}(\Omega)$.
Come back to the case $n=4$, in [1] the Hopf algebra
$\mathcal{A}_{\mathbb{R}}(\Omega)$ is also endowed with a particular quasi-
triangular structure through a $R$-matrix, say
$\mathcal{R}:\mathbb{R}^{4}\otimes\mathbb{R}^{4}\rightarrow\mathbb{R}^{4}\otimes\mathbb{R}^{4}$,
given by $\mathcal{R}_{a}=\tau+a(\Omega^{-1})^{t}\otimes\Omega$, where
$a\in\mathbb{R}\setminus\\{0\\}$ and $\tau$ is the flip map.
Indeed, for $a\neq 0$, we have the following homogeneous defining relations of
$\mathcal{A}_{\mathbb{R}}(\Omega)$:
$\mathcal{R}^{i_{1}i_{2}}_{k_{1}k_{2}}t^{k_{1}}_{j_{1}}t^{k_{2}}_{j_{2}}=t^{i_{1}}_{k_{1}}t^{i_{2}}_{k_{2}}\mathcal{R}^{k_{1}k_{2}}_{j_{1}j_{2}},\qquad
i_{l},j_{l}=0,1,2,3,\quad l=1,2,$
so that such $\mathcal{R}$ is a $R$-matrix because it satisfy the following,
well-known Yang-Baxter equation
$\mathcal{R}_{12}\mathcal{R}_{13}\mathcal{R}_{23}=\mathcal{R}_{23}\mathcal{R}_{13}\mathcal{R}_{12}$
when and only when $a\in\mathbb{R}\setminus\\{0\\}$ verify the braid relation
$a(1+a\Omega^{ij}\Omega_{ij}+a^{2})=0$, equivalent (since $a\neq 0$) to
$a+a^{-1}+\Omega^{ij}\Omega_{ij}=a+a^{-1}+tr(\Omega^{-1}\Omega^{t})=0$.
Thus, we have a $R$-matrix for $\mathcal{A}_{\mathbb{R}}(\Omega)$, given by
$\mathcal{R}=\tau+a(\Omega^{-1})^{t}\otimes\Omega$ with
$a\in\mathbb{R}\setminus\\{0\\}$ such that
$a+a^{-1}+\Omega^{ij}\Omega_{ij}=0$.
In [14], it is proved that the representation category of
$\mathcal{A}_{\mathbb{R}}(\Omega)$ (in $\mathbb{R}^{n},\ \ n\geq 2$) is
monoidally equivalent to the representation category of the quantum group
$\mathcal{O}_{a}(SL_{2}(\mathbb{R}))$ of functions over $SL_{2}(\mathbb{R})$,
if $a\in\mathbb{R}\setminus\\{0\\}$ verify the above braid relation, so that
$Comod(\mathcal{A}_{\mathbb{R}}(\Omega))\cong^{\otimes}Comod(O_{q}(SL_{2}(\mathbb{R}))$.
Moreover, in the § 5. of [14] it is also presented the following isomorphic
classification of the Hopf algebra $\mathcal{A}_{\mathbb{R}}(\Omega)$: if
$\Omega$ and $\Omega^{\prime}$ are non-degenerate bilinear forms respectively
in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ with $n,m\geq 2$, then
$\mathcal{A}_{\mathbb{R}}(\Omega)$ and
$\mathcal{A}_{\mathbb{R}}(\Omega^{\prime})$ are isomorphic if and only if
$m=n$ and there exists $M\in GL_{n}(\mathbb{R})$ such that
$\Omega^{\prime}=M^{t}\Omega M$.
Then, in the § 6 of [14], the Author determines the possible Hopf
$\ast$-algebra structures and CQG (compact quantum group) algebra structures
on $\mathcal{A}_{\mathbb{C}}(\Omega)$ (that is, in the complex case).
Following the results of [14], T. Aubriot, in [15], studies the possible
Galois and bi-Galois objects over $\mathcal{A}_{\mathbb{R}}(\Omega)$.
At last, the paper [1] finishes with some remarks; in particular, the Authors
notices that, in dimension $n\geq 3$, there is no $\Omega$ such that
$\mathcal{A}_{\mathbb{R}}(\Omega)$ be commutative, that is to say, a such Hopf
algebra is necessarily non-commutative.
On the other hand, we remember that in $\mathbb{R}^{4}$ may be establish a
standard canonical complex structure as follows.
Respect to the canonical base of $\mathbb{R}^{4}$, if $J_{0}\in End\
(\mathbb{R}^{4})$ is defined putting $J_{0}(e_{j})=e_{n+j}$ for $1\leq j\leq
2$ and $J_{0}(e_{j})=-e_{j-n}$ for $3\leq j\leq 4$, then it follows that such
a $J_{0}$ is a complex structure888Since $J_{0}^{2}=-id_{\mathbb{R}^{4}}$. on
$\mathbb{R}^{4}$, and if $\mathbb{R}^{4}_{\mathbb{C}}(J_{0})$ is the resulting
linear complex space structure induced by $J_{0}$ on $\mathbb{R}^{4}$, then we
have the canonical isomorphism
$\mathbb{R}^{4}_{\mathbb{C}}(J_{0})\cong\mathbb{C}^{2}$.
From here, it is possible to construct the following faithful representation
$\rho:M^{(2,2)}(\mathbb{C})\rightarrow M^{(4,4)}(\mathbb{R})$ defined by
$\rho(A+iB)=\left(\begin{array}[]{cc}A&-B\\\ B&A\end{array}\right),$
that it is a $\mathbb{R}$-algebra monomorphism such that
$\rho(iH)=J_{0}\rho(H)$ for any $H\in M^{(2,2)}(\mathbb{C})$, extending the
usual immersion999For any $n\geq 2$, we remember that there exists a well-
known immersion $GL_{n}(\mathbb{C})\hookrightarrow GL_{2n}(\mathbb{R})$.
$GL_{2}(\mathbb{C})\hookrightarrow GL_{4}(\mathbb{R})$.
Hence, if $\Omega_{ij}\in M^{(4,4)}(\mathbb{R})$ of $(2^{\prime})$, is such
that $\Omega_{ij}\in\rho(M^{(2,2)}(\mathbb{C}))$, let
$\tilde{\Omega}_{ij}=\rho^{-1}(\Omega_{ij})\in M^{(2,2)}(\mathbb{C})$; whence,
we may identifies $\Omega_{ij}$ with $\tilde{\Omega}_{ij}$, that it is a non-
degenerate (if such is $\Omega_{ij}$) bilinear form of $\mathbb{C}^{2}$.
Therefore, if $\mathcal{A}_{\mathbb{C}}(\tilde{\Omega})$ is the Hopf algebra
associated to $\tilde{\Omega}$, then it is immediate to prove that
$\mathcal{A}_{\mathbb{R}}(\Omega)\cong\mathcal{A}_{\mathbb{C}}(\tilde{\Omega})$.
In [1], § 6., there is a complete classification of the moduli space of
$\tilde{\Omega}$, according to the rank of $\tilde{\Omega}$. Precisely
* •
if $rk\ \tilde{\Omega}=0$, then there is only one orbit of which one
representative element is $\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)$, this case corresponding to $SL_{2}(\mathbb{C})$ with
$R$-matrix the identity $R_{0}$ of $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$;
* •
if $rk\ \tilde{\Omega}=1$, then there is only one orbit of which one
representative element is $\left(\begin{array}[]{cc}0&-1\\\
1&\lambda\end{array}\right)$ with $\lambda\neq 0$ (these are all equivalent
among them), this case corresponding to the so called Manin’s jordanian (that
it is a special quantum deformation of $SL_{2}(\mathbb{C})$; see [3]), with
equivalent $R$-matrices $R_{\lambda}$ such that $\lim_{\lambda\rightarrow
0}R_{\lambda}=R_{0}$;
* •
if $rk\ \tilde{\Omega}=2$, then there are many orbits, each represented by
$\tilde{\Omega}_{q}=\left(\begin{array}[]{cc}0&-1\\\ q&0\end{array}\right)$
for every $q\in\mathbb{C}\setminus\\{0,1\\}$, with
$\mathcal{A}_{\mathbb{C}}(\tilde{\Omega}_{q})\cong SL_{2,q}(\mathbb{C})$,
$R$-matrix corresponding to that of $M_{2,q}(\mathbb{C})$ (quantum deformation
of $M^{(2,2)}(\mathbb{C})$; see [3]), and
$\mathcal{Q}_{\mathbb{C}}(\tilde{\Omega}_{q})$ corresponding to the Manin
plane (that it is the natural quantum space for the coaction of
$SL_{2,q}(\mathbb{C})$).
The considerations of this paper, may have physical interpretations in view of
the possible physical meaning of $\Omega$ (and of $\tilde{\Omega}$, when
$\tilde{\Omega}$ exists).
$\bf References.$
[1] M. Dubois-Violette, G. Launer, ”The quantum group of a non-degenerate
bilinear form”, Physics Letters B, 245(2) (1990) 175-177.
[2] S. Majid, Foundations of quantum group theory, Cambridge University Press,
Cambridge, 1995.
[3] Yu.I. Manin, Quantum groups and Non-Commutative Geometry, Publications du
CRM de l’Univesité de Montréal, Montréal, 1988.
[4] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, W.H. Freeman and
Company, San Francisco, 1973.
[5] R.K. Sachs, H. Wu, General Relativity for Mathematicians, Springer-Verlag,
New York, 1977.
[6] J. Stewart, Advanced General Relativity, Cambridge University Press,
Cambridge, 1991.
[7] R.M. Wald, General Relativity, University of Chicago Press, Chicago, 1984.
[8] M. Dubois-Violette, ”On the theory of quantum groups”, Letters in
Mathematical Physics, 19 (1990) 121-126.
[9] M. Francaviglia, Relativistic Theories, Quaderni del GNFM-CNR, Firenze,
1988.
[10] L. Nobili, Astrofisica Relativistica, CLEUP Editrice, Padova, 2003.
[11] J. Madore, An Introduction to Noncommutative Geometry and its Physical
Applications, LMS 206, Cambridge University Press, Cambridge, 1998.
[12] M. Dubois-Violette, ”Multilinear Forms and Graded Algebras”, Journal of
Algebra, 317 (2007) 198-225.
[13] M. Dubois-Violette, ”Graded algebras and multilinear forms”, C.R. Acad.
Sci. Paris, Ser. I, 341 (2005) 719-724.
[14] J. Bichon, ”The representation category of the quantum group of a non-
degenerate bilinear form”, Comm. Alg., 31 (2003) 4831-4851.
[15] T. Aubriot, ”On the classification of Galois objects over the quantum
group of a nondegenerate bilinear form”, Man. Math., 122 (2007) 119-135.
[16] D. Lovelock, ”The four-dimensionality of space and the Einstein’s
tensor”, Journal of Mathematical Physics, 13 (6) (1972) 874-876.
[17] T. Frankel, Gravitational Curvature, W.H. Freeman and Comp., San
Francisco, 1979.
|
arxiv-papers
| 2011-02-09T14:08:28 |
2024-09-04T02:49:16.880657
|
{
"license": "Public Domain",
"authors": "Giuseppe Iurato",
"submitter": "Giuseppe Iurato",
"url": "https://arxiv.org/abs/1102.1866"
}
|
1102.1889
|
# Ologs: a categorical framework for knowledge representation
David I. Spivak Mathematics, MIT, Cambridge, MA 02139 dspivak@math.mit.edu
and Robert E. Kent Ontologos rekent@ontologos.org
###### Abstract.
In this paper we introduce the olog, or ontology log, a category-theoretic
model for knowledge representation (KR). Grounded in formal mathematics, ologs
can be rigorously formulated and cross-compared in ways that other KR models
(such as semantic networks) cannot. An olog is similar to a relational
database schema; in fact an olog can serve as a data repository if desired.
Unlike database schemas, which are generally difficult to create or modify,
ologs are designed to be user-friendly enough that authoring or reconfiguring
an olog is a matter of course rather than a difficult chore. It is hoped that
learning to author ologs is much simpler than learning a database definition
language, despite their similarity. We describe ologs carefully and illustrate
with many examples. As an application we show that any primitive recursive
function can be described by an olog. We also show that ologs can be aligned
or connected together into a larger network using functors. The various
methods of information flow and institutions can then be used to integrate
local and global world-views. We finish by providing several different avenues
for future research.
This project was supported by Office of Naval Research grant: N000141010841
and a generous contribution by Clark Barwick, Jacob Lurie, and the
Massachusetts Institute of Technology Department of Mathematics
###### Contents
1. 1 Introduction
2. 2 Types, aspects, and facts
3. 3 Instances
4. 4 Communication between ologs
5. 5 More expressive ologs I
6. 6 More expressive ologs II
7. 7 Further directions
## 1\. Introduction
Scientists have a pressing need to organize their experiments, their data,
their results, and their conclusions into a framework such that this work is
reusable, transferable, and comparable with the work of other scientists. In
this paper, we will discuss the “ontology log” or olog as a possibility for
such a framework. Ontology is the study of what something is, i.e the nature
of a given subject, and ologs are designed to record the results of such a
study. The structure of ologs is based on a branch of mathematics called
category theory. An olog is roughly a category that models a given real-world
situation.
The main advantages of authoring an olog rather than writing a prose
description of a subject are that
* •
an olog gives a precise formulation of a conceptual world-view,
* •
an olog can be formulaically converted into a database schema,
* •
an olog can be extended as new information is obtained,
* •
an olog written by one author can be easily and precisely referenced by
others,
* •
an olog can be input into a computer and “meaningfully stored”, and
* •
different ologs can be compared by functors, which in turn generate automatic
terminology translation systems.
The main disadvantage to using ologs over prose, aside from taking more space
on the page, is that writing a good olog demands a clarity of thought that
ordinary writing or conversation can more easily elide. However, the
contemplation required to write a good olog about a subject may have
unexpected benefits as well.
A category is a mathematical structure that appears much like a directed
graph: it consists of objects (often drawn as nodes or dots, but here drawn as
boxes) and arrows between them. The feature of categories that distinguishes
them from graphs is the ability to declare an equivalence relation on the set
of paths. A functor is a mapping from one category to another that preserves
the structure (i.e. the nodes, the arrows, and the equivalences). If one views
a category as a kind of language (as we shall in this paper) then a functor
would act as a kind of translating dictionary between languages. There are
many good references on category theory, including [LS], [Sic], [Pie], [BW1],
[Awo], and [Mac]; the first and second are suited for general audiences, the
third and fourth are suited for computer scientists, and the fifth and sixth
are suited for mathematicians (in each class the first reference is easier
than the second).
A basic olog, defined in Section 2, is a category in which the objects and
arrows have been labeled by English-language phrases that indicate their
intended meaning. The objects represent types of things, the arrows represent
functional relationships (also known as aspects, attributes, or observables),
and the commutative diagrams represent facts. Here is a simple olog about an
amino acid called arginine ([W]):
(7) $\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{\parbox{72.26999pt}{\raggedright an amino acid found in
dairy\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
A}}{{\framebox{\parbox{36.135pt}{arginine}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$hasisis$\textstyle{\stackrel{{\scriptstyle
E}}{{\framebox{\parbox{65.04256pt}{\raggedright an electrically-charged side
chain\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
X}}{{\framebox{\parbox{65.04256pt}{an amino
acid}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$hashashas$\textstyle{\stackrel{{\scriptstyle
R}}{{\framebox{a side chain}}}}$$\textstyle{\stackrel{{\scriptstyle
N}}{{\framebox{\parbox{72.26999pt}{an amine
group}}}}}$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{72.26999pt}{a carboxylic acid}}}}}$
The idea of representing information in a graph is not new. For example the
Resource Descriptive Framework (RDF) is a system for doing just that [CM]. The
key difference between a category and a graph is the consideration of paths,
and that two paths from $A$ to $B$ may be declared identical in a category
(see [Spi3]). For example, we can further declare that in Diagram (7), the
diagram
(12)
commutes, i.e. that the two paths
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R}$
are equivalent, which can be translated as follows. Let $A$ be a molecule of
arginine. On the one hand $A$, being an amino acid, has a side chain; on the
other hand $A$ has an electrically-charged side-chain, which is of course a
side chain. We seem to have associated two side-chains to $A$, but in fact
they both refer to the same physical thing, the same side-chain. Thus, the two
paths $A\rightarrow R$ are deemed equivalent. The fact that this equivalence
may seem trivial is not an indictment of the category idea but instead
reinforces its importance — we must be able to indicate obvious facts within a
given situation because what is obvious is the most essential.
While many situations can be modeled using basic ologs (categories), we often
need to encode more structure. For this we will need so-called sketches. An
olog will be defined as a finite limit, finite colimit sketch (see [BW2]),
meaning we have the ability to encode objects (“types”), arrows (“aspects”),
commutative diagrams (“facts”), as well as finite limits (“layouts”) and
finite colimits (“groupings”).
Throughout this paper, whenever we refer to “the author” of an olog we am
referring to the fictitious person who created it. We will refer to ourselves,
David Spivak and Robert Kent, as “we” so as not to confuse things.
###### Warning 1.0.1.
The author of an olog has a world-view, some fragment of which is captured in
the olog. When person A examines the olog of person B, person A may or may not
“agree with it.” For example, person B may have the following olog
a marriage includesincludes a mana woman
which associates to each marriage a man and a woman. Person A may take the
position that some marriages involve two men or two women, and thus see B’s
olog as “wrong.” Such disputes are not “problems” with either A’s olog or B’s
olog, they are discrepancies between world-views. Hence, throughout this
paper, a reader R may see a displayed olog and notice a discrepancy between
R’s world-view and our own, but R should not worry that this is a problem.
This is not to say that ologs need not follow rules, but instead that the
rules are enforced to ensure that an olog is structurally sound, rather than
that it “correctly reflects reality,” whatever that may mean.
### 1.1. Plan of this paper
In this paper, we will define ologs and give several examples. We will state
some rules of “good practice” which help one to author ologs that are
meaningful to others and easily extendable. We will begin in Section 2 by
laying out the basics: types as objects, aspects as arrows, and facts as
commutative diagrams. In Section 3, we will explain how to attach “instance”
data to an olog and hence realize ologs as database schemas. In Section 4, we
will discuss meaningful constraints betweeen ologs that allow us to develop a
higher-dimensional web of information called an information system, and we
will discuss how the various parts of such a system interact via information
channels. In Sections 5 and 6, we will extend the olog definition language to
include “layouts” and “groupings”, which make for more expressive ologs; we
will also describe two applications, one which explicates the computation of
the factorial function, and the other which defines a notion from pure
mathematics (that of pseudo-metric spaces). Finally, in Section 7, we will
discuss some possible directions for future research.
For the remainder of the present section, we will explain how ologs relate to
existing ideas in the field of knowledge representation.
### 1.2. The semantic advantage of ologs: modularity
The difference between ologs and prose is modularity: small conceptual pieces
can form large ideas, and these pieces work best when they are reusable. The
same phenomenon is true throughout computer science and mathematics. In
programming languages, modularity brings not only vast efficiency to the
writing of programs but enables an “abstraction barrier” that keeps the ideas
clean. In mathematics, the most powerful results are often simple lemmas that
are reusable in a wide variety of circumstances.
Web pages that consist of prose writing are often referred to as information
silos. The idea is that a silo is a “big tube of stuff” which is not organized
in any real way. Links between web pages provide some structure, but such a
link does not carry with it a precise method to correlate the information
within the two pages. Similarly in science, one author may reference another
paper, but such a reference carries very little structure — it just points to
a silo.
Ologs can be connected with links which are much richer than the link between
two silos could possibly be. Individual concepts and connections within one
olog can be “functorially aligned” with concepts and connections in another. A
functor creates a precise connection between the work of one author and the
work of another so that the precise nature of the comparison is not left to
the reader’s imagination but explicitly specified. The ability to incorporate
mathematical precision into the sharing of ideas is a central feature of
ologs.
### 1.3. Relation to other models
There are many languages for knowledge representation (KR). For example, there
are database languages such as SQL, ontology languages such as RDF and OWL,
the language of Semantic Nets, and others (see [Bor]). One may ask what makes
the olog concept different or better than the others.
The first response is that ologs are closely related to the above ideas.
Indeed, all of these KR models can be “categorified” (i.e. phrased in the
language of category theory) and related by functors, so that many of the
ideas align and can be transferred between the different systems. In fact, as
we will make clear in Section 3, ologs are almost identical to the categorical
model of databases presented in [Spi2].
However, ologs have advantages over many existing KR models. The first
advantage arises from the notion of commutative diagrams (which allow us to
equate different paths through the domain, see Section 2.3) and of limits and
colimits (which allow us to lay out and group things, see Sections 5 and 6).
The additional expressivity of ologs give them a certain semantic clarity and
interoperability that cannot be achieved with graphs and networks in the usual
sense. The second advantage arises from the notion of olog morphisms, which
allow the definition of meaningful constraints between ologs. With this in
hand, we can integrate a set of similar ologs into a single information
system, and go on to define information fusion. This will be discussed further
Section 4.
In the remainder of this section we will provide a few more details on the
relationship between ologs and each of the above KR models: databases,
RDF/OWL, and semantic nets. The reader who does not know or care much about
other systems of knowledge representation can skip to Section 1.4.
#### 1.3.1. Ologs and Databases
A database is a system of tables, each table of which consists of a header of
columns and a set of rows. A table represents a type of thing $T$, each column
represents an attribute of $T$, and each row represents an example of $T$. An
attribute is itself a “type of thing”, so each column of a table points to
another table.
The relationship between ologs and databases is that every box $B$ in an olog
represents a type of thing and every arrow $B\rightarrow X$ emanating from $B$
represents an attribute of $B$ (whose results are of type $X$). Thus the boxes
and arrows in an olog correspond to tables and their columns in a database.
The rows of each table in a database will correspond to “instances” of each
type in an olog. Again, this will be made more clear in Section 3 or one can
see [Spi2] or [Ken5].
The point is that every olog can serve as a database schema, and the schemas
represented by ologs range from simple (just objects and arrows) to complex
(including commutative diagrams, products, sums, etc.). However, whereas
database schemas are often prescriptive (“you must put your data into this
format!”), ologs are usually descriptive (“this is how I see things”). One can
think of an olog as an interface between people and databases: an olog is
human readable, but it is also easily converted to a database schema upon
which powerful applications can be put to work. Of course, if one is to use an
olog as a database schema, it will become prescriptive. However, since the
intention of each object and arrow is well-documented (as its label), schema
evolution would be straightforward. Moreover, the categorical structure of
ologs allows for functorial data migration by which one can transfer the
instance data from an older schema to the current one (see [Spi2]).
#### 1.3.2. Ologs and RDF / OWL
In [Spi2], the first author explained how a categorical database can be
converted into an RDF triple store using the Grothendieck construction. The
main difference between a categorical database schema (or an olog) and an RDF
schema is that one cannot specify commutativity in an RDF schema. Thus one
cannot express things like “the woman parent of a person $x$ is the mother of
$x$.” Without this expressivity, it is hard to enforce much rigor, and thus
RDF data tends to be too loose for many applications.
OWL schemas, on the other hand, can express many more constraints on classes
and properties. We have not yet explored the connection, nor compared the
expressive power, of ologs and OWL. However, they are significantly different
systems, most obviously in that OWL relies on logic where ologs rely on
category theory.
#### 1.3.3. Semantic Nets
On the surface, ologs look the most like semantic networks, or concept webs,
but there are important differences between the two notions. First, arrows in
a semantic network need not indicate functions; they can be relations. So
there could be an arrow $\ulcorner$a
father$\urcorner$$\xrightarrow{\textnormal{has}}$$\ulcorner$a child$\urcorner$
in a semantic network, but not in an olog (see Section 2.2.3 for how the same
idea is expressible in an olog). There is a nice category of sets and
relations, often denoted Rel, but this category is harder to reason about than
is the ordinary category of sets and functions (often denoted ${\bf Set}$).
Thus, as mentioned above, semantic networks are categorifiable (using Rel),
but this underlying formalism does not appear to play a part in the study or
use of semantic networks. However, some attempt to integrate category theory
and neural nets has been made, see [HC].
Moreover, commutative diagrams and other expressive abilities held by ologs
are not generally part of the semantic network concept (see [Sow1]). For these
reasons, semantic networks tend to be brittle: minor changes can have
devastating effects. For example, if two semantic networks are somehow synced
up and then one is changed, the linkage must be revised or may be altogether
broken. Such a disaster is often avoided if one uses categories: because
different paths can be equivalent, one can simply add new ideas (types and
aspects) without changing the semantic meaning of what was already there. As
section 4.4 demonstates with an extended example, conceptual graphs, which are
a popular formalism for semantics nets, can be linearized to ologs, thereby
gaining in precision and expressibility.
### 1.4. Acknowledgements
#### 1.4.1. David Spivak’s acknowledgments
I would like to thank Mathieu Anel and Henrik Forssell for many pleasant and
quite useful conversations. I would also like to thank Micha Breakstone for
his help on understanding the relationship between ologs and linguistics.
Finally I would like to thank Dave Balaban for helpful suggestions on this
document itself.
#### 1.4.2. Robert Kent’s acknowledgments
I would like to thank the participants in the Standard Upper Ontology working
group for many interesting, spirited, rewarding and enlightening discussions
about knowledge representation in general and ontologies in particular; I
especially want to thank Leo Obrst, Marco Schorlemmer and John Sowa from that
group. I want to thank Jon Barwise for leading the development of the theory
of information flow. I want to thank Joseph Goguen for leading the development
of the theory of institutions, and for pointing out the common approach to
knowledge representation used by both the Information Flow Framework and the
theory of institutions.
## 2\. Types, aspects, and facts
In this section we will explain basic ologs, which involve types, aspects, and
facts. A basic olog is a category in which each object and arrow has been
labeled by text; throughout this paper we will assume that text to be written
in English.
The purpose of this section is to show how one can convert a real-world
situation into an olog. It is probably impossible to explain this process
precisely in words. Instead, we will explain mainly by example. We will give
“rules of good practice” that lead to good ologs. While these rules are not
strictly necessary, they help to ensure that the olog is properly formulated.
As the Dalai Lama says, “Learn the rules so you know how to break them
properly.”
### 2.1. Types
A type is an abstract concept, a distinction the author has made. We represent
each type as a box containing a singular indefinite noun phrase. Each of the
following four boxes is a type:
(17)
Each of the four boxes in (17) represents a type of thing, a whole class of
things, and the label on that box is what one should call each example of that
class. Thus $\ulcorner$a man$\urcorner$ does not represent a single man, but
the set of men, each example of which is called “a man”111In other words,
types in ologs are intentional, rather than extensional — the label on a type
describes its intention. The extension of a type will be captured by instance
data; see Section 3 .. Similarly, the bottom right-hand box in (17) represents
an abstract type of thing, which probably has more than a million examples,
but the label on the box indicates a common name for each such example.
Typographical problems emerge when writing a text-box in a line of text, e.g.
the text-box a man seems out of place here, and the more in-line text-boxes
one has in a given paragraph, the worse it gets. To remedy this, we will
denote types which occur in a line of text with corner-symbols, e.g. we will
write $\ulcorner$a man$\urcorner$ instead of a man.
#### 2.1.1. Types with compound structures
Many types have compound structures; i.e. they are composed of smaller units.
Examples include
(20)
It is good practice to declare the variables in a “compound type”, as we did
in the last two cases of (20). In other words, it is preferable to replace the
first box above with something like
$\stackrel{{\scriptstyle}}{{\framebox{\parbox{57.81621pt}{a man $m$ and a
woman $w$}}}}\hskip 14.45377pt\textnormal{or}\hskip
14.45377pt\stackrel{{\scriptstyle}}{{\framebox{\parbox{79.49744pt}{\raggedright
a pair $(m,w)$ where $m$ is a man and $w$ is a woman\@add@raggedright}}}}$
so that the variables $(m,w)$ are clear.
###### Rules of good practice 2.1.1.
A type is presented as a text box. The text in that box should
1. (i)
begin with the word “a” or “an”;
2. (ii)
refer to a distinction made and recognizable by the author;
3. (iii)
refer to a distinction for which instances can be documented;
4. (iv)
not end in a punctuation mark;
5. (v)
declare all variables in a compound structure.
The first, second, and third rules ensure that the class of things represented
by each box appears to the author as a well-defined set; see Section 3 for
more details. The fourth and fifth rules encourage good “readability” of
arrows, as will be discussed next in Section 2.2.
We will not always follow the rules of good practice throughout this document.
We think of these rules being followed “in the background” but that we have
“nicknamed” various boxes. So $\ulcorner$Steve$\urcorner$ may stand as a
nickname for $\ulcorner$a thing classified as Steve$\urcorner$ and
$\ulcorner$arginine$\urcorner$ as a nickname for $\ulcorner$a molecule of
arginine$\urcorner$.
### 2.2. Aspects
An aspect of a thing $x$ is a way of viewing it, a particular way in which $x$
can be regarded or measured. For example, a woman can be regarded as a person;
hence “being a person” is an aspect of a woman. A man has a height (say, taken
in inches), so “having a height (in inches)” is an aspect of a man. In an
olog, an aspect of $A$ is represented by an arrow $A\rightarrow B$, where $B$
is the set of possible “answers” or results of the measurement. For example
when observing the height of a man, the set of possible results is the set of
integers, or perhaps the set of integers between 20 and 120.
(23) (26)
We will formalize the notion of aspect by saying that aspects are functional
relationships.222In type theory, what we here call aspects are called
functions. Since our types are not fixed sets (see Section 3), we preferred a
term that was less formal. Suppose we wish to say that a thing classified as
$X$ has an aspect $f$ whose result set is $Y$. This means there is a
functional relationship called $f$ between $X$ and $Y$, which can be denoted
$f\colon X\rightarrow Y$. We call $X$ the domain of definition for the aspect
$f$, and we call $Y$ the set of result values for $f$. For example, a man has
a height in inches whose result is an integer, and we could denote this by
$h\colon M\rightarrow{\bf Int}$. Here, $M$ is the domain of definition for
height and ${\bf Int}$ is the set of result values.
A set may always be drawn as a blob with dots in it. If $X$ and $Y$ are two
sets, then a a function from $X$ to $Y$, denoted $f\colon X\rightarrow Y$ can
be presented by drawing arrows from dots in blob $X$ to dots in blob $Y$.
There are two rules:
1. (i)
each arrow must emanate from a dot in $X$ and point to a dot in $Y$;
2. (ii)
each dot in $X$ must have precisely one arrow emanating from it.
Given an element $x\in X$, the arrow emanating from it points to some element
$y\in Y$, which we call the image of $x$ under $f$ and denote $f(x)=y$.
Again, in an olog, an aspect of a thing $X$ is drawn as a labeled arrow
pointing from $X$ to a “set of result values.” Let us concentrate briefly on
the arrow in (23). The domain of definition is the set of women (a set with
perhaps 3 billion elements); the set of result values is the set of persons (a
set with perhaps 6 billion elements). We can imagine drawing an arrow from
each dot in the “woman” set to a unique dot in the “person” set. No woman
points to two different people, nor to zero people — each woman is exactly one
person — so the rules for a functional relationship are satisfied. Let us now
concentrate briefly on the arrow in (26). The domain of definition is the set
of men, the set of result values is the set of integers
$\\{20,21,22,\ldots,119,120\\}$. We can imagine drawing an arrow from each dot
in the “man” set to a single dot in the “integer” set. No man points to two
different heights, nor can a man have no height: each man has exactly one
height. Note however that two different men can point to the same height.
#### 2.2.1. Invalid aspects
We tried above to clarify what it is that makes an aspect “valid”, namely that
it must be a “functional relationship.” In this subsection we will present two
arrows which on their face may appear to be aspects, but which on closer
inspection are not functional (and hence are not valid as aspects).
Consider the following two arrows:
(29) (32)
A person may have no children or may have more than one child, so the first
arrow is invalid: it is not functional because it does not satisfy rule (2)
above. Similarly, if we drew an arrow from each mechanical pencil to each
piece of lead it uses, it would not satisfy rule (2) above. Thus neither of
these is a valid aspect.
Of course, in keeping with Warning 1.0.1, the above arrows may not be wrong
but simply reflect that the author has a strange world-view or a strange
vocabulary. Maybe the author believes that every mechanical pencil uses
exactly one piece of lead. If this is so, then $\textnormal{$\ulcorner$a
mechanical
pencil$\urcorner$}\xrightarrow{\textnormal{uses}}\textnormal{$\ulcorner$a
piece of lead$\urcorner$}$ is indeed a valid aspect! Similarly, suppose the
author meant to say that each person was once a child, or that a person has an
inner child. Since every person has one and only one inner child (according to
the author), the map $\textnormal{$\ulcorner$a
person$\urcorner$}\xrightarrow{\textnormal{has as inner
child}}\textnormal{$\ulcorner$a child$\urcorner$}$ is a valid aspect. We
cannot fault the author for such a view, but note that we have changed the
name of the label to make its intention more explicit.
#### 2.2.2. Reading aspects and paths as English phrases
Each arrow (aspect) $X\xrightarrow{f}Y$ can be read by first reading the label
on its source box (domain of definition) $X$, then the label on the arrow $f$,
and finally the label on its target box (set of result values) $Y$. For
example, the arrow
(11) $\textstyle{\stackrel{{\scriptstyle}}{{\framebox{a
book}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has as first
author$\textstyle{\stackrel{{\scriptstyle}}{{\framebox{a person}}}}$
is read “a book has as first author a person”, a valid English sentence.
Sometimes the label on an arrow can be shortened or dropped altogether if it
is obvious from context. We will discuss this more in Section 2.3 but here is
a common example from the way we write ologs.
(16) $\textstyle{\stackrel{{\scriptstyle
A}}{{\framebox{\parbox{86.72377pt}{\raggedright a pair $(x,y)$ where $x$ and
$y$ are
integers\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\scriptstyle{y}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{an integer}}}}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{an integer}}}}$
Neither arrow is readable by the protocol given above (e.g. “a pair $(x,y)$
where $x$ and $y$ are integers $x$ an integer” is not an English sentence),
and yet it is obvious what each map means. For example, given the pair
$(8,11)$ which belongs in box $A$, application of arrow $x$ would yield $8$ in
box $B$. The label $x$ can be thought of as a nickname for the full name
“yields, via the value of $x$,” and similarly for $y$. We do not generally use
the full name for fear that the olog would become cluttered with text.
One can also read paths through an olog by inserting the word “which” after
each intermediate box. For example the following olog has two paths of length
3 (counting arrows in a chain):
(21)
---
a childisa personhas as parentshas, as
birthday$\textstyle{\stackrel{{\scriptstyle}}{{\framebox{\parbox{57.81621pt}{\raggedright
a pair $(w,m)$ where $w$ is a woman and $m$ is a
man\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$w$a
womana dateincludesa year
The top path is read “a child is a person, which has as parents a pair $(w,m)$
where $w$ is a woman and $m$ is a man, which yields, via the value of $w$, a
woman.” The reader should read and understand the content of the bottom path.
#### 2.2.3. Converting non-functional relationships to aspects
There are many relationships that are not functional, and these cannot be
considered aspects. Often the word “has” indicates a relationship — sometimes
it is functional as in $\textnormal{$\ulcorner$a
person$\urcorner$}\xrightarrow{\textnormal{ has }}\textnormal{$\ulcorner$a
stomach$\urcorner$}$, and sometimes it is not, as in $\textnormal{$\ulcorner$a
father$\urcorner$}\xrightarrow{\textnormal{has}}\textnormal{$\ulcorner$a
child$\urcorner$}$. (Obviously, a father may have more than one child.) A
quick fix would be to replace the latter by $\textnormal{$\ulcorner$a
father$\urcorner$}\xrightarrow{\textnormal{has}}\textnormal{$\ulcorner$a set
of children$\urcorner$}$. This is ok, but the relationship between
$\ulcorner$a child$\urcorner$ and $\ulcorner$a set of children$\urcorner$ then
becomes an issue to deal with later. There is another way to indicate such
“non-functional” relationships.
In mathematics, a relation between sets $A_{1},A_{2}$, and so on through
$A_{n}$ is defined to be a subset of the Cartesian product
$R\subseteq A_{1}\times A_{2}\times\cdots\times A_{n}.$
The set $R$ represents those sequences $(a_{1},a_{2},\ldots,a_{n})$ that are
so-related. In an olog, we represent this as follows
|
---|---
$\textstyle{\framebox{$R$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\framebox{$A_{1}$}}$$\textstyle{\framebox{$A_{2}$}}$$\textstyle{\cdots}$$\textstyle{\framebox{$A_{n}$}}$
For example,
---
$\textstyle{\stackrel{{\scriptstyle R}}{{\framebox{\parbox{115.63243pt}{a
sequence $(p,a,j)$ where $p$ is a paper, $a$ is an author of $p$, and $j$ is a
journal in which $p$ was
published}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{a}$$\scriptstyle{j}$$\textstyle{\stackrel{{\scriptstyle
A_{1}}}{{\framebox{a paper}}}}$$\textstyle{\stackrel{{\scriptstyle
A_{2}}}{{\framebox{an author}}}}$$\textstyle{\stackrel{{\scriptstyle
A_{3}}}{{\framebox{a journal}}}}$
Whereas $A_{1}\times A_{2}\times A_{3}$ includes all possible triples
$(p,a,j)$ where $a$ is a person, $p$ is a paper, and $j$ is a journal, it is
obvious that not all such triples are found in $R$. Thus $R$ represents a
proper subset of $A_{1}\times A_{2}\times A_{3}$.
###### Rules of good practice 2.2.1.
An aspect is presented as a labeled arrow, pointing from a source box to a
target box. The arrow text should
1. (i)
begin with a verb;
2. (ii)
yield an English sentence, when the source-box text followed by the arrow text
followed by the target-box text is read;
3. (iii)
refer to a functional dependence: each instance of the source type should give
rise to a specific instance of the target type;
### 2.3. Facts
In this section we will discuss facts and their relationship to “path
equivalences.” It is such path equivalences, which exist in categories but do
not exist in graphs, that make category theory so powerful. See [Spi3] for
details.
Given an olog, the author may want to declare that two paths are equivalent.
For example consider the two paths from $A$ to $C$ in the olog
(26) $\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{a
person}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has
as parents has as mother $\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{57.81621pt}{\raggedright a pair $(w,m)$ where $w$ is a
woman and $m$ is a
man\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\checkmark}$$w$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{a woman}}}}$
We know as English speakers that a woman parent is called a mother, so these
two paths $A\rightarrow C$ should be equivalent. A more mathematical way to
say this is that the triangle in Olog (26) commutes.
A commutative diagram is a graph with some declared path equivalences. In the
example above we concisely say “a woman parent is equivalent to a mother.” We
declare this by defining the diagonal map in (26) to be the composition of the
horizontal map and the vertical map.
We generally prefer to indicate a commutative diagram by drawing a check-mark,
$\checkmark$, in the region bounded by the two paths, as in Olog (26).
Sometimes, however, one cannot do this unambiguously on the 2-dimensional
page. In such a case we will indicate the commutative diagrams (fact) by
writing an equation. For example to say that the diagram
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{D}$
commutes, we could either draw a checkmark inside the square or write the
equation $f;g=h;i$ above it. Either way, it means that “$f$ then $g$” is
equivalent to “$h$ then $i$”.
#### 2.3.1. More complex facts
Recording real-world facts in an olog can require some creativity. Whereas a
fact like “the brother of ones father is ones uncle” is recorded as a simple
commutative diagram, others are not so simple. We will try to show the range
of expressivity of commutative diagrams in the following two examples.
###### Example 2.3.2.
How would one record a fact like “a truck weighs more than a car”? We suggest
something like this:
---
$\textstyle{\stackrel{{\scriptstyle B_{1}}}{{\framebox{a
truck}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\scriptstyle{\checkmark}$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{43.36243pt}{a physical
object}}}}}$$\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{a truck $t$ and
a car
$c$}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{c}$$\scriptstyle{t\mapsto
x,\;\;c\mapsto y}$$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{\parbox{79.49744pt}{a pair $(x,y)$ where $x$ and $y$ are
physical objects and $x$ weighs more than
$y$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\scriptstyle{y}$$\textstyle{\stackrel{{\scriptstyle
B_{2}}}{{\framebox{a
car}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\scriptstyle{\checkmark}$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{43.36243pt}{a physical object}}}}}$
where both top and bottom commute. This olog exemplifies the fact that simple
sentences sometimes contain large amounts of information. While the long map
may seem to suffice to convey the idea “a truck weighs more than a car,” the
path equivalences (declared by check-marks) serve to ground the idea in more
basic types. These other types tend to be useful for other purposes, both
within the olog and when connecting it to others.
#### 2.3.3. Specific facts at the olog level
Another fact one might wish to record is that “John Doe’s weight is 150 lbs.”
This is established by declaring that the following diagram commutes:
(33) | | |
---|---|---|---
John Doe$\scriptstyle{\checkmark}$has as weight (in pounds)is150isa personhas
as weight (in pounds)a real number
If one only had the top line, it would be less obvious how to connect its
information with that of other ologs. (See Section 4 for more on connecting
different ologs).
Note that the top line in Diagram (33) might also be considered as existing at
the “data level” rather than at the “olog level.” In other words, one could
see John Doe as an “instance” of $\ulcorner$a person$\urcorner$, rather than
as a type in and of itself, and similarly see 150 as an instance of
$\ulcorner$a real number$\urcorner$. This idea of an olog having a “data
level” is the subject of the Section 3.
###### Rules of good practice 2.3.4.
A fact is the declaration that two paths (having the same source and target)
in an olog are equivalent. Such a fact is either presented as a checkmark
between the two paths (if such a check-mark is unambiguous) or by an equation.
Every such equivalence should be declared; i.e. no fact should be considered
too obvious to declare.
## 3\. Instances
The reader at this point hopefully sees an olog as a kind of “concept map,”
and it is one, albeit a concept map with a formal structure (implicitly coming
from category theory) and specific rules of good practice. In this section we
will show that one can also load an olog with data. Each type can be assigned
a set of instances, each aspect will map the instances of one type to
instances of the other, and each fact will equate two such mappings. We give
examples of these ideas in Section 3.1.
In Section 3.2, we will show that in fact every olog can also serve as the
layout for a database. In other words, given an olog one can immediately
generate a database schema, i.e. a system of tables, in any reasonable data
definition language such as that of SQL. The tables in this database will be
in one-to-one correspondence with the types in the olog. The columns of a
given table will be the aspects of the corresponding type, i.e. the arrows
whose source is that type. Commutative diagrams in the olog will give
constraints on the data.
In fact, this idea is the basic thesis in [Spi2], even though the word olog
does not appear in that paper. There it was explained that a category
${\mathcal{C}}$ naturally can be viewed as a database schema and that a
functor $I\colon{\mathcal{C}}\rightarrow{\bf Set}$, where ${\bf Set}$ is the
category of sets, is a database state. Since an olog is a drawing of a
category, it is also a drawing of a database schema. The current section is
about the “states” of an olog, i.e. the kinds of data that can be captured by
it.
### 3.1. Instances of types, aspects, and facts
Recall from Section 2 that basic ologs consist of types, displayed as boxes;
aspects, displayed as arrows; and facts, displayed as equations or check-
marks. In this section we discuss the instances of these three basic
constructions. The rules of good practice (2.1.1, 2.2.1, and 2.3.4) were
specifically designed to simplify the process of finding instances.
#### 3.1.1. Instances of types
According to Rules 2.1.1, each box in an olog contains text which should refer
to a distinction made and recognizable by the author for which instances can
be documented. For example if my olog contains a box
(34) $\displaystyle\stackrel{{\scriptstyle}}{{\framebox{\parbox{93.95122pt}{a
pair $(p,c)$ where $p$ is a person, $c$ is a cat, and $p$ has petted $c$}}}}$
then I must have some concept of when this situation occurs. Every time I
witness a new person-cat petting, I document it. Whether this is done in my
mind, in a ledger notebook, or on a computer does not matter; however using a
computer would probably be the most self-explanatory. Imagine a computer
program in which one can create ologs. Clicking a text box in an olog results
in it “opening up” to show a list of documented instances of that type. If one
is reading the CBS news olog and clicks on the box $\ulcorner$an episode of 60
Minutes$\urcorner$, he or she should see a list of all episodes of the TV show
“60 Minutes.” If we wish to document a new person-cat petting incident we
click on the box in (34) and add this new instance.
#### 3.1.2. Instances of aspects
According to Rules 2.2.1, each arrow in an olog should be labeled with text
that refers to a functional relationship between the source box and the target
box. A functional relationship $f\colon A\rightarrow B$ between finite sets
$A$ and $B$ can always be written as a 2-column table: the first column is
filled with the instances of type $A$ and the second column is filled with
their $f$-values, which are instances of type $B$.
For example, consider the aspect
(35) $\displaystyle\framebox{a
moon}\xrightarrow{\textnormal{orbits}}\framebox{a planet}$
We can document some instances of this relationship using the following table:
(43)
Clearly, this table of instances can be updated as more moons are discovered
by the author (be it by telescope, conversation, or research).
The correspondence between aspect (35) and Table (43) makes it clear that
ologs can serve to hold data which exemplifies the author’s world-view. In
Section 3.2, we will show that ologs (which have many aspects and facts) can
serve as bona fide database schemas.
#### 3.1.3. Instances of facts
Recall the following olog:
$\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{a
person}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has
as parentshas as mother$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{57.81621pt}{\raggedright a pair $(w,m)$ where $w$ is a
woman and $m$ is a
man\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
✓ $w$$\textstyle{\stackrel{{\scriptstyle C}}{{\framebox{a woman}}}}$
and consider the following instances of the three aspects in it:
has as parents
---
a person | a pair $(w,m)$ …
Cain | (Eve, Adam)
Abel | (Eve, Adam)
Chelsey | (Hillary, Bill)
$w$
---
a pair $(w,m)$ … | a woman
(Eve, Adam) | Eve
(Hillary, Bill) | Hillary
(Margaret, Samuel) | Margaret
(Emily, Kris) | Emily
has as mother
---
a person | a woman
Cain | Eve
Abel | Eve
Chelsey | Hillary
When we declare that the diagram in (26) commutes (using the check-mark), we
are saying that for every instance of $\ulcorner$a person$\urcorner$ (of which
we have three: Cain, Abel, and Chelsey), the two paths to $\ulcorner$a
woman$\urcorner$ give the same answers. Indeed, for Cain the two paths are:
1. (i)
Cain $\mapsto$ (Eve, Adam) $\mapsto$ Eve;
2. (ii)
Cain $\mapsto$ Eve;
and these answers agree. If one changed any instance of the word “Eve” to the
word “Steve” in one of the tables in (LABEL:dia:instances_of_facts), some pair
of paths would fail to agree. Thus the “fact” that the diagram in (26)
commutes ensures that there is some internal consistency between the meaning
of parents and the meaning of mother, and this consistency must be born out at
the instance level.
All of this will be formalized in Section 3.2.2.
### 3.2. The relationship between ologs and databases
Recall from Section 3.1.1 that we can imagine creating an olog on a computer.
The user creates boxes, arrows, and compositions, hence creating a category
${\mathcal{C}}$. Each text-box $x$ in the olog can be “clicked” by the
computer mouse, an action which allows the user to “view the contents” of $x$.
The result will be a set of things, which we might call $I(x)\in{\bf Set}$,
whose elements are things of type $x$. So clicking on the box $\ulcorner$a
man$\urcorner$ one sees $I(\textnormal{$\ulcorner$a man$\urcorner$})$, the set
of everything the author has documented as being a man. For each aspect
$f\colon x\rightarrow y$ of $x$, the user can see a function from the set
$I(x)$ to $I(y)$, perhaps as a 2-column table as in
(LABEL:dia:instances_of_facts).
The type $x$ may have many aspects, which we can put together into a single
multi-column table. Its columns are the aspects of $x$, and its rows are the
elements of $I(x)$. Consider the following olog, taken from [Spi2] where it
was presented as a database schema.
(51) |
---|---
employeeworks inmanagerfirst namelast namedepartmentsecretarynamestring
The type $\ulcorner$Employee$\urcorner$ has four aspects, namely manager
(valued in $\ulcorner$Employee$\urcorner$), works in (valued in
$\ulcorner$department$\urcorner$), and first name and last name (valued in
$\ulcorner$string$\urcorner$). As a database, each type together with its
aspects form a multi-column table, as in the following example.
###### Example 3.2.1.
We can convert Olog (51) into a database schema. Each box represents a table,
each arrow out of a box represents a column of that table. Here is an example
state of that database.
(108)
Note that every arrow $f\colon x\rightarrow y$ of Olog (51) is represented in
Database (108) as a column of table $x$, and that every cell in that column
can be found in the Id column of table $y$. For example, every cell in the
“works in” column of table employee can be found in the Id column of table
department.
The point is that ologs can be drawn to represent a world-view (as in Section
2), but they can also store data. Rules 1,2, and 3 in 2.1.1 align the
construction of an olog with the ability to document instances for each of its
types.
#### 3.2.2. Instance data as a set-valued functor
Let ${\mathcal{C}}$ be an olog. Section 3 so far has described instances of
types, aspects, and facts and how all of these come together into a set of
interconnected tables. The assignment of a set of instances to each type and a
function to each aspect in ${\mathcal{C}}$, such that the declared facts hold,
is called an assignment of instance data for ${\mathcal{C}}$. More precisely,
instance data on ${\mathcal{C}}$ is a functor ${\mathcal{C}}\rightarrow{\bf
Set}$, as in Definition 3.2.3.
###### Definition 3.2.3.
Let ${\mathcal{C}}$ be a category (olog) with underlying graph
$|{\mathcal{C}}|$, and let ${\bf Set}$ denote the category of sets. An
instance of ${\mathcal{C}}$ (or an assignment of instance data for
${\mathcal{C}}$) is a functor $I\colon{\mathcal{C}}\rightarrow{\bf Set}$. That
is, it consists of
* •
a set $I(x)$ for each object (type) $x$ in ${\mathcal{C}}$,
* •
a function $I(f)\colon I(x)\rightarrow I(y)$ for each arrow (aspect) $f\colon
x\rightarrow y$ in ${\mathcal{C}}$, and
* •
for each fact (path-equivalence or equation) 333If we let
$f=f_{1}{\,;\,}f_{2}{\,;\,}\cdots{\,;\,}f_{n}$ and
$f^{\prime}=f^{\prime}_{1}{\,;\,}f^{\prime}_{2}{\,;\,}\cdots{\,;\,}f^{\prime}_{m}$,
then we often write $(f=f^{\prime})\colon i\rightarrow j$ to denote the fact
that these paths are equivalent.
$f_{1}{\,;\,}f_{2}{\,;\,}\cdots{\,;\,}f_{n}=f^{\prime}_{1}{\,;\,}f^{\prime}_{2}{\,;\,}\cdots{\,;\,}f^{\prime}_{m}$
declared in ${\mathcal{C}}$, an equality of functions
$I(f_{1}){\,;\,}I(f_{2}){\,;\,}\cdots{\,;\,}I(f_{n})=I(f^{\prime}_{1}){\,;\,}I(f^{\prime}_{2}){\,;\,}\cdots{\,;\,}I(f^{\prime}_{m}).$
For more on this viewpoint of categories and functors, the reader can consult
[Spi3].
## 4\. Communication between ologs
The world is inherently heterogeneous. Different individuals 444By an
individual we mean either an individual person acting on their own, a
community acting as a single entity, a software agent, etc. Later in this
section we will use the notion of a community acting as a distributed
collection of linked, yet independent, individuals. in the world naturally
have different world-views — each individual has its own perspective on the
world. The conceptual knowledge (information resources) of an individual
represents its world-view, and is encoded in an ontology log, or olog,
containing the concepts, relations, and observations that are important to
that individual. An olog is a formal specification of an individual’s world-
view in a language representing the concepts and relationships used by that
individual. In addition to the formulation of an expressive language, a
specification needs to contain axioms (facts) that constrain the possible
interpretations of that language.
Since the ologs of different individuals are encoded in different languages,
the important need to merge disparate ologs into a more general representation
is difficult, time-consuming and expensive. The solution is to develop
appropriate communication between individuals to allow interoperability of
their ologs. Communication can occur between individuals when there is some
commonality between their world-views. It is this commonality that allows one
individual to benefit from the knowledge and experience of another. In this
section we will discuss how to formulate these channels of communication,
thereby describing a generalized and practical technique for merging ologs.
The mathematical concept that makes it all work is that of a functor. A
functor is a mapping from one category to another that preserves all the
declared structure. Whereas in Definition 3.2.3 we defined a functor from an
olog to $\mathrmbf{Set}$, here we will be discussing functors from one olog to
another.
Suppose we have two ologs, ${\mathcal{C}}$ and ${\mathcal{D}}$, that represent
the world-views of two individuals. A functor
$F\colon{\mathcal{C}}\rightarrow{\mathcal{D}}$ is basically a way of matching
each type (box) of ${\mathcal{C}}$ to a type of ${\mathcal{D}}$, and each
aspect (arrow) in ${\mathcal{C}}$ to an aspect (or path of aspects) in
${\mathcal{D}}$. Once ologs are aligned in this way, communication can occur:
the two individuals know what each other is talking about. In fact,
mathematically we can show that instance data held in ${\mathcal{C}}$ can be
transformed (in coherent ways) to instance data held in ${\mathcal{D}}$, and
vice versa (see [Spi2]). In simple terms, once individuals understand each
other in a certain domain (be it social, mathematical, etc.), they can
communicate their views about it.
While the basic idea is not hard, the details can be a bit technical. This
section is written in a more formal and logical style, and is decidedly more
difficult than the others. For this section only, we assume the reader is
familiar with the notion of fibered categories, colimits in the category ${\bf
Cat}$ of categories, etc. We return to our more informal style in Section 5,
where we discuss how an individual can author a more expressive olog.
### 4.1. Categories and their presentations
We never defined categories in this paper, but we defined ologs and said that
the two notions amounted to the same thing. Thus, we implied that a category
consists of the following: a set of objects, a set of arrows (each pointing
from one object to another), and a congruence relation on paths.555A
congruence relation on paths is an equivalence relation on paths that respects
endpoints and is closed under composition from left and right (see the axioms
in 115). This differs from the standard definition of categories (see [Mac]),
which replaces our congruence relation with a composition rule and
associativity law (obtained by taking the categorical quotient). One could say
that an olog is a presentation of a category by generators (objects and
arrows) and relations (path congruences). Any category can be resolved and
presented in such a way, which we will call a specification. Likewise any
functor can be resolved and presented as a morphism between specifications.
666We take an agnostic approach to foundations here. With the presentation
form, we show how categories and functors are definable in terms of sets and
functions, indicating how category theoretic concepts could be defined in
terms of set theory. However, we fully understand that $\mathbf{Set}$, the
category of sets and functions, is but one example of a topos, indicating how
set theoretic concepts could be defined in terms of category theory.
In fact, this presentation form for categories (and the analogous one for
functors) is preferable for our work on communication between ologs, because
it separates the strictly graphical part of an olog (its types and aspects,
regarded as the olog language) from the propositional part (its facts,
regarded as the olog formalism). This presentation form is standard in the
institutions [GB] and information flow [BS] communities, since it separates
the mechanism of flow from the content of flow; in this case the formal
content. Our work here applies the general theories of institutions and
information flow to the specific logical system that underlies categories and
functors,777For the expert, this refers to the sketch logical system Sk, in
its various manifestations. demonstrating how this logical system can be used
for knowledge representation. Using the presentation forms for categories and
functors, we show how communication between individuals is effected by the
flow of information along channels.
### 4.2. The architecture underlying information systems
We think of a community of people, businesses, etc. in terms of the ologs of
each individual participant together with the information channels that
connect them. These channels are functors between ologs, which allow
communication to occur. The heterogeneity of multiple differing world-views
connected through such links can lead to a flexibility and robustness of
interaction. For example, heterogeneity allows for multiple schemas to be
employed in the design of database systems in particular, and multiple
languages to be employed in the design of knowledge representation systems in
general.
For any olog, consider the underlying graph of types and aspects. We regard
this graph as being the language of the olog, 888Section 4.4 indicates how
natural languages can be encoded into ologs. with the facts of the olog being
a subset of all the possible assertions that one can make within this
language. Any two ologs with the same underlying graph of types and aspects
have the same language, and since the facts of each olog are expressed in the
same language, they can be “understood” by each other without translation. As
such, we think of the collection of all ologs with the same language
(underlying graph) as forming a homogeneous context, with the ologs ordered in
a specialization-generalization hierarchy.
Whereas an olog represents (the world-view of) a single individual, an
information system (of ologs) represents a community of separate, independent
and distributed individuals. Here we consider an information system to be a
diagram of ologs of some shape $\mathrmbf{I}$; that is, a collection of ologs
and constraints indexed by a base category $\mathrmbf{I}$. The parts of the
system represent either the ologs of the various individuals in the system or
common grounds needed for communication between the individuals. Each part of
the system specifies its world-view as facts expressed in terms of its
language. The system is heterogeneous, since each part has a separate language
for the expression of its world-view. The morphisms between the parts are the
alignment (constraint) links defining the common grounds.
As will be made clear in a moment, there is an underlying distributed system
consisting of the language (underlying graph) for each component part of the
information system and a translation (graph morphism) for each alignment link.
We can think of this distributed system as an underlying system of languages
linked by translating dictionaries. This distributed system determines an
information channel with core language (graph) and component translation links
(graph morphisms) along which the specifications of each component part can
flow to the core. We can think of this core as a universal language for the
whole system and the channel as a translation mechanism from parts to whole.
At the core, the direct flow of the component specifications are joined
together (unioned) and allowed to interact through entailment. The result of
this interaction can then be distributed back to the component parts, thereby
allowing the separate parts of an information system to interoperate.
In this section, we will make all this clear and rigorous. As mentioned above,
we will work with category presentations (here called specifications) rather
than categories. We will discuss the homogeneous contexts called fibers in
detail and give the axioms of satisfaction. We will then discuss how morphisms
between graphs (the translating dictionaries between the ologs) allow for
direct and inverse information flow between these homogeneous fiber contexts.
Finally, we discuss specifications (also known as theories) and the lattice of
theories construction for ontologies.
In Section 4.3 we will discuss how the information in ologs can be aligned by
the use of common grounds. This alignment will result in the creation of
information systems, which are systems of ologs connected together along
functors. We will discuss how to take the information contained in each olog
of a heterogeneous system and integrate it all into a single whole, called the
fusion olog. Finally we will discuss how the consequence of bringing all this
information together, and allowing it to interact, can be transferred back to
each part of the system (individual olog) as a set of local facts entailed by
remote ologs, allowing for a kind of interoperability between ologs. In
Section 4.4 we will discuss conceptual graphs and their relationship to ologs.
#### 4.2.1. Fibers
A graph $G$ contains types as nodes and aspects as edges. The graphs
underlying an olog is considered its language. Any category $\mathcal{C}$ has
an underlying graph $|\mathcal{C}|$. In particular, $|\mathrmbf{Set}|$ is the
graph underlying the category of sets and functions. Olog (12) has an
underlying graph containing the three types $\ulcorner$person$\urcorner$,
$\ulcorner$person-pair$\urcorner$ and $\ulcorner$woman$\urcorner$ and the
three aspects ‘has a parent’, ‘woman’ and ‘has as mother’. Olog (17) has an
underlying graph containing the three types $\ulcorner$employee$\urcorner$,
$\ulcorner$department$\urcorner$, and $\ulcorner$string$\urcorner$ and the six
aspects ‘manager’, ‘works in’, ‘secretary’, ‘name’, ‘first name’ and ‘last
name’. Let $\mathrmbfit{eqn}(G)$ denote the set of all facts (equations) that
are possible to express using the types and aspects of $G$. A
$G$-specification is a set $E\subseteq\mathrmbfit{eqn}(G)$ consisting of some
of the facts expressible in $G$. The singleton set with the one fact that “the
female parent of a person is his/her mother” is a specification for the graph
of Olog (12). The set with the two facts that “the manager has the same
department as any employee” and “the secretary of a department is an employee
in that department” is a specification for the graph of Olog (17). Let
$\mathrmbfit{spec}(G)$ denote the collection of all $G$-specifications ordered
by inclusion $E_{1}\subseteq E_{2}$.
#### 4.2.2. Satisfaction
It will be useful here to define an instance of a graph $G$, instead of an
instance of a category $\mathcal{C}$. An instance of a graph populates the
graph by assigning instance data to it. An instance of a graph $G$ is a graph
morphism $D\colon G\rightarrow|\mathrmbf{Set}|$ mapping each type $x$ in $G$
to a set $D(x)$ of instances and mapping each aspect $e\colon x\rightarrow y$
in $G$ to an instance function $D(e)\colon D(x)\rightarrow D(y)$. Using
database terminology, we also call $D$ a key diagram, since it gives the set
of row identifiers (primary keys) of tables and the cell contents defined by
key maps.
A key diagram $D\colon G\rightarrow|\mathrmbf{Set}|$ satisfies (is a model of)
a $G$-fact $\epsilon\in\mathrmbfit{eqn}(G)$ (see Definition 3.2.3), symbolized
$D\models_{G}\epsilon$, when we have an equality of functions
$D^{\ast}(\epsilon_{0})=D^{\ast}(\epsilon_{1})$. We also say that $\epsilon$
(holds in) is true when interpreted in $D$. An identity $(f=_{G}f)\colon
i\rightarrow j$ holds in all key diagrams (hence, is a tautology), and vice-
versa for any set $A\in|\mathrmbf{Set}|$ a constant key diagram
$\Delta(A)\colon G\rightarrow|\mathrmbf{Set}|$ satisfies any fact
$\epsilon\in\mathrmbfit{eqn}(G)$. A key diagram $D\colon
G\rightarrow|\mathrmbf{Set}|$ satisfies (is a model of) a $G$-specification
$E$, symbolized $D\models_{G}E$, when it satisfies every fact in the
specification. For any graph $G$, a $G$-specification $E$ entails a $G$-fact
$\epsilon$, denoted by $E\vdash_{G}\epsilon$, when any model of the
specification satisfies the fact. The consequence
$E^{\scriptscriptstyle\bullet}$ of a $G$-specification $E$ is the set of all
entailed equations. The consequence operator $(-)^{\scriptscriptstyle\bullet}$
is a closure operator, and the consequence of a specification is a congruence.
For any $G$-specification $E$, entailment satisfies the following axioms.
(115) (basic) If $E$ contains the equation $\epsilon$, then $E$ entails
$\epsilon$. (reflexive) $E$ entails the equations $(f=_{G}f)\colon
i\rightarrow j$ for any path $f\colon i\rightarrow j$. (symmetric) If $E$
entails the equation $(f_{1}=_{G}f_{2})\colon i\rightarrow j$, then $E$
entails the equation $(f_{2}=_{G}f_{1})\colon i\rightarrow j$. (transitive) If
$E$ entails the two equations $(f_{1}=_{G}f_{2})\colon i\rightarrow j$ and
$(f_{2}=_{G}f_{3})\colon i\rightarrow j$, then $E$ entails the equation
$(f_{1}=_{G}f_{3})\colon i\rightarrow j$. (compositional) If $E$ entails the
two equations $(f_{1}=_{G}f_{2})\colon i\rightarrow j$ and
$(g_{1}=_{G}g_{2})\colon j\rightarrow k$, then $E$ entails the equation
$(f_{1}{\,;\,}g_{1}=_{G}f_{2}{\,;\,}g_{2})\colon i\rightarrow k$. (bi-closed)
If $E$ entails the equation $(g_{1}=_{G}g_{2})\colon j\rightarrow k$, then $E$
entails the equations $(f{\,;\,}g_{1}=_{G}f{\,;\,}g_{2})\colon i\rightarrow k$
and $(g_{1}{\,;\,}h=_{G}g_{2}{\,;\,}h)\colon j\rightarrow l$ for any left
composable path $f\colon i\rightarrow j$ and any right composable path
$h\colon k\rightarrow l$.
These are converted to inference rules in Table 1. To construct
$E^{\scriptscriptstyle\bullet}$, we first take the reflexive, symmetric, and
transitive closure $E^{\ast}$ of $E$ (so that $E^{\ast}$ is a
$G$-specification and also the smallest equivalence relation containing $E$),
and then we get $E^{\scriptscriptstyle\bullet}$ by closing up under
composition on left and right. We extend specification inclusion with the
entailment order, where $E_{1}\leq_{G}E_{2}$ when $E_{1}$ entails each
equation in $E_{2}$; that is, when $E_{1}^{\scriptscriptstyle\bullet}\supseteq
E_{2}$ or equivalently when $E_{1}^{\scriptscriptstyle\bullet}\supseteq
E_{2}^{\scriptscriptstyle\bullet}$. The statement “$E_{1}\leq_{G}E_{2}$”
asserts that $E_{1}$ is at least as specialized as $E_{2}$. The entailment
order ${\langle{\mathrmbfit{spec}(G),\leq_{G}}\rangle}$, which is a
specialization-generalization order, represents a local version of the
“lattice of theories” construction of Sowa [Sow2] (see Section 4.2.5). The
opposite entailment order
$\mathrmbfit{fbr}(G)={\langle{\mathrmbfit{spec}(G),\geq_{G}}\rangle}$ is
called the fiber order.999For consistency in discussion, we follow the
terminology of formal concept analysis [GW], information flow [BS] and the
theory of institutions [GB]. This includes the polarity induced by concept
lattices and the directionality of infomorphisms. In the lattice101010This is
a complete preorder, loosely called a “lattice”. $\mathrmbfit{spec}(G)$, the
meet is union $\wedge=\cup$ and the join is intersection $\vee=\cap$; whereas
in the lattice $\mathrmbfit{fbr}(G)$, the join is union $\vee=\cup$ and the
meet is intersection $\wedge=\cap$. Any specification $E$ is entailment
equivalent to its consequence $E\cong E^{\scriptstyle\bullet}$. A
specification $E$ is closed when it is equal to its consequence
$E=E^{\scriptstyle\bullet}$. There is a one-one correspondence between closed
$G$-specifications and categories over graph $G$. The conceptual intent of a
key diagram $D$, implicit in satisfaction, is the closed specification
$\mathrmbfit{int}(D)$ consisting of all facts satisfied by the key diagram.
Hence, $D\models_{G}E$ iff $E\subseteq\mathrmbfit{int}(D)$ iff
$\mathrmbfit{int}(D)\leq_{G}E$.111111This is the first step in the
algebraization of Tarski’s “semantic definition of truth” [Ken4].
#### 4.2.3. Elementary flow
A graph morphism $H\colon G_{1}\rightarrow G_{2}$ maps the types and aspects
of $G_{1}$ to the types and aspects of $G_{2}$. Graph morphisms are the
translations between ologs. A functor
$\mathcal{F}\colon\mathcal{C}_{1}\rightarrow\mathcal{C}_{2}$ has an underlying
graph morphism
$|\mathcal{F}|\colon|\mathcal{C}_{1}|\rightarrow|\mathcal{C}_{2}|$. For any
graph morphism $H\colon G_{1}\rightarrow G_{2}$, there is a fact function
$\mathrmbfit{eqn}(H)\colon\mathrmbfit{eqn}(G_{1})\rightarrow\mathrmbfit{eqn}(G_{2})$
that maps a $G_{1}$-equation $(f_{1}=_{G_{1}}f^{\prime}_{1})\colon
i_{1}\rightarrow j_{1}$ to the $G_{2}$-equation
$(H^{\ast}(f_{1})=_{G_{2}}H^{\ast}(f^{\prime}_{1}))\colon H(i_{1})\rightarrow
H(j_{1})$, and a key diagram functor
$\mathrmbfit{dgm}(H)\colon\mathrmbfit{dgm}(G_{2})\rightarrow\mathrmbfit{dgm}(G_{1})$
that maps a key diagram $D_{2}\colon G_{2}\rightarrow|\mathrmbf{Set}|$ to the
key diagram $H\circ D_{2}\colon G_{2}\rightarrow|\mathrmbf{Set}|$.121212The
composition of graph morphisms is written in diagrammatic order. The fact
function is the fundamental unit of information (formal) flow for ologs, and
the key diagram functor is the fundamental unit of semantic flow for
ologs.131313This is so, at the abstraction of institutions [Ken3]. Formal flow
is adjoint to semantic flow — satisfaction is invariant under flow:
$\mathrmbfit{dgm}(H)(D_{2})\models_{G_{1}}\epsilon_{1}$ iff
$D_{2}\models_{G_{2}}\mathrmbfit{eqn}(H)(\epsilon_{1})$ for any graph morphism
$H\colon G_{1}\rightarrow G_{2}$, source fact $\epsilon_{1}$ and target
diagram $D_{2}$. Specifications can be moved along graph morphisms by
extending the fact (equation) function. For any graph morphism $H\colon
G_{1}\rightarrow G_{2}$, define the direct flow operator
$\mathrmbfit{dir}(H)={\wp}\mathrmbfit{eqn}(H):\mathrmbfit{spec}(G_{1})\rightarrow\mathrmbfit{spec}(G_{2})$141414The
symbol $\wp$ denotes the power-set operator. and the inverse flow operator
$\mathrmbfit{inv}(H)=\mathrmbfit{eqn}(H)^{-1}((\mbox{-})^{\scriptscriptstyle\bullet}):\mathrmbfit{spec}(G_{2})\rightarrow\mathrmbfit{spec}(G_{1})$.
Direct and inverse flow are adjoint monotonic functions
${\langle{\mathrmbfit{dir}(H)\dashv\mathrmbfit{inv}(H)}\rangle}\colon\mathrmbfit{fbr}(G_{1})\rightarrow\mathrmbfit{fbr}(G_{2})$
w.r.t. fiber order: $\mathrmbfit{dir}(H)(E_{1})\geq_{G_{2}}E_{2}\text{
\text@underline{iff} }E_{1}\geq_{G_{1}}\mathrmbfit{inv}(H)(E_{2})$. For any
graph morphism $H\colon G_{1}\rightarrow G_{2}$, any $G_{1}$-specification
$E_{1}$, and any $G_{2}$-specification $E_{2}$, entailment satisfies the
following axioms.
(direct flow) | If $E_{1}$ entails the equation $(f=_{G_{1}}f^{\prime})\colon i\rightarrow j$, then $\mathrmbfit{dir}(H)(E_{1})$ entails the equation $(H^{\ast}(f_{1})=_{G_{2}}H^{\ast}(f^{\prime}_{1}))\colon H(i_{1})\rightarrow H(j_{1})$.
---|---
(inverse flow) | If $E_{2}$ entails the equation $(H^{\ast}(f)=_{G_{2}}H^{\ast}(f^{\prime}))\colon H(i)\rightarrow H(j)$, then $\mathrmbfit{inv}(H)(E_{2})$ entails the equation $(f=_{G_{1}}f^{\prime})\colon i\rightarrow j$.
These are converted to inference rules in Table 1. A graph morphism $H\colon
G_{1}\rightarrow G_{2}$ defines a consequence operator
${(\mbox{-})}^{{\scriptscriptstyle\blacklozenge}_{H}}=\mathrmbfit{dir}(H)\circ\mathrmbfit{inv}(H)$
on the fiber preorder $\mathrmbfit{fbr}(G_{1})$, where
$E_{1}\geq_{G_{1}}E_{1}^{\scriptstyle\bullet}\geq_{G_{1}}E_{1}^{{\scriptscriptstyle\blacklozenge}_{H}}$.
#### 4.2.4. Specifications
A specification $\mathcal{S}={\langle{G,E}\rangle}$ is an indexed notion
consisting of a graph $G$ and a $G$-specification $E\in\mathrmbfit{spec}(G)$.
It is sometimes convenient to use the symbol ‘$\mathcal{S}$’ in place of
‘$E$’; for example, to say that “$\mathcal{S}\in\mathrmbfit{spec}(G)$”. A
category $\mathcal{C}$ can be resolved and presented as a specification
$\mathrmbfit{spec}(\mathcal{C})={\langle{G,E}\rangle}$ consisting of the
underlying graph $G=|\mathcal{C}|$ containing the types and aspects of
$\mathcal{C}$ and the collection $E$ of all facts that hold in $\mathcal{C}$.
In the other direction, any specification $\mathcal{S}$ induces a (quotient)
category $\mathrmbfit{cat}(\mathcal{S})$. Olog (12) and Olog (17) are
described as specifications in Section 4.2.1. A specification morphism
$H\colon{\langle{G_{1},E_{1}}\rangle}\rightarrow{\langle{G_{2},E_{2}}\rangle}$
is a graph morphism $H\colon G_{1}\rightarrow G_{2}$ that preserves
entailment: $E_{1}\vdash_{G_{1}}\epsilon_{1}$ implies
$E_{2}\vdash_{G_{2}}\mathrmbfit{eqn}(H)(\epsilon_{1})$ for any
$\epsilon_{1}\in\mathrmbfit{eqn}(G_{1})$; or equivalently that satisfies the
adjointness conditions, $\mathrmbfit{dir}(H)(E_{1})\geq_{G_{2}}E_{2}\text{
\text@underline{iff} }E_{1}\geq_{G_{1}}\mathrmbfit{inv}(H)(E_{2})$. Being a
graph morphism, it maps types to types and aspects to aspects. Moreover, it
also maps facts in $E_{1}$ to facts in $E_{2}$; that is, it preserves all the
declared structure. A functor
$\mathcal{F}\colon\mathcal{C}_{1}\rightarrow\mathcal{C}_{2}$ can be resolved
and presented as a specification morphism
$\mathcal{F}\colon\mathrmbfit{spec}(\mathcal{C}_{1})\rightarrow\mathrmbfit{spec}(\mathcal{C}_{2})$.
Hence, the presentation form for a functor does exactly what the functor does.
The fibered category of specifications $\mathrmbf{Spec}$ has specifications as
objects and specification morphisms as morphisms. Thus, it is defined in terms
of information flow. There is an underlying graph functor
$\mathrmbfit{gph}\colon\mathrmbf{Spec}\rightarrow\mathrmbf{Gph}$ from
specifications to graphs ${\langle{G,E}\rangle}\mapsto G$. The subcategory
over any fixed graph $G$ is the fiber $\mathrmbfit{fbr}(G)$; because of the
opposite orientation, we say that “the category of specifications points
downward in the concept lattice”. Throughout this section we identify ologs
with specifications and olog morphisms with specification morphisms.
#### 4.2.5. The lattice of theories construction
Sowa’s “lattice of theories” construction (LOT) describes a modular framework
for ontologies [Sow2]. The Olog formalism follows the approach to LOT
described in [IFF2].151515The IFF term ‘theory’ is replaced by the Olog term
’specification’ or ’olog’. In the Olog formalism, LOT is locally represented
by the entailment preorders $\mathrmbfit{spec}(G)$, and globally represented
by the category of specifications $\mathrmbf{Spec}$. We follow the discussion
in section 6.5 “Theories, Models and the World” of Sowa [Sow2]. From each olog
(specification) in the “lattice of theories”, the entailment ordering defines
paths to the more generalized ologs above and the more specialized ologs
below. Sowa defines four ways for moving along paths from one olog to another:
contraction, expansion, revision and analogy.
Contraction:
Any olog can be contracted or reduced to a smaller, simpler olog, moving
upward in the preorder $\mathrmbfit{spec}(G)$, by deleting one or more facts.
Expansion:
Any olog can be expanded, moving downward in the preorder
$\mathrmbfit{spec}(G)$, by adding one or more facts.
Revision:
A revision step is composite, moving crosswise in the preorder
$\mathrmbfit{spec}(G)$; it uses a contraction step to discard irrelevant
details, followed by an expansion step to added new facts.
Analogy:
Unlike contraction and expansion, which move to nearby ologs in an entailment
preorder $\mathrmbfit{spec}(G)$, analogy moves to an olog in a remote
entailment preorder in the category $\mathrmbf{Spec}$ via the flow along an
underlying graph morphism $H\colon G_{1}\rightarrow G_{2}$ by systematically
renaming the types and aspects that appear in the facts: any olog $E_{1}$ in
$\mathrmbfit{spec}(G_{1})$ is moved (by systematic renaming) to the olog
$\mathrmbfit{dir}(H)(E_{1})$ in $\mathrmbfit{spec}(G_{2})$.
According to Sowa, the various methods used in nonmonotonic logic and the
operators for belief revision correspond to movement through the lattice of
theories.
### 4.3. Alignment and integration of information systems
#### 4.3.1. Common ground
Given the world-views of two individuals, as represented by ologs
$\mathcal{S}_{1}={\langle{G_{1},E_{1}}\rangle}$ and
$\mathcal{S}_{2}={\langle{G_{2},E_{2}}\rangle}$, there is little hope that one
of them completely contains the other (even after allowing for renaming of
types and aspects), and there is correspondingly little chance of finding a
meaningful olog morphism between the two. Instead, in order to communicate the
two individuals could attempt to find a common ground, a third olog
$\mathcal{S}={\langle{G,E}\rangle}$ and meaningful morphisms161616Roughly
speaking, an olog morphism $F\colon\mathcal{C}\rightarrow\mathcal{D}$ is
meaningful when for each type $X$ in ${\mathcal{C}}$, every intended instance
of $X$ in ${\mathcal{C}}$ would be considered an instance of $F(X)$ by the
author of ${\mathcal{D}}$ (in which case we say the intention for types is
respected), and in a similar way the intention for aspects is respected.
Precisely speaking, if $I\colon{\mathcal{C}}\rightarrow{\bf Set}$ and
$J\colon{\mathcal{D}}\rightarrow{\bf Set}$ are instance data for
${\mathcal{C}}$ and ${\mathcal{D}}$, then $F$ is meaningful relative to $I$
and $J$ if one can exhibit a natural transformation $\mu\colon I\Rightarrow
F\circ J$ as in [Spi2]. $H_{1}\colon\mathcal{S}\rightarrow\mathcal{S}_{1}$ and
$H_{2}\colon\mathcal{S}\rightarrow\mathcal{S}_{2}$.171717A common ground olog
is also called a reference ontology in knowledge representation. This
connection is a 1-dimensional knowledge network
$\mathcal{S}_{1}\xleftarrow{H_{1}}\mathcal{S}\xrightarrow{H_{2}}\mathcal{S}_{2}$
of shape $\bullet\leftarrow\bullet\rightarrow\bullet$ called a span (in
$\mathrmbf{Spec}$), where each node is an olog and each edge is a morphism
between ologs. The requirements of this span are that
$\mathrmbfit{dir}(H_{1})(E)\geq_{G_{1}}E_{1}$ and
$\mathrmbfit{dir}(H_{2})(E)\geq_{G_{2}}E_{2}$, two requirements involving
local flow. Equivalently, that
$E\geq_{G}\mathrmbfit{inv}(H_{1})(E_{1})\vee_{G}\mathrmbfit{inv}(H_{2})(E_{2})$.
The latter precise expression can be rendered in natural language as “the
world-view of the common ground is contained in the combined world-views of
the two individuals”. The various local direct/inverse flows allow world-views
to be compared. Such a common ground can be expanded and improved over time.
The basic idea is that one individual can attempt to explain a new idea (type,
aspect or fact) to another in terms of the common ground. Then the other
individual can either interpret this idea as they already have, learn from it
(i.e. freely add it to their olog), or reject it. We view an olog morphism
$H_{1}\colon\mathcal{S}_{1}\rightarrow\mathcal{S}_{2}$ as an atomic constraint
(alignment) link between $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$.181818This is
so, at the abstraction of institutions [Ken3]. We view a common ground span
$\mathcal{S}_{1}\xleftarrow{H_{1}}\mathcal{S}\xrightarrow{H_{2}}\mathcal{S}_{2}$
as a molecular constraint between $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$,
which is weakest when $\mathcal{S}=\emptyset$ and strongest when
$\mathcal{S}_{1}=\mathcal{S}=\mathcal{S}_{2}$.
#### 4.3.2. Systems of ologs
In the general case, more than two individuals will share a common ground. For
example, companies that do business together may have a common-ground olog as
part of a legal contract; or, the various participants at a conference will
have some common understanding of the topic of that conference. In fact, for
any finite set of ologs
$\mathbb{X}=\\{{\mathcal{S}}_{1},{\mathcal{S}}_{2},\ldots,{\mathcal{S}}_{n}\\}$,
there should be a common ground world-view (even if empty), say
$\mathcal{S}_{\mathbb{X}}$. If $\mathbb{Y}\subseteq\mathbb{X}$ is a subset,
then there should be a map
$\mathcal{S}_{\mathbb{X}}\rightarrow\mathcal{S}_{\mathbb{Y}}$ because any
common understanding held by the individuals in $\mathbb{X}$ is held by the
individuals in $\mathbb{Y}$. For example, the triangular-shaped diagram
(126)
represents three individuals $\\{1,2,3\\}$, their ologs
$\\{{\mathcal{S}}_{1},{\mathcal{S}}_{2},{\mathcal{S}}_{3}\\}$, their pair-wise
commonality ologs
$\\{{\mathcal{S}}_{12},{\mathcal{S}}_{13},{\mathcal{S}}_{23}\\}$, and their
three-way commonality olog ${\mathcal{S}}_{123}$. This diagram, which stands
for the interaction between individuals $\\{1,2,3\\}$, does not stand alone,
but is part of an intricate web of other ologs and alignment constraints. In
particular, individuals 1 and 3 may be part of some different interacting
group, say of individuals $\\{1,3,6,7\\}$, and hence the right edge of the
diagram would be part of some tetrahedron-shaped diagram with vertices
$\\{1,3,6,7\\}$. If we take the point-of-view that “a collection of ologs
representing the world-views of various individuals” is a system, then we can
think of the ologs as being the types of that system, the morphisms connecting
the ologs as being the aspects of that system, with the shape of a system
being its underlying graph. In essence, we can apply ologs to themselves. In
the system represented by diagram (126), there are seven types
$\\{{\mathcal{S}}_{1},{\mathcal{S}}_{2},{\mathcal{S}}_{3},{\mathcal{S}}_{12},{\mathcal{S}}_{13},{\mathcal{S}}_{23},{\mathcal{S}}_{123}\\}$
and nine aspects
$\\{\cdots,{\mathcal{S}}_{123}\rightarrow{\mathcal{S}}_{13},\dots\\}$, and the
shape looks like this
---
In addition, we can introduce certain facts to represent the meaning of that
system and then enforce those facts.
A distributed system is a diagram (functor)
$\mathcal{G}\colon\mathrmbf{I}\rightarrow\mathrmbf{Gph}$ of shape
$\mathrmbf{I}$ within the ambient category $\mathrmbf{Gph}$. As such, it
consists of an indexed family $\\{G_{n}\mid n\in\mathrmbf{I}\\}$ of graphs
together with an indexed family $\\{G_{e}\colon G_{n}\rightarrow
G_{m}\mid(e\colon n\rightarrow m)\in\mathrmbf{I}\\}$ of graph morphisms. Let
$\mathrmbf{Dist}(\mathrmbf{I})$ denote the collection of distributed systems
of shape $\mathrmbf{I}$. An information system is a diagram
$\mathcal{S}\colon\mathrmbf{I}\rightarrow\mathrmbf{Spec}$ of shape
$\mathrmbf{I}$ within the ambient category $\mathrmbf{Spec}$. As such, it
consists of an indexed family
$\\{\mathcal{S}_{n}={\langle{G_{n},E_{n}}\rangle}\mid n\in\mathrmbf{I}\\}$ of
ologs together with an indexed family
$\\{\mathcal{S}_{e}\colon\mathcal{S}_{n}\rightarrow\mathcal{S}_{m}\mid(e\colon
n\rightarrow m)\in\mathrmbf{I}\\}$ of olog morphisms. Some of these ologs
might represent the world-views of various individuals, whereas others could
be common grounds; also included might be portals between individual ologs and
common grounds, as in the CG example of Section 4.4. Let
$\mathrmbf{Info}(\mathrmbf{I})$ denote the collection of information systems
of shape $\mathrmbf{I}$. An information system $\mathcal{S}$ with component
ologs $\mathcal{S}_{n}={\langle{G_{n},E_{n}}\rangle}$ has an underlying
distributed system $\mathcal{G}$ of the same shape with component graphs
$G_{n}$ for $n\in\mathrmbf{I}$. For any distributed system $\mathcal{G}$, let
$\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$ denote the collection of
information systems over $\mathcal{G}$ of shape $\mathrmbf{I}$. There is a
pointwise entailment order
$\mathcal{S}\leq^{\mathrmbf{I}}_{\mathcal{G}}\mathcal{S}^{\prime}$ on
$\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$ when component ologs satisfy
the same entailment ordering $E_{n}\leq_{G_{n}}E^{\prime}_{n}$ for
$n\in\mathrmbf{I}$, and by taking the coproduct there is a pointwise
entailment order on
$\mathrmbf{Info}(\mathrmbf{I})=\coprod_{\mathcal{G}\in\mathrmbf{Dist}(\mathrmbf{I})}\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$.
A constant distributed system $\Delta(G)\in\mathrmbf{Dist}(\mathrmbf{I})$ is a
distributed system $\Delta(G)\colon\mathrmbf{I}\rightarrow\mathrmbf{Gph}$ with
the same language $G$ for any index $n\in\mathrmbf{I}$. Any constant
distributed system defines join and meet monotonic functions
$\bigvee^{\mathrmbf{I}}_{G},\bigwedge^{\mathrmbf{I}}_{G}:\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(G))\rightarrow\mathrmbfit{fbr}(G)$
mapping an information system
$\mathcal{S}\in\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(G))$ to the join and
meet ologs $\bigvee\mathcal{S}=\bigcup_{n\in\mathrmbf{I}}E_{n}$ and
$\bigwedge\mathcal{S}=\bigcap_{n\in\mathrmbf{I}}E_{n}$ in
$\mathrmbfit{fbr}(G)$. The join monotonic function is adjoint to the constant
monotonic function
$\Delta^{\mathrmbf{I}}_{G}:\mathrmbfit{fbr}(G)\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(G))$
that distributes an olog $\mathcal{S}^{\prime}\in\mathrmbfit{fbr}(G)$ to the
various locations $n\in\mathrmbf{I}$ forming a constant information system
$\Delta(\mathcal{S}^{\prime})\in\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(G))$,
since $\bigvee\mathcal{S}\geq_{G}\mathcal{S}^{\prime}$ iff
$\mathcal{S}\geq^{\mathrmbf{I}}_{\Delta(G)}\Delta(\mathcal{S}^{\prime})$ for
any system $\mathcal{S}\in\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(G))$ and any
olog $\mathcal{S}^{\prime}\in\mathrmbfit{fbr}(G)$.
#### 4.3.3. System morphisms
Just as ologs are linked by morphisms, information systems are also linked by
morphisms. For these there is the new complication of shape. In this paper we
define fixed-shape system moorphisms, but a more general definition would
allow the shape to vary. A distributed system morphism
$\theta\colon\mathcal{G}\Rightarrow\mathcal{G}^{\prime}$ in
$\mathrmbf{Dist}(\mathrmbf{I})$ consists of a collection $\\{\theta_{n}\colon
G_{n}\rightarrow G^{\prime}_{n}\mid n\in\mathrmbf{I}\\}$ of component graph
morphisms, which are systematically coordinated in the sense that they satisfy
the naturality conditions $G_{e}\circ\theta_{m}=\theta_{n}\circ
G^{\prime}_{e}$ for any indexing link $e\colon n\rightarrow m$ in
$\mathrmbf{I}$. A direct flow operator
$\mathrmbfit{dir}_{\mathrmbf{I}}(\theta):\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G}^{\prime})$
along $\theta$ can be define, which maps an information system
$\mathcal{S}\in\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$ to an
information system
$\mathrmbfit{dir}_{\mathrmbf{I}}(\theta)(\mathcal{S})\in\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G}^{\prime})$
defined by
$\mathrmbfit{dir}_{\mathrmbf{I}}(\theta)(\mathcal{S})_{n}=\mathrmbfit{dir}(\theta_{n})(E_{n})$
for $n\in\mathrmbf{I}$.191919Well-defined, since
$\mathrmbfit{dir}(G^{\prime}_{e})(\mathrmbfit{dir}(\theta_{n})(E_{n}))=\mathrmbfit{dir}(\theta_{m})(\mathrmbfit{dir}(G_{e})(E_{n}))\geq_{m}\mathrmbfit{dir}(\theta_{m})(E_{m})$.
An inverse flow operator
$\mathrmbfit{inv}_{\mathrmbf{I}}(\theta):\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G}^{\prime})\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$
can similarly be defined. Direct and inverse flow are adjoint monotonic
functions
${\langle{\mathrmbfit{dir}_{\mathrmbf{I}}(\theta)\dashv\mathrmbfit{inv}_{\mathrmbf{I}}(\theta)}\rangle}:\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G}^{\prime})$,
since
$\mathrmbfit{dir}_{\mathrmbf{I}}(\theta)(\mathcal{S})\geq^{\mathrmbf{I}}_{\mathcal{G}^{\prime}}\mathcal{S}^{\prime}$
iff
$\mathcal{S}\geq^{\mathrmbf{I}}_{\mathcal{G}}\mathrmbfit{inv}_{\mathrmbf{I}}(\theta)(\mathcal{S}^{\prime})$.
An information system morphism
$\theta\colon\mathcal{S}\Rightarrow\mathcal{S}^{\prime}$ in
$\mathrmbf{Info}(\mathrmbf{I})$ consists of a collection
$\\{\theta_{n}\colon\mathcal{S}_{n}\rightarrow\mathcal{S}^{\prime}_{n}\mid
n\in\mathrmbf{I}\\}$ of component olog morphisms, which are systematically
coordinated and preserve alignment in the sense that they satisfy the
naturality conditions
$\mathcal{S}_{e}\circ\theta_{m}=\theta_{n}\circ\mathcal{S}^{\prime}_{e}$ for
any indexing link $e\colon n\rightarrow m$ in $\mathrmbf{I}$; equivalently,
$\theta$ is a morphism between the underlying distributed systems
$\theta\colon\mathcal{G}\Rightarrow\mathcal{G}^{\prime}$ and the direct flow
of $\mathcal{S}$ is at least as general as $\mathcal{S}^{\prime}$:
$\mathrmbfit{dir}_{\mathrmbf{I}}(\theta)(\mathcal{S})\geq^{\mathrmbf{I}}_{\mathcal{G}^{\prime}}\mathcal{S}^{\prime}$.
The ordering
$\mathcal{S}\geq^{\mathrmbf{I}}_{\mathcal{G}}\mathcal{S}^{\prime}$ is an
information system morphism
$\theta\colon\mathcal{S}\Rightarrow\mathcal{S}^{\prime}$ with identity
component translations $\theta_{n}=\mathrmit{id}_{G_{n}}$ for each index
$n\in\mathrmbf{I}$.
#### 4.3.4. Channels
We continue with our systems point-of-view. Since we have represented the
whole system as a diagram $\mathcal{S}$ of parts (ologs) $\mathcal{S}_{n}$
with part-part relations (alignment constraints)
$\mathcal{S}_{n}\rightarrow\mathcal{S}_{m}$, we also want to represent the
whole system as an olog $\mathcal{C}$ with part-whole relations
$\mathcal{S}_{n}\rightarrow\mathcal{C}$.202020The theory of part-whole
relations is called mereology. It studies how parts are related to wholes, and
how parts are related to other parts within a whole. An information channel
${\langle{\gamma\colon\mathcal{M}\Rightarrow\Delta(C),C}\rangle}$ consists of
an indexed family $\\{\gamma_{n}\colon G_{n}\rightarrow C\mid
n\in\mathrmbf{I}\\}$ of graph morphisms called flow links with a common target
graph $C$ called the core of the channel. A channel
${\langle{\gamma,C}\rangle}$ covers a distributed system $\mathcal{G}$ of
shape $\mathrmbf{I}$ when the part-whole relationships respect the alignment
constraints (are consistent with the part-part relationships):
$\gamma_{n}=G_{e}\circ\gamma_{m}$ for each indexing morphism $e\colon
n\rightarrow m$ in $\mathrmbf{I}$. A covering channel is a distributed system
morphism $\gamma\colon\mathcal{G}\Rightarrow\Delta(C)$ in
$\mathrmbf{Dist}(\mathrmbf{I})$ from distributed system $\mathcal{G}$ to
constant distributed system
$\Delta(C)\colon\mathrmbf{I}\rightarrow\mathrmbf{Gph}$. Such a channel defines
a direct flow operator
$\mathrmbfit{dir}_{\mathrmbf{I}}(\gamma):\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(C))$
and an inverse flow operator
$\mathrmbfit{inv}_{\mathrmbf{I}}(\gamma):\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(C))\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$.
For any two covering channels ${\langle{\gamma^{\prime},C^{\prime}}\rangle}$
and ${\langle{\gamma,C}\rangle}$ over the same distributed system
$\mathcal{G}$, a refinement
$H\colon{\langle{\gamma^{\prime},C^{\prime}}\rangle}\rightarrow{\langle{\gamma,C}\rangle}$
is a graph morphism between cores $H\colon C^{\prime}\rightarrow C$ that
respects the part-whole relationships of the two channels:
$\gamma^{\prime}_{n}\circ H=\gamma_{n}$ for $n\in\mathrmbf{I}$. In such a
situation, we say the channel ${\langle{\gamma^{\prime},C^{\prime}}\rangle}$
is a refinement of the channel ${\langle{\gamma,C}\rangle}$. A channel
${\langle{\iota,\coprod\mathcal{G}}\rangle}$ is a minimal
cover212121Information flow terminology [BS]. or optimal(ly refined covering)
channel of a distributed system $\mathcal{G}$ when it covers $\mathcal{G}$ and
for any other covering channel ${\langle{\gamma,C}\rangle}$ there is a unique
refinement $[\gamma,C]\colon\coprod\mathcal{G}\rightarrow C$ from
${\langle{\iota,\coprod\mathcal{G}}\rangle}$ to ${\langle{\gamma,C}\rangle}$.
#### 4.3.5. System flow
In order to represent an information system
$\mathcal{S}=\\{\mathcal{S}_{n}\xrightarrow{\mathcal{S}_{e}}\mathcal{S}_{m}\\}$
as a single olog $\coprod\mathcal{S}$, called the fusion of $\mathcal{S}$,
with part-whole relations $\mathcal{S}_{n}\rightarrow\coprod\mathcal{S}$, we
follow the colimit theorem of [TBG] by recognizing the following three
properties.
* •
Optimal channels exist for any distributed system $\mathcal{G}$.
* •
$\mathrmbfit{fbr}(G)$ is a complete preorder for any graph $G$, loosely called
a “lattice”.
* •
For any graph morphism $H\colon G_{1}\rightarrow G_{2}$, direct and inverse
flow are adjoint monotonic functions
${\langle{\mathrmbfit{dir}(H),\mathrmbfit{inv}(H)}\rangle}\colon\mathrmbfit{fbr}(G_{1})\rightarrow\mathrmbfit{fbr}(G_{2})$.
Let $\mathcal{G}\in\mathrmbf{Dist}(\mathrmbf{I})$ be a distributed system of
shape $\mathrmbf{I}$ with optimal channel
${\langle{\iota,\coprod\mathcal{G}}\rangle}$. The optimal core
$\widehat{\mathcal{G}}=\coprod\mathcal{G}$ is called the sum of the
distributed system $\mathcal{G}$, and the optimal channel components (graph
morphisms) $\\{\iota_{n}\colon G_{n}\rightarrow\coprod\mathcal{G}\mid
n\in\mathrmbf{I}\\}$ are called flow links. There is a direct system flow
monotonic function (see Figure 1)
$\mathrmbfit{dir}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}=\mathrmbfit{dir}_{\mathrmbf{I}}(\iota)\cdot{\vee^{\mathrmbf{I}}_{\hat{\mathcal{G}}}}\colon\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})\rightarrow\mathrmbfit{fbr}(\widehat{\mathcal{G}})$.
Direct system flow has two steps: (i) direct (fixed shape) system flow of an
information system along the optimal channel
($\mathrmbf{Dist}(\mathrmbf{I})$-morphism)
$\iota\colon\mathcal{G}\Rightarrow\Delta(\widehat{\mathcal{G}})$ and (ii)
lattice join combining the contributions of the parts into a whole. In the
opposite direction, there is an inverse system flow monotonic function (see
Figure 1)
$\mathrmbfit{inv}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}={\Delta^{\mathrmbf{I}}_{\hat{\mathcal{G}}}}\cdot\mathrmbfit{inv}_{\mathrmbf{I}}(\iota)\colon\mathrmbfit{fbr}(\widehat{\mathcal{G}})\rightarrow\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$.
Inverse system flow has two steps: (i) mapping an olog with core language
$\widehat{\mathcal{G}}$ to a constant information system over
$\Delta(\widehat{\mathcal{G}})$ with shape $\mathrmbf{I}$ by distributing the
olog to the locations $n\in\mathrmbf{I}$, and (ii) inverse (fixed shape)
system flow of this constant information system back along the optimal channel
$\iota\colon\mathcal{G}\Rightarrow\Delta(\widehat{\mathcal{G}})$. Direct
system flow is adjoint to inverse system flow
${\langle{\mathrmbfit{dir}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}\dashv\mathrmbfit{inv}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}}\rangle}\colon\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})\rightarrow\mathrmbfit{fbr}(\widehat{\mathcal{G}})$,
since the composition components are adjoint. For any distributed system
$\mathcal{G}\in\mathrmbf{Dist}(\mathrmbf{I})$ with optimal core
$\widehat{\mathcal{G}}=\coprod\mathcal{G}$, any information system
$\mathcal{S}\in\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$, and any olog
$\widehat{\mathcal{S}}\in\mathrmbfit{fbr}(\widehat{\mathcal{G}})$, entailment
satisfies the following axioms.
(direct flow) | If $E_{n}$ entails the equation $(f=_{G_{n}}f^{\prime})\colon i\rightarrow j$, then $\mathrmbfit{dir}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}(\mathcal{S})$ entails the equation $(\iota_{n}^{\ast}(f)=_{\hat{\mathcal{G}}}\iota_{n}^{\ast}(f^{\prime}))\colon\iota_{n}(i)\rightarrow\iota_{n}(j)$ for any $n\in\mathrmbf{I}$.
---|---
(inverse flow) | If $\widehat{\mathcal{S}}$ entails the equation $(\iota_{n}^{\ast}(f)=_{\hat{\mathcal{G}}}\iota_{n}^{\ast}(f^{\prime}))\colon\iota_{n}(i)\rightarrow\iota_{n}(j)$, then $\mathrmbfit{inv}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}(\widehat{\mathcal{S}})_{n}$ entails the equation $(f=_{G_{n}}f^{\prime})\colon i\rightarrow j$ for any $n\in\mathrmbf{I}$.
These are converted to inference rules in Table 1.
$\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$$\mathrmbfit{info}_{\mathrmbf{I}}(\Delta(\widehat{\mathcal{G}}))$$\mathrmbfit{fbr}(\widehat{\mathcal{G}})$$\coprod\mathcal{S}$$\ni$$\mathcal{S}$$\in$$\in$$\mathcal{S}^{\scriptscriptstyle\blacklozenge}$$\mathrmbfit{dir}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}$$\mathrmbfit{inv}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}$$\mathrmbfit{dir}_{\mathrmbf{I}}(\iota)$$\mathrmbfit{inv}_{\mathrmbf{I}}(\iota)$$\vee^{\mathrmbf{I}}_{\hat{\mathcal{G}}}$$\Delta^{\mathrmbf{I}}_{\hat{\mathcal{G}}}$$\dashv$$\dashv$
Figure 1. System Flow
Information flow can be used to compute the fusion olog for an information
system and to define the consequence of an information system. Fusion is
direct system flow, and consequence is the composition of direct and inverse
system flow. Let $\mathcal{S}\in\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$
be any information system. The fusion
$\coprod\mathcal{S}=\mathrmbfit{dir}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}(\mathcal{S})={\langle{\coprod\mathcal{G},\bigvee_{n\in\mathrmbf{I}}\mathrmbfit{dir}(\iota_{n})(E_{n})}\rangle}\in\mathrmbfit{fbr}(\widehat{\mathcal{G}})$
is an olog that represents the whole system in a centralized fashion
[Ken2],[Ken3]. The consequence
$\mathcal{S}^{\scriptscriptstyle\blacklozenge}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}=\mathrmbfit{inv}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}(\mathrmbfit{dir}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}(\mathcal{S}))=\mathrmbfit{inv}_{{\langle{\mathrmbf{I},\mathcal{G}}\rangle}}(\coprod\mathcal{S})=\\{\mathrmbfit{inv}(\iota_{n})(\coprod\mathcal{S})\mid
n\in\mathrmbf{I}\\}\in\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$ is an
information system that represents the whole system in a distributed fashion
[Ken3]. It is inverse flow of the fusion olog along the optimal channel,
transfering the entailed facts of the whole system to the component parts.
The consequence operator $(\mbox{-})^{\scriptscriptstyle\blacklozenge}$, which
is defined on information systems, is a closure operator on the complete
preorder $\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$, and by taking the
coproduct it is a closure operator on the complete preorder
$\mathrmbf{Info}(\mathrmbf{I})=\coprod_{\mathcal{G}\in\mathrmbf{Dist}(\mathrmbf{I})}\mathrmbfit{info}_{\mathrmbf{I}}(\mathcal{G})$
: (increasing) $\mathcal{S}\geq\mathcal{S}^{\scriptscriptstyle\blacklozenge}$,
(monotonic) $\mathcal{S}\geq\mathcal{S}^{\prime}$ implies
$\mathcal{S}^{\scriptscriptstyle\blacklozenge}\geq\mathcal{S}^{\prime\scriptscriptstyle\blacklozenge}$
and (idempotent)
$\mathcal{S}^{\scriptscriptstyle\blacklozenge\blacklozenge}=\mathcal{S}^{\scriptscriptstyle\blacklozenge}$.
222222By allowing system shape to vary, channels can be generalized to
morphisms of distributed systems. Then a notion of relative fusion (direct
system flow) can be defined in terms of left Kan extension, and a notion of
relative system consequence can be defined as the composition of direct
followed by inverse system flow. Pointwise entailment order $\leq$ on
$\mathrmbf{Info}(\mathrmbf{I})$ is only a preliminary order, since it does not
incorporate interactions between system component parts. System entailment
order $\preceq$ on $\mathrmbf{Info}(\mathrmbf{I})$ is defined by
$\mathcal{S}_{1}\preceq\mathcal{S}_{2}$ when
$\mathcal{S}_{1}^{\scriptscriptstyle\blacklozenge}\leq\mathcal{S}_{2}^{\scriptscriptstyle\blacklozenge}$;
equivalently,
$\mathcal{S}_{1}^{\scriptscriptstyle\blacklozenge}\leq\mathcal{S}_{2}$.
Pointwise order is stronger than system entailment order:
$\mathcal{S}_{1}\leq\mathcal{S}_{2}$ implies
$\mathcal{S}_{1}\preceq\mathcal{S}_{2}$. This is a specialization-
generalization order. Any information system $\mathcal{S}$ is entailment
equivalent to its consequence
$\mathcal{S}\cong\mathcal{S}^{\scriptscriptstyle\blacklozenge}$. An
information system $\mathcal{S}$ is closed when it is equal to its consequence
$\mathcal{S}=\mathcal{S}^{\scriptscriptstyle\blacklozenge}$.
The whole effect of taking the system consequence may be greater than the sum
of its parts, in the sense that
$\mathcal{S}_{n}\geq_{n}\mathcal{S}_{n}^{{\scriptscriptstyle\blacklozenge}_{\iota_{n}}}\geq_{n}\bigvee_{m}\mathrmbfit{inv}(\iota_{n})(\mathrmbfit{dir}(\iota_{m})(\mathcal{S}_{m}))\geq_{n}\mathcal{S}^{\scriptscriptstyle\blacklozenge}_{n}$
for any $n\in\mathrmbf{I}$, since separate parts may have a productive
interaction at the channel core. A final part of an information system is a
part with no non-trivial constraint links from it. (The graphical subsystem
beneath) nonfinal parts are necessary for the alignment of information
systems, resulting in the equivalencing of types and aspects through
quotienting. However, because of the covering condition
$\iota_{n}=G_{e}\circ\iota_{m}$ and the entailment order
$\mathrmbfit{dir}(G_{e})(E_{n})\geq_{m}E_{m}$ for constraint links
$\mathcal{S}_{e}\colon\mathcal{S}_{n}\rightarrow\mathcal{S}_{m}$, only the
fact(ual) content of final parts of information systems are necessary to
compute the system fusion and consequence.
equivalence: | (reflexive) | | $(f=_{G}f)\colon i\rightarrow j$
---
| (symmetric) | | $(f_{1}=_{G}f_{2})\colon i\rightarrow j$
---
$(f_{2}=_{G}f_{1})\colon i\rightarrow j$
| (transitive) | | $(f_{1}=_{G}f_{2})\colon i\rightarrow j$, $(f_{2}=_{G}f_{3})\colon i\rightarrow j$
---
$(f_{1}=_{G}f_{3})\colon i\rightarrow j$
algebra: | (compositional) | | $(f_{1}=_{G}f_{2})\colon i\rightarrow j$, $(g_{1}=_{G}g_{2})\colon j\rightarrow k$
---
$(f_{1}{\;;\;}g_{1}=_{G}f_{2}{\;;\;}g_{2})\colon i\rightarrow k$
| (bi-closed) | | $(g_{1}=_{G}g_{2})\colon j\rightarrow k$
---
$(f{\,;\,}g_{1}=_{G}f{\;;\;}g_{2})\colon i\rightarrow k$,
$(g_{1}{\;;\;}h=_{G}g_{2}{\,;\,}h)\colon j\rightarrow l$
morphic flow: | (direct) | | $(f_{1}=_{G_{1}}f^{\prime}_{1})\colon i_{1}\rightarrow j_{1}$
---
$(H^{\ast}(f_{1})=_{G_{2}}H^{\ast}(f^{\prime}_{1}))\colon H(i_{1})\rightarrow
H(j_{1})$
| (inverse) | | $(H^{\ast}(f_{1})=_{G_{2}}H^{\ast}(f^{\prime}_{1}))\colon H(i_{1})\rightarrow H(j_{1})$
---
$(f_{1}=_{G_{1}}f^{\prime}_{1})\colon i_{1}\rightarrow j_{1}$
system flow: | (direct) | | $(f=_{G_{n}}f^{\prime})\colon i\rightarrow j$
---
$(\iota_{n}^{\ast}(f)=_{\hat{\mathcal{G}}}\iota_{n}^{\ast}(f^{\prime}))\colon\iota_{n}(i)\rightarrow\iota_{n}(j)$
| (inverse) | | $(\iota_{n}^{\ast}(f)=_{\hat{\mathcal{G}}}\iota_{n}^{\ast}(f^{\prime}))\colon\iota_{n}(i)\rightarrow\iota_{n}(j)$
---
$(f=_{G_{n}}f^{\prime})\colon i\rightarrow j$
Table 1. Inference Rules
#### 4.3.6. General examples
Here are some examples of system fusion/consequence.
* •
An information system $\mathcal{S}$ with a constant underlying distributed
system, $G_{i}=G$ for all $n\in\mathrmbf{I}$, gathers together all the
component parts of the information system and forms their consequence. It has
identity flow links $\\{\iota_{n}=\mathrmit{id}_{G}\colon G\rightarrow
G=\coprod\mathcal{G}\mid n\in\mathrmbf{I}\\}$, component join fusion
$\coprod\mathcal{S}=\bigvee_{n\in\mathrmbf{I}}\mathcal{S}_{n}={\langle{G,\bigcup_{n\in\mathrmbf{I}}E_{n}}\rangle}$,
and constant system consequence
$\mathcal{S}^{\scriptscriptstyle\blacklozenge}_{n}=\left(\bigvee_{n^{\prime}\in\mathrmbf{I}}\mathcal{S}_{n^{\prime}}\right)^{\scriptscriptstyle\bullet}$
for all $n\in\mathrmbf{I}$.
* •
A discrete information system
$\mathcal{S}=\\{\mathcal{S}_{n}={\langle{G_{n},E_{n}}\rangle}\mid
n\in\mathrmbf{I}\\}$ with no constraint links
$G_{e}\colon\mathcal{S}_{n}\rightarrow\mathcal{S}_{m}$ for $n\neq m$, has
coproduct injection flow links $\iota_{n}\colon
G_{n}\rightarrow\mbox{\Large$+$}_{n\in\mathrmbf{I}}\,G_{n}$, non-restricting
fusion, and inverse flow projecting back to individual component consequence
$\mathcal{S}^{\scriptscriptstyle\blacklozenge}_{n}=\mathcal{S}_{n}^{\scriptscriptstyle\bullet}$
for all $n\in\mathrmbf{I}$. No alignment (constraint) links means no
interaction.
* •
An information system
$\mathcal{S}=\\{\mathcal{S}_{1}\xleftarrow{H_{1}}\mathcal{S}\xrightarrow{H_{2}}\mathcal{S}_{2}\\}$
consisting of a single common ground $\mathcal{S}={\langle{G,E}\rangle}$
between two component ologs $\mathcal{S}_{1}={\langle{G_{1},E_{1}}\rangle}$
and $\mathcal{S}_{2}={\langle{G_{2},E_{2}}\rangle}$, with underlying
distributed system (span)
$\mathcal{G}=\\{G_{1}\xleftarrow{H_{1}}G\xrightarrow{H_{2}}G_{2}\\}$, has
pushout injection flow links
$G_{1}\xrightarrow{\iota_{1}}\coprod\mathcal{G}\xleftarrow{\iota_{2}}G_{2}$,
direct image union fusion
$\coprod\mathcal{S}={\langle{\coprod\mathcal{G},\mathrmbfit{dir}(\iota_{1})(E_{1})\cup\mathrmbfit{dir}(\iota_{2})(E_{2})}\rangle}$,
and system consequence components
$\mathcal{S}^{\scriptscriptstyle\blacklozenge}_{n}={\langle{G_{n},\mathrmbfit{inv}(\iota_{n})(\mathrmbfit{dir}(\iota_{1})(E_{1})\cup\mathrmbfit{dir}(\iota_{2})(E_{2}))}\rangle}$
for $n=1,2$. The flow links will quotient any types and aspects that are
connected through the common ground allowing for the approprate interaction in
the fusion consequence
$(\mathrmbfit{dir}(\iota_{1})(E_{1})\cup\mathrmbfit{dir}(\iota_{2})(E_{2}))^{\scriptscriptstyle\bullet}$,
then the inverse flow will reconnect this with the component types and
aspects.
### 4.4. Conceptual graphs
The conceptual graph formalism (CG) for knowledge representation [Sow2], was
initially formulated to represent database systems (DBS), but is now used in
natural language processing (NLP) and first-order logic (FOL). Verbs in NLP
can often be represented relationally by star(-shaped conceptual) graphs. For
example, the sentence “John is going to Boston by bus” might be represented by
the conceptual graph
(131)
In a sentence of natural language, thematic roles are semantic descriptions of
the way (the entities described by) a noun phrase functions with respect to
(the action of) the verb. These entities are the participants in the occurrent
expressed by the verb. For the action of ‘going’ in the above sentence there
are three participants and hence three thematic roles. ‘John’ plays the role
of the agent of the action, a ‘Bus’ is the instrument used in the action and
‘Boston’ is the destination of the action. Translations using thematic roles
can be used to align two ontologies with respect to a common ground. A CG-
style translation of conceptual graph (131) would replace the verb relation
‘going’ with a concept ‘Go’ and replace the edges that form the signature of
the ‘going’ relation with binary relations for the three roles ‘agent’,
‘instrument’ and ‘destination’.
(138)
However, the case relations that semantically describe the thematic roles
should be viewed as functional in nature; that is, for any instance of the
action of a sentence’s verb there is a unique entity described by a noun
phrase of the sentence. When this semantics is respected, the translation to
thematic roles becomes a process of “linearization”, which is best described
abstractly as: (1) the identification of relation types with entity types, (2)
the translation of a sorted multiarity relation to a span of functions, one
function for each role, and (3) the functional interpretation of thematic
roles.
The Olog formalism, which also represents DBS and NLP, is a version of
equational logic. Both the Olog and CG formalisms were designed as graphical
representations. However, the CG formalism is binary and relational, whereas
the Olog formalism is unary and functional. The CG formalism is binary since
it has two kinds of type, concepts and relations; it is relational in the way
it interprets edges. The Olog formalism is unary since it has only one kind of
type, the abstract concept; it is functional in the way it interprets aspects
(edges). However, much of the semantics of the CG formalism can be transformed
to the Olog formalism by the process of linearization232323The linearization
process works for any binary/relational knowledge representation, such as CGs,
entity-relationship data modelling [JRW], relational database systems [Ken5]
or the Information Flow Framework [IFF1]. In the entity-relationship data
modelling, $n$-ary relationship links are replaced by $n$-ary spans of aspects
and attributes are included as types., thereby gaining in efficiency and
conciseness. For example, conceptual graph (131) can be linearized to the olog
graph242424$\ulcorner$1$\urcorner$ is the universal type to which all types
have a unique aspect.
(143)
Since olog aspects are interpreted functionally, the functional nature of
thematic roles is respected. In this manner, the olog formalism could be used
to replace the CG representation of ontologies. For example, a community
(acting as an individual) could build its ontology $\mathcal{C}$ from ground
up by aligning it with some top-level reference ontology $\mathcal{T}$ (such
as in the appendix of [Sow2]), thereby importing some formal semantics from
$\mathcal{T}$. The following fragment demonstrates how this works.
Assume that ontology $\mathcal{T}$ contains the concept of “spatial process”
as represented by the general concept type $\ulcorner$Spatial-
Process$\urcorner$ with aspects $\textnormal{$\ulcorner$Spatial-
Process$\urcorner$}\xrightarrow{\text{agent}}\textnormal{$\ulcorner$Agent$\urcorner$}$,
$\textnormal{$\ulcorner$Spatial-
Process$\urcorner$}\xrightarrow{\text{inst}}\textnormal{$\ulcorner$Vehicle$\urcorner$}$
and $\textnormal{$\ulcorner$Spatial-
Process$\urcorner$}\xrightarrow{\text{dest}}\textnormal{$\ulcorner$Location$\urcorner$}$.
At some stage assume that the community ontology $\mathcal{C}$ has specified
the concept type orderings
$\textnormal{$\ulcorner$Person$\urcorner$}\leq\textnormal{$\ulcorner$Agent$\urcorner$}$,
$\textnormal{$\ulcorner$Bus$\urcorner$}\leq\textnormal{$\ulcorner$Vehicle$\urcorner$}$
and
$\textnormal{$\ulcorner$City$\urcorner$}\leq\textnormal{$\ulcorner$Location$\urcorner$}$
with corresponding injective aspects
$\textnormal{$\ulcorner$Person$\urcorner$}\xrightarrow{\text{is}}\textnormal{$\ulcorner$Agent$\urcorner$}$,
$\textnormal{$\ulcorner$Bus$\urcorner$}\xrightarrow{\text{is}}\textnormal{$\ulcorner$Vehicle$\urcorner$}$
and
$\textnormal{$\ulcorner$City$\urcorner$}\xrightarrow{\text{is}}\textnormal{$\ulcorner$Location$\urcorner$}$.
At the next stage it could define a concept type $\ulcorner$C$\urcorner$ with
aspects
$\textnormal{$\ulcorner$C$\urcorner$}\xrightarrow{\text{person}}\textnormal{$\ulcorner$Person$\urcorner$}$,
$\textnormal{$\ulcorner$C$\urcorner$}\xrightarrow{\text{bus}}\textnormal{$\ulcorner$Bus$\urcorner$}$
and
$\textnormal{$\ulcorner$C$\urcorner$}\xrightarrow{\text{city}}\textnormal{$\ulcorner$City$\urcorner$}$,
and link it with the reference ontology concept $\ulcorner$Spatial-
Process$\urcorner$ by specifying a connecting aspect
$\textnormal{$\ulcorner$C$\urcorner$}\xrightarrow{\text{process}}\textnormal{$\ulcorner$Spatial-
Process$\urcorner$}$ and asserting the facts
‘$\mbox{person}{\;;\;}\mbox{is}=\mbox{process}{\;;\;}\mbox{agent}$’,
‘$\mbox{bus}{\;;\;}\mbox{is}=\mbox{process}{\;;\;}\mbox{vehicle}$’ and
‘$\mbox{city}{\;;\;}\mbox{is}=\mbox{process}{\;;\;}\mbox{location}$’.252525The
symbol ‘;’ denotes concatenation or formal composition. In the more expressive
ologs with joins (Section 5), the process concept of “going to city by bus”
can then be defined as the pullback of the “spatial process” concept: here,
the concept type $\ulcorner$Go$\urcorner$ with aspects
$\textnormal{$\ulcorner$Go$\urcorner$}\xrightarrow{\text{person}}\textnormal{$\ulcorner$Person$\urcorner$}$,
$\textnormal{$\ulcorner$Go$\urcorner$}\xrightarrow{\text{bus}}\textnormal{$\ulcorner$Bus$\urcorner$}$
and
$\textnormal{$\ulcorner$Go$\urcorner$}\xrightarrow{\text{city}}\textnormal{$\ulcorner$City$\urcorner$}$
is pulled back along the above injective aspects, resulting in the injective
aspect
$\textnormal{$\ulcorner$Go$\urcorner$}\xrightarrow{\text{is}}\textnormal{$\ulcorner$Spatial-
Process$\urcorner$}$ with corresponding concept type ordering
$\textnormal{$\ulcorner$Go$\urcorner$}\leq\textnormal{$\ulcorner$Spatial-
Process$\urcorner$}$. As a result, the concept $\ulcorner$C$\urcorner$ has the
new mediating aspect $\mbox{C}\xrightarrow{\text{going}}\mbox{Go}$, which
satisfies the fact ‘$\mbox{going}{\;;\;}\mbox{is}=\mbox{process}$’. In this
manner the community ontology $\mathcal{C}$ has been enlarged.
$\mathcal{C}$$\mathcal{T}$CpersonbuscitygoingprocessGopersonbuscityis$\textstyle{{\cdot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Spatial-
Processagentinstdest$\textstyle{{\cdot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$PersonisBusisCityis$\textstyle{{\cdot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$AgentVehicleLocation
$\displaystyle\overset{\underbrace{\rule{300.0pt}{0.0pt}}}{\mbox{\Large\rule{0.0pt}{16.0pt}$\mathcal{P}$}}$
We assume that community ontology $\mathcal{C}$ and reference ontology
$\mathcal{T}$ are combined into a portal ontology $\mathcal{P}$ with portal
link $\mathcal{C}\xrightarrow{P}\mathcal{P}$ and alignment link
$\mathcal{T}\xrightarrow{A}\mathcal{P}$. If some other ontology
$\mathcal{C}^{\prime}$ is built up and aligned in the same fashion, then
$\mathcal{T}$ is being used as a common ground, and we have a ‘W’-shaped
information system
(148)
with portals $\mathcal{P}$ and $\mathcal{P}^{\prime}$ being the final parts.
This ‘W’-shaped information system uses the sketch institution Sk for ologs.
It can be compared to the ‘W’-shaped information system in [Ken1], which uses
the information flow IF institution for (local) logics.
## 5\. More expressive ologs I
In this section and the next (5 and 6) we will introduce limits and colimits
within the context of ologs. These will allow authors to build ologs that are
quite expressive. For example we can declare one type to be the union or
intersection of other types. We do not assume mathematical knowledge beyond
that of sets and functions, which were loosely defined in Section 2.2.
However, the reader may benefit by consulting a reference on category theory,
such as [Awo].
The basic ologs discussed in previous sections are based on the mathematical
notion of categories, whereas the olog presentation language we will discuss
in this section and the next are based on general sketches (see [Mak]). The
difference is in what can be expressed: in basic ologs we can declare types,
aspects, and facts, whereas in general ologs we can express ideas like
products and sums, as we will see below.
We will begin by discussing layouts, which will be represented categorically
by “finite limits”. As usual, the english terminology (layout) is not precise
enough to express the notion we mean it to express (limit). Intuitively, a
limit can be thought of as a system: it is a collection of units, each of a
specific type, such that these units have compatible aspects. These will
include types like $\ulcorner$a man and a woman with the same last
name$\urcorner$. In Section 6 we will discuss groupings, which will be
represented by colimits. These will include types like $\ulcorner$a thing that
is either a pear or a watermelon$\urcorner$.
### 5.1. Layouts
A dictionary might define the word layout as something like “a structured
arrangement of items within certain limits; a plan for such arrangement.” In
other words, we can lay out or specify the need for a set of parts, each of a
given type, such that the parts fit together well. This idea roughly
corresponds to the notion of limits in category theory, especially limits in
the category of sets. Given a diagram of sets and functions, its limit is the
set of ways to accordingly choose one element from each. For example, we could
have a type $\ulcorner$a car and a driver$\urcorner$, which category-
theoretically is a product, but which we are calling a “layout” — a compound
type whose parts are “laid out.” Of course, the term layout is insufficient to
express the precise meaning of limits, but it will have to do for now. To
understand limits, one really only need understand pullbacks and products.
These will be the subjects of Sections 5.2 and 5.3, or one can see [Awo] for
more details.
### 5.2. Pullbacks
Given three objects and two arrows arranged as to the left, the pullback is
the commutative square to the right:
Given:
$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D}$
the pullback is drawn:
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D.}$
We write $A=B\times_{D}C$ and say “$A$ is the pullback of $B$ and $C$ over
$D$.” The question is, what does it signify? We will begin with some examples
and then give a precise definition.
###### Example 5.2.1.
We will now give four examples to motivate the definition of pullback. In the
first example, (157), both $B$ and $C$ will be subtypes of $D$, and in such
cases the pullback will be their intersection. In the next two examples (166
and 175), only $B$ will be a subtype of $D$, and in such cases the pullback
will be the “corresponding subtype of $C$” (as should make sense upon
inspection). In the last example (184), neither $B$ nor $C$ will be a subtype
of $D$. In each line below, the pullback of the diagram to the left is the
diagram to the right. The reader should think of the left-hand olog as a kind
of problem to which the new box $A$ in the right-hand olog is a solution.
(157) $\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{50.58878pt}{\raggedright a loyal
customer\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{50.58878pt}{\raggedright a wealthy
customer\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a customer}}}}$ $\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C}}{{\framebox{\parbox{65.04256pt}{\raggedright a customer that
is wealthy and
loyal\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$isis$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{50.58878pt}{\raggedright a loyal
customer\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{50.58878pt}{\raggedright a wealthy
customer\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a customer}}}}$ (166) $\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{blue}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a
person}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ has as
favorite color $\textstyle{\stackrel{{\scriptstyle D}}{{\framebox{a color}}}}$
$\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C}}{{\framebox{\parbox{65.04256pt}{\raggedright a person whose
favorite color is
blue\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is
has as favorite color $\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{blue}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a
person}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ has as
favorite color $\textstyle{\stackrel{{\scriptstyle D}}{{\framebox{a color}}}}$
(175) $\textstyle{\stackrel{{\scriptstyle C}}{{\framebox{a
woman}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a dog}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
has as owner $\textstyle{\stackrel{{\scriptstyle D}}{{\framebox{a person}}}}$
$\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C}}{{\framebox{\parbox{65.04256pt}{\raggedright a dog whose owner
is a
woman\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is
has as owner $\textstyle{\stackrel{{\scriptstyle C}}{{\framebox{a
woman}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a dog}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
has as owner $\textstyle{\stackrel{{\scriptstyle D}}{{\framebox{a person}}}}$
(184) $\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{50.58878pt}{\raggedright a piece of
furniture\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{50.58878pt}{\raggedright a space in our
house\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a width}}}}$ $\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C}}{{\framebox{\parbox{79.49744pt}{\raggedright a pair $(f,s)$
where $f$ is a piece of furniture and $s$ is a space in our house, and where
$f$ and $s$ have the same
width\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$s$$f$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{50.58878pt}{\raggedright a piece of
furniture\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{50.58878pt}{\raggedright a space in our
house\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a width}}}}$
See Example 5.2.3 for a justification of these, in light of Definition 5.2.2.
The following is the definition of pullbacks in the category of sets. For an
olog, the instance data are given by sets (at least in this paper, see Section
3), so this definition suffices for now. See [Awo] for more details on
pullbacks.
###### Definition 5.2.2.
Let $B,C,$ and $D$ be sets, and let $f\colon B\rightarrow D$ and $g\colon
C\rightarrow D$ be functions. The pullback of
$B\xrightarrow{f}D\xleftarrow{g}C$, denoted $B\times_{D}C$, is defined to be
the set
$B\times_{D}C:=\\{(b,c)\;|\;b\in B,c\in C,\textnormal{ and }f(b)=g(c)\\}$
together with the obvious maps $B\times_{D}C\rightarrow B$ and
$B\times_{D}C\rightarrow C$, which send an element $(b,c)$ to $b$ and to $c$,
respectively. In other words, the pullback of
$B\xrightarrow{f}D\xleftarrow{g}C$ is a commutative square
$\textstyle{B\times_{D}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D.}$
###### Example 5.2.3.
In Example 5.2.1 we gave four examples of pullbacks. For each, we will
consider $B\xrightarrow{f}D\xleftarrow{g}C$ to be sets and functions as in
Definition 5.2.2 and explain how the set $A$ follows that definition, i.e. how
its label fits with the set $B\times_{D}C=\\{(b,c)\;|\;b\in B,c\in
C,\textnormal{ and }f(b)=g(c)\\}$.
In the case of (157), the set $B\times_{D}C$ should consist of pairs $(w,l)$
where $w$ is a wealthy customer, $l$ is a loyal customer, and $w$ is equal to
$l$ (as customers). But if $w$ and $l$ are the same customer then $(w,l)$ is
just a customer that is both wealthy and loyal, not two different customers.
In other words, an instance of the pullback is a customer that is both loyal
and wealthy, so the label of $A$ fits.
In the case of (166), the set $B\times_{D}C$ should consist of pairs $(p,b)$
where $p$ is a person, $b$ is the color blue, and the favorite color of $p$ is
equal to $b$ (as colors). In other words, it is a person whose favorite color
is blue, so the label of $A$ fits. If desired, one could instead label $A$
with $\ulcorner$a pair $(p,b)$ where $p$ is a person, $b$ is blue, and the
favorite color of $p$ is $b$$\urcorner$.
In the case of (175), the set $B\times_{D}C$ should consist of pairs $(d,w)$
where $d$ is a dog, $w$ is a woman, and the owner of $d$ is equal to $w$ (as
people). In other words, it is a dog whose owner is a woman, so the label of
$A$ fits. If desired, one could instead label $A$ with $\ulcorner$a pair
$(d,w)$ where $d$ is a dog, $w$ is a woman, and the owner of $d$ is
$w$$\urcorner$.
In the case of (184), the set $B\times_{D}C$ should consist of pairs $(f,s)$
where $f$ is a piece of furniture, $s$ is a space in our house, and the width
of $f$ is equal to the width of $s$. This is fits perfectly with the label of
$A$.
#### 5.2.4. Using pullbacks to classify
To distinguish between two things, one must find a common aspect of the two
things for which they have differing results. For example, a pen is different
from a pencil in that they both use some material to write (a common aspect),
but the two materials they use are different. Thus the material which a
writing implement uses is an aspect of writing implements, and this aspect
serves to segregate or classify them. We can think of three such writing-
materials: graphite, ink, and pigment-wax. For each, we will make a layout in
the olog below:
| | |
---|---|---|---
$\textstyle{\stackrel{{\scriptstyle
A_{1}=B\times_{D}C_{1}}}{{\framebox{\parbox{72.26999pt}{a writing implement
that uses
graphite}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$usesis$\textstyle{\stackrel{{\scriptstyle
C_{1}}}{{\framebox{graphite}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
A_{2}=B\times_{D}C_{2}}}{{\framebox{\parbox{72.26999pt}{a writing implement
that uses
ink}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$usesis$\textstyle{\stackrel{{\scriptstyle
C_{2}}}{{\framebox{ink}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
A_{3}=B\times_{D}C_{3}}}{{\framebox{\parbox{72.26999pt}{a writing implement
that uses pigment-
wax}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$usesis$\textstyle{\stackrel{{\scriptstyle
C_{3}}}{{\framebox{pigment-
wax}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a writing
implement}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$uses$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a writing material}}}}$
One could also replace the label of box $A_{1}$ with “a pencil”, the label of
box $A_{2}$ with “a pen”, and the label of box $A_{3}$ with “a crayon”; in so
doing, the layouts at the top would define a pencil, a pen, and a crayon to be
a writing implement that uses respectively graphite, ink, and pigment-wax.
#### 5.2.5. Building pullbacks on pullbacks
There is a theorem in category theory which states the following. Suppose
given two commutative squares
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lrcorner}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{6}$
such that the right-hand square (3,4,5,6) is a pullback. It follows that if
the left-hand square (1,2,3,4) is a pullback then so is the big rectangle
(1,2,5,6). It also follows that if the big rectangle (1,2,5,6) is a pullback
then so is the left-hand square (1,2,3,4). This fact can be useful in
authoring ologs.
For example, the type $\ulcorner$a cellphone that has a bad battery$\urcorner$
is vague, but we can lay out precisely what it means using pullbacks:
$\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C}}{{\framebox{\parbox{72.26999pt}{a cellphone that has a bad
battery}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
C=D\times_{F}E}}{{\framebox{a bad
battery}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
E=F\times_{H}G}}{{\framebox{\parbox{36.135pt}{less than 1
hour}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
G}}{{\framebox{\parbox{36.135pt}{between 0 and
1}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a
cellphone}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a
battery}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ remains
charged for $\textstyle{\stackrel{{\scriptstyle
F}}{{\framebox{\parbox{43.36243pt}{a duration of
time}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ in hours yields
$\textstyle{\stackrel{{\scriptstyle H}}{{\framebox{\parbox{43.36243pt}{a range
of numbers}}}}}$
The category-theoretic fact described above says that since $A=B\times_{D}C$
and $C=D\times_{F}E$, it follows that $A=B\times_{F}E$. That is, we can decuce
the definition “a cellphone that has a bad battery is defined as a cellphone
that has a battery which remains charged for less than one hour.” In other
words, $A=B\times_{F}E$.
### 5.3. Products
Given a set of types (boxes) in an olog, one can select one instance from
each. All the ways of doing just that comprise what is called the product of
these types. For example, if $A=\textnormal{$\ulcorner$a number between 1 and
10$\urcorner$}$ and $B=\textnormal{$\ulcorner$a letter between x and
z$\urcorner$}$, the product includes a total of 30 elements, including
$(4,z)$. We are ready for the definition.
###### Definition 5.3.1.
Given sets $A,B$, their product, denoted $A\times B$, is the set
$A\times B=\\{(a,b)\;|\;a\in A\textnormal{ and }b\in B\\}.$
There are two obvious projection maps $A\times B\rightarrow A$ and $A\times
B\rightarrow B$, sending the pair $(a,b)$ to $a$ and to $b$ respectively.
###### Example 5.3.2.
In Example 5.2.1, (184) we presented the idea of a piece of furniture that was
the same width as a space in the house. What if we say that $\ulcorner$a nice
furniture placement$\urcorner$ is any space that is between 1 and 8 inches
bigger than a piece of furniture? We can use a combination of products and
pullbacks to create the appropriate type.
| |
---|---|---
| |
$\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C}}{{\framebox{\parbox{72.26999pt}{a nice furniture
placement}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{79.49744pt}{\raggedright a pair of widths
$(w_{1},w_{2})$ such that $1\leq w_{2}-w_{1}\leq
8$\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
B_{1}}}{{\framebox{\parbox{50.58878pt}{\raggedright a piece of
furniture\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\checkmark}$has$\textstyle{\stackrel{{\scriptstyle
D_{1}}}{{\framebox{a width}}}}$$\textstyle{\stackrel{{\scriptstyle
B=B_{1}\times B_{2}}}{{\framebox{\parbox{79.49744pt}{\raggedright a pair
$(f,s)$ where $f$ is a piece of furniture and $s$ is a space in the
house\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{f}$$\scriptstyle{f\mapsto
w_{1},\;\;s\mapsto w_{2}}$$\textstyle{\stackrel{{\scriptstyle D=D_{1}\times
D_{2}}}{{\framebox{\parbox{72.26999pt}{a pair of widths
$(w_{1},w_{2})$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w_{1}}$$\scriptstyle{w_{2}}$$\textstyle{\stackrel{{\scriptstyle
B_{2}}}{{\framebox{\parbox{50.58878pt}{\raggedright a space in the
house\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\checkmark}$has$\textstyle{\stackrel{{\scriptstyle
D_{2}}}{{\framebox{a width}}}}$
Here $B$ and $D$ are products and $A$ is a pullback. This olog lays out what
it means to be “a nice furniture placement” using products. The bottom
horizontal aspect $B\rightarrow D$ is an example of a map obtained by the
“universal property of products”; see Section 5.6.
#### 5.3.3. Products of more (or fewer) types
The product of two sets $A$ and $B$ was defined in 5.3.1. One may also take
the product of three sets $A,B,C$ in a similar way, so the elements are
triples $(a,b,c)$ where $a\in A,b\in B,$ and $c\in C$. In fact this idea holds
for any number of sets. It even makes sense to take the product of one set
(just $A$) or no sets! The product of one set is itself, and the product of no
sets is the singleton set $\\{*\\}$. For more on this, see Section 5.5 or
[Mac].
### 5.4. Declaring an injective aspect
A function is called injective if different inputs always yield different
outputs. For example the function that doubles every integer ($x\mapsto 2x$)
is injective, whereas the function that squares every integer ($x\mapsto
x^{2}$) is not because $3^{2}=(-3)^{2}$. An example of an injective aspect is
$\textnormal{$\ulcorner$a
woman$\urcorner$}\xrightarrow{\textnormal{is}}\textnormal{$\ulcorner$a
person$\urcorner$}$ because different women are always different as people. An
example of a non-injective aspect is $\textnormal{$\ulcorner$a
person$\urcorner$}\xrightarrow{\textnormal{has as
father}}\textnormal{$\ulcorner$a person$\urcorner$}$ because different people
may have the same father.
The easiest way to indicate that an aspect is injective is to use a “hook
arrow” as in $f\colon A\hookrightarrow B$, instead of a regular arrow $f\colon
A\rightarrow B$, to denote it. For example, the first map is injective (and
specified as such with a hook-arrow), but the second is not in the olog:
a person hasa personality can be classified as being of a personality type
The author of this olog believes that no two people can have precisely the
same personality (though they may have the same personality type).
We include injective aspects in this section because it turns out that
injectivity can also be specified by pullbacks. See [nL1] for details.
### 5.5. Singletons types
A singleton set is a set with one element; it can be considered the “empty
product.” In other words if we denote $A^{n}=A\times A\times\cdots A$ (where
$A$ is written $n$ times), then $A^{0}$ is the empty product and is a
singleton set. One can specify that a certain type has only one instance by
annotating it with $A=\\{*\\}$ in the olog. For example the olog
$\textstyle{\stackrel{{\scriptstyle
A=\\{*\\}}}{{\framebox{God}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{a good thing}}}}$
says that the author considers $\ulcorner$God$\urcorner$ to be single. As a
more concrete example, the intersection of $\\{x\in{\mathbb{R}}\;|\;x\geq
0\\}$ and $\\{y\in{\mathbb{R}}\;|\;x\leq 0\\}$ is a singleton set, as
expressed in the olog
$\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C=\\{*\\}}}{{\framebox{\parbox{72.26999pt}{a real number $z$ such
that $z\geq 0$ and $z\leq
0$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x=z}$$\scriptstyle{y=z}$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{72.26999pt}{a real number $x$ such that $x\geq
0$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{72.26999pt}{a real number $y$ such that $y\leq
0$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a real number}}}}$
The fact that $A=B\times_{D}C$ and $A=\\{*\\}$ are declared indicates that
there is only one possible instance of a real number that is in both $B$ and
$C$.
### 5.6. The universal property of layouts
We cannot do the notion of universal properties justice in this paper, but the
basic idea is as follows. Suppose that ${\mathcal{D}}$ is an olog, that
$D_{1},D_{2}$ are types in it, and that $D=D_{1}\times D_{2}$ (together with
its projection maps $p_{1}\colon D\rightarrow D_{1}$ and $p_{2}\colon
D\rightarrow D_{2}$) is their product.
(191)
The so-called universal property of products should be thought of as “an
existence and uniqueness” claim in ${\mathcal{D}}$. Namely, for any type $X$
with maps $f\colon X\rightarrow D_{1}$ and $g\colon X\rightarrow D_{2}$, there
is exactly one possible map $m\colon X\rightarrow D$ such that the facts
$f=m;p_{1}$ and $g=m;p_{2}$ hold.
(198)
This may sound esoteric, but consider the following example.
The following olog is similar to the one in Example 5.3.2
$\textstyle{\stackrel{{\scriptstyle
B_{1}}}{{\framebox{\parbox{50.58878pt}{\raggedright a piece of
furniture\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
C_{1}}}{{\framebox{a
width}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\checkmark}$
is, in inches $\textstyle{\stackrel{{\scriptstyle D_{1}}}{{\framebox{a
number}}}}$$\textstyle{\stackrel{{\scriptstyle B=B_{1}\times
B_{2}}}{{\framebox{\parbox{79.49744pt}{\raggedright a pair $(f,s)$ where $f$
is a piece of furniture and $s$ is a space in the
house\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{f}$$\textstyle{\stackrel{{\scriptstyle
D=D_{1}\times D_{2}}}{{\framebox{\parbox{72.26999pt}{a pair of numbers
$(w_{1},w_{2})$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w_{1}}$$\scriptstyle{w_{2}}$$\textstyle{\stackrel{{\scriptstyle
B_{2}}}{{\framebox{\parbox{50.58878pt}{\raggedright a space in the
house\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has$\textstyle{\stackrel{{\scriptstyle
C_{2}}}{{\framebox{a
width}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\checkmark}$
is, in inches $\textstyle{\stackrel{{\scriptstyle D_{2}}}{{\framebox{a
number}}}}$
Here the only unlabeled map is the horizontal one $B\rightarrow D$; how can we
get away with leaving it unlabeled? How does a piece of furniture and a space
in the house yield a pair of numbers? The answer is that $B$ has a map to
$D_{1}$ (the path across the top) and a map to $D_{2}$ (the path across the
bottom), and hence the universal property of products gives a unique arrow
$B\rightarrow D$ such that the two facts indicated by checkmarks hold. (In
terms of (191) and (198) we are using $X=B$.) In other words, there is exactly
one way to take a piece of furniture and a space in the house and yield a pair
of numbers if we enforce that the first number is the width in inches of the
piece of furniture and the second number is the width in inches of the space
in the house.
At this point we hope it is clear that the universal property of products is a
useful and constructive one. We will not describe the other universal
properties (either for pullbacks, singletons, or any colimits); as mentioned
above they can be found in [Awo].
## 6\. More expressive ologs II
In this section we will describe various colimits, which are in some sense
dual to limits. Whereas limits allow one to “lay out” a team consisting of
many different interacting or non-interacting parts, colimits allow one to
“group” different types together. For example, whereas the product of
$\ulcorner$a number between 1 and 10$\urcorner$ and $\ulcorner$a letter
between x and z$\urcorner$ has 30 elements (such as $(3,y)$), the coproduct of
these two types has 13 elements (including 4). Just as “layout” is a too weak
a word to capture the essence of limits, “grouping” is too weak a word to
capture the essence of colimits, but it will have to do.
We will start by describing coproducts or “disjoint unions” in Section 6.1.
Then we will describe pushouts in Section 6.2, wherein one can declare some
elements in a union to be equivalent to others. There is a category-theoretic
duality between coproducts and products and between pushouts and pullbacks. It
extends to a duality between surjections and injections and a duality between
empty types and singleton types, the subject of Sections 6.3 and 6.4. The
interested reader can see [Awo] for details.
### 6.1. Coproducts
Coproducts are also called “disjoint unions.” If $A$ and $B$ are sets with no
members in common, then the coproduct of $A$ and $B$ is their union. However,
if they have elements in common, one must include both copies in $A\amalg B$
and differentiate between them. Here is a definition.
###### Definition 6.1.1.
Given sets $A$ and $B$, their coproduct, denoted $A\amalg B$, is the set
$A\amalg B=\\{(a,``A")\;|\;a\in A\\}\cup\\{(b,``B")\;|\;b\in B\\}.$
There are two obvious inclusion maps $A\rightarrow A\amalg B$ and
$B\rightarrow A\amalg B$, sending $a$ to $(a,``A")$ and $b$ to $(b,``B")$,
respectively.
If $A$ and $B$ have no elements in common, then the one can drop the $``A"$
and “$B$” labels without changing the set $A\amalg B$ in a substantial way.
Here are two examples that should make the coproduct idea clear.
###### Example 6.1.2.
In the following olog the types $A$ and $B$ are disjoint, so the coproduct
$C=A\amalg B$ is just the union.
$\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{a
person}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
C=A\amalg B}}{{\framebox{a person or a
cat}}}}$$\textstyle{\stackrel{{\scriptstyle B}}{{\framebox{a
cat}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is
###### Example 6.1.3.
In the following olog, $A$ and $B$ are not disjoint, so care must be taken to
differentiate common elements.
$\textstyle{\stackrel{{\scriptstyle
A}}{{\framebox{\parbox{50.58878pt}{\raggedright an animal that can
fly\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$labeled
“A” is$\textstyle{\stackrel{{\scriptstyle C=A\amalg
B}}{{\framebox{\parbox{93.95122pt}{an animal that can fly (labeled ``A") or an
animal that can swim (labeled ``B")}}}}}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{65.04256pt}{\raggedright an animal that can
swim\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$labeled
“B” is
Since ducks can both swim and fly, each duck is found twice in $C$, once
labeled as a flyer and once labeled as a swimmer. The types $A$ and $B$ are
kept disjoint in $C$, which justifies the name “disjoint union.”
### 6.2. Pushouts
Pushouts can express unions in which an overlap is declared. They can also
express “quotients,” where different objects can be declared equivalent. Given
three objects and two arrows arranged as to the left, the pushout is drawn as
the commutative square to the right:
Given:
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{f}$$\textstyle{C}$$\textstyle{B}$
the pushout is drawn:
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{D.}$
We write $D=B\amalg_{A}C$ and say “$D$ is the pushout of $B$ and $C$ along
$A$.” The question is, what does it signify?
The idea is that an instance of the pushout $B\amalg_{A}C$ is any instance of
$B$ or any instance of $C$, but where some instances are considered equivalent
to others. That is, for any instance of $A$, its $B$-aspect is considered the
same as its $C$-aspect. This is formalized in Definition 6.2.2 after being
exemplified in Example 6.2.1.
###### Example 6.2.1.
In each example below, the diagram to the right is the pushout of the diagram
to the left. The new object, $D$, is the union of $B$ and $C$, but instances
of $A$ are equated to their $B$ and $C$ aspects. This will be discussed after
the two diagrams.
(207)
---
$\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{\parbox{50.58878pt}{a cell
in the
shoulder}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$isis$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{43.36243pt}{a cell in the
arm}}}}}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{50.58878pt}{a cell in the torso}}}}}$
$\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{\parbox{50.58878pt}{a cell
in the
shoulder}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$isis$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{43.36243pt}{a cell in the
arm}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{50.58878pt}{a cell in the
torso}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
D=B\amalg_{A}C}}{{\framebox{\parbox{57.81621pt}{a cell in the torso or
arm}}}}}$ (216)
---
$\textstyle{\stackrel{{\scriptstyle
A}}{{\framebox{\parbox{57.81621pt}{\raggedright a college mathematics
course\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$yieldsis$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{57.81621pt}{an utterance of the phrase ``too
hard"}}}}}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{43.36243pt}{\raggedright a college
course\@add@raggedright}}}}}$ $\textstyle{\stackrel{{\scriptstyle
A}}{{\framebox{\parbox{57.81621pt}{\raggedright a college mathematics
course\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$yieldsis$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{57.81621pt}{an utterance of the phrase ``too
hard"}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{43.36243pt}{\raggedright a college
course\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle\parbox{43.36243pt}{\vspace{.1in}\tiny$D=B\\!\amalg_{A}\\!C$}}}{{\framebox{\parbox{72.26999pt}{\raggedright
a college course, where every mathematics course is replaced by an utterance
of the phrase ``too hard"\@add@raggedright}}}}}$
In Olog (207), the shoulder is seen as part of the arm and part of the torso.
When taking the union of these two parts, we do not want to “double-count” the
shoulder (as would be done in the coproduct $B\amalg C$, see Example 6.1.3).
Thus we create a new type $A$ for cells in the shoulder, which are considered
the same whether viewed as cells in the arm or cells in the body. In general,
if one wishes to take two things and glue them together, the glue serves as
$A$ and the two things serve as $B$ and $C$, and the union (or grouping) is
the pushout $B\amalg_{A}C$.
In Olog (216), if every mathematics course is simply “too hard,” then when
reading off a list of courses, each math course will not be read aloud but
simply read as “too hard.” To form $D$ we begin by taking the union of $B$ and
$C$, and then we consider everything in $A$ to be the same whether one looks
at it as a course or as the phrase “too hard.” The math courses are all
blurred together as one thing. Thus we see that the power to equate different
things can be exercised with pushouts.
###### Definition 6.2.2.
Let $A,B,$ and $C$ be sets and let $f\colon A\rightarrow B$ and $g\colon
A\rightarrow C$ be functions. The pushout of
$B\xleftarrow{f}A\xrightarrow{g}C$, denoted $B\amalg_{A}C$, is the quotient of
$B\amalg C$ (see Definition 6.1.1) by the equivalence relation generated by
declaring $b\sim c$ (i.e. $b$ is equivalent to $c$) if: $b\in B,c\in C$, and
there exists $a\in A$ with $f(a)=b$ and $g(a)=c$.
### 6.3. Declaring a surjective aspect
A function $f\colon A\rightarrow B$ is called surjective if every value in $B$
is the image of something in the domain $A$. For example, the function which
subtracts 1 from every integer ($x\mapsto x-1$) is surjective, because every
integer has a successor; whereas the function that doubles every integer
($x\mapsto 2x$) is not surjective because odd numbers are not mapped to. The
aspect is $\textnormal{$\ulcorner$a published
paper$\urcorner$}\xrightarrow{\textnormal{was published
in}}\textnormal{$\ulcorner$an established journal$\urcorner$}$ is surjective
because every established journal has had at least one paper published in it.
The aspect is $\textnormal{$\ulcorner$a published
paper$\urcorner$}\xrightarrow{\textnormal{has as first
author}}\textnormal{$\ulcorner$a person$\urcorner$}$ is not surjective because
not every person is the first author of a published paper.
The easiest way to indicate that an aspect is surjective is to denote it with
a “two-headed arrow” as in $f\colon A\twoheadrightarrow B$. For example, the
second map is surjective (and indicated with a two-headed arrow) in the olog
a personhasa personality can be classified as being of
$\textstyle{\stackrel{{\scriptstyle}}{{\framebox{\parbox{79.49744pt}{\raggedright
a documented personality type\@add@raggedright}}}}}$
Here the first aspect is not considered surjective, presumably because the
author imagines personalities had by no person.
We include surjective aspects in this section because it turns out that
surjectivity can also be specified by pushouts. See [nL2] for details.
### 6.4. Empty types
The empty set is a set with no elements; it can be considered the “empty
coproduct.” In other words if we denote $n*A=A\amalg A\amalg\cdots\amalg A$
(where $A$ is written $n$ times), then $0*A$ is the empty coproduct and is the
empty set. One can declare a type to be empty by annotating it with
$A=\emptyset$ in the olog. For example the olog
$\framebox{$\stackrel{{\scriptstyle A=\emptyset}}{{\framebox{a supernatural
being}}}$}$
says that the set of supernatural beings is empty. As a more concrete example,
the intersection of positive numbers and negative numbers is empty, as
expressed in the olog
$\textstyle{\stackrel{{\scriptstyle
A=B\times_{D}C=\emptyset}}{{\framebox{\parbox{72.26999pt}{a real number $z$
such that $z<0$ and
$z>0$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle
C}}{{\framebox{\parbox{72.26999pt}{a real number $x$ such that
$x>0$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{72.26999pt}{a real number $y$ such that
$y<0$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a real number}}}}$
### 6.5. Images
In what remains of Section 6, we will discuss how the ideas of this section
and the previous (Section 5) can be used together to create quite expressive
ologs. First we will discuss how each aspect $f\colon A\rightarrow B$ has an
“image,” the subset of $B$ that are “hit” by $f$. Then, in Sections 6.6 and
6.7, we will discuss how ologs can express all primitive recursive functions
and many other mathematical concepts.
Consider the olog
(221) $\textstyle{\stackrel{{\scriptstyle
X}}{{\framebox{\parbox{115.63243pt}{a pair $(p,c)$ where $p$ is a person, $c$
is a computer, and $p$ owns
$c$}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{c}$$\textstyle{\stackrel{{\scriptstyle
Y}}{{\framebox{a person}}}}$$\textstyle{\stackrel{{\scriptstyle
Z}}{{\framebox{a computer}}}}$
Some people own more than one computer, and some computers are owned by more
than one person. Some computers are not owned by a person, and some people do
not own a computer. The purpose of this section is to show how to use ologs to
capture ideas such as “a person who owns a computer” and “a computer that is
owned by a person”. These are called the images of $p$ and $c$ respectively.
Every aspect has an image, and these are quite important for human
understanding. For example the image of the map $\textnormal{$\ulcorner$a
person$\urcorner$}\xrightarrow{\textnormal{has as
father}}\textnormal{$\ulcorner$a person$\urcorner$}$ is the type $\ulcorner$a
father$\urcorner$. In other words, a father is defined to be a person $x$ for
which there is some other person $y$ such that $x$ is the father of $y$.
The image of a function $f\colon A\rightarrow B$ is a commutative diagram
(fact)
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{f_{s}}$$\scriptstyle{\checkmark}$$\textstyle{B}$$\textstyle{{\bf
im}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{i}}$
where $f_{s}$ is surjective and $f_{i}$ is injective (see Sections 6.3 and
5.4). We indicate that a type is the image of a map $f$ by annotating it with
Im$(f)$, as in the following olog:
$\textstyle{\stackrel{{\scriptstyle A}}{{\framebox{a
child}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$has
as parents$f$$\textstyle{\stackrel{{\scriptstyle
B}}{{\framebox{\parbox{108.405pt}{a pair $(w,m)$ where $w$ is a woman and $m$
is a
man}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\checkmark}$$\scriptstyle{m}$$\textstyle{\stackrel{{\scriptstyle
C={\bf Im}(f)}}{{\framebox{a
father}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$is$\scriptstyle{\checkmark}$$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a man}}}}$
Hopefully it is also clear that $\ulcorner$a person who owns a
computer$\urcorner$ and $\ulcorner$a computer that is owned by a
person$\urcorner$ are the images of $p\colon X\rightarrow Y$ and $c\colon
X\rightarrow Z$ (respectively) in Olog (221).
Using the label Im$(f)$ is the easiest way to indicate an image, although one
can also do so categorically using limits and colimits. See [Mac, Chapter
VIII] for details.
### 6.6. Application: Primitive recursion
We have already seen how ologs can be used to express a conceptual
understanding of a situation (all the ologs thus far exemplify this idea). In
this section we hope to convince the reader that ologs are also able to
express certain computations. In particular we will show by example that
primitive recursive functions (like factorial or fibonacci) can be expressed
by ologs. In this way, we can to put computation and knowledge representation
together into the same framework. It would be quite valuable to strengthen
this connection by showing that Ologs (or an extension thereof) can express
any recursive function (i.e. simulate Turing machines). This is an open
research possibility.
###### Example 6.6.1.
In this example we will present an olog that can represent the “Factorial
function,” often denoted $n\mapsto n!$, where for example the factorial of $4$
is $24$. Recall that a natural number is any nonnegative whole number:
$0,1,2,3,4,\ldots$.
$f(n)=n!$ $\underline{s;p=\textnormal{id}_{A}}\hskip 14.45377pt\underline{s;q=d;f}\hskip 14.45377pt\underline{i_{0};f=\omega}\hskip 14.45377pt\underline{i_{1};f=s;m}$ | |
---|---|---
$\textstyle{\stackrel{{\scriptstyle
A}}{{\framebox{\parbox{72.26999pt}{\raggedright a positive natural
number\@add@raggedright}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{d}$$\scriptstyle{i_{1}}$$\textstyle{\stackrel{{\scriptstyle
B=A\times D}}{{\framebox{\parbox{86.72377pt}{a pair $(p,q)$ where $p$ is a
positive natural number and $q$ is a natural
number}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{m}$$\textstyle{\stackrel{{\scriptstyle
C=A\amalg E}}{{\framebox{a natural
number}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\stackrel{{\scriptstyle
D}}{{\framebox{a natural number}}}}$$\textstyle{\stackrel{{\scriptstyle
E}}{{\framebox{zero}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{0}}$$\scriptstyle{\omega}$
The idea of this olog is to convey the factorial function as follows. A
natural number is either zero or positive. Every positive natural number $n$
has a decrement, $n-1$. The factorial of zero is 1. The factorial of a
positive number $n$ is obtained by multiplying $n$ by the factorial of $n-1$.
To more explicitly describe the above olog, we must describe its intended
instances. Hopefully the instances of each type ($A$ through $E$) are self-
explanatory, so we will describe the grouping, the layout, the aspects, and
the facts. The set of natural numbers is the disjoint union of zero and the
set of positive natural numbers and the maps $i_{0}$ and $i_{1}$ are the
inclusions into the coproduct, which explains the grouping $C=A\amalg E$. The
layout $B=A\times D$ is self-explanatory, and the maps $p$ and $q$ are the
projections from the product. The map $d$ is the decrement map $n\mapsto n-1$,
the map $\omega$ sends $0$ to $1$, the map $m$ is multiplication
$(n,n^{\prime})\mapsto n*n^{\prime}$. Once $m$, $d$, and $\omega$ are so-
defined, the first two facts ($s;p=\textnormal{id}_{A}$ and $s;q=d;f$) specify
that $s$ sends $n$ to the pair $(n,f(d(n)))$, and the second two facts specify
that $f$ sends $0$ to $1$ and sends a positive number $n$ to
$m(s(n))=m(n,f(d(n)))$, i.e. $n$ goes to the product $n*(n-1)!$.
The above olog defines the factorial function ($f$) in terms of itself, which
is the hallmark of primitive recursion. Note, however, that this same olog can
compute many things besides the factorial function. That is, nothing about the
olog says that the instances of $\ulcorner$Zero$\urcorner$ is the set
$\\{0\\}$, that $\omega$ sends $0$ to $1$, that $d$ is the decrement function,
or that $m$ is multiplication — changing any of these will change $f$ as a
function. For example, the same olog can be used to compute “triangle numbers”
(e.g. f(4)=1+2+3+4=10) by simply changing the instances of $\omega$ and $m$ in
the obvious ways (use $\omega=0,m=+$ rather than $\omega=1,m=*)$). For a
radical departure, fix any forest (set of graphical trees) $F$, let
$E=\textnormal{$\ulcorner$zero$\urcorner$}$ represent its set of roots, $A$
the other nodes, $\omega$ the constant 0 function, $d$ the parent function,
and $m$ sending $(p,d(p))$ to $f(d(p))+1$. Then for each tree in $F$ and each
node $n$ in that tree, the function $f$ will send $n$ to its height on the
tree.
Primitive recursion is a powerful technique for deriving new functions from
the repetition of others using a kind of “while loop.” The general form of
primitive recursive functions can be found in [BBJ], and it is not hard to
imitate Example 6.6.1 for the general case.
### 6.7. Application: defining mathematical concepts
In this subsection we hope to convince the reader that many mathematical
concepts can be defined by ologs. This should not seem like much of a stretch:
ologs describe relationships between sets, so we rely on the maxim that all of
mathematics can be formulated within set theory. To make the idea explicit,
however, we will recall the definition of pseudo-metric space (in 6.7.1) and
then provide an olog with the same content (in 230).
###### Definition 6.7.1.
Let ${\mathbb{R}}_{\geq 0}$ denote the set of non-negative real numbers. A
pseudo-metric space is a pair $(X,\delta)$ where $X$ is a set and
$\delta\colon X\times X\rightarrow{\mathbb{R}}_{\geq 0}$ is a function with
the following properties for all elements $x,y,z\in X$:
1. (1)
$\delta(x,x)=0$;
2. (2)
$\delta(x,y)=\delta(y,x)$; and
3. (3)
$\delta(x,z)\leq\delta(x,y)+\delta(y,z)$.
(230)
As long as the instances for the right-hand side of this olog are
mathematically correct (i.e. we assign $4$ the set of non-negative real
numbers), this olog has the same content as Definition 6.7.1. One can use
ologs to define usual metric spaces (in which Property (1) in Definition 6.7.1
is strengthened), but it would have taken too much space here.
It should be clear that ologs provide a more precise and explicit description
of any concept, relying less on the grammar of English and more on the
mathematical “grammar” of sets and functions. Assumptions are exposed as all
the working parts of an object need to be explicitly documented. Thus an olog
is likely to be instantly readable by a theorem prover such as Coq ([Coq]), at
least if one creates the olog within an appropriate Olog-Coq interface API.
Moreover, various parts of this olog may be reusable in other contexts, and
hence connect pseudo-metric spaces into a web of neighboring definitions and
theorems.
In fact, once a corpus of mathematics has been written in olog form, evidence
of conjectures not yet proven could be written down as instance data. For
example, one could record every known prime as instances of a type
$\ulcorner$prime$\urcorner$ and a machine could automatically check that
Goldbach’s conjecture (written as an olog containing
$\ulcorner$prime$\urcorner$ as a type) holds for all example “so far.” With
definitions, theorems, and examples all written in the same computer-readable
language of ologs, one may hope for much more advanced searching and knowledge
retrieval by humans. For example, one could formulate very precise questions
as database queries and use SQL on the database corresponding to a given olog
(see Section 3.2).
## 7\. Further directions
Ologs are basically categories which have text labels to explain their
intended semantic. As such there are many directions to explore ranging from
quite theoretical to quite practical. Here we consider three main classes:
extending the theory of ologs, studying communication with ologs, and
implementing ologs in the real world.
### 7.1. Extending the theory of ologs
In this paper we began by discussing basic ologs, which are rich enough to
capture the semantic of many situations. In Sections 5 and 6 we added more
expressivity to ologs to allow one to encode ideas such as intersections,
unions, and images. However, ologs could be even more expressive. One could
add “function types” (also known as exponentials); add a “subobject classifier
type,” which could allow for negation and complements as well as power-sets;
or even add fixed sets (like the set of Strings) to the language of ologs.
This is not too hard (using sketches, see [Mak]); the reason we did not
include them in this paper was more because of space than any other reason.
Another generalization would be to allow the instances of an olog to take
values in a category other than ${\bf Set}$. For example, one could have an
instance-space rather than an instance-set, e.g. it is clear that the
instances of the type $\ulcorner$a point on the unit circle$\urcorner$
constitute a topological space. One could similarly argue that the instances
of the type $\ulcorner$a human invention$\urcorner$ have a topology or metric
as well (e.g. as an invention, the cellphone is closer to the telephone than
it is to artificial flavoring). Instance data on an olog ${\mathcal{C}}$
corresponds to a functor ${\mathcal{C}}\rightarrow{\bf Set}$ in this paper,
but it is quite easy to replace ${\bf Set}$ with a different category such as
${\bf Top}$ (the category of topological spaces), and this may have
interesting uses in data modeling.
In Section 6.7, we explicitly showed that pseudo-metric spaces (and we stated
further that metric spaces) can be presented by ologs. It would be interesting
to see if theorems could also be proven entirely within the context of ologs.
If so, a teacher could first sketch a mathematical proof as a small or sparse
olog ${\mathcal{C}}$, and then use a functor
${\mathcal{C}}\rightarrow{\mathcal{D}}$ to rigorously “zoom in” on that proof
so that the sketch becomes a full-fledged proof (as the maps in
${\mathcal{C}}$ are factored into understandable units in ${\mathcal{D}}$).
If ologs are to be viable venues in which to discuss results in mathematics,
then they should be capable of describing all recursion, not just primitive
recursion (as in Section 6.6). We do not yet have an understanding for how
this can be done. If recursion can be fully defined with the ologs described
above, it would be interesting to see it written out; if not, it would be
interesting to understand what basic idea could be gracefully added to ologs
so that recursion becomes expressible.
In a different direction, one could test the expressive power of ologs by
defining simple games, like Tic Tac Toe or Chess, using ologs. It would be
impressive to define a vocabulary for writing games and a program which could
automatically convert an olog-defined game into a playable computer game. This
would show that the same theory that we have seen express ideas about
fatherhood and factorials can also be used to invent games and program
computers.
### 7.2. Studying communication with ologs
As discussed in Section 4, ologs can be connected by functors into networks
that are not just 2-way, but $n$-way. These communication networks should be
studied: what kinds of information can pass, how reliable is it, how quickly
can it spread, etc. This may be applicable in fields from economics to
psychology to sociology. Such research may use results from established
mathematics such as Network Coding Theory (see [YLC]).
In [SA], we study how two or more entities (described as ologs) can
communicate new ideas (not just new instance data) to each other. It would be
interesting to see how well this “communication protocol” works in practice,
and whether it can be theoretically automated. Furthermore, this communication
protocol and any theoretical automation of it should be implemented on a
computer to see if different database schemas can be meaningfully integrated
with minimal human assistance.
It may be possible to train children to create ologs about their interests or
about a given lesson. These ologs would show how the child actually perceives
something, which would probably be fascinating. By our experience and that of
people we have taught, the process of building an olog usually leads to a
clarification of the concepts involved. Moreover, a class project to connect
the ologs of different students and between the students and the teacher, may
have excellent pedagogical benefits.
Finally, it may be interesting to study “local truth” vs. “global truth” in
a network of ologs. Functorial connections between ologs can allow for
translation of ideas between members of a group, but there may be ideas which
do not extend globally, just as a Möbius band does not admit a global
orientation. That is, given three parties on the Möbius band, any pair can
agree on a compass orientation, but there is no choice that the three can
simultaneously agree on. Similarly, whether or not it is possible to construct
a global language which extends all the existing local ones could be
determined if these local languages and their connections were entered into a
computer olog system.
### 7.3. Implementing ologs in the real world
Once ologs are implemented on computers, and once people learn how to author
good ologs, much is possible. One advantage comes in searching the information
space. Currently when we search for a concept (say in Google or on our hard
drive), we can only describe the concept in words and hope that those words
are found in a document describing the concept. That is, search is always
text-based. Better would be if the concept is meaningfully interconnected in a
web of concepts (an olog) that could be navigated in a meaningful (as opposed
to text-based) way.
Indeed, this is the semantic web vision: When internet data is machine-
readable, search becomes much more powerful. Currently, we rely on RDF
scrapers that scour web pages for $\langle$subject, predicate, object$\rangle$
sentences and store them in RDF format, as though each such sentence is a
fact. Since people are inputting their data as prose text, this may be the
best available method for now; however, it is quite inaccurate (e.g. often 15%
of the facts are wrong, a number which can lead to degeneration of deductive
reasoning – see [MBCH]). If ideas could be put on the internet such that they
compatibly made sense to both human and computer, it would give a huge boost
to the semantic web. We believe that ologs can serve as such a human-computer
interface.
While it is often assumed that because we all speak the same language we all
must mean the same things by it, this is simply not true. The age-old question
about whether “blue for me” is the same as “blue for you” is applicable to
every single word and idiom in our language. There is no easy way to sync up
different people’s perceptions. If communication is to be efficient,
agreements must be fairly explicit and precise, and this precision demands a
rigor that is simply unavailable in English prose. It is available in a
network of ologs (as described in Section 4).
For example, the laws of the United States are hopelessly complex. Residents
of the US are required to obey the laws. However, unlike the rules of the
Scholastic Aptitude Test (SAT), which take 10 minutes for the proctor to read
aloud, the laws of the US are never really expressed — the most important
among them are hopefully picked up by cultural osmosis. If an olog was created
which had enough detail that laws could be written in that format, then a
woman could research for herself whether her landlord was required to fix her
refrigerator or whether this was her responsibility. It may prove that the
olog of laws is internally inconsistent, i.e. that it is impossible for a
person to satisfy all the laws — such an analysis, if performed, could
fundamentally change our outlook on the legal system.
The same goes for science; information written up in articles is much less
accessible than information that is entered into an ontology. However, the
dream of a single universal ontology is untenable ([Min]). Instead we must
allow each lab or institute to create its own ontology, and then require
citations to be functorial olog connections, rather than mere silo-to-silo
pointers. Thus, a network of ologs should be created to represent the
understanding of the modern scientific community as a multi-faceted whole.
Another impetus for a scientist to write an olog about the study at hand is
that, once an olog is made, it can be instantly converted to a database schema
which the scientist can use to input all the data pertaining to this study.
Indeed, if some data did not fit within this schema, then the olog must have
been insufficient to begin with and should be modified to fully describe the
experiment. If scientists work this way, then the separation between them and
database modelers can be reduced or eliminated (the scientist assumes the
database modeling role with little additional burden). Moreover, if functorial
connections are established between the ologs of different labs, then data can
be meaningfully shared along those connections, and ideas written in the
language of one lab’s olog can be translated automatically into the language
of the other’s. The speed and accuracy of scientific research should improve.
## References
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|
arxiv-papers
| 2011-02-09T15:49:49 |
2024-09-04T02:49:16.888912
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David I. Spivak, Robert E. Kent",
"submitter": "David Spivak",
"url": "https://arxiv.org/abs/1102.1889"
}
|
1102.1925
|
SNSN-323-63
$b\rightarrow s\gamma$ and $b\rightarrow d\gamma$ (B factories)
Wenfeng Wang
University of Notre Dame Du Lac, South Bend, IN 46556, USA
> The photon spectrum in $B\rightarrow X_{s,d}\gamma$ decay, where $X_{s}(d)$
> is any strange (non-strange) hadronic state, is studied using data samples
> of
> $e^{+}e^{-}\rightarrow\Upsilon(4S)\to{{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
> decays collected by the BABAR and Belle experiments. Here I present the
> latest measurements of the branching fraction and spectral moments from
> $B\to X_{s}\gamma$ decays by Belle and the direct $C\\!P$ asymmetry
> $A_{CP}(B\to X_{s+d}\gamma)$ measured at BABAR. The determination of
> $|V_{td}/V_{ts}|^{2}$ is also presented.
> PRESENTED AT
>
>
>
>
> 6th International Workshop on the CKM Unitarity Triangle(CKM2010),
## 1 Introduction
The electromagnetic radiative process $b\rightarrow q\gamma$ ($q=s,d$)
proceeds at leading order via the loop diagram in the Standard Model (SM).
Here the SM predication of the inclusive rate
$\Gamma({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})$
can be equated with the precisely calculable partonic rate
$\Gamma({{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})$
at the level of a few percent [1] (heavy quark duality). An extraordinary
theoretical effort has led to a precision SM prediction for the branching
fraction at the next-to-next-to-leading order (four-loop), $\mbox{\rm
BR}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=(3.15\pm
0.23)\times 10^{-4}$ ($E_{\gamma}>1.6\mathrm{\,Ge\kern-1.00006ptV}$) [2],
where $E_{\gamma}$ is the photon energy measured in the rest frame of the
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ meson. The possibility
for new heavy particles to enter into the loop at leading order could cause
significant deviations from the SM prediction. A recent review can be seen in
[3].
New physics can also significantly enhance the direct $C\\!P$ asymmetry for
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$
and
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$
decay [4] without changing the branching fraction. We define
$A_{CP}=\frac{\Gamma({{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})-\Gamma({{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{s}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{d}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})}{\Gamma({{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})+\Gamma({{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{d}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{d}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})}$
(1)
which is $\sim 10^{-6}$ in the SM, with nearly exact cancellation of opposite
asymmetries for
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$
and
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$.
Thus any non-zero measurements of this joint asymmetry is an indication of new
physics.
The shape of the photon energy spectrum, which is insensitive to non-SM
physics [5], can be used to determine the Heavy Quark Expansion
(HQE)parameters, $m_{b}$ and $\mu_{\pi}^{2}$, related to the mass and momentum
of the ${{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ quark within the
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ meson. These parameters
can be used to reduce the error in the extraction of the CKM matrix elements
$V_{cb}$ and $V_{ub}$ from the inclusive semi-leptonic
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$-meson decays,
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{c}\ell\nu$ and
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{u}\ell\nu$ ($\ell=e$ or $\mu$) [6] .
The inclusive rate for $b\rightarrow d\gamma$ is suppressed compared to
$b\rightarrow s\gamma$ by a factor $|V_{td}/V_{ts}|^{2}$ in the SM. This ratio
can also be obtained from the $B_{d}$ and $B_{s}$ mixing frequencies [7]. New
physics effects would enter in different ways in mixing and radiative decays.
Measurements of $|V_{td}/V_{ts}|$ using the exclusive modes
$B\rightarrow(\rho,\omega)\gamma$ and $B\to K^{*}\gamma$ [8, 9] are now well-
established, with theoretical uncertainties of 7% [10]. A measurement of
inclusive $b\rightarrow d\gamma$ relative to $b\rightarrow s\gamma$ would
determine $|V_{td}/V_{ts}|$ with reduced theoretical uncertainties. We
parametrize the inclusive ratio (following [11]) by:
${{\mbox{\rm BR}(b\rightarrow d\gamma)}\over{\mbox{\rm BR}(b\rightarrow
s\gamma)}}=\zeta^{2}\left|{{V_{td}}\over{V_{ts}}}\right|^{2}(1+\Delta R)$ (2)
where $\zeta$ accounts for any remaining SU(3) breaking and $\Delta R$
accounts for weak annihilation in $B^{+}$ decays.
B factories, BABAR [12] and Belle [13], already accumulated more than one
billion of
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
events, which allows BABAR and Belle collaborations to perform precision
measurements on $b\to s\gamma$ and $b\to d\gamma$ processes [14]. Here I
summarize the latest experimental achievements on the above inclusive
processes.
## 2 Direct CP asymmetry in $B\rightarrow X_{s,d}\gamma$
The result presented***The branching fraction of $B\to X_{s}\gamma$ and its
spectra shape from same analysis will be present in near future. is based on a
data sample of
${{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{+}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{-}}\mspace{-0.6mu}}}}}}}}}\to{{{}{{}{{{{{{{}{}{{\Upsilon}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}}{{}{{{{{{{}{}{{\left({4S}\right)}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}\to{{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
collisions collected with the BABAR detector at the PEP-II asymmetric-energy
${{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{+}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{-}}\mspace{-0.6mu}}}}}}}}}$
collider. The on-resonance integrated luminosity is 347 $fb^{-1}$ and 36
$fb^{-1}$ of off-resonance data, taken 40 $\mathrm{\,Me\kern-1.00006ptV}$
below the
${{{}{{}{{{{{{{}{}{{\Upsilon}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}}{{}{{{{{{{}{}{{\left({4S}\right)}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}$
resonance energy, are used to estimate the continuum background
(${}{{}{{{{{{{}{}{{q}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}{}{}{{}{{}{{{{{{{}{}{{\overline{q}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}:q={{}{{}{{{{{{{}{}{{u}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{c}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}},{{}{{}{{{{{{{}{}{{\tau}_{\mspace{-2.0mu}{}}^{+}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\tau}_{\mspace{-2.0mu}{}}^{-}}\mspace{-0.6mu}}}}}}}}}$).
Figure 1: Left: The photon spectrum in $347fb^{-1}$ of data after background
subtraction. The inner error bars are statistical only, while the outer
include both statistical and systematic errors in quadrature; Right:
Measurements of $A_{CP}$
(${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$),
with statistical and systematic errors. The three published results, top to
bottom, are from references [15].
The analysis begins by requiring a high-energy photon, characteristic of $B\to
X_{s}\gamma$ decays, while photons from $\pi^{0}$ and $\eta$ are vetoed. The
background from continuum events is significantly suppressed by charged lepton
tagging and by exploiting the more jet-like topology of the $q\overline{q}$ or
$\tau^{+}\tau^{-}$ events compared to the isotropic $B\overline{B}$ decays.
The remaining continuum backgrounds are estimated with off-resonance data. The
non signal
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
background arises predominantly from
${{{{}{}{{\pi}_{\mspace{-2.0mu}{}}^{0}}\mspace{-0.6mu}}}}$,${{{{}{}{{\eta}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$
decay but also from decays of other light mesons, mis-reconstructed electrons
and hadrons, which are estimated using Monte Carlo simulation and corrected
the data and MC difference using appropriate control samples. Figure 1 shows
the observed photo spectrum after subtracting off-resonance data and the
corrected
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
backgrounds. Two prior selected control regions,
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
control ($1.53<E^{*}_{\gamma}<1.8\mathrm{\,Ge\kern-1.00006ptV}$) and Continuum
control ($2.9<E^{*}_{\gamma}<3.5\mathrm{\,Ge\kern-1.00006ptV}$), are used to
validate the background estimation. In the
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
control region we find $1252\pm 272(stat.)\pm 841(syst.)$ events, dominated by
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
background with a small signal contribution component ( 200-400 events
depending on models); the continuum region yields s $-100\pm 138(stat.)$
events, consistent with zero which showing good estimation of off-resonance
subtraction.
The direct CP asymmetry, $A_{CP}$
(${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$)
is measured by dividing the signal sample into
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$ and
${{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
decays according to the charge of the lepton tag to measure
$A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=\frac{N^{+}-N^{-}}{N^{+}+N^{-}}$,
where $N^{+(-)}$ are the positively (negatively) tagged signal yields. The
asymmetry must be corrected for the dilution due to the mistag fraction
$\omega$,
$A_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=\frac{1}{1-2\omega}A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}).$
the missing fraction $\omega$ is found to be $0.131\pm 0.007$, from
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}^{0}}\mspace{-0.6mu}}}}}}}}}-{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{0}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
oscillation, the fraction of events with wrong-sign leptons from the
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ decay chain and the
similar fraction due to misidentification of hadrons as leptons.
The theoretical SM predictions for a near-zero asymmetry do not require the
entire spectrum to be measured. To reduce the sensitivity to background, the
signal region is restricted to
$2.1<E^{*}_{\gamma}<2.8\mathrm{\,Ge\kern-1.00006ptV}$. In this selected energy
region, the tagged signal yields are $N^{+}=2623\pm 158(stat.)$ and
$N^{-}=2397\pm 151(stat.)$ giving an asymmetry of
$A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=0.045\pm
0.044\ .$ Finally the
$A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})$
is corrected for mistagging and bias to give $A_{CP}=0.056\pm 0.060(stat.)\pm
0.018(syst.)$, where the systematic error is mainly from non signal
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
background and the lepton tagging efficiency. The result is consistent with no
observed asymmetry, consistent with SM expectation and previous measurements.
A comparison of the result to published measurements is shown in figure 1. The
current measurement is the most precise to date.
## 3 Branching fraction and moments of $b\rightarrow s\gamma$
Currently the most precise measurement of the inclusive $B\to X_{s}\gamma$
branching fractions has been done by Belle [16]. The data consists of a sample
of 605 $fb^{-1}$ taken on the $\Upsilon(4S)$ resonance. Another 68 $fb^{-1}$
sample was taken at an energy 60 MeV below the resonance.
The signal spectrum is extracted by collecting all high energy photons,
vetoing those originating from $\pi^{0}$ and $\eta$ decays to two photons. The
non $B\overline{B}$ background, mainly $e^{+}e^{-}\rightarrow q\overline{q}$
($q=u,d,s,c$) events, is subtracted using the off-resonance sample. The
remaining background from $B\overline{B}$ are subtracted using Monte-Carlo
simulated distributions normalized using data control samples.
The analysis proceeds in two different streams, with lepton tag (LT) and
without (MAIN). Two samples give similar sensitivity to the signal while being
largely statistically independent. After these selection criteria, $41.1\times
10^{5}$ ( $24.6\times 10^{4}$) and $3.5\times 10^{5}$ ($0.9\times 10^{4}$)
photon candidates survive in the MAIN (LT) stream of the on- and off-resonance
data samples, respectively.
The photon candidates from $B\overline{B}$ background is divided into six
categories:(i) $\pi^{0}$; (ii)$\eta$; (iii) other real photons from decays of
$\omega$, $\eta^{\prime}$ and $J/\psi$ mesons; (iv) mis-identified calorimeter
clusters from $K^{0}_{L}$ and $\overline{n}$; (v) electrons misidentified as
photons and; (vi) beam background. Each category is checked using appropriate
control samples as described in Ref.[17]. Each background yield, scaled by the
described procedures, is subtracted from the data spectrum. The photon energy
ranges 1.4-1.7 GeV and 2.8-4.0 GeV were chosen a prior as control regions to
test the integrity of the background subtraction since in the low energy
region the little signal expected is negligible with respect to the
uncertainty on the background, and no signal is possible in the high energy
region above the kinematic limit. The yield in the high energy region are
$1245\pm 4349$ and $292\pm 410$ candidates in the MAIN and LT stream,
respectively, while corresponding yields in the low energy region are
$-1629\pm 3071$ and $-745\pm 623$, respectively.
To obtain the true spectrum, a three-step unfolding procedure is used to
correct the raw spectrum. The procedure does not distinguish between $B\to
X_{s}\gamma$ and $B\to X_{d}\gamma$. Assuming the shape of the corresponding
photon energy spectra are equivalent, the contribution of $B\to X_{d}\gamma$
is subtracted using the ratio $R_{d/s}=(4.5\pm 0.3)\%$. Boost corrections,
obtained from MC simulation, are used to derive the measurements in the rest
frame of the $B$ meson. The two streams, MAIN and LT, are combined taking the
correlation into count.
The measured branching fraction in the $B$-meson rest frame is $BF(B\to
X_{s}\gamma)=(3.45\pm 0.15\pm 0.40)\times 10^{-4}$ for the photon energy range
from 1.7 GeV to 2.8 GeV. The most accurate measurement is given in the photon
energy range 2.0 GeV to 2.8 GeV, $BF(B\to X_{s}\gamma)=(3.02\pm 0.10\pm
0.11)\times 10^{-4}$. Here the errors are statistical and systematic,
respectively. The measured branching fractions are in agreement with the
latest theoretical calculation. The measured spectral moments can be used to
reduce the uncertainty on $|V_{ub}|$ [18, 19].
## 4 $b\rightarrow d\gamma$
Here we present the first significant observation of the $b\to d\gamma$
transition in the hadronic mass range
$M(X_{d})>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, used in the
determination of $|V_{td}/V_{ts}|$ via the ratio of inclusive widths.Inclusive
$B\to X_{s}\gamma$ and $B\to X_{d}\gamma$ rates are extrapolated from the
measurements of the partial decay rates of seven exclusive final states in the
hadronic mass ranges
$0.5<M(X_{d})<1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$1.0<M(X_{d})<2.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Here the $X_{d}$
includes $\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}$ $\pi^{0}$, $\pi^{+}\pi^{-}$
$\pi^{+}$, $\pi^{+}\pi^{-}$ $\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}$ $\pi^{+}$
$\pi^{0}$ and $\pi^{+}\eta$; while the $X_{s}$ includes $K^{+}$ $\pi^{-}$,
$K^{+}$ $\pi^{0}$, $K^{+}$ $\pi^{+}\pi^{-}$, $K^{+}$ $\pi^{-}$ $\pi^{0}$
$K^{+}$ $\pi^{-}$ $\pi^{+}\pi^{-}$, $K^{+}$ $\pi^{-}$ $\pi^{+}$ $\pi^{0}$ and
$K^{+}\eta$. We combine these measurements and make a model-dependent
extrapolation to higher hadronic mass to obtain an inclusive branching
fraction (${\cal B})$ for $b\to(s,d)\gamma$. These measurements use a sample
of $471\times 10^{6}$
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
pairs collected by the BABAR experiment.
The signal yields in the data for the combination of all seven decay modes are
determined from two-dimensional extended maximum likelihood fits to the
$\Delta E^{*}$ and $m_{\rm ES}$ distributions after all event selections,
where $\Delta E^{*}=E^{*}_{B}-E^{*}_{\rm beam}$, $E^{*}_{B}$ is the energy of
the $B$ meson candidate and $E^{*}_{\rm beam}$ is the beam energy, and
$\mbox{$m_{\rm ES}$}=\sqrt{E^{*2}_{\rm beam}-{\vec{p}}_{B}^{\;*2}}$,
${\vec{p}}_{B}^{\;*}$ is the momentum of the $B$ candidate. Table 1 gives the
signal yields, efficiencies and partial branching fractions.
Table 1: Signal yields ($N_{S}$), efficiencies ($\epsilon$). partial branching fractions ($BF$) and inclusive branching fractions ($\cal{B})$ for the measured decay modes. The first error is statistical the the second systematic (including error from extrapolation to missing decay modes, for the inclusive $\cal{B}$). | $M(X_{s})0.4-1.0$ | $M(X_{d})0.4-1.0$ | $M(X_{s})1.0-2.0$ | $M(X_{d})1.0-2.0$ (GeV/$c^{2}$)
---|---|---|---|---
$N_{S}$ | $804\pm 33$ | $35\pm 9$ | $990\pm 42$ | $56\pm 14$
$\epsilon$ | 4.5% | 3.1% | 1.6% | 1.9%
$BF(\times 10^{-6})$ | $18.9\pm 0.8\pm 0.8$ | $1.2\pm 0.3\pm 0.1$ | $65.7\pm 2.8\pm 5.9$ | $3.2\pm 0.8\pm 0.5$
${\cal{B}}(\times 10^{-6})$ | $38.3\pm 1.6\pm 1.5$ | $1.3\pm 0.3\pm 0.1$ | $192\pm 80\pm 45$ | $7.9\pm 2.0\pm 3.3$
$\frac{{\cal{B}}(b\to d\gamma)}{{\cal{B}}(b\to s\gamma)}$ | $0.0033\pm 0.009\pm 0.003$ | -
To obtain inclusive ${\cal B}(b\to s\gamma)$ and ${\cal B}(b\to d\gamma)$ we
need to correct the partial $\cal B$ values in Table 1 for the fractions of
missing final states. After correcting for the 50% of missing decay modes with
neutral kaons, the low mass $B\to X_{s}\gamma$ measurement is found to be
consistent with previous measurements of the rate for $B\to K^{*}\gamma$ [14].
For the low mass $B\to X_{d}\gamma$ region, we correct for the small amount of
non-reconstructed $\omega$ final states ($\omega\to\pi^{0}\gamma$ and others),
and find a partial branching fraction consistent with previous measurements of
$\mbox{\rm BR}(B\to(\rho,\omega)\gamma)$ [14]. We assume that non-resonant
decays do not contribute in this region.
In the high mass region, the missing fractions depend on the fragmentation of
the hadronic system and are expected to be different for $X_{d}$ and $X_{s}$.
We explore the uncertainty in the correction for missing modes by considering
several alternative models. The resulting missing fractions vary by up to 50%
relative to the nominal model. We therefore independently vary final states
with $\geq 5$ stable hadrons, or with $\geq 2\pi^{0}$ or $\eta$ mesons, by
$\pm$50%.
Combining the two mass regions, taking into account a partial cancellation of
the missing fraction errors in the ratio of $b\to d\gamma$ to $b\to s\gamma$ ,
we find ${\cal B}(b\to d\gamma)/{\cal B}(b\to s\gamma)=0.040\pm
0.009(stat.)\pm 0.010(syst.)$ in the mass range
$M(X)<2.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. For the unmeasured region
$M(X)>2.0$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ the differences between
$b\to s\gamma$ and $b\to d\gamma$ are small and almost completely cancel in
the ratio.
Conversion of the ratio of inclusive branching fractions to the ratio
$|V_{td}/V_{ts}|$ is done according to [11], which requires $\overline{\rho}$
and $\overline{\eta}$ as input. However, since these are partially determined
from previous measurements of $|V_{td}/V_{ts}|$, we instead re-express
$\overline{\rho}$ and $\overline{\eta}$ in terms of the independent CKM angle
$\beta$. This procedure yields a value of $|V_{td}/V_{ts}|=0.199\pm
0.022(stat.)\pm 0.024(syst.)\pm 0.002(th.)$ competitive with more model-
dependent determinations from the measurement of the exclusive modes
$B\to(\rho,\omega)\gamma$ and $B\to K^{*}\gamma$ [8, 9].
## 5 Summary
Here I summarized the experiment progresses on the inclusive $b\to s\gamma$
and $b\to d\gamma$ after the last CKM workshop. Belle measured the inclusive
branching fraction in the $B$-meson rest frame, $BF(B\to X_{s}\gamma)=(3.45\pm
0.15\pm 0.40)\times 10^{-4}$, for the photon energy range from 1.7 GeV to 2.8
GeV. BABAR presents the most precise direct CP asymmetry measurement to date,
its preliminary result is consistent with SM prediction. BABARalso measured
the ratio of $b\rightarrow d\gamma$ over $b\rightarrow s\gamma$ using seven
exclusive modes, providing the independent determination of $|V_{td}/V_{ts}|$.
ACKNOWLEDGEMENTS
I am grateful to the wonderful works from BABAR and Belle collaborations.
Thanks also to the organizers of the CKM2010 for all efforts in making this
venue successful.
## References
* [1] I. I. Y. Bigi et al., Phys. Lett. B 293 430 (1992).
* [2] M. Misiak et al., Phys. Rev. Lett. 98 022002 (2007).
* [3] T. Hurth et al., Ann.Rev.Nucl.Part.Sci.60:645,2010.
* [4] T. Hurth et al., Nucl. Phys. B704 56 (2005).
* [5] A.L. Kagan, M. Neubert, Eur. Phys. J. C7 (1999) 5-27.
* [6] C.W. Bauer et al, Phys. Rev. D70 094017 (2004); B.O. Lange et al.,Phys. Rev. D72 073006 (2005); C.W. Bauer et al., Phys. Rev. D67 054012 (2003); P. Gambino et al., JHEP, 10 058 (2007).
* [7] A. Abulencia et al., [CDF Collaboration], Phys. Rev. Lett. 97, 242003 (2006).
* [8] D. Mohapatra et al. [Belle Collaboration], Phys. Rev Lett. 96, 221601 (2006).
* [9] B. Aubert et al. [BABAR Collaboration], Phys. Rev Lett. 98, 151802 (2007).
* [10] P. Ball, G. Jones and R. Zwicky, Phys. Rev. D 75, 054004 (2007).
* [11] A. Ali, H. Asatrian & C. Greub, Phys. Lett. B 429, 87 (1998). K. Nakamura et al., Particle Data Group, J. Phys. G 37 075021 (2010).
* [12] B. Aubert et al., [BABAR Collaboration], Nucl. Instrum. Methods A 479, 1 (2002).
* [13] S. KuroKawa et al., Bell Collaboration, Nucl. Instrum. Methods A 479, 117 (2002).
* [14] K. Nakamura et al. (Particle Data Group). J. Phys. G 37, 075021 (2010).
* [15] B. Aubert et al.,Phys. Rev. Lett. 97 171803 (2006); B. Aubert et al.,Phys. Rev. D77 051103 (2008); T. E. Coan et al.,Phys. Rev. Lett. 86 5661 (2001).
* [16] A. Limosani et al. (Belle Collaboration). Phys. Rev. Lett. bf 103 241801 (2009);
* [17] P. Koppenburg et al. Phys. Rev. Lett. 93, 061803 (2004).
* [18] F. U. Bernlochner et al., arXiv:1102.0210.
* [19] Heavy Flavor Averaging Group, E. Barberio et al., arXiv:0704.3575 (hep-ex) (2007).
|
arxiv-papers
| 2011-02-09T18:15:31 |
2024-09-04T02:49:16.901703
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenfeng Wang",
"submitter": "Wang Wenfeng",
"url": "https://arxiv.org/abs/1102.1925"
}
|
1102.1930
|
# Quasiparticle interference in antiferromagnetic parent compounds of Fe-based
superconductors.
I.I. Mazin1, Simon A.J. Kimber2, and Dimitri N. Argyriou2 1Code 6393, Naval
Research Laboratory, Washington, DC 20375, USA 2 Helmholtz-Zentrum Berlin für
Materialien und Energie, Hahn-Meitner Platz 1, Berlin 14109, Germany
###### Abstract
Recently reported quasiparticle interference imaging in underdoped
Ca(Fe1-xCox)2As2 shows pronounced C2 asymmetry that is interpreted as an
indication of an electronic nematic phase with a unidirectional electron band,
dispersive predominantly along the $b$-axis of this orthorhombic material. On
the other hand, even more recent transport measurements on untwinned samples
show near isotropy of the resistivity in the $ab$ plane, with slightly larger
conductivity along $a$ (and not $b$). We show that in fact both sets of data
are consistent with the calculated ab initio Fermi surfaces, which has a
decisevly broken C4, and yet similar Fermi velocity in both directions. This
reconciles completely the apparent contradiction between the conclusions of
the STM and the transport experiments.
###### pacs:
74.20.Pq,74.25.Jb,74.70.Xa
The Fe-based superconductors present a new paradigm for high-$T_{C}$
superconductivity as here Cooper-pairs appear to emerge upon chemical doping
from a metallic ground state as opposed from a Mott insulator as found in the
celebrated High-$T_{C}$ cupratesLee . Despite this difference of parent ground
state of the Fe- and Cu-based superconductors, similarities lie in that in
both cases superconductivity emerges after the suppression of static ordered
magnetismI . Although band theory has correctly predicted the unusual
antiferromagnetic (AFM) order in the parent compounds of the Fe-based
superconductors, it consistently overestimates the tendency to magnetism and
underestimates the electronic mass, so there is no doubt that electronic
interactions can not be ignored in quantitative descriptions, and that they
play a different role compared to cuprates. The exact role of correlations,
especially once the parent phase of the Fe-superconductors is doped, has been
the focus of much debate and controversy.
An almost universal feature of the Fe-superconductors is that in the parent
phases, there is a tetragonal to orthorhombic structural phase transition that
is closely associated with the onset of antiferromagnetic orderReview . Upon
chemical doping $x$, the onset of the structural and magnetic transitions
($T_{S}$ and $T_{N}$ respectively) decrease with $x$ and superconductivity
emerges. The physical nature of the cross over from antiferromagnetic order to
superconductivity varies between specific materials. In some cases both
$T_{S}$ and $T_{N}$ coincide while in others $T_{S}$ is a few degrees higher
than $T_{N}$Review .
Band structure calculations have suggested that the AFM ordering is
accompanied by a strong restructuring of the Fermi surface, with the Fermi
surface area being reduced by roughly an order of magnitude. This has been
confirmed by optical and Hall measurements that register a drastic reduction
of the carrier concentration in the AFM stateOpt_Hall . The calculated AFM
Fermi surface consists of several small pockets, which are arranged in the
Brillouin zone in a way that strongly breaks the tetragonal symmetry, but each
of them is rather isotropic8 . This led to a prediction of small transport
anisotropy. An alternative point of view, that associates the orthorhombic
transition with orbital (charge) degrees of freedom, suggests a double
exchange (metallic) ferromagnetic interaction along one crystallographic
direction and a superexchange along the other direction. This picture is also
consistent with the observed AFM order and naturally suggests a metallic
conductivity along the ferromagnetic chains and a substantially reduced
conductivity in the other direction.
Recent experiments on detwinned single crystals support the former point of
view: they demonstrate a small anisotropy with the AFM direction being $more,$
not $less$ metallic. However, transport measurments are integrated probes, and
also involve possibly anisotropic scattering rate, therefore experiments
directly probing the topology of the Fermi surface in the AFM state are highly
desirable.
One such experiment has been recently performed by Chuang et al.1 . They have
reported quasiparticle interference (QPI) imaging of a lightly cobalt doped
sample of CaFe2As2 compound. They interpreted their result in terms of a
quasi-1D (“unidirectional”) electronic structure, metallic only along the FM,
consistent with above-mentioned orbital picture. On the the other hand, their
argumentation was rather indirect, based largly on the fact that directly
measured dispersion of the QPI maxima (which was indeed 1D) coincded with the
ARPES-measured band dispersion along the the same direction.
In this paper we show that in reality the data of Ref. 1, are consistent with
the calculated ab initio Fermi surfaces, and not with the implied in that work
1D bands. This reconciles completely the apparent contradiction between the
conclusions of Ref. 1, and the transport measurements on untwinned samples.
The reported STM examination shows a QPI pattern in the momentum space that
breaks completely the $C_{4}$ symmetry, the main features being two bright
spots along the $y$ (crystallographic $b)$ direction, with no counterparts
along $x$ (note that $y$ is the $ferromagnetic$ direction, and $x$ in the
antiferromagnetic one). Ref. 1, insists “that the scattering interference
modulations are strongly unidirectional, which should occur if the k-space
band supporting them is nematic”. However it should be kept in mind that this
occurs in that part of the phase diagram where the long-range
antiferromagnetic order is fully established, as reflected by the fact that
the lattice symmetry is orthorhombic, and the $C_{2}$ symmetry is already
completely broken. Indeed the size of the orthorhombic distortion is not
“minute”, as Ref. 1, posits, with $b/a$ $\sim 1$%, and is instead comparable
with distortions seen in various iron oxides systems. For instance, in the
Verwey transition the Fe-O bond dilation is $\sim$0.6% with Fe atoms in the
same tetrahedral symmetry as in the ferropnictide superconductors6 , and this
is usually considered to be a strong distortion. Similarly, in the
antiferromagnetic phase of FeO, where the cubic symmetry is completely broken,
the structural effect is also on the same order7 .
Since the sample under study is orthorhombic it is misleading to call its
electronic structure nematic, as the lattice orthorhombic distortion here is
substantial. Nematic phases are frequently found in organic matter. The
defining characteristic of these phases is orientational order in the absence
of long range positional order, resulting in distinctive uniaxial physical
properties. It has also been proposed that nematic order exists in some
electronic systems, and may even play a role in mediating high temperature
superconductivity4 . Borzi et al5 demonstrated the presence of another
interesting phase in Sr3Ru2O7 at millikelvin temperatures and high magnetic
fields, which has also been called nematic. In this case, the crystallographic
planes were shown to remain strictly tetragonal (withing 0.01%) with $C_{4}$
structural symmetry, while a pronounced $C_{2}$ asymmetry in electronic
properties was measured. This breaking of the electronic symmetry compared to
that of the underlying lattice is now conventionally referred to as electronic
nematicity (in fact, even in those cases one has to be careful to distinguish
between nematic physics and simply an unusually weak electron-lattice
coupling, but this goes beyond the scope of this paper, and in any event is
not a concern for Fe pnictides where this coupling is strong).
Since the tetragonal symmetry is decisively broken at the onset of the
magnetic order in this ferropnictide, it is clear that the symmetry of the
electronic structure defining the structural distortion is also completely
broken. What is more important is that while the observed QPI pattern does
violate the $C_{4}$ symmetry, it is clearly not one-dimensional, in the sense
that it varies equally strongly along $k_{x}$ and $k_{y}$ directions. Thus,
interpretation of the data in terms of a 1D electron band does not appear to
be possible. To understand this experiment one needs to start with a realistic
model for the electronic structure and actually calculate the QPI pattern.
Such calculation has recently been presented by Knolle $et$ $al$11 . They used
a weak-coupling theory that interprets tha antiferromagnetic state as
resulting from a spin-Peierls transition, with a correspondingly small
magnetic moment. Knolle $et$ $al$ have been able to describe qualitatively the
experimental data obtained by Chuang $et$ $al$ in the sense that their
calculated QPI pattern strongly breaks the $C_{2}$ symmetry, while the band
dispersion, on average, remains fairly isotropic in plane. Note that one
should not be looking for a quantitative interpretation, since the STM
experiment in question did not detect any Ca atoms on the surface, so the
sample surface is likely charged with up to 0.5 hole per Fe, and thus any bulk
calculation can only be applied to this experiment in a qualitative way.
Besides, it was recently shownSS that Fe pnictide systems feature surface
states quite different from the bulk that should undoubtedly affect the STM
spectra.
However, this result, as mentioned, has been obtained in a weak coupling
limit, corresponding to small magnetization, while in this system the ordered
magnetic moments are on the order of 1 $\mu_{B},$ and local moments even
larger2 ; local ; 12 . Not surprisingly, their Fermi surface is rather far
from that measure recently on untwinned samples by Wang et alDessau , while
the LDA Fermi surface reproduces it quite wellnote . Indeed, this is a known
problem in the weak coupling approach: while being physically justified for
the paramagnetic parts of the phase diagram, the Fe magnetism in the ordered
phases is driven by the strong local Hund rule coupling, and not by the Fermi
surface nesting, as assumed in the weak copling models.
Therefore we have calculated the QPI images for antiferromagnetic CaFe2As2
entirely from first principlesnote2, using the Local Density Approximation
(LDA) magnetic moment (somewhat larger that the experimental moment at zero
doping). We used the standard linear augmmented plane wave method as
implemented in the WIEN2k codeW2k . The corresponding Fermi surface is shown
in Fig. 1. We see that the magnetism has a drastic effect on the Fermiology,
and the resulting Fermi sirfaces are completely breaking the $C_{4}$ symmetry.
Apart from small quasi-2D tubular pockets, originating from Dirac cones, there
is one hole pocket around Z (0,0,$\pi/c$ or 2$\pi/a$,0,0) and two electron
pockets between Z and 0,$\pi/b$,$\pi/c$. It is immediately obvious that the QP
scattering between these pockets must exhibit strong interference for
scattering along $b,$ but not $a.$
Figure 1: (Color online) Calculated LDA Fermi surface for CaFe2As2 in the
antiferromagnetic state. Figure 2: (Color online) Quasiparticle interference
pattern (in arbitrary units) for zero bias and qz$\sim$0, calculated using the
same electronic structure as in Fig. 1 and Eq. 1.
Indeed, we have calculated the QPI function $Z,$ using the known expression
(Ref. 9, , Eq. S9)
$|Z(\mathbf{q},E^{\prime}\mathbf{)|}^{2}\mathbf{\propto}\int\frac{dE^{\prime}}{E-E^{\prime}}\sum_{\mathbf{k}}\delta(E-E_{\mathbf{k}})\delta(E^{\prime}-E_{\mathbf{k+q}}),$
(1)
where we assumed a constant inpurity scattering rate and a constant tunneling
matrix elements. This approximation is sufficient for a qualitative or
semiquantitative comparison. As explained above, given that the surface in the
experiment in question was charged compared to the bulk, a quantitative
comparison is meaningless.
A calculated pattern (there is some dependence on $q_{z}$ and on $E,$ but we
are interested in the qualitative features only) are shown in Fig.2. One can
see iimediately that, very similarly to the patterns obtained in Ref. 1, , two
sharp maxima appear at $\mathbf{q}=0,\pm\xi,0,$ where $\xi\sim\pi/4b$. The
origin of these QPI features is obvious from the Fermi surface (Fig. 1). Note
that these LDA calculations have no adjustable parameters, and yet are in
excellent qualitative agreement with the QPI images.
It is also worth noting that while the calculated Fermi surfaces completely
break the tetragonal symmetry, which is fully reflected in the QPI images, the
individual pockets are very three-dimensional, so that the calculated
conductivity is comparable for all three directions8 . While experimentally
there is up to a 20% $a/b$ charge transport anisotropy8 close to tetragonal
to orthorhombic phase boundary in CaFe2As2, it is much less than what would be
predicted for a quasi 1D electronic band, and of the opposite sign10 .
It may be worth at this point to explain at some length while a quantitative
comparison between a Fourier transform of a tunneling current map, and
theoretical calculations, whether ours or any other, is impossible at this
stage. Quasiparticle interference, as discussed in many papers, manifests
itself in tunneling in a very indirect way. In a sense, it is a multistage
process. First, a defect existing near the metal surface, is sdreened by the
conducting electrons. This creates Friedel oscillations in the real space.
This oscillations are formed by all electrons (mostly those near the Fermi
surface, but not only). In a multiband system, it includes electrons
originated from different atomic orbitals, such as $xy,$ $xz,$ $yz,$ $z^{2}$
and $x^{2}-y^{2}.$ As is well known in the theory of tunelling, the rate at
which electrons tunnel through vacuum depends drastically on their orbital
symmetry, especially on their parity (see, e.g., Ref. EPL ). Indeed tunelling
through a wide barrier mainly proceeds through electrons with zero momentum
projection onto the interface plane (such electrons have to travel the
shortest lengths in the subbarrier regime). If such electrons belong to an odd
2D representation (for d-electrons, all but $z^{2},$ if $z$ is the normal
direction), the tunneling rate is suppressed. This effect is well known in
spintronics, where it can drastically change the current spin polarization. On
the other hand, for a thin barrier the tunneling conductance depends on the
number of the conductivity channels, which is given by the density of states
(DOS) times normal velocity. In both cases, it is not just the density of
quasiparticles, as assumed in Eq. 1 (and in Ref. 11 ), but the DOS weighed by
a strongly k-dependent, unknown function.
Nothing is known about the nature of the scattering centers, producing the
above mentioned Friedel oscillations. In this particular experiment they may
be magnetic or nonmagnetic defects, twin domain boundaries, antiphase domain
boundaries, remaining surface Ca ions, and more. Some of these scatterers are
strongly anisotropic by nature, others are strongly dependent on the orbital
character. We have dropped the scattering matrix elements completely form our
consideration. Knolle $et$ $al$11 instead have chosen a specific model for
the scattering centers. We believe that without any knowledge about the actual
scattering centers in the system any QPI using a particular model is more
obscuring the actual physics, compared to the simplest constant matrix
elements approximation, rather than clarifying it.
Finally, there are several issues specific for this particular experiment: (1)
unknown, but strongly different from the bulk, charge state. As opposed to
Ba122, and Sr122, where 1/2 of the alkaline earth atoms stay on the surface,
providing charge neutrality, in Ca122 STM does not detect any Ca on the
surface, suggesting a strongly charged surface. A corollary of that is
appearence of a surface reconstruction (as indeed observed), of a surface
relaxation, and, importantly (since tunneling proceeds largely through the
surface states), of surface bands (as demonstrated, for instance, in Ref. sb .
While the above considerations preclude a quantitative comparison and
extracting quantitative analysis of the experiment in question, we see,
particularly when comparing our calculations with those of Knolle $et$ $al$11
, that the $C_{2}$ QPI structure observed in Ref. 1, is a very universal
consequence of the long-range stripe-type antiferromagnetic ordering. Indeed,
Knolle $et$ $al$ calculations were built upon a besically incorrect band
structure and fermi surfaces, an used a weak coupling nesting scenario for the
antiferromagnetism, while in reality the magnetism in pnictides is a strong
coupling phenomenos; yet, their calculations produced a “unidirectional” QPI
pattern just as well. Together with the strong-coupling LDA calculations, this
span a large range of possible models, indicating that the $C_{4}$ symmetry is
strongly broken in QPI images with simply by virtue of the long range AFM
order, whatever the the origin of this order.
Last but not least, we can also predict, from our calculations, that this
symmetry will be also broken, although the peaks are likely to be
substantially broaden, in the truly $nematic$ phase (see review 2 for a
discussion), that is to say, the phase between the long-range magnetic
transition and the structural orthorhombic transition.
## References
* (1) T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni,S. L. Bud’ko, G. S. Boebinger, P. C. Canfield, J. C. Davis, Science, 327, 181 (2010).
* (2) M.A. Tanatar, E. C. Blomberg, A. Kreyssig, M. G. Kim, N. Ni, A. Thaler, S. L. Bud’ko, P. C. Canfield, A. I. Goldman, I. I. Mazin, and R. Prozorov, Phys. Rev. B 81, 184508 (2010).
* (3) P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)
* (4) I. I. Mazin, Nature, 464, 183 (2010).
* (5) D.C. Johnston, Adv. in Phys., 59, 803 (2010)
* (6) W. Z. Hu, J. Dong, G. Li, Z. Li, P. Zheng, G. F. Chen, J. L. Luo, and N. L. Wang, Phys. Rev. Lett. 101, 257005 (2008); L. Fang, H. Luo, P. Cheng, Z. Wang, Y. Jia, G. Mu, B. Shen, I. I. Mazin, L. Shan, C. Ren, and H.-H. Wen, Phys. Rev. B 80, 140508 (R) (2009); F. Rullier-Albenque, D. Colson, A. Forget, and H. Alloul, Phys. Rev. Lett. 103, 057001 2009
* (7) J. P. Wright, J. P. Attfield, and P. G. Radaelli, Phys. Rev. Lett. 87, 266401 (2001).
* (8) D.G. Isaak, R. E. Cohen, M. J. Mehl, and D. J. Singh Phys. Rev. B 47, 7720 (1993).
* (9) S.A. Kivelson, E. Fradkin, and V.J. Emery, Nature 393, 550 (1998).
* (10) R. A. Borzi, R. A. Borzi, S. A. Grigera, J. Farrell, R. S. Perry, S. J. S. Lister, S. L. Lee, D. A. Tennant, Y. Maeno, A. P. Mackenzie, Science, 315, 214 (2006)
* (11) J. Knolle, I. Eremin, A. Akbari, R. Moessner, Phys. Rev. Lett. 104, 257001 (2010)
* (12) E. van Heumen, J. Vuorinen, K. Koepernik, F. Massee, Y. Huang, M. Shi, J. Klei, J. Goedkoop, M. Lindroos, J. van den Brink, M. S. Golden, arXiv:1009.3493 (unpublished).
* (13) I.I. Mazin and J. Schmalian, Physica C, 469, 614-627 (2009).
* (14) M.D. Johannes, I.I. Mazin, D.S. Parker, Phys. Rev. B 82, 024527 (2010)
* (15) M.D. Johannes and I.I. Mazin, Phys. Rev. B 79, 220510(R) (2009)
* (16) Q. Wang, Z. Sun, E. Rotenberg, F. Ronning, E.D. Bauer, H. Lin, R.S. Markiewicz, M. Lindroos, B. Barbiellini, A. Bansil, D.S. Dessau, arXiv:1009.0271 (unpublished).
* (17) Compared to Ref. Dessau , both $\beta$1 and $\beta$2 bands are present, their nontrivial crescent shape is reproduced, their size and location (around the Z point) are consistent with the calculation. Not that these $\beta$ pockets are mainly responsible for the QPI peak in our Fig. 2\. The flattish $\gamma$ pocket is also in excellent agreement with the calculation, although in the experiment it is split into $\gamma$3 and $\gamma$4 (probably an effect of the surface reconstruction). The claimed experimental bands ($\alpha$1, $\alpha$2, $\gamma$1 and $\gamma$2) along Z-X are quite messy. The calculations predict small pockets there located roughly where ARPES sees some bands. These are formed by the famous “Dirac cones”. The only feature that does not find any correspondence in the calculation is the long segment “$\gamma$2” stretched along $k_{y}$. This may be a surface state similar to those discovered in Ref. SS, (note that this band is drawn rather speculatively, the corresponding signal is really weak). This agreement is even more impressive given that the Fermi surface reproduced here was published 7 months ago (Ref. 8, ), well before any untwinned ARPES data became known. from 1D bands.
* (18) P. Blaha et al., computer code WIEN2K, Technische Universität Wien, Austria, 2001;
* (19) T. Hanaguri, T. Hanaguri, Y. Kohsaka, M. Ono, M. Maltseva, P. Coleman, I. Yamada, M. Azuma, M. Takano, K. Ohishi, and H. Takagi, Science 323, 923 (2009).
* (20) Measurement of charge transport in Co-doped BaFe2As2 reported by J-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon, Z. Islam, Y. Yamamoto, and I. R. Fisher, Science 329, 824 (2010), show a larger a/b anisotropy close to the orthorhombic transition, which we believe not to be representative of the CaFe2As2 upon which the STM measurements of Ref. 1, were taken. Regardless, the values of a factor of 2 anisotropy for doped samples in that study still remain too low to be associated with the large anisotropies expected from a unidirections band structure suggested in Ref. 1, , and are of the opposite sign.
* (21) I. I. Mazin, Europhys. Lett., 55, 404, 2001
* (22) E. van Heumen, J. Vuorinen, K. Koepernik, F. Massee, Y. Huang, M. Shi, J. Klei, J. Goedkoop, M.i Lindroos, J. van den Brink, M. S. Golden, http://arxiv.org/abs/1009.3493 (unpublished)
|
arxiv-papers
| 2011-02-09T18:32:54 |
2024-09-04T02:49:16.906185
|
{
"license": "Public Domain",
"authors": "I.I. Mazin, Simon A.J. Kimber, and Dimitri N. Argyriou",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/1102.1930"
}
|
1102.1933
|
# Magnetoelastic Coupling in the Phase Diagram of Ba1-xKxFe2As2
S. Avci Materials Science Division, Argonne National Laboratory, Argonne, IL
60439, USA O. Chmaissem Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA Department of Physics, Northern Illinois
University, DeKalb, IL 60115, USA E. A. Goremychkin S. Rosenkranz J.-P.
Castellan D. Y. Chung I. S. Todorov J. A. Schlueter H. Claus Materials
Science Division, Argonne National Laboratory, Argonne, IL 60439, USA M. G.
Kanatzidis Materials Science Division, Argonne National Laboratory, Argonne,
IL 60439, USA Department of Chemistry, Northwestern University, Evanston, IL
60208-3113, USA A. Daoud-Aladine D. Khalyavin ISIS Pulsed Neutron and Muon
Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United
Kingdom R. Osborn Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439, USA ROsborn@anl.gov
###### Abstract
We report a high resolution neutron diffraction investigation of the coupling
of structural and magnetic transitions in Ba1-xKxFe2As2. The tetragonal-
orthorhombic and antiferromagnetic transitions are suppressed with potassium-
doping, falling to zero at $x\lesssim 0.3$. However, unlike Ba(Fe1-xCox)2As2,
the two transitions are first-order and coincident over the entire phase
diagram, with a biquadratic coupling of the two order parameters. The phase
diagram is refined showing that the onset of superconductivity is at $x=0.133$
with all three phases coexisting until $x\gtrsim 0.24$.
Phase competition is an essential ingredient of superconductivity in the iron
arsenides and related compounds. The superconducting phase emerges when
antiferromagnetism has been suppressed either by hole or electron
dopingWen:2008p7965 ; Rotter:2008p11893 , applied
pressureTorikachvili:2008p11685 , or disorderWadati:2010p34986 , but the
nature of the phase boundary from antiferromagnetism to superconductivity is
not universal. In the so-called ‘1111’ system, LaFeAsO1-xFx, it has been
reported that there is a sharp first-order transition at $x\sim 0.045$ from
the antiferromagnetic phase to the superconducting phase, but there are
conflicting reports of phase coexistence in isostructural compounds containing
other rare earth ionsZhao:2008p13557 ; Drew:2009p19227 ; Sanna:2009p29496 . On
the other hand, in the ‘122’ systems with the parent compound BaFe2As2, both
hole and electron doping produce a gradual suppression of the
antiferromagnetism leading to the onset of superconductivity with some overlap
of the two phases.
Antiferromagnetism is also associated with a structural phase transition from
tetragonal to orthorhombic symmetry that occurs at a temperature either just
above or coincident with the onset of magnetic orderdelaCruz:2008p8095 ;
Pratt:2009p29514 ; Rotter:2008p11893 ; Fernandes:2010p32347 . It has been
proposed that the structural distortion involves a change in the orbital
configurationShimojima:2010p32066 producing an electronic nematic phase that
is either a precursor of, or is driven by, antiferromagnetic correlations.
This has led to considerable interest in the role of possible nematic
fluctuations in the normal phase of the nominally tetragonal
superconductorsFang:2008p8200 ; Chuang:2010p31929 ; Fernandes:2010p34989 .
Investigations of the interplay of magnetism, orbital order, and
superconductivity are therefore important in unravelling the origin of
unconventional superconductivity in these compounds.
When investigating the phase diagram of doped materials, it is a challenge to
separate effects due to chemical inhomogeneity from those due to intrinsic
phase separationMukhopadhyay:2009p31071 . In the ‘122’ compounds, comparisons
of bulk diffraction with local probes, such as NMR and $\mu$SR, have led to
two different conclusions for the electron-doped compounds, Ba(Fe1-xCox)2As2,
and the hole-doped compounds, Ba1-xKxFe2As2. In the case of electron-doping,
there is evidence of true phase coexistence in the underdoped compounds, with
a coupling of the antiferromagnetic and superconducting order
parametersPratt:2009p29514 ; Julien:2009p30867 . On the other hand, in the
case of hole-doping, local probes have indicated that there may be phase
separation i.e., the antiferromagnetic and superconducting phases occur in
separate mesoscopic domains within the crossover regionJulien:2009p30867 ;
Park:2009p20194 . A theoretical analysis of this phase competition concludes
that both phase diagrams can be consistent with a superconducting order
parameter of $s_{\pm}$ symmetryFernandes:2010p32347 ; Mazin:2008p11687 ,
whether there is true phase coexistence below a tetracritical point or phase
separation close to a first-order bicritical line.
In this paper, we report a reexamination of the phase diagram of
Ba1-xKxFe2As2, one of the most challenging of the iron pnictide
superconductors to synthesize. Discrepancies in the published phase diagrams,
with antiferromagnetism being suppressed at dopant concentrations varying from
$x=0.25$Johrendt:2009p28737 to 0.4Chen:2009p15806 , reflect the difficulty of
controlling the stoichiometry owing to the high volatility of potassium.
Because of this, most research has been conducted on Ba(Fe1-xCox)2As2 and
other transition-metal-doped compounds. Nevertheless, it is important to study
Ba1-xKxFe2As2, partly to investigate any assymmetry between electron and hole
doping in the phase diagram, but also because potassium substitution is
intrinsically cleaner, since there is no disorder in the superconducting
Fe2As2 planes themselves. By optimizing the homogeneity of potassium-doped
samples, we have been able to show that the superconducting phase starts at
$x=0.133\pm 0.002$ with evidence of phase coexistence, rather than phase
separation, up to $x\sim 0.24$. Using high-resolution neutron powder
diffraction, we observe that the structural and antiferromagnetic transition
temperatures are coincident and first-order over the range $0\leqslant
x\leqslant 0.24$, with biquadratic coupling at all $x$, a highly unusual form
of magnetoelastic coupling that has implications for the nature of the ordered
state.
Figure 1: Magnetization of Ba1-xKxFe2As2 for $x=0.15$, 0.175, 0.2, 0.21, 0.24,
and 0.3, measured using a SQUID magnetometer. Samples with $x<0.15$ showed no
superconductivity above a temperature of 0.3 K.
In order to overcome the high vapor pressure and reactivity of potassium metal
and the formation of more stable K/As binary by-products in the synthesis of
Ba1-xKxFe2As2, we examined all reasonable combinations of reaction parameters
(e.g., starting materials, reaction containers, temperature, and heating
times, etc.) before establishing the optimal conditions to produce high
quality homogeneous samples with sharp magnetic and superconducting
transitions. Samples were prepared using a stoichiometric mixture of binary
BaAs, KAs, and FeAs powders prepared in a N2-filled glove box. The mixtures
were loaded in alumina tubes and pre-heated at 500 - 800∘C. The pre-annealed
mixtures were then ground and loaded in niobium, which were then placed inside
quartz tubes. Heating the materials at 1000∘C for 24 to 48 h followed by
cooling to room temperature over 12 hours resulted in black polycrystalline
powders. Homogeneity of the samples was ensured by repeating this process
multiple times. X-ray diffraction, magnetic susceptibility, and ICP elemental
analysis were all used to control and monitor the progress of the sample
quality during and after synthesis. High quality samples were successfully
synthesized to cover the entire phase diagram of the Ba1-xKxFe2As2 series from
$0\leqslant x\leqslant 1.0$, with increments of $\Delta x=0.025$ from
$0.1\leqslant x\leqslant 0.25$, close to the superconducting phase boundary.
The neutron powder diffraction measurements were carried out on the High
Resolution Powder Diffractometer (HRPD) at the ISIS Pulsed Neutron Source,
whose resolution of 10-4 is extremely sensitive to inhomogeneous line-
broadening. The high quality of our samples was demonstrated by the constant
width of reflections in both undoped and doped compounds, e.g. FWHM$\sim
0.0037(3)$ Å for the (220) peak. SQUID (Quantum Design) magnetization
measurements were used to determine the superconducting transition
temperatures, Tc, and the Néel temperatures, TN. The peak in the first
derivative of the magnetization produced values of TN that were in good
agreement with the neutron diffraction measurements over the entire phase
diagram.
Figure 2: Variation of lattice constants $a$ and $b$ with temperature in
Ba1-xKxFe2As2for $x=0$, 0.1, 0.21 and 0.3.
The magnetization measurements showed no evidence of superconductivity above
300 mK for any of the samples with $0\leqslant x\leqslant 0.125$. Bulk
superconductivity is first seen at $x=0.15$ with a Tc of 4 K, and then
increases more rapidly with potassium concentration than previously seen,
peaking at 38 K for $x=0.4$ before it decreases again to 3 K for the end
member, KFe2As2. The magnetization of the underdoped compounds in Fig. 1 shows
that well-defined superconducting transitions are observed even when Tc is
varying rapidy with $x$, where the results would be most sensitive to
composition fluctuations. Using linear regression of the underdoped region, we
estimate the critical concentration for superconductivity to be $x=0.133\pm
0.002$. The complete phase diagram is discussed later.
Figure 3: Temperature dependence of unit cell volumes for Ba1-xKxFe2As2 with
$x=0$, 0.1, 0.21 and 0.3. The solid lines are fits below Ts to the quadratic
temperature dependence typical of conventional thermal expansion, which is
obeyed for $x=0.3$. The insets magnify the region close to Ts.
Rietveld refinements of Ba1-xKxFe2As2 confirmed the earlier reports of a
structural transition from the tetragonal ThCr2Si2-type structure of space
group $I4/mmm$ to the orthorhombic symmetry of the $\beta$-SrRh2As2-type
structure of space group $Fmmm$Rotter:2008p12981 . The structural transition
temperature, Ts decreases with potassium doping from 140 K at $x=0$ to 80 K at
$x=0.24$, and is completely suppressed at $x=0.3$, below the value reported by
Chen et alChen:2009p15806 , but in reasonable agreement with Johrendt et
alJohrendt:2009p28737 . Fig. 2 shows the temperature dependence of the
orthorhombic splitting for $x=0$, 0.1, and 0.21, and the absence of any
splitting for $x=0.3$.
The high $d$-spacing resolution on HRPD allows extremely small volume
anomalies to be observed at Ts for all values of $x$ (Fig. 3), a clear
signature that the structural phase transitions are first-order. Although the
equivalent transitions were also observed to be first-order in
SrFe2As2Jesche:2008p14067 and CaFe2As2Goldman:2008p13171 , a previous neutron
study concluded that the transition in BaFe2As2 was second-order with 3D
critical fluctuations above Ts and an anomalously small 2D critical exponent
of $\beta=0.103$ belowWilson:2009p23312 . They attributed this behavior to a
3D to 2D crossover in the immediate vicinity of the transition. However, they
did not rule out that the transition was weakly first-order and subsequent
x-ray and heat capacity measurements on a sample prepared with longer
annealing times identified a small first-order jumpRotundu:2010p35102 . The
HRPD data provide clear evidence that the transition is first-order and that
this characterizes the transition over the entire phase diagram.
The neutron powder diffraction data also reveals the presence of weak magnetic
Bragg reflections below the Néel temperatures for all the orthorhombic
samples. The peaks indexed as (121) and (103), with $d$-spacings of 2.45 Å and
3.43 Å respectively, are consistent with the previously identified spin
density wave orderRotter:2008p12981 . The magnetic structure was refined using
the symmetry of the magnetic space group $F_{c}mm^{\prime}m^{\prime}$. In this
model, the removal of time reversal symmetry from the last two mirror planes
(perpendicular to the $b$ and $c$ axes) resulted in an arrangement in which
the Fe magnetic moments are antiferromagnetically coupled along the $x$ and
$z$ direction but ferromagnetically coupled along the $y$ axis. The Fe
magnetic moment refines to 0.75(3) $\mu_{B}$ at 1.7 K for the parent BaFe2As2
material. A linearly decreasing magnetic moment was observed upon increasing
the K content from $x=0.1$ to $0.24$. No magnetic peaks are observed beyond
this limiting value.
Figure 4: Refined magnetic moments (blue circles) and orthorhombic order
parameter (red squares) as a function of temperature for x=0, 0.1, 0.15 and
0.21 samples. Solid lines are guide to the eye.
Fig. 4 shows a comparison of the temperature dependence of the refined
magnetic moment and the orthorhombic order parameter, defined by the
expression $\delta=(a-b)/(a+b)$, where $a$ and $b$ are the in-plane
orthorhombic lattice parameters. Although the statistical precision of the
magnetic order parameter, $M$, is much less than the orthorhombic order
parameter, it is clear that they have identical temperature dependences at all
compositions. The data are in clear contradiction to an earlier NMR report
that the two transitions are distinct at finite $x$Urbano:2010p34275 , so it
is worth emphasizing that the two order parameters are determined from the
same diffraction data, although their refined values are not coupled; the
magnetic moment is determined by the integrated intensity of the magnetic
Bragg peaks and the orthorhombicity is determined by the splittings of
structural Bragg peaks. We can therefore draw two unambiguous conclusions from
the data. First, the transition temperatures for both structural and
antiferromagnetic order are identical and, second, that the two order
parameters are strongly coupled.
When the two transitions are coincident, they are predicted to be first-order
in Ginzburg-Landau treatments of the magnetoelastic couplingBergman:1976p36257
; Barzykin:2009p21345 ; Cano:2010p34098 . Cano et al show that a linear-
quadratic magnetoelastic coupling generates an effective shear stress in the
magnetically ordered phaseCano:2010p34098 , driving a structural distortion if
Ts would fall below TN in the absence of coupling. When the uncoupled Ts is
greater than TN, as in Ba(Fe1-xCox)2As2 and other transition-metal-doped
compounds, the two transitions can be distinctCanfield:2010p34239 .
On the other hand, the fact that $M\propto\delta$ in Ba1-xKxFe2As2 implies a
biquadratic couplingWilson:2009p23312 . It is unclear why the linear-quadratic
term is not relevant but, as a consequence, neither order parameter can be
considered as secondary to the other. Wilson et al proposed that the unusual
coupling was due to the accidental proximity to a tetracritical
pointWilson:2009p23312 , but our data show that it persists over an extended
region of the phase diagram. This suggests that there may be a deeper
connection between the two order parameters, as proposed, for example, by
Cvetkovic and Tesanovic who postulate the existence of a ”mother” instability
driving a combined spin/charge/orbital-density-waveCvetkovic:2009p24266 .
Figure 5: Phase diagram of Ba1-xKxFe2As2 with the superconducting critical
temperatures (Tc) and Néel temperatures (TN), determined from magnetization
measurements, and the combined antiferromagnetic/orthorhombic (AF/O)
transition temperatures (Ts), determined from neutron diffraction. Solid lines
are guides to the eye. The phase boundary separating the mixed
AF/O-superconducting phase from the purely superconducting phase, shown by the
dotted line, is not known experimentally, but is illustrated with a positive
slope as discussed in the text.
The complete phase diagram, compiled from both the neutron diffraction and
magnetization data, is shown in Fig. 5, where we note that the error bars are
all smaller than the size of the points. The antiferromagnetic/orthorhombic
(AF/O) phase overlaps with superconductivity from $x=0.133$ to $\sim 0.3$. We
do not currently have any measurements between $0.24\leqslant x\leqslant 0.3$
so the precise nature of the mixed phase boundary still needs to be
determined. Nevertheless, we note that there is clear evidence at both
$x=0.21$ (Fig. 4) and 0.24 (not shown) that there is a slight depression of
the structural and magnetic order parameters on entering the superconducting
phase (Fig. 4 inset). Although the statistical accuracy of the magnetic order
parameter is not sufficient on its own, the orthorhombic order parameter is
measured with much higher precision and shows that the biquadratically-coupled
order parameters compete with the superconducting order parameter within the
superconducting phase. This issue was addressed by Fernandes et al where they
point out that such competition implies that the phase boundary within the
superconducting phase must have a positive slopeFernandes:2010p32347 . This
has been drawn schematically in Fig. 5, although the exact slope has not been
determined experimentally.
Finally, the coupling of the two order parameters throws light on the nature
of the phase coexistence. The magnetization data in Fig. 1 shows that we have
bulk superconductivity in all samples for $x\geqslant 0.15$, whereas the
neutron diffraction data shows that the decrease in the AF/O order parameter
on entering the superconducting phase is less than 5%. This is clearly
inconsistent with a mesoscopic phase separation, which would imply a
significant reduction in the volume fraction of the AF/O phase below Tc. Our
results are much more consistent with the microscopic phase coexistence
inferred in Ba(Fe1-xCox)2As2. The discrepancy with earlier NMR and $\mu$SR
data could be a result of improved control over chemical homogeneity in the
current samples, although we will have to repeat the local probe measurements
on our own samples to confirm this.
In summary, we have determined the phase diagram of Ba1-xKxFe2As2 using high
resolution neutron powder diffraction and SQUID magnetization measurements.
The magnetic and structural phase transitions at low doping are coincident and
first-order, with a strong biquadratic coupling of the magnetic structure to
the nuclear lattice. This unusual form of magnetoelastic coupling across an
extended region of the phase diagram, including within the superconducting
phase, may indicate that both order parameters are more strongly coupled than
implied by conventional theories of spin density waves and orbital
orderCvetkovic:2009p24266 .
We acknowledge valuable discussions with I. Paul and A. Cano. Work supported
by U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences, under contract No. DE-AC02-06CH11357.
## References
* (1) H.-H. Wen et al., EPL 82, 17009 (2008).
* (2) M. Rotter, M. Pangerl, M. Tegel, and D. Johrendt, Angewandte Chemie 47, 7949 (2008).
* (3) M. S. Torikachvili, S. Bud’ko, N. Ni, and P. Canfield, Phys Rev Lett 101, 057006 (2008).
* (4) H. Wadati, I. Elfimov, and G. A. Sawatzky, Phys Rev Lett 105, 157004 (2010).
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* (11) T. Shimojima et al., Phys Rev Lett 104, 057002 (2010).
* (12) C. Fang et al., Phys Rev B 77, 224509 (2008).
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* (15) S. Mukhopadhyay et al., New J Phys 11, 055002 (2009).
* (16) M.-H. Julien et al., EPL 87, 37001 (2009).
* (17) J. T. Park et al., Phys Rev Lett 102, 117006 (2009).
* (18) I. Mazin, D. Singh, M. Johannes, and M. H. Du, Phys Rev Lett 101, 057003 (2008).
* (19) D. Johrendt and R. Pöttgen, Physica C 469, 332 (2009).
* (20) H. Chen et al., EPL 85, 17006 (2009).
* (21) M. Rotter, M. Tegel, and D. Johrendt, Phys Rev Lett 101, 107006 (2008).
* (22) A. Jesche et al., Phys Rev B 78, 180504 (2008).
* (23) A. I. Goldman et al., Phys Rev B 78, 100506 (2008).
* (24) S. Wilson et al., Phys Rev B 79, 184519 (2009).
* (25) C. R. Rotundu et al., Phys Rev B 82, 144525 (2010).
* (26) R. R. Urbano et al., Phys Rev Lett 105, 107001 (2010).
* (27) D. Bergman and B. Halperin, Phys. Rev. B 13, 2145 (1976).
* (28) V. Barzykin and L. P. Gor’kov, Phys Rev B 79, 134510 (2009).
* (29) A. Cano, M. Civelli, I. Eremin, and I. Paul, Phys Rev B 82, 020408 (2010).
* (30) P. Canfield and S. Bud’ko, Annual Review of Condensed Matter Physics 1, 27 (2010).
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|
arxiv-papers
| 2011-02-09T18:34:00 |
2024-09-04T02:49:16.910753
|
{
"license": "Public Domain",
"authors": "Sevda Avci, Omar Chmaissem, Eugene A. Goremychkin, Stephan Rosenkranz,\n John-Paul Castellan, Duck-Young Chung, Ilya S. Todorov, John A. Schlueter,\n Helmut Claus, Mercouri G. Kanatzidis, Aziz Daoud-Aladine, Dmitry Khalyavin,\n Raymond Osborn",
"submitter": "Ray Osborn",
"url": "https://arxiv.org/abs/1102.1933"
}
|
1102.1998
|
# Fidelity of Physical Measurements
Thomas B. Bahder Aviation and Missile Research, Development, and Engineering
Center,
US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A.
###### Abstract
The fidelity (Shannon mutual information between measurements and physical
quantities) is proposed as a quantitative measure of the quality of physical
measurements. The fidelity does not depend on the true value of unknown
physical quantities (as does the Fisher information) and it allows for the
role of prior information in the measurement process. The fidelity is general
enough to allow a natural comparison of the quality of classical and quantum
measurements. As an example, the fidelity is used to compare the quality of
measurements made by a classical and a quantum Mach-Zehnder interferometer.
###### pacs:
PACS number 07.60.Ly, 03.75.Dg, 06.20.Dk, 07.07.Df
## I Introduction
In any experiment, we want to determine the value of one or more physical
quantities, say $x$, which can be one or more numbers. However, in most
experiments the actual quantities measured are not $x$ but some other
quantities $y$. We infer the quantities $x$ from the measured quantities $y$,
by using a conditional probability, $P(y|x)$, that specifies the statistical
relation between $x$ and $y$. The quantities $x$ and $y$ can be both discrete,
both continuous, or any combination thereof. The conditional probability,
$P(y|x)$, gives the probability of a measurement outcome $y$ given that the
value of the physical quantity is $x$. The probability $P(y|x)$ completely
characterizes the measurement process, independent whether the system is
classical or quantum. In general, the conditional probability also depends on
one or more additional quantities, $\xi$, thereby having the form,
$P(y|x,\xi)$, where the quantities $\xi$ label the state of the system at the
time of measurement and the type of measurement that is performed.
As an example, consider an experiment in which we want to measure the
wavelength, $\lambda$, of a light signal in units of nanometer. It may happen
that an experimenter has designed an apparatus to measure the wavelength
$\lambda$, however the apparatus actually measures an electric current, $I$,
in units of ampere. The conditional probability that describes this
measurement apparatus is $P_{1}(I|\lambda)$. Consider now a second
experimentalist that designs another apparatus to measure the wavelength
$\lambda$ using an alternate method. In this alternate method, the apparatus
may actually measure a voltage in units of volt. This second apparatus is
characterized by a conditional probability $P_{2}(V|\lambda)$, where $V$ is
the measured voltage. The question is: which apparatus is the best for
measuring the wavelength $\lambda$?
In this letter, I propose to answer this question by comparing the fidelity
(defined below) of each apparatus (measurement method). The apparatus with the
highest value of fidelity provides, on average, the best measurement of
$\lambda$. The fidelity also takes into account our prior information about
$\lambda$ through the prior probability distribution, $p(\lambda)$, see
discussion below.
Historically, the quality of measurements of a quantity $x$ has been discussed
in terms of parameter estimation Cramér (1958); Helstrom (1967, 1976); Holevo
(1982); Braunstein and Caves (1994); Braunstein et al. (1996); Barndorff-
Nielsen and Gill (2000); Barndorff-Nielsen et al. (2003). For example, for a
single quantity $x$, the Fisher information provides an upper bound on the
variance, $\sigma^{2}_{x}$, of an unbiased estimator, $\hat{x}$, of the
parameter $x$ through the Cramér-Rao inequalityCramér (1958); Cover and Thomas
(2006)
$\sigma_{x}^{2}\geq\frac{1}{{F_{c}(x)}}$ (1)
where $F_{c}(x)$ is the classical Fisher information, defined by
$F_{c}(x)=\sum\limits_{y}{\frac{1}{{P(y|x)}}\,\left[{\frac{{\partial
P(y|x)}}{{\partial x}}}\right]^{2}}$ (2)
The quantity $F_{c}(x)$ depends on the type of measurement that is performed.
For a quantum system, by maximizing over all possible measurement types,
Braunstein and Caves Braunstein and Caves (1994); Braunstein et al. (1996)
have shown that a quantum Fisher information exists, such that $F_{q}(x)\geq
F_{c}(x)$, and therefore an improved lower bound for $\sigma_{x}^{2}$ can be
obtained by replacing $F_{c}(x)$ with $F_{q}(x)$ in Eq. (1). The quantum
Fisher information, $F_{q}(x)$, is defined by
$F_{q}\left(x\right)={\rm tr}\left(\hat{\rho}_{x}\hat{\Lambda}_{x}^{2}\right)$
(3)
where $\hat{\Lambda}_{x}$ is the symmetric logarithmic derivative (SLD) that
is implicitly given by Helstrom (1967, 1976); Holevo (1982); Braunstein and
Caves (1994); Braunstein et al. (1996); Barndorff-Nielsen and Gill (2000);
Barndorff-Nielsen et al. (2003).
$\frac{{\partial\hat{\rho}_{x}}}{{\partial
x}}=\frac{1}{2}\left[{\hat{\Lambda}_{x}\,\hat{\rho}_{x}+\hat{\rho}_{x}\hat{\Lambda}_{x}}\right]$
(4)
For a quantum measurement, the conditional probabilities can be obtained from
$P(y|x)={\rm tr}\left(\hat{\rho}_{x}\,\hat{\Pi}\left(y\right)\right)$ (5)
where the state is specified by the density matrix, $\hat{\rho}_{x}$, and the
measurements are defined by the positive-operator valued measure (POVM),
$\hat{\Pi}\left(y\right)$. For classical measurements, the conditional
probabilities $P(y|x)$ can be obtained from a model of the classical
measurement process, which may include phenomenological parameters
characterizing noise in the measurements.
The above description of the quality of measurements based on Fisher
information is not satisfactory for two reasons. First, the classical or
quantum Fisher information, $F_{c}(x)$ or $F_{q}(x)$, may depend on the true
value of the parameter $x$ if dissipation is present in the quantum system
Braunstein et al. (1996); Olivares and Paris (2009); Gaiba and Paris (2009);
Bahder (2011a). Of course, the true value of the parameter $x$ is unknown.
Second, the Fisher information does not take into account the prior
information of the observer. As an example, consider a child and an adult
reading the same printed page of a book. Each of them may obtain a certain
amount of information from the same printed page. However, the child may
obtain less information from the printed page than the adult because the adult
has more prior experience in the subject. The Fisher information has no
provision to take into account the observer’s prior information.
## II Shannon Mutual Information as Fidelity of Measurement
I propose to use the fidelity as a quantitative measure of the quality of any
physical experiment. The fidelity is the Shannon mutual information Shannon
(1948); Cover and Thomas (2006) between the measurements, $y$, and the
physical quantities, $x$, defined by
$\small
H[\xi]=\sum_{y}\sum_{x}P(y|x,\xi)P(x)\log_{2}\left[\frac{P(y|x,\xi)}{\sum_{x^{\prime}}P(y|x^{\prime},\xi)P(x^{\prime})}\right]$
(6)
where $P(y|x,\xi)$ is the conditional probability density of measuring $y$,
given that the true value of the quantity is $x$, and given the state of the
system and measurement type are specified by one or more quantities $\xi$. The
fidelity gives the information (in bits) transferred between the quantity of
interest, $x$, and the measurement result, $y$, for each use of the
measurement apparatus. The conditional probabilities $P(y|x)$ can be obtained
from a model (see below), or, from statistics of repeated experiments.
Using the fidelity to characterize the quality of a measurement apparatus does
not suffer from the two objections to using the Fisher information, which were
described above. The fidelity does not depend on the true value of the
quantity $x$, because it is an average over all values of $x$ and $y$, using
the conditional probabilities $P(y|x)$. Furthermore, the fidelity depends on
our prior knowledge about $x$ through the prior probability distribution
$P(x)$. (If either $x$ or $y$, or both, are continuous quantities, then the
respective sums in Eq. (6) are to be replaced by integrals.)
The fidelity in Eq.(6) is a completely general way to characterize the quality
of any classical or quantum measurement experiment. The classical or quantum
measurement apparatus is a channel through which information flows from the
phenomena, which is characterized by the value of the physical quantity $x$,
to the measurements $y$.
The fidelity gives the quality of the inference about the value of $x$ from
the measurement of $y$. However the fidelity does not give an estimate of the
value of $x$. The value of the quantity $x$ can be inferred from the
probability distribution for $x$, using Bayes’ rule
$P(x|y,\xi)=\frac{{P(y|x,\xi)P(x,\xi)}}{{\sum\limits_{x}{P(y|x,\xi)P(x,\xi)}}}$
(7)
where I have included the dependence on other parameters $\xi$. The value of
the quantity $x$ can be estimated, for example, by taking the mean of the
distribution given by Eq. (7). Once we have made an estimate of the value of
$x$, we can improve on this estimate by making recursive measurements. The
distribution for the value of $x$, $P(x|y,\xi)$ given in Eq. (7), can be used
as our new prior probability distribution, setting $P(x)=P(x|y,\xi)$ in
Eq.(6). Furthermore, the fidelity can be maximized with respect to $\xi$ for
the next measurement, using our current state of knowledge, represented by
$P(x|y,\xi)$. In this way, we can optimize a measurement device (classical or
quantum) to give the best possible measurement in the next measurement cycle.
The fidelity has already been used to discuss the quality of phase
determination in quantum interferometers Bahder and Lopata (2006a, b); Simon
et al. (2008); Bahder (2011a) and to discuss the sensitivity to rotation of
classical Sagnac gyroscopes Bahder (2011b).
## III Comparison of Classical and Quantum Measurements
The fidelity can be used to determine which experiment (apparatus) provides a
better measurement of a given physical quantity. As an example, I compare the
fidelity of a quantum and a classical measuring device. Specifically, I
compare a classical and a quantum Mach-Zehnder (M-Z) interferometer, each of
which can be used to determine the phase $\phi$ in one arm of the
interferometer. For the classical M-Z interferometer, the direct measurement
is the energy in each of the output ports, $E_{c}$ and $E_{d}$, which are
continous variables, see discussion below. For a quantum M-Z interferometer,
the direct measurement is the integer number of photons in the output ports,
$n_{c}$ and $n_{d}$. So in the notation above, the phase $\phi$ plays the role
of $x$ and the measurements $y$ are the pair of numbers, $(E_{c},E_{d})$, see
Eq. (6).
Consider a quantum Mach-Zehnder interferometer with input ports labeled “a”
and “b” and output ports labeled “c” and “d”. Assume we input the state
$|\alpha\rangle_{a}\otimes|0\rangle_{b}=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{(n!)^{1/2}}|n\rangle_{a}\otimes|0\rangle_{b}$
(8)
which consists of a coherent state input into port “a” and vacuum input into
port “b”, where $a$ and $b$ label the modes, $|n\rangle$ is a Fock state of
$n$ photons, and the complex parameter $\alpha$ specifies the average photon
number and photon number variance,
$|\alpha|^{2}=\langle\hat{n}\rangle=\langle(\Delta\hat{n})^{2}\rangle$. The
probability that $n_{c}$ and $n_{d}$ photons are output in ports “c” and “d”,
respectively, is Bahder and Lopata (2006a)
$\small
P\left(\left.n_{c},n_{d}\right|\phi,\alpha\right)=\frac{e^{-|\alpha|^{2}}}{n_{c}!n_{d}!}|\alpha|^{2\left(n_{c}+n_{d}\right)}\sin^{2n_{c}}\left(\frac{\phi}{2}\right)\cos^{2n_{c}}\left(\frac{\phi}{2}\right)$
(9)
where $\phi$ is the phase shift in one arm of the interferometer. The average
energy of this coherent state is $\langle E\rangle=\langle
n\rangle\hbar\omega=\hbar\omega|\alpha|^{2}$ with energy spread $\Delta
E=\hbar\omega|\alpha|=\sqrt{\hbar\omega}\,\langle E\rangle^{1/2}$. The
discrete energies output in port “c” and “d” are $E_{c}=n_{c}\hbar\omega$ and
$E_{d}=n_{d}\hbar\omega$, respectively. The joint probability density for
measuring energy $E_{c}$ and $E_{d}$ output in ports “c” and “d”,
respectively, is given by
$p(E_{c},E_{d}|\phi,\bar{E})=\frac{P\left(\left.n_{c,}n_{d}\right|\phi,\alpha\right)}{(\hbar\omega)^{2}}$
(10)
where $P\left(\left.n_{c,}n_{d}\right|\phi,\alpha\right)$ is given in Eq. (9)
and I use the notation for the average energy $\bar{E}=\langle E\rangle$.
Using Eq. (9) in Eq. (6), and assuming no prior knowledge about phase, taking
$p(\phi)=1/(2\pi)$, I find the fidelity for the quantum M-Z interferometer
with coherent state input to be:
$H_{\rm
coh}(|\alpha|^{2})=\frac{e^{-|\alpha|^{2}}}{2\pi}\sum_{n_{c}=0}^{\infty}\sum_{n_{d}=0}^{\infty}\frac{|\alpha|^{2\left(n_{c}+n_{d}\right)}}{n_{c}!n_{d}!}\int_{-\pi}^{+\pi}d\phi\,\sin^{2n_{c}}\left(\frac{\phi}{2}\right)\cos^{2n_{d}}\left(\frac{\phi}{2}\right)\log_{2}\left[\frac{\pi\left(n_{c}+n_{d}\right)!}{\Gamma\left(n_{c}+\frac{1}{2}\right)\Gamma\left(n_{d}+\frac{1}{2}\right)}\sin^{2n_{c}}\left(\frac{\phi}{2}\right)\cos^{2n_{d}}\left(\frac{\phi}{2}\right)\right]$
(11)
which is only a function of the parameter $|\alpha|^{2}$.
Now consider a classical M-Z interferometer with a finite-duration pulse of
monochromatic light of energy $E_{n}$ input into port “a” and vacuum input
into port “b”. I assume that the pulse has a duration in time sufficiently
long that I can describe the pulse as having a single frequency and energy
$E_{n}$. We want to determine the phase $\phi$, however, the direct
measurement consists of the energies in the output ports, $E_{c}$ and $E_{d}$.
The classical M-Z interferometer is defined by energies $E_{c}$ and $E_{d}$
output in ports “c” and “d”, respectively
$E_{c}=E_{n}\sin^{2}\left(\frac{\phi}{2}\right),\hskip
28.90755ptE_{d}=E_{n}\cos^{2}\left(\frac{\phi}{2}\right)$ (12)
where $\phi$ is the phase in one arm of the M-Z interferometer. In order to
compute the fidelity of the classical M-Z interferometer, we need to define a
classical measurement model. In the quantum interferometer described above in
Eqs. (8)—(11), the input state had a spread in energy $\Delta E$ due to photon
number fluctuations. For the case of the classical M-Z interferometer, I
assume a probability distribution, $P_{a}\left(E_{n}\right)$, of closely-
spaced discrete input energies, $E_{n}=n\,\delta E$, where
$n=0,1,2,\cdots,N_{E}$, where $N_{E}$ is the number of energies in the energy
grid, and $\delta E$ is an arbitrary discrete energy step, which can be taken
to zero later. I also take the phase as a discrete set of $2N_{\phi}$ possible
values, $\phi_{k}=\pi k/N_{\phi}$, for $k=\\{0,\pm 1,\pm 2,\cdots,\pm
N_{\phi}-1,N_{\phi}\\}$, where $\Delta\phi=\phi_{k+1}-\phi_{k}$ is the uniform
grid spacing of phase. Therefore, I take the conditional probability for a
classical measurement outcome, $(E_{c},E_{d})$, to be
$P\left(E_{c},\left.E_{d}\right|\phi_{k}\right)=\sum_{E_{n}}P\left(E_{c},\left.E_{d}\right|\phi_{k},E_{n}\right)\,P_{a}\left(E_{n}\right)$
(13)
where $E_{c}=E_{c}(n,k)=n\,\delta E\sin^{2}(\phi_{k}/2)$ and
$E_{d}=E_{d}(n,k)=n\,\delta E\cos^{2}(\phi_{k}/2)$, are discrete energies
measured in output ports “c” and “d”, respectively, given the energy
$E_{n}=n\,\delta E$ is input in port “a” and vacuum is input in port “b”.
Equation (13) is a special case of the identity between conditional
probabilities, $P(B|C)=\sum_{A}P(B|A,C)\,P(A|C)$. Note that the discrete
energies $E_{c}(n,k)$ and $E_{d}(n,k)$ depend on a two indices, $n$ and $k$,
each of which have different ranges, depending on the energy grid and the
phase grid. The probability distribution for input energies is normalized,
$\sum_{n=0}^{\infty}P_{a}\left(E_{n}\right)=1$. The conditional probability of
measuring discrete energies $E_{c}$ and $E_{d}$ in the output ports, given
monochromatic input energy $E_{m}=m\,\delta E$ and phase $\phi_{l}$, can be
written as a product of Krönecker delta functions:
$P\left(E_{c}(n,k),E_{d}(n^{\prime},k^{\prime})|\,\phi_{l},E_{m}\right)=\delta_{n,m}\,\delta_{k,l},\,\delta_{n^{\prime},m}\,\delta_{k^{\prime},l}$
(14)
where where $n,n^{\prime},m=0,1,2,\cdots,N_{E}$ and $k,k^{\prime},l=\\{0,\pm
1,\pm 2,\cdots,\pm N_{\phi}-1,N_{\phi}\\}$. Using Bayes’ rule in Eq. (7), the
phase probability distribution is given by
$P\left(\phi_{l}|E_{c}(n,k),E_{d}(n,k)\right)=\delta_{k,l}$ (15)
In the limit $\Delta\phi\to 0$ of a continuous phase variable, the phase
probability density,
$p\left(\phi\left|E_{c}\right.,E_{d}\right)\equiv\left.P\left(\phi_{l}|E_{c}(n,k),E_{d}(n,k)\right)\right/\Delta\phi$,
is given by the right side of Eq. (21), which gives two values of phase for
each classical measurement outcome $(E_{c},E_{d})$.
Using Eq. (13) and (14) in Eq. (6), I find the fidelity of this classical M-Z
interferometer to be
$H=\frac{2\pi}{\Delta\phi}\log_{2}\left(\frac{2\pi}{\Delta\phi}\right)$ (16)
where $2\pi/\Delta\phi$ is the number of phase points in the range
$-\pi<\phi\leq+\pi$. The fidelity in Eq. (16) is independent of the input
energy probability distribution $P_{a}\left(E_{n}\right)$. As
$\Delta\phi\rightarrow 0$, the fidelity in Eq. (16) diverges because there are
no fluctuations or energy measurement errors built into the classical
measurement model in Eq. (13) and (14). This classical measurement model
assumes that energy measurements are arbitrarily precise. In reality, there is
noise in energy measurements that limits the phase resolution, leading to a
non-zero value $\Delta\phi$ that makes the fidelity $H$ finite.
An improvement over the classical measurement model in Eq. (13) and (14) can
be made by assuming that the probability of classical energy measurement is
not sharp but instead has some statistical error of order $\Delta$ due to
unmodelled physical processes. The value of the phenomenological parameter
$\Delta$ can be obtained from experiments by taking $\Delta$ equal to the
standard deviation of classical energy measurements in the M-Z interferometer.
As an improved classical measurement model, I take the probability density for
measuring energy $E_{c}$ and $E_{d}$ in output ports “c” and “d” respectively,
as
Figure 1: The fidelity is plotted (in units of bits) for quantum (red line)
and classical (blue line) interferometers, $H_{\rm coh}(\eta)$ and $H_{\rm
class}(\eta)$, respectively, vs. $\eta$, where $\eta$ is the dimensionless
energy in units of photon number. No prior knowledge of phase was assumed,
taking $p(\phi)=1/(2\pi)$. The quantum M-Z interferometer has a higher
fidelity than the classical M-Z interferometer, showing that the quantum
interferometer gives more information on phase $\phi$ than the classical
interferometer.
$\displaystyle p\left(\left.E_{c}\right|\phi,E\right)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}p\left(\left.E_{c}\right|\phi,E,\varepsilon_{c}\right)p_{\Delta}\left(\varepsilon_{c}\right)d\varepsilon_{c}$
$\displaystyle p\left(\left.E_{d}\right|\phi,E\right)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}p\left(\left.E_{d}\right|\phi,E,\varepsilon_{d}\right)p_{\Delta}\left(\varepsilon_{d}\right)d\varepsilon_{d}$
(17)
where $p\left(\left.E_{c}\right|\phi,E,\varepsilon_{c}\right)$ is the
conditional probability that energy $E_{c}$ is measured in port “c”, given the
phase $\phi$, the input energy $E$ and the error in the classical measurement
$\varepsilon_{c}$, with analogous definitions for port “d”. Here,
$p_{\Delta}(\varepsilon)$ is a normal probability distribution that introduces
errors into the measurement of energies $E_{c}$ and $E_{d}$:
$p_{\Delta}(\varepsilon)=\frac{1}{\sqrt{2\pi}\Delta}\exp\left({-\frac{\varepsilon^{2}}{2\Delta^{2}}}\right)$
(18)
Equation (17) is the result of an identity for conditional probabilities. In
view of the classical output relations in Eq. (12), I take the conditional
probability density for measuring the output energies $E_{c}$ and $E_{d}$ to
be defined in terms of Dirac $\delta$ functions:
$\displaystyle p\left(\left.E_{c}\right|\phi,E,\varepsilon_{c}\right)$
$\displaystyle=$
$\displaystyle\delta(E_{c}-E\sin^{2}\left(\frac{\phi}{2}\right)-\varepsilon_{c})$
$\displaystyle p\left(\left.E_{d}\right|\phi,E,\varepsilon_{d}\right)$
$\displaystyle=$
$\displaystyle\delta(E_{d}-E\cos^{2}\left(\frac{\phi}{2}\right)-\varepsilon_{d})$
(19)
Taking the product of the distributions in Eq. (17) leads to the conditional
joint probability density for a classical measurement outcome,
$(E_{c},E_{d})$, given the phase is $\phi$ and monochromatic energy $E$ is
input:
$\displaystyle
p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)=p\left(\left.E_{c}\right|\phi,E\right)\,p\left(\left.E_{d}\right|\phi,E\right)$
$\displaystyle=$ $\displaystyle
p_{\Delta}\left(E_{c}-E\sin^{2}\left(\frac{\phi}{2}\right)\right)p_{\Delta}\left(E_{d}-E\cos^{2}\left(\frac{\phi}{2}\right)\right)$
(20)
In Eq. (20), the energies $E_{c}$ and $E_{d}$ are continuous variables, as is
the phase $\phi$. From Bayes’ rule in Eq. (7), assuming no prior information
on phase, therefore taking $p(\phi)=1/2\pi$, the phase probability density is
$p\left(\phi\left|E_{c}\right.,E_{d},E,\Delta\right)=\frac{p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)}{\int_{-\pi}^{+\pi}p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)\,d\phi}$
(21)
The phase probability density,
$p\left(\phi\left|E_{c}\right.,E_{d},E,\Delta\right)$, has two peaks, and as
$\Delta\rightarrow 0$ it approaches the sum of two $\delta$ functions:
$\displaystyle p\left(\phi\left|E_{c}\right.,E_{d},E,\Delta\right)$
$\displaystyle\to$
$\displaystyle\frac{1}{2}\left[\,\,\delta\left(\phi-2\arctan\sqrt{\frac{E_{c}}{E_{d}}}\right)\right.$
(22)
$\displaystyle\left.+\delta\left(\phi+2\arctan\sqrt{\frac{E_{c}}{E_{d}}}\right)\,\,\right]$
Trivially changing the sums to integrals in the definition of fidelity in Eq.
(6), and using Eq. (20), leads to the fidelity for a classical M-Z
interferometer:
$\displaystyle H_{\rm
class}(E,\Delta)=\int_{-\infty}^{+\infty}dE_{c}\int_{-\infty}^{+\infty}dE_{d}\int_{-\pi}^{+\pi}d\phi\times$
$\displaystyle
p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)p(\phi)\log_{2}\left(\frac{p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)}{p\left(E_{c},E_{d},E,\Delta\right)}\right)\,\,\,\,$
(23)
where
$p\left(E_{c},E_{d},E,\Delta\right)=\int_{-\pi}^{+\pi}p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)p(\phi)d\phi$
(24)
and $p\left(E_{c},\left.E_{d}\right|\phi,E,\Delta\right)$ is given by Eq.
(20), and $p(\phi)$ is the probability representing our prior knowledge about
$\phi$. Equation (23) gives the fidelity of a classical M-Z interferometer
with monochromatic input energy $E$ and errors in energy measurements of order
$\Delta$. The errors, $\varepsilon_{c}$ and $\varepsilon_{d}$, in energy
measurements, $E_{c}$ and $E_{d}$, can be imagined as due to unmodelled noise
(e.g., shot noise) in the measurements. As $\Delta\rightarrow 0$, the
measurements have no noise (errors) and the fidelity $H_{\rm
class}(E,\Delta)\rightarrow\infty$, compare with Eq. (16) for the case
$\Delta\phi\to 0$.
The fidelity for the classical interferometer, $H_{\rm class}(E,\Delta)$ in
Eq. (23), depends on two parameters, $E$ and $\Delta$, and on our knowledge of
$\phi$ given by the prior phase distribution, $p(\phi)$. The fidelity for the
quantum interferometer, $H_{\rm coh}(|\alpha|^{2})$ in Eq. (11), depends on
only one parameter, $\eta$, where we assumed no prior knowledge about phase by
taking $p(\phi)=1/(2\pi)$. A direct comparison of the fidelity of the quantum
and classical M-Z interferometers can be made by assuming in the classical
case in Eq. (23) no prior knowledge of the phase taking $p(\phi)=1/(2\pi)$,
and taking the energy $E=\hbar\omega|\alpha|^{2}\equiv\hbar\omega\eta$ and
energy width $\Delta=\sqrt{\hbar\omega E}\equiv\hbar\omega\sqrt{\eta}$, which
gives the same energy width for the measurements of the classical M-Z
interferometer as for the coherent input state of the quantum M-Z
interferometer. For the classical M-Z interferometer, $\eta$ is the
monochromatic input energy in units of photon energy, $\hbar\omega$. For the
quantum interferometer, $\eta$ is the average energy $\langle E\rangle$ of the
input coherent state in units of photon energy, $\hbar\omega$. With this
parametrization, the fidelity of the classical M-Z interferometer, $H_{\rm
class}(\hbar\omega\eta,\hbar\omega\sqrt{\eta})$, depends only on $\eta$, and
can be directly compared to the fidelity of the quantum interferometer,
$H_{\rm coh}(\eta)$, see Fig. 1. It is clear that, for a single use of the
interferometer, the quantum measurement (apparatus) has a higher fidelity
(provides more bits of information about the phase) than the classical
measurement.
## IV Conclusion
Two objections have been raised against using the Fisher information as a
measure of the quality of measurements. First, the Fisher information may
depend on the unknown physical quantity (parameter to be determined), which
may occur when dissipation is present. Second, the Fisher information does not
take into account prior information about the parameter. Consequently, I
proposed the use of fidelity (Shannon mutual information between measurements
and physical quantities) in Eq. (6) as a quantitative measure of the quality
of physical measurements. The fidelity does not depend on the value of the
unknown physical quantity because it is an average over all probability
distributions of that quantity. Also, the fidelity takes into account an
observer’s prior information through the prior probability distribution,
$P(x)$, see Eq. (6). The dependence on prior information also allows us to
update recursively our information during repeated experiments. Also, the
fidelity can be maximized with respect to (classical or quantum) measurements,
parameters in the experiment, and with respect to (classical or quantum) input
states. Finally, the fidelity is general enough to quantitatively compare the
quality of classical and quantum measurements, or to compare two different
experiments that attempt to determine the same physical quantity. As an
example of this, I have considered a quantum M-Z interferometer with a
coherent state input into one port, and I have compared it with a classical
interferometer with phenomenological error in measuring the energy in the
output ports. For the range of parameters considered, see Fig. 1, the quantum
M-Z interferometer has higher fidelity than the classical interferometer,
indicating that, for each measurement the quantum M-Z interferometer provides
more bits of information on the phase than the classical M-Z interferometer.
The fidelity allows a quantitative comparison of the quality of these two
types of measurements. Finally, non-ideal aspects of experiments, such as non-
deterministic state creation, absorption, and errors in measurements (e.g.,
photon counting errors or energy measurement errors) can be included in the
fidelity by using the formalism that was developed in Ref. Bahder (2011a).
## References
* Cramér (1958) H. Cramér, _Mathematical Methods of Statistics_ (Princeton University Press, Princeton, 1958), eighth printing.
* Helstrom (1967) C. W. Helstrom, Phys. Lett. A 25, 101 (1967).
* Helstrom (1976) C. W. Helstrom, _Quantum Detection and Estimation Theory_ (Academic Press, New York, 1976).
* Holevo (1982) A. S. Holevo, _Probabilistic and Statistical Aspects of Quantum Theory_ (North-Holland, Amsterdam, 1982).
* Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
* Braunstein et al. (1996) S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. of Phys. 247, 135 (1996).
* Barndorff-Nielsen and Gill (2000) O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen. 33, 4481 (2000).
* Barndorff-Nielsen et al. (2003) O. E. Barndorff-Nielsen, R. D. Gill, and P. E. Jupp, J. Roy. Stat. Soc. B 65, 775 (2003), URL http://arxiv.org/abs/quant-ph/0307191.
* Cover and Thomas (2006) T. M. Cover and J. A. Thomas, _Elements of Information Theory_ (J. Wiley & Sons, Inc., Hoboken, New Jersey, 2006), second edition ed.
* Olivares and Paris (2009) S. Olivares and M. G. A. Paris, J. Phys. B 42, 055506 (2009).
* Gaiba and Paris (2009) R. Gaiba and M. G. A. Paris, Phys. Lett. A 373, 934 (2009).
* Bahder (2011a) T. B. Bahder, accepted for publication in Phys. Rev. A (2011a), URL http://arxiv.org/abs/1012.5293.
* Shannon (1948) C. E. Shannon, The Bell System Technical Journal 27, 379 (1948).
* Bahder and Lopata (2006a) T. B. Bahder and P. A. Lopata, Phys. Rev. A 74, 051801R (2006a), URL http://arxiv.org/abs/quant-ph/0602123.
* Bahder and Lopata (2006b) T. B. Bahder and P. A. Lopata, in _The 8th International Conference on Quantum Communication, Measurement, and Computing_ (Tsukuba, Japan, 2006b), pp. 369–372, URL http://xxx.lanl.gov/abs/quant-ph/0701243.
* Simon et al. (2008) D. S. Simon, A. V. Sergienko, and T. B. Bahder, Phys. Rev. A 78, 053829 (2008).
* Bahder (2011b) T. B. Bahder (2011b), URL http://arxiv.org/abs/1101.4634.
|
arxiv-papers
| 2011-02-09T22:26:06 |
2024-09-04T02:49:16.916299
|
{
"license": "Public Domain",
"authors": "Thomas B. Bahder",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/1102.1998"
}
|
1102.2024
|
# Slow-light probe of Fermi pairing through an atom-molecule dark state
H. Jing1,2, Y. Deng1, and P. Meystre2 1Department of Physics, Henan Normal
University, Xinxiang 453007, China
2B2 Institute, Department of Physics and College of Optical Sciences, The
University of Arizona, Tucson, Arizona 85721
###### Abstract
We consider the two-color photooassociation of a quantum degenerate atomic gas
into ground-state diatomic molecules via a molecular dark state. This process
can be described in terms of a lambda level scheme that is formally analogous
to the situation in electromagnetically-induced transparency (EIT) in atomic
systems, and therefore can result in slow light propagation. We show that the
group velocity of the light field depends explicitly on whether the atoms are
bosons or fermions, as well as on the existence or absence of a pairing gap in
the case of fermions, so that the measurement of the group velocity realizes a
non-destructive diagnosis of the atomic state and the pairing gap.
###### pacs:
03.75.Fi, 03.75.Ss, 42.50.Gy, 74.20.-z
## I INTRODUCTION
Degenerate atomic Fermi gases have attracted much interest in recent years,
well past the confines of traditional atomic, molecular and optical (AMO)
physics BCS . The existence of correlated Fermi pairs results in a number of
effects that can be explored particularly well in these systems, due in
particular to the control of two-body interactions provided by Feshbach
resonances. These include detailed studies of the crossover from Bardeen-
Cooper-Schrieffer (BCS) superfluidity to Bose-Einstein condensation (BCS) BCS
, of crystalline and supersolid phases solid , as well as spin-charge
separation or spin drag drag , to mention by a few examples. However, in
absence of any obvious change of density profile, the detection of Fermi
pairing is challenging, in sharp contrast to the familiar BEC transition of
bosons. A long-standing goal remains therefore to develop methods to
efficiently detect the pairing signature of fermionic systems and other
related exotic phases. Approaches toward this goal have focused on the
measurement of atomic density-density correlations via the resonant or non-
resonant optical response of the fermionic atoms laser , including methods of
radio-frequency spectroscopy Chin , photoemission spectroscopy emi , and Raman
spectroscopy cote . Alternative methods, like scanning tunneling microscopy
inf or acoustic attenuation molprob , are also actively pursued.
In parallel to these developments, rapid experimental advances have resulted
in the coherent formation of ultracold molecules from Bose or Fermi atoms mol
. The stable formation of diatomic molecules from laser-cooled alkali atoms
has been achieved by using magnetic Feshbach resonances and optical
photoassociation (PA) techniques. By applying an all-optical PA method,
molecules associated from ultracold atoms can be successfully transferred into
their rovibrational ground state mol2 .
A key component of the two-color PA method is the existence of an atom-
molecule dark state, as first demonstrated by Winkler $\it{et~{}al.}$ AMDS .
The underlying quantum interference and slow light propagation were also
observed for ultracold sodium atoms by Turner $\it{et~{}al.}$ Tur , hinting at
the possibility to study the quantum control of light through cold reactions
mol ; mol2 ; AMDS ; Tur ; HJ , quantum state transfer from light to molecules
HJ ; Letok , as well as high-precision diagnostics of Fermi gases via PA
spectroscopy cote .
In this paper we show that the slow light propagation associated with the
existence of that dark state provides a relatively simple nondestructive probe
of Fermi pairing, without the need for additional excitations (atom-to-atom,
atom-ion-to-molecule, or molecule-to-molecule) or for laser imaging of the
populations of transferred particles. This proposed method finds its
motivation in a previous work Meiser which showed that the statistical
properties of the molecular field formed from ultracold atoms depends strongly
on the statistical properties of these atoms. In particular, it was found that
for short times, the number of molecules created scales as the square $N^{2}$
of the number of atoms in case of an atomic Bose-Einstein condensate, but as
$N$ for a normal Fermi gas at zero temperature, a manifestation of the
independence of all atomic pairs in that case. For a paired Fermi gas, the
situation is intermediate between these two extremes: the molecules are formed
at a higher rate than for a normal Fermi gas, and the maximum number of
molecules is larger, approaching the BEC situation for strong pairing.
The main result of the present analysis is that a related situation occurs
when considering the dark-state propagation of a photoassociating light field:
in contrast to the case where photoassociation originates from a condensate of
bosonic atoms, and where the inverse group velocity $v_{g}^{-1}$ of the light
field is known to scale as $N^{2}$, we find that for a normal Fermi gas at
$T=0$ it scales as $N$. A paired Fermi system represents an intermediate
situation, as was the case in Ref. Meiser . It follows that the group velocity
is a direct measure of the pairing gap $\Delta.$ This simple all-optical
method is also expected to prove useful in probing e.g. polaron-to-molecule
transitions and atom-molecule vortex states polaron by photoassociating a
spin-imbalanced or a rotating Fermi gas. We remark that this proposal involves
the use of tunable atom-molecule interactions and as such is fundamentally
different from approaches based on single-atom excitations laser ; f-eit .
The paper is organized as follows. Section II describes our model and
calculates the slow light group velocity of a quantized optical field that
propagates in a normal Fermi gas and helps photoassociating atoms into
molecules via a dark state intermediate level. Section III evaluates the
effect of a Fermi pairing gap on that velocity and shows that it depends
strongly on the magnitude of the gap. Finally Section IV is a conclusion and
outlook.
## II Normal Fermi Gas
We first consider the two-color photoassociation of a homogeneous, normal
degenerate Fermi gas with no pairing. The entrance channel atoms, the
intermediate state $|m\rangle$ and the closed channel bosonic molecules are
characterized by the annihilation operators $\hat{c}_{{\bf{k}}\sigma}$,
$\hat{m}_{{\bf{k+k^{\prime}}}}$ and $\hat{a}$, respectively, where ${\bf{k}}$
and ${\bf{k}}^{\prime}$ are wave numbers and $\sigma$ labels the fermionic
spin. We assume that the PA between atomic pairs and excited molecules in
state $|m\rangle$ is driven by an optical field that is treated quantum
mechanically at that point, and the field that drives the molecules to their
ground state $|g\rangle$ is classical, with Rabi frequency $\Omega(t)$ (see
Fig. 1).
Figure 1: Schematic of two-color PA in an ultracold degenerate Fermi gas with
or without Cooper pairing.
At the simplest level the Hamiltonian of this system can be expressed as
($\hbar=1$)
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{{\bf{k}},\sigma}\frac{\epsilon_{\bf{k}}}{2}\hat{c}_{{\bf{k}}\sigma}^{\dagger}\hat{c}_{{\bf{k}}\sigma}+g\sum_{{\bf{k}},{\bf{k^{\prime}}}}\left(\hat{\cal
E}\hat{m}_{{\bf{k+k^{\prime}}}}^{\dagger}\hat{c}_{{\bf{k}}\uparrow}\hat{c}_{{\bf{k^{\prime}}}\downarrow}+\text{h.c.}\right)$
(1) $\displaystyle+$
$\displaystyle\sum_{\bf{k,k^{\prime}}}\left[\delta\hat{m}_{{\bf{k+k^{\prime}}}}^{\dagger}\hat{m}_{{\bf{k+k^{\prime}}}}+\Omega(\hat{a}\hat{m}_{{\bf{k+k^{\prime}}}}^{\dagger}+\text{h.c.})\right],$
where $g$ is the atom-molecule coupling constant, $\delta$ is the detuning
between the frequency of the quantized photoassociation field and the
frequency difference between the atomic fermions and the molecular state
$|m\rangle$ – we neglect the dispersion in fermionic energies
$\epsilon_{{\bf{k}}}$ for simplicity – and $\Omega(t)$ is the Rabi frequency
of the classical field, taken to be real without lack of generality. The
$s$-wave collisions between fermionic atoms, between molecules, and between
atoms and molecules are ignored for a dilute gas.
For simplicity, we restrict ourselves to the association of atom pairs with
opposite momenta (${\bf{k=-k^{\prime}}}$) and opposite spin, in which case the
intermediate molecules can be also described in terms a single-mode bosonic
field when concentrating on short-time dynamics, see e.g. Refs. Meiser ;
Holland . With these simplifying assumption this system is formally analogous
to the situation of EIT in atomic lambda systems, and as such can result in
slow light propagation.
The quantized optical field $\hat{E}(z,t)$, of carried frequency $\nu$, is
given by
$\hat{E}(z,t)=\sqrt{\frac{\hbar\nu}{2\epsilon_{0}L}}\hat{\cal
E}(z,t)\exp\left[i\frac{\nu}{c}(z-ct)\right],$
where $L$ is the quantization length. It satisfies the commutation relation
$[\hat{E}(z,t),\hat{E}^{\dagger}(z^{\prime},t)]=\frac{\nu}{\epsilon_{0}}\delta(z-z^{\prime}).$
Within the slowly-varying-amplitude approximation, the propagation equation of
the field envelope $\hat{\cal E}(z,t)$ is given by
$\left(\frac{\partial}{\partial t}\\!+\\!c\frac{\partial}{\partial
z}\right)\hat{\cal
E}(z,t)=igL\sum_{{\bf{k}}}\hat{c}_{{\bf{-k}}\downarrow}^{\dagger}(z,t)\hat{c}_{{\bf{k}}{\uparrow}}^{\dagger}(z,t)\hat{m}(z,t).$
(2)
In the following we consider the regime of weak excitations, where the atomic
population remains essentially undepleted. The initial state of the atom-
molecule system is taken as
$|\psi(0)\rangle=|F\rangle\otimes|0\rangle_{m}\otimes|0\rangle_{a},$
where $|0\rangle_{m}$, and $|0\rangle_{g}$ denote the vacuum state for the
molecules and
$|F\rangle=\prod_{k}\hat{c}_{-{\bf{k}}\downarrow}^{\dagger}\hat{c}_{{\bf{k}}\uparrow}^{\dagger}|0\rangle,$
and the product is taken up to the Fermi surface, a step appropriate for
temperatures much below the Fermi temperature Meiser . Introducing the pseudo-
spin operators
$\displaystyle\hat{s}_{{\bf{k}}}^{+}$ $\displaystyle=$
$\displaystyle(\hat{s}_{{\bf{k}}}^{-})^{\dagger}=\hat{c}_{{\bf{-k}}\downarrow}^{\dagger}\hat{c}_{{\bf{k}}{\uparrow}}^{\dagger},$
$\displaystyle\hat{s}_{{\bf{k}}}^{z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\hat{c}_{{\bf{k}}\uparrow}^{\dagger}\hat{c}_{{\bf{k}}\uparrow}+\hat{c}_{{\bf{-k}}\downarrow}^{\dagger}\hat{c}_{{\bf{-k}}\downarrow}-1\right),$
(3)
which satisfy the commutation relations
$[\hat{s}_{{\bf{k}}}^{+},\hat{s}_{{\bf{k}}^{\prime}}^{-}]=2\delta_{{\bf{kk^{\prime}}}}\hat{s}_{{\bf{k}}}^{z},\,\,\,\,\,[\hat{s}_{{\bf{k}}}^{z},\hat{s}_{{\bf{k}}^{\prime}}^{\pm}]=\pm\delta_{{\bf{kk^{\prime}}}}\hat{s}_{{\bf{k}}}^{\pm},$
(4)
and the collective operators
$\displaystyle\hat{S}_{\pm}$ $\displaystyle=$
$\displaystyle\sum_{{\bf{k}}}\hat{s}_{{\bf{k}}}^{\pm},$
$\displaystyle\hat{S}_{z}$ $\displaystyle=$
$\displaystyle\sum_{{\bf{k}}}\hat{s}_{{\bf{k}}}^{z}=\frac{N}{2}-\hat{a}^{\dagger}\hat{a}-\hat{m}^{\dagger}\hat{m},$
$\displaystyle\hat{{\bf{S}}}^{2}$ $\displaystyle=$
$\displaystyle\hat{S}_{+}\hat{S}_{-}+\hat{S}_{z}(\hat{S}_{z}-1),$ (5)
with the conserved total number of atomic pairs and molecules
$\displaystyle N$ $\displaystyle=$
$\displaystyle\sum_{k}\left(\hat{c}_{{\bf{k}}\uparrow}^{\dagger}\hat{c}_{{\bf{k}}\uparrow}+\hat{c}_{-{\bf{k}}\downarrow}^{\dagger}\hat{c}_{-{\bf{k}}\downarrow}\right)/2+(\hat{a}^{\dagger}\hat{a}+\hat{m}^{\dagger}\hat{m})$
(6) $\displaystyle=$
$\displaystyle(\hat{S}_{z}+N/2)+(\hat{a}^{\dagger}\hat{a}+\hat{m}^{\dagger}\hat{m}),$
yields for the Hamiltonian $\hat{H}_{\mathcal{N}}$ the simplified form
$\hat{H}=\sum_{{\bf{k}}}\epsilon_{\bf{k}}\hat{s}^{z}_{{\bf{k}}}+\delta\hat{m}^{\dagger}\hat{m}+\left(g\hat{\cal
E}\hat{m}^{\dagger}\hat{S}_{-}+\Omega\hat{m}^{\dagger}\hat{a}+{\rm
h.c.}\right).$ (7)
The resulting Heisenberg equations of motion are, by approximating all
$\epsilon_{{\bf{k}}}$’s as the Fermi energy $\epsilon_{F}$,
$\displaystyle i\frac{d{\hat{S}}_{z}}{dt}$ $\displaystyle=$ $\displaystyle
g\hat{\cal E}^{\dagger}\hat{m}\hat{S}_{+}-g\hat{\cal
E}\hat{m}^{\dagger}\hat{S}_{-},$ $\displaystyle i\frac{d{\hat{S}}_{-}}{dt}$
$\displaystyle=$ $\displaystyle\epsilon_{F}\hat{S}_{-}-2g\hat{\cal
E}^{\dagger}\hat{m}\hat{S}_{z},$ $\displaystyle i\frac{d{\hat{S}}_{+}}{dt}$
$\displaystyle=$ $\displaystyle-\epsilon_{F}\hat{S}_{-}+2g\hat{\cal
E}\hat{m}^{\dagger}\hat{S}_{z},$ $\displaystyle i\frac{d\hat{m}}{dt}$
$\displaystyle=$ $\displaystyle g\hat{\cal
E}\hat{S}_{-}+\delta\hat{m}+\Omega\hat{a},$ $\displaystyle
i\frac{d\hat{a}}{dt}$ $\displaystyle=$ $\displaystyle\Omega\hat{m},$
$\displaystyle i\frac{d\hat{\cal E}}{dt}$ $\displaystyle=$ $\displaystyle
g\hat{m}\hat{S}_{-}.$ (8)
In the following we consider the resonant situation $\delta=0$ and the limit
of weak excitations. By setting $d\hat{m}/dt\rightarrow 0$, we have then in
the lowest nonvanishing order of the excited molecular state Lukin ; HJ ,
$\displaystyle\hat{a}$ $\displaystyle=$ $\displaystyle-{(g/{\Omega})\hat{\cal
E}}\hat{S}_{-},$ $\displaystyle\hat{m}$ $\displaystyle=$
$\displaystyle-i(g/{\Omega}){\hat{S}_{-}}\frac{\partial}{\partial
t}(\frac{\hat{\cal E}}{\Omega}).$ (9)
The propagation of the field $\hat{\cal E}(z,t)$ is then governed by the
equation
$\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z}\right)\hat{\cal E}(z,t)=-\frac{g^{2}LN}{\Omega}\frac{\partial}{\partial
t}\left(\frac{\hat{\cal E}}{\Omega}\right),$ (10)
where we have used
$\hat{{\bf{S}}}^{2}|F\rangle=S(S+1)|F\rangle=\frac{N}{2}\left(\frac{N}{2}+1\right)|F\rangle,$
(11)
and the weak excitation approximation
$\langle\hat{S}_{+}\hat{S}_{-}\rangle=(-n_{a}^{2}+n_{a}N-n_{a})+N\sim N.$ (12)
Equation (10) can be recast as
$\displaystyle\left(\frac{\partial}{\partial
t}+\frac{c}{1+\beta_{f}}\frac{\partial}{\partial z}\right)\hat{\cal
E}(z,t)=\frac{\beta_{f}}{1+\beta_{f}}\left(\frac{1}{\Omega}\frac{\partial\Omega}{\partial
t}\right)\hat{\cal E}.$ (13)
where
$\beta_{f}\equiv\frac{g^{2}LN}{\Omega^{2}}.$ (14)
That is, the group velocity of the field $\hat{\cal E}(z,t)$ is
$v_{g}=\frac{c}{1+\beta_{f}}=c\cos^{2}\theta,$ (15)
with
$\theta=\tan^{-1}(g\sqrt{LN}/\Omega).$ (16)
As mentioned in the introduction, the scaling of $\beta_{f}$ with $N$ should
be contrasted with the situation for a pure condensate of bosonic atoms, in
which case HJ
$\beta_{f}\rightarrow\beta_{b}=\frac{g^{2}LN^{2}}{\Omega^{2}}=N\beta_{f}.$
(17)
As was the case in the analysis of molecule formation of Ref. Meiser , this
difference is due to the fact that for a Bose-Einstein condensate the
photoassociation is a collective atomic effect, while in a normal Fermi gas
the atom pairs act independently from each other.
We remark that the form of $v_{g}$ is independent of whether the field
${\hat{\cal E}}(z,t)$ is treated classically or quantum mechanically (see the
related experiment of Ref. Tur ). The quantized description used here is
primarily to facilitate a direct comparison with the bosonic atom-molecule
system of Ref. HJ . Note however that Eqs. (9) shows that the statistical
properties of the closed-channel molecules are determined by the states of
both the optical field and the Fermi atoms, hinting at the possibility of
quantum control of the closed-channel molecules, e.g. by applying a squeezed
PA field HJ .
The next section expands these considerations to the case of a paired Fermi
gas, which is then expected to represent an intermediate situation between
these two extremes. We show that this is indeed the case, and as a result,
measuring the group velocity of the photoassociating field provides a direct
measure of the pairing gap.
## III Pairing and Group Velocity
In order to account for the impact of Cooper pairing on the group velocity
$v_{g}$ we include attractive pairing interactions into Eq. (1) in the usual
fashion via the Hamiltonian Meiser ; Holland
$\hat{H}_{\rm
BCS}=\hat{H}-U\sum_{k,k^{\prime}}\hat{s}_{k}^{+}\hat{s}_{k^{\prime}}^{-}.$
(18)
The BCS ground state is found as usual by minimizing $\langle{\hat{H}}_{\rm
BCS}-\mu{\hat{N}}\rangle$ , where $\mu$ is the chemical potential, using the
ansatz
$|{\rm
BCS}\rangle=\prod_{k}(u_{{\bf{k}}}+v_{{\bf{k}}}{\hat{s}}_{{\bf{k}}}^{+})|0\rangle,$
(19)
with the result
$\left(\begin{array}[]{c}u_{{\bf{k}}}^{2}\\\
v_{{\bf{k}}}^{2}\end{array}\right)=\frac{1}{2}\left(1\mp\frac{\xi_{{\bf{k}}}}{\sqrt{\xi_{{\bf{k}}}^{2}+|\Delta|^{2}}}\right)$
(20)
where $\eta_{{\bf{k}}}=\sqrt{\xi_{{\bf{k}}}^{2}+|\Delta|^{2}}$ is the mean-
field quasiparticle energy, $\xi_{{\bf{k}}}=\epsilon_{{\bf{k}}}-\mu$ is the
kinetic energy of the atoms measured from the Fermi surface, and
$\Delta=U\sum_{{\bf{k}}}u_{{\bf{k}}}v_{{\bf{k}}}=\frac{U}{2}\sum_{{\bf{k}}}\frac{\Delta}{\sqrt{\xi_{{\bf{k}}}^{2}+|\Delta|^{2}}}$
(21)
is the gap parameter.
The interaction Hamiltonian (18) does not modify the equations of motion for
the operators $\hat{m}$, $\hat{a}$ and $\hat{\cal E}$. In the present context,
its main effect in the weak excitation limit is to replace
$\langle{\hat{S}}_{+}{\hat{S}}_{-}\rangle$ by
$\langle{\hat{S}}_{+}{\hat{S}}_{-}\rangle=\sum_{\bf{k}}v_{\bf{k}}^{2}+\sum_{{\bf{k}}\neq{\bf{k}}^{\prime}}u_{\bf{k}}v_{\bf{k}}u_{{\bf{k}}^{\prime}}v_{{\bf{k}}^{\prime}}\simeq
N+\left(\frac{\Delta}{U}\right)^{2}.$ (22)
Within the weak-coupling limit of BCS theory, $\epsilon_{\bf{k}}$ and
$\xi_{{\bf{k}}}$ are approximately independent of the wave vector
${{\bf{k}}}$, $\epsilon_{\bf{k}}\rightarrow\epsilon_{F}$ and
$\xi_{{\bf{k}}}\rightarrow\xi$, where $\epsilon_{F}$ is the Fermi energy
Ohashi . In that case the group velocity becomes
$v_{g,\Delta}=\frac{c}{1+\beta_{\Delta}},$ (23)
where
$\beta_{f}\rightarrow\beta_{\Delta}=\beta_{f}\left(1+\frac{N\Delta^{2}}{4\xi^{2}+4\Delta^{2}}\right),$
(24)
indicating that it now depends on both $N$ and the pairing gap $\Delta$.
Figure 2: Dimensionless relative time delay $T_{d}$ (scaled by $L/v_{g}$) as a
function of $N$ and the dimensionless pairing gap $\Delta/\xi$.
This is illustrated in Fig. 2, which shows the time delay
$T_{d}=\frac{L}{v_{g,\Delta}}-\frac{L}{v_{g}}=\frac{L\beta_{\Delta}}{c}$ (25)
experienced by a short photoassociating light pulse as a function of $N$ and
the pairing gap $\Delta$, relative to the delay in the absence of gap. For
large values of $\Delta$, we have $v_{g,\Delta}\sim N^{-2}$, approaching the
case of a bosonic atom-molecule dark-state medium HJ , with a gap-dependent
enhancement factor that is determined precisely by the ratio of the molecule
population $N_{a}(\Delta)$ and $N_{a}$ in the presence or absence of a pairing
gap,
$\zeta=1+\frac{N\Delta^{2}}{4(\xi^{2}+\Delta^{2})}=\frac{N_{a,\Delta}}{N_{a}},$
(26)
see Fig. 3. That is, the variation in group velocity originates directly from
the PA-induced atom-molecule superpositions in the $\Lambda$ level scheme of
Fig. 1.
Figure 3: Relative molecule population $\zeta^{-1}=N_{a}/N_{a}(\Delta)$ as a
function of $N$ and the dimensionless paring gap $\Delta/\xi$.
## IV Conclusion
In conclusion, we have shown that the two-color photoassociation of fermionic
atoms into bosonic molecules via a dark-state transition results in a group
velocity of the photoassociating field that can be slowed significantly, in
complete analogy with the situation of EIT in lambda three-level atomic
systems. That velocity $v_{g}$ depends not only on whether the atoms are
bosonic or fermionic, with an associated $N^{2}$ versus $N$ dependence, but
also on the possible pairing of the fermionic atoms resulting from attractive
two-body interactions. As such, a measure of the propagation delay of the
photoassociating light pulse ${\hat{\cal E}}(z,t)$ provides a direct
measurement of the pairing gap $\Delta$. This nondestructive ${\it in~{}situ}$
diagnostic technique, which provides clear evidence of Fermi pairing in the
weakly interacting BCS regime, supports and extends the idea of using Raman
spectroscopy cote to extract the pairing parameters, but differs from
proposals based solely on the use of atomic transitions f-eit .
In order to estimate the pairing-induced optical time delay of the propagating
pulse, we consider the typical values $g\sim 100\mathrm{KHz}$, $\Omega\sim
1\mathrm{MHz}$, $N=10^{5}$, $L=1\mathrm{mm}$, and $\gamma_{m}\sim
16\mathrm{MHz}$, $\gamma_{a}\sim 600\mathrm{Hz}$ cote . These values give for
the bosonic sample a group velocity of $v_{g}\sim 3\mathrm{km}\cdot s^{-1}$,
that is, a significant slowing down of the light pulse. For the normal Fermi
gas, the significantly less favorable scaling of $v_{g}$ with $N$ instead of
$N^{2}$ gives $v_{g}\sim 0.5c$, the rather small change that is expected to be
challenging to observe. Finally, for paired fermionic atoms we find $v_{g}\sim
300\mathrm{km}\cdot{\rm s}^{-1}$ for $\Delta/\xi=0.2$, and $v_{g}\sim
15\mathrm{km}\cdot s^{-1}$ for $\Delta/\xi=2$, a change of two to three orders
of magnitude compared to the case of a normal Fermi gas. As already mentioned,
for an increasing gap $v_{g}$ rapidly approaches the bosonic case. Note that
shorter samples lead to a reduction in delay time $T_{d}$ that scales as
$L^{2}$, as readily seen from Eqs. (14) and (25).
Our discussion ignores the decay of molecular states. However, it can be
readily shown that after including these decay terms, the group velocity of
the signal is still in the form of Eq. (15), but with the substitution
$\Omega\rightarrow\sqrt{\Omega^{2}+\gamma_{m}\gamma_{a}}$ HJ . In practice,
the PA pulse duration $\tau$ should satisfy $\tau\ll\gamma_{a}^{-1}\sim
1.67\mathrm{ms}$, a condition that can be fulfilled in current experiments
cote ; d1 ; d2 ; d3 .
Future work will improve the sample description by incorporating its spatial
profile in a more realistic multi-mode model, with a more detailed description
of the two-body physics. In this context it will also be interesting to
consider cavity-induced transparency with a degenerate Fermi gas Search . A
significantly more challenging problem will involve the situation of strong
pair fluctuations at the BEC-BCS crossover Hu . Finally, we note that the use
of non-classical associating light fields may also allow one to consider the
correlations of the transmitted field and/or a possible molecule-photon
entanglement as probes of the Fermi pairing or perhaps of other exotic phases.
###### Acknowledgements.
This work is supported by the U.S. National Science Foundation, by the U.S.
Army Research Office, and by the NSFC.
## References
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* (5) C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag, and R. Grimm, ibid. 305, 1128 (2004); J. Kinnunen, M. Rodríguez, and P. Törmä, ibid. 305, 1131 (2004).
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|
arxiv-papers
| 2011-02-10T02:57:26 |
2024-09-04T02:49:16.923181
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui Jing, Y. Deng, and P. Meystre",
"submitter": "H Jing",
"url": "https://arxiv.org/abs/1102.2024"
}
|
1102.2116
|
# Two particle correlations with photon and $\pi^{0}$ triggers with ALICE
Yaxian Mao1,2, for the ALICE Collaboration 1 Key Laboratory of Quark $\&$
Lepton Physics (Huazhong Normal University), Ministry of Education, Wuhan
430079, China 2 Laboratoire de Physique Subatomique et de Cosmologie, Grenoble
38026, France
###### Abstract
Comparing the measurements of the hadronic final state from partonic showers
in proton-proton and heavy-ion collisions will reveal the modifications
generated by the medium on partons produced in hard scatterings. This can be
achieved by selecting the hard processes in which there is a direct photon in
the final state. The experimental technique consists in tagging events with a
well identified high energy direct photon and in measuring the azimuthal angle
correlation with charged hadrons. To establish a reference measurement for
heavy-ion collisions, proton-proton collision data collected with ALICE have
been analyzed. Preliminary results are presented together with photon and
$\pi^{0}$-charged hadrons correlations showing the characteristic di-jet
pattern from where the partonic momentum $k_{T}$ is extracted.
###### keywords:
Triggers, Azimuthal Correlation, Isolation Cut, $k_{T}$
###### PACS:
21.10.Hw, 25.75.Gz, 21.10.Hw, 12.38.Mh
††thanks: Supported by NSFC (10875051, 10635020 and 10975061), the Key Project
of Chinese Ministry of Education (306022), the Program of Introducing Talents
of Discipline to Universities of China (QLPL200909, B08033 and CCNU09C01002)
## 1 Introduction
High energy heavy-ion collisions enable the study of strongly interacting
matter under extreme conditions. At sufficiently high collision energies
Quantum-Chromodynamics (QCD) predicts that hot and dense deconfined matter,
commonly referred to as the Quark-Gluon Plasma (QGP), is formed. The
experiment ALICE [1] at the CERN Large Hadron Collider (LHC) [2], allows the
study of the QCD matter in a new energy domain.
High $p_{T}$ partons produced in the initial stage of the collisions, have
been identified as a valuable probe of the medium. They are only observed
indirectly, as a collimated jet of hadrons originating from the hadronization
of the partonic shower. Comparing the properties of the jet fragmentation in
proton-proton and heavy-ion collisions will reveal the modifications induced
by the medium on the hard scattered partons. Ideally, one needs to know the
4-momentum of the parton when it has been produced in the hard scattering and
after it has been modified by the medium. This can be achieved by selecting
particular hard processes in which there is a photon in the final state. Since
the photon does not interact with the medium, its 4-momentum is not modified
and thus provides a measure of the hard scattered parton emitted back-to-back
with the photon. Measuring the hadrons opposite to the photon is thus a
promising way to measure the jet fragmentation and imbalance between photon
and hadrons to quantify the modifications due to the medium.
To establish a reference measurement for heavy-ion collisions, proton-proton
collision data collected with ALICE in 2010 have been analyzed with the
ultimate goal to construct the direct photon-charged hadron correlations.
Minimum bias data have been collected in pp collisions at center of mass
$\sqrt{s}~{}=7~{}$TeV. The present results have been obtained by analyzing
about 160 million events. The preliminary result is presented together with
inclusive photon-charged hadrons correlation and $\pi^{0}$-charged hadrons
spectra all showing the characteristic di-jet pattern from where the momentum
imbalance $k_{T}$ is extracted.
## 2 Trigger selection
The experimental technique consists in tagging events with a leading trigger
and measuring the distribution of hadrons associated to this leading trigger
from the same event. Such a measurement requires an excellent photon and
$\pi^{0}$ identification and the measurement of charged and neutral hadrons
with good $p_{T}$ resolution. In ALICE, the electromagnetic calorimeters, PHOS
($|\Delta\eta|<0.12$ and $\Delta\phi$ =100o) and EMCal ($|\Delta\eta|<0.7$ and
$\Delta\phi$ =100o) [3], are capable to measure photons with high efficiency
and resolution. In the calorimeters, electromagnetic particles are detected as
clusters of cells in the calorimeters. Roughly we have identified $\pi^{0}$
candidate as a pair of clusters with invariant mass around the $\pi^{0}$ mass
between 110 and 160 MeV/c2, and single clusters are identified as inclusive
photon candidates. No particle identification has been applied yet so that the
single cluster sample may contain a sizable fraction of charged particles
which develop a shower in the calorimeters or high-$p_{T}$ merged $\pi^{0}$
cluster which can not be reconstructed by invariant mass. The central tracking
system (ITS and TPC), covering the pseudo-rapidity $-0.9\leq\eta\leq+0.9$ and
the full azimuth, is used for charged track measurements, contributes to the
direct photon identification by applying the isolation technique.
Three different trigger particles have been selected for the correlation
measurements: (i) the charged trigger is chosen as the track with highest
transverse momentum among all the tracks from the same event, (ii) the photon
cluster trigger is defined as the calorimeter cluster with highest energy and
no charged track from the same event has momentum larger than photon cluster,
(iii) the $\pi^{0}$ trigger is selected as the cluster pair within the
appropriate invariant mass range and with the highest transverse momentum in
the event.
## 3 Azimuthal Correlation
The azimuthal correlation between the trigger particles (charged particle,
single cluster) and charged hadrons are shown in Fig. 2. The near side
($\Delta\phi=0$) and away side ($\Delta\phi=\pi$) peaks are clearly observed.
The correlation with cluster triggers shows larger di-jet peaks reflecting the
fact that the neutral trigger selection enhances the probability that the
trigger is the leading particle of the jet fragmentation compared to the less
restrictive charged trigger selection. The azimuthal correlations from
inclusive photon clusters and $\pi^{0}$ triggers show quite similar shapes
(Fig. 2), indicating that most of the inclusive photon clusters are $\pi^{0}$
decay photons.
Figure 1: Relative azimuthal angle distribution
$\Delta\phi=\phi_{trig}-\phi_{h^{\pm}}$ for charged trigger and inclusive
cluster trigger with $p_{T}^{trig}>5$ GeV/c in pp collisions at $\sqrt{s}$ = 7
TeV.
Figure 2: Azimuthal correlation distributions for inclusive cluster trigger
and $\pi^{0}$ triggers on the trigger particles with $p_{T}>5$ GeV/c in pp
collisions at $\sqrt{s}$ = 7 TeV.
By selecting isolated triggers, i.e. the trigger satisfies: the sum of the
transverse momentum of the hadrons inside a cone with size $R=0.4$ around the
trigger candidate carries less than 10 % of the trigger’s transverse momentum,
we can enrich the sample with direct photons or single hadron jets. The near
side peak is suppressed by construction, whereas the away side peak remains
and a slight difference is due to the imperfect isolation parameters used in
the analysis (Fig. 4) indicating the existence of di-jet events with one of
the jet being a hard fragmenting jet or eventually a direct photon.
## 4 $k_{T}$ extraction
Because of the hadronization, we do not have direct access to the parton
kinematics and therefore can measure neither the fragmentation function nor
the magnitude of partonic transverse momentum $k_{T}$ which modifies the ideal
$2\rightarrow 2$ kinematics. However, the isolated photon/$\pi^{0}$ triggered
correlation could be used to extract the partonic level kinematics to the
extend that the Leading Order kinematics dominates, as suggested by the PHENIX
analysis [4]:
$\frac{<z_{t}>}{\hat{x}_{h}}\sqrt{<k_{T}^{2}>}=\frac{1}{x_{h}}\sqrt{<p_{out}^{2}>-<j_{T_{y}}^{2}>(1+x_{h}^{2})}\;$
(1)
where $z_{t}=\frac{p_{T}^{trig}}{\hat{p}_{T}^{trig}}$ is the trigger
fragmentation variable and
$\hat{x}_{h}=\frac{\hat{p}_{T}^{assoc}}{\hat{p}_{T}^{trig}}$ is the ratio
between away and near side hard scattered partons, $x_{h}$ is similar to
$\hat{x}_{h}$ but at the hadronic level, and $j_{T_{y}}$ is the projection of
trigger particle deviates from the parton before fragmentation (see detail in
[4]). The values of $\sqrt{<k_{T}^{2}>}$ are determined by measuring the width
of the away side peak $\sqrt{<p_{out}^{2}>}$, using the fitting function
described in [5]. The fitted away side peak width shows in Fig. 4, the width
is weakly depend on the trigger $p_{T}$. The isolated trigger represents the
hard scattered parton direction approximately
($\hat{p}_{T}^{trig}~{}\simeq~{}p_{T}^{trig}$), therefore, $z_{t}~{}\approx$ 1
and $j_{T_{y}}~{}\approx~{}$0\. The $\sqrt{<k_{T}^{2}>}$ values we have
measured for $p_{T}^{trig}>5~{}$GeV/c and $p_{T}^{assoc}>1~{}$GeV/c is
consistent with another measurement obtained from charged di-hardon
correlations in ALICE. The measured value agrees the extrapolated value at LHC
energies from available worldwide data [6].
Figure 3: Azimuthal correlation distributions for inclusive cluster triggers
with $p_{T}>5GeV/c$ before and after isolation cut (IC) selection: $R=0.4$,
$\varepsilon=0.1$
Figure 4: Fitted width of the away side peak on the azimuthal correlation
distribution with cluster triggers before and after isolation selection in
EMCAL.
However this preliminary analysis does not allow to draw any conclusion other
than these results indicate the expected behavior. Exciting physics will
certainly come with the final analysis of large statistics within well-
calibrated detectors and all efficiency corrections.
## References
* [1] K. Aamodt et al 2008, JINST 3 S08002.
* [2] L. Evans and P. Bryant (editors) 2008, JINST 3 S08001.
* [3] ALICE Collaboration 1999, http://alice.web.cern.ch/Alice/TDR/.
* [4] A. Adare et al., 2006 Phys. Rev. D 74 072002\.
* [5] A. Adare et al., 2010 Phys. Rev. D 82 072001\.
* [6] Y. X. Mao et al., 2008, Eur. Phys. J. C 57 613; QM2009 poster.
|
arxiv-papers
| 2011-02-10T13:44:38 |
2024-09-04T02:49:16.929352
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yaxian Mao (for the ALICE Collaboration)",
"submitter": "Yaxian Mao",
"url": "https://arxiv.org/abs/1102.2116"
}
|
1102.2119
|
# Two particle correlations: a probe of the LHC QCD medium
Yaxian Mao1,2 for the ALICE Collaboration 1 Laboratoire de Physique
Subatomique et de Cosmologie, Grenoble 38026, France 2 Key Laboratory of Quark
& Lepton Physics (Huazhong Normal University), Ministry of Education, Wuhan
430079, China maoyx@iopp.ccnu.edu.cn
###### Abstract
The properties of $\gamma$–jet pairs emitted in heavy-ion collisions provide
an accurate mean to perform a tomographic measurement of the medium created in
the collision through the study of the medium modified jet properties. The
idea is to measure the distribution of hadrons emitted on the opposite side of
the direct photon. The feasibility of such measurements is studied by applying
the approach on the simulation data, we have demonstrated that this method
allows us to measure, with a good approximation, both the jet fragmentation
and the back-to-back azimuthal alignment of the direct photon and the jet.
Comparing these two observables measured in pp collisions with the ones
measured in AA collisions reveals the modifications induced by the medium on
the jet structure and consequently allows us to infer the medium properties.
In this contribution, we discuss a first attempt of such measurements applied
to real proton-proton data from the ALICE experiment.
## 1 Introduction
Quantum ChromoDynamics (QCD) [1] is a theory of the strong interaction, the
fundamental force describing the interactions of quarks and gluons making up
hadrons. The QCD calculations performed on a lattice, indicate that a phase
transition from normal hadronic matter to partonic matter, the Quark-Gluon
Plasma (QGP), will occur beyond a critical temperature of $T_{\rm
c}\sim~{}170$ MeV [2]. By colliding heavy ions at ultra relativistic energies,
this new state of matter can be created and its properties, such as the
equation of state, the degrees of freedom and the transport properties can be
measured.
The phase diagram has been explored in various regions with heavy-ion
collisions at continuously increasing kinetic energies. Experiments at CERN’s
Super Proton Synchrotron (SPS) [3] concluded on the indirect evidence of a
”new state of matter”. Current experiments at Brookhaven National Laboratory’s
Relativistic Heavy Ion Collider (RHIC) [4] have found that matter does not
behave as an ideal gas of free quarks and gluons predicted by theory, but,
rather, as an almost perfect fluid. The new experiment ALICE [5] at CERN’s
Large Hadron Collider (LHC), will push further the study of the QCD medium.
Thanks to the huge step in collision energy ($\sqrt{s_{NN}}=5.5~{}TeV$ in Pb-
Pb collisions), LHC will open new avenues for the exploration of matter under
extreme conditions of temperature and density. Since the hot QCD medium will
be formed at higher temperatures than at RHIC, the deconfined phase will last
longer and more readily modify our experimental probes, allowing for a more
accurate study of this new state matter.
Hard scattered partons produced in initial stage of the collisions, have been
identified as a valuable probe of the medium. Indeed, medium properties can be
inferred from the modifications experienced by the partonic shower inside the
medium. Partons are only observed indirectly, as a collimated jet of hadrons
coming from the fragmentation of the partonic shower [6]. Comparing the
measurements of the jet fragmentation in proton-proton and heavy-ion
collisions will reveal the modifications produced by the medium on the hard
scattered partons. Ideally, one needs to know the 4-momentum of the parton
when it has been produced in the hard scattering and after it has been
modified by the medium. This can be achieved by selecting particular hard
processes in which there is a photon in the final state. Since the photon does
not interact with the medium, its 4-momentum is not modified and thus provides
a measure of the hard scattered parton emitted back-to-back with the photon.
Measuring the hadrons opposite to the photon is thus a promising way to
measure the jet fragmentation and misalignment between photon and hadrons to
quantify the modifications due to the medium.
## 2 Approach Validation with Monte-Carlo Data
The experimental technique consists in tagging events with a well identified
high energy direct photon and measuring the distribution of hadrons emitted
oppositely to the photon as a function of the parameter
$x_{E}=-\vec{p}_{T}^{h}\cdot\vec{p}_{T}^{\gamma}/\mid p_{T}^{\gamma}\mid^{2}$.
Such a measurement requires an excellent direct photon identification and the
measurement of charged and neutral hadrons with good $p_{T}$ resolution. In
ALICE, the electromagnetic calorimeters, PHOS ($|\Delta\eta|<0.12$ and
$\Delta\phi$ =100o) and EMCal ($|\Delta\eta|<0.7$ and $\Delta\phi$ =100o) [7,
8], are capable to measure photons with high efficiency and resolution [9].
The central tracking system (ITS and TPC), covers the pseudorapidity
$-0.9\leq\eta\leq+0.9$ and the full azimuth, is helpful for direct photon
extraction with the isolation technique.
We have first established the feasibility of $\gamma$–hadrons correlation
measurement with ALICE detectors using Monte-Carlo data. As a first result
[10] of this study, PYTHIA [11] generator is used to simulate $pp$ collisions
at $\sqrt{s}~{}=~{}14~{}$TeV containing a 2$\rightarrow$2 process with a
direct photon inside PHOS acceptance. we have demonstrated that this
measurement allows us to determine, both the jet fragmentation distribution
and the back-to-back azimuthal alignment of the direct photon and the jet.
However because of the limited acceptance covered by the calorimeters, the
measurement is restricted by statistics to photon with energies below 50 GeV.
This kinematic region is particularly interesting because jets of such low
energy loose a large fraction of their energy while traversing the medium,
rendering the medium modification most visible. In addition, because jets with
energy below 50 GeV can hardly be reconstructed in the heavy-ion environment,
the photon tagging technique provides a sensitive measurement of jets in this
kinematic range. Systematic errors due to the improper identification of
direct photons remain, within this kinematic range, lower than statistical
errors from our study [10].
To quantify the medium modification, the photon–hadrons correlation
distribution has been studied with events generated in $pp$ collisions at
$\sqrt{s}~{}=~{}5.5~{}$TeV containing a 2$\rightarrow$2 process with a direct
photon inside EMCal acceptance. They have been generated by PYTHIA [11]
generator and qPYTHIA, which includes a parton energy loss model [12] with the
medium transport parameter $\hat{q}$ =50 GeV2/fm for photon energies between 5
and 200 GeV. At this stage of the study, the heavy-ion collision background
has not yet been taken into account. Direct photons are identified with the
isolation technique requiring no hadronic activity around the direct photon
candidate inside a given cone size [13]. Hadrons detected in the azimuthal
range $\pi/2<\Delta\phi<3\pi/2$ relative to the photon were used to construct
the correlation function. The contribution of hadrons from the underlying
event was calculated from the hadrons emitted in the same azimuthal hemisphere
as the photon.
The relative azimuthal angle, $\Delta\phi=\phi_{\gamma}-\phi_{h}$, between the
direct photon and charged hadrons is strongly peaked at $\pi$ as expected for
the 2$\rightarrow$2 process (Fig. 2). When medium effects are simulated
(qPYTHIA), the $\Delta\phi$ distribution becomes broader. The broadening can
be related to the medium transport parameter $\hat{q}$. However, the effect is
quite small which will make the measurement in the heavy-ion environment quite
challenging. A stronger signal is expected to be observed in the
photon–hadrons distribution from heavy-ion collisions when compared to the
distribution from pp collisions. The resulting photon-triggered hadrons
distributions, after subtraction of underlying events, are shown in Fig. 2,
normalized to the number of trigger particles found in corresponding
generation. The statistical errors are estimated from the annual yield of
photon events with $p_{T}$ larger than 30 GeV we anticipate to collect during
one PbPb run at nominal luminosity [5]. The distribution exhibits the expected
suppression at high $x_{E}$, due to the enegy loss of the hard scatered parton
and the enhancement at low $x_{E}$ due to the fragmentation of soft gluons
radiated in the medium.
Figure 1: Relative azimuthal angle distribution
$\Delta\phi=\phi_{\gamma}-\phi_{hadron}$ for $\gamma$-jet events in pp
collisions at $\sqrt{s}$ = 5.5 TeV.
Figure 2: $\gamma$-hadron correlation distributions in quenched and unquenched
PYTHIA events as a function of $ln(1/x_{E})$.
## 3 Two Particle Correlations in pp@7TeV
Minimum bias data have been collected in pp collisions at center of mass
$\sqrt{s}~{}=7~{}$TeV. We have analyzed about 35 million events in a first
attempt to measure $\gamma$-hadrons correlations. The trigger particle is
selected as the one with the highest transverse momentum measured either in
the central tracking system or in the electromagnetic calorimeters. In the
calorimeters, electromagnetic particles are detected as clusters of hit
calorimeter cells. Roughly we have identified $\pi^{0}$ candidate as a pair of
clusters which invariant mass matches the $\pi^{0}$ mass range, $135\pm 15$
MeV, and single clusters (which do no pair with another cluster) as direct
photon candidates. No particle identification has been applied yet so that the
single cluster sample contains a sizable fraction of charged particles which
develop a shower in the calorimeters.
The azimuthal correlation between the trigger particle (charged particle,
$\pi^{0}$ candidate, single cluster) and the charged hadrons are shown in Fig.
4. The near side ($\Delta\phi=0$) and away side ($\Delta\phi=\pi$) peak are
clearly observed. Note, however, that these distributions have not been
corrected for efficiency. It is interesting to remark that at this very
preliminary stage of the analysis we find that the underlying event background
level, outside the peaks region, is independent of the type of trigger
particles, giving some confidence in the measurement. By applying an isolation
selection on the trigger candidate, where hadron activity carries less than 30
% transverse momentum of the trigger candidate inside a cone with size $R=0.4$
required, the probability of direct photon or single particle jets in the
sample enhances. Comparing the azimuthal correlation with and without
isolation selection, obviously, a suppression of the near side peak is
observed, but the away side peak is almost unaffected, as expected (Fig. 4).
However this preliminary analysis does not allow to draw any conclusion other
that these results indicate the expected behaviour. The isolation parameters
are not well adjusted and especially in our case only charged tracks are
considered in our isolation cone due to the limited calorimeter acceptance (40
% EMCAL and 60 % PHOS have been installed so far).
Figure 3: Relative azimuthal angle distribution
$\Delta\phi=\phi_{trigger}-\phi_{hadron}$ in pp collisions at $\sqrt{s}$ = 7
TeV.
Figure 4: Azimuthal correlation distributions before and after isolation cut
(IC) on the trigger particles with $p_{T}>5GeV/c$
## 4 Summary and outlook
The feasibility to measure $\gamma$–hadrons correlation in pp collisions and
medium modification effect in PbPb collisions with ALICE has been evaluated.
Such a measurement provides an exclusive observable sensitive to the
properties of the medium formed in heavy-ion collisions. So far only a
preliminary analysis has been performed on a small fraction of the data
collected by ALICE. Exciting physics will certainly come with the final
analysis of large statistics with well calibrated detectors.
## Acknowledgement
The work is partially supported by the NSFC (10875051, 10635020 and 10975061),
the Key Project of Chinese Ministry of Education (306022), the Program of
Introducing Talents of Discipline to Universities of China (QLPL200909, B08033
and CCNU09C01002).
## 5 References
## References
* [1] D. J. Gross and F. Wilczec 1973, Phys. Rev. D 8 3633\.
* [2] E. V. Shuryak 1980, Phys. Rep. 61 71\.
* [3] U. W. Heinz and M. Jacob 2000, arXiv:nucl-th/0002042.
* [4] RHIC White paper 2005, Nucl. Phys. A 757 1\.
* [5] K. Aamodt et al 2008, JINST 3 S08002.
* [6] A. Morsch 2007,Nucl. Phys. A 783 427\.
* [7] ALICE Collaboration 1999, http://alice.web.cern.ch/Alice/TDR/.
* [8] T. M. Cormier 2004, Eur. Phys. J C 34 s333.
* [9] Y. X. Mao et al., 2008, Chinese Physics C 32(07).
* [10] Y. X. Mao et al. 2008, Eur. Phys. J. C 57 613\.
* [11] T. Sjostrand et al. 2001, Comput. Phys. Commun. 135 238\.
* [12] N. Armesto et al. 2009, hep-ph/0906.0754; Eur. Phys. J C 63 679\.
* [13] G. Conesa et al. 2007 Nucl. Instr. and Meth. Nucl. Res. A 580 1446\.
|
arxiv-papers
| 2011-02-10T14:01:30 |
2024-09-04T02:49:16.933278
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yaxian Mao (for the ALICE Collaboration)",
"submitter": "Yaxian Mao",
"url": "https://arxiv.org/abs/1102.2119"
}
|
1102.2123
|
00footnotetext: Received 25 November 2009
# QGP tomography with photon tagged jets in ALICE††thanks: This work is
supported partly by the NSFC (10875051 and 10635020), the State Key
Development Program of Basic Research of China (2008CB317106), the Key Project
of Chinese Ministry of Education (306022 and IRT0624) and the Programme of
Introducing Talents of Discipline to Universitiesnder of China: B08033
Yaxian Mao1,3 Yves Schutz2 maoyx@iopp.ccnu.edu.cn Daicui Zhou1 Christophe
Furget3 Gustavo Conesa Balbastre4 1 (Institute of Particle Physics, Huazhong
Normal University, Wuhan 430079, China )
2 (CERN, Geneva 23, Switzerland)
3 (Laboratoire de Physique Subatomique et de Cosmologie, CNRS/IN2P3, Grenoble
38026, France)
4 (Laboratori Nazionali di Frascati, INFN, Frascati, Italy)
###### Abstract
$\gamma+$jet events provide a tomographic measurement of the medium formed in
heavy ion collisions at LHC energies. Tagging events with a well identified
high $p_{T}$ direct photon and measuring the correlation distribution of
hadrons emitted oppositely to the photon, allows us to determine, with a good
approximation, both the jet fragmentation function and the back-to-back
azimuthal misalignement of the direct photon and the jet. Comparing these two
observables measured in $pp$ collisions with the ones measured in $AA$
collisions will reveal the modifications of the jet structure induced by the
medium formed in $AA$ collisions and consequently will infer the medium
properties.
###### keywords:
direct photon, QGP, jet structure, tomography, path length
###### pacs:
2
4.85.+p, 25.75.Bh, 25.75.Cj, 25.75.Nq
## 1 Introduction
The Large Hadron Collider (LHC) at CERN, will collide heavyions at
unprecedented high energies, exceeding by a factor 30 the energy available at
RHIC [1]. The main objective of ALICE (A Large Ion Collider Experiment) [2],
is to study matter under extreme conditions of energy density to gain a better
understanding of the fundamental properties of the strong interaction. In
particular, ALICE will explore the Quark-Gluon Plasma (QGP), the state of
deconfined matter predicted by QCD [3]. The medium formed in heavy-ion
collisions can be best probed by hard scattered partons produced in
2$\rightarrow$2 QCD processes at the leading order (LO) including in the final
state a hard direct photon (Compton scattering: q + g $\rightarrow\gamma$+q
and quark annihilation: q+$\bar{q}\rightarrow\gamma$+g) . On one hand, the
4-momentum of the scattered parton is modified while traversing the medium,
and on the other hand, the scattered photon does not interact, thus providing
a reference for the 4-momentum of the partner parton. Hence, from the
modification experienced by the hard scattered partons, measured though photon
tagged jets, the medium properties can be inferred. In particular, since these
hard scattering processes sample the entire collision volume, the final state
hadronic observables provide a real tomographic probe of the medium [4].
Several algorithms [5] have been developed to identify $\gamma$-jet events in
p–p and Pb–Pb collisions, demonstrating the feasibility of such measurements
with the ALICE detectors. However, the jet identification remains challenging
in the heavy-ion environment in particular for the energies
$E_{\gamma}\sim~{}30$ GeV where $\gamma$-jet events are measurable in ALICE
with sufficient statistics. An equivalent approach is to measure direct-
photon–hadrons correlation [6].
In the following, we have first established the intrinsic properties ($k_{T}$)
of $\gamma$-jet events expected in pp collisions at LHC energies. Then we
discuss the nucleus-nucleus (AA) collision case, in particular, we explore the
possibility to select $\gamma$-jet events as a function of their localization
in the medium to validate the tomographic approach.
## 2 $\gamma$-hadron topololigy in pp collisions
At leading order perturbative QCD, a pair of hard-scattered partons emerges
exactly back-to-back in the center of mass of the partonic system. Due to the
finite size of the proton, however, it was found that each of the colliding
parton carries initial transverse momentum with respect to the colliding axis,
originally described as ”intrinsic $k_{T}$”. Beyond the leading order, initial
and final state radiations (ISR/FSR) will generate additional transverse
momentum. Therefore, the resulting total transverse momentum of the outgoing
parton pair causes an acoplanar and a momentum imbalance, $<k_{T}>$ [7]. It is
measured as the net transverse momentum of the outgoing parton-pair
$<p_{T}>_{pair}~{}=~{}\sqrt{2}\cdot<k_{T}>$. It is anticipated that medium
effects will generate additional transverse momentum resulting in a broadening
of $k_{T}$ . This transverse momentum broadening can be directly related to
the transport parameter $\hat{q}$, which describes the transverse momentum
transferred from the medium to the traversing parton [8].
Using the PYTHIA event generator [9], we have established the collision energy
dependence of $<k_{T}>$ from $\gamma$-jet events, by taking available data
from different experiments measurements [10] and extrapolate to the LHC
energies. The dependence is $<p_{T}>_{pair}=A\cdot\log(B\cdot\sqrt{s})$ with
$A=2.064\pm 0.171$ and $B=0.164\pm 0.045$.
To study the dependence of $<k_{T}>$ with the transverse momentum of the hard
scattering, we have generated $\gamma$-jet and jet-jet events with PYTHIA
generator in different $p_{T}$ bins with collision energy 14 TeV, within
$k_{T}$ setting predicted above and ISR/FSR on. The averaged $<p_{T}>_{pair}$
versus the transverse momentum, shows a weakly linear dependence.
## 3 Medium modification by heavy ion collisions
The tomography measurement can be performed by selecting $\gamma$-h pairs with
different values of the parameter $x_{E}$ =
-$\vec{p}_{T}^{h}\cdot\vec{p}_{T}^{\gamma}/\mid p_{T}^{\gamma}\mid^{2}$. This
criteria can effectively control hadron emission from different regions of the
medium and therefore extract the corresponding jet modification parameters
[4].
To simulate the medium induced energy loss, we used the Monte-Carlo model
QPYTHIA [11], which combines an energy loss mechanism [12] and a realistic
description of the collision geometry [13]. The HIJING [14] generator was used
to simulate the underlying events of heavy-ion collisions and PYTHIA to
simulate pp collisions. Three samples of $\gamma$-jet events were generated
with photon energy larger than 20 GeV. A first sample of pp collisions at 5.5
TeV generated with PYTHIA provides the baseline. The second sample consists in
similar events modified by QPYTHIA merged with central collision events . The
last sample is obtained by merging the PYTHIA events and peripheral collision
events.
Tagging events with a direct photon well identified [15] by the ALICE
calorimeters and measuring the distribution of hadrons emitted oppositely to
the photon as a function of $x_{E}$, allows us to determinate the jet
fragmentation function [6]. The underlying event is subtracted by correlating
the isolated photon with charged hadrons emitted on the same side as the
photon, in the azimuthal range $-\pi/2<\Delta\phi<\pi/2$. To quantify the
medium modification, $I_{AA}$ is calculated (Fig. 3)
$\displaystyle I_{AA}(x_{E})=\frac{CF_{AA}}{CF_{pp}}\;$ (1)
as the ratio of $\gamma$-hadrons correlation distribution measured in AA and
pp collisions. The expected enhancement at low $x_{E}$ and suppression at high
$x_{E}$ for central collision is observed, whereas, $I_{AA}$ is equal to 1 for
peripheral collisions, where quenching effects are absent.
The nuclear modification factor $I_{AA}$ for $\gamma$-hadrons correlation
distribution in central and peripheral Pb+Pb collisions at
$\sqrt{s_{NN}}=~{}$5.5 TeV.
To illustrate the selectivity of the tomographic measurement, the length, L,
the jet travels inside the medium is calculated. Fig. 3 indicates that most
high $p_{T}$ leading particles are preferentially produced at the surface
(small L), while low $p_{T}$ leading particles are produced inside the whole
volume (large L), which demonstrates the L dependence of of the $\gamma$
tagged charged hadron production for 2 different $x_{E}$ regions.
The probability of the leading particles production as a function of medium
length L.
We have then studied the L dependence of the medium modification factor
$I_{AA}$ (Fig. 3) by selecting different $x_{E}$ regions. For large $x_{E}$
particles, an obvious suppression is observed, and the suppression is stronger
with increasing the medium length. For small $x_{E}$, the opposite behavior is
obtained as an enhancement ($I_{AA}>$ 1). This result implies that
$\gamma$-hadrons correlation could be used to probe volume versus surface
emission by selecting $\gamma$-jet events with different $x_{E}$ values.
However such L dependence will be challenging to measure in the experiments.
The nuclear modification factor $I_{AA}$ distribution as a function of medium
length L by selecting different regions of $x_{E}$ on correlation
distribution.
## 4 Conclusions
$\gamma$+jet studies are widely recognized as a powerfull tool to characterize
QGP. The ”$\gamma$+jet tomography” study will enable us to extract jet
quenching parameters in different regions of the dense medium via measurement
of the nuclear modification factor of $\gamma$-hadrons correlation.
###### Acknowledgements.
We especially thank Prof.Xin-Nian Wang, Prof.Andreas Morsh, Prof.Peter Jacobs
and Dr.Yuri Kharlov for their enthustic and fruitful discussions, also the
full PWG4 workgroup in ALICE collabration.
## References
* [1] http://www.bnl.gov/rhic.
* [2] http://aliceinfo.cern.ch/Collaboration/.
* [3] R. J. Fries and B. Muller, Eur. Phys. J. C 34 (2004) S279 .
* [4] H. Zhang, J. F. Owens, E. Wang and X. N. Wang, nucl-th/0902.4000v1.
* [5] G. Conesa, et al., Nucl. Instr. and Meth. A 585 (2008) 28
* [6] Y. X. Mao, et al., Eur. Phys. J. C 57 (2008) 316-319.
* [7] M. Della Negra et al., Nucl. Phys. B 127 (1977) 1.
* [8] C. A. Salgado and U. A. Wiedemann, Phys. Rev. D 68 (2003) 014008.
* [9] T. Sjostrand et al., JHEP 0605 (2006) 026, hep-ph/0603175 (2006).
* [10] S. S. Adler, et al., Phys. Rev. D 74 (2006) 072002.
* [11] N. Armesto, G. Corcella, L. Cunqueiro and C. A. Salgado, hep-ph/0906.0754; hep-ph/0907.1014.
* [12] X. N. Wang, et al., Phys. Rev. C 55 (1997) 3047.
* [13] R. J. Glauber and G. Matthiae, Nucl. Phys. B 21 (1970) 135.
* [14] M. Gyulassy and X. N. Wang, nucl-th/9502021v1 (1995).
* [15] Y. X. Mao, et al., Chinese Physics C 32 (2008) 07.
|
arxiv-papers
| 2011-02-10T14:14:23 |
2024-09-04T02:49:16.936888
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yaxian Mao, Yves Schutz, Daicui Zhou, Christophe Furget and Gustavo\n Conesa Balbastre",
"submitter": "Yaxian Mao",
"url": "https://arxiv.org/abs/1102.2123"
}
|
1102.2134
|
# Well-Quasi-Ordering of Matrices under Schur Complement and Applications to
Directed Graphs
Mamadou Moustapha Kanté Clermont-Université, Université Blaise Pascal, LIMOS,
CNRS
Complexe Scientifique des Cézeaux 63173 Aubiére Cedex, France
mamadou.kante@isima.fr
###### Abstract
In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric
Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field
$\mathbb{F}$, any infinite sequence $M_{1},M_{2},\ldots$ of (skew) symmetric
matrices over $\mathbb{F}$ of bounded _$\mathbb{F}$ -rank-width_ has a pair
$i<j$, such that $M_{i}$ is isomorphic to a principal submatrix of a
_principal pivot transform_ of $M_{j}$. We generalise this result to _$\sigma$
-symmetric matrices_ introduced by Rao and myself in [The Rank-Width of Edge-
Coloured Graphs, arXiv:0709.1433v4]. (Skew) symmetric matrices are special
cases of $\sigma$-symmetric matrices. As a by-product, we obtain that for
every infinite sequence $G_{1},G_{2},\ldots$ of directed graphs of bounded
rank-width there exist a pair $i<j$ such that $G_{i}$ is a _pivot-minor_ of
$G_{j}$. Another consequence is that non-singular principal submatrices of a
$\sigma$-symmetric matrix form a _delta-matroid_. We extend in this way the
notion of representability of delta-matroids by Bouchet.
###### keywords:
rank-width; sigma-symmetry; edge-coloured graph; well-quasi-ordering;
principal pivot transform; pivot-minor.
††journal:
## 1 Introduction
_Clique-width_ [6] is a graph complexity measure that emerges in the works by
Courcelle et al. (see for instance the book [7]). It extends _tree-width_ [21]
in the sense that graph classes of bounded tree-width have bounded clique-
width, but the converse is false (distance hereditary graphs have clique-width
at most $3$ and unbounded tree-width). Clique-width has similar algorithmic
properties as tree-width and seems to be the right complexity measure for the
investigations of polynomial time algorithms in dense graphs for a large set
of NP-complete problems [7]. It is then important to identify graph classes of
bounded clique-width. Unfortunately, contrary to tree-width, there is no known
polynomial time algorithm that checks if a given graph has clique-width at
most $k$, for fixed $k\geq 4$ (for $k\leq 3$, see the algorithm by Corneil et
al. [5]). Furthermore, clique-width is not monotone with respect to _graph
minor_ (cliques have clique-width $2$) and is only known to be monotone with
respect to the _induced subgraph_ relation which is not a well-quasi-order on
graph classes of bounded clique-width (cycles have clique-width at most $4$
and are not well-quasi-ordered by the induced subgraph relation).
In their investigations for a recognition algorithm for graphs of clique-width
at most $k$, for fixed $k$, Oum and Seymour [20] introduced the complexity
measure _rank-width_ of undirected graphs. Rank-width and clique-width of
undirected graphs are equivalent in the sense that a class of undirected
graphs has bounded rank-width if and only if it has bounded clique-width. But,
if rank-width shares with clique-width its same algorithmic properties (see
for instance [8]), it has better combinatorial properties.
1. 1.
There exists a cubic-time algorithm that checks whether an undirected graph
has rank-width at most $k$, for fixed $k$ [13].
2. 2.
Rank-width is monotone with respect to the _pivot-minor_ relation. This
relation generalises the notion of graph minor because if $H$ is a minor of
$G$, then $I(H)$, the _incidence graph_ of $H$, is a pivot-minor of $I(G)$.
Undirected graphs of rank-width at most $k$ are characterised by a finite list
of undirected graphs to exclude as pivot-minors [17].
3. 3.
Furthermore, rank-width is related to the _branch-width_ of binary matroids.
Branch-width of matroids plays an important role in the project by Geelen et
al. [12] aiming at extending techniques in the Graph Minors Project to
representable matroids over finite fields in order to prove that representable
matroids over finite fields are well-quasi-ordered by _matroid minors_. Such a
result would answer positively Rota’s Conjecture [12]. It turns out that the
branch-width of a binary matroid is one more than the rank-width of its
fundamental graphs and a fundamental graph of a minor of a matroid
$\mathcal{M}$ is a pivot-minor of a fundamental graph of $\mathcal{M}$.
It is then relevant to ask whether undirected graphs are well-quasi-ordered by
the pivot-minor relation. This would imply that binary matroids are well-
quasi-ordered by matroid minors, and hence the _Graph Minor Theorem_ [22].
This would also help understand the structure of graph classes of bounded
clique-width and of many dense graph classes where the Graph Minor Theorem
fails to explain their structure. Geelen et al. have successfully adapted many
techniques in the Graph Minors Project [23] and obtained generalisations of
some results in the Graph Minors Projects to representable matroids over
finite fields (see the survey [12]). Inspired by the links between rank-width
and branch-width of binary matroids, Oum [18] adapted the techniques by Geelen
et al. and proved that undirected graphs of bounded rank-width are well-quasi-
ordered by the pivot-minor relation. As for the Graph Minors Project, this
seems to be a first step towards a Graph Pivot-Minor Theorem.
However, rank-width has a drawback: it is defined in Oum’s works only for
undirected graphs. But, clique-width was originally defined for graphs
(directed or not, with edge-colours or not). Hence, one would know about the
structure of (edge-coloured) directed graphs of bounded clique-width. Rao and
myself [14] we have defined a notion of rank-width, called _$\mathbb{F}$
-rank-width_, for $\mathbb{F}^{*}$-graphs, _i.e._ , graphs with edge-colours
from a field $\mathbb{F}$, and explained how to use it to define a notion of
rank-width for graphs (directed or not, with edge-colours or not). Moreover,
the notion of rank-width of undirected graphs is a special case of it.
$\mathbb{F}$-rank-width is equivalent to clique-width and all the known
results, but the well-quasi-ordering theorem by Oum [18], concerning the rank-
width of undirected graphs have been generalised to the $\mathbb{F}$-rank-
width of $\mathbb{F}^{*}$-graphs. We complete the tableau in this paper by
proving a well-quasi-ordering theorem for $\mathbb{F}^{*}$-graphs of bounded
$\mathbb{F}$-rank-width, and hence for directed graphs.
In [19] Oum noticed that the _principal pivot transform_ introduced by Tucker
[25] can be used to obtain a well-quasi-ordering theorem for (skew) symmetric
matrices over finite fields of bounded $\mathbb{F}$-rank-width. This result
unifies his own result on the well-quasi-ordering of undirected graphs of
bounded rank-width by pivot-minor[18], the well-quasi-ordering by matroid
minor of matroids representable over finite fields of bounded branch-width
[11] and the well-quasi-ordering by graph minor of undirected graphs of
bounded tree-width [21]. In order to prove the well-quasi-ordering theorem for
$\mathbb{F}^{*}$-graphs of bounded $\mathbb{F}$-rank-width, we will adapt the
techniques used by Oum in [19] to _$\sigma$ -symmetric_ matrices. The notion
of $\sigma$-symmetric matrices were introduced by Rao and myself in [14] and
subsumes the notion of (skew) symmetric matrices. Oum’s proof can be
summarised into two steps.
* (i)
He first developed a theory about the notion of _lagrangian chain-groups_ ,
which are generalisations of _isotropic systems_ [1] and of _Tutte chain-
groups_ [26]. Tutte chain-groups are another characterisation of representable
matroids, and isotropic systems are structures that extend some properties of
$4$-regular graphs and of circle graphs. Isotropic systems played an important
role in the proof of the well-quasi-ordering of undirected graphs of bounded
rank-width by pivot-minor. As for Tutte chain groups and isotropic systems,
lagrangian chain groups are vector spaces equipped with a bilinear form. Oum
introduced a notion of minor for lagrangian chain groups that subsumes the
matroid minor and the notion of minor of isotropic systems. He also defined a
connectivity function for lagrangian chain groups that generalises the
connectivity function of matroids and allows to define a notion of _branch-
width_ for them. He then proved that lagrangian chain-groups of bounded
branch-width are well-quasi-ordered by lagrangian chain groups minor.
* (ii)
He secondly proved that to any lagrangian chain-group, one can associate a
(skew) symmetric matrix and vice-versa. These matrices are called _matrix
representations_ of lagrangian chain-groups. He can thus formulate the well-
quasi-ordering theorem of lagrangian chain-groups in terms of (skew) symmetric
matrices.
We will follow the same steps. We will extend the notion of lagrangian chain-
groups to make it compatible with $\sigma$-symmetric matrices. Then, we prove
that these lagrangian chain-groups admit representations by $\sigma$-symmetric
matrices.
The paper is organised as follows. We present some notations needed throughout
the paper in Section 2. Chain groups are revisited in Section 3. Section 4 is
devoted to the links between chain groups and $\sigma$-symmetric matrices. The
main theorem (Theorem 29) of the paper is presented in Section 4. Applications
to directed graphs and more generally to edge-coloured graphs is presented in
Section 5. An old result by Bouchet [3] states that non-singular principal
submatrices of a (skew) symmetric matrix form a _delta-matroid_. We extend
this result to $\sigma$-symmetric matrices and obtain a new notion of
representability of delta-matroids in Section 6.
## 2 Preliminaries
For two sets $A$ and $B$, we let $A\setminus B$ be the set $\\{x\in A\mid
x\notin B\\}$. The power-set of a set $V$ is denoted by $2^{V}$. We often
write $x$ to denote the set $\\{x\\}$. We denote by $\mathbf{N}$ the set
containing zero and the positive integers. If $f:A\to B$ is a function, we let
$\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,X}}$, the restriction of $f$ to
$X\subseteq A$, be the function $\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,X}}:X\to B$ where for every $a\in X,\
\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,X}}(a):=f(a)$. For a finite set $V$, we
say that the function $f:2^{V}\to\mathbf{N}$ is _symmetric_ if for any
$X\subseteq V,\leavevmode\nobreak\ f(X)=f(V\setminus X)$; $f$ is _submodular_
if for any $X,Y\subseteq V$, $f(X\cup Y)+f(X\cap Y)\leq f(X)+f(Y)$.
We denote by $+$ and $\cdot$ the binary operations of any field and by $0$ and
$1$ the identity elements of $+$ and $\cdot$ respectively. Fields are denoted
by the symbol $\mathbb{F}$ and finite fields of order $q$ by $\mathbb{F}_{q}$.
We recall that finite fields are commutative. For a field $\mathbb{F}$, we let
$\mathbb{F}^{*}$ be the set $\mathbb{F}\setminus\\{0\\}$. We refer to [16] for
our field terminology.
We use the standard graph terminology, see for instance [9]. A _directed
graph_ $G$ is a couple $(V_{G},E_{G})$ where $V_{G}$ is the set of vertices
and $E_{G}\subseteq V_{G}\times V_{G}$ is the set of edges. A directed graph
$G$ is said to be _undirected_ if $(x,y)\in E_{G}$ implies $(y,x)\in E_{G}$.
For a directed graph $G$, we denote by $G[X]$, called the subgraph of $G$
induced by $X\subseteq V_{G}$, the directed graph $(X,E_{G}\cap(X\times X))$.
The degree of a vertex $x$ in an undirected graph $G$ is the cardinal of the
set $\\{y\mid xy\in E_{G}\\}$. Two directed graphs $G$ and $H$ are
_isomorphic_ if there exists a bijection $h:V_{G}\to V_{H}$ such that
$(x,y)\in E_{G}$ if and only if $(h(x),h(y))\in E_{H}$. We call $h$ an
_isomorphism_ between $G$ and $H$. All directed graphs are finite and can have
loops.
A _tree_ is an acyclic connected undirected graph. A _cubic tree_ is a tree
such that the degree of each vertex is either $1$ or $3$. For a tree $T$ and
an edge $e$ of $T$, we let $T\textrm{-}e$ denote the graph
$(V_{T},E_{T}\setminus\\{e\\})$.
A _layout_ of a finite set $V$ is a pair $(T,\mathcal{L})$ of a cubic tree $T$
and a bijective function $\mathcal{L}$ from the set $V$ to the set
$\operatorname{L}_{T}$ of vertices of degree $1$ in $T$. For each edge $e$ of
$T$, the connected components of $T\textrm{-}e$ induce a bipartition
$(X_{e},V\setminus X_{e})$ of $\operatorname{L}_{T}$, and thus a bipartition
$(X^{e},V\backslash
X^{e})=(\mathcal{L}^{-1}(X_{e}),\mathcal{L}^{-1}(V\setminus X_{e}))$ of $V$.
Let $f:2^{V}\to\mathbf{N}$ be a symmetric function and $(T,\mathcal{L})$ a
layout of $V$. The _$f$ -width of each edge $e$ of $T$_ is defined as
$f(X^{e})$ and the _$f$ -width of $(T,\mathcal{L})$_ is the maximum $f$-width
over all edges of $T$. The _$f$ -width of $V$_ is the minimum $f$-width over
all layouts of $V$. The notions of layout and of $f$-width are commonly called
_branch-decomposition_ and _branch-width_ of $f$. However, this terminology is
not appropriate since $f$ is only a measure for the cuts
$(\mathcal{L}^{-1}(X_{e}),\mathcal{L}^{-1}(V\setminus X_{e}))$ and other
measures could be used with the same layout.
### 2.1 Well-Quasi-Order
We review in this section the _well-quasi-ordering_ notion. A binary relation
is a _quasi-order_ if it is reflexive and transitive. A quasi-order $\preceq$
on a set $\mathcal{U}$ is a _well-quasi-order_ , and the elements of
$\mathcal{U}$ are _well-quasi-ordered_ by $\preceq$, if for every infinite
sequence $x_{0},x_{1},\ldots$ in $\mathcal{U}$ there exist $i<j$ such that
$x_{i}\preceq x_{j}$. The notion of well-quasi-ordering is flourishing and
there exist several equivalent definitions of the well-quasi-ordering notion.
For instance, a quasi-order $\preceq$ on a set $\mathcal{U}$ is a well-quasi-
order if and only if $\mathcal{U}$ contains no infinite antichain and no
infinite strictly decreasing sequence. One consequence of this
characterisation is that every $\preceq$-closed set $X$ of $\mathcal{U}$,
_i.e._ , if $y\in X$ and $x\preceq y$ then $x\in X$, is characterised by a
finite list $Forb(X)$ such that $x\in X$ if and only if there is no $z\in
Forb(X)$ with $z\preceq x$. Hence, the well-quasi-ordering notion is an
interesting tool for characterising graph classes. There exist several well-
quasi-ordering theorems in the literature, see for instance [9, Chapter 12]
for some of them.
### 2.2 Sesqui-Morphism
We recall the notion of _sesqui-morphism_ introduced in [14] in order to
extend the notion of rank-width to directed graphs. Let $\mathbb{F}$ be a
field and $\sigma:\mathbb{F}\to\mathbb{F}$ a bijection. We recall that
$\sigma$ is an involution if $\sigma\circ\sigma$ is the identity. We call
$\sigma$ a _sesqui-morphism_ if $\sigma$ is an involution, and the function
$\tilde{\sigma}:=[x\mapsto\sigma(x)/\sigma(1)]$ is an automorphism. It is
worth noticing that if $\sigma:\mathbb{F}\to\mathbb{F}$ is a sesqui-morphism,
then $\sigma(0)=0$ and for every $a,b\in\mathbb{F}$,
$\sigma(a+b)=\sigma(a)+\sigma(b)$. Moreover, $\tilde{\sigma}$ is an
involution. The next proposition summarises some properties of sesqui-
morphisms.
###### Proposition 1
Let $\sigma:\mathbb{F}\to\mathbb{F}$ be a sesqui-morphism. Then, for all
$a,b,a_{i}\in\mathbb{F}$, $c\in\mathbb{F}^{*}$ and all $n\in\mathbf{N}$,
$\displaystyle\sigma(-a)$ $\displaystyle=-\sigma(a)$ (1)
$\displaystyle\sigma(a_{1}\cdot a_{2}\cdots a_{n})$
$\displaystyle=\frac{\sigma(a_{1})\cdot\sigma(a_{2})\cdots\sigma(a_{n})}{\sigma(1)^{n-1}}$
(2) $\displaystyle\sigma(a^{n})$
$\displaystyle=\frac{\sigma(a)^{n}}{\sigma(1)^{n-1}}$ (3)
$\displaystyle\sigma(a^{-n})$
$\displaystyle=\frac{\sigma(1)^{n+1}}{\sigma(a)^{n}}$ (4)
$\displaystyle\sigma\left(\frac{a}{c}\right)$
$\displaystyle=\frac{\sigma(1)\cdot\sigma(a)}{\sigma(c)}$ (5)
$\displaystyle\sigma\left(\frac{a\cdot b}{c}\right)$
$\displaystyle=\frac{\sigma(a)\cdot\sigma(b)}{\sigma(c)}$ (6)
Proof. Equation (1) is trivial since
$\sigma(a)+\sigma(-a)=\sigma(a-a)=\sigma(0)=0$.
Equation (2) will be proved by induction. The case $n=2$ is trivial since
$\tilde{\sigma}$ is an automorphism. Assume $n>2$. Then,
$\displaystyle\sigma(a_{1}\cdot a_{2}\cdots a_{n})$
$\displaystyle=\sigma(a_{1}\cdot a_{2}\cdots
a_{n-1})\cdot\frac{\sigma(a_{n})}{\sigma(1)}$
$\displaystyle=\frac{\sigma(a_{1})\cdot\sigma(a_{2})\cdots\sigma(a_{n-1})}{\sigma(1)^{n-2}}\cdot\frac{\sigma(a_{n})}{\sigma(1)}$
This proves the equation. Equation (3) is a direct consequence of Equation (2)
since $\sigma(a^{n})=\sigma(\underbrace{a\cdots a}_{n})$.
Since $\sigma(a^{-n})=\tilde{\sigma}(a^{-n})\cdot\sigma(1)$, Equation (4)
follows from this equality
$\tilde{\sigma}(a^{-n})=\frac{1}{\tilde{\sigma}(a^{n})}$. Equations (5) and
(6) are consequences of Equations (2)-(4). ∎
Examples of sesqui-morphisms are the identity automorphism (called _symmetric
sesqui-morphism_) and the function $[x\mapsto-x]$ (called _skew-symmetric
sesqui-morphism_). The next proposition states that they are the only ones in
prime fields.
###### Proposition 2
Let $p$ be a prime number and let $\sigma:\mathbb{F}_{p}\to\mathbb{F}_{p}$ be
a function. Then, $\sigma$ is a sesqui-morphism if and only if $\sigma$ is
symmetric or skew-symmetric.
Proof. Assume $\sigma:\mathbb{F}_{p}\to\mathbb{F}_{p}$ is a sesqui-morphism.
It is well-known that the only automorphism in $\mathbb{F}_{p}$, $p$ prime, is
the identity. Hence, $\tilde{\sigma}(a)=a$ for all $a\in\mathbb{F}_{p}$. Thus,
$\sigma(a)=a\cdot\sigma(1)$, and hence, $1=\sigma(\sigma(1))=\sigma(1)^{2}$.
Therefore, $\sigma(1)=\pm 1$.∎
Along this paper, sesqui-morphisms will be denoted by the Greek letter
$\sigma$, and then we will often omit to say "let
$\sigma:\mathbb{F}\to\mathbb{F}$ be a sesqui-morphism".
### 2.3 Matrices and $\mathbb{F}$-Rank-Width
For sets $R$ and $C$, an _$(R,C)$ -matrix_ is a matrix where the rows are
indexed by elements in $R$ and columns indexed by elements in $C$. If the
entries are over a field $\mathbb{F}$, we call it an $(R,C)$-matrix over
$\mathbb{F}$. For an $(R,C)$-matrix $M$, if $X\subseteq R$ and $Y\subseteq C$,
we let ${M}[{X},{Y}]$ be the submatrix of $M$ where the rows and the columns
are indexed by $X$ and $Y$ respectively. Along this paper matrices are denoted
by capital letters, which will allow us to write $m_{xy}$ for ${M}[{x},{y}]$
when it is possible. The matrix rank-function is denoted $\operatorname{rk}$.
We will write $M[X]$ instead of $M[X,X]$ and such submatrices are called
_principal submatrices_. The transpose of a matrix $M$ is denoted by $M^{t}$,
and the inverse of $M$, if it exists, _i.e._ , if $M$ is _non-singular_ , is
denoted by $M^{-1}$. The _determinant_ of $M$ is denoted by $\det(M)$. A
$(V_{1},V_{1})$-matrix $M$ is said _isomorphic_ to a $(V_{2},V_{2})$-matrix
$N$ if there exists a bijection $h:V_{1}\to V_{2}$ such that
$m_{xy}=n_{\scriptsize h(x)h(y)}$. We refer to [15] for our linear algebra
terminology.
For a sesqui-morphism $\sigma:\mathbb{F}\to\mathbb{F}$, a $(V,V)$-matrix $M$
over $\mathbb{F}$ is said _$\sigma$ -symmetric_ if $m_{yx}=\sigma(m_{xy})$ for
all $x,y\in V$. Examples of $\sigma$-symmetric matrices are (skew) symmetric
matrices with $\sigma$ being the (skew) symmetric sesqui-morphism. From
Proposition 2 they are the only $\sigma$-symmetric matrices over prime fields.
A $(V,V)$-matrix $M$ is said _$(\sigma,\epsilon)$ -symmetric_ if
$\epsilon(x)\cdot m_{xy}=\epsilon(y)\cdot\sigma(m_{yx})$ for all $x,y\in V$,
$\epsilon:V\to\\{-1,+1\\}$ being a function. If $\sigma$ is the (skew)
symmetric sesqui-morphism, $(\sigma,\epsilon)$-matrices are called matrices of
_symmetric type_ in [3]. It is worth noticing that a matrix is
$\sigma$-symmetric if and only if it is $(\sigma,\epsilon)$-symmetric with
$\epsilon$ a constant function.
We recall now the notion of _$\mathbb{F}$ -rank-width_ of
$(\sigma,\epsilon)$-matrices. It will be used to extend the notion of rank-
width to directed graphs. The _$\mathbb{F}$ -cut-rank_ function of a
$(\sigma,\epsilon)$-symmetric $(V,V)$-matrix $M$ is the function
$\operatorname{cutrk}^{{\mathbb{F}}}_{M}:2^{V}\to\mathbf{N}$ where
$\operatorname{cutrk}^{{\mathbb{F}}}_{M}(X)=\operatorname{rk}({M}[{X},{V\setminus
X}])$ for all $X\subseteq V$. From Proposition 15 and Theorem 22, the function
$\operatorname{cutrk}^{{\mathbb{F}}}_{M}$ is symmetric and submodular (a more
direct proof for $\sigma$-symmetric matrices can be found in [14], but it can
be easily adapted to $(\sigma,\epsilon)$-symmetric matrices). The
_$\mathbb{F}$ -rank-width_ of a $(\sigma,\epsilon)$-symmetric $(V,V)$-matrix
$M$ is the $\operatorname{cutrk}^{{\mathbb{F}}}_{M}$-width of $V$.
If $G$ is an undirected graph, then its adjacency matrix $A_{G}$ over
$\mathbb{F}_{2}$ is $\sigma_{1}$-symmetric, with $\sigma_{1}$ the identity
automorphism on $\mathbb{F}_{2}$. One easily checks that the rank-width of $G$
[17] is exactly the $\mathbb{F}_{2}$-rank-width of $A_{G}$.
Let $M$ be a matrix of the form $\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)$ where $A:=M[X]$ is non-singular. The _Schur
complement of $A$ in $M$_, denoted by $M/A$, is $D-C\cdot A^{-1}\cdot B$. Oum
proved the following.
###### Theorem 3 ([19])
Let $\mathbb{F}$ be a finite field and $k$ a positive integer. For every
infinite sequence $M_{1},M_{2},\ldots$ of symmetric or skew-symmetric matrices
over $\mathbb{F}$ of $\mathbb{F}$-rank-width at most $k$, there exist $i<j$
such that $M_{i}$ is isomorphic to a principal submatrix of $M_{j}/A$ for some
non-singular principal submatrix $A$ of $M_{j}$.
This theorem unifies in a single one the well-quasi-ordering theorems in [11,
18, 21]. We will show that this theorem still holds in the case of
$(\sigma,\epsilon)$-symmetric matrices that are not necessarily (skew)
symmetric. As a by product, we will get a well-quasi-ordering theorem for
directed graphs. In order to do so, we will adapt the same techniques as Oum’s
proof.
## 3 Chain Groups Revisited
_Chain groups_ were introduced by Tutte [26] for matroids and were also
studied by Bouchet in his series of papers dealing with circle graphs and
eulerian circuits of $4$-regular graphs (see for instance [1, 2, 3]).
The key point in the proof of Theorem 3 is to associate to each (skew)
symmetric matrix a chain group and then use the well-quasi-ordering theorem on
chain groups. We will revise the definitions by Oum so that to associate to
each $(\sigma,\epsilon)$-symmetric matrix a chain group. All the vector spaces
manipulated have finite dimension. The dimension of a vector space $W$ is
denoted by $\dim(W)$. If $f:W\to V$ is a linear transformation, we denote by
$Ker(f)$ the set $\\{u\in W\mid f(u)=0\\}$ and $Im(f)$ the set $\\{f(u)\in
V\mid u\in W\\}$. It is worth noticing that both are vector spaces. For a
vector space $K$, we let $K^{*}:=K\setminus\\{0\\}$.
For a field $\mathbb{F}$ and sesqui-morphism $\sigma:\mathbb{F}\to\mathbb{F}$,
we let $\mathbb{K}_{\sigma}$ be the $2$-dimensional vector space
$\mathbb{F}^{2}$ over $\mathbb{F}$ equipped with the application
$\mathbf{b}_{\sigma}:\mathbb{K}_{\sigma}\times\mathbb{K}_{\sigma}\to\mathbb{F}$
where $\mathbf{b}_{\sigma}(\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right),\left(\begin{smallmatrix}c\\\
d\end{smallmatrix}\right))=\sigma(1)\cdot a\cdot\sigma(d)-b\cdot\sigma(c)$.
The application $\mathbf{b}_{\sigma}$ is not bilinear, however it is linear
with respect to its left operand, which is enough for our purposes. It is
worth noticing that if $\sigma$ is skew-symmetric (or symmetric), then
$\mathbf{b}_{\sigma}$ is what is called $b^{+}$ (or $b^{-}$) in [19]. The
following properties are easy to obtain from the definition of
$\mathbf{b}_{\sigma}$.
###### Property 4
Let $u,v,w\in\mathbb{K}_{\sigma}$ and $k\in\mathbb{F}$. Then,
$\displaystyle\mathbf{b}_{\sigma}(u+v,w)$
$\displaystyle=\mathbf{b}_{\sigma}(u,w)+\mathbf{b}_{\sigma}(v,w),$
$\displaystyle\mathbf{b}_{\sigma}(u,v+w)$
$\displaystyle=\mathbf{b}_{\sigma}(u,v)+\mathbf{b}_{\sigma}(u,w),$
$\displaystyle\mathbf{b}_{\sigma}(k\cdot u,v)$
$\displaystyle=k\cdot\mathbf{b}_{\sigma}(u,v),$
$\displaystyle\mathbf{b}_{\sigma}(u,k\cdot v)$
$\displaystyle=\tilde{\sigma}(k)\cdot\mathbf{b}_{\sigma}(u,v).$
$\displaystyle\sigma(\mathbf{b}_{\sigma}(u,v))$
$\displaystyle=\frac{-1}{\sigma(1)^{2}}\cdot\mathbf{b}_{\sigma}(v,u).$
###### Property 5
Let $u\in\mathbb{K}_{\sigma}$.
1. (i)
If $\mathbf{b}_{\sigma}(u,v)=0$ for all $v\in\mathbb{K}_{\sigma}$, then $u=0$.
2. (ii)
If $\mathbf{b}_{\sigma}(v,u)=0$ for all $v\in\mathbb{K}_{\sigma}$, then $u=0$.
Let $W$ be a vector space over $\mathbb{F}$ and $\varphi:W\times
W\to\mathbb{F}$ a function. If $\varphi$ satisfies equalities in Property 4,
we call it a _$\sigma$ -sesqui-bilinear form_. It is called a _non-degenerate_
$\sigma$-sesqui-bilinear form if it also satisfies Property 5.
Let $W$ be a vector space over $\mathbb{F}$ equipped with $\varphi$ a non-
degenerate $\sigma$-sesqui-bilinear form. A vector $u$ is said _isotropic_ if
$\varphi(u,u)=0$. A subspace $L$ of $W$ is called _totally isotropic_ if
$\varphi(u,v)=0$ for all $u,v\in L$. For a subspace $L$ of $W$, we let
$L^{\bot}:=\\{v\in W\mid\varphi(u,v)=0$ for all $u\in L\\}$. It is worth
noticing that if $L$ is totally isotropic, then $L\subseteq L^{\bot}$. The
following theorem is a well-known theorem in the case where $\varphi$ is a
non-degenerate bilinear form.
###### Theorem 6
Let $W$ be a vector space over $\mathbb{F}$ equipped with a non-degenerate
$\sigma$-sesqui-bilinear form $\varphi$. Then,
$\dim(L)+\dim(L^{\bot})=\dim(W)$ for any subspace $L$ of $W$.
Proof. The proof is a standard one. We denote by $W^{*}$ the set of linear
transformations $[W\to\mathbb{F}]$. It is well-known that $W^{*}$ is a vector
space. Let $\varphi_{R}:W\to W^{*}$ such that
$\varphi_{R}(u):=[w\mapsto\varphi(w,u)]$. From Property 4, $\varphi_{R}$ is
clearly a linear transformation. Let $\alpha$ be a restriction of
$\varphi_{R}$ to $L$. By a well-known theorem in linear algebra,
$\dim(L)=\dim(Ker(\alpha))+\dim(Im(\alpha))$.
By definition, $Ker(\alpha)=\\{u\in L\mid\varphi(w,u)=0$ for all $w\in W\\}$,
which is equal to $\\{0\\}$ since $\varphi$ is non-degenerate. Hence,
$\dim(Ker(\alpha))=0$, _i.e._ , $\dim(L)=\dim(Im(\alpha))$.
If we let $Im(\alpha)^{\circ}:=\\{v\in W\mid\theta(v)=0$ for all $\theta\in
Im(\alpha)\\}$, we know by a theorem in linear algebra that
$\dim(Im(\alpha))+\dim(Im(\alpha)^{\circ})=\dim(W^{*})$. But,
$\displaystyle Im(\alpha)^{\circ}$ $\displaystyle=\\{v\in W\mid\alpha(w)(v)=0\
\textrm{for all}\ w\in L\\}$ $\displaystyle=\\{v\in W\mid\varphi(v,w)=0\
\textrm{for all}\ w\in L\\}=L^{\bot}.$
Hence, $\dim(L)=\dim(W^{*})-\dim(L^{\bot})=\dim(W)-\dim(L^{\bot})$ since
$\dim(W^{*})=\dim(W)$. ∎
As a consequence, we get that $L=(L^{\bot})^{\bot}$. And, if $L$ is totally
isotropic, then $2\cdot\dim(L)\leq\dim(W)$.
Let $V$ be a finite set and $K$ a vector space over $\mathbb{F}$. A _$K$
-chain on $V$_ is a function $f:V\to K$. We let $K^{V}$ be the set of
$K$-chains on $V$. It is well-known that $K^{V}$ is a vector space over
$\mathbb{F}$ by letting $(f+g)(x):=f(x)+g(x)$ and $(k\cdot f)(x):=k\cdot f(x)$
for all $x\in V$ and $k\in\mathbb{F}$, and by setting the $K$-chain $[x\mapsto
0]$ as the zero vector. It is worth noticing that
$\dim(K^{V})=\dim(K)\cdot|V|$. If $K$ is equipped with a non-degenerate
$\sigma$-sesqui-bilinear form $\varphi$, we let
$\langle,\rangle_{\varphi}:K^{V}\times K^{V}\to\mathbb{F}$ be such that for
all $f,g\in K^{V}$,
$\displaystyle\langle f,g\rangle_{\varphi}$ $\displaystyle:=\sum\limits_{x\in
V}\varphi(f(x),g(x)).$
It is straightforward to verify that $\langle,\rangle_{\varphi}$ is a non-
degenerate $\sigma$-sesqui-bilinear form. (We will often write
$\langle,\rangle$ for convenience when the context is clear.) Subspaces of
$K^{V}$ are called _$K$ -chain groups on $V$_. A $K$-chain group $L$ on $V$ is
said _lagrangian_ if it is totally isotropic and $\dim(L)=|V|$.
A _simple isomorphism_ from a $K$-chain group $L$ on $V$ to a $K$-chain group
$L^{\prime}$ on $V^{\prime}$ is a bijection $\mu:V\to V^{\prime}$ such that
$L=\\{f\circ\mu\mid f\in L^{\prime}\\}$ where $(f\circ\mu)(x)=f(\mu(x))$ for
all $x\in V$. In this case we say that $L$ and $L^{\prime}$ are _simply
isomorphic_.
From now on, we are only interested in $\mathbb{K}_{\sigma}$-chain groups on
$V$. Recall that $\mathbb{K}_{\sigma}$ is the $2$-dimensional vector space
$\mathbb{F}^{2}$ over $\mathbb{F}$ equipped with the $\sigma$-sesqui-bilinear
form $\mathbf{b}_{\sigma}$. The following is a direct consequence of
definitions and Theorem 6.
###### Lemma 7
If $L$ is a totally isotropic $\mathbb{K}_{\sigma}$-chain group on $V$, then
$\dim(L)\leq|V|$. If $L$ is lagrangian, then $L=L^{\bot}$.
###### Lemma 8
Let $u,v\in\mathbb{K}_{\sigma}$ and assume $u\neq 0$ is isotropic. If
$\mathbf{b}_{\sigma}(u,v)=0$, then $v=c\cdot u$ for some $c\in\mathbb{F}$.
Proof. Since $\mathbf{b}_{\sigma}$ is non-degenerate, there exists
$u^{\prime}\in\mathbb{K}_{\sigma}$ such that
$\mathbf{b}_{\sigma}(u,u^{\prime})\neq 0$. In this case, $\\{u,u^{\prime}\\}$
is a basis for $\mathbb{K}_{\sigma}$ (Property 4). Hence, there exist
$c,d\in\mathbb{F}$ such that $v=c\cdot u+d\cdot u^{\prime}$. Therefore,
$\displaystyle\mathbf{b}_{\sigma}(u,v)$
$\displaystyle=\frac{\sigma(c)}{\sigma(1)}\cdot\mathbf{b}_{\sigma}(u,u)+\frac{\sigma(d)}{\sigma(1)}\cdot\mathbf{b}_{\sigma}(u,u^{\prime})=\frac{\sigma(d)}{\sigma(1)}\cdot\mathbf{b}_{\sigma}(u,u^{\prime}).$
Since $\mathbf{b}_{\sigma}(u,u^{\prime})\neq 0$ and
$\mathbf{b}_{\sigma}(u,v)=0$, we have that $\sigma(d)=0$, _i.e._ , $d=0$. ∎
We now introduce _minors_ for $\mathbb{K}_{\sigma}$-chain groups on $V$. If
$f$ is a $\mathbb{K}_{\sigma}$-chain on $V$, then $Sp(f):=\\{x\in V\mid
f(x)\neq 0\\}$. If $L\subseteq\mathbb{K}_{\sigma}^{V}$ and $X\subseteq V$, we
let $L_{\mid X}:=\\{\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,X}}\mid f\in L\\}$ and $L^{\mid
X}:=\\{\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,X}}\mid f\in L$ and $Sp(f)\subseteq
X\\}$. For $\alpha\in\mathbb{K}_{\sigma}^{*}$ and $X\subseteq V$, we let
$L\operatorname{\parallel}\limits_{\alpha}X$ be the
$\mathbb{K}_{\sigma}$-chain group
$\displaystyle L\operatorname{\parallel}\limits_{\alpha}X$
$\displaystyle:=\\{\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus X)}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus X)}}{{f\,\smash{\vrule
height=3.88889pt,depth=2.12502pt}}_{\,(V\setminus X)}}{{f\,\smash{\vrule
height=2.77777pt,depth=2.12502pt}}_{\,(V\setminus X)}}\mid f\in L\ \textrm{and
$\mathbf{b}_{\sigma}(f(x),\alpha)=0$ for all}\ x\in X\\}$
on $V\setminus X$. A pair $\\{\alpha,\beta\\}\subseteq\mathbb{K}_{\sigma}^{*}$
is said _minor-compatible_ if
$\mathbf{b}_{\sigma}(\alpha,\alpha)=\mathbf{b}_{\sigma}(\beta,\beta)=0$ and
$\\{\alpha,\beta\\}$ forms a basis for $\mathbb{K}_{\sigma}$. For a minor-
compatible pair $\\{\alpha,\beta\\}$, a $\mathbb{K}_{\sigma}$-chain group on
$V\setminus(X\cup Y)$ of the form
$L\operatorname{\parallel}\limits_{\alpha}X\operatorname{\parallel}\limits_{\beta}Y$
is called an _$\alpha\beta$ -minor_ of $L$.
One easily verifies that
$L\operatorname{\parallel}\limits_{\alpha}X\operatorname{\parallel}\limits_{\alpha}Y=L\operatorname{\parallel}\limits_{\alpha}(X\cup
Y)$, and
$L\operatorname{\parallel}\limits_{\alpha}X\operatorname{\parallel}\limits_{\beta}Y=L\operatorname{\parallel}\limits_{\beta}Y\operatorname{\parallel}\limits_{\alpha}X$.
Hence, we have the following which is already proved in [19] for a special
case of $\\{\alpha,\beta\\}$.
###### Proposition 9
Let $\\{\alpha,\beta\\}$ be minor-compatible. An $\alpha\beta$-minor of an
$\alpha\beta$-minor of $L$ is an $\alpha\beta$-minor of $L$.
We now prove that $\alpha\beta$-minors of lagrangian
$\mathbb{K}_{\sigma}$-chain groups are also lagrangian. The proofs are the
same as in [19]. We include some of them that we expect can convince the
reader that the proofs are not different.
###### Proposition 10
Let $\\{\alpha,\beta\\}$ be minor-compatible. An $\alpha\beta$-minor of a
totally isotropic $\mathbb{K}_{\sigma}$-chain group $L$ on $V$ is totally
isotropic.
Proof. Let
$L^{\prime}:=L\operatorname{\parallel}\limits_{\alpha}X\operatorname{\parallel}\limits_{\beta}Y$
be an $\alpha\beta$-minor of $L$ on $V^{\prime}:=V\setminus(X\cup Y)$. Let
$f^{\prime},g^{\prime}\in L^{\prime}$ and let $f,g\in L$ such that
$f^{\prime}=\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}$ and
$g^{\prime}=\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.53944pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.53944pt}}_{\,V^{\prime}}}$. By Lemma 8, for all $x\in
X\cup Y$, $\mathbf{b}_{\sigma}(f(x),g(x))=0$. Hence, $\sum\limits_{x\in
V}\mathbf{b}_{\sigma}(f(x),g(x))=\sum\limits_{x\in
V^{\prime}}\mathbf{b}_{\sigma}(f(x),g(x))=\langle
f^{\prime},g^{\prime}\rangle$. Therefore, $\langle
f^{\prime},g^{\prime}\rangle=0$. ∎
###### Lemma 11
Let $L$ be a $\mathbb{K}_{\sigma}$-chain group on $V$ and $X\subseteq V$.
Then, $\dim(L_{\mid X})+\dim(L^{\mid(V\setminus X)})=\dim(L)$
Proof. Let $\varphi:L\to L_{\mid X}$ be the linear transformation that maps
any $f\in L$ to $\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=5.55557pt,depth=1.65279pt}}_{\,X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.16167pt}}_{\,X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.16167pt}}_{\,X}}$. We have clearly $L_{\mid
X}=Im(\varphi)$. For any $f\in Ker(\varphi)$, we have $f(x)=0$ for all $x\in
X$. Hence, $L^{\mid(V\setminus X)}=Ker(\varphi)$. This concludes the lemma. ∎
For any $x\in V$ and $\gamma\in\mathbb{K}_{\sigma}^{*}$, we let $x^{\gamma}$
be the $\mathbb{K}_{\sigma}$-chain on $V$ such that
$\displaystyle x^{\gamma}(z)$ $\displaystyle:=\begin{cases}\gamma&\textrm{if
$z=x$},\\\ 0&\textrm{otherwise}.\end{cases}$
The following admits a similar proof as the one in [19, Proposition 3.6].
###### Proposition 12
Let $L$ be a $\mathbb{K}_{\sigma}$-chain group on $V$, $x\in V$ and
$\gamma\in\mathbb{K}_{\sigma}^{*}$. Hence,
$\displaystyle\dim(L\operatorname{\parallel}\limits_{\gamma}x)$
$\displaystyle=\begin{cases}\dim(L)&\textrm{if $x^{\gamma}\in
L^{\bot}\setminus L$},\\\ \dim(L)-2&\textrm{if $x^{\gamma}\in L\setminus
L^{\bot}$},\\\ \dim(L)-1&\textrm{otherwise}.\end{cases}$
###### Corollary 13
Let $\\{\alpha,\beta\\}$ be minor-compatible. If $L$ is a totally isotropic
$\mathbb{K}_{\sigma}$-chain group on $V$ and $L^{\prime}$ is an
$\alpha\beta$-minor of $L$ on $V^{\prime}$, then
$|V^{\prime}|-\dim(L^{\prime})\leq|V|-\dim(L)$.
Proof. By induction on $|V\setminus V^{\prime}|$. Since $L$ is totally
isotropic, for all $x\in V\setminus V^{\prime}$, we cannot have neither
$x^{\alpha}\in L\setminus L^{\bot}$ nor $x^{\beta}\in L\setminus L^{\bot}$.
Hence, $\dim(L)-\dim(L\operatorname{\parallel}\limits_{\alpha}x)\in\\{0,1\\}$
and $\dim(L)-\dim(L\operatorname{\parallel}\limits_{\beta}x)\in\\{0,1\\}$ by
Proposition 12. Hence, if $|V\setminus V^{\prime}|=1$, we are done.
If $|V\setminus V^{\prime}|>1$, let $x\in V\setminus V^{\prime}$. Hence,
$L^{\prime}$ is an $\alpha\beta$-minor of
$L\operatorname{\parallel}\limits_{\alpha}x$ or
$L\operatorname{\parallel}\limits_{\beta}x$. By inductive hypothesis,
$|V^{\prime}|-\dim(L^{\prime})\leq|V\setminus
x|-\dim(L\operatorname{\parallel}\limits_{\alpha}x)$ or
$|V^{\prime}|-\dim(L^{\prime})\leq|V\setminus
x|-\dim(L\operatorname{\parallel}\limits_{\beta}x)$. And since, $|V\setminus
x|-\dim(L\operatorname{\parallel}\limits_{\alpha}x)\leq|V|-\dim(L)$ and
$|V\setminus
x|-\dim(L\operatorname{\parallel}\limits_{\beta}x)\leq|V|-\dim(L)$, we are
done. ∎
###### Proposition 14
Let $\\{\alpha,\beta\\}$ be minor-compatible. An $\alpha\beta$-minor of a
lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$ is lagrangian.
Proof. Let $L^{\prime}$ be an $\alpha\beta$-minor of $L$ on $V^{\prime}$. By
Proposition 10, $L^{\prime}$ is totally isotropic, hence
$\dim(L^{\prime})\leq|V^{\prime}|$. By Corollary 13,
$|V^{\prime}|-\dim(L^{\prime})\leq 0$ since $\dim(L)=|V|$ ($L$ lagrangian).
Hence, $\dim(L^{\prime})\geq|V^{\prime}|$. ∎
We now define the connectivity function for lagrangian
$\mathbb{K}_{\sigma}$-chain groups. Let $L$ be a lagrangian
$\mathbb{K}_{\sigma}$-chain group on $V$. For every $X\subseteq V$, we let
$\lambda_{L}(X):=|X|-\dim(L^{\mid X})$. Since $L^{\mid X}$ is totally
isotropic, $\dim(L^{\mid X})\leq|X|$, and hence $\lambda_{L}(X)\geq 0$.
###### Proposition 15 ([19])
Let $L$ be a lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$. Then,
$\lambda_{L}$ is symmetric and submodular.
The proof of Proposition 15 uses the fact that
$2\cdot\lambda_{L}(X)=\dim(L)-\dim(L^{\mid X})-\dim(L^{\mid(V\setminus X)})$
and the following theorem by Tutte.
###### Theorem 16 ([19])
If $L$ is a $\mathbb{K}_{\sigma}$-chain group on $V$ and $X\subseteq V$, then
$(L_{\mid X})^{\bot}=(L^{\bot})^{\mid X}$.
The branch-width of a lagrangian $\mathbb{K}_{\sigma}$-chain group $L$ on $V$,
denoted by $\operatorname{bwd}(L)$, is then defined as the $\lambda_{L}$-width
of $V$.
We can now state the well-quasi-ordering of lagrangian
$\mathbb{K}_{\sigma}$-chain groups of bounded branch-width under
$\alpha\beta$-minor. Let us first enrich the $\alpha\beta$-minor to labelled
$\mathbb{K}_{\sigma}$-chain groups on $V$. Let $(Q,\preceq)$ be a well-quasi-
ordered set. A _$Q$ -labelling_ of a lagrangian $\mathbb{K}_{\sigma}$-chain
group $L$ on $V$ is a mapping $\gamma_{L}:V\to Q$. A _$Q$ -labelled_
lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$ is a couple
$(L,\gamma_{L})$ where $L$ is a lagrangian $\mathbb{K}_{\sigma}$-chain group
on $V$ and $\gamma_{L}$ a $Q$-labelling of $L$. A $Q$-labelled lagrangian
$\mathbb{K}_{\sigma}$-chain group $(L^{\prime},\gamma_{L^{\prime}})$ on
$V^{\prime}$ is an _$(\alpha\beta,Q)$ -minor_ of a $Q$-labelled lagrangian
$\mathbb{K}_{\sigma}$-chain group $(L,\gamma_{L})$ on $V$ if $L^{\prime}$ is
an $\alpha\beta$-minor of $L$ and $\gamma_{L^{\prime}}(x)\preceq\gamma_{L}(x)$
for all $x\in V^{\prime}$. $(L,\gamma_{L})$ is _simply isomorphic_ to
$(L^{\prime},\gamma_{L^{\prime}})$ if there exists a simple isomorphism $\mu$
from $L$ to $L^{\prime}$ and $\gamma_{L}=\gamma_{L^{\prime}}\circ\mu$. The
following is more or less proved in [19].
###### Theorem 17
Let $\mathbb{F}$ be a finite field and $k$ a positive integer, and let
$\\{\alpha,\beta\\}$ be minor-compatible. Let $(Q,\preceq)$ be a well-quasi-
ordered set and let $(L_{1},\gamma_{L_{1}}),(L_{2},\gamma_{L_{2}}),\ldots$ be
an infinite sequence of $Q$-labelled lagrangian
$\mathbb{K}_{\sigma_{i}}$-chain groups having branch-width at most $k$. Then,
there exist $i<j$ such that $(L_{i},\gamma_{L_{i}})$ is simply isomorphic to
an $(\alpha\beta,Q)$-minor of $(L_{j},\gamma_{L_{j}})$.
Theorem 17 is proved in [19] for $\alpha=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right),\ \beta=\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)$ and $\langle,\rangle_{\mathbf{b}_{\sigma_{i}}}$
being a (skew) symmetric bilinear form. However, the proof uses only the
axioms in Properties 4 and 5, and Theorem 6. The other necessary ingredients
are Lemmas 7, 8 and 11, Proposition 12, and Theorem 16. We refer to [19] for
the technical details. It is important that the reader keeps in mind that even
if $\mathbf{b}_{\sigma}$ is not a bilinear form, it shares with the bilinear
forms in [19] the necessary properties for proving Theorem 17.
## 4 Representations of $\mathbb{K}_{\sigma}$-Chain Groups by
$(\sigma,\epsilon)$-Symmetric Matrices
In this section we will use Theorem 17 to obtain a similar result for
$(\sigma,\epsilon)$-symmetric matrices. We recall that we use the Greek letter
$\sigma$ for sesqui-morphisms, and if $\mathbb{F}$ is a field, then we let
$\mathbb{K}_{\sigma}$ be the $2$-dimensional vector space $\mathbb{F}^{2}$
over $\mathbb{F}$ equipped with the $\sigma$-sesqui-bilinear form
$\mathbf{b}_{\sigma}$. We will associate with each
$(\sigma,\epsilon)$-symmetric matrix a lagrangian $\mathbb{K}_{\sigma}$-chain
group. These matrices are called _matrix representations_. We also need to
relate $\alpha\beta$-minors of lagrangian $\mathbb{K}_{\sigma}$-chain groups
to principal submatrices of their matrix representations, and relate
$\mathbb{F}$-rank-width of $(\sigma,\epsilon)$-symmetric matrices to branch-
width of lagrangian $\mathbb{K}_{\sigma}$-chain groups. We follow similar
steps as in [19].
Let $\epsilon:V\to\\{-1,+1\\}$ be a function. We say that two
$\mathbb{K}_{\sigma}$-chains $f$ and $g$ on $V$ are _$\epsilon$
-supplementary_ if, for all $x\in V$,
1. (i)
$\mathbf{b}_{\sigma}(f(x),f(x))=\mathbf{b}_{\sigma}(g(x),g(x))=0$,
2. (ii)
$\mathbf{b}_{\sigma}(f(x),g(x))=\epsilon(x)\cdot\sigma(1)$ and
3. (iii)
$\mathbf{b}_{\sigma}(g(x),f(x))=-\epsilon(x)\cdot\sigma(1)^{2}$.
For any $c\in\mathbb{F}^{*}$, we let $c^{*}:=\left(\begin{smallmatrix}c\\\
0\end{smallmatrix}\right)$, $c_{*}:=\left(\begin{smallmatrix}0\\\
c\end{smallmatrix}\right)$, $\widetilde{c^{*}}:=\left(\begin{smallmatrix}0\\\
\sigma(c^{-1})\end{smallmatrix}\right)$ and
$\widetilde{c_{*}}:=\left(\begin{smallmatrix}-\sigma(1)\cdot\sigma(c)^{-1}\\\
0\end{smallmatrix}\right)$.
As a consequence of the following easy property, we get that for any
$\epsilon:V\to\\{-1,+1\\}$, we can construct $\epsilon$-supplementary
$\mathbb{K}_{\sigma}$-chains on $V$.
###### Property 18
For any $c\in\mathbb{F}^{*}$ and $\epsilon\in\\{-1,+1\\}$, we have
$\displaystyle\begin{cases}\mathbf{b}_{\sigma}\left(\epsilon\cdot
c^{*},\widetilde{c^{*}}\right)&=\epsilon\cdot\sigma(1)\\\
\mathbf{b}_{\sigma}\left(\widetilde{c^{*}},\epsilon\cdot
c^{*}\right)&=-\epsilon\cdot\sigma(1)^{2}\end{cases}$ $\displaystyle\
\textrm{and}\ \begin{cases}\mathbf{b}_{\sigma}\left(\epsilon\cdot
c_{*},\widetilde{c_{*}}\right)&=\epsilon\cdot\sigma(1)\\\
\mathbf{b}_{\sigma}\left(\widetilde{c_{*}},\epsilon\cdot
c_{*}\right)&=-\epsilon\cdot\sigma(1)^{2}\end{cases}$
The following associates with each $(\sigma,\epsilon)$-symmetric
$(V,V)$-matrix a lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$.
###### Proposition 19
Let $M$ be a $(\sigma,\epsilon)$-symmetric $(V,V)$-matrix over $\mathbb{F}$,
and let $f$ and $g$ be $\epsilon$-supplementaty $\mathbb{K}_{\sigma}$-chains
on $V$. For every $x\in V$, we let $f_{x}$ be the $\mathbb{K}_{\sigma}$-chain
on $V$ such that, for all $y\in V$,
$\displaystyle f_{x}(y)$ $\displaystyle:=\begin{cases}m_{xx}\cdot
f(x)+g(x)&\textrm{if $y=x$},\\\ m_{xy}\cdot
f(y)&\textrm{otherwise}.\end{cases}$
Then, the $\mathbb{K}_{\sigma}$-chain group on $V$ denoted by $(M,f,g)$ and
spanned by $\\{f_{x}\mid x\in V\\}$ is lagrangian.
Proof. It is enough to prove that for all $x,y$, $\langle
f_{x},f_{y}\rangle=0$ and the $f_{x}$’s are linearly independent.
For all $x,y\in V$ and all $z\in V\setminus\\{x,y\\}$,
$\mathbf{b}_{\sigma}(f_{x}(z),f_{y}(z))=\mathbf{b}_{\sigma}(m_{xz}\cdot
f(z),m_{yz}\cdot
f(z))=m_{xz}\cdot\sigma(m_{yz})\cdot\sigma(1)^{-1}\cdot\mathbf{b}_{\sigma}(f(z),f(z))=0$.
Hence for all $x,y\in V$,
$\displaystyle\langle f_{x},f_{y}\rangle$
$\displaystyle=\mathbf{b}_{\sigma}\left(f_{x}(x),f_{y}(x)\right)+\mathbf{b}_{\sigma}\left(f_{x}(y),f_{y}(y)\right)$
$\displaystyle=\mathbf{b}_{\sigma}\left(m_{xx}\cdot f(x)+g(x),m_{yx}\cdot
f(x)\right)+\mathbf{b}_{\sigma}\left(m_{xy}\cdot f(y),m_{yy}\cdot
f(y)+g(y)\right)$
$\displaystyle=\sigma(m_{yx})\cdot\sigma(1)^{-1}\cdot\mathbf{b}_{\sigma}\left(g(x),f(x)\right)+m_{xy}\cdot\mathbf{b}_{\sigma}\left(f(y),g(y)\right)$
$\displaystyle=\sigma(1)\cdot\left(\epsilon(y)\cdot
m_{xy}-\epsilon(x)\cdot\sigma(m_{yx})\right)$ $\displaystyle=0.$
It remains to prove that the $f_{x}$’s are linearly independent. Assume there
exist constants $c_{x}$ such that $\sum\limits_{x\in V}c_{x}\cdot f_{x}=0$.
Hence, for all $y\in V$, $\mathbf{b}_{\sigma}\left(f(y),\sum\limits_{x\in
V}c_{x}\cdot f_{x}(y)\right)=0$. But for all $x\in V$ and all $y\in V\setminus
x$, $\mathbf{b}_{\sigma}\left(f(y),c_{x}\cdot f_{x}(y)\right)=0$. Hence, for
all $y\in V$, $\mathbf{b}_{\sigma}\left(f(y),\sum\limits_{x\in V}c_{x}\cdot
f_{x}(y)\right)=\mathbf{b}_{\sigma}(f(y),c_{y}\cdot
f_{y}(y))=\epsilon(y)\cdot\sigma(c_{y})$, _i.e._ , $\sigma(c_{y})=0$. Hence,
we conclude that $c_{y}=0$ for all $y\in V$, _i.e._ , the $f_{x}$’s are
linearly independent. ∎
If a lagrangian $\mathbb{K}_{\sigma}$-chain group $L$ is simply isomorphic to
$(M,f,g)$, we call $(M,f,g)$ a _matrix representation of $L$_. One easily
verifies from the definition of $(M,f,g)$, that for all non zero
$\mathbb{K}_{\sigma}$-chains $h\in(M,f,g)$, we do not have
$\mathbf{b}_{\sigma}(h(x),f(x))=0$ for all $x\in V$. We now make precise this
property.
A $\mathbb{K}_{\sigma}$-chain $f$ on $V$ is called an _eulerian chain_ of a
lagrangian $\mathbb{K}_{\sigma}$-chain group $L$ on $V$ if:
1. (i)
for all $x\in V$, $f(x)\neq 0$ and $\mathbf{b}_{\sigma}(f(x),f(x))=0$, and
2. (ii)
there is no non-zero $\mathbb{K}_{\sigma}$-chain $h$ in $L$ such that
$\mathbf{b}_{\sigma}(h(x),f(x))=0$ for all $x\in V$.
The proof of the following is the same as in [19].
###### Proposition 20 ([19])
Every lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$ has an eulerian
chain.
Proof. By induction on the size of $V$. We let $\alpha:=c^{*}$ and
$\beta:=\widetilde{c^{*}}$ for some $c\in\mathbb{F}^{*}$. Let $L$ be a
lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$. If $V=\\{x\\}$, then
$\dim(L)=1$, hence either $x^{\alpha}$ or $x^{\beta}$ is an eulerian chain.
Assume $|V|>1$ and let $V^{\prime}:=V\setminus x$ for some $x\in V$. Hence,
both $L\operatorname{\parallel}\limits_{\alpha}x$ and
$L\operatorname{\parallel}\limits_{\beta}x$ are lagrangian. By inductive
hypothesis, there exist $f^{\prime}$ and $g^{\prime}$ such that $f^{\prime}$
(resp. $g^{\prime}$) is an eulerian chain of
$L\operatorname{\parallel}\limits_{\alpha}x$ (resp.
$L\operatorname{\parallel}\limits_{\beta}x$).
Let $f$ and $g$ be $\mathbb{K}_{\sigma}$-chains on $V$ such that
$f(x)=\alpha$, $g(x)=\beta$, and $f^{\prime}=\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}$ and
$g^{\prime}=\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.53944pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.53944pt}}_{\,V^{\prime}}}$. We claim that either $f$
or $g$ is an eulerian chain of $L$. Otherwise, there exist non-zero
$\mathbb{K}_{\sigma}$-chains $h$ and $h^{\prime}$ in $L$ such that
$\mathbf{b}_{\sigma}(h(x),f(x))=0$ and
$\mathbf{b}_{\sigma}(h^{\prime}(x),g(x))=0$ for all $x\in V$. Hence, we have
$\mathbf{b}_{\sigma}(\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}(x),f^{\prime}(x))=0$ and
$\mathbf{b}_{\sigma}(\mathchoice{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=4.77779pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=3.66667pt,depth=1.53944pt}}_{\,V^{\prime}}}(x),g^{\prime}(x))=0$ for
all $x\in V^{\prime}$. Therefore, $\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}=\mathchoice{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=4.77779pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=3.66667pt,depth=1.53944pt}}_{\,V^{\prime}}}=0$, otherwise there is a
contradiction because $\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}\in
L\operatorname{\parallel}\limits_{\alpha}x$ and
$\mathchoice{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=4.77779pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=3.66667pt,depth=1.53944pt}}_{\,V^{\prime}}}\in
L\operatorname{\parallel}\limits_{\beta}x$ by construction of $f$ and $g$.
Thus, $h(x)\neq 0$ and $h^{\prime}(x)\neq 0$, and $\langle
h,h^{\prime}\rangle=\mathbf{b}_{\sigma}(h(x),h^{\prime}(x))$. By Lemma 8, we
have $h(x)=d\cdot\alpha$ and $h^{\prime}(x)=d^{\prime}\cdot\beta$ for some
$d,d^{\prime}\in\mathbb{F}^{*}$. Hence, $\langle
h,h^{\prime}\rangle=d\cdot\sigma(d^{\prime})\neq 0$, which contradicts the
totally isotropy of $L$.∎
The next proposition shows how to construct a matrix representation of a
lagrangian $\mathbb{K}_{\sigma}$-chain group.
###### Proposition 21
Let $L$ be a lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$. Let
$\epsilon:V\to\\{-1,+1\\}$, and let $f$ and $g$ be $\epsilon$-supplementary
with $f$ being an eulerian chain of $L$. For every $x\in V$, there exists a
unique $\mathbb{K}_{\sigma}$-chain $f_{x}\in L$ such that
1. (i)
$\mathbf{b}_{\sigma}(f(y),f_{x}(y))=0$ for all $y\in V\setminus x$,
2. (ii)
$\mathbf{b}_{\sigma}(f(x),f_{x}(x))=\epsilon(x)\cdot\sigma(1)$.
Moreover, $\\{f_{x}\mid x\in V\\}$ is a basis for $L$. If we let $M$ be the
$(V,V)$-matrix such that
$m_{xy}:=\mathbf{b}_{\sigma}(f_{x}(y),g(y))\cdot\sigma(1)^{-1}\cdot\epsilon(y)$,
then $M$ is $(\sigma,\epsilon)$-symmetric and $(M,f,g)$ is a matrix
representation of $L$.
Proof. The proof is the same as the one in [19]. We first prove that
$\mathbb{K}_{\sigma}$-chains verifying statements (i) and (ii) exist. For
every $x\in V$, let $g_{x}$ be the $\mathbb{K}_{\sigma}$-chain on $V$ such
that $g_{x}(x)=f(x)$ and $g_{x}(y)=0$ for all $y\in V\setminus x$. We let $W$
be the $\mathbb{K}_{\sigma}$-chain group spanned by $\\{g_{x}\mid x\in V\\}$.
The dimension of $W$ is clearly $|V|$. Let $L+W=\\{h+h^{\prime}\mid h\in L,\
h^{\prime}\in W\\}$. We have $L\cap W=\\{0\\}$ because $f$ is eulerian to $L$.
Hence, $\dim(L+W)=2\cdot|V|$, _i.e._ , $\mathbb{K}_{\sigma}^{V}=L+W$. For each
$x\in V$, let $h_{x}\in\mathbb{K}_{\sigma}^{V}$ such that $h_{x}(x)=g(x)$ and
$h_{x}(y)=0$ for all $y\in V\setminus x$. Hence, there exist $f_{x}\in L$ and
$g^{\prime}_{x}\in W$ such that $h_{x}=f_{x}+g^{\prime}_{x}$. We now prove
that these $f_{x}$’s verify statements (i) and (ii). Let
$g^{\prime}_{x}=\sum\limits_{z\in V}c_{z}\cdot g_{z}$. For all $x\in V$ and
all $y\in V\setminus x$,
$\displaystyle\mathbf{b}_{\sigma}(f(x),f_{x}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(f(x),h_{x}(x)-g^{\prime}_{x}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(f(x),h_{x}(x))-\mathbf{b}_{\sigma}(f(x),g^{\prime}_{x}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(f(x),g(x))-\mathbf{b}_{\sigma}(f(x),c_{x}\cdot
f(x))$ $\displaystyle=\epsilon(x)\cdot\sigma(1)$ and
$\displaystyle\mathbf{b}_{\sigma}(f(y),f_{x}(y))$
$\displaystyle=\mathbf{b}_{\sigma}(f(y),h_{x}(y))-\mathbf{b}_{\sigma}(f(y),c_{y}\cdot
g_{y}(y))$
$\displaystyle=\mathbf{b}_{\sigma}(f(y),0)-\mathbf{b}_{\sigma}(f(y),c_{y}\cdot
f(y))=0.$
We now prove that each $f_{x}$ is unique. Assume there exist $f_{x}$’s and
$f^{\prime}_{x}$’s verifying statements (i) and (ii). For each $x\in V$, we
have
$\mathbf{b}_{\sigma}(f(x),f_{x}(x)-g(x))=\mathbf{b}_{\sigma}(f(x),f_{x}(x))-\mathbf{b}_{\sigma}(f(x),g(x))=0$.
Similarly, $\mathbf{b}_{\sigma}(f(x),f^{\prime}_{x}(x)-g(x))=0$. Hence, by
Lemma 8, $f_{x}(x)=c\cdot f(x)+g(x)$ and $f^{\prime}_{x}(x)=c^{\prime}\cdot
f(x)+g(x)$ for $c,c^{\prime}\in\mathbb{F}^{*}$. We let
$h^{\prime}_{x}=f_{x}-f^{\prime}_{x}$ which belongs to $L$. Therefore, for all
$z\in V$, we have $\mathbf{b}_{\sigma}(f(z),h^{\prime}_{x}(z))=0$. And since
$f$ is eulerian to $L$, we have $h^{\prime}_{x}=0$, _i.e._ ,
$f_{x}=f^{\prime}_{x}$.
By using the same technique as in the proof of Proposition 19, one easily
proves that $\\{f_{x}\mid x\in V\\}$ is linearly independent. It remains to
prove that $M:=(m_{xy})_{x,y\in V}$ with
$m_{xy}=\mathbf{b}_{\sigma}(f_{x}(y),g(y))\cdot\sigma(1)^{-1}\cdot\epsilon(y)$
is $(\sigma,\epsilon)$-symmetric and $L=(M,f,g)$.
We recall that $f(x)$ is isotropic for all $x\in V$. By statement (i) and
Lemma 8, for all $x\in V$ and all $y\in V\setminus x$, we have
$f_{x}(y)=c_{xy}\cdot f(y)$ for some $c_{xy}\in\mathbb{F}$. Hence,
$m_{xy}=c_{xy}$. Similarly, we have $f_{x}(x)=c_{xx}\cdot f(x)+g(x)$ for some
$c_{xx}\in\mathbb{F}$, _i.e._ , $m_{xx}=c_{xx}$. It is thus clear that
$L=(M,f,g)$. We now show that $M$ is $(\sigma,\epsilon)$-symmetric. Since $L$
is isotropic, we have for all $x,y\in V$, $\langle
f_{x},f_{y}\rangle=\mathbf{b}_{\sigma}(f_{x}(x),f_{y}(x))+\mathbf{b}_{\sigma}(f_{x}(y),f_{y}(y))=0$.
But,
$\displaystyle\mathbf{b}_{\sigma}(f_{x}(x),f_{y}(x))+\mathbf{b}_{\sigma}(f_{x}(y),f_{y}(y))$
$\displaystyle=\mathbf{b}_{\sigma}(m_{xx}\cdot f(x)+g(x),m_{yx}\cdot f(x))+$
$\displaystyle\qquad\qquad\qquad\mathbf{b}_{\sigma}(m_{xy}\cdot
f(y),m_{yy}\cdot f(y)+g(y))$
$\displaystyle=\sigma(m_{yx})\cdot\sigma(1)^{-1}\cdot\mathbf{b}_{\sigma}(g(x),f(x))+m_{xy}\cdot\mathbf{b}_{\sigma}(f(y),g(y))$
$\displaystyle=\sigma(1)\cdot(\epsilon(y)\cdot
m_{xy}-\epsilon(x)\cdot\sigma(m_{yx}))$
Hence, $\epsilon(y)\cdot m_{xy}=\epsilon(x)\cdot\sigma(m_{yx})$. ∎
From Proposition 19 (resp. 21), to every every $(\sigma,\epsilon)$-symmetric
$(V,V)$-matrix (resp. lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$) one
can associate a lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$ (resp. a
$(\sigma,\epsilon)$-symmetric $(V,V)$-matrix). The next theorem relates the
branch-width of a lagrangian $\mathbb{K}_{\sigma}$-chain group on $V$ to the
$\mathbb{F}$-rank-width of its matrix-representations. Its proof is present in
[19], but we give it for completeness.
###### Theorem 22 ([19])
Let $(M,f,g)$ be a matrix representation of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$. For every $X\subseteq V$, we
have $\operatorname{cutrk}^{{\mathbb{F}}}_{M}(X)=\lambda_{L}(X)$.
Proof. We let $\\{f_{x}\mid x\in V\\}$ be the basis of $L$ given in
Proposition 19. Let $A:={M}[{X},{V\setminus X}]$. It is well-known in linear
algebra that $\operatorname{rk}(A)=\operatorname{rk}(A^{t})=|X|-n(A^{t})$
where $n(A^{t})$ is $\dim\left(\\{p\in\mathbb{F}^{X}\mid A^{t}\cdot
p=0\\}\right)=\dim\left(\\{p\in\mathbb{F}^{X}\mid p^{t}\cdot A=0\\}\right)$.
Let $\varphi:\mathbb{F}^{V}\to L$ be such that $\varphi(p):=\sum\limits_{x\in
V}p(x)\cdot f_{x}$. It is clear that $\varphi$ is a linear transformation and
is therefore an isomorphism. Hence,
$\displaystyle\dim(L^{\mid X})$ $\displaystyle=\dim\left(\\{h\in L\mid
Sp(h)\subseteq X\\}\right)$ $\displaystyle=\dim\left(\varphi^{-1}\left(\\{h\in
L\mid Sp(h)\subseteq X\\}\right)\right)$
$\displaystyle=\dim\left(\\{p\in\mathbb{F}^{V}\mid\sum\limits_{x\in
V}p(x)\cdot f_{x}(y)=0\ \textrm{for all}\ y\in V\setminus X\\}\right).$
Now, let $p\in\mathbb{F}^{V}$ such that
$\mathchoice{{\varphi(p)\,\smash{\vrule
height=6.00002pt,depth=2.12502pt}}_{\,X}}{{\varphi(p)\,\smash{\vrule
height=6.00002pt,depth=2.12502pt}}_{\,X}}{{\varphi(p)\,\smash{\vrule
height=4.20001pt,depth=1.4875pt}}_{\,X}}{{\varphi(p)\,\smash{\vrule
height=3.0pt,depth=1.16167pt}}_{\,X}}\in L^{\mid X}$. Then, for all $y\in
V\setminus X$, $\varphi(p)(y)=0$, _i.e._ ,
$\mathbf{b}_{\sigma}(f(y),\varphi(p)(y))=0$. But,
$\varphi(p)(y)=\sum\limits_{x\in V}p(x)\cdot f_{x}(y)$. And, since
$\mathbf{b}_{\sigma}(f(y),f_{x}(y))=0$ for all $x\neq y$, we have
$\mathbf{b}_{\sigma}(f(y),\varphi(p)(y))=\mathbf{b}_{\sigma}(f(y),p(y)\cdot
f_{y}(y))=\sigma(p(y))\cdot\epsilon(y)$, _i.e._ , $p(y)=0$. Hence,
$\displaystyle\dim(L^{\mid X})$
$\displaystyle=\dim\left(\\{p\in\mathbb{F}^{X}\mid\sum\limits_{x\in
X}p(x)\cdot m_{xy}=0\ \textrm{for all}\ y\in V\setminus X\\}\right)$
$\displaystyle=\dim\left(\\{p\in\mathbb{F}^{X}\mid p^{t}\cdot A=0\\}\right)$
$\displaystyle=n(A^{t})$
Since, $\lambda_{L}(X)=|X|-\dim(L^{\mid X})$, we can conclude that
$\operatorname{cutrk}^{{\mathbb{F}}}_{M}(X)=\lambda_{L}(X)$. ∎
It remains now to relate $\alpha\beta$-minors of lagrangian
$\mathbb{K}_{\sigma}$-chain groups to principal submatrices of their matrix
representations. For doing so, we need to prove some technical lemmas. For
$X\subseteq V$, we let $P_{X}$ and $I_{X}$ be the non-singular diagonal
$(V,V)$-matrices where
$\displaystyle P_{X}[x,x]$ $\displaystyle:=\begin{cases}\sigma(-1)&\textrm{if
$x\in X$},\\\ 1&\textrm{otherwise},\end{cases}$ and $\displaystyle\quad
I_{X}[x,x]:=\begin{cases}-1&\textrm{if $x\in X$},\\\
1&\textrm{otherwise.}\end{cases}$
If $M$ is a matrix of the form $\left(\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right)$ where $\alpha:=M[X]$ is non-singular,
the _principal pivot transform_ of $M$ at $X$, denoted by $M*X$, is the matrix
$\displaystyle\centering\begin{pmatrix}\alpha^{-1}&\alpha^{-1}\cdot\beta\\\
-\gamma\cdot\alpha^{-1}&\ \ M/\alpha\end{pmatrix}.\@add@centering$
The principal pivot transform was introduced by Tucker [25] in an attempt to
understand the linear algebraic structure of the _simplex method_ by Dantzig.
It appeared to have wide applicability in many domains; without being
exhaustive we can cite linear algebra [24], graph theory [3] and biology [4].
###### Proposition 23
Let $(M,f,g)$ be a matrix representation of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$. Let $X\subseteq V$ such that
$M[X]$ is non-singular. Let $f^{\prime}$ and $g^{\prime}$ be
$\mathbb{K}_{\sigma}$-chains on $V$ such that, for all $x\in V$,
$\displaystyle f^{\prime}(x)$ $\displaystyle:=\begin{cases}f(x)&\textrm{if
$x\notin X$},\\\ g(x)&\textrm{otherwise},\end{cases}$ and $\displaystyle\quad
g^{\prime}(x)$ $\displaystyle:=\begin{cases}g(x)&\textrm{if $x\notin X$},\\\
\sigma(-1)\cdot f(x)&\textrm{otherwise}.\end{cases}$
Then, $(P_{X}\cdot(M*X),f^{\prime},g^{\prime})$ is a matrix representation of
$L$.
Proof. Let $\epsilon$ be such that $M$ is $(\sigma,\epsilon)$-symmetric,
_i.e._ , $f$ and $g$ are $\epsilon$-supplementary. Let us first show that
$f^{\prime}$ and $g^{\prime}$ are $\epsilon$-supplementary. Since for all
$x\notin X$, we have $f^{\prime}(x)=f(x)$ and $g^{\prime}(x)=g(x)$, we need to
verify the properties of $\epsilon$-supplementary for the $x\in X$. For each
$x\in X$, we have:
$\displaystyle\mathbf{b}_{\sigma}(f^{\prime}(x),f^{\prime}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(g(x),g(x))=0$
$\displaystyle\mathbf{b}_{\sigma}(g^{\prime}(x),g^{\prime}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(\sigma(-1)\cdot f(x),\sigma(-1)\cdot f(x))$
$\displaystyle=\mathbf{b}_{\sigma}(f(x),f(x))=0$
$\displaystyle\mathbf{b}_{\sigma}(f^{\prime}(x),g^{\prime}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(g(x),\sigma(-1)\cdot f(x))$
$\displaystyle=\frac{-1}{\sigma(1)}\cdot\mathbf{b}_{\sigma}(g(x),f(x))=\epsilon(x)\cdot\sigma(1)$
$\displaystyle\mathbf{b}_{\sigma}(g^{\prime}(x),f^{\prime}(x))$
$\displaystyle=\mathbf{b}_{\sigma}(\sigma(-1)\cdot f(x),g(x))$
$\displaystyle=-\sigma(1)\cdot\mathbf{b}_{\sigma}(f(x),g(x))=-\epsilon(x)\cdot\sigma(1)^{2}$
Hence, $f^{\prime}$ and $g^{\prime}$ are $\epsilon$-supplementary. It remains
to show that $f^{\prime}$ is eulerian to $L$. For each $x\in V$, we let
$f_{x}$ be the $\mathbb{K}_{\sigma}$-chain on $V$ such that
$\displaystyle f_{x}(y)$ $\displaystyle:=\begin{cases}m_{xy}\cdot
f(y)&\textrm{if $y\neq x$},\\\ m_{xx}\cdot
f(x)+g(x)&\textrm{otherwise}\end{cases}$
By Propositions 19 and 21 the set $\\{f_{x}\mid x\in V\\}$ is a basis for $L$.
Let $h\in L$ such that $\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))=0$ for all
$y\in V$. Let $h=\sum\limits_{z\in V}c_{z}\cdot f_{z}$. For all $y\notin X$,
we have
$\displaystyle\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$
$\displaystyle=\mathbf{b}_{\sigma}\left(\sum\limits_{z\in V}\left(c_{z}\cdot
m_{zy}\cdot f(y)\right)+c_{y}\cdot g(y),f(y)\right)$
$\displaystyle=\mathbf{b}_{\sigma}(c_{y}\cdot g(y),f(y))$
$\displaystyle=-c_{y}\cdot\epsilon(y)\cdot\sigma(1)^{2}.$
Hence, $c_{y}=0$ for all $y\notin X$. If $y\in X$, then
$\displaystyle\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$
$\displaystyle=\mathbf{b}_{\sigma}\left(\sum\limits_{z\in X}\left(c_{z}\cdot
m_{zy}\cdot f(y)\right)+c_{y}\cdot g(y),g(y)\right)$
$\displaystyle=\sum\limits_{z\in X}\left(c_{z}\cdot
m_{zy}\cdot\mathbf{b}_{\sigma}(f(y),g(y))\right)$
$\displaystyle=\sigma(1)\cdot\epsilon(y)\cdot\sum\limits_{z\in X}c_{z}\cdot
m_{zy}.$
And for $\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$ to being $0$, we must have
$\sum\limits_{z\in X}\left(c_{z}\cdot m_{zy}\right)=0$. But, since $M[X]$ is
non-singular, we have $\sum\limits_{z\in X}\left(c_{z}\cdot m_{zy}\right)=0$
for all $y\in X$ if and only if $c_{z}=0$ for all $z\in X$. Therefore, we have
$h=0$, _i.e._ , $f^{\prime}$ is eulerian.
By Proposition 21 there exists a unique matrix $M^{\prime}$ such that
$L=(M^{\prime},f^{\prime},g^{\prime})$. We will show that
$M^{\prime}=P_{X}\cdot(M*X)$. Assume
$M=\left(\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right)$ with $\alpha:=M[X]$. Let $I_{f}$ and
$I_{\bar{f}}$ be respectively $(X,X)$ and $(V\setminus X,V\setminus
X)$-diagonal matrices with diagonal entries being the $f(x)$’s. We define
similarly, $I_{g}$ and $I_{\bar{g}}$, but diagonal entries are $g(x)$’s. We
let $A$ be the $(V,V)$-matrix, where $a_{xy}:=f_{x}(y)$. Hence,
$\displaystyle A$ $\displaystyle=\begin{pmatrix}\alpha\cdot
I_{f}+I_{g}&\beta\cdot I_{\bar{f}}\\\ \gamma\cdot I_{f}&\delta\cdot
I_{\bar{f}}+I_{\bar{g}}\end{pmatrix}.$
The row space of $A$ is exactly $L$. Let $B$ be the non-singular
$(V,V)$-matrix
$\displaystyle\begin{pmatrix}\alpha^{-1}&0\\\ -\gamma\cdot\alpha^{-1}&\quad
I\end{pmatrix}.$
Therefore,
$\displaystyle B\cdot A$ $\displaystyle=\begin{pmatrix}\alpha^{-1}\cdot
I_{g}+I_{f}&\alpha^{-1}\cdot\beta\cdot I_{\bar{f}}\\\
-\gamma\cdot\alpha^{-1}\cdot
I_{g}&(\delta-\gamma\cdot\alpha^{-1}\cdot\beta)\cdot
I_{\bar{f}}+I_{\bar{g}}\end{pmatrix}.$
Let $A^{\prime}:=P_{X}\cdot B\cdot A$, and for each $x\in V$, let
$f^{\prime}_{x}$ be the $\mathbb{K}_{\sigma}$-chain on $V$ with
$f^{\prime}_{x}(y):=a^{\prime}_{xy}$. From above, we have that
$\\{f^{\prime}_{x}\mid x\in V\\}$ is a basis for $L$. Let
$C:=P_{X}\cdot(M*X)$. Then, for every $x,y\in V$, we have
$\displaystyle f^{\prime}_{x}(y)$ $\displaystyle=\begin{cases}c_{xy}\cdot
f(y)&\textrm{if $y\neq x$ and $y\notin X$},\\\ c_{xy}\cdot g(y)&\textrm{if
$y\neq x$ and $y\in X$},\\\ c_{xx}\cdot f(x)+g(x)&\textrm{if $y=x\notin
X$},\\\ c_{xx}\cdot g(x)+\sigma(-1)\cdot f(x)&\textrm{if $y=x\in
X$}.\end{cases}$ Hence,
$\displaystyle\mathbf{b}_{\sigma}(f^{\prime}(y),f^{\prime}_{x}(y))$
$\displaystyle=\begin{cases}\mathbf{b}_{\sigma}(f(y),c_{xy}\cdot
f(y))&\textrm{if $y\neq x$ and $y\notin X$},\\\
\mathbf{b}_{\sigma}(g(y),c_{xy}\cdot g(y))&\textrm{if $y\neq x$ and $y\in
X$},\\\ \mathbf{b}_{\sigma}(f(x),c_{xx}\cdot f(x)+g(x))&\textrm{if $y=x\notin
X$},\\\ \mathbf{b}_{\sigma}(g(x),c_{xx}\cdot g(x)+\sigma(-1)\cdot
f(x))&\textrm{if $y=x\in X$}.\end{cases}$
Hence, for all $x\in V$ and all $y\in V\setminus x$, we have
$\mathbf{b}_{\sigma}(f^{\prime}(x),f^{\prime}_{x}(x))=\epsilon(x)\cdot\sigma(1)$
and $\mathbf{b}_{\sigma}(f^{\prime}(y),f^{\prime}_{x}(y))=0$. Therefore, by
Propositions 19 and 21 $\\{f^{\prime}_{x}\mid x\in V\\}$ is the basis
associated with $(M^{\prime},f^{\prime},g^{\prime})$ and
$M^{\prime}=C=P_{X}\cdot(M*X)$. ∎
###### Proposition 24
Let $(M,f,g)$ be a matrix representation of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$ and let $Z\subseteq V$. Let
$f^{\prime}$ and $g^{\prime}$ be $\mathbb{K}_{\sigma}$-chains on $V$ such that
$\displaystyle f^{\prime}(x)$ $\displaystyle:=\begin{cases}-f(x)&\textrm{if
$x\in Z$},\\\ f(x)&\textrm{otherwise},\end{cases}$ and $\displaystyle\quad
g^{\prime}(x)$ $\displaystyle:=\begin{cases}-g(x)&\textrm{if $x\in Z$},\\\
g(x)&\textrm{otherwise}.\end{cases}$
Then, $(I_{Z}\cdot M,f,g^{\prime})$ and $(M\cdot I_{Z},f^{\prime},g)$ are
matrix representations of $L$.
Proof. Let $\epsilon:V\to\\{+1,-1\\}$ be such that $M$ is
$(\sigma,\epsilon)$-symmetric, _i.e._ , $f$ and $g$ are
$\epsilon$-supplementary. Let $\\{f_{x}\mid x\in V\\}$ be the basis of $L$
associated with $f$ and $g$ by Proposition 19. One easily verifies that
$f^{\prime}$ and $g$, and $f$ and $g^{\prime}$ are
$\epsilon^{\prime}$-supplementary with $\epsilon^{\prime}(x)=-\epsilon(x)$ if
$x\in Z$, otherwise $\epsilon^{\prime}(x)=\epsilon(x)$. Moreover, $f^{\prime}$
is eulerian (because $f$ is eulerian). By Proposition 21, there exist unique
$f^{\prime}_{x}$’s and $f^{\prime\prime}_{x}$’s such that
$(M^{\prime},f^{\prime},g)$ and $(M^{\prime\prime},f,g^{\prime})$ are matrix
representations of $L$ with
$m^{\prime}_{xy}:=\mathbf{b}_{\sigma}(f^{\prime}_{x}(y),g^{\prime}(y))\cdot\sigma(1)^{-1}\cdot\epsilon^{\prime}(y)$
and
$m^{\prime\prime}_{xy}:=\mathbf{b}_{\sigma}(f^{\prime\prime}_{x}(y),g(y))\cdot\sigma(1)^{-1}\cdot\epsilon^{\prime}(y)$.
One easily checks that $\\{-f_{x}\mid x\in Z\\}\cup\\{f_{x}\mid x\in
V\setminus Z\\}$ is the basis of $L$ associated with $f$ and $g^{\prime}$ by
Proposition 21. It remains to prove that $M^{\prime}=M\cdot I_{Z}$. If $x,y\in
Z$, then
$m^{\prime}_{xy}=\mathbf{b}_{\sigma}(-f_{x}(y),-g(y))\cdot(-\epsilon(y))\cdot\sigma(1)^{-1}=-m_{xy}$.
If $x\in Z$ and $y\notin Z$, then
$m^{\prime}_{xy}=\mathbf{b}_{\sigma}(-f_{x}(y),g(y))\cdot\epsilon(y)\cdot\sigma(1)^{-1}=-m_{xy}$.
If $x,y\notin Z$, then
$m^{\prime}_{xy}=\mathbf{b}_{\sigma}(f_{x}(y),g(y))\cdot\epsilon(y)\cdot\sigma(1)^{-1}=m_{xy}$.
And finally if $x\notin Z$ and $y\in Z$,
$m^{\prime}_{xy}=\mathbf{b}_{\sigma}(f_{x}(y),-g(y))\cdot(-\epsilon(y))\cdot\sigma(1)^{-1}=m_{xy}$.
Therefore, $M^{\prime}=I_{Z}\cdot M$.
It is straightforward to check that $\\{f_{x}\mid x\in V\\}$ is the basis of
$L$ associated with $f^{\prime}$ and $g$ by Proposition 21. Then,
$f^{\prime\prime}_{x}=f_{x}$. Let $x\in V$. We have clearly that
$m^{\prime\prime}_{xy}=m_{xy}$ for all $y\in V\setminus Z$. Let now $y\in Z$.
Hence,
$m^{\prime\prime}_{xy}=-\mathbf{b}_{\sigma}(f_{x}(y),g(y))\cdot\epsilon(y)\cdot\sigma(1)^{-1}=-m_{xy}$.
Hence, $M^{\prime\prime}=M\cdot I_{Z}$. ∎
A pair $(p,q)$ of non-zero scalars in $\mathbb{F}$ is said $\sigma$-compatible
if $p^{-1}=\sigma(q)\cdot\sigma(1)^{-1}$ (equivalently
$q^{-1}=\sigma(p)\cdot\sigma(1)^{-1}$). That means that $(q,p)$ is also
$\sigma$-compatible. It is worth noticing that if $(p,q)$ is
$\sigma$-compatible, then $(p^{-1},q^{-1})$ is also $\sigma$-compatible. A
pair $(P,Q)$ of non-singular diagonal $(V,V)$-matrices is said
$\sigma$-compatible if $(p_{xx},q_{xx})$ is $\sigma$-compatible for all $x\in
V$. For instance the pair $(P_{X},P_{X}^{-1})$ is $\sigma$-compatible.
###### Proposition 25
Let $(M,f,g)$ be a matrix representation of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$ and let $(P,Q)$ be a
$\sigma$-compatible pair of diagonal $(V,V)$-matrices. Let $f^{\prime}$ and
$g^{\prime}$ be $\mathbb{K}_{\sigma}$-chains on $V$ such that for all $x\in
V$, $f^{\prime}(x):=q_{xx}\cdot f(x)$ and $g^{\prime}(x):=p_{xx}\cdot g(x)$.
Then, $(P\cdot M\cdot Q^{-1},f^{\prime},g^{\prime})$ is a matrix
representation of $L$.
Proof. Let $\epsilon:V\to\\{+1,-1\\}$ such that $M$ is
$(\sigma,\epsilon)$-symmetric, _i.e._ , $f$ and $g$ are
$\epsilon$-supplementary. It is a straightforward computation to check that
$f^{\prime}$ and $g^{\prime}$ are $\epsilon$-supplementary
$\mathbb{K}_{\sigma}$-chains on $V$. Moreover, $f^{\prime}$ is eulerian to $L$
(because $f$ is). By Proposition 21, there exists a unique basis
$\\{f^{\prime}_{x}\mid x\in V\\}$ of $L$ such that
$(M^{\prime},f^{\prime},g^{\prime})$ is a matrix representation of $L$ with
$m^{\prime}_{xy}:=\mathbf{b}_{\sigma}(f^{\prime}_{x}(y),g^{\prime}(y))\cdot\epsilon(y)\cdot\sigma(1)^{-1}$.
Let $\\{f_{x}\mid x\in V\\}$ be the basis of $L$ associated with $f$ and $g$
by Proposition 19.
For each $x\in V$, we clearly have
$\mathbf{b}_{\sigma}(f^{\prime}(y),p_{xx}\cdot f_{x}(y))=q_{yy}\cdot
q_{xx}^{-1}\cdot\mathbf{b}_{\sigma}(f(y),f_{x}(y))$ for all $x,y\in V$.
Therefore, for all $x\in V$ and all $y\in V\setminus x$, we have
$\displaystyle\mathbf{b}_{\sigma}(f^{\prime}(x),p_{xx}\cdot f_{x}(x))$
$\displaystyle=\epsilon(x)\cdot\sigma(1),$
$\displaystyle\mathbf{b}_{\sigma}(f^{\prime}(y),p_{xx}\cdot f_{x}(y))$
$\displaystyle=0.$
Hence, by Proposition 21 $f^{\prime}_{x}=p_{xx}\cdot f_{x}$. Then, for each
$x,y\in V$, we have
$\displaystyle m^{\prime}_{xy}$ $\displaystyle=\mathbf{b}_{\sigma}(p_{xx}\cdot
f_{x}(y),p_{yy}\cdot g(y))\cdot\epsilon(y)\cdot\sigma(1)^{-1}$
$\displaystyle=p_{xx}\cdot\sigma(p_{yy})\cdot\sigma(1)^{-1}\cdot\left(\mathbf{b}_{\sigma}(f_{x}(y),g(y))\cdot\epsilon(y)\cdot\sigma(1)^{-1}\right)=p_{xx}\cdot
q_{yy}^{-1}\cdot m_{xy}.$
Hence, $(P\cdot M\cdot Q^{-1},f^{\prime},g^{\prime})$ is a matrix
representation of $L$. ∎
We call $(M,f,g)$ a _special matrix representation_ of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$ if
$f(x),g(x)\in\\{c^{*},c_{*}\mid c\in\mathbb{F}^{*}\\}$ for all $x\in V$. A
special case of the following is proved in [19].
###### Lemma 26
Let $(M,f,g)$ be a special matrix representation of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$. Let $f^{\prime}$ be a
$\mathbb{K}_{\sigma}$-chain on $V$ such that
$f^{\prime}(x)\in\\{c^{*},c_{*}\mid c\in\mathbb{F}^{*}\\}$ for all $x\in V$.
Then, $f^{\prime}$ is eulerian if and only if $M[X]$ is non-singular with
$X:=\\{x\in V\mid f^{\prime}(x)\neq c\cdot f(x)$ for some
$c\in\mathbb{F}^{*}\\}$.
Proof. (Proof already present in [19].) Let $\\{f_{x}\mid x\in V\\}$ be the
basis of $L$ associated with $f$ and $g$ from Proposition 19. For each $y\in
X$, there exists $d_{y}\in\mathbb{F}^{*}$ such that
$\displaystyle f^{\prime}(y)$ $\displaystyle=\begin{cases}d_{y}\cdot
f(y)&\textrm{if $y\notin X$},\\\ d_{y}\cdot g(y)&\textrm{if $y\in
X$}.\end{cases}$
Assume that $M[X]$ is non-singular and let $h\in L$ such that
$\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))=0$ for all $y\in V$. Let
$h=\sum\limits_{z\in V}c_{z}\cdot f_{z}$. For all $y\notin X$, we have
$\displaystyle\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$
$\displaystyle=\mathbf{b}_{\sigma}\left(\sum\limits_{z\in V}\left(c_{z}\cdot
m_{zy}\cdot f(y)\right)+c_{y}\cdot g(y),d_{y}\cdot f(y)\right)$
$\displaystyle=\mathbf{b}_{\sigma}(c_{y}\cdot g(y),d_{y}\cdot f(y))$
$\displaystyle=-c_{y}\cdot\sigma(d_{y})\cdot\epsilon(y)\cdot\sigma(1).$
Hence, $c_{y}=0$ for all $y\notin X$. If $y\in X$, then
$\displaystyle\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$
$\displaystyle=\mathbf{b}_{\sigma}\left(\sum\limits_{z\in X}\left(c_{z}\cdot
m_{zy}\cdot f(y)\right)+c_{y}\cdot g(y),d_{y}\cdot g(y)\right)$
$\displaystyle=\sum\limits_{z\in X}\left(c_{z}\cdot
m_{zy}\cdot\frac{\sigma(d_{y})}{\sigma(1)}\cdot\mathbf{b}_{\sigma}(f(y),g(y))\right)$
$\displaystyle=(\sigma(d_{y})\cdot\epsilon(y))\cdot\sum\limits_{z\in
X}c_{z}\cdot m_{zy}.$
For $\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$ to being $0$, we must have
$\sum\limits_{z\in X}\left(c_{z}\cdot m_{zy}\right)=0$. But, since $M[X]$ is
non-singular, we have $\sum\limits_{z\in X}\left(c_{z}\cdot m_{zy}\right)=0$
for all $y\in X$ if and only if $c_{z}=0$ for all $z\in X$. Therefore, we have
$h=0$, _i.e._ , $f^{\prime}$ is eulerian.
Assume now that $M[X]$ is singular. Hence, there exist $c_{z}$ for $z\in X$,
not all zero, such that for all $y\in X$, $\sum\limits_{z\in
X}\left(c_{z}\cdot m_{zy}\right)=0$. Let $h:=\sum\limits_{z\in X}c_{z}\cdot
f_{z}$, which is not zero. Hence, for each $y\notin X$,
$\displaystyle\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$
$\displaystyle=\frac{\sigma(d_{y})}{\sigma(1)}\cdot\mathbf{b}_{\sigma}\left(\sum\limits_{z\in
X}\left(c_{z}\cdot f_{z}(y)\right),f(y)\right)$
$\displaystyle=\frac{\sigma(d_{y})}{\sigma(1)}\cdot\left(\sum\limits_{z\in
X}\left(c_{z}\cdot m_{zy}\cdot\mathbf{b}_{\sigma}(f(y),f(y))\right)\right)=0$
For each $y\in X$,
$\displaystyle\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))$
$\displaystyle=\frac{\sigma(d_{y})}{\sigma(1)}\cdot\mathbf{b}_{\sigma}\left(\sum\limits_{z\in
X}\left(c_{z}\cdot f_{z}(y)\right),g(y)\right)$
$\displaystyle=\frac{\sigma(d_{y})}{\sigma(1)}\cdot\left(\sum\limits_{z\in
X}\left(c_{z}\cdot m_{zy}\cdot\mathbf{b}_{\sigma}(f(y),g(y))\right)\right)$
$\displaystyle=\sigma(d_{y})\cdot\epsilon(y)\cdot\left(\sum\limits_{z\in
X}c_{z}\cdot m_{zy}\right)=0$
Since $h$ is not zero and $\mathbf{b}_{\sigma}(h(y),f^{\prime}(y))=0$ for all
$y\in V$, $f^{\prime}$ is not eulerian. ∎
We now relate special matrix representations of a lagrangian
$\mathbb{K}_{\sigma}$-chain group with the ones of its $\alpha\beta$-minors.
###### Lemma 27
Let $\\{\alpha,\beta\\}\subseteq\\{c^{*},c_{*}\mid c\in\mathbb{F}^{*}\\}$ be
minor-compatible. Let $(M,f,g)$ be a special matrix representation of a
lagrangian $\mathbb{K}_{\sigma}$-chain group $L$ on $V$, and let $x\in V$.
Then, $(M[V\setminus x],\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f\,\smash{\vrule
height=3.88889pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{f\,\smash{\vrule
height=2.77777pt,depth=2.12502pt}}_{\,(V\setminus
x)}},\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{g\,\smash{\vrule
height=2.41112pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{g\,\smash{\vrule
height=1.72221pt,depth=2.12502pt}}_{\,(V\setminus x)}})$ is a special matrix
representation of $L\operatorname{\parallel}\limits_{\alpha}x$ if
$f(x)=c\cdot\alpha$, otherwise of $L\operatorname{\parallel}\limits_{\beta}x$.
Proof. We can assume by symmetry that $f(x)=c\cdot\alpha$. Let $\\{f_{x}\mid
x\in V\\}$ be the basis of $L$ associated with $f$ and $g$ from Proposition
19.
For all $y\in V\setminus x$, we have $f_{y}(x)=m_{yx}\cdot c\cdot\alpha$.
Hence, $f_{y}\in L\operatorname{\parallel}\limits_{\alpha}x$ for all $y\in
V\setminus x$. We claim that the set $\\{\mathchoice{{f_{y}\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f_{y}\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f_{y}\,\smash{\vrule
height=3.88889pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{f_{y}\,\smash{\vrule
height=2.77777pt,depth=2.12502pt}}_{\,(V\setminus x)}}\mid y\in V\setminus
x\\}$ is linearly independent. Suppose the contrary and let
$h:=\sum\limits_{y\in V\setminus x}c_{y}\cdot f_{y}\in L$ with
$\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{h\,\smash{\vrule
height=3.88889pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{h\,\smash{\vrule
height=2.77777pt,depth=2.12502pt}}_{\,(V\setminus x)}}=0$. Hence,
$h(x)=\sum\limits_{y\in V\setminus x}\left(c_{y}\cdot m_{yx}\cdot
c\cdot\alpha\right)$ and $h(y)=0$ for all $y\in V\setminus x$. Therefore,
$\mathbf{b}_{\sigma}(h(z),f(z))=0$ for all $z\in V$, contradicting the
eulerian of $f$. By Proposition 14,
$L\operatorname{\parallel}\limits_{\alpha}x$ is lagrangian, _i.e._ ,
$\dim(L\operatorname{\parallel}\limits_{\alpha}x)=|V\setminus x|$, hence
$\\{\mathchoice{{f_{y}\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f_{y}\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f_{y}\,\smash{\vrule
height=3.88889pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{f_{y}\,\smash{\vrule
height=2.77777pt,depth=2.12502pt}}_{\,(V\setminus x)}}\mid y\in V\setminus
x\\}$ is a basis for $L\operatorname{\parallel}\limits_{\alpha}x$. But, this
is actually the basis of $(M[V\setminus x],\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{f\,\smash{\vrule
height=3.88889pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{f\,\smash{\vrule
height=2.77777pt,depth=2.12502pt}}_{\,(V\setminus
x)}},\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.97502pt}}_{\,(V\setminus x)}}{{g\,\smash{\vrule
height=2.41112pt,depth=2.12502pt}}_{\,(V\setminus x)}}{{g\,\smash{\vrule
height=1.72221pt,depth=2.12502pt}}_{\,(V\setminus x)}})$ from Proposition 19.
∎
We have then the following.
###### Proposition 28
Let $\\{\alpha,\beta\\}\subseteq\\{c^{*},c_{*}\mid c\in\mathbb{F}^{*}\\}$ be
minor-compatible. Let $L$ and $L^{\prime}$ be lagrangian
$\mathbb{K}_{\sigma}$-chain groups on $V$ and $V^{\prime}$ respectively. Let
$(M,f,g)$ and $(M^{\prime},f^{\prime},g^{\prime})$ be special matrix
representations of $L$ and $L^{\prime}$ respectively with $f(x):=\pm\alpha,\
g(x):=\beta$ for all $x\in V$, and $f^{\prime}(x):=\pm\alpha,\
g^{\prime}(x):=\beta$ for all $x\in V^{\prime}$. If
$L^{\prime}=L\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y$,
then $M^{\prime}=\big{(}(M/M[A])[V^{\prime}]\big{)}\cdot I_{Z}$ with
$A\subseteq X$ and $Z:=\\{x\in V^{\prime}\mid f^{\prime}(x)=-f(x)\\}$.
Proof. If $X=\emptyset$, then by Lemma 27
$(M[V^{\prime}],\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}},\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.53944pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.53944pt}}_{\,V^{\prime}}})$ is a special matrix
representation of $L^{\prime}$. By hypothesis,
$g^{\prime}=\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.53944pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.53944pt}}_{\,V^{\prime}}}$. If we let $Z:=\\{x\in
V^{\prime}\mid f^{\prime}(x)=-f(x)\\}$, then by Proposition 24
$(M[V^{\prime}]\cdot I_{Z},f^{\prime},g^{\prime})$ is a special matrix
representation of $L^{\prime}$. Therefore, $M^{\prime}=M[V^{\prime}]\cdot
I_{Z}$ by Proposition 21. We can now assume that $X\neq\emptyset$ and is
minimal with the property that there exists $Y$ such that
$L^{\prime}=L\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y$.
We claim that $M[X]$ is non-singular. Assume the contrary and let $f_{1}$ be
the $\mathbb{K}_{\sigma}$-chain on $V$ where $f_{1}(x)=f(x)$ if $x\notin X$,
and $f_{1}(x)=g(x)$ otherwise. By Lemma 26, $f_{1}$ is not eulerian. Hence,
there exists $h\in L$ a non-zero $\mathbb{K}_{\sigma}$-chain on $V$ such that
$\mathbf{b}_{\sigma}(h(x),f_{1}(x))=0$ for all $x\in V$. Then,
$\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}\in L^{\prime}$. And since
$\mathchoice{{f_{1}\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f_{1}\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f_{1}\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f_{1}\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}=\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}=f^{\prime}$, we have
$\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}=0$ ($f^{\prime}$ is
eulerian). Moreover, there exists $z\in X$ such that $h(z)\neq 0$, otherwise
it contradicts the fact that $f$ is eulerian (recall that for all $y\in
V\setminus X,\ f_{1}(y)=f(y)$). By Lemma 8, we have $h(z)=c_{z}\cdot\beta$,
$c_{z}\in\mathbb{F}^{*}$. Let $h^{\prime}\in L$ such that
$\mathchoice{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=4.77779pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=3.66667pt,depth=1.53944pt}}_{\,V^{\prime}}}\in L^{\prime}$. Then,
$\mathbf{b}_{\sigma}(h^{\prime}(z),\beta)=0$, and hence
$\mathbf{b}_{\sigma}(h(z),h^{\prime}(z))=0$. Thus by Lemma 8,
$h^{\prime}(z)=c_{h^{\prime}}\cdot h(z)$. Hence,
$\mathchoice{{(h^{\prime}-c_{h^{\prime}}\cdot h)\,\smash{\vrule
height=6.80002pt,depth=2.12502pt}}_{\,V^{\prime}}}{{(h^{\prime}-c_{h^{\prime}}\cdot
h)\,\smash{\vrule
height=6.80002pt,depth=2.12502pt}}_{\,V^{\prime}}}{{(h^{\prime}-c_{h^{\prime}}\cdot
h)\,\smash{\vrule
height=4.77779pt,depth=1.55833pt}}_{\,V^{\prime}}}{{(h^{\prime}-c_{h^{\prime}}\cdot
h)\,\smash{\vrule height=3.66667pt,depth=1.55833pt}}_{\,V^{\prime}}}\in
L\operatorname{\parallel}\limits_{\beta}(X\setminus
z)\operatorname{\parallel}\limits_{\alpha}(Y\cup z)$. But, we have
$\mathchoice{{(h^{\prime}-c_{h^{\prime}}\cdot h)\,\smash{\vrule
height=6.80002pt,depth=2.12502pt}}_{\,V^{\prime}}}{{(h^{\prime}-c_{h^{\prime}}\cdot
h)\,\smash{\vrule
height=6.80002pt,depth=2.12502pt}}_{\,V^{\prime}}}{{(h^{\prime}-c_{h^{\prime}}\cdot
h)\,\smash{\vrule
height=4.77779pt,depth=1.55833pt}}_{\,V^{\prime}}}{{(h^{\prime}-c_{h^{\prime}}\cdot
h)\,\smash{\vrule
height=3.66667pt,depth=1.55833pt}}_{\,V^{\prime}}}=\mathchoice{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=6.80002pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=4.77779pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h^{\prime}\,\smash{\vrule
height=3.66667pt,depth=1.53944pt}}_{\,V^{\prime}}}$ because
$\mathchoice{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{h\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}=0$. Therefore,
$L\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y\subseteq
L\operatorname{\parallel}\limits_{\beta}(X\setminus
z)\operatorname{\parallel}\limits_{\alpha}(Y\cup z)$. By Proposition 14,
$\dim(L\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y)=|V^{\prime}|$
and $\dim(L\operatorname{\parallel}\limits_{\beta}(X\setminus
z)\operatorname{\parallel}\limits_{\alpha}(Y\cup z))=|V\setminus(X\setminus
z)\setminus(Y\cup z)|=|V^{\prime}|$. Hence,
$L\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y=L\operatorname{\parallel}\limits_{\beta}(X\setminus
z)\operatorname{\parallel}\limits_{\alpha}(Y\cup z)$. This contradicts the
assumption that $X$ is minimal. Hence, $M[X]$ is non-singular.
Let $M_{1}:=P_{X}\cdot(M*X)$. By Proposition 23, there exist $f_{2}$ and
$g_{2}$ such that $L=(M_{1},f_{2},g_{2})$. By Lemma 27, $(M_{1}[V\setminus
X],\mathchoice{{f_{2}\,\smash{\vrule
height=5.55557pt,depth=2.01506pt}}_{\,V\setminus X}}{{f_{2}\,\smash{\vrule
height=5.55557pt,depth=2.01506pt}}_{\,V\setminus X}}{{f_{2}\,\smash{\vrule
height=3.88889pt,depth=1.43933pt}}_{\,V\setminus X}}{{f_{2}\,\smash{\vrule
height=2.77777pt,depth=1.43933pt}}_{\,V\setminus
X}},\mathchoice{{g_{2}\,\smash{\vrule
height=3.44444pt,depth=2.01506pt}}_{\,V\setminus X}}{{g_{2}\,\smash{\vrule
height=3.44444pt,depth=2.01506pt}}_{\,V\setminus X}}{{g_{2}\,\smash{\vrule
height=2.41112pt,depth=1.43933pt}}_{\,V\setminus X}}{{g_{2}\,\smash{\vrule
height=1.72221pt,depth=1.43933pt}}_{\,V\setminus X}})$ is a matrix
representation of $L\operatorname{\parallel}\limits_{\beta}X$. Notice that
$\mathchoice{{f_{2}\,\smash{\vrule
height=5.55557pt,depth=2.01506pt}}_{\,V\setminus X}}{{f_{2}\,\smash{\vrule
height=5.55557pt,depth=2.01506pt}}_{\,V\setminus X}}{{f_{2}\,\smash{\vrule
height=3.88889pt,depth=1.43933pt}}_{\,V\setminus X}}{{f_{2}\,\smash{\vrule
height=2.77777pt,depth=1.43933pt}}_{\,V\setminus
X}}=\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.01506pt}}_{\,V\setminus X}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.01506pt}}_{\,V\setminus X}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.43933pt}}_{\,V\setminus X}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.43933pt}}_{\,V\setminus X}}$ and
$\mathchoice{{g_{2}\,\smash{\vrule
height=3.44444pt,depth=2.01506pt}}_{\,V\setminus X}}{{g_{2}\,\smash{\vrule
height=3.44444pt,depth=2.01506pt}}_{\,V\setminus X}}{{g_{2}\,\smash{\vrule
height=2.41112pt,depth=1.43933pt}}_{\,V\setminus X}}{{g_{2}\,\smash{\vrule
height=1.72221pt,depth=1.43933pt}}_{\,V\setminus
X}}=\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.01506pt}}_{\,V\setminus X}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.01506pt}}_{\,V\setminus X}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.43933pt}}_{\,V\setminus X}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.43933pt}}_{\,V\setminus X}}$. By Lemma 27,
$(M_{1}[V^{\prime}],\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}},\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.53944pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.53944pt}}_{\,V^{\prime}}})$ is a special matrix
representation of
$L\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y$.
But, $f^{\prime}=\pm\mathchoice{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=5.55557pt,depth=2.00412pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=3.88889pt,depth=1.53944pt}}_{\,V^{\prime}}}{{f\,\smash{\vrule
height=2.77777pt,depth=1.53944pt}}_{\,V^{\prime}}}$ and
$g^{\prime}=\mathchoice{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=3.44444pt,depth=2.00412pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=2.41112pt,depth=1.53944pt}}_{\,V^{\prime}}}{{g\,\smash{\vrule
height=1.72221pt,depth=1.53944pt}}_{\,V^{\prime}}}$. Let $Z:=\\{x\in
V^{\prime}\mid f^{\prime}(x)=-f(x)\\}$. By Proposition 24,
$(M_{1}[V^{\prime}]\cdot I_{Z},f^{\prime},g^{\prime})$ is a special matrix
representation of $L^{\prime}$. Therefore, $M^{\prime}=M_{1}[V^{\prime}]\cdot
I_{Z}$ by Proposition 21. And, the fact that
$M_{1}[V^{\prime}]=(M/M[X])[V^{\prime}]$ finishes the proof. ∎
We are now ready to prove the principal result of the paper.
###### Theorem 29
Let $\mathbb{F}$ be a finite field and $k$ a positive integer. For every
infinite sequence $M_{1},M_{2},\ldots$ of
$(\sigma_{i},\epsilon_{i})$-symmetric $(V_{i},V_{i})$-matrices over
$\mathbb{F}$ of $\mathbb{F}$-rank-width at most $k$, there exist $i<j$ such
that $M_{i}$ is isomorphic to $\big{(}(M_{j}/M_{j}[A])[V^{\prime}]\big{)}\cdot
I_{Z}$ with $A\subseteq V_{j}\setminus V^{\prime}$ and $Z\subseteq
V^{\prime}$.
Proof. Let $\alpha:=c^{*}$ and $\beta:=\widetilde{c^{*}}$ for some
$c\in\mathbb{F}^{*}$. Since the set of sesqui-morphisms over $\mathbb{F}$ is
finite, we can assume by taking a sub-sequence that each matrix $M_{i}$ is
$(\sigma,\epsilon_{i})$-symmetric, for some sesqui-morphism
$\sigma:\mathbb{F}\to\mathbb{F}$. For each $i$, let $f_{i}$ and $g_{i}$ be
$\mathbb{K}_{\sigma}$-chains on $V_{i}$ with
$f_{i}(x):=\epsilon_{i}(x)\cdot\alpha$ and $g_{i}(x):=\beta$ for all $x\in
V_{i}$. Let $L_{i}$ be $(M_{i},f_{i},g_{i})$. By Theorem 17, there exist $i<j$
such that $L_{i}$ is simply isomorphic to an $\alpha\beta$-minor of $L_{j}$.
Let $X,Y\subseteq V_{j}$ such that $L_{i}$ is simply isomorphic to
$L_{j}\operatorname{\parallel}\limits_{\beta}X\operatorname{\parallel}\limits_{\alpha}Y$.
Let $V^{\prime}:=V_{j}\setminus(X\cup Y)$. By Proposition 28, $M_{i}$ is
isomorphic to $\big{(}(M_{j}/M_{j}[A])[V^{\prime}]\big{)}\cdot I_{Z}$ with
$A\subseteq X$ and $Z\subseteq V^{\prime}$.∎
Since each symmetric (or skew-symmetric) $(V,V)$-matrix is a
$(\sigma,\epsilon)$-symmetric $(V,V)$-matrix with $\epsilon(x)=1$ for all
$x\in V$, and $\sigma$ being symmetric (or skew-symmetric), Theorem 3 is a
corollary of Theorem 29. It is worth noticing as noted in [19] that the well-
quasi-ordering results in [11, 17, 21] are corollaries of Theorem 3, hence of
Theorem 29. We give some other corollaries about graphs in the next section.
## 5 Applications to Graphs
Clique-width was defined by Courcelle et al. [6] for graphs (directed or not,
with edge-colours or not). But, the notion of rank-width introduced by Oum and
Seymour in [20] and studied by Oum (see for instance [17, 18]) concerned only
undirected graphs. Rao and myself we generalised in [14] the notion of rank-
width to directed graphs, and more generally to edge-coloured graphs. We give
well-quasi-ordering theorems for directed graphs and edge-coloured graphs.
### 5.1 The Case of Edge-Coloured Graphs
Let $C$ be a (possibly infinite) set that we call the _colours_. A _$C$
-coloured graph_ $G$ is a tuple $(V_{G},E_{G},\ell_{G})$ where $(V_{G},E_{G})$
is a directed graph and $\ell_{G}:E_{G}\to 2^{C}\setminus\\{\emptyset\\}$ is a
function. Its associated _underlying graph_ $\mathpzc{u}(G)$ is the directed
graph $(V_{G},E_{G})$. Two $C$-coloured graphs $G$ and $H$ are isomorphic if
there is an isomorphism $h$ between $\mathpzc{u}(G)$ and $\mathpzc{u}(H)$ such
that for every $(x,y)\in E_{G}$, $\ell_{G}((x,y))=\ell_{H}((h(x),h(y))$. We
call $h$ an _isomorphism_ between $G$ and $H$. It is worth noticing that an
edge-uncoloured graph can be seen as an edge-coloured graph where all the
edges have the same colour.
The notion of rank-width of $C$-coloured graphs is based on the
$\mathbb{F}$-rank-width of $(\sigma,\epsilon)$-symmetric matrices. Let
$\mathbb{F}$ be a field. An _$\mathbb{F}^{*}$ -graph_ $G$ is an
$\mathbb{F}^{*}$-coloured graph where for every edge $(x,y)\in E_{G}$, we have
$\ell_{G}((x,y))\in\mathbb{F}^{*}$, _i.e._ , each edge has exactly one colour
in $\mathbb{F}^{*}$. It is clear that every directed graph is an
$\mathbb{F}_{2}^{*}$-graph. One interesting point is that every
$\mathbb{F}^{*}$-graph $G$ can be represented by a $(V_{G},V_{G})$-matrix
$M_{G}$ over $\mathbb{F}$, that generalises the adjacency matrix of directed
graphs, such that
$\displaystyle{M_{G}}[{x},{y}]:=\begin{cases}\ell_{G}((x,y))&\textrm{if
$(x,y)\in E_{G}$},\\\ 0&\textrm{otherwise}.\end{cases}$
If $M_{G}$ is $(\sigma,\epsilon)$-symmetric, we call $G$ a
_$(\sigma,\epsilon)$ -symmetric $\mathbb{F}^{*}$-graph_. It is worth noticing
that in this case $\mathpzc{u}(G)$ is undirected. Not all
$\mathbb{F}^{*}$-graphs are $(\sigma,\epsilon)$-symmetric, however we have the
following.
###### Proposition 30 ([14])
Let $\mathbb{F}$ be a finite field. Then, one can construct a sesqui-morphism
$\sigma:\mathbb{F}^{2}\to\mathbb{F}^{2}$ where $\mathbb{F}^{2}$ is an
algebraic extension of $\mathbb{F}$ of order $2$. Moreover, for every
$\mathbb{F}^{*}$-graph $G$, one can associate a $\sigma$-symmetric
$(\mathbb{F}^{2})^{*}$-graph $\widetilde{G}$ such that for every
$\mathbb{F}^{*}$-graphs $G$ and $H$, $\widetilde{G}$ and $\widetilde{H}$ are
isomorphic if and only if $G$ and $H$ are isomorphic.
In order to define a notion of rank-width for $C$-coloured graphs, we proceed
as follows. For a $C$-coloured graph $G$, let $\Pi(G)\subseteq 2^{C}$ be the
set of subsets of $C$ appearing as colours of edges in $G$.
1. 1.
take an injection $i:\Pi(G)\to\mathbb{F}^{*}$ for a large enough finite field
$\mathbb{F}$ and let $G^{\prime}$ be the $\mathbb{F}^{*}$-graph obtained from
$G$ by replacing each edge colour $A\subseteq C$ by $i(A)$. If the
$\mathbb{F}^{*}$-graph $G^{\prime}$ is $(\sigma,\epsilon)$-symmetric for some
sesqui-morphism $\sigma:\mathbb{F}\to\mathbb{F}$, then define the
$\mathbb{F}$-rank-width of $G$ as the $\mathbb{F}$-rank-width of
$M_{G^{\prime}}$. Otherwise,
2. 2.
take $\widetilde{G^{\prime}}$ from Proposition 30.
$M_{\widetilde{G^{\prime}}}$ is $\sigma$-symmetric for some
$\sigma:\mathbb{F}^{2}\to\mathbb{F}^{2}$. The _$\mathbb{F}^{2}$ -rank-width_
of $G$ will be defined as the $\mathbb{F}^{2}$-rank-width of
$M_{\widetilde{G^{\prime}}}$.
The choice of the injection in step (1) above is not unique and leads to
different representations of $C$-coloured graphs, and then different
parameters. However, as proved in [14], the parameters are equivalent.
Therefore, in order to investigate the structure of $C$-coloured graphs, we
can concentrate our efforts in $(\sigma,\epsilon)$-symmetric
$\mathbb{F}^{*}$-graphs. The authors in [14] did only consider
$\sigma$-symmetric graphs. We relax this constraint because we may have some
$\mathbb{F}^{*}$-graphs which are $(\sigma,\epsilon)$-symmetric but are not
$\sigma^{\prime}$-symmetric at all, for all sesqui-morphisms
$\sigma^{\prime}:\mathbb{F}\to\mathbb{F}$. Examples of such graphs are
$\mathbb{F}^{*}$-graphs $G$ where $M_{G}$ is obtained from a
$\sigma$-symmetric matrix by multiplying some rows and/or columns by $-1$.
All the results, but the well-quasi-ordering theorem, concerning the rank-
width of undirected graphs are generalised in [14] to the $\mathbb{F}$-rank-
width of $\sigma$-symmetric loop-free $\mathbb{F}^{*}$-graphs. These results
extend easily to $(\sigma,\epsilon)$-symmetric $\mathbb{F}^{*}$-graphs. We
prove here two well-quasi-ordering theorems for $(\sigma,\epsilon)$-symmetric
$\mathbb{F}^{*}$-graphs. For that, we will derive from the principal pivot
transform two notions of pivot-minor: one that preserves the loop-freeness and
one that does not.
We recall that a pair $(P,Q)$ of non-singular diagonal $(V,V)$-matrices is
$\sigma$-compatible if $p_{xx}^{-1}=\sigma(q_{xx})\cdot\sigma(1)^{-1}$
(equivalently $q_{xx}^{-1}=\sigma(p_{xx})\cdot\sigma(1)^{-1}$) for all $x\in
V$, and for $X\subseteq V$, $P_{X}$ and $I_{X}$ are the non-singular diagonal
$(V,V)$-matrices where
$\displaystyle P_{X}[x,x]$ $\displaystyle:=\begin{cases}\sigma(-1)&\textrm{if
$x\in X$},\\\ 1&\textrm{otherwise},\end{cases}$ and $\displaystyle\quad
I_{X}[x,x]:=\begin{cases}-1&\textrm{if $x\in X$},\\\
1&\textrm{otherwise.}\end{cases}$
###### Definition 31 ($\sigma$-loop-pivot complementation)
Let $G$ be a $(\sigma,\epsilon)$-symmetric $\mathbb{F}^{*}$-graph and let
$X\subseteq V_{G}$ such that $M_{G}[X]$ is non-singular. An
$\mathbb{F}^{*}$-graph $G^{\prime}$ is a _$\sigma$ -loop-pivot complementation
of $G$ at $X$_ if $M_{G^{\prime}}:=I_{Z}\cdot P\cdot P_{X}\cdot(M*X)\cdot
Q^{-1}\cdot I_{Z^{\prime}}$ for some $Z,Z^{\prime}\subseteq V_{G}$, and
$(P,Q)$ a pair of $\sigma$-compatible diagonal $(V_{G},V_{G})$-matrices.
An $\mathbb{F}^{*}$-graph $G^{\prime}$ is _$\sigma$ -loop-pivot equivalent_ to
$G$ if $G^{\prime}$ is obtained from $G$ by applying a sequence of
$\sigma$-loop-pivot complementations. An $\mathbb{F}^{*}$-graph $H$ is a
$\sigma$-loop-pivot-minor of $G$ if $H$ is isomorphic to
$G^{\prime}[V^{\prime}],\ V^{\prime}\subseteq V_{G}$, where $G^{\prime}$ is
$\sigma$-loop-pivot equivalent to $G$.
The $\sigma$-loop-pivot complementation does not clearly preserve the loop-
freeness. A corollary of Theorem 22, and Propositions 23, 24 and 25 is the
following.
###### Corollary 32
1. 1.
Let $G$ be a $(\sigma,\epsilon)$-symmetric $\mathbb{F}^{*}$-graph. If
$G^{\prime}$ is $\sigma$-loop-pivot equivalent to $G$, then $G^{\prime}$ is
$(\sigma,\epsilon^{\prime})$-symmetric for some
$\epsilon^{\prime}:V_{G}\to\\{+1,-1\\}$.
2. 2.
Let $G$ and $G^{\prime}$ be respectively $(\sigma,\epsilon)$ and
$(\sigma,\epsilon^{\prime})$-symmetric $\mathbb{F}^{*}$-graphs. If
$G^{\prime}$ is $\sigma$-loop-pivot equivalent to $G$, then
$\operatorname{rwd}^{{\mathbb{F}}}(G^{\prime})=\operatorname{rwd}^{{\mathbb{F}}}(G)$.
If $G^{\prime}$ is a $\sigma$-loop-pivot-minor of $G$, then
$\operatorname{rwd}^{{\mathbb{F}}}(G^{\prime})\leq\operatorname{rwd}^{{\mathbb{F}}}(G)$.
We now introduce a variant of the $\sigma$-loop-pivot complementation that
preserves the loop-freeness and prove that Corollary 32 still holds.
###### Definition 33 ($\sigma$-pivot complementation)
Let $G$ be a $(\sigma,\epsilon)$-symmetric loop-free $\mathbb{F}^{*}$-graph
and let $X\subseteq V_{G}$ such that $M_{G}[X]$ is non-singular. A loop-free
$\mathbb{F}^{*}$-graph $H$ is a _$\sigma$ -pivot complementation of $G$ at
$X$_ if $M_{H}$ is obtained from $M_{G^{\prime}}$, $G^{\prime}$ a
$\sigma$-loop-pivot complementation of $G$ at $X$, by replacing each diagonal
entry by $0$.
A loop-free $\mathbb{F}^{*}$-graph $G^{\prime}$ is _$\sigma$ -pivot
equivalent_ to $G$ if $G^{\prime}$ is obtained from $G$ by applying a sequence
of $\sigma$-pivot complementations. A loop-free $\mathbb{F}^{*}$-graph $H$ is
a $\sigma$-pivot-minor of $G$ if $H$ is isomorphic to
$G^{\prime}[V^{\prime}],\ V^{\prime}\subseteq V_{G}$, where $G^{\prime}$ is
$\sigma$-pivot equivalent to $G$.
It is clear that the $\sigma$-pivot complementation preserves the loop-
freeness. The proof of the following is straightforward.
###### Proposition 34
Let $(M,f,g)$ be a matrix representation of a lagrangian
$\mathbb{K}_{\sigma}$-chain group $L$ on $V$ and let $M^{\prime}$ be obtained
from $M$ by replacing each diagonal entry by $0$. Let $g^{\prime}$ be the
$\mathbb{K}_{\sigma}$-chain on $V$ with $g^{\prime}(x):=m_{xx}\cdot
f(x)+g(x)$. Then, $(M^{\prime},f,g^{\prime})$ is a matrix representation of
$L$.
The following is hence true.
###### Corollary 35
1. 1.
Let $G$ be a $(\sigma,\epsilon)$-symmetric loop-free $\mathbb{F}^{*}$-graph.
If $G^{\prime}$ is $\sigma$-pivot equivalent to $G$, then $G^{\prime}$ is
$(\sigma,\epsilon^{\prime})$-symmetric for some
$\epsilon^{\prime}:V_{G}\to\\{+1,-1\\}$.
2. 2.
Let $G$ and $G^{\prime}$ be respectively $(\sigma,\epsilon)$ and
$(\sigma,\epsilon^{\prime})$-symmetric loop-free $\mathbb{F}^{*}$-graphs. If
$G^{\prime}$ is $\sigma$-pivot equivalent to $G$, then
$\operatorname{rwd}^{{\mathbb{F}}}(G^{\prime})=\operatorname{rwd}^{{\mathbb{F}}}(G)$.
If $G^{\prime}$ is a $\sigma$-pivot-minor of $G$, then
$\operatorname{rwd}^{{\mathbb{F}}}(G^{\prime})\leq\operatorname{rwd}^{{\mathbb{F}}}(G)$.
As corollaries of Theorem 29, we have the following well-quasi-ordering
theorems for $\mathbb{F}^{*}$-graphs.
###### Theorem 36
Let $\mathbb{F}$ be a finite field and $k$ a positive integer. For every
infinite sequence $G_{1},G_{2},\ldots$ of
$(\sigma_{i},\epsilon_{i})$-symmetric $\mathbb{F}^{*}$-graphs of
$\mathbb{F}$-rank-width at most $k$, there exist $i<j$ such that $G_{i}$ is
isomorphic a $\sigma$-loop-pivot-minor of $G_{j}$.
Proof. Let $M_{G_{1}},M_{G_{2}},\ldots$ be the infinite sequence of
$(\sigma_{i},\epsilon_{i})$-symmetric $(V_{G_{i}},V_{G_{i}})$-matrices over
$\mathbb{F}$ associated with the infinite sequence $G_{1},G_{2},\ldots$. By
definition,
$\operatorname{rwd}^{{\mathbb{F}}}(G_{i})=\operatorname{rwd}^{{\mathbb{F}}}(M_{G_{i}})$.
From Theorem 29, there exist $i<j$ such that $M_{G_{i}}$ is isomorphic to
$\big{(}(M_{G_{j}}/M_{G_{j}}[A])[V^{\prime}]\big{)}\cdot I_{Z}$ with
$A,V^{\prime},Z\subseteq V_{G_{j}}$. But, that means that $G_{i}$ is
isomorphic to a $\sigma$-loop-pivot-minor of $G_{j}$. ∎
###### Theorem 37
Let $\mathbb{F}$ be a finite field and $k$ a positive integer. For every
infinite sequence $G_{1},G_{2},\ldots$ of
$(\sigma_{i},\epsilon_{i})$-symmetric loop-free $\mathbb{F}^{*}$-graphs of
$\mathbb{F}$-rank-width at most $k$, there exist $i<j$ such that $G_{i}$ is
isomorphic to a $\sigma$-pivot-minor of $G_{j}$.
Proof. Let $M_{G_{1}},M_{G_{2}},\ldots$ be the infinite sequence of
$(\sigma_{i},\epsilon_{i})$-symmetric $(V_{G_{i}},V_{G_{i}})$-matrices over
$\mathbb{F}$ associated with the infinite sequence $G_{1},G_{2},\ldots$. By
definition,
$\operatorname{rwd}^{{\mathbb{F}}}(G_{i})=\operatorname{rwd}^{{\mathbb{F}}}(M_{G_{i}})$.
From Theorem 29, there exist $i<j$ such that $M_{G_{i}}$ is isomorphic to
$((M_{G_{j}}/M_{G_{j}}[A])[V^{\prime}])\cdot I_{Z}$ with
$A,V^{\prime},Z\subseteq V_{G_{j}}$. Since, $G_{i}$ is loop-free, this means
that the diagonal entries of
$\big{(}(M_{G_{j}}/M_{G_{j}}[A])[V^{\prime}]\big{)}\cdot I_{Z}$ are equal to
$0$. Hence, $(M_{G_{j}}*A)[V^{\prime}]$ has only zero in its diagonal entries.
Then, $G_{i}$ is isomorphic to a $\sigma$-pivot-minor of $G_{j}$. ∎
### 5.2 A Specialisation to Directed Graphs
We discuss in this section a corollary about directed graphs. Let us first
recall the rank-width notion of directed graphs. We recall that
$\mathbb{F}_{4}$ is the finite field of order four. We let
$\\{0,1,\mathbb{a},{\mathbb{a}^{2}}\\}$ be its elements with the property that
$1+\mathbb{a}+{\mathbb{a}^{2}}=0$ and $\mathbb{a}^{3}=1$. Moreover, it is of
characteristic $2$. We let $\sigma_{4}:\mathbb{F}_{4}\to\mathbb{F}_{4}$ be the
automorphism where $\sigma_{4}(\mathbb{a})={\mathbb{a}^{2}}$ and
$\sigma_{4}({\mathbb{a}^{2}})=\mathbb{a}$. It is clearly a sesqui-morphism.
For every directed graph $G$, let
$\widetilde{G}:=(V_{G},E_{G}\cup\\{(y,x)|(x,y)\in
E_{G}\\},\ell_{\widetilde{G}})$ be the
$\operatorname{\mathbb{F}_{4}}^{*}$-graph where for every pair of vertices
$(x,y)$:
$\displaystyle\ell_{\widetilde{G}}((x,y))$
$\displaystyle:=\begin{cases}1&\textrm{if $(x,y)\in E_{G}\ \textrm{and}\
(y,x)\in E_{G}$},\\\ \mathbb{a}&\textrm{$(x,y)\in E_{G}\ \textrm{and}\
(y,x)\notin E_{G}$},\\\ {\mathbb{a}^{2}}&\textrm{$(y,x)\in E_{G}\
\textrm{and}\ (x,y)\notin E_{G}$},\\\ 0&\textrm{otherwise}.\end{cases}$
It is straightforward to verify that $\widetilde{G}$ is $\sigma_{4}$-symmetric
and there is a one-to-one correspondence between directed graphs and
$\sigma_{4}$-symmetric $\mathbb{F}_{4}^{*}$-graphs. The _rank-width_ of a
directed graph $G$, denoted by
$\operatorname{rwd}^{{\operatorname{\mathbb{F}_{4}}}}(G)$, is the
$\operatorname{\mathbb{F}_{4}}$-rank-width of $\widetilde{G}$ [14]. One easily
verifies that if $G$ is an undirected graph, then the rank-width of $G$ is
exactly the $\mathbb{F}_{4}$-rank-width of $\widetilde{G}$.
A directed graph $H$ is _loop-pivot equivalent_ (resp. _pivot equivalent_) to
a directed graph $G$ if $\widetilde{H}$ is $\sigma_{4}$-loop-pivot equivalent
(resp. $\sigma_{4}$-pivot equivalent) to $\widetilde{G}$; and $H$ is a _loop-
pivot-minor_ (resp. _pivot-minor_) of $G$ if $\widetilde{H}$ is a
$\sigma_{4}$-loop-pivot minor (resp. $\sigma_{4}$-pivot minor) of
$\widetilde{G}$. Since there is a one-to-one correspondence between
$\sigma_{4}$-symmetric $\mathbb{F}_{4}^{*}$-graphs and directed graphs, loop-
pivot equivalence (resp. pivot-equivalence) and loop-pivot minor (resp. pivot-
minor) are well-defined in directed graphs. Figure 1 shows an example of loop-
pivot complementation and pivot complementation.
$x_{2}$$x_{2}$$x_{3}$$x_{4}$$x_{1}$$x_{5}$$x_{6}$$x_{5}$$x_{3}$$x_{6}$$x_{4}$$x_{1}$
Figure 1: (a) A directed graph $G$. (b) The directed graph obtained after a
pivot-complementation of $G$ at $\\{x_{2},x_{5}\\}$. If you apply a loop-
pivot-complementation of $G$ at $\\{x_{2},x_{5}\\}$, you obtain the graph in
(b) with a loop at $x_{1}$.
As a consequence of Theorems 36 and 37 we have the following which generalises
[18, Theorem 4.1].
###### Theorem 38
Let $k$ be a positive integer.
1. 1.
For every infinite sequence $G_{1},G_{2},\ldots$ of directed graphs of rank-
width at most $k$, there exist $i<j$ such that $G_{i}$ is isomorphic to a
loop-pivot-minor of $G_{j}$.
2. 2.
For every infinite sequence $G_{1},G_{2},\ldots$ of loop-free directed graphs
of rank-width at most $k$, there exist $i<j$ such that $G_{i}$ is isomorphic
to a pivot-minor of $G_{j}$.
## 6 Delta-Matroids and Chain Groups
In this section we discuss some consequences of results in Sections 3 and 4
about _delta-matroids_. If $V$ is a finite set, then $\mathcal{F}\subseteq
2^{V}$ is said to satisfy the _symmetric exchange axiom_ if:
> (SEA) for $F,F^{\prime}\in\mathcal{F}$, for $x\in F\triangle F^{\prime}$,
> there exists $y\in F^{\prime}\triangle F$ such that
> $F\triangle\\{x,y\\}\in\mathcal{F}$.
A _set system_ is a pair $(V,\mathcal{F})$ where $V$ is finite and
$\emptyset\neq\mathcal{F}\subseteq 2^{V}$. A _delta-matroid_ is a set-system
$(V,\mathcal{F})$ such that $\mathcal{F}$ satisfies (SEA); the elements of
$\mathcal{F}$ are called _feasible sets_. Delta-matroids were introduced in
[2], and as for matroids, are characterised by the validity of a greedy
algorithm. We recall that a set system $\mathcal{M}:=(V,\mathcal{B})$ is a
_matroid_ if $\mathcal{B}$, called the set of _bases_ , satisfy the following
_Exchange Axiom_
> (EA) for $B,B^{\prime}\in\mathcal{B}$, for $x\in B\setminus B^{\prime}$,
> there exists $y\in B^{\prime}\setminus B$ such that
> $B\triangle\\{x,y\\}\in\mathcal{B}$.
It is worth noticing that a matroid is also a delta-matroid (see [2, 3, 10]
for other examples of delta-matroids).
For a set system $\mathcal{S}=(V,\mathcal{F})$ and $X\subseteq V$, we let
$\mathcal{S}\triangle X$ be the set system $(V,\mathcal{F}\triangle X)$ where
$\mathcal{F}\triangle X:=\\{F\triangle X\mid F\in\mathcal{F}\\}$. We have that
$\mathcal{F}\triangle X$ satisfies (SEA) if and only if $\mathcal{F}$
satisfies (SEA). Hence, $\mathcal{S}$ is a delta-matroid if and only if
$\mathcal{S}\triangle X$ is. A delta-matroid $\mathcal{S}=(V,\mathcal{F})$ is
said _equivalent_ to a delta-matroid
$\mathcal{S}^{\prime}=(V,\mathcal{F}^{\prime})$ if there exists $X\subseteq V$
such that $\mathcal{S}=\mathcal{S}^{\prime}\triangle X$. If $M$ is a
$(V,V)$-matrix over a field $\mathbb{F}$, we let $\mathcal{S}(M)$ be the set
system $(V,\mathcal{F}(M))$ where $\mathcal{F}(M):=\\{X\subseteq V\mid M[X]$
is non-singular$\\}$. The following is due to Bouchet [3].
###### Theorem 39 ([3])
Let $M$ be a matrix over $\mathbb{F}$ of symmetric type, _i.e._ , $M$ is
$(\sigma,\epsilon)$-symmetric with $\sigma$ (skew) symmetric. Then,
$\mathcal{S}(M)$ is a delta matroid.
Delta-matroids equivalent to $\mathcal{S}(M)$, for some matrix $M$ over
$\mathbb{F}$ of symmetric type, are called _representable over $\mathbb{F}$_
[3]. A slight modification of the proof given in [10] extends Theorem 39 to
all $(\sigma,\epsilon)$-symmetric matrices.
###### Theorem 40
Let $M$ be a $(\sigma,\epsilon)$-symmetric $(V,V)$-matrix over $\mathbb{F}$.
Then, $\mathcal{S}(M)$ is a delta matroid.
Let us recall the following from Tucker.
###### Theorem 41 ([25])
Let $M$ be a $(V,V)$-matrix such that $M[X]$ is non-singular. For any
$Z\subseteq V$, we have
$\displaystyle\det((M{*}X)[Z])$ $\displaystyle=\pm\frac{\det(M[Z\triangle
X])}{\det(A)}.$
Proof of Theorem 40. Let $X,Y\subseteq V$ such that $M[X]$ and $M[Y]$ are non-
singular. Let $x\in X\triangle Y$. Let $M^{\prime}:=P_{X}\cdot(M*X)$. By
Theorem 41, $M^{\prime}[Z]$ is non-singular if and only if $M[Z\triangle X]$
is non-singular. Assume $m^{\prime}_{xx}\neq 0$, then if we take $y:=x$, we
have that $M[X\triangle\\{x\\}]$ is non-singular. Suppose that
$m^{\prime}_{xx}=0$. Since $M^{\prime}[X\triangle Y]$ is non-singular, there
exists $y\in X\triangle Y$ such that $m^{\prime}_{xy}\neq 0$ and because
$M^{\prime}$ is $(\sigma,\epsilon)$-symmetric, $m^{\prime}_{yx}\neq 0$. Hence,
$M^{\prime}[\\{x,y\\}]$ is non-singular, _i.e._ ,
$M^{\prime}[X\triangle\\{x,y\\}]$ is non-singular. ∎
A consequence of Theorem 40 is that we can extend the notion of
representability of delta-matroids by the following.
> A delta-matroid is _representable over $\mathbb{F}$_ if it is equivalent to
> $\mathcal{S}(M)$ for some $(\sigma,\epsilon)$-symmetric matrix $M$ over
> $\mathbb{F}$.
It is worth noticing from Proposition 2 that over prime fields this notion of
representability is the same as the one defined by Bouchet [3]. We now discuss
some other corollaries. First, if $M$ is a $(\sigma,\epsilon)$-symmetric
$(V,V)$-matrix, then for any $X\subseteq V$ such that $M[X]$ is non-singular,
$\mathcal{S}(M)\triangle X=\mathcal{S}(M^{\prime})$ for any
$M^{\prime}:=I_{Z}\cdot P\cdot P_{X}\cdot(M*X)\cdot Q^{-1}\cdot
I_{Z^{\prime}}$ for some $Z,Z^{\prime}\subseteq V$, and $(P,Q)$ a pair of
$\sigma$-compatible diagonal $(V,V)$-matrices.
Lemma 26 characterises non-singular principal submatrices of
$(\sigma,\epsilon)$-symmetric matrices in terms of eulerian
$\mathbb{K}_{\sigma}$-chains of their associated lagrangian
$\mathbb{K}_{\sigma}$-chain groups. One can derive from this a
characterisation of representable delta-matroids in terms of lagrangian
$\mathbb{K}_{\sigma}$-chain groups.
One can derive from Theorem 29 a well-quasi-ordering theorem for representable
delta-matroids as follows. Let the _branch-width_ of a delta-matroid
$\mathcal{S}$ representable over $\mathbb{F}$ as
$\min\\{\operatorname{rwd}^{{\mathbb{F}}}(M)\mid\mathcal{S}(M)$ is equivalent
to $\mathcal{S}\\}$. A delta-matroid $\mathcal{S}^{\prime}$ is a _minor_ of a
delta-matroid $\mathcal{S}=(V,\mathcal{F})$ if there exist $X,Y\subseteq V$
such that $\mathcal{S}^{\prime}=(V\setminus(X\cup Y),\\{(F\triangle
X)\setminus Y\mid F\in\mathcal{F}\\})$. An extension of [19, Theorem 7.3] is
the following.
###### Theorem 42
Let $\mathbb{F}$ be a finite field and $k$ a positive integer. Every infinite
sequence $\mathcal{S}_{1},\mathcal{S}_{2},\ldots$ of delta-matroids
representable over $\mathbb{F}$ of branch-width at most $k$ has a pair $i<j$
such that $\mathcal{S}_{i}$ is isomorphic to a minor of $\mathcal{S}_{j}$.
Proof. Let $M_{1},M_{2},\ldots$ be $(\sigma_{i},\epsilon_{i})$-symmetric
matrices over $\mathbb{F}$ such that, for every $i$, $\mathcal{S}_{i}$ is
equivalent to $\mathcal{S}(M_{i})$ and the branch-width of $\mathcal{S}_{i}$
is equal to the $\mathbb{F}$-rank-width of $M_{i}$. By Theorem 29, there exist
$i<j$ such that $M_{i}$ is isomorphic to $(M_{j}/M_{j}[A])[V^{\prime}]\cdot
I_{Z}$ with $A\subseteq V_{j}\setminus V^{\prime}$ and $Z\subseteq
V^{\prime}\subseteq V_{j}$. Hence, $\mathcal{S}_{i}$ is isomorphic to a minor
of $\mathcal{S}_{j}$. ∎
We conclude by some questions. It is well-known that columns of a matrix over
a field yields a matroid. It would be challenging to characterise matrices
whose non-singular principal submatrices yield a delta-matroid. Currently,
there is no connectivity function for delta-matroids. Another challenge is to
find a connectivity function for delta-matroids that subsumes the connectivity
function of matroids and such that if a delta-matroid is equivalent to
$\mathcal{S}(M)$, then the branch-width of $\mathcal{S}(M)$ is proportional to
the $\mathbb{F}$-rank-width of $M$.
We would like to thank S. Oum for letting at our disposal a first draft of
[19], which was of great help for our understanding of the problem. We thank
also B. Courcelle and the anonymous referee for their helpful comments. The
author is supported by the DORSO project of “Agence Nationale Pour la
Recherche”.
## References
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* [19] S. Oum. Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices. arXiv:1007.3807v1. Submitted, 2010.
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|
arxiv-papers
| 2011-02-10T14:52:45 |
2024-09-04T02:49:16.942704
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mamadou Moustapha Kant\\'e",
"submitter": "Mamadou Moustapha Kant\\'e",
"url": "https://arxiv.org/abs/1102.2134"
}
|
1102.2144
|
# Dynamics of cylindrical droplets on flat substrate: Lattice Boltzmann
modeling versus simple analytic models
Nasrollah Moradi nasrollah.moradi@rub.de ICAMS, Ruhr-Universität Bochum,
Stiepeler Strasse 129, 44801 Bochum, Germany Fathollah Varnik ICAMS, Ruhr-
Universität Bochum, Stiepeler Strasse 129, 44801 Bochum, Germany Max-Planck
Institut für Eisenforschung, Max-Planck Str. 1, 40237 Düsseldorf, Germany
Ingo Steinbach ICAMS, Ruhr-Universität Bochum, Stiepeler Strasse 129, 44801
Bochum, Germany
(August 27, 2024)
###### Abstract
The steady state motion of cylindrical droplets under the action of external
body force is investigated both theoretically and via lattice Boltzmann
simulation. As long as the shape-invariance of droplet is maintained, the
droplet’s center-of-mass velocity linearly scales with both the force density
and the square of droplet radius. However, a non-linear behavior appears as
the droplet deformation becomes significant. This deformation is associated
with the drop elongation occurring at sufficiently high external forcing. Yet,
independent of either the force density or the droplet size, the center-of-
mass velocity is found to be linear in terms of the inverse of dynamic
viscosity. In addition, it is shown that the energy is mainly dissipated in a
region near the substrate particularly close to the three phase contact line.
The total viscous dissipation is found to be proportional to both the square
of force density and the inverse of dynamic viscosity. Moreover, the
dependence of the center-of-mass velocity on the equilibrium contact angle is
investigated. A simple analytic model is provided reproducing the observed
behavior.
Keywords : _droplet dynamics, steady state, viscous dissipation, lattice
Boltzmann modeling_
## I Introduction
Individual droplets play a key role in many biological systems Wolgemuth .
Droplet behavior is also crucial for numerous industrial applications such as
in automobile manufacturing and drug production as well as glass industry.
Consequently, understanding the underling physics behind droplet behavior and
finding novel applications are currently an active field of research Quere ;
QuereAnnu . Recently, study of microdroplets has received lots of attentions
both experimentally and by numerical modeling. For example, droplet spreading
on chemically and topographically patterned substrates, droplet evaporation,
and wetting properties of superhydrophobic surfaces have been extensively
studied in the literature Varnik2 ; Markus ; Lipowsky ; Seemann ; Lenz ;
Dorrer ; Reyssat1 ; Ajdari . Particularly, controlling droplet motion is
essential for many industrial purposes ranging from microfluidic devices to
fuel cells and inkjet printing Reyssat .
The equilibrium contact angle of a droplet, $\theta_{\text{eq}}$, placed on a
perfectly flat and homogeneous solid substrate is given by the Young equation,
$\cos\theta_{\text{eq}}=(\sigma_{SV}-\sigma_{SL})/\sigma_{LV}$, where
$\sigma_{LV}$, $\sigma_{SL}$, and $\sigma_{SV}$ are the surface tensions of
liquid-vapor, solid-liquid and solid-vapor, respectively Young . However, in
the case of moving drops, the advancing contact angle is often found to be
larger than the receding deGennes . This deference can be considered as a
measure of droplet deformation and it may appear in characterizing of droplet
velocity Kim . A droplet may move due to a wettability or temperature gradient
Brochard ; Varnik1 ; Thiele . Recently, we were successful to report a
spontaneous droplet motion on a substrate topographically patterned with a
step-wise gradient of pillars Nasrollah . Obviously, a droplet may also move
under the action of a body force, e.g., a falling drop on an inclined surface
under the gravitational forcing. Depending on the material parameters of the
considered system such as $\eta$, dynamic viscosity, and $\theta_{\text{eq}}$
as well as superhydrophobicity of the substrate, droplets perform a sliding,
rolling, or tank treading motion or a combination thereof Hodges ; Aussillous
; Mahadevan . Associated to a very high external forcing (i.e. sufficiently
large velocity), drops may highly be elongated (pearling) and, further up,
they exhibit a cuspid tail that emits smaller drops Podgorski . Introducing
slippage at solid boundary is another issue that helps to characterize droplet
motion Muller .
Here, we concentrate on the steady state motion of cylindrical drops. However,
despite the apparent simplicity of the problem, several issues, such as
dependence of center-of-mass velocity, $U_{\text{cm}}$, and the dissipation
loss on the material parameters and external forcing as fully as the role of
droplet deformation are still not well understood Muller . The steady state is
reached due to the balance between the rate at which energy is imparted onto
the droplet and the rate of energy dissipation. In general, there are
different possible mechanisms for energy dissipation within a moving droplet:
the viscous dissipation due to the velocity gradients, dissipation at the
vicinity of the three phase contact line, and the dissipation in the precursor
film which may form around the droplet in contact with a solid Quere ;
deGennes . Since, the numerical model used in the present studies does not
take account of precursor film, we will focus on the effects related to
dissipation only. This includes both the bulk of the drop as well as the
vicinity of the contact line. Interestingly, as long as external force is
sufficiently weak or –equivalently– the droplet volume is sufficiently low so
that the drop approximately maintains its equilibrium shape during motion, the
dependence of the center-of-mass velocity on external force and on the droplet
volume can be easily worked out via simple rescaling of the relevant
parameters. In particular, we find that the steady state drop velocity is
directly proportional to $gR_{\text{eff}}^{2}/\eta$, where $g$ is the external
body force (equivalent of the gravitational acceleration), $R_{\text{eff}}$
the effective drop radius and $\eta$ the shear viscosity. Deviation from this
simple behavior is observed in the case of strong droplet deformation.
However, since dynamic viscosity does not affect the droplet shape, the drop
velocity remains proportional to $1/\eta$ even in the strongly deformed limit.
Using numerical simulations, we also calculate the local energy dissipation
inside the droplet. It is observed that the main dissipation takes place
within a volume below the drop’s center-of-mass. Based on this observation, we
propose a simple model which successfully captures the dependence of drop
velocity on equilibrium contact angle.
## II Numerical Model
Because of complicated nature of fluid flows, tractable analytical approaches
are often limited to simplified systems. In addition, experimental studies are
available only for a restricted range of parameters. In this context, computer
simulations can help to bridge the gap between analytical approaches and
experiments. In the past two decades, the lattice Boltzmann (LB) method
McNamara1988 ; Higuera1989a ; Higuera1989 ; Benzi1992 ; Qian1992 ; Rothman ;
Succi2001 ; Wolf-Gladrow2000 has proved itself as a powerful Navier-Stokes
solver for simulating a wide range of complex fluidic systems.
We employ a free-energy-based two-phase lattice Boltzmann (LB) model to solve
the discrete Boltzmann equation (DBE) for the van der Walls fluid with the BGK
approximation. A detailed description of the model can be found in references
Lee1 ; Lee2 . For the sake of completeness, however, a brief overview of the
model is provided here. The DBE with external force F can be written as
$\frac{\partial f_{\alpha}}{\partial
t}+\mathbf{e}_{\alpha}\cdot\mathbf{\nabla}f_{\alpha}=-\frac{f_{\alpha}-f_{\alpha}^{eq}}{\lambda}+\frac{(\mathbf{e_{\alpha}}-\mathbf{u})\cdot\mathbf{F}}{\rho
c_{s}^{2}}f_{\alpha}^{eq}.$ (1)
In the above, $f_{\alpha}$, $\mathbf{e}_{\alpha}$ and $\mathbf{u}$ are
particle distribution function, the microscopic particle velocity and the
macroscopic velocity, respectively. The parameter $\rho$ stands for the fluid
density, $\lambda$ is the relaxation time and $c_{s}$ denotes the sound speed.
The non dimensional relaxation time $\tau=\lambda/\delta t$ is related to
kinematic viscosity by $\nu=\tau c_{s}^{2}\delta t$. The equilibrium
distribution function, $f_{\alpha}^{eq}$, is given by
$f_{\alpha}^{eq}=w_{\alpha}\rho\left[1+\frac{\mathbf{e_{\alpha}\cdot\mathbf{u}}}{c_{s}^{2}}+\frac{(\mathbf{e_{\alpha}\cdot\mathbf{u})^{2}}}{2c_{s}^{4}}-\frac{\mathbf{u}\cdot\mathbf{u}}{2c_{s}^{2}}\right],$
(2)
where $w_{\alpha}$is a weighing factor. In order to eliminate the parasitic
currents, the averaged external force experienced by each particle
$\mathbf{F}$ is chosen in the potential form
$\mathbf{F}=\mathbf{\nabla}\rho
c_{s}^{2}-\rho\mathbf{\nabla}(\mu_{0}-\kappa\mathbf{\nabla}^{2}\rho),$ (3)
where $\mu_{0}$ is the chemical potential and $\kappa$ the gradient parameter.
The equilibrium properties of the present model can be obtained from a free-
energy functional consisting of a volume and a surface part,
$\Psi=\int_{V}\left(E_{0}(\rho)+\frac{\kappa}{2}|\mathbf{\nabla}|^{2}\right)dV-\int_{S}(\phi_{1}\rho_{s})dS,$
(4)
where $V$ is the system volume and $S$ the surface area of the substrate. The
bulk energy density, $E_{0}$, can be approximated by
$E_{0}(\rho)=\beta(\rho-\rho_{\text{V}})^{2}((\rho-\rho_{\text{L}})^{2})$ in
which $\beta$ is a constant and both $\rho_{\text{L}}$ and $\rho_{\text{V}}$
are saturation densities in liquid and vapor phase, respectively. The gradient
parameter and the liquid-vapor surface tension can be computed as
$\kappa=\beta D^{2}(\rho_{\text{L}}-\rho_{\text{V}})^{2}/8$ and
$\sigma=(\rho_{\text{L}}-\rho_{\text{V}})^{3}\sqrt{2\kappa\beta}/6$,
respectively. The interface thickness $D$, $\beta$, $\rho_{\text{L}}$, and
$\rho_{\text{V}}$ are input parameters. The second integral in Eq. (3) is the
contribution of solid-liquid interfaces in the total free energy $\Psi$. At
equilibrium, there are two solutions that satisfy $\phi_{1}=\pm\sqrt{2\kappa
E_{0}(\rho)}$. Minimizing the free energy functional $\Psi$ leads to an
equilibrium boundary condition for the spatial derivative of fluid density in
the direction normal to the substrate $\partial\bot\rho=-\phi_{1}/\kappa$. The
parameter $\phi_{1}$ is related to $\theta_{\text{eq}}$ via
$\displaystyle\phi_{1}$
$\displaystyle=\frac{\sqrt{2\kappa\beta}}{2}(\rho_{\text{L}}-\rho_{\text{V}})^{2}\text{sgn}\left(\frac{\pi}{2}-\theta_{\text{eq}}\right)$
(5)
$\displaystyle\times\left\\{\cos\left(\frac{\alpha}{3}\right)\left[1-\cos\left(\frac{\alpha}{3}\right)\right]\right\\}^{1/2},$
where $\alpha=\text{arccos(sin}\theta_{\text{eq}})^{2}$.
Figure 1: (color on-line) Difference between rescaled velocity fields,
$\hat{u}_{2}(\hat{y},\hat{z})-\hat{u}_{1}(\hat{y},\hat{z})$, for two different
values of $g$ (left), $\eta$ (middle) and $R_{\text{eff}}$ (right) as
indicated. Other control parameters of the simulation are as follows:
$\eta=0.16$ and $R_{\text{eff}}=19.7$ (left panel); $g=10^{-7}$ and
$R_{\text{eff}}=19.7$ (middle panel) and finally $g=5\times 10^{-8}$ and
$\eta=0.16$ (right panel). The difference between rescaled velocity fields in
computed after a shift operation such that the center-of-mass of the droplets
coincide with one another.
The advantage of this model for the current study is both the possibility of
achieving a high density ratio and, as it was already mentioned, the
elimination of parasitic currents at the liquid-vapor interface. It is
important to note that the elimination of the spurious currents is an
important step towards a reliable description of fluid dynamics inside a
droplet. Simulating a two-phase system with a high density ratio, on the other
hand, not only is more realistic but also allows to significantly reduce the
finite size effect related to the dissipation loss in the vapor phase.
In our LB simulations, the bounce-back rule is imposed at solid boundaries.
For the open boundaries (in the $x$ and $z$-directions), the periodic boundary
condition is applied. A body force, $\rho g$, is applied to the liquid phase
along the $z$-direction. The body force, however, monotonously decreases
through the interface and it vanishes in the gas phase. This accounts for the
fact that the gas remains inert (static equilibrium) in the limit of zero
droplet size. All the quantities in this paper are given in dimensionless LB
units. The parameter $\beta$, the interface thickness $D$ and the saturation
densities are fixed to $0.01$, $5$, $1$, and $0.01$, respectively. This choice
of the parameters leads to a surface free energy of $\sigma\simeq 0.004$. Note
that, in order to focus on situations, which can be easily controlled in real
experiments, we do not change surface tension or liquid density in our
simulations. Depending to the case of interest, the parameters $\tau$,
$\theta_{\text{eq}}$, $R_{\text{eff}}$, and $g$ lie in the ranges
$[0.02,1.6]$, $[35^{\circ},140^{\circ}]$, $[22,75]$ and $[10^{-7},10^{-5}]$ in
the order given. Typically, we use a simulation box of size $L_{x}\times
L_{y}\times L_{z}$ $=$ $2\times 120\times 120$ lattice nodes. However, for
large droplets, we increase the size of the simulation box (in the $y$ and
$z$-directions) ensuring that there are no finite size effects. The volume of
droplet is given by $V=SL_{x}$ where $S$ is the surface of droplet’s cross-
section normal to the $x$-direction. For the cylindrical geometry considered
in this study, we define the droplet’s effective radius as
$R_{\text{eff}}=(S/\pi)^{1/2}$.
## III A simple scaling relation
Here, we investigate the effect of external forcing on the steady state motion
of cylindrical drops on a flat surface. In addition, the influence of system
parameters such as droplet size $R_{\text{eff}}$, viscosity $\eta$ as well as
equilibrium contact angle $\theta_{\text{eq}}$ on the steady state velocity of
the droplet’s center-of-mass is addressed.
By driving a droplet via an external body force, we mimic a real situation in
which a droplet moves downward on an inclined surface due to the gravity. The
external force does work on droplet with a rate equal to the total force
applied on the droplet multiplied by the droplet’s center-of-mass velocity,
$g\rho VU_{\text{cm}}$. In the steady state, this energy is entirely
transferred into dissipation. On the other hand, the total viscous dissipation
is given by
$\int_{V}S_{ij}\sigma_{ij}dV=\int_{V}\sigma_{ij}\sigma_{ij}/(2\eta)dV$, where
the strain rate and stress tensors are given by $S_{ij}=(\partial
u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i})/2$ and $\sigma_{ij}=2\eta
S_{ij}$, the later relation being valid for non-diagonal (shear) components of
a Newtonian fluid (note that, due to the incompressibility of the liquid
phase, the diagonal components of the strain rate and stress tensors are not
relevant here). One thus obtains
$g\rho VU_{\text{cm}}=\frac{1}{2\eta}\int\sigma_{ij}\sigma_{ij}dV=2\eta\int
S_{ij}S_{ij}dV.$ (6)
As long as the shape of the droplet does not change, it is reasonable to take
$R_{\text{eff}}$ as a characteristic length. We also chose $U_{\text{cm}}$ as
a characteristic velocity and introduce dimensionless quantities such as
$\hat{x}=x_{\alpha}/R_{\text{eff}}$ and
$\hat{u}_{\alpha}=u_{\alpha}/U_{\text{cm}}$. Using these rescaled quantities,
the strain rate tensor can also be written as
$S_{ij}=U_{\text{cm}}/R_{\text{eff}}(\partial\hat{u}_{i}/\partial\hat{x}_{j}+\partial\hat{u}_{j}/\partial\hat{x}_{i})/2=U_{\text{cm}}/R_{\text{eff}}\hat{S}_{ij}$.
Inserting this into Eq. (6) yields
$g\rho VU_{\text{cm}}=\frac{2\eta
U_{\text{cm}}^{2}V}{R_{\text{eff}}^{2}}\int\hat{S}_{ij}\hat{S}_{ij}d\hat{V}.$
(7)
where we also made the volume element dimensionless ($dV=Vd\hat{V}$). The
important step is now to assume that the rescaled velocity field within the
droplet does not change upon a variation of the external force, drop radius or
viscosity provided that the shape of the droplet remains constant. With this
assumption, the integral in Eq. (7) becomes a constant ‘shape factor’ and one
obtains
$U_{\text{cm}}\propto\frac{g\rho R_{\text{eff}}^{2}}{\eta}.$ (8)
It is noteworthy that, in the above model, the dependence of $U_{\text{cm}}$
on $R_{\text{eff}}^{2}$ arises from the rescaling of the strain rate tensor
$S^{2}_{ij}$ only. In particular, it remains valid regardless of the
dimensionality of the space. Interestingly, when expressed in terms of droplet
volume, $V$, the spatial dimension, $d$, does play a role. This is simply a
consequence of the fact that $R_{\text{eff}}\propto V^{1/d}$. In particular,
$U_{\text{cm}}\propto V$ in 2D, while $U_{\text{cm}}\propto V^{2/3}$ in 3D.
In order to test the above assumption of the scale invariance, we have
performed a series of lattice Boltzmann simulations while varying $g$, $\eta$
and $R_{\text{eff}}$ in a range where droplet shape remains unchanged. The
simulated velocity fields are then compared with one another by first
rescaling the relevant velocity and length scales (see the text below Eq. (6))
and then plotting the difference of the thus obtained velocity fields.
Results of such an analysis are illustrated in Fig. 1. As seen from this
figure, the rescaled velocity fields are very close to each other almost in
the entire droplet with deviations in the vicinity of the three phase contact
line. Noting that these deviations (being at most of the order of 10%) are
limited to a small fraction of the droplet’s volume, the relative contribution
of these deviations to the integral in the right hand side of Eq. (7) becomes
quite negligible in all the cases shown. Obviously, the assumption of a scale
invariant velocity field is a good approximation to the actual flow behavior
in the studied parameter range. Equation (8) is thus expected to well describe
our data as long as droplet shape is unaltered.
The presence of a parameter range for the validity of Eq. (8) is evidenced in
Fig. 2, where the center-of-mass velocity, $U_{\text{cm}}$, is depicted versus
force density, $g$, for different droplet sizes. As seen from this figure, the
range of the validity of scaling relation Eq. (8) extends to larger $g$ as
droplet size decreases. Conversely, the larger the droplet, the earlier the
onset of significant deviations. A similar trend is also observed in Fig. 3,
where droplet size is varied as control parameter for three different choices
of $g$.
Figure 2: (Color on-line) $U_{\text{cm}}$ versus body force $g$ for different
values of the effective droplet radius $R_{\text{eff}}$ as specified. A linear
behavior is visible at sufficiently low $g$. The range of the validity of this
linear regime is progressively restricted as droplet radius increases. In the
right panel, $\eta U_{\text{cm}}/(g\rho R_{\text{eff}}^{2})$ is plotted versus
$g$ for exactly the same data as in the left panel. In all these simulations,
shear viscosity and equilibrium contact angle are set to $\eta=0.16$ and
$\theta_{\text{Y}}=90^{\circ}$.
Figure 3: (Color on-line) $U_{\text{cm}}$ versus $R_{\text{eff}}^{2}$ for
different choices of the body force $g$ as indicated. Again, a linear behavior
is visible at sufficiently low $R_{\text{eff}}$. The range of the validity of
the linear behavior shrinks upon a raise of the body force. Following the same
idea as in the right panel of Fig. 2, we plot in the right panel $\eta
U_{\text{cm}}/(g\rho R_{\text{eff}}^{2})$ versus $R_{\text{eff}}^{2}$ for
exactly the same data as in the left panel. In all these simulations, shear
viscosity and equilibrium contact angle are set to $\eta=0.16$ and
$\theta_{\text{Y}}=90^{\circ}$.
In the present study, the shape of droplet is determined by the competition
between the surface force and the total body force. For a cylindrical drop of
the cross sectional radius $R_{\text{eff}}$ and axial length $L_{x}$, this
leads to $\sigma_{\text{LV}}L_{x}\leq g\rho R_{\text{eff}}^{2}L_{x}$ as a
condition for a significant deformation. By introducing the Bond number,
$Bo=\rho gR_{\text{eff}}^{2}/\sigma_{\text{LV}}$, one sees that strong
deformation is expected for $Bo\geq 1$. Within prefactors of the order of
unity, the same condition for drop deformation is also obtained in the case of
a spherical droplet (to see this, replace $L_{x}$ by $R_{\text{eff}}$). It
must be emphasized here, that this criterion is based on a scaling argument
and the precise value of the Bond number for the transition from undeformed to
a deformed state may be different from unity. What is essential here is the
fact that a higher Bond number leads to a higher degree of deformation. In the
case of our simulations, for example, slight but observable deformation occurs
already for a Bond number as low as 0.25 (Fig. 4 b) with a significant
increase in the deformation state as Bo increases from 0.25 to 0.72 (Fig. 4c).
Droplet shapes and the corresponding momentum fields are shown in Fig. 4 for
three typical values of $g$. As seen from the left panel of this figure, for a
sufficiently weak body force (here $g=10^{-7}$), the deformation of the
droplet is quite negligible but it becomes important upon an increase of $g$
(middle and right panels).
(a) (b) (c)
Figure 4: Droplet shape and the corresponding momentum field in the center-of-
mass frame for three different values of the driving force $g$. As $g$
increases, the deformation becomes more pronounced. In the left panel, the
deformation is negligible and the droplet’s center-of-mass velocity obeys the
simple relation Eq. (8) for driving forces below the specified value. The
middle panel marks the onset of deviations from Eq. (8) and the left panel is
well beyond the validity of this simple scaling relation. A rolling motion is
clearly visible regardless of the deformation state of droplet. In all the
cases shown, the droplet’s effective radius, dynamic viscosity and equilibrium
contact angle are fixed to $R_{\text{eff}}=19.7$, $\eta=0.16$ and
$\theta_{\text{Y}}=90^{\circ}$, respectively. Recalling that
$\sigma_{\text{LV}}=0.004$ and $\rho_{\text{L}}=1.0$, the Bond number from
left to right reads $Bo=g\rho R_{\text{eff}}^{2}/\sigma_{\text{LV}}\approx
1\times 10^{-2},\;\;0.25$ and $0.72$.
Furthermore, Fig. 4 also shows the momentum field inside the droplet providing
direct evidence for the existence of rolling motion in the center-of-mass
frame of reference. Thus, an observer moving with the droplet’s center-of-mass
will confirm the presence of a well established rolling motion inside the
droplet regardless of its deformation state. This rolling motion is associated
to the tendency of droplet to minimize its total dissipation loss Mahadevan ;
Aussillous . Interestingly, similar rolling motion are also observed in
molecular dynamics simulations of polymeric liquids Muller .
We close this section by addressing the effect of viscosity on
$U_{\text{cm}}$. For this purpose, we mention that a change in viscosity only
affects the time scale of the entire simulation. In particular, a variation of
viscosity has no influence on the shape of droplet. Consequently, we expect
$U_{\text{cm}}\propto 1/\eta$ regardless of the deformation state of droplet.
This expectation is confirmed in Fig. 5, where $U_{\text{cm}}$ versus $1/\eta$
is shown for droplets with different degrees of deformation.
Figure 5: (Color on-line) $U_{\text{cm}}$ versus $1/\eta$ for droplets with
different degrees of deformation. The labeles (a)-(c) refer to deformation
states shown in Fig. 4, respectively (the velocity in the case (a) has been
multiplied by a factor of 10 for better visibility). In all the cases shown, a
perfect linear variation is seen in accordance with Eq. (8). In the left
panel, we plot simulation results for two different choice of $R_{\text{eff}}$
and $g$ but keeping the product $R_{\text{eff}}^{2}g$ almost unchanged. In
this case, the velocities of both droplets fall onto a single line which also
confirms the validity of Eq. (8). $\theta_{\text{eq}}$ is fixed to
$90^{\circ}$ in all cases.
## IV Local viscous dissipation
In this section, we provide a detailed analysis of the local dissipation rate,
$\phi(\bm{r})=\sigma^{2}(\bm{r})/2\eta$ inside droplet. As will be shown
hereafter, the insight gained via these investigations enables us to propose a
simple model capable of accounting for the dependence of the total dissipation
rate, $\Phi_{T}=\int\phi(\bm{r})d^{3}\bm{r}$, on the contact angle,
$\theta_{\text{Y}}$. Equating this to the work done by the external force then
yields a relation between the droplet’s center-of-mass velocity and the
equilibrium contact angle.
It is noteworthy that, unlike conventional Navier-Stokes solvers, the lattice
Boltzmann method does not require —although allows for— the computation of
velocity gradients to obtain the local stress tensor. Rather, it offers the
unique possibility of obtaining the stress tensor _locally_ via the non-
equilibrium part of the populations. In this regard, particular attention has
been payed to a correct implementation of the stress computation Markus1 .
In order to figure out at which parts of droplet the energy is mainly
dissipated, we compute local dissipation rate along the three lines labeled by
A, B, and C in the panel (a) of Fig. 6. The variation of $\phi$ along these
lines is depicted in the next panels of Fig. 6. We first note that $\phi$ is
negligible in the gas phase, which is often the case in real experiments due
to the low vapor pressure. Furthermore —as a comparison of the panels (a), (b)
and (c) reveals— the strongest dissipation occurs in the vicinity of the three
phase contact line (panel (c)), which is roughly two orders of magnitude
larger than the dissipation rate inside droplet (panel (b)). This behavior can
be rationalized due to the fact that large velocity gradients occur near the
substrate particularly in the vicinity of the three phase contact line Yeomans
. However, one must realize that bulk dissipation acts in a larger domain than
the dissipation close to the contact line and thus may eventually dominate the
overall dissipation rate if the droplet is sufficiently large.
An interesting feature, relevant for our further analysis is the fact that
viscous dissipation inside droplet is mainly localized to regions below the
droplet’s center-of-mass (panel (a) in Fig. 6). This idea is further evidenced
in Fig. 7 (a), where we plot the viscous dissipation integrated along a
horizontal line, $\Phi(y)=\int_{0}^{Lz}\phi(y,z)dz$ as a function of vertical
position $y$ (distance from the substrate). Indeed, as expected, viscous
dissipation mainly occurs in a region specified by $y<Y_{\text{cm}}$. This
finding is further underlined by showing in Fig. 7 (b) the relative
contribution, $\Phi_{\text{R}}(y)$, to total dissipation within a region
restricted between the substrate and a horizontal line at $y$,
$\Phi_{R}(y)=\int_{0}^{y}\Phi(y^{\prime})dy^{\prime}/\int_{0}^{Ly}\Phi(y^{\prime})dy^{\prime})$.
It is visible from Fig. 7 (b) that $96\%$ of total dissipation occurs in a
region specified by $y<Y_{\text{cm}}$.
(a)(b)(c)(d)
Figure 6: (a) Illustration of the typical shape of a droplet and the lines
along which viscous dissipation is determined (‘CM’ stands for the center-of-
mass). The system parameters are $g=10^{-7}$, $R_{\text{eff}}=28.2$,
$\eta=0.16$, and $\theta_{\text{Y}}=90^{\circ}$. (b-d) The variation of local
viscous dissipation rate, $\phi=\sigma^{2}(\bm{r})/(2\eta)$, along lines A, B
and C as indicated. Note that $\phi$ is almost negligible in the gas phase. It
has a large value close to the substrate, but rapidly decreases far from the
substrate. It is also highly enlarged in the vicinity of triple contact line.
Figure 7: (Color on-line) variation of the local dissipation rate $\Phi$ and
the relative dissipation rate $\Phi_{R}$ as a function of $y$, corresponding
to the droplet shown in Fig. 6 in the left and right, respectively. The main
contribution to the total dissipation $\Phi_{T}$ occurs close to the substrate
in a region given by $y<Y_{\text{cm}}$.
Before working out an important consequence of this observation, we first
check whether it remains valid upon a variation of shear viscosity and driving
force. Inserting Eq. (8) in the right hand side of Eq. (7), it is seen that
the total dissipation rate is expected to obey
$\Phi_{\text{T}}\propto\frac{2g^{2}\rho^{2}R_{\text{eff}}^{2+d}}{\eta}.$ (9)
In order to verify Eq. (9), we determine $\Phi(y)$ for different values of
body force and viscosity. Typical plots of the thus obtained results are shown
in Figs. 8 and 9. These data clearly underline the validity of Eq. (9) within
the studied range of parameters.
Figure 8: Left: Dissipation rate integrated along a horizontal line at $y$,
$\Phi(y)$. Right: The same quantity as in the left panel but divided by
$g^{2}$. The observed master curve supports the validity of Eq. (9).
,
Figure 9: A similar plot as in Fig. 8 but now for various fluid viscosities
$\eta$. Here, the right panel depicts $\Phi(y)$-data from the left panel
multiplied by $\eta$. Again, the validity of Eq. (9) is supported by the
master curve.
In addition to supporting the validity of Eq. (9), the data shown in Figs. 8
and 9 provide further evidence for the fact that most part of dissipation
occurs in the region below the droplet’s center-of-mass. Based on this
observation, we propose a simple relation allowing to describe the dependence
of droplet velocity on contact angle.
Our simple analytic model is based on scaling arguments. To proceed, we start
with the energy balance equation for a cylindrical droplet of axial length
$L_{x}$ in the steady state. Using the translation invariance with respect to
the $x$-coordinate, one can write $g\rho\pi
R_{\text{eff}}^{2}L_{x}U_{\text{cm}}=L_{x}(\eta/2)\int_{0}^{H}\int\dot{\gamma}^{2}dzdy$,
where $\dot{\gamma}$ is the local shear rate. Since the energy is almost
completely dissipated in a region below $Y_{\text{cm}}$, we can safely
restrict the upper limit of the integration to $Y_{\text{cm}}$ and rewrite the
energy ballance equation as $2g\rho\pi
R_{\text{eff}}^{2}U_{\text{cm}}=\eta\int_{0}^{Y_{\text{cm}}}\int\dot{\gamma}^{2}dzdy$
(see Fig. 7). Neglecting droplet deformation, the droplet’s cross-section is a
circular segment with a base contact angle of $\theta_{\text{eq}}$. Here, we
assume that $\dot{\gamma}$ simply scales as $U_{\text{cm}}/Y_{\text{cm}}$
throughout the droplet. This may appear as a crude approximation, but it
allows to obtain a solvable analytic expression. Furthermore, we approximate
the surface of the droplet below $Y_{\text{cm}}$ as that of a rectangle of
height $Y_{\text{cm}}$ and length $l_{z}$. Adopting this, the right hand side
of the energy balance equation can now be estimated by
$(\eta/2)l_{z}Y_{\text{cm}}\times(U_{\text{cm}}/Y_{\text{cm}})^{2}=(\eta/2)l_{z}U_{\text{cm}}^{2}/Y_{\text{cm}}$.
Thus, one obtains $2g\rho\pi R_{\text{eff}}^{2}U_{\text{cm}}=\eta
l_{z}U_{\text{cm}}^{2}/Y_{\text{cm}}$, which then yields
$U_{\text{cm}}=g\rho\pi R_{\text{eff}}^{2}Y_{\text{cm}}/(l_{z}\eta)$. For the
considered geometry, the quantity $Y_{\text{cm}}/l_{z}$, is only a function of
$\theta_{\text{eq}}$. Taking this into account, we finally arrive at
$U_{\text{cm}}=C\frac{g\rho
R_{\text{eff}}^{2}}{\eta}\left[\dfrac{4\text{sin}^{2}\theta_{\text{eq}}}{3(2\theta_{\text{eq}}-\text{sin}2\theta_{\text{eq}})}-\text{cot}\theta_{\text{eq}}\right].$
(10)
The validity of the model has been tested in Fig. 10 for two different droplet
radii. This simple model reproduces well the simulation results.
Interestingly, the fitting prefactor, $C$, for both investigated droplet sizes
are very close to each other ($0.56$ and $0.57$) showing the consistency of
the model.
We would like to emphasize that the present approach is different from
conventional approaches, where the integration is taken over the entire volume
of droplet. Following the conventional route, one would obtain a different
expression, $U_{\text{cm}}=C(g\rho
R_{\text{eff}}^{2}/\eta)(1-\text{cos}\theta_{\text{eq}})^{2}/(\theta_{\text{eq}}-\text{sin}\theta_{\text{eq}}\text{cos}\theta_{\text{eq}})$,
which, as shown in Fig. 10, is not successful in capturing the observed
behavior.
It is noteworthy that an extension of Eq. (10) to $3D$ can simply be obtained
by writing the energy ballance equation in $3D$ and replacing the
corresponding expression for $Y_{\text{cm}}/l_{z}$ by that of a spherical cap.
Figure 10: Droplet velocity versus equilibrium contact angle for two different
droplet volumes as indicated. Full solid lines are best fit results to Eq.
(10) while dashed lines give best fit results to $U_{\text{cm}}=C(g\rho
R_{\text{eff}}^{2}/\eta)(1-\text{cos}\theta_{\text{eq}})^{2}/(\theta_{\text{eq}}-\text{sin}\theta_{\text{eq}}\text{cos}\theta_{\text{eq}})$
(see the text). The force density and dynamic viscosity are fixed to
$g=10^{-7}$ and $\eta=0.16$ respectively for both droplets.
## V Conclusion
We use a two-phase lattice Boltzmann method to study the dynamics of
cylindrical droplets on a flat substrate under the action a gravity-like
external force density. Starting from the energy ballance equation, we first
drive a simple analytic relation, Eq. (8), indicating that —as long as the
shape-invariance of droplet is maintained— droplet’s center-of-mass velocity,
linearly scales with force density, and the square of the droplet radius. At
strong body forces or large droplet volumes, deviations from Eq. (8) are
observed. A survey of droplet shape within our simulations suggest that
droplet deformation is indeed the main cause of observed deviations from the
simple scaling relation. Interestingly, however, the droplet’s center-of-mass
velocity remains proportional to the inverse of the dynamic viscosity
regardless of droplet’s deformation state. This is in line with the idea that
viscosity merely affects the time scale of the problem with no influence on
droplet shape. A detailed study of the local dissipation inside droplet is
also provided. A results of these investigations is that dissipation mainly
occurs close to the three phase contact line and within a region below the
droplet’s center-of-mass. Using the latter observation, we propose a simple
analytic expression accounting for the dependence of droplet velocity on the
equilibrium contact angle. Results of computer simulations confirm the
validity of this simple model.
## VI Acknowledgments
We would like to thank Dmitry Medvedev and Markus Gross for insightful
discussions. M.G. is also acknowledged for providing us a version of his LB
code. N.M. gratefully acknowledges the grant provided by the Deutsche
Forschungsgemeinschaft (DFG) under the number Va 205/3-2. ICAMS gratefully
acknowledges funding from ThyssenKrupp AG, Bayer MaterialScience AG,
Salzgitter Mannesmann Forschung GmbH, Robert Bosch GmbH, Benteler Stahl/Rohr
GmbH, Bayer Technology Services GmbH and the state of North-Rhine Westphalia
as well as the European Commission in the framework of the European Regional
Development Fund (ERDF).
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|
arxiv-papers
| 2011-02-10T15:26:08 |
2024-09-04T02:49:16.951899
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nasrollah Moradi, Fathollah Varnik and Ingo Steinbach",
"submitter": "Nasrollah Moradi",
"url": "https://arxiv.org/abs/1102.2144"
}
|
1102.2289
|
Heavy quarks in the presence of higher derivative corrections from AdS/CFT
K. Bitaghsir Fadafan
Physics Department, Shahrood University of Technology,
P.O.Box 3619995161, Shahrood, Iran
E-mails: bitaghsir@shahroodut.ac.ir
Abstract
We use the gauge-string duality to study heavy quarks in the presence of
higher derivative corrections. These corrections correspond to the finite
coupling corrections on the properties of heavy quarks in a hot plasma. In
particular, we study the effects of these corrections on the energy loss and
the dissociation length of a quark-antiquark pair. We show that the calculated
energy loss of heavy quarks through the plasma increases. We also find in
general that the dissociation length becomes shorter with the increase of
coupling parameters of higher curvature terms.
###### Contents
1. 1 Introduction
2. 2 Energy loss of heavy quark at finite coupling
1. 2.1 Set up of calculations
2. 2.2 Positive couplings
3. 2.3 non-positive couplings
4. 2.4 analytic solution
3. 3 dissociation length of quark-antiquark pair at finite coupling
1. 3.1 dissociation length from AdS/CFT
2. 3.2 Numerical Solutions
4. 4 Conclusion
5. 5 Review of Quasi-topological gravity
## 1 Introduction
The experiments of Relativistic Heavy Ion Collisions (RHIC) have produced a
strongly-coupled quark$-$gluon plasma (QGP)[1]. There are no known
quantitative methods to study strong coupling phenomena in QCD which are not
visible in perturbation theory (except by lattice simulation). A new method
for studying different aspects of QGP is the $AdS/CFT$ correspondence [2, 3,
4, 5]. This method has yielded many important insights into the dynamics of
strongly-coupled gauge theories. It has been used to investigate
hydrodynamical transport quantities in various interesting strongly-coupled
gauge theories where perturbation theory is not applicable [6]. Methods based
on $AdS/CFT$ relate gravity in $AdS_{5}$ space to the conformal field theory
on the four-dimensional boundary. It was shown that an $AdS$ space with a
black brane is dual to a conformal field theory at finite temperature.
The universality of the ratio of shear viscosity $\eta$ to entropy density $s$
[7, 8, 9, 10] for all gauge theories with Einstein gravity dual raised the
tantalizing prospect of a connection between string theory and RHIC. The
results were obtained for a class of gauge theories whose holographic duals
are dictated by classical Einstein gravity. Recently, $\frac{\eta}{s}$ has
been studied for a class of CFTs in flat space with higher derivative
corrections [11, 13, 12, 15, 16, 14]. In these studies, the effects of $R^{2}$
corrections to the gravitational action in AdS space have been computed and it
was shown that the conjecture lower bound on the $\frac{\eta}{s}$ can be
violated. For example, in the Reissner$-$Nordström$-$AdS black brane solution
in Gauss$-$Bonnet gravity, the $\frac{\eta}{s}$ bound is violated and the
Maxwell charge slightly reduces the deviation [16]. Regarding this study and
motivated by the vastness of the string landscape [18], we explored the
modification of the jet quenching parameter and drag force on a moving heavy
quark in the strongly-coupled plasma in [19].
Recently, a new higher derivative theory of gravity in five-dimensional
spacetime which contains not only the Gauss-Bonnet term but also a curvature-
cubed interaction introduced [33, 34]. This theory is known as quasi-
topological gravity theory which is thought to be dual to the large $N$ limit
of some conformal field theory without supersymmetry. Unlike Lovelock gravity,
this cubic term is not purely topological. Therefor it would be useful to
consider curvature-cubed terms as the higher derivative corrections and
investigate behavior of the heavy quarks by means of the $AdS/CFT$
correspondence. Holographic investigation of Quasi-topological gravity have
been done in [35]. Also it was shown that the lower bound of the ratio of
shear viscosity to density entropy can be violated in this background [36].
In this paper we use the $AdS/CFT$ correspondence to study effect of higher
derivative corrections to the properties of the heavy quarks.111 In general,
we do not know about forms of higher derivative corrections in string theory,
but it is known that due to the string landscape one expects that generic
corrections can occur. One should notice that string theory contains higher
derivative corrections from stringy or quantum effects, and such corrections
correspond to $1/\lambda$ and $1/N$ corrections. In the case of
$\mathcal{N}=4$ super Yang$-$Mills theory, the dual corresponds to the type
$\amalg B$ string theory on $AdS_{5}\times S^{5}$ background. The leading
order corrections in $1/\lambda$ arise from stringy corrections to the low
energy effective action of type $\amalg B$ supergravity, $\alpha^{\prime
3}R^{4}$.
Employing numerical methods, we investigate energy loss and dissociation
length of heavy quarks in section 2 and 3, respectively. One finds that the
energy loss of heavy quarks increases by increasing higher derivative
corrections. Also the dissociation length becomes shorter with the increase of
coupling parameters of higher curvature terms. We summarize the effects of
these corrections in the last section. In the appendix, we give a brief review
of [33].
## 2 Energy loss of heavy quark at finite coupling
In this section, we investigate the finite-coupling corrections to the energy
loss of a moving heavy quark in the Super Yang-Mills plasma using the AdS/CFT.
These corrections are related to the curvature corrections to the AdS black
brane solution.
The effect of curvature-squared corrections to the drag force on a moving
heavy quark in the Super Yang-Mills plasma is investigated in [39]. It is
shown that the corrections to the drag force depend on the velocity of heavy
quark. This dependance is such that for $v>v_{c}$ including the corrections
increase the drag force. This means that at the critical velocity $v_{c}$ the
curvature-squared corrections have the minimum effect on the drag force. For
the particular case of Gauss-Bonnet gravity, we do not expect a critical
velocity [39]. Also in this background, the drag force is larger than the
$\mathcal{N}=4$ case if $\lambda$ (Gauss-Bonnet gravity constant) is positive
while it is smaller than the $\mathcal{N}=4$ case if $\lambda$ is negative.
Now we continue with considering curvature-cubic corrections. Our purpose is
finding a general rule for considering higher derivative terms. We use the
proposal of [33, 34] and study new higher derivative theory of gravity in
five-dimensional spacetime which contains not only the Gauss-Bonnet term but
also a curvature-cubed interaction.
We should emphasize that in the case of these corrections, one can not predict
a result for $\mathcal{N}=4$ SYM because the first higher derivative
correction in weakly curved type IIB backgrounds enters at order
${\cal{R}}^{4}$. These corrections on the drag force have been studied in [40]
and it was found that the drag force for a heavy quark moving through
$\mathcal{N}=4$ SYM plasma is generally enhanced by the leading correction due
to finite ’t Hooft coupling. We will compare our results with this observation
and interestingly find a general rule for curvature corrections.
### 2.1 Set up of calculations
In the framework of $AdS/CFT$, an external quark is represented as a string
dangling from the boundary of $AdS_{5}-$Schwarzschild and a dynamical quark is
represented as a string ending on flavor D7-brane and extending down to some
finite radius in $AdS$ black brane background. We consider the $AdS$ black
hole solution in quasi-topological gravity 222one finds a quick review of this
background in the appendix. as
$ds^{2}=r^{2}\left(-N^{2}\,f(r)dt^{2}+d\vec{x}^{2}\right)+\frac{dr^{2}}{r^{2}\,f(r)},$
(1)
notice that we work in units where the radius of $AdS$ is one. Here $r$
denotes the radial coordinate of the black brane geometry and $t,\vec{x}$
label the directions along the boundary at the spatial infinity. In these
coordinates the event horizon is located at $r_{h}$ and it is found by solving
$f(r_{h})=0$ equation. The boundary is located at infinity and the geometry
will be as asymptotically $AdS$. The constant $N^{2}$ specifies the speed of
light of the boundary gauge theory and one can choose it to be unity. We name
$f(r)$ at the boundary where $r\rightarrow\infty$, as $f_{\infty}$ and one
finds that
$N^{2}=\frac{1}{f_{\infty}},$ (2)
The temperature of the hot plasma is given by the Hawking temperature of the
black hole
$T=\frac{N\,r_{h}}{\pi}.$ (3)
The relevant string dynamics is captured by the Nambu-Goto action
$S=-\frac{1}{2\pi\alpha^{\prime}}\int d\tau d\sigma\sqrt{-det\,g_{ab}},$ (4)
where the coordinates $(\sigma,\tau)$ parameterize the induced metric $g_{ab}$
on the string world-sheet and $X^{\mu}(\sigma,\tau)$ is a map from the string
world-sheet into the space-time. Defining $\dot{X}=\partial_{\tau}X$,
$X^{\prime}=\partial_{\sigma}X$, and $V\cdot W=V^{\mu}W^{\nu}G_{\mu\nu}$ where
$G_{\mu\nu}$ is the AdS black hole solution in Quasi-topological gravity (1),
then
$-det\,g_{ab}=(\dot{X}\cdot X^{\prime})^{2}-(X^{\prime})^{2}(\dot{X})^{2}.$
(5)
We follow [37, 38] and focus on the dual configuration of the external quark
moving in the $x$ direction on the plasma. The string in this case, trails
behind its boundary endpoint as it moves at constant speed $v$ in the $x$
direction
$x(r,t)=vt+\xi(r),\,\,\,\,\,y=0,\,\,\,z=0.$ (6)
One finds the lagrangian in the static gauge $(\sigma=r,\tau=t)$ as follows
$\mathcal{L}=\sqrt{-det\,g_{ab}}=N^{2}+r^{4}N^{2}f(r)x^{\prime
2}-\frac{\dot{x}^{2}}{f(r)},$ (7)
The equation of motion for $\xi$ implies that $\frac{\partial
L}{\partial\xi^{\prime}}$ is a constant. We name this constant as $\Pi_{\xi}$
and solve this relation for $\xi^{\prime}$, the result is
$\xi^{\prime
2}=\frac{\left(\frac{\Pi_{\xi}^{2}}{f(r)}\right)\left(-N^{2}f(r)+v^{2}\right)}{r^{4}N^{2}f(r)\left(-r^{4}N^{2}f(r)+\Pi_{\xi}^{2}\right)}.$
(8)
We are interested in a string that stretches from the boundary to the horizon.
In such a string, $\xi^{\prime 2}$ remains positive everywhere on the string.
Hence both numerator and denominator change sign at the same point and with
this condition, one finds the constant of motion $\Pi_{\xi}$ in terms of the
critical value of $r_{c}$ as follows
$\Pi_{\xi}=v\,r_{c}^{2},$ (9)
The drag force that is experienced by the heavy quark is calculated by the
current density for momentum along $x^{1}$ direction. After straightforward
calculations, the drag force is easily simplified in terms of $\Pi_{\xi}$
$F=-\frac{1}{2\pi\alpha^{\prime}}\Pi_{\xi}.$ (10)
As a result, to find the drag force one should find the constant of motion,
$\Pi_{\xi}$ from (9). Numerator and denominator in (8) change sign at $r_{c}$
and it can be found by solving this equation
$f(r_{c})-\frac{v^{2}}{N^{2}}=0.$ (11)
As it is clear in the appendix, Gauss-Bonnet coupling and curvature-cubed
interaction constant are $\lambda$ and $\mu$, respectively and the precise
form of $f(r)$ depends on $\lambda$ and $\mu$. It was found that there are
three different $AdS$ black hole solutions in quasi-topological gravity which
are determined by $f_{1}(r),f_{2}(r)$ and $f_{3}(r)$ in (32). Then for
different values of coupling constant $\lambda$ and $\mu$, one should choose
appropriate form of $f(r)$ from (32) and solve (11). However (11) is
complicated one can solve it numerically. Then, we assume different values for
$\mu$ and $\lambda$ and discuss behavior of the drag force in terms of these
coupling constants.
### 2.2 Positive couplings
We assume both coupling parameters $\mu$ and $\lambda$ are positive. As
pointed out in [33], for this case, only $f_{3}(r)$ in (32) leads to a stable
AdS black hole solution. The drag force versus the velocity of the heavy quark
has been plotted in Fig. 1. In the right and left plots of this figure, Gauss-
Bonnet coupling constant is $\lambda=0.01$ and $\lambda=0.20$, respectively.
Also different values of cubic-curvature coupling interaction are assumed.
As one finds from [40], by increasing $\lambda$ the value of the drag force
increases. This behavior of the drag force is clearly seen in these plots. One
finds that by increasing Gauss-Bonnet coupling constant from $\lambda=0.01$ to
$\lambda=0.20$, the drag force increases. In the plots of Fig. 1, one finds
that by increasing $\mu$ the value of the drag force also increases. Though at
the small velocities, the cubic-curvature interactions have minimum effect on
the drag force. As a result, the main effect of increasing cubic-curvature
coupling constant is increasing the drag force value. This is the same as the
case of $R^{2}$ and $R^{4}$ case [39, 40]. We should check this result in the
case of non-positive $\mu$ and $\lambda$.
,
Figure 1: The drag force versus the velocity of the heavy quark for _positive_
values of cubic-curvature coupling $\mu$ at fixed _positive_ Gauss-Bonnet
coupling constant.
### 2.3 non-positive couplings
Now we intend to study the effect of the non-positive coupling constants
$(\lambda,\mu)$ to the drag force. Three distinct $AdS$ black hole backgrounds
are discussed in (32). These solutions for different regimes of the parameter
space of $(\lambda,\mu)$ are discussed in the table 1 of [33]. As it is
explained in this table, to study the non-positive coupling constants, one
needs $f_{1}(r),f_{2}(r)$ and $f_{3}(r)$ from (32). In the case of positive
$\mu$ and negative $\lambda$, only $f_{3}(r)$ in (32) leads to a stable AdS
black hole solution. In Fig. 2, we assume $\lambda=-0.2$ and
$\mu=+0.01,+0.02,+0.25$ and plot the drag force versus the velocity of the
heavy quark. Also here, one finds that by increasing $\mu$ the value of drag
force increases. One should notice that at the small velocities, the cubic-
curvature corrections have the minimum effects. Therefor we confirm the
previous result. If one assumes negative $\mu$ and positive $\lambda$, also
finds that for larger $\mu$ the value of the drag force becomes larger.
### 2.4 analytic solution
Fortunately, we find an analytic result for the drag force in the special case
of $\mu=-\frac{\lambda^{2}}{3}$ which corresponds to $p=0$ in (34), as
$F_{R^{2}+R^{3}}=-\frac{1}{2\pi\alpha^{\prime}}\left(\frac{\sqrt{3}\,\pi^{2}\,T^{2}\,v}{N^{\frac{1}{2}}\sqrt{-v^{6}\,\lambda^{2}+3v^{4}\,\lambda\,N-3\,v^{2}\,N^{2}+3N^{3}}}\right),$
(12)
where $N$ is defined in (2).
It would be interesting to compare the drag force in the presence of higher
derivative corrections with the case of $\mathcal{N}=4$ strongly-coupled SYM
plasma $F_{\mathcal{N}=4}$. The authors of [38, 37] have obtained
$F_{\mathcal{N}=4}=-\left(\frac{\pi\,\sqrt{\tilde{\lambda}}\,T_{0}^{2}}{2}\right)\,\frac{v}{\sqrt{1-v^{2}}}.$
(13)
where $\tilde{\lambda}$ is ’t Hooft coupling333Notice that
$\alpha^{\prime-2}=\tilde{\lambda}$. and $T_{0}$ is the temperature of AdS
black hole solution without any corrections. Let us consider the case of
$\lambda\rightarrow 0$ in (12). In this limit, one does not consider any
correction in the action (24) and finds that the drag force is nothing but the
drag force in the case of $\mathcal{N}=4$ strongly-coupled SYM plasma
$F_{\mathcal{N}=4}$.
The analytical result in (12), shows the effect of the higher derivative
corrections on the drag force. It is clearly seen that the corrections appear
in the denominator of (12) and as a result the drag force increases.
Figure 2: The drag force versus the velocity of the heavy quark for _positive_
values of cubic-curvature coupling $\mu$ at fixed _negative_ Gauss-Bonnet
coupling constant $\lambda$.
## 3 dissociation length of quark-antiquark pair at finite coupling
In this section we investigate the effect of the higher derivative terms to
the dissociation length of quark-antiquark pair.
In the usual fashion, the two endpoints of the classical open string at the
boundary are seen as a quark and antiquark pair which may be considered as a
meson [31]. Based on lattice results and experiments, it is found that the
meson shows interesting behavior as the temperature of the plasma increases.
It is known that heavy quark bound states can survive in a QGP to temperatures
higher than the confinement/deconfinement transition [32]. Thermal properties
of _static_ quark-antiquark systems have been studied in [20, 21] in an AdS-
Schwarzschild black hole setting using the AdS/CFT correspondence. In [29], a
rotating quark-antiquark in the presence of higher derivative corrections is
studied. In the case of Gauss-Bonnet corrections, it is shown that as the
Gauss-Bonnet coupling constant $\lambda$ increases the string endpoints become
less separated i.e. the radius of the rotating open string at the boundary
decreases but the tip of the U-shaped string does not change considerably.
The heavy quark potential in the presence of curvature-squared corrections is
calculated in [41]. It is shown that the potential can be calculated as a
power series in $LT<<1$, where $T$ is the temperature of the hot plasma. One
finds that at fixed temperature, as the Gauss-Bonnet coupling constant
$\lambda$ increases the interquark distance $L$ decreases. It would be
interesting to investigate this observation in the case of higher derivative
corrections. To do this, we consider Quasi-topological gravity and study
effect of curvature-cubed corrections to the dissociation length. Because of
the complicated feature in this background, we use numerical methods.
### 3.1 dissociation length from AdS/CFT
To find the dissociation length, one should study the heavy quark potential,
$V_{q\bar{q}}(L)$,444We call quark-antiquark potential in (15) as ”heavy quark
potential”. where $L$ is the distance between two quarks [28]. One finds that
the heavy quark potential can be _negative_ , _positive_ or _zero_. If
$V_{q\bar{q}}(L)<0$, the dominant string configuration becomes the one for the
U-shaped string which can be interpreted as a heavy meson. When
$V_{q\bar{q}}(L)>0$, the heavy meson dissociates to two free quarks and the
string configuration changes. This phenomena happens at special length $L=d$
which is obtained from $V_{q\bar{q}}(L=d)=0$. We call $d$ as a dissociation
length. Thus by studying the heavy quark potential in the quasi-topological
gravity, we will find the effect of higher derivative terms to this quantity.
The heavy quark potential is given by the expectation value of the following
static Wilson loop
$W(C)=\frac{1}{N}Tr\,P\,e^{i\,\int A_{\mu}dx^{\mu}},$ (14)
where $C$ denotes a closed loop in spacetime and the trace is over the
fundamental representation of $SU(N)$ group. We consider a rectangular loop
along the time coordinate $t$ and spatial extension $L$. The static heavy
quark potential is related to the expectation value of this rectangular Wilson
loop in the limit of $t\rightarrow\infty$,
$\langle W(C)\rangle\sim e^{-t\,V_{q\bar{q}}(L)},$ (15)
This expectation value can be calculated from $AdS/CFT$ correspondence [20,
21]. In this set up, one should consider an infinitely massive quark in the
fundamental representation of $SU(N)$ group in $\mathcal{N}=4$ Yang-Mills
gauge theory. This quark is dual to a classical string hanging down to the
horizon from a probe brane at the boundary. The classical string hanging in
the bulk space and connecting two endpoints has a characteristic U-shaped. We
name $r_{*}$ as the tip of the U-shaped string and we let it to define the
nearest point between the string and the horizon of the black hole; i.e.
$r_{*}>r_{h}$. Let us emphasize that for non-physical states we would have
$r_{*}<r_{h}$ [20].
The dynamics of the U-shaped string is given by the Euclidean version of the
Nambu-Goto action in (4). To calculate the heavy quark potential, one has to
subtract the infinite self-energy of two independent heavy quarks and from the
$AdS/CFT$ correspondence. These massive quarks are dual to two straight
strings that extend from the probe brane at the boundary to the horizon. The
regularized action is shown by $\bigtriangleup S$ and it is related to the
expectation value of Wilson loop in (15) by this equation
$\langle W(C)\rangle\sim e^{-\bigtriangleup S},$ (16)
As a result, the heavy quark potential is
$V_{q\bar{q}}(L)=\frac{\bigtriangleup S}{t}.$ (17)
The heavy quark potential in the vacuum and in the strongly coupled $N=4$ SYM
gauge theory was found in [20]
$V_{q\bar{q}(L)}=-\frac{4\pi^{2}\sqrt{\lambda}}{\Gamma(1/4)^{2}}\left(\frac{1}{L}\right).$
(18)
We consider $X^{\mu}=(t,x,0,0,r(x))$ for the coordinates of U-shaped string in
the static gauge $\sigma=x,\,\tau=t$. As a result, the Euclidean version of
Nambu-Goto action in (4) can be found as
$S=\frac{N\,t}{2\pi\alpha^{\prime}}\int\,dx\sqrt{r^{4}\,f(r)+r^{\prime 2}},$
(19)
Notice that $r$ depends on $x$. The Hamiltonian density of this action is
constant and it is
$H=-\frac{N\,t}{2\pi\alpha^{\prime}}\frac{r^{4}\,f(r)}{\sqrt{r^{4}\,f(r)+r^{\prime
2}}},$ (20)
This constant is found at special point $r(0)=r_{*}$, where
$r^{\prime}_{*}=0$, as
$H=-\frac{N\,t}{2\pi\alpha^{\prime}}\sqrt{r_{*}^{4}\,f(r_{*})}.$ (21)
Then it is possible to find $L$ as follows
$\frac{L}{2}=\int_{r_{*}}^{\infty}\,dr\left(\frac{1}{r^{4}\,f(r)\left(\frac{r^{4}\,f(r)}{r_{*}^{4}\,f(r_{*})}-1\right)}\right)^{1/2}.$
(22)
Finally, the heavy quark potential is given by
$V_{q\bar{q}}(L)=\frac{\,N}{\pi\alpha^{\prime}}\,\int_{r_{*}}^{\infty}\,dr\left(\left(\frac{\frac{r^{4}\,f(r)}{r_{*}^{4}f(r_{*})}}{\frac{r^{4}\,f(r)}{r_{*}^{4}f(r_{*})}-1}\right)^{\frac{1}{2}}-1\right)-\frac{\,N}{\pi\alpha^{\prime}}\,\int_{r_{h}}^{r_{*}}\,dr.$
(23)
We intend to study the effect of the higher derivative corrections to the
heavy quark potential in (23) and the interquark distance in (22). For
different values of coupling constants $(\lambda,\mu)$, one should consider
three distinct $AdS$ black hole backgrounds which are discussed in (32).
However, we can not solve (23) and (22) analytically and we have to resort to
numerical methods. Also the coefficient $\frac{N\,}{\pi\alpha^{\prime}}$ does
not play any role in our physical discussion.
### 3.2 Numerical Solutions
We illustrate behavior of $V_{q\bar{q}}(L)$ as a function of $L$ at fixed
temperature $(r_{h}=1)$ in Fig. 3. It is clearly seen that there is a maximal
interquark distance, $L_{max}$. It has been shown that for $L<L_{max}$ there
are two kinds of strings; long strings and short strings [42, 43, 44, 45].
These strings correspond to the upper and lower parts of $V_{q\bar{q}}(L)$ in
Fig. 3, respectively. The stability analysis has shown that short strings are
favorable [42, 43, 44, 45]. One concludes that only the lower part is
physical[21].
By analyzing Fig. 3., we investigate behavior of the dissociation length for
different values for $\mu$ and $\lambda$. In this figure, the heavy quark
potential (23) is plotted versus the interquark distance (22). We take that
different values of cubic-curvature coupling $\mu$ while the Gauss-Bonnet
coupling constant $\lambda$ is fixed in each frame. Notice that in this case
the corresponding black hole backgrounds are specified by $f_{3}$. In this
figure, from left to right the Gauss-Bonnet coupling constant $\lambda$ is
increasing, $\lambda=-0.2,0.01$ and $0.2$. By increasing Gauss-Bonnet coupling
constant, the dissociation length of meson decreases. This phenomena has been
found also in the case of a rotating meson [29].
Figure 3: The heavy quark potential versus the interquark distance for
different values of cubic-curvature coupling $\mu$ at fixed Gauss-Bonnet
coupling constant $\lambda$. Left:$\lambda=-0.2$. Middle:$\lambda=0.01$.
Right: $\lambda=0.2$.
What is the effect of increasing cubic-curvature coupling $\mu$ while Gauss-
Bonnet coupling $\lambda$ is fixed? In each plot of Fig. 3, $\lambda$ is fixed
and $\mu$ is increasing. For example in the left plot of this figure
$\lambda=-0.20$ and $\mu=0.01,0.20,0.25$ and $0.28$. One can see that the
interquark distance decreases by increasing $\mu$. This observation is clearly
seen in the middle and right plots of Fig. 3, too. Therefor by increasing
cubic-curvature coupling, the dissociation length of meson decreases.
As we pointed out, there are three distinct AdS black hole backgrounds which
correspond to $f_{1}(r),f_{2}(r)$ and $f_{3}(r)$ in (32). In the case of
$\lambda<0$ and $\mu<0$ one should consider $f_{1}(r)$. We show the heavy
quark potential versus the interquark distance in the left plot of Fig. 4. In
this plot, $\lambda=-0.9$ and from right to left curve $\mu$ is increasing
from $-0.2,-0.1$ to $-0.01$. As before, one finds that the dissociation length
decreases by increasing the cubic-curvature constant $\mu$. One should notice
that the rate of decreasing is not so large. In the case of $\lambda>0$ and
$\mu<0$, one should choose $f_{2}(r)$ to investigate behavior of the heavy
quark potential versus the interquark distance. We show the result in the
right plot of Fig. 4. In this plot $\lambda=0.20$ and $\mu$ is increasing from
$-0.01$ to $-0.0001$. It is clearly seen that by increasing $\mu$, the
dissociation length decreases. This observation is consistent with what we see
in Fig. 3.
Figure 4: The heavy quark potential versus the interquark distance.
Left:$\lambda=-0.9$ and from right to left $\mu=-0.2,\,-0.1,-0.01$. Right:
$\lambda=0.2$ and from right to left $\mu=-0.01,-0.0001$.
One infers that as Gauss-Bonnet coupling constant $\lambda$ increases the
interquark distance decreases. Also at fixed $\lambda$ by increasing cubic-
curvature constant $\mu$ the interquark distant decreases. As a result,
including the higher derivative corrections decrease the dissociation length.
## 4 Conclusion
The higher derivative corrections on the gravity side correspond to finite
coupling corrections on the gauge theory side. The main motivation to consider
these corrections comes from the fact that string theory contains higher
derivative corrections arising from stringy effects. On the gauge theory side,
computations are exactly valid when the ’t Hooft coupling constant goes to
infinity ($\tilde{\lambda}=g_{YM}^{2}N\rightarrow\infty$). An understanding of
how these computations are affected by finite $\lambda$ corrections may be
essential for more precise theoretical predictions.
Although AdS/CFT correspondence is not directly applicable to QCD, one expects
that results obtained from closely related non-abelian gauge theories should
shed qualitative (or even quantitative) insights into analogous questions in
QCD. This has motivated much work devoted to study various properties of
thermal SYM theories like the hydrodynamical transport quantities. In this
paper we have studied the energy loss and the interquark-antiquark distance in
the presence of higher derivative terms. We have considered the cubic-
curvature terms which is known as quasi-topological gravity.
We calculated the energy loss of heavy quark and from numerical analysis,
found that the drag force increases. Fortunately, we found an analytical
result in (12) which confirms our result.
We introduced the heavy quark potential in (15). As it is seen from Fig. 3,
the heavy quark potential is _negative_ , _positive_ or _zero_. By studying
the zero case, we investigated effect of higher derivative terms in quasi-
topological gravity on the dissociation length. We found that the interquark-
antiquark distance becomes shorter with the increase of coupling parameters of
higher curvature terms. This result is consistent with the case of rotating
quark-antiquark pair in [29]. We can therefore conclude that the higher
curvature corrections make the dissociation length shorter. Interestingly, the
subleading term of the strong coupling expansion of the heavy quark potential
in a $\mathcal{N}=4$ SYM plasma is studied in [46]. It is also found that this
correction reduces the magnitude of the heavy quark potential and leads to a
smaller screening radius.
## Acknowledgment
We would like to thank E. Azimfard for helpful discussions and specially thank
M. Ali-Akbari and M. Sohani for reading the manuscript and useful comments.
## 5 Review of Quasi-topological gravity
In this appendix we give a brief review of the quasi-topological gravity in
five-dimensional spacetime [33]. The bulk action is given by
$I=\frac{1}{16\pi G_{5}}\int dx^{5}\sqrt{-g}\left(R-\Lambda+\frac{\lambda
L^{2}}{2}\chi_{4}+\frac{7L^{4}\mu}{8}Z_{5}\right),$ (24)
where $\lambda$ and $\mu$ are Gauss-Bonnet coupling and curvature-cubed
interaction constant, respectively. The negative cosmological constant is
related to radius of AdS space by $\Lambda=-\frac{12}{L^{2}}$. The curvature-
squared interaction is given by $\chi_{4}$ as
$\chi_{4}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$
(25)
and $Z_{5}$ is the new curvature-cubed interaction
$\displaystyle Z_{5}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}^{\,\,\,\,\,\,\rho\sigma}R_{\rho\sigma}^{\,\,\,\,\,\,\alpha\beta}R_{\alpha\beta}^{\,\,\,\,\,\,\mu\nu}+\frac{1}{14}\left(21R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,R-120\,R_{\mu\nu\rho\sigma}R^{\mu\nu\rho}_{\,\,\,\,\,\,\,\,\,\alpha}R^{\sigma\alpha}\right.$
(26)
$\displaystyle\left.144\,R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma}+128R_{\mu}^{\,\,\,\,\nu}R_{\nu}^{\,\,\,\,\rho}R_{\rho}^{\,\,\,\,\mu}-108R_{\mu}^{\,\,\,\,\nu}R_{\nu}^{\,\,\,\,\mu}R+11\,R^{3}\right).$
The planar AdS black hole solutions for different values of the coupling
constants were found in [33]. The solution, in units where the radius of $AdS$
is one, is
$ds^{2}=r^{2}\left(-N^{2}\,f(r)dt^{2}+d\vec{x}^{2}\right)+\frac{dr^{2}}{r^{2}\,f(r)},$
(27)
where $f(r)$ is determined by roots of the following equation
$1-f(r)+\lambda f(r)^{2}+\mu f(r)^{3}=\frac{r_{h}^{4}}{r^{4}}.$ (28)
Here $r$ denotes the radial coordinate of the black brane geometry and
$t,\vec{x}$ label the directions along the boundary at the spatial infinity.
In these coordinates the event horizon is located at $f(r_{h})=0$ where
$r_{h}$ is found by solving this equation. The boundary is located at infinity
and the geometry will be as asymptotically AdS . The constant $N^{2}$
specifies the speed of light of the boundary gauge theory and one can choose
it to be unity. We name $f(r)$ at the boundary where $r\rightarrow\infty$, as
$f_{\infty}$ and one finds that
$N^{2}=\frac{1}{f_{\infty}},$ (29)
One also finds from (28) that $f_{\infty}$ satisfies
$1-f_{\infty}+\lambda f_{\infty}^{2}+\mu f_{\infty}^{3}=0.$ (30)
The temperature of the hot plasma is given by the Hawking temperature of the
black hole
$T=\frac{N\,r_{h}}{\pi}.$ (31)
Authors in [33], solved (28) and found $f(r)$ for different values of coupling
constants $\lambda$ and $\mu$. It is shown that there are three different
solutions of (28) in the $\mu-\lambda$ plane:
$\displaystyle f_{1}(r)$ $\displaystyle=$ $\displaystyle
u+v-\frac{\lambda}{3\mu},$ $\displaystyle f_{2}(r)$ $\displaystyle=$
$\displaystyle-\frac{u+v}{2}+i\,\frac{\sqrt{3}}{2}(u-v)-\frac{\lambda}{3\mu},$
$\displaystyle f_{3}(r)$ $\displaystyle=$
$\displaystyle-\frac{u+v}{2}-i\,\frac{\sqrt{3}}{2}(u-v)-\frac{\lambda}{3\mu},$
(32)
where
$u=(q+\sqrt{q^{2}-p^{3}})^{\frac{1}{3}},\,\,\,\,\,\,v=(q-\sqrt{q^{2}-p^{3}})^{\frac{1}{3}},$
(33)
and
$p=\frac{3\mu+\lambda^{2}}{9\mu^{2}},\,\,\,q=-\frac{2\lambda^{3}+9\mu\lambda+27\mu^{2}\left(1-\frac{r_{h}^{4}}{r^{4}}\right)}{54\mu^{3}}.$
(34)
There is a relation between Gauss-Bonnet coupling constant $\lambda$ and
cubic-curvature coupling constant $\mu$ as follows
$\mu=\frac{2}{27}-\frac{\lambda}{3}\pm\frac{2}{27}\left(1-3\lambda\right)^{\frac{3}{2}}.$
(35)
which shows the upper and lower bound on the cubic-curvature interaction
coupling. There is a special case $p=0$ in (34) which corresponds to
$\mu=-\frac{\lambda^{2}}{3}$. $f(r)$ is also found at this point.
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|
arxiv-papers
| 2011-02-11T05:52:39 |
2024-09-04T02:49:16.960227
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. Bitaghsir Fadafan",
"submitter": "Kazem Bitaghsir Fadafan",
"url": "https://arxiv.org/abs/1102.2289"
}
|
1102.2304
|
# Commuting powers and exterior degree of finite groups
Peyman Niroomand School of Mathematics and Computer Science
Damghan University of Basic Sciences
Damghan, Iran p$\\_$niroomand@yahoo.com , Rashid Rezaei Department of
Mathematics, Faculty of Mathematical Sciences
Malayer University
Post Box: 657719–95863, Malayer, Iran ras$\\_$rezaei@yahoo.com and Francesco
G. Russo Department of Mathematics
University of Palermo, via Archirafi 34, 90123, Palermo, Italy
francescog.russo@yahoo.com
###### Abstract.
In [P. Niroomand, R. Rezaei, On the exterior degree of finite groups, Comm.
Algebra 39 (2011), 335–343] it is introduced a group invariant, related to the
number of elements $x$ and $y$ of a finite group $G$, such that $x\wedge
y=1_{{}_{G\wedge G}}$ in the exterior square $G\wedge G$ of $G$. This number
gives restrictions on the Schur multiplier of $G$ and, consequently, large
classes of groups can be described. In the present paper we generalize the
previous investigations on the topic, focusing on the number of elements of
the form $h^{m}\wedge k$ of $H\wedge K$ such that $h^{m}\wedge
k=1_{{}_{H\wedge K}}$, where $m\geq 1$ and $H$ and $K$ are arbitrary subgroups
of $G$.
###### Key words and phrases:
$m$–th relative exterior degree, commutativity degree, exterior product, Schur
multiplier, dihedral groups, generalized quaternion groups.
###### 2010 Mathematics Subject Classification:
Primary: 20J99, 20D15; Secondary: 20D60; 20C25.
## 1\. Non–abelian tensor product, homological algebra and commutativity
degree
All the groups, which are considered in the paper, are supposed to be finite.
Some technical notions of homological algebra should be recalled from [8, 9,
10] in order to formulate our topic of investigation in an appropriate way.
For any group $G$ we can construct functorially a classifying space $B(G)$
with the following properties.
* 1)
The topological space $B(G)$ is a connected CW-complex.
* 2)
The fundamental group $\pi_{1}(B(G))$ of $B(G)$ is isomorphic to $G$.
* 3)
The higher homotopy groups $\pi_{n}(B(G))$ are trivial for $n\geq 2$.
The singular homology groups of any space $X$, with coefficients in the
abelian group $\mathbb{Z}$, will be denoted by $H_{n}(X)$. Since the homology
groups $H_{n}(B(G))$ depend only on the group $G$, we can write
$H_{n}(G)=H_{n}(B(G))$, for all $n\geq 0$. For each normal subgroup $H$ in $G$
we functorially construct a space $B(G,H)$ as follows. The natural
homomorphism $G\rightarrow G/H$ induces a map $f:B(G)\rightarrow B(G/H)$. Let
$M(f)$ denote the mapping cylinder of this map. Note that $B(G)$ is a subspace
of $M(f)$, and that $M(f)$ is homotopy equivalent to $B(G/H)$. We take
$B(G,H)$ to be mapping cone of the cofibration $B(G)\rightarrow M(f)$. The
cofibration sequence $B(G)\rightarrow M(f)\rightarrow B(G,H)$ yields a natural
long exact homology (Mayer–Vietoris) sequence $\ldots\rightarrow
H_{n+1}(G/H)\rightarrow H_{n+1}(B(G,H))\rightarrow H_{n}(G)\rightarrow
H_{n}(G/H)\rightarrow\ldots$ for $n\geq 0$. It can be shown that
$H_{1}(B(G,H))=0$ and $H_{2}(B(G,H))\simeq H/[H,G]$. The classifying space
$B(F)$ of a free group $F$ is one-dimensional, and so $H_{n}(F)=0$ for $n\geq
2$ and it is easy to check that $H_{1}(G)\simeq G/[G,G]=G^{ab}$ and
$H_{2}(G)\simeq\ker\psi\simeq M(G)$, where $G=F/R$ is a presentation of $G$
from $F$ and $R$ and $\psi:R/[R,F]\rightarrow F/[F,F]$ is a natural
homomorphism and $M(G)$ is the Schur multiplier of $G$. Now it is meaningful
to define the Schur multiplier of the pair of groups $(G,H)$ as the set
$M(G,H)=H_{3}(B(G,H))$. We can generalize more. By a triple we mean a group
$G$ with two normal subgroups $H$ and $K$. A homomorphism of triples
$(G,H,K)\rightarrow(G^{\prime},H^{\prime},K^{\prime})$ is a group homomorphism
$G\rightarrow G^{\prime}$ that sends $H$ into $H^{\prime}$ and $K$ into
$K^{\prime}$. The Schur multiplier of the triple $(G,H,K)$ is a functorial
abelian group $M(G,H,K)$ whose principal feature is a natural exact sequence
$H_{3}(G,H)\rightarrow H_{3}(G/H,HK/K)\rightarrow M(G,H,K)\rightarrow
M(G,K)\rightarrow M(G/H,HK/H)\rightarrow H\cap K/[H\cap K,G][H,K]\rightarrow
K/[K,G]\rightarrow KH/H[K,G]\rightarrow 0$ in which, by definition,
$H_{3}(G,H)=H_{4}(B(G,H))$. The definition of $M(G,H,K)$ is in terms of the
mapping cone $B(G,H,K)$ of the canonical cofibration $B(G,K)\rightarrow
B(G/K,HK/H)$. An analogy with the case of pairs allows us to define
$M(G,H,K)=H_{4}(B(G,H,K))$.
The Schur multiplier of a triple is related to an important construction,
which we recall as in [10, Section 3] and [3, 4]. A group $G$ acts by
conjugation on its normal subgroups $H$ and $K$ via the rule
${}^{g}x=gxg^{-1}$, for $g$ in $G$ and $x$ in $H$ or $K$, and the exterior
product $H\wedge K$ is defined as the group generated by the symbols $h\otimes
k$, subject to the relations:
(1.1) $hh^{\prime}\otimes k=(~{}^{h}h^{\prime}\otimes~{}^{h}k)\ (h\otimes k),\
\ \ \ kk^{\prime}\otimes h=(k\otimes h)\ (~{}^{k}h\otimes~{}^{k}k^{\prime}),\
\ \ \ y\otimes y=1,$
where $h,h^{\prime}\in H$, $k,k^{\prime}\in K$ and $y\in H\cap K$. Briefly,
$h\wedge k$ denotes $h\otimes k$ satisfying all the above relations. The map
$\kappa^{\prime}:h\wedge k\in H\wedge K\mapsto[h,k]=hkh^{-1}k^{-1}\in[H,K]$
turns out to be a group epimorphism, whose kernel $\ker\kappa^{\prime}$ is
abelian. Furthermore, $\ker\kappa^{\prime}\simeq M(G,H,K)$ whenever $G=HK$
(see [10, Theorem 6.1]). Omitting the relation $y\otimes y=1$, it is similarly
defined the non–abelian tensor product $H\otimes K$ of $H$ and $K$. By
analogy, the map $\kappa:h\otimes k\in H\otimes
K\mapsto[h,k]=hkh^{-1}k^{-1}\in[H,K]$ turns out to be a group epimorphism,
whose kernel $\ker\kappa=J(G,H,K)$ is again abelian. We note that $J(G,H,K)$
is related to the fundamental group of a covering space and has significant
interest in algebraic topology (see [3, 4, 8, 9, 10]).
The above information are summarized below, where $G=HK$ (with $H$ and $K$
normal in $G$).
(1.2) $\begin{CD}1@>{}>{}>J(G,H,K)@>{}>{}>H\otimes
K@>{\kappa}>{}>[H,K]@>{}>{}>1\\\ @V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>M(G,H,K)@>{}>{}>H\wedge K@>{\kappa^{\prime}}>{}>[H,K]@>{}>{}>1.\\\
\end{CD}$
From the results in [3, 4, 8, 9, 10], (1.2) is commutative with central
extensions as rows and natural epimorphisms $\pi:h\otimes k\in J(G,H,K)\mapsto
h\wedge k\in M(G,H,K)$, $\epsilon:h\otimes k\in H\otimes K\mapsto h\wedge k\in
H\wedge K$ as columns. Of course, if $G=H=K$, then $M(G)$ is the Schur
multiplier of $G$, $H\otimes K=G\otimes G$ is the non–abelian tensor square of
$G$ and, in particular, $G^{ab}\otimes_{\mathbb{Z}}G^{ab}$ is the usual tensor
square of an abelian group.
It may be helpful to recall that the actions of $H$ on $K$ induce an action
$a\in H*K\ \longmapsto\ ^{a}(h\otimes k)=\ ^{a}h\ \otimes\ ^{a}k\in H\otimes
K$, which allows us to see $H\otimes K$ as a suitable homomorphic image of the
central product $H*K$. In this context, if $x\in G$, the exterior centralizer
of $x$ in $G$ is the set $C_{G}^{\wedge}(x)=\\{a\in G\ |\ a\wedge
x=1_{{}_{G\wedge G}}\\}$, which turns out to be a subgroup of $G$ and the
exterior center of $G$ is the set $Z^{\wedge}(G)=\\{g\in G\ |\ 1_{{}_{G\wedge
G}}=g\wedge y\in G\wedge G,\forall y\in G\\}={\underset{x\in
G}{\bigcap}}C_{G}^{\wedge}(x)$ which is a subgroup of the center $Z(G)$ of
$G$. Further details can be found in [8, 9, 16, 18]. Very briefly, we mention
that the interest in studying $C_{G}^{\wedge}(x)$ and $Z^{\wedge}(G)$ is due
to the fact that they allow us to decide whether $G$ is a capable group or
not, that is, whether $G$ is isomorphic to $E/Z(E)$ for some group $E$ or not.
[2] and [1, Chapter 21] illustrate that capable groups are well–known and
classified.
Now we recall from [6, 7, 11, 12, 13, 14, 15, 19] that the commutativity
degree of $G$ is the ratio
(1.3) $d(G)=\frac{|\\{(x,y)\in G\times G\ |\
[x,y]=1\\}|}{|G|^{2}}=\frac{1}{|G|^{2}}\underset{x\in
G}{\sum}|C_{G}(x)|=\frac{k(G)}{|G|},$
where $k(G)$ is the number of the $G$–conjugacy classes $[x]_{G}=\\{x^{g}\ |\
g\in G\\}$ that constitute $G$. There is a wide production on $d(G)$ and its
generalizations in the last decades. For instance, given an arbitrary subgroup
$H$ of $G$, it was introduced in [11] the $n$-th relative nilpotency degree of
$G$
(1.4) $d^{(n)}(H,G)=\frac{|\\{(h_{1},\ldots,h_{n},g)\in H^{n}\times G\ |\
[h_{1},\ldots,h_{n},g]=1\\}|}{|H|^{n}\ |G|}=\frac{1}{|H|^{n}\
|G|}\underset{h_{1},\ldots,h_{n}\in H}{\sum}|C_{G}([h_{1},\ldots,h_{n}])|.$
It is clearly a generalization of $d(G)$, and, in case $n=1$, it was proposed
the further generalization
(1.5) $d(H,K)=\frac{|\\{(h,k)\in H\times K\ |\ [h,k]=1\\}|}{|H|\
|K|}=\frac{1}{|H|\ |K|}\sum_{h\in H}|C_{K}(h)|=\frac{k_{K}(H)}{|H|}$
in [6], where $H$ is a normal subgroup of $G$, $K$ is an arbitrary subgroup of
$G$ and $k_{K}(H)$ is the number of the $K$–conjugacy classes
$[h]_{K}=\\{h^{k}\ |\ k\in K\\}$ that constitute $H$.
We will focuse on a recent contribution in [17], where it is introduced the
exterior degree of $G$
(1.6) $d^{\wedge}(G)=\frac{|\\{(x,y)\in G\times G\ |\ x\wedge y=1_{{}_{G\wedge
G}}\\}|}{|G|^{2}},$
which can be written by [17, Lemma 2.2] as
(1.7)
$d^{\wedge}(G)=\frac{1}{|G|}\sum^{k(G)}_{i=1}\frac{|C^{\wedge}_{G}(x_{i})|}{|C_{G}(x_{i})|}.$
In analogy, given two arbitrary subgroups $H$ and $K$ of $G$, we define for
$m\geq 1$ the $m$-th relative exterior degree of $H$ and $K$ in $G$
(1.8) $d^{\wedge}_{m}(H,K)=\frac{|\\{(h,k)\in H\times K\ |\ h^{m}\wedge
k=1_{{}_{H\wedge K}}\\}|}{|H|\ |K|}.$
In particular, $d^{\wedge}_{m}(G)=d^{\wedge}_{m}(G,G)$ is the $m$-th exterior
degree of $G$ and, of course, $d^{\wedge}_{1}(G,G)=d^{\wedge}(G)$ so that it
is meaningful to generalize the bounds in [17]. We also note that for $H=G$
and $m=1$ there are results on $d^{\wedge}(G,K)$ in [18]. While the
commutativity degree represents the probability that two randomly picked
elements of $G$ are commuting, the $n$-th relative nilpotency degree is a
variation on this theme. By analogy with the operator $\wedge$, the $m$-th
relative exterior degree is a variation on the theme of the exterior degree,
involving the powers of $x$ and the single element $y$. We will study the
effects of $d^{\wedge}_{m}(H,K)$ on the structure of $G$ in the successive
sections.
## 2\. Basic properties
An immediate observation is that we may rewrite $d^{\wedge}_{m}(H,K)$ as:
(2.1) $d^{\wedge}_{m}(H,K)=\frac{1}{|H|\ |K|}\sum_{h\in
H}|C^{\wedge}_{K}(h^{m})|.$
Assume that $H$ is normal in $G$ and $C_{1}\ldots,C_{k_{K}(H)}$ are the
$K$–conjugacy classes that constitute $H$. It follows that
(2.2) $|H|\ |K|\ d^{\wedge}_{m}(H,K)=\sum_{h\in
H}|C^{\wedge}_{K}(h^{m})|=\sum^{k_{K}(H)}_{i=1}\sum_{h\in
C_{i}}|C^{\wedge}_{K}(h^{m})|=\sum^{k_{K}(H)}_{i=1}|K:C_{K}(h_{i})|\
|C^{\wedge}_{K}(h^{m}_{i})|$
$=\sum^{k_{K}(H)}_{i=1}\frac{|K|}{|C_{K}(h^{m}_{i})|}\
\frac{|C_{K}(h^{m}_{i})|}{|C_{K}(h_{i})|}\ |C^{\wedge}_{K}(h^{m}_{i})|=|K|\
\sum^{k_{K}(H)}_{i=1}\left(\frac{|C_{K}(h^{m}_{i})|}{|C_{K}(h_{i})|}\right)\
\frac{|C^{\wedge}_{K}(h^{m}_{i})|}{|C_{K}(h^{m}_{i})|}=|K|\
\sum^{k_{K}(H)}_{i=1}\alpha(m,i)\frac{|C^{\wedge}_{K}(h^{m}_{i})|}{|C_{K}(h^{m}_{i})|},$
where $\alpha(m,i)$ is the index of $|C_{K}(h_{i})|$ in $|C_{K}(h^{m}_{i})|$
and then a natural number. The assumption that $H$ has to be normal in $G$ is
done in order to have an entire conjugacy class which is fixed under the
action of $K$ on $H$. It may be helpful for the rest of the paper to define
the group
(2.3) $L(m,i;h,K)=\frac{C_{K}(h^{m}_{i})}{C^{\wedge}_{K}(h^{m}_{i})}.$
###### Lemma 2.1.
Let $H$ be a normal subgroup of a group $G$ and $K$ be a subgroup of $G$. Then
(2.4)
$d^{\wedge}_{m}(H,K)=\frac{1}{|H|}\sum^{k_{K}(H)}_{i=1}\alpha(m,i)\frac{|C^{\wedge}_{K}(h^{m}_{i})|}{|C_{K}(h^{m}_{i})|}=\frac{1}{|H|}\
\sum^{k_{K}(H)}_{i=1}\frac{\alpha(m,i)}{|L(m,i;h,K)|}.$
In particular, if $G=HK$ and $K$ is normal in $G$, then $L(m,i;h,K)$ is
isomorphic to a subgroup of $M(G,H,K)$.
###### Proof.
The first part follows from (2.2). Now assume that $G=HK$ for $H$ and $K$
normal in $G$. The exact sequence (1.2) implies that for all
$i=1,\ldots,k_{K}(H)$ the map $x\in C_{K}(h_{i}^{m})\mapsto h_{i}^{m}\wedge
x\in M(G,H,K)$ is a homomorphism of groups. On another hand, its kernel is
$C^{\wedge}_{K}(h_{i}^{m})$, and, consequently, $L(m,i;h,K)$ is isomorphic to
a subgroup of $M(G,H,K)$. ∎
The sequence $d^{\wedge}_{m}(H,K)$ is monotone in the sense of the next
result. We should do an assumption on $m$ of being of prime power order. This
will be necessary (but not sufficient) to have the subgroup lattice of a
cyclic group which is a chain.
###### Proposition 2.2.
Let $H$ and $K$ be subgroups of $G$ and $p$ be a prime divisor of $|H|$. Then
there exists an integer $r\geq 1$ such that
(2.5) $d^{\wedge}_{p^{r-1}}(H,G)\geq d^{\wedge}_{p^{r-1}}(H,K)\geq
d^{\wedge}_{p^{r-2}}(H,K)\geq\ldots\geq d^{\wedge}_{p}(H,K)\geq
d^{\wedge}(H,K).$
###### Proof.
Let $h\in H$ be of order $p^{r}$ for some integer $r\geq 1$. Then
$\\{1\\}=\langle h^{p^{r}}\rangle\leq\langle
h^{p^{r-1}}\rangle\leq\ldots\leq\langle h\rangle$ implies
$C^{\wedge}_{K}(\\{1\\})=K\geq
C^{\wedge}_{K}(h^{p^{r-1}})=C^{\wedge}_{K}(\langle
h^{p^{r-1}}\rangle)\geq\ldots\geq C^{\wedge}_{K}(h^{p})=C^{\wedge}_{K}(\langle
h^{p}\rangle)\geq C^{\wedge}_{K}(h)=C^{\wedge}_{K}(\langle h\rangle)$.
Therefore
(2.6) $\sum_{h\in H}|C^{\wedge}_{K}(h)|\leq\sum_{h\in
H}|C^{\wedge}_{K}(h^{p})|\leq\ldots\leq\sum_{h\in
H}|C^{\wedge}_{K}(h^{p^{r-1}})|,$
from which we deduce
(2.7) $d^{\wedge}(H,K)\leq d^{\wedge}_{p}(H,K)\leq\ldots\leq
d^{\wedge}_{p^{r-1}}(H,K).$
On another hand,
(2.8) $|H|\ |G|\ d^{\wedge}_{p^{r-1}}(H,G)=\sum_{h\in
H}|C^{\wedge}_{G}(h^{p^{r-1}})|\geq\sum_{h\in
H}|C^{\wedge}_{K}(h^{p^{r-1}})|=|H|\ |K|\ d^{\wedge}_{p^{r-1}}(H,K).$
∎
Among groups with trivial Schur multiplier there are important classes of
groups. For instance, a cyclic group $C=\langle c\rangle$ has $|M(C)|=1$ by
[1, Lemma 21.1]; a metacyclic group of the form $D=\langle a,b\ |\
a^{p^{n}}=b^{p}=1,b^{-1}ab=a^{1+p^{n-1}}\rangle$ (where $n\geq 3$ if $p=2$)
has also $|M(D)|=1$ by [1, Theorem 1.2 and Lemma 21.2]; finally, looking at
[5], several sporadic simple groups have trivial Schur multiplier. In our
context, we are interested to see what happens to $d^{\wedge}_{m}(H,K)$ when
$M(G,H,K)$ is trivial. Immediately, we find the next consequence of Lemma 2.1.
###### Corollary 2.3.
Let $G=HK$ for two normal subgroups $H$ and $K$ of $G$ with $H$ of exponent
$p^{r}-1$ for some $r\geq 1$ and some prime $p$. If $M(G,H,K)$ is trivial,
then $\alpha(p^{r},i)=|L(p^{r},i;h,K)|=1$.
###### Proof.
By Lemma 2.1, $|L(p^{r},i;h,K)|=1$. The fact that $H$ has exponent $p^{r}-1$
implies $h^{p^{r}-1}_{i}=1$, that is, $h^{p^{r}}_{i}=h_{i}$ for all
$i=1,\ldots,k_{K}(H)$, and then $\alpha(p^{r},i)=1$. ∎
We can refine the condition at infinity of $r$, by looking at Proposition 2.2,
and we have the following result.
###### Corollary 2.4.
Let $H$ be a normal subgroup of a group $G$, $K$ a subgroup of $G$ and $p$ a
prime divisor of $|H|$. Then ${\underset{r\rightarrow 0}{\lim}}\
d^{\wedge}_{p^{r}}(H,K)=d^{\wedge}(H,K)$. Furthermore, if
${\underset{r\rightarrow\infty}{\lim}}\frac{\alpha(p^{r},i)}{|L(p^{r},i;h,K)|}=1$
and the action of $K$ on $H$ induces just one orbit, then
${\underset{r\rightarrow\infty}{\lim}}\
d^{\wedge}_{p^{r}}(H,K)\leq\frac{1}{p}$. In particular,
$d(H,K)={\underset{r\rightarrow\infty}{\lim}}\
d^{\wedge}_{p^{r}}(H,K)=\frac{1}{p}$, provided that $|H|=p$.
###### Proof.
The first part of the result follows from Proposition 2.2.
Lemma 2.1 and the assumptions imply
(2.9)
${\underset{r\rightarrow\infty}{\lim}}d^{\wedge}_{p^{r}}(H,K)={\underset{r\rightarrow\infty}{\lim}}\frac{1}{|H|}\sum^{k_{K}(H)}_{i=1}\frac{\alpha(p^{r},i)}{|L(p^{r},i;h,K)|}=\frac{1}{|H|}{\underset{r\rightarrow\infty}{\lim}}\sum^{k_{K}(H)}_{i=1}\frac{\alpha(p^{r},i)}{|L(p^{r},i;h,K)|}$
$=\frac{1}{|H|}\ \sum^{k_{K}(H)}_{i=1}{\underset{r\rightarrow\infty}{\lim}}\
\frac{\alpha(p^{r},i)}{|L(p^{r},i;h,K)|}=\frac{k_{K}(H)}{|H|}=d(H,K).$
The choice of $p$ implies $\frac{1}{|H|}\leq\frac{1}{p}$ and therefore
${\underset{r\rightarrow\infty}{\lim}}\
d^{\wedge}_{p^{r}}(H,K)\leq\frac{k_{K}(H)}{p}$. In particular, if the action
of $K$ on $H$ induces just one orbit, then $k_{K}(H)$ is just one and so
${\underset{r\rightarrow\infty}{\lim}}\
d^{\wedge}_{p^{r}}(H,K)\leq\frac{1}{p}$. The rest follows clearly from (2.9).
∎
With respect to direct products there is a sort of natural splitting for
$d^{\wedge}_{m}(H,K)$ and this is shown below.
###### Proposition 2.5.
If $A,B,C,D$ are subgroups of a group $G$ such that $(|A|,|B|)=(|C|,|D|)=1$,
then
(2.10) $d^{\wedge}_{m}(A\times B,C\times D)=d^{\wedge}_{m}(A,C)\cdot
d^{\wedge}_{m}(B,D).$
###### Proof.
(2.11) $|A\times B|\ |C\times D|\ d^{\wedge}_{m}(A\times B,C\times D)=|A|\
|B|\ |C|\ |D|\ d^{\wedge}_{m}(A\times B,C\times D)$ (2.12) $=\sum_{(a,b)\in
A\times B}|C^{\wedge}_{C\times D}((a^{m},b^{m}))|=\left(\sum_{a\in
A}|C^{\wedge}_{C}(a^{m})|\right)\ \left(\sum_{b\in
B}|C^{\wedge}_{D}(b^{m})|\right)=|A|\ |C|\ d^{\wedge}_{m}(A,C)\ |B|\ |D|\
d^{\wedge}_{m}(B,D).$
∎
In particular, [17, Lemma 2.10] can be found as a special case of the previous
result. Another general property is encountered when we go to form quotients
and for $m=1$ it can be found in [17, Proposition 2.6]. Before to describe it,
we introduce the set $Z^{\wedge}(H,K)=\\{h\in H\ |\ h\wedge k=1{{}_{H\wedge
K}}\ \forall k\in K\\}$, where $H$ and $K$ are normal subgroups of $G$, acting
upon each other by conjugation. $Z^{\wedge}(H,K)$ is largely described in [18]
when $G=H$ and it is easy to check that $Z^{\wedge}(H,K)$ is a subgroup of
$H$, and, in particular, $Z^{\wedge}(G,G)=Z^{\wedge}(G)$.
###### Proposition 2.6.
If $H$ and $K$ are two subgroups of $G$ containing a normal subgroup $N$ of
$G$, then $d^{\wedge}_{m}(H,K)\leq d^{\wedge}_{m}(H/N,K/N).$ The equality
holds, if $N\subseteq Z^{\wedge}(H,K)$.
###### Proof.
(2.13) $|H|\ |K|\ d^{\wedge}_{m}(H,K)=\sum_{h\in
H}|C^{\wedge}_{K}(h^{m})|=\sum_{hN\in H/N}\sum_{n\in
N}|C^{\wedge}_{K}(h^{m}n)|=\sum_{hN\in H/N}\sum_{n\in
N}\frac{|C^{\wedge}_{K}(h^{m}n)N|}{|N|}\ |C^{\wedge}_{K}(h^{m}n)\cap N|$
(2.14) $\leq\sum_{hN\in H/N}\sum_{n\in N}|C^{\wedge}_{K/N}(h^{m}N)|\
|C^{\wedge}_{K}(h^{m}n)\cap N|=\sum_{hN\in H/N}|C^{\wedge}_{K/N}(h^{m}N)|\
\sum_{n\in N}|C^{\wedge}_{K}(h^{m}n)\cap N|$ (2.15) $\leq|N|^{2}\sum_{hN\in
H/N}|C^{\wedge}_{K/N}(h^{m}N)|=|H|\ |K|\ d^{\wedge}(H/N,K/N).$
We find always an exact sequence
(2.16) $\begin{CD}1@>{}>{}>N\wedge K@>{\varphi}>{}>H\wedge
K@>{\epsilon}>{}>(H/N)\wedge(K/N)@>{}>{}>1\end{CD}$
where $\iota:n\in N\mapsto\iota(n)\in H$ is the natural embedding of $N$ into
$H$, $\varphi:n\wedge k\in N\wedge K\mapsto\iota(n)\wedge h\in H\wedge K$ and
$\epsilon:h\wedge k\in H\wedge K\mapsto hN\wedge kN\in(H/N)\wedge(K/N)$ is
induced by the natural epimorphisms of $H$ onto $H/N$ and of $K$ onto $K/N$.
If $N\subseteq Z^{\wedge}(H,K)$, then $\mathrm{Im}\ \varphi=1_{{}_{N\wedge
K}}$ and (2.16) implies $H/N\wedge K/N\simeq H\wedge K$ so that $|N|^{2}\
|\\{(hN,kN)\in H/N\times K/N\ |\ h^{m}N\wedge
kN=1_{{}_{(H/N)\wedge(K/N)}}\\}|=|\\{(h,k)\in H\times K\ |\ h^{m}\wedge
k=1_{{}_{H\wedge K}}\\}|$, hence
$d^{\wedge}_{m}(H,K)=d^{\wedge}_{m}(H/N,K/N)$. ∎
A general restriction is the following.
###### Theorem 2.7.
Let $G=HK$ for two normal subgroups $H$ and $K$. Then for all $m\geq 1$
(2.17) $\beta(m)\ \frac{d(H,K)}{|M(G,H,K)|}\leq
d^{\wedge}_{m}(H,K)\leq\gamma(m)\ d(H,K),$
where $\beta(m)=\mathrm{min}\\{\alpha(m,i)\ |\ i=1,\ldots,k_{K}(H)\\}$ and
$\gamma(m)=\mathrm{max}\\{\alpha(m,i)\ |\ i=1,\ldots,k_{K}(H)\\}$.
###### Proof.
Keeping in mind Lemma 2.1 and noting that
$C_{K}(h^{m}_{i})/C^{\wedge}_{K}(h^{m}_{i})$ is isomorphic to a subgroup of
$M(G,H,K)$, we have $|C^{\wedge}_{K}(h^{m}_{i})|/|C_{K}(h^{m}_{i})|\geq
1/|M(G,H,K)|$. Therefore
(2.18) $d^{\wedge}_{m}(H,K)=\frac{1}{|H|}\sum^{k_{K}(H)}_{i=1}\alpha(m,i)\
\frac{|C^{\wedge}_{K}(h^{m}_{i})|}{|C_{K}(h^{m}_{i})|}\geq\frac{\beta(m)\
k_{K}(H)}{|H|\ |M(G,H,K)|}=\beta(m)\ \frac{\ d(H,K)}{|M(G,H,K)|}.$
On another hand, again Lemma 2.1 implies
(2.19) $d^{\wedge}_{m}(H,K)=\frac{1}{|H|}\sum^{k_{K}(H)}_{i=1}\alpha(m,i)\
\frac{|C^{\wedge}_{K}(h^{m}_{i})|}{|C_{K}(h^{m}_{i})|}\leq\gamma(m)\
\frac{k_{K}(H)}{|H|}=\gamma(m)\ d(H,K).$
∎
In [8, 16, 17] it was noted that a group $G$ such that $Z^{\wedge}(G)=Z(G)$
has strong structural restrictions; among these it was noted in [17] that
$d^{\wedge}_{1}(G)=d^{\wedge}(G)=d(G)$. We find something of similar in the
next result.
###### Corollary 2.8.
Let $G=HK$ for two normal subgroups $H$ and $K$. If $M(G,H,K)$ is trivial and
$H$ has exponent $m-1$, then $d^{\wedge}_{m}(H,K)=d(H,K)$ for all $m\geq 1$.
###### Proof.
Since $M(G,H,K)=1$ is trivial, the lower bound in (2.17) is reduced to
$\beta(m)\ d(H,K)$. $H$ has exponent $m-1$ and then, using the notations of
Lemma 2.1, $h^{m-1}_{i}=1$, that is, $h^{m}_{i}=h_{i}$, for all
$i=1,\ldots,k_{K}(H)$. Consequently,
$\alpha(m,i)=\alpha(m)=\beta(m)=\gamma(m)=1$. Then (2.17) becomes $d(H,K)\leq
d^{\wedge}_{m}(H,K)\leq d(H,K)$ and the result follows. ∎
Another consequence of Theorem 2.7 is related to Proposition 2.2.
###### Corollary 2.9.
Let $G=HK$ for two normal subgroups $H$ and $K$ and $p$ be a prime divisor of
$|H|$. Then there exists an integer $r\geq 1$ such that
(2.20) $\frac{\gamma(p^{r-1})}{p}\ k_{K}(H)\geq d^{\wedge}_{p^{r-1}}(H,K)\geq
d^{\wedge}_{p^{r-2}}(H,K)\geq\ldots\geq\beta(p^{r-1})\ \frac{\
d(H,K)}{|M(G,H,K)|}.$
###### Proof.
It is enough to apply Theorem 2.7 and Proposition 2.2. ∎
In a certain sense, Corollary 2.4 continues to be true without restrictions on
$m$. This is illustrated below.
###### Proposition 2.10.
Let $G=HK$ for two normal subgroups $H$ and $K$ such that $[H,K]\not=1$. Then
$d^{\wedge}_{m}(H,K)\leq\gamma(m)\ \frac{2p-1}{p^{2}}$, where $p$ is the
smallest prime dividing $|G|$ and $|K|$. In particular, if $H$ has exponent
$m-1$, then $d^{\wedge}_{m}(H,K)\leq\frac{2}{p}$.
###### Proof.
From the choice of $p$ we deduce
$|C^{\wedge}_{K}(H)|\leq|C_{K}(H)|\leq\frac{|K|}{p}$. Now
$|C_{K}(H)|-|C^{\wedge}_{K}(H)|\leq\frac{|K|}{p}$. The upper bound in Theorem
2.7 allows us to continue as in [6, Corollary 3.9] and so
(2.21) $d^{\wedge}_{m}(H,K)\leq\gamma(m)\
d(H,K)\leq\gamma(m)\frac{2p-1}{p^{2}}.$
In particular, if $H$ has exponent $m-1$, then the argument in Corollary 2.8
implies $\gamma(m)=1$, hence
(2.22) $\frac{2p-1}{p^{2}}=\frac{2}{p}-\frac{1}{p^{2}}\leq\frac{2}{p}.$
∎
The following result justifies the interest for the numerical restrictions on
$d^{\wedge}_{m}(H,K)$, which have been the subject of most of the previous
bounds. These allow us to describe the position of some subgroups in the whole
group, when we consider some special values of the $m$-th relative exterior
degree.
###### Corollary 2.11.
Let $H$ be a normal subgroup of $G$ of exponent $m-1$ and $K$ be a normal
subgroup of $G$ such that $G=HK$ and $M(G,H,K)$ is trivial. If
$d^{\wedge}_{m}(H,K)=\frac{2p-1}{p^{2}}$ for some prime $p$, then $p$ divides
$|G|$. If $p$ is the smallest prime divisor of $|G|$, then
$|H:C_{H}(K)|=|K:C_{K}(H)|=p$ and, hence, $H\not=K$. In particular, if
$d^{\wedge}_{m}(H,K)=\frac{3}{4}$, then $|H:C_{H}(K)|=|K:C_{K}(H)|=2$.
###### Proof.
Corollary 2.8 implies $d^{\wedge}_{m}(H,K)=d(H,K)$ for all $m\geq 1$. The rest
follows from [6, Proposition 3.1]. ∎
## 3\. Dihedral groups and generalized quaternion groups
Thanks to the results in Section 2 and to those in [11, 12, 14, 15, 17, 18],
we want to have a closer look at the class of dihedral groups and at that of
generalized quaternion groups. Their structure is described in [1, Theorem
1.2]: these groups possess a cyclic group of index 2 and are metacyclic. We
will be quite general and recall that
(3.1) $D_{2n}=C_{n}\rtimes C_{2}=\langle a,b\ |\
a^{n}=b^{2}=1,b^{-1}ab=a^{-1}\rangle$
is the dihedral group of order $2n$, where $n\geq 1$. Assume
$D_{2n}=\\{1,a,a^{2},\ldots,a^{n-1},b,ab,a^{2}b,\ldots,a^{n-1}b\\}$ and
$t=\gcd(m,n)$. Since $Z^{\wedge}(G)=1$ and $(a^{\frac{in}{t}})^{m}=1$ for
$0\leq i\leq t-1$, for $t$ elements of $D_{2n}$ we have
$|C^{\wedge}_{D_{2n}}(x^{m})|=2n$ and for $n-t$ elements
$|C^{\wedge}_{D_{2n}}(x^{m})|=n$. Now, if $m$ is odd, then
$|C^{\wedge}_{D_{2n}}((a^{j}b)^{m})|=2$ for $0\leq j\leq n-1$ and so
$d^{\wedge}_{m}(D_{2n})=\frac{n^{2}+nt+2n}{4n^{2}}.$ If $m$ is even, then
$(a^{j}b)^{m}=1$ and so $|C^{\wedge}_{D_{2n}}((a^{j}b)^{m})|=2n$, therefore
$d^{\wedge}_{m}(D_{2n})=\frac{3n^{2}+nt}{4n^{2}}.$ Summarizing,
(3.2)
$d^{\wedge}_{m}(D_{2n})=\left\\{\begin{array}[]{lcl}\frac{3n+\gcd(m,n)}{4n},&&\mathrm{if}\
m\ \mathrm{is\ even},\\\ \frac{n+\gcd(m,n)+2}{4n},&&\mathrm{if}\ m\
\mathrm{is\ odd}.\end{array}\right.$
A similar computation can be made for
(3.3) $Q_{n}=\langle a,b\ |\ a^{n}=b^{2}=(ab)^{2}\rangle,$
which is called generalized quaternion group of order $4n$. Here, as done for
$D_{2n}$, we find that
(3.4)
$d^{\wedge}_{m}(Q_{n})=\left\\{\begin{array}[]{lcl}\frac{3n+\gcd(m,n)}{4n},&&\mathrm{if}\
m\ \mathrm{is\ even},\\\ \frac{n+\gcd(m,n)+2}{4n},&&\mathrm{if}\ m\
\mathrm{is\ odd}.\end{array}\right.$
From [17, Examples 3.1 and 3.2],
$d^{\wedge}(D_{2n})=d(D_{2n})=d^{\wedge}(Q_{n})=d(Q_{n})$ for all $n\geq 1$
and we have just shown that for all $m\geq 1$ (and for all $n\geq 1$)
$d^{\wedge}_{m}(D_{2n})=d^{\wedge}_{m}(Q_{n}).$ We note that
$|M(D_{2n})|\not=1$ and $|M(Q_{n})|=1$ and then we cannot apply Corollary 2.8,
but (3.2) and (3.4) show, in some sense, that the thesis of Corollary 2.8 is
still true. We note also that (3.2) and (3.4) agree with [11, Example 3.11]
and [18, Section 4].
## References
* [1] Y. Berkovich, Groups of Prime Power Order, Vol.1, de Gruyter, Berlin, 2008.
* [2] F. R. Beyl, U. Felgner and P. Schmid, On groups occurring as center factor groups, J. Algebra 61 (1979), 161–177.
* [3] R. Brown, D. L. Johnson and E. F. Robertson, Some computations of non-abelian tensor products of groups, J. Algebra 111 (1987), 177–202.
* [4] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311–335.
* [5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, The Atlas of Finite Simple Groups, Oxford University Press, Oxford, 1985.
* [6] A. K. Das and R. K. Nath, On the generalized relative commutative degree of a finite group, Int. Electr. J. Algebra 7 (2010), 140–151.
* [7] A. K. Das and R. K. Nath, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo 59 (2010), 137–142.
* [8] G. Ellis, Tensor products and $q$-crossed modules, J. London Math. Soc. 2 (1995), 241–258.
* [9] G. Ellis, A bound for the derived Frattini subgroups of a prime-power group, Proc. Amer. Math. Soc. 126 (1998), 2513–2523.
* [10] G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures 6 (1998), 355–371.
* [11] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra 35 (2007), 4183–4197.
* [12] A. Erfanian, R. Rezaei, F. G. Russo, Relative $n$-isoclinism classes and relative $n$-th nilpotency degree of finite groups, E–print, Cornell University Library, arXiv:1003.2306, 2010.
* [13] R. M. Guralnick and G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300 (2006), 509–528.
* [14] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995), 847–869.
* [15] P. Lescot, Central extensions and commutativity degree, Comm. Algebra 29 (2001), 4451–4460.
* [16] P. Niroomand and F. G. Russo, A note on the exterior centralizer, Arch. Math. (Basel) 93 (2009), 505–512.
* [17] P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335–343.
* [18] P. Niroomand and R. Rezaei, The exterior degree of a pair of finite groups, E–print, Cornell University Library, arXiv:1101.4312v1, 2011.
* [19] D. J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82 (1979), 237–247.
|
arxiv-papers
| 2011-02-11T09:04:06 |
2024-09-04T02:49:16.966868
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Peyman Niroomand (Damghan University, Damghan, Iran), Rashid Rezaei\n (University of Malayer, Malayer, Iran) and Francesco G. Russo (Universita'\n degli Studi di Palermo, Palermo, Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1102.2304"
}
|
1102.2390
|
# Bounds for sectional genera of varieties invariant under Pfaff fields
Maurício Corrêa Jr Departamento de Matemática, Universidade Federal de
Viçosa-UFV, Avenida P.H. Rolfs, 36571-000 Brazil mauricio.correa@ufv.br and
Marcos Jardim Instituto de Matemática, Estatística e Computação Científica
Universidade Estadual de Campinas
Rua Sérgio Buarque de Holanda, 651
Campinas, SP, Brazil
CEP 13083-859 jardim@ime.unicamp.br
###### Abstract.
We establish an upper bound for the sectional genus of varieties which are
invariant under Pfaff fields on projective spaces.
###### 1991 Mathematics Subject Classification:
Primary: 32S65; Secondary: 37F75, 58A17
Partially supported by CNPq.
Partially supported by the CNPq grant number 305464/2007-8 and the FAPESP
grant number 2005/04558-0.
## 1\. Introduction
In [20] P. Painlevé asked the following question: _“Is it possible to
recognize the genus of the general solution of an algebraic differential
equation in two variables which has a rational first integral?_ ” In [16],
Lins Neto has constructed families of foliations with fixed degree and local
analytic type of the singularities where foliations with rational first
integral of arbitrarily large degree appear. In other words, such families
show that Painlevé’s question has a negative answer.
However, one can obtain an affirmative answer to Painlevé’s question provided
some additional hypotheses are made. The problem of bounding the genus of an
invariant curve in terms of the degree of a foliation on $\mathbb{P}^{n}$ has
been considered by several authors, see for instance [6, 8]. In [3], Campillo,
Carnicer and de la Fuente showed that if $C$ is a reduced curve which is
invariant by a one-dimensional foliation ${\mathcal{F}}$ on $\mathbb{P}^{n}$
then
(1) $\frac{2p_{a}(C)-2}{\deg(C)}\leq\deg({\mathcal{F}})-1+a,$
where $p_{a}(C)$ is the arithmetic genus of $C$ and $a$ is an integer obtained
from the concrete problem of imposing singularities to projective
hypersurfaces. For instance, if $C$ has only nodal singularities then $a=0$,
and thus formula $(\ref{ccf})$ follows from [11]. This bound has been improved
by Esteves and Kleiman in [8].
Painlevé’s question is related to the problem posed by Poincaré in [23] of
bounding the degree of algebraic solutions of an algebraic differential
equation on the complex plane. Nowadays, this problem is known as Poincaré’s
Problem. Many mathematicians have been working on it and on some of its
generalizations, see for instance the papers by Cerveau and Lins Neto [6],
Carnicer [4], Pereira [21], Soares [24], Brunella and Mendes [2], Esteves and
Kleiman [8], Cavalier and Lehmann [5], and Zamora [28].
In [8], Esteves and Kleiman extended Jouanolou’s work on algebraic Pfaff
systems on a nonsingular scheme $V$. Essentially, an algebraic Pfaff system is
a singular distribution. More precisely, an algebraic Pfaff system of rank $r$
on a nonsingular scheme $X$ of pure dimension $n$ is, according to Jouanolou
[13, pp. $136$-$38$], a nonzero map $u:E\rightarrow\Omega_{X}^{1}$ where $E$
is a locally free sheaf of constant rank $r$ with $1\leq r\leq n-1$. Esteves
and Kleiman introduced the notion of a _Pfaff field_ on $V$, which is a
nontrivial sheaf map $\eta:\Omega_{V}^{k}\to L$, where $L$ is a invertible
sheaf on $V$, and the integer $1\leq k\leq n-1$ is called the rank of $\eta$.
A subvariety $X\subset V$ is said to be invariant under $\eta$ if the map
$\eta$ factors through the natural map $\Omega^{k}_{V}|_{X}\to\Omega^{k}_{X}$.
A Pfaff system on $V$ induces, via exterior powers and the perfect pairing of
differential forms, a Pfaff field on $V$. However, the converse is not true;
see [8, Section 3] for more details.
In this paper, we establish new upper bounds for the sectional genera of
nonsingular projective varieties which are invariant under Pfaff fields on
$\mathbb{P}^{n}$.
First, we use the hypothesis of stability (in the sense of Mumford–Takemoto)
of the tangent bundle of $X$ to establish an upper bound for the sectional
genus in terms of the degree and the rank of a Pfaff field.
More precisely, our first main result is the following. Let
$g(X,\mathcal{O}_{X}(1))$ denote the sectional genus of $X$ with respect to
the line bundle $\mathcal{O}_{X}(1)$ associated to the hyperplane section.
###### Theorem 1.
Let $X$ be a nonsingular projective variety of dimension $m$ which is
invariant under a Pfaff field ${\mathcal{F}}$ of rank $k$ on $\mathbb{P}^{n}$;
assume that $m\geq k$. If the tangent bundle $\Theta_{X}$ is stable, then
(2)
$\frac{2g(X,\mathcal{O}_{X}(1))-2}{\deg(X)}\leq\dfrac{\deg({\mathcal{F}})-k}{{m-1\choose
k-1}}+m-1.$
To the best of our knowledge, this is the first time that the stability of the
tangent bundle is used to obtain such bounds. Notice that the left-hand side
of inequality (2) does not change when we take generic linear sections
$\mathbb{P}^{l}\subset\mathbb{P}^{n}$, while the right-hand side gets larger,
and so the bound becomes worse. This means that the above result is a truly
higher dimensional one.
Examples of projective varieties with stable tangent bundle are Calabi–Yau
[27], Fano [9, 12, 22, 25] and complete intersection [22, 26] varieties.
In the critical case when the rank of Pfaff field ${\mathcal{F}}$ is equal to
the dimension of the invariant variety $X$, we show that one can substitute
for the stability condition the conditions of $X$ being Gorenstein and smooth
in codimension $1$, i.e. $\mathrm{codim}(Sing(X),X)\geq 2$.
###### Theorem 2.
Let $X\subset\mathbb{P}^{n}$ be a Gorenstein projective variety nonsingular in
codimension $1$, which is invariant under a Pfaff field ${\mathcal{F}}$ on
$\mathbb{P}^{n}$ whose rank is equal to the dimension of $X$. Then
(3) $\frac{2g(X,\mathcal{O}_{X}(1))-2}{\deg(X)}\leq\deg({\mathcal{F}})-1,$
This generalizes a bound obtained by Campillo, Carnicer and de la Fuente in
[3, Theorem 4.1 (a)]. As an application, we improve upon a bound obtained by
Cruz and Esteves [7, Corollary 4.5], see Section 5.
This note is organized as follows. First, in order to make this presentation
as self-contained as possible, we provide all the necessary definitions in
Section 2. The proofs of our main results along with some further consequences
are given in Sections 4 and 3.
## 2\. Background material
We work over the field of complex numbers. Let $(X,L)$ be a Gorenstein
projective variety $X$ of dimension $n$ equipped with a very ample line bundle
$L$; recall that, since $X$ is Gorenstein, the canonical divisor $K_{X}$ is a
Cartier divisor.
###### Definition 1.
The _sectional genus_ of $X$ with respect to $L$, denoted $g(X,L)$, is defined
by the formula:
$2g(X,L)-2=(K_{X}+(\dim(X)-1)L)\cdot L^{\dim(X)-1}.$
This quantity has the following geometric interpretation. Suppose that $X$ is
nonsingular, and let $H_{1},\dots,H_{n-1}$ be general elements in the linear
system $|L|$. By Bertini’s Theorem, the curve $X_{n-1}=H_{1}\cap\cdots\cap
H_{n-1}$ is nonsingular. Then $g(X,L)$ coincides with the geometric genus of
$X_{n-1}$, see [10, Remark 2.5].
###### Definition 2.
Let $(V,L)$ be a nonsingular polarized algebraic variety. A Pfaff field
${\mathcal{F}}$ of rank $k$ on $V$ is a nonzero global section of
$\bigwedge^{k}\Theta_{V}\otimes N$, where $\Theta_{V}$ is the tangent bundle
and $N$ is a line bundle, where $0<k<n$. The degree of ${\mathcal{F}}$ with
respect to $L$ is defined by the formula
$\deg_{L}({\mathcal{F}})=\deg_{L}(N)+k\deg_{L}(L)$, where the degree of a line
bundle $N$ relative to $L$ is given by $\deg_{L}(N)=N\cdot L^{\dim(V)-1}$.
Since the ambient space is nonsingular, our definition is equivalent to the
one introduced in [8, Section 3]. In fact, since
$\bigwedge^{k}\Theta_{V}\otimes
N\simeq\mathcal{H}om(\Omega^{k}_{V},N)\simeq\mathcal{H}om(N^{*},\bigwedge^{k}\Theta_{V})$,
a Pfaff field can also be regarded either as a map
$\xi_{{\mathcal{F}}}:N^{*}\rightarrow\bigwedge^{k}\Theta_{V}$ or as a map
$\xi_{{\mathcal{F}}}^{\vee}:\Omega^{k}_{V}\rightarrow N$. The present
definition emphasizes the existence of a global section of
$\bigwedge^{k}\Theta_{V}\otimes N$, which will play a central role in our
arguments.
###### Definition 3.
The _singular set_ of ${\mathcal{F}}$ is given by
$Sing({\mathcal{F}})=\\{x\in V;\ \xi_{{\mathcal{F}}}(x)\ \ $is not
injective$\\}=\\{x\in V;\ \xi_{{\mathcal{F}}}^{\vee}(x)\ \ $is not
surjective$\\}$.
For instance, a Pfaff field of rank $k$ on $\mathbb{P}^{n}$ is a section of
$\bigwedge^{k}\Theta_{\mathbb{P}^{n}}\otimes\mathcal{O}_{\mathbb{P}^{n}}(s)$,
and $\deg_{\mathcal{O}_{\mathbb{P}^{n}}(1)}({\mathcal{F}})=s+k$.
More generally, if $\mathrm{Pic}(V)\simeq\mathbb{Z}$ and
$L:=\mathcal{O}_{V}(1)$ is the positive generator of $\mathrm{Pic}(V)$, then a
Pfaff field of rank $k$ on $V$ is a section of
$\bigwedge^{k}\Theta_{V}\otimes\mathcal{O}_{V}(s)$, for some $s\in\mathbb{Z}$.
Thus, $\deg_{L}({\mathcal{F}})=(s+k)\deg(V)$, where $\deg(V)=\deg_{L}(L)$. If
we define $d_{{\mathcal{F}}}:=s+k$ we have
$\deg_{L}({\mathcal{F}})=d_{{\mathcal{F}}}\cdot\deg(V).$
Alternatively, a Pfaff field can also be defined as a global section of
$\Omega^{n-k}_{V}\otimes N^{\prime}$, where $N^{\prime}=N\otimes K_{V}^{-1}$.
If $V$ is nonsingular, this definition is equivalent to the one above.
Let $X\subset V$ be a closed subscheme of dimension larger than or equal to
the rank of a Pfaff field ${\mathcal{F}}$. Following [8, Section 3], we
introduce the following definition.
###### Definition 4.
We say $X$ is invariant under ${\mathcal{F}}$ if $X\not\subset
Sing({\mathcal{F}})$ and there exists a morphism of sheaves
$\phi:\Omega_{X}^{k}\rightarrow N|_{X}$ such that the following diagram
$\textstyle{\Omega_{V}^{k}|_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{{\mathcal{F}}}^{\vee}|_{X}}$$\textstyle{N|_{X}}$$\textstyle{\Omega_{X}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$
commutes.
Applying the functor $\mathcal{H}om(\cdot,\mathcal{O}_{X})$ to the above
diagram, we get the following commutative diagram:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.90613pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-12.89082pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{N^{*}|_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-20.5479pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{\phi^{\vee}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.04025pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.19534pt\raise-14.0479pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\xi_{{\mathcal{F}}}|_{X}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.64119pt\raise-29.15137pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
54.27475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-14.90613pt\raise-41.09581pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\Omega_{X}^{k})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.90613pt\raise-41.09581pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.90613pt\raise-41.09581pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bigwedge^{k}\Theta_{V}|_{X}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Therefore, $X$ is invariant under ${\mathcal{F}}$ if
$\xi_{{\mathcal{F}}}|_{X}$ induces a nonzero global section of
$(\Omega_{X}^{k})^{\vee}\otimes N|_{X}$.
Our two main results are concerned only with the case when
$V={\mathbb{P}^{n}}$; but we would like to conclude this section with two
general propositions.
Let $E$ be a torsion-free sheaf on $V$. The ratio $\mu_{L}(E)=\deg_{L}(E)/{\rm
rk}(E)$ is called the slope of $E$, where
$\deg_{L}(E)=\deg_{L}((\Lambda^{r}E)^{\vee\vee})$ and $r={\rm rk}(E)$. Recall
that $E$ is _semistable_ (in the sense of Mumford–Takemoto) if every torsion-
free subsheaf $E^{\prime}$ of $E$ satisfies
$\mu_{L}(E^{\prime})\leq\mu_{L}(E)$. Furthermore, $E$ is _stable_ if the
strict inequality is satisfied for proper subsheaves. Further details can be
found in [14, Sections V.6 and V.7].
###### Proposition 5.
If $\Theta_{V}$ is stable, then the following inequality holds:
$\deg_{L}({\mathcal{F}})\geq{\rm
rk}({\mathcal{F}})\left(\deg_{L}(V)+\frac{\deg_{L}(K_{V})}{\dim(V)}\right).$
If $V={\mathbb{P}^{n}}$ the above inequality becomes $\deg({\mathcal{F}})\geq
0$. Bott’s formula [19, page 8] implies the existence of a rank $k$ Pfaff
field of degree $0$ for each $k$, hence in this case the bound given above is
sharp.
###### Proof.
The stability of $\Theta_{V}$ implies that $\bigwedge^{k}\Theta_{V}$ is
semistable with slope equal to $k\mu_{L}(\Theta_{V})$ [1, Corollary 1.6]. As
observed above, a Pfaff field ${\mathcal{F}}$ of rank $k$ induces a map
$\xi_{{\mathcal{F}}}:N^{*}\to\bigwedge^{k}\Theta_{V}$, so from the
semistability of $\bigwedge^{k}\Theta_{V}$ we conclude that $-\deg_{L}(N)\leq
k\mu_{L}(\Theta_{V})=-k\deg_{L}(K_{V})/\dim(V)$. The stated inequality follows
easily. ∎
If $D$ is a divisor on an algebraic variety $V$ with
$\mathrm{Pic}(V)\simeq\mathbb{Z}$, then
$\mathcal{O}_{V}(D)=\mathcal{O}_{V}(d_{D})$, for some $d_{D}\in\mathbb{Z}$. In
this case, we denote $\kappa(V)=d_{K_{V}}$.
###### Proposition 6.
Let $V$ be a $n$-dimensional nonsingular algebraic variety with
$\mathrm{Pic}(V)\simeq\mathbb{Z}$. Let $X$ be a $k$-dimensional nonsingular
complete intersection of hypersurfaces $D_{1},\dots,D_{n-k}$ on $V$. If $X$ is
invariant under a Pfaff field ${\mathcal{F}}$ of rank $k$ on $V$, then
$d_{D_{1}}+\cdots+d_{D_{n-k}}\leq d_{{\mathcal{F}}}-k-\kappa(V).$
###### Proof.
Since $X$ is invariant by ${\mathcal{F}}$ we have that
$H^{0}(X,\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{V}(d_{{\mathcal{F}}}-k)|_{X})\neq\\{0\\}$,
then
$\deg(\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{V}(d_{{\mathcal{F}}}-k)|_{X})\geq
0$. Let $\mathcal{O}_{V}(D_{i})$ be the line bundle associated to the
hypersurface $D_{i}$, $i=1,\dots,n-k$. We have the following adjunction
formula
$\bigwedge^{k}\Theta_{X}=\bigwedge^{n}\Theta_{V}|_{X}\otimes\mathcal{O}_{V}(-D_{1})|_{X}\otimes\cdots\mathcal{O}_{V}(-D_{n-k})|_{X}.$
Therefore
$\bigwedge^{k}\Theta_{X}=\mathcal{O}_{V}(-\kappa(V)-d_{D_{1}}-\cdots-
d_{D_{n-k}})|_{X}$, thus
$\deg(\mathcal{O}_{V}(d_{{\mathcal{F}}}-k-\kappa(V)-d_{D_{1}}-\cdots-
d_{D_{n-k}})|_{X})=\deg(\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{V}(d_{{\mathcal{F}}}-k)|_{X})\geq
0.$
∎
## 3\. Proof of Theorem 1
We recall that the stability of $\Theta_{X}$ implies that
$\bigwedge^{k}\Theta_{X}$ is semistable. Since $X$ is invariant under
${\mathcal{F}}$, we can conclude that
$H^{0}(X,\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{X}(d-k))\neq\\{0\\}$, with
$d=\deg({\mathcal{F}})$. It then follows from the semistability of
$\bigwedge^{k}\Theta_{X}$ that
$\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{X}(d-k)$ is also semistable, thus
(4) $\deg(\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{X}(d-k))\geq 0.$
On the other hand, note that
(5) $\deg(\bigwedge^{k}\Theta_{X})=-{\dim(X)-1\choose k-1}\deg(K_{X}).$
Let $i:X\rightarrow\mathbb{P}^{n}$ be the embedding, and set, as usual,
$\mathcal{O}_{X}(1)=i^{*}\mathcal{O}_{\mathbb{P}^{n}}(1)$. Now, we consider
the following difference, using 5:
$(2g(X,\mathcal{O}_{X}(1))-2)-\left[\frac{\mathcal{O}_{X}(d-k)}{{m-1\choose
k-1}}+(m-1)\mathcal{O}_{X}(1)\right]\cdot\mathcal{O}_{X}(1)^{m-1}=$
$-\left(-K_{X}+\frac{\mathcal{O}_{X}(d-k)}{{m-1\choose
k-1}}\right)\cdot\mathcal{O}_{X}(1)^{m-1}=-\frac{\deg(\bigwedge^{k}\Theta_{X}\otimes\mathcal{O}_{X}(d-k))}{{m-1\choose
k-1}}.$
It follows from (4) that the difference must be less than or equal to zero,
hence
$\begin{array}[]{ccl}2g(X,\mathcal{O}_{X}(1))-2&\leq&\left[\frac{\mathcal{O}_{X}(d-k)}{{m-1\choose
k-1}}+(m-1)\mathcal{O}_{X}(1)\right]\cdot\mathcal{O}_{X}(1)^{m-1}\\\ \\\
&\leq&\deg(X)\left(\dfrac{d-k}{{m-1\choose k-1}}+m-1\right).\end{array}$
This completes the proof of Theorem 1.
Let us now consider applications of Theorem 1 to a few particular cases.
First, specializing to the case when the invariant variety is Fano with Picard
number one, i.e., $\deg(K_{X})<0$ and $\rho(X)=rank(NS(X))=1$, where $NS(X)$
is the Néron–Severi group of $X$.
###### Corollary 7.
Let $X$ be a nonsingular Fano variety, with Picard number one, and let
$\mathcal{O}_{X}(1):=K_{X}^{-1}$. If $X$ is invariant under a Pfaff field
${\mathcal{F}}$ of rank $k=\dim(X)$, then
$\deg_{K_{X}^{-1}}(X)\leq k^{k}(\deg({\mathcal{F}})+2)^{k},$
where $\deg_{K_{X}^{-1}}(X)$ is the degree of $X$ with respect to the
anticanonical polarization.
###### Proof.
Indeed, in this case we have
$2g(X,K_{X}^{-1})-2=(k-2)\deg_{K_{X}^{-1}}(X).$
Thus, it follows from Theorem 1 that $k\leq\deg({\mathcal{F}})+1$. On the
other hand, it follows from [18] that $d(X)\leq k+1$ and
$\deg_{K_{X}^{-1}}(X)\leq(d(X)k)^{k}$ , where $d(X)$ is the least positive
integer $d$ for which $X$ can be covered by rational curves of (anticanonical)
degree at most $d$, see [18, Subsection 1.3]. ∎
Finally, we also consider the case when the invariant variety is Calabi–Yau,
i.e. $K_{X}=0$.
###### Corollary 8.
If $X$ is Calabi–Yau and invariant by ${\mathcal{F}}$ then ${\rm
rk}({\mathcal{F}})\leq\deg({\mathcal{F}})$.
In other words, Pfaff fields of small degree do not admit invariant Calabi–Yau
varieties.
## 4\. Proof of Theorem 2
First, let us briefly recall the construction of the so-called _canonical map_
$\gamma_{X}:\Omega^{k}_{X}\rightarrow\omega_{X},$ where $\omega_{X}$ is the
dualizing sheaf of $X$, as it was done in [7, Section 3].
Let $X$ be a reduced projective variety of pure dimension $k$, and let
$X_{1},\dots,X_{s}$ be its irreducible components. For each $i=1,\dots,s$,
consider Kunz’s sheaf $\widetilde{\omega}_{X_{i}}$ of regular differential
forms of $X_{i}$, see [15]. By definition, the canonical map $\gamma_{X}$ is
the composition
$\begin{array}[]{ccccccccc}\Omega^{k}_{X}&\stackrel{{\scriptstyle\widetilde{\tau}}}{{\longrightarrow}}&\bigoplus_{i=1}^{s}\Omega^{k}_{X_{i}}&\stackrel{{\scriptstyle(\gamma_{1},\dots,\gamma_{s})}}{{\longrightarrow}}&\bigoplus_{i=1}^{s}\widetilde{\omega}_{X_{i}}&\stackrel{{\scriptstyle(\zeta_{1},\dots,\zeta_{s})}}{{\longrightarrow}}&\bigoplus_{i=1}^{s}\omega_{X_{i}}&\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}&\omega_{X},\end{array}$
where $\widetilde{\tau}$ and $\tau$ are the maps induced by restriction, for
each $i=1,\dots,s$ the map
$\gamma_{i}:\Omega^{k}_{X_{i}}\rightarrow\widetilde{\omega}_{X_{i}}$ is the
canonical class of $X_{i}$, constructed by Lipman in [17], which is an
isomorphism on the nonsingular locus of $X_{i}$. Moreover,
$\zeta_{i}:\widetilde{\omega}_{X_{i}}\rightarrow\omega_{X_{i}}$ is a
isomorphism on $X_{i}$, since it follows from [17, Theorem 0.2B] that
$\widetilde{\omega}_{X_{i}}$ is dualizing. Therefore, $\gamma_{X}$ is an
isomorphism on the nonsingular locus $X_{0}:=X-Sing(X)$. Thus the map
$\widetilde{\gamma_{X}}=\gamma_{X}^{\vee}\otimes{\mathbf{1}}_{\mathcal{O}_{X}(d-k)}:\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k)\rightarrow(\Omega^{k}_{X})^{\vee}\otimes\mathcal{O}_{X}(d-k)$
is also an isomorphism when restricted to $X_{0}$.
Now assume that $X\subset\mathbb{P}^{n}$ is a Gorenstein variety of pure
dimension $k$ such that $\operatorname{{codim}}(Sing(X),X)\geq 2$. Then the
sheaf $\omega_{X}^{\vee}$ is locally-free, hence, in particular, reflexive.
Moreover, from [14, Proposition 5.21], we also conclude that
$\omega_{X}^{\vee}$ is normal.
If $X$ is invariant under a Pfaff field ${\mathcal{F}}$ on $\mathbb{P}^{n}$ of
rank $k$ and degree $d$, then we have a nonzero global section
$\zeta_{{\mathcal{F}}}$ of
$(\Omega^{k}_{X})^{\vee}\otimes\mathcal{O}_{X}(d-k)$; consider its restriction
$\zeta_{{\mathcal{F}},0}=\zeta_{{\mathcal{F}}}|_{X_{0}}$ to $X_{0}$. Composing
it with the the inverse of $\widetilde{\gamma_{X}}|_{X_{0}}$, the restriction
of the map $\widetilde{\gamma_{X}}$ to $X_{0}$, we obtain a section
$\widetilde{\gamma_{X}}|_{X_{0}}(\zeta_{{\mathcal{F}},0})\in
H^{0}(X_{0},\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k)|_{X_{0}}).$
However, $\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k)|_{X_{0}}$ is a normal
sheaf, so the above section extends to a global section of
$\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k)$. In particular,
$H^{0}(X,\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k))\neq\\{0\\}$, therefore
(6) $\deg(\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k))\geq 0.$
Let $K_{X}$ be a Cartier divisor such that
$\mathcal{O}_{X}(K_{X})=\omega_{X}$.
Now, consider the following difference
$(2g(X,\mathcal{O}_{X}(1))-2)-[\mathcal{O}_{X}(d-k)+(k-1)\mathcal{O}_{X}(1)]\cdot\mathcal{O}_{X}(1)^{k-1}=$
$-\left(K_{X}^{-1}+\mathcal{O}_{X}(d-k)\right)\cdot\mathcal{O}_{X}(1)^{k-1}=-\deg(\omega_{X}^{\vee}\otimes\mathcal{O}_{X}(d-k))\leq
0.$
## 5\. Complete intersection invariant varieties
We specialize to the case when the invariant variety $X$ is a complete
intersection.
First, we notice that the inequality of Theorem 1 is not sharp in general. To
see this, let $X$ be a nonsingular complete intersection variety of dimension
$m$ and multidegree $(d_{1},\dots,d_{n-m})$, which is invariant under a
$k$-dimensional Pfaff field ${\mathcal{F}}$ on $\mathbb{P}^{n}$; assume that
$m\geq k$. It follows from [22, Corollary 1.5] that $\Theta_{X}$ is stable and
one can apply Theorem 1 to obtain the following inequality:
$d_{1}+\cdots+d_{n-m}\leq\dfrac{\deg({\mathcal{F}})-k}{{m-1\choose k-1}}+n+1.$
Setting $m=n-1$ and $k=1$, the inequality reduces to
$d_{1}\leq\deg({\mathcal{F}})+n$. However, Soares has shown, under the same
circumstances, that $d_{1}\leq\deg({\mathcal{F}})+1$ [24, Theorem B].
In the critical case $\dim(X)={\rm rank}({\mathcal{F}})$, Theorem 2 gives us
the following Corollary.
###### Corollary 9.
Let $X$ be a $k$-dimensional complete intersection variety of multidegree
$(d_{1},\dots,d_{n-k})$ such that either $X$ is nonsingular in codimension
$1$. If $X$ is invariant under a Pfaff field ${\mathcal{F}}$ of rank $k$ on
$\mathbb{P}^{n}$, then
$d_{1}+\cdots+d_{n-k}\leq\deg({\mathcal{F}})+n-k+1.$
###### Proof.
From the adjunction formula for dualizing sheaves one obtains
$2g(X,\mathcal{O}_{X}(1))-2=\deg(X)\left(d_{1}+\cdots+d_{n-k}-n+k-2\right).$
By Theorem 2, this is less than or equal to $(\deg({\mathcal{F}})-1)\deg(X)$,
and the desired inequality follows easily. ∎
It follows from [7, Corollary 4.5] that if $X$ and ${\mathcal{F}}$ are as
above, then
$d_{1}+\cdots+d_{n-k}\leq\left\\{\begin{array}[]{ll}\deg({\mathcal{F}})+n-k,&\hbox{if
}\ \rho\leq 0\\\ \\\ \deg({\mathcal{F}})+n-k+\rho,&\hbox{if
}\rho>0\end{array}\right.$
where $\rho:=\sigma+n-k+1-d_{1}-\cdots-d_{n-k}$, with $\sigma$ denoting the
Castelnuovo–Mumford regularity of the singular locus of $X$. Therefore,
Corollary 9 allows us to conclude that if $X$ is nonsingular in codimension
$1$, then one can take $\rho=1$, regardless of $\sigma$.
## References
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* [7] J. D. A. S. Cruz and E. Esteves, _Regularity of subschemes invariant under Pfaff fields on projective spaces_ , To appear in Comment. Math. Helv. (2011).
* [8] E. Esteves and S. Kleiman, _Bounds on leaves of one-dimensional foliations_ , Bull. Braz. Mat. Soc. (NS) 34 (2003), 145–169.
* [9] R. Fahlaoui, _Stabilité du fibre tangent des surfaces de del Pezzo_ , Math. Ann. 283 (1989), 171–176.
* [10] Y. Fukuma, _On the sectional geometric genus of quasi-polarized varieties I_ , Comm. Algebra 32 (2004), 1069–1100.
* [11] J. Garcia, _Multiplicity of a foliation on projective spaces along an integral curve_ , Rev. Mat. Univ. Complut. Madrid 6 (1993), 207–217.
* [12] J.M. Hwang, _Stability of tangent bundles of low dimensional Fano manifolds with Picard number 1_ , Math. Ann. 312 (1998), 599–606.
* [13] J. P. Jouanolou, _Equations de Pfaff algébriques_ , Lecture Notes in Mathematics 708, Springer, 1979.
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* [19] O. Okonek, M. Schneider and H. Spindler, _Vector bundles on complex projective spaces_ , Boston: Birkhauser (1980)
* [20] P. Painlevé, _Sur les intégrales algébrique des équations differentielles du premier ordre_ and _Mémoire sur les équations différentielles du premier ordre_ , Oeuvres de Paul Painlevé; Tome II, Éditions du Centre National de la Recherche Scientifique, 15, quai Anatole-France, 75700, Paris, 1974.
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* [23] H. Poincaré, _Sur l’integration algébric des Équations différentiales du primier ordre et du premier degré I and II_ , Rendiconti del Circolo Matematico di Palermo 5 (1891), 161-191; 11 (1897), 193-239.
* [24] M. G. Soares, _The Poincaré problem for hypersurfaces invariant by one-dimensional foliations_ , Inv. Math. 128 (1997), 495–500.
* [25] A. Steffens, _On the stability of the tangent bundle of Fano manifolds_. Math. Ann. 304 (1996), 635–643.
* [26] S. Subramanian, _Stability of the tangent bundle and existence of a Kähler-Einstein metric_ , Math. Ann. 291 (1991), 573–577.
* [27] H. Tsuji, _Stability of tangent bundles of minimal algebraic varieties_ , Topology 27 (1988), 429–442.
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|
arxiv-papers
| 2011-02-11T16:50:47 |
2024-09-04T02:49:16.972841
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maur\\'icio Corr\\^ea JR., Marcos Jardim",
"submitter": "Mauricio Corr\\^ea J.R",
"url": "https://arxiv.org/abs/1102.2390"
}
|
1102.2665
|
# Energy transfer process in gas models of Lennard-Jones interactions
Jinghua Yang Yong Zhang Jiao Wang Hong Zhao zhaoh@xmu.edu.cn Department of
Physics and Institute of Theoretical Physics and Astrophysics,
Xiamen University, Xiamen 361005, China.
###### Abstract
We perform simulations to investigate how the energy carried by a molecule
transfers to others in an equilibrium gas model. For this purpose we consider
a microcanonical ensemble of equilibrium gas systems, each of them contains a
tagged molecule located at the same position initially. The ensuing transfer
process of the energy initially carried by the tagged molecule is then exposed
in terms of the ensemble-averaged energy density distribution. In both a 2D
and a 3D gas model with Lennard-Jones interactions at room temperature, it is
found that the energy carried by a molecule propagates in the gas
ballistically, in clear contrast with the Gaussian diffusion widely assumed in
previous studies. A possible scheme of experimental study of this issue is
also proposed.
###### pacs:
05.60.Cd, 51.10.+y, 51.20.+d
One important task of statistical mechanics is to understand various transport
processes. A well-known successful example is the self-diffusion of gas
molecules: Due to Einstein’s 1905 work, it has been widely accepted that a
particle in a gas undergoes the Brownian motion Einstein , which can be
modeled essentially with the random walk smoluchowski and the resulting
probability distribution function (PDF) follows the diffusion equation
Einstein ; smoluchowski . Because of its fundamental importance for various
scientific disciplines, the study of Einstein’s theory has never ceased. Very
recently, the direct experimental measurement of the instantaneous velocity of
a Brownian particle in a gas has been realized, and the random walk picture
was confirmed with high precision scieceexpress .
Another question of fundamental importance is how the energy carried by a
molecule transfers with time, which is a key step towards understanding the
macroscopic energy transport. In general, the existing theories, taking for
example the Helfand theory helfand , approached this issue by simply extending
the random walk picture of a Brownian particle, and predicted a Gaussian
energy density distribution as well. However, it should be pointed out that
whether random walk is the underlying mechanism of the energy dispersion has
never been examined experimentally nor numerically in a gas or more generally
in fluids.
Unlike in the study of the self-diffusion (or mass diffusion) where the
position of a particle can be traced accurately expt-1 ; expt-2 ; expt-3 ;
expt-4 (and now even its instantaneous velocity can be measured scieceexpress
), a key difficulty in the study of the energy transfer is that it is hard to
trace the energy transferred from particle to particle. The mass diffusion can
be explored by focusing on the trace of an individual particle, but by nature,
the energy transfer is a collective behavior involving all the molecules
related by the transferred energy.
In this work we perform an equilibrium molecular dynamics investigation to
explore how the energy of a molecule may transfer in a gas. We will restrict
ourselves to a 2D gas model, but it has been verified that in its 3D
counterpart the results remain qualitatively the same. We assume that the gas
is composed of only one kind of molecule with diameter $\sigma$ and mass $m$.
The setup consists of a square of area $S$ with periodic boundary conditions
and $N$ molecules moving inside. The interaction between molecules is given by
the Lennard-Jones potential and the Hamiltonian of the system reads
$H=\sum_{i}^{N}H_{i}=\sum_{i}^{N}\\{\frac{\mathbf{p}_{i}^{2}}{2m}+\sum_{j=1(\neq
i)}^{N}{2\varepsilon[(\frac{\sigma}{r_{ij}})^{12}-(\frac{\sigma}{r_{ij}})^{6}]}\\},$
(1)
where $\varepsilon$ is a constant governing the interaction strength and
$r_{ij}$ denotes the distance between molecules $i$ and $j$. In our
calculations the dimensionless parameters are set to be $\sigma=1$, $m=1$,
$\varepsilon=1$ and the Boltzmann constant $k_{B}=1$; In addition, the number
of the molecules $N=2500$ and the area $S=200\times 200$ are adopted. Another
important parameter is the temperature, which is fixed at $T=2.5$, a value
that corresponds to the room temperature with other adopted parameter values.
To make the simulations more efficient, the potential energy between two
molecules is approximated by zero when their distance is larger than
$r_{c}=3.5$, as conventionally adopted in the molecular dynamics studies of
gases. Given these, the evolution of a system can be simulated
straightforwardly. In our calculations the 7-order Runge-Kutta algorithm runge
with step 0.01 is employed.
Figure 1: The PDF $\rho_{m}(\mathbf{r},t)$ of the tagged molecule at time
$t=0$ (a) and $t=15$ (b). (For the sake of presentation, the coarse-grained
results over a grid of squares of size $4\times 4$ are plotted.) The
intersection of the plot in (b) with plane $y=0$ is shown in (c) for a close
look, where $\rho_{m}(x,y,t)\equiv\rho_{m}(\mathbf{r},t)$. (d)-(f) and (g)-(i)
are the same as (a)-(c) but for the ensemble averaged energy density
distribution $\rho_{e}(\mathbf{r},t)$ and the spatiotemporal correlation
function $c_{e}(\mathbf{r},t)$ of the energy fluctuation respectively. The
ensemble average for all the three cases is evaluated over $3\times 10^{8}$
systems.
The ensemble is prepared through the following three steps: (i) First, a
“seed” equilibrium system of $N$ molecules at temperature $T$ is prepared by
evolving a system for a long enough time ($>1\times 10^{6}$) from a properly
assigned random state; (ii) Then a molecule is chosen and its position is set
to be the origin by translating the coordinate system. In this way we build an
equilibrium system with one molecule initially localized at the origin. The
molecule at the origin is hereinafter referred to as “the tagged molecule”. By
assigning respectively all $N$ molecules to be the tagged one in this way,
i.e. setting their initial positions to be at the origin one by one, a
subensemble of $N$ equilibrium systems, each has a tagged molecule initially
residing on the origin, is thus prepared. (iii) By repeating step (i) to have
different realizations of the seed equilibrium system followed by (ii) to
generate the corresponding subensemble, we then build the whole ensemble whose
member number can be large enough (up to $\sim 10^{8}$) for satisfying
statistic results. It is worth noting that building the ensemble in such a way
is not new; a similar idea was once employed by Helfand to establish the
energy diffusion theory of fluids helfand .
To investigate that as the system evolves, how a molecule (represented by the
tagged molecule) diffuses and how the energy it carries initially spreads over
the whole system, we will study in particular the following three processes
with the ensemble prepared:
A. The self-diffusion, or mass diffusion process, which can be accessed by
calculating the PDF, denoted by $\rho_{m}(\mathbf{r},t)$, of the tagged
molecule. It is defined as
$\rho_{m}(\mathbf{r},t)\equiv\langle\delta[\mathbf{r}-{\mathbf{r}}_{1}(t)]\rangle,$
(2)
where $\langle\cdot\rangle$ denotes the ensemble average, ${\bf{r}}_{i}(t)$
denotes the position of molecule $i$ at time $t$ and number one molecule
represents the tagged molecule. As initially the tagged molecule is located at
the origin, i.e., ${\bf{r}}_{1}(0)=\mathbf{0}$, we have
$\rho_{m}(\mathbf{r},0)=\delta(\bf{r})$.
B. The dispersion of the energy initially carried by the tagged molecule. For
this aim we consider the ensemble averaged energy density distribution (EAEDD)
of the systems, i.e.
$\langle E(\mathbf{r},t)\rangle=\langle
H_{1}(t)\delta[\mathbf{r}-\mathbf{r}_{1}(t)]\rangle+\langle\sum_{j=2}^{N}H_{j}(t)\delta[\mathbf{r}-\mathbf{r}_{j}(t)]\rangle.$
(3)
Initially, as the origin is occupied exclusively by the tagged molecule, we
have $\langle
H_{1}(0)\delta[{\bf{r}}-{\bf{r}}_{1}(0)]\rangle=\widetilde{E}\delta(\mathbf{r})$,
where $\widetilde{E}\equiv\langle H_{j}\rangle$ is the average energy of a
molecule in the gas. On the other hand, as the rest area of the space
($\mathbf{r}\neq\mathbf{0}$) is occupied uniformly by other $N-1$ molecules,
the EAEDD they contribute to, i.e. the second term on the r.h.s of Eq. (3),
equals a constant $\eta\equiv\widetilde{E}(N-1)/S$. Hence initially $\langle
E(\mathbf{r},0)\rangle$ is characterized by a peak at the origin with a flat
background. As the system evolves, while the portion of $\langle
E(\mathbf{r},t)\rangle$ contributed by the tagged molecule may spread out from
its initial $\delta$-function, that by the other $N-1$ molecules remains to be
$\eta$. Therefore we can use the reformed distribution
$\rho_{e}(\mathbf{r},t)\equiv(\langle
E(\mathbf{r},t)\rangle-\eta)/\widetilde{E}$ to capture the dispersion of the
energy initially carried by the tagged molecule. This is the key technique of
this work; with it the energy transferred to others from a $single$ molecule
can thus be traced by checking the EAEDD of the $whole$ system, making it
possible to study the former conveniently.
C. The energy fluctuation correlation. The spatiotemporal correlation function
of the energy fluctuation zhao is a useful tool Lepri ; Dhar in tracing how
the energy transfers notelead ; lead . When the energy initially concentrated
at the origin (the cause) is transferred to position $\mathbf{r}$ at time $t$,
it will induce a solid correlation (the effect); we have to expose the
correlation induced exclusively by this causality. As the gas model we study
here is a microcanonical system with the total energy conserved, we have at
any time $\tau$ that $\sum_{j=1}^{N}\Delta H_{j}(\tau)=0$ and thus $\Delta
H_{1}(\tau)\sum_{j=1}^{N}\Delta H_{j}(\tau)=\Delta H_{1}(\tau)\Delta
H_{1}(\tau)+\sum_{j=2}^{N}\Delta H_{1}(\tau)\Delta H_{j}(\tau)=0$. Here
$\Delta H_{j}(\tau)\equiv H_{j}(\tau)-\widetilde{E}$. Considering the ensemble
average, we then have $\langle\Delta H_{1}(\tau)\Delta
H_{j}(\tau)\rangle=\langle\Delta H_{1}^{2}(\tau)\rangle/(N-1)$ since the gas
is homogeneous, which indicates that there is a trivial correlation between
any two molecules induced by the conservation of the energy. To get rid of it
we consider instead $\langle\Delta H_{1}(0)\sum_{j=1}^{N}\Delta
H_{j}(t)\delta[\mathbf{r}-\mathbf{r}_{j}(t)]\rangle$; As initially (at $t=0$)
it has a center of $\delta$-function form $\langle\Delta H_{1}(0)\Delta
H_{1}(0)\delta(\mathbf{r)}\rangle$ and a flat background
$\mu\equiv-\langle\Delta H_{1}(0)\Delta H_{1}(0)\rangle N/(N-1)S$, we
accordingly define the spatiotemporal correlation function as
$c_{e}(\mathbf{r},t)\equiv\langle\Delta H_{1}(0)\sum_{j=1}^{N}\Delta
H_{j}(t)\delta[\mathbf{r}-\mathbf{r}_{j}(t)]\rangle-\mu$ to explore the
correlation at position $\mathbf{r}$ and time $t$ induced by the initial
energy fluctuation $\Delta H_{1}(0)$.
The main results are summarized in Fig. 1. First of all Fig. 1 (a)-(c) are for
$\rho_{m}(\mathbf{r},t)$, the PDF of the tagged molecule. Initially it is a
$\delta$-function as seen in Fig. 1(a), in agreement with the fact that the
tagged molecule is located at the origin at the beginning. Later it develops
into a Gaussian distribution [Fig. 1(b)-(c)]. As a double check we have also
studied the squared displacement of the tagged molecule and found that it
depends on time linearly after a transient time of $t\approx 10$; i.e.,
$\langle|\mathbf{r}|^{2}(t)\rangle=4Dt$ with $D=7.08\pm 0.01$ as suggested by
the best linear fitting $\langle|\mathbf{r}|^{2}(t)\rangle$ over $10<t<80$.
These results are clear evidence that the self-diffusion of a molecule is
normal in our system.
Fig. 1 (d)-(f) show the energy transfer behavior. The $\delta$-function seen
in Fig. 1 (d) indicates that the energy we are interested in is initially
located at the origin. However, in its later development, a distinctive
difference from the self-diffusion of a molecule can be identified: Instead of
a Gaussian distribution, $\rho_{e}(\mathbf{r},t)$ features a growing “crater”,
i.e., a ring ridge (where $\rho_{e}(\mathbf{r},t)>0$) moves outwards leaving
behind a dip (where $\rho_{e}(\mathbf{r},t)<0$) in center [see Fig. 1(f)].
Fig. 1 (g)-(i) show the correlation function $c_{e}(\mathbf{{r}},t)$. It can
be found that $c_{e}(\mathbf{{r}},t)$ reveals the same features of the energy
transfer process seen in $\rho_{e}(\mathbf{{r}},t)$. The crater structure in
$c_{e}(\mathbf{{r}},t\mathbf{)}$ appears at the same position and expands
outward with the same velocity as that in $\rho_{e}(\mathbf{{r}},t)$. As in
the center dip region of $\rho_{e}(\mathbf{{r},}t)$ we have $\langle\Delta
H_{j}(t)\rangle<0$ [see Fig. 1(f)], which implies $\langle\Delta
H_{1}(0)\Delta H_{j}(t)\rangle>0$, the center peak in $c_{e}(\mathbf{{r},}t)$
is consistent with the dip in $\rho_{e}(\mathbf{{r},}t)$. It is interesting to
note that between the crater and the center peak there is a small region
showing a negative correlation; This feature is different from that observed
in the lattice models, where $c_{e}(\mathbf{{r}},t\mathbf{)}$ is always
positive and as a consequence can be employed to represent the PDF of the
energy diffusion zhao .
Figure 2: (a) The time dependence of the radii characterizing the crater
structure in $\rho_{e}({\bf r},t)$, corresponding to the top ring of the ridge
(red bullets) and the opening of the center dip (blue stars) respectively. The
best fittings (dashed lines) suggest their expanding speeds $\nu_{H}\approx
1.1\nu_{s}$ and $\nu_{\eta}\approx 0.7\nu_{s}$. (b) The time dependence of the
positive and negative potion of energy, i.e. $E_{+}$ (red bullets) and
$E_{-}$(blue stars), corresponding to the integrated energy density
distribution over region $|\mathbf{r}|>r_{\eta}$ and $|\mathbf{r}|<r_{\eta}$,
respectively.
Now let us have a closer look at $\rho_{e}(\mathbf{r},t)$, the main result of
this paper. Two key geometric parameters characterizing the crater structure
of $\rho_{e}(\mathbf{r},t)$ as shown in Fig. 1(e)-(f), denoted by $r_{\eta}$
and $r_{H}$ respectively, are the radius of the intersection ring on which
$\rho_{e}(\mathbf{r},t)=0$ and that of the ring of the ridge top where
$\rho_{e}(\mathbf{r},t)$ takes the maximum value [see Fig. 1(f)]. It is found
that they both depend on time linearly [see Fig. 2(a)], but however, the speed
of the ridge top, represented by $\nu_{H}\equiv dr_{H}/dt$, is different from
that of the opening of the dip $\nu_{\eta}\equiv dr_{\eta}/dt$: The best
linear fitting results suggest $\nu_{H}\approx 1.6\nu_{\eta}$. Just as a
comparison, it is interesting to note that the two speeds are comparable to
the speed of sound, a macroscopic characteristic; i.e., $\nu_{H}\approx
1.1\nu_{s}$ and $\nu_{\eta}\approx 0.7\nu_{s}$, where
$\nu_{s}=\sqrt{c_{p}k_{B}T}/\sqrt{c_{v}m}$ is the sound speed of the 2D ideal
gas of the same molecular mass and density and at the same temperature as our
model. On the other hand, as $\langle E(\mathbf{r},t)\rangle-\eta$ describes
how the initial energy carried by the tagged molecule transfers away, the
interesting fact that $\rho_{e}(\mathbf{r},t)$ has a negative center suggests
that during this process some energy of the neighboring molecules is brought
away in addition. This additional portion of energy is given by $-E_{-}$ where
$E_{-}\equiv\widetilde{E}\int_{|\mathbf{r}|<r_{\eta}}\rho_{e}(\mathbf{r},t)d\mathbf{r}$.
Similarly, the total positive energy carried by the bulk of the ridge is given
by
$E_{+}\equiv\widetilde{E}\int_{|\mathbf{r}|>r_{\eta}}\rho_{e}(\mathbf{r},t)d\mathbf{r}$.
Due to the conversation of the energy we have always
$E_{+}+E_{-}=\widetilde{E}$. Fig. 2 (b) shows the time dependence of $E_{-}$
and that of $E_{+}$; initially $E_{-}$($E_{+}$) decreases (increases) but
after a transition time it reaches a constant. In other words, eventually the
total energy brought away by the ridge is a constant and is larger than the
energy initially the tagged molecule carries. Together with the results of
$\nu_{H}$ and $\nu_{\eta}$, they suggest clearly that rather than the Gaussian
diffusion, the energy carried by a molecule propagates away ballistically in
our gas model.
Why the energy transfer and the molecule self-diffusion are so different can
be actually understood within the framework of the random walk theory.
According to this theory, a normal diffusion occurs if the random walker loses
its memory of the previous states completely, otherwise an abnormal diffusion
may take place instead. In our gas system, if we focus on the tagged molecule,
we may find that due to its frequent collisions with others, its memory of the
initial direction of motion suffers a quick loss. This memory loss process can
be measured by the decay of the autocorrelation function
$A(t)\equiv\langle\mathbf{p}_{1}(0)\cdot\mathbf{p}_{1}(t)\rangle$ of the
tagged molecule. (Here $\mathbf{p}_{1}(t)$ denotes the momentum of the tagged
molecule.) Indeed, as shown in Fig. 3(a), $A(t)$ decreases exponentially with
time, hence the motion of the tagged molecule is equivalent to that of a
random walker.
However, the information of the initial moving direction of the tagged
molecule is well remembered by the $whole$ system. When the energy of the
tagged molecule transfers to others, the memory of its initial state may
transfer to the surrounding molecules as well. To show this we consider the
correlation function
$C(t)\equiv\sum_{j=1}^{N}\langle\mathbf{p}_{1}(0)\cdot\mathbf{p}_{j}(t)\rangle$
as a measure of the total memory (note that $C(t)$ is evaluated over the whole
system). Dividing the momentum of a molecule, say molecule $j$, into two
parts; i.e.,
$\mathbf{p}_{j}(t)=\mathbf{p}_{j}^{\prime}(t)+\mathbf{p}_{j}^{\prime\prime}(t)$
, where $\mathbf{p}_{j}^{\prime}(t)$ and $\mathbf{p}_{j}^{\prime\prime}(t)$
represent respectively the momentum transferred to molecule $j$ from the
tagged molecule and other molecules, we then have
$C(t)=\sum_{j=1}^{N}\langle\mathbf{p}_{1}(0)\cdot\mathbf{p}_{j}^{\prime}(t)\rangle=\langle\mathbf{p}_{1}(0)\cdot\mathbf{p}_{1}(0)\rangle$.
This is because first
$\langle\mathbf{p}_{1}(0)\cdot\mathbf{p}_{j}^{\prime\prime}(t)\rangle=0$ as
$\mathbf{p}_{j}^{\prime\prime}(t)$ is independent of $\mathbf{p}_{1}(0)$ and
second $\sum_{j=1}^{N}\mathbf{p}_{j}^{\prime}(t)=\mathbf{p}_{1}(0)$ as the
momentum $\mathbf{p}_{1}(0)$ is conserved in the system. As a result $C(t)$ is
in fact a time-independent constant, suggesting that though the initial moving
direction will be forgotten quickly by the tagged molecule itself, it will be
well remembered by the whole system in future.
This fact implies that the energy transfer process studied here cannot be a
Markov process. To verify that the memory is kept during the energy
transferring process, we reproduce Fig. 1(e) in an alternative way: In
preparing a gas system in our ensemble as described in Step (ii), we rotate
additionally the coordinate system so that the initial moving direction of the
tagged molecule is along axis $y$. This is equivalent to considering a subset
of Helfand’s subensemble where the momentum direction of the tagged molecule
is specified as well. Fig. 3 (b) shows the result, from which we can see that
$\rho_{e}(\mathbf{r},t)$ is obviously anisotropic and suggests clearly the
initial moving direction of the tagged molecule.
Figure 3: (a) The autocorrelation function $A(t)$ of the momentum of the
tagged molecule decreases exponentially with time; (b) The same as Fig. 1(e),
but the direction of the initial velocity of the tagged molecule is set to be
along axis $y$.
In summary, rather than the Brownian motion, our simulation study of a 2D gas
at room temperature shows that the energy carried by a molecule would
propagate away in a ballistic wavelike manner. Considering the ensemble
average, the profile of the transferred energy is found to be characterized by
a ring ridge and a dip in center, and both expand outwards with constant
speeds comparable to the speed of sound. The ballistic propagation of the
energy is also confirmed by the spatiotemporal correlation function of the
energy fluctuation.
We emphasize that our study investigates the energy dispersion of a single
molecule in equilibrium state, hence in nature the observed ballistic
characteristic is distinct from the nonequilibrium macroscopic waves such as
the sound wave and the heat wave HW . For example, the heat wave is a
macroscopic relaxing phenomenon, it may decay in fluids due to viscosity and
heat conduction, but the energy density distribution in the present study does
not decay [see Fig. 2(b)]. The heat wave can also exist in lattice systems HC1
, but again it only survives for a finite time due to decaying effect. The
properties of the energy dispersion studied here are also in clear contrast to
those of the macroscopic heat conduction sustained by the temperature
gradient; e.g., we have also studied the 3D counterpart of our gas model, and
both a 2D and a 3D hard-disc gas model, but obtained qualitatively the same
results. However, the heat conduction may have a dramatic dependence on the
dimensionality in momentum-conserved systems HC2 . In spite of these
difference, the energy dispersion properties of a molecule must have
underlying implications to various macroscopic energy transport behavior. In
this respect the mode coupling theory of hydrodynamics, which has been shown
powerful in bridging the microscopic and macroscopic descriptions of fluids
hansen , may provide deep insights.
Finally we would like to suggest a possible laboratory testing scheme of the
energy transfer process bases on the energy correlation function discussed
(see Fig. 1 (g)-(i) for example simulation results). The requisite laboratory
technique is the measurement of the simultaneous position and velocity of a
particle immersed in a gas (fluid) note . Given this, a sample of $\Delta
H_{1}(0)\cdot\Delta H_{j}(t)$ can be obtained by making the measurement to a
certain immersed particle at a time and to another after time $t$. Repeating
this data collecting process till a sufficient large amount of samples are
available; $c_{e}(\mathbf{r},t)$ can then be evaluated.
This work is supported by the National Natural Science Foundation of China
under Grants No. 10775115, No. 10975115, and No. 10925525; and the National
Basic Research Program of China (973 Program) under Grant No 2007CB814800.
## References
* (1) A. Einstein, Annalen Der Physik 17, 549-560 (1905).
* (2) M.R. von Smoluchowski, Annalen Der Physik 21, 756-780 (1906).
* (3) T. Li, S. Kheifets, D. Medellin, and M.G. Raizen, Science, science.1189403 (2010).
* (4) E. Helfand, Phys. Rev. 119, 1 (1960).
* (5) B. Lukić, S. Jeney, C. Tischer, A.J. Kulik, L. Forró, and E.L. Florin, Phys. Rev. Lett. 95, 160601 (2005).
* (6) Y. Han, A.M. Alsayed, M. Nobili, J. Zhang, T.C. Lubensky, and A.G. Yodh, Science 314, 626-630 (2006).
* (7) J. Blum, S. Bruns, D. Rademacher, A. Voss, B. Willenberg, and M. Krause, Phys. Rev. Lett. 97, 230601 (2006).
* (8) I. Chavez, R. Huang, K. Henderson, E.L. Florin, and M.G. Raizen, Rev Sci Instrum. 79, 105104 (2008).
* (9) J.R. Dormand and P.J. Prince, Celestial Mech. Dyn. Astron., 18, 223 (1978).
* (10) H. Zhao, Phys. Rev. Lett. 96, 140602 (2006).
* (11) L. Delfini, S. Denisov, S. Lepri, R. Livi, P.K. Mohanty, and A. Politi, The European Physical Journal - Special Topics 146, 21-35 (2007).
* (12) A. Dhar and J.L. Lebowitz, Phys. Rev. Lett. 100, 134301 (2008).
* (13) The energy fluctuation correlation can also be investigated in the Fourier space by using the technique developed in lead ; here we restrict ourselves in real space in order to make a comparison with $\rho_{m}({\bf r},t)$ and $\rho_{e}({\bf r},t)$.
* (14) T. Bryk and I. Mryglod, Phys. Rev. E 63, 051202 (2001).
* (15) D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, 41 (1989); ibid 62, 375 (1990).
* (16) O.V. Gendelman and A.V. Savin, Phys. Rev. E 81, 020103(R) (2010).
* (17) G. Basile, C. Bernardin, and S. Olla, Phys. Rev. Lett. 96, 204303 (1996); K. Saito and A. Dhar, Phys. Rev. Lett. 104, 00601 (2010); D. Xiong, J. Wang, Y. Zhang, and H. Zhao, Phys. Rev. E 82, 030101(R) (2010).
* (18) See for example Theory of Simple Liquids (3rd ed., Academic Press, London, 2006) by J. P. Hansen and I. R. McDonald and references therein.
* (19) This required technique has been advanced in a resent laboratory study scieceexpress .
|
arxiv-papers
| 2011-02-14T03:02:58 |
2024-09-04T02:49:16.980937
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinghua Yang, Yong Zhang, Jiao Wang, Hong Zhao",
"submitter": "Zhao Hong",
"url": "https://arxiv.org/abs/1102.2665"
}
|
1102.2784
|
# A New Limit on Planck Scale Lorentz Violation from Gamma Ray Burst
Polarization
Floyd W. Stecker Astrophysics Science Division, NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA floyd.w.stecker@nasa.gov
###### Abstract
Constraints on possible Lorentz invariance violation (LIV) to first order in
$E/M_{\rm Planck}$ for photons in the framework of effective field theory
(EFT) are discussed, taking cosmological factors into account. Then, using the
reported detection of polarized soft $\gamma$-ray emission from the
$\gamma$-ray burst GRB041219a that is indicative of an absence of vacuum
birefringence, together with a very recent improved method for estimating the
redshift of the burst, we derive constraints on the dimension 5 Lorentz
violating modification to the Lagrangian of an effective local QFT for QED.
Our new constraints are more than five orders of magnitude better than recent
constraints from observations of the Crab Nebula. We obtain the upper limit on
the Lorentz violating dimension 5 EFT parameter $|\xi|$ of $2.4\times
10^{-15}$, corresponding to a constraint on the dimension 5 standard model
extension parameter, $k^{(5)}_{(V)00}\leq 4.2\times 10^{-34}$ GeV-1.
###### keywords:
Lorentz invariance; quantum gravity; gamma-rays; gamma-ray bursts
††journal: Astroparticle Physics
## 1 Introduction
Because of the problems associated with merging relativity with quantum
theory, it has long been felt that relativity will have to be modified in some
way in order to construct a quantum theory of gravitation. Since the Lorentz
group is unbounded at the high boost (or high energy) end, in principle it may
be subject to modifications in the high boost limit [1, 2]. There is also a
fundamental relationship between the Lorentz transformation group and the
assumption that space-time is scale-free, since there is no fundamental length
scale associated with the Lorentz group. However, as noted by Planck [3],
there is a potentially fundamental scale associated with gravity, viz., the
Planck scale $\lambda_{Pl}=\sqrt{G\hbar/c^{3}}\sim 10^{-35}$ m, corresponding
to an energy (mass) scale of $M_{Pl}=\hbar c/\lambda_{Pl}\sim 10^{19}$ GeV.
In recent years, there has been much interest in testing Lorentz invariance
violating terms that are of first order in $E/M_{Pl}$, since such terms vanish
at very low energy and are amenable to testing at higher energies. In
particular, tests using high energy astrophysics data have proved useful in
providing constraints on Lorentz invariance violation (LIV) (e.g., see reviews
in Refs. [4] and [5]).
## 2 Vacuum Birefringence
Important fundamental constraints on LIV come from searches for the vacuum
birefringence effect predicted within the framework of the effective field
theory (EFT) analysis of [6]. (See also Ref. [7]). Within this framework,
applying the Bianchi identities to the leading order Maxwell equations in
vacua, a mass dimension 5 operator term is derived of the form
${\Delta\cal{L}}_{\gamma}={{\xi}\over{M_{Pl}}}{n^{a}F_{ad}n\cdot\partial(n_{b}\tilde{F}^{bd})}.$
(1)
It is shown in Ref. [6] that the expression given in Equation (1) is the only
dimension 5 modification of the free photon Lagrangian that preserves both
rotational symmetry and gauge invariance. This leads to a modification in the
dispersion relation proportional to $\xi(\omega/M_{Pl})=\xi(E/M_{Pl})$
111adopting the conventions $\hbar=1$ and the low energy speed of light $c=1$.
with the new dispersion relation given by
$\omega^{2}~{}=~{}k^{2}\pm\xi\,k^{3}/M_{Pl}.\\\ $ (2)
Some models of quantized space-time suggest $\xi$ should be $\cal{O}$(1),
(see, e.g., Ref. [8]). The sign in the photon dispersion relation corresponds
to the helicity, i.e., right or left circular polarization. Equation (2)
indicates that photons of opposite circular polarization have different phase
velocities and therefore travel with different speeds. The effect on photons
from a distant linearly polarized source can be constructed by decomposing the
linear polarization into left and right circularly polarized states. It is
then apparent that this leads to a rotation of the linear polarization
direction through an angle
$\theta(t)=\left[\omega_{+}(k)-\omega_{-}(k)\right]t_{P}/2~{}\simeq~{}\xi
k^{2}t_{P}/2M_{Pl}$ (3)
for a plane wave with wave-vector $k$, where $\xi k/M_{Pl}\ll 1$ and where
$t_{P}$ is the propagation time.
Observations of polarized radiation from distant sources can thus be used to
place an upper bound on $\xi$. The vacuum birefringence constraint arises from
the fact that if the angle of polarization rotation (3) were to differ by more
than $\pi/2$ over the energy range covered by the observation the
instantaneous polarization at the detector would fluctuate sufficiently for
the net polarization of the signal to be suppressed well below any observed
value. The difference in rotation angles for wave-vectors $k_{1}$ and $k_{2}$
is
$\Delta\theta=\xi(k_{2}^{2}-k_{1}^{2})L_{P}/2M_{Pl},$ (4)
where we have replaced the propagation time $t_{P}$ by the propagation
distance $L_{P}$ from the source to the detector.
If polarization is detected from a source at redshift $z$, this yields the
constraint
$|\xi|<{{\pi
M_{Pl}}\over{\int\displaylimits_{0}^{z}dz^{\prime}{[k_{2}(z^{\prime})^{2}-k_{1}(z^{\prime})^{2}]|dL_{P}(z^{\prime})/dz^{\prime}|}}}$
(5)
where $k_{1,2}(z^{\prime})=(1+z^{\prime})\cdot k_{1,2}(z^{\prime}=0)$ and
$\Bigl{|}{dL_{P}\over{dz^{\prime}}}\Bigr{|}={{c}\over{H_{0}}}{{1}\over{(1+z^{\prime})\sqrt{\Omega_{\Lambda}+(1+z^{\prime})^{3}\Omega_{m}}}}.$
(6)
Defining
${\cal{D}}={{c}\over{H_{0}}}\int\displaylimits_{0}^{z}dz^{\prime}{{(1+z^{\prime})}\over{\sqrt{\Omega_{\Lambda}+(1+z^{\prime})^{3}\Omega_{m}}}}$
(7)
it follows from equations (5)-(7) and the definitions of $k_{1,2}(z^{\prime})$
that
$|\xi|<{{\pi M_{Pl}}\over{{\cal{D}}(k_{2}^{2}-k_{1}^{2})}},$ (8)
with the standard cosmological values [9] of $\Omega_{m}=0.27$,
$\Omega_{\Lambda}=0.73$, and $H_{0}$ = 71 km s-1 Mpc-1 (1 Mpc = $3.09\times
10^{22}$ m). Figure 1 shows the function ${\cal{D}}(z)$ as defined in Equation
(7).
Figure 1: A linear plot of the integral $\cal{D}$ as defined in Equation (7),
given as a function of redshift, $z$.
## 3 Previous Constraints
A previous bound of $|\xi|\lesssim 2\times 10^{-4}$, was obtained by Gleiser
and Kozameh [10] using the observed 10% polarization of ultraviolet light from
a galaxy at distance of around 300 Mpc. Fan et al. used the observation of
polarized UV and optical radiation at several wavelengths from the
$\gamma$-ray bursts (GRBs) GRB020813 at a redshift $z=1.3$ and GRB021004
$z=2.3$ to get a constraint of $|\xi|\lesssim 2\times 10^{-7}$ [11]. Jacobson
et al. [12] used a report of polarized $\gamma$-rays observed [13] in the
prompt emission from the $\gamma$-ray burst GRB021206 in the energy range 0.15
to 2 MeV using the RHESSI detector [14] to place strong limits on $\xi$.
However, this claimed polarization detection has been refuted [15, 16].
Kostelecký and Mewes [17] have shown that the EFT model parameter $\xi$ can be
related to the model independent isotropic dimension 5 standard model
extension (SME) parameter $k^{(5)}_{(V)00}$. They derive the relation
$k^{(5)}_{(V)00}=3\sqrt{4\pi}\xi/5M_{Pl},$ (9)
which we use in this paper. Their upper limit of $1\times 10^{-32}$ GeV-1,
obtained by assuming a lower limit on the redshift of these bursts of $z=0.1$,
then corresponds to the constraint $\xi<6\times 10^{-14}$.222Ref. [18] gives a
table of similar limits on $k^{(5)}_{(V)00}$ with citations.
More recently, Maccione et al. have derived a constraint of $|\xi|\lesssim
9\times 10^{-10}$ using observations of polarized hard X-rays from the Crab
Nebula detected by the INTEGRAL satellite [19].
It is clear from Equation (5) that the larger the distance of the polarized
source, and the larger the energy of the photons from the source, the greater
the sensitivity to small values of $\xi$. In that respect, the ideal source to
study would be polarized X-rays or $\gamma$-rays from a GRB with a known
redshift at a deep cosmological distance [12].
## 4 A New Treatment
Unfortunately, despite the many GRBs that have been detected and have known
host galaxy spectral redshifts, none of these bursts have measured
$\gamma$-ray polarization. However, in this paper we take a new approach,
deriving an estimated redshift for GRB041219a. This is a GRB with reported
polarization but no spectral redshift measurement.
Polarization at a level of 63(+31,-30)% to 96(+39,-40)% in the soft
$\gamma$-ray energy range has been detected by analyzing data from the
spectrometer on INTEGRAL for GRB041219a in the 100 to 350 keV energy range
[20]. It should be noted that that a systematic effect that might mimic
polarization in the analysis could not definitively be excluded. This GRB does
not have an associated host galaxy spectral redshift.
Useful relations have been recently obtained where known spectral redshifts of
GRBs are statistically correlated with various observational parameters of the
bursts such as luminosity, the Band function [21] parameter $E_{peak}$, rise
time, lag time and variability of a burst (Ref. [22] and references therein).
A detailed treatment of these correlations is given in Ref. [22]. By deriving
updated luminosity correlations for a very large number of GRBs, they find the
tightest correlation is the luminosity-$E_{peak}$ correlation. Using the
relation given in Ref. [22],
$\log L=51.75+1.35\log[(1+z)E_{peak}/300{\rm keV}]$ (10)
and the iterative method described in Ref. [23], and taking $E_{peak}$ = 170
keV and a peak fluance of $5.7\times 10^{-4}$ erg cm-2 [20], we derive a value
for $z$ for GRB041219a of $0.23\pm 0.03$. Taking a lower limit of 0.2 for the
redshift and taking $k_{2}$ = 350 keV/c and $k_{1}$ = 100 keV/c in Equation
(5), we find a new, most accurate cosmological constraint on $|\xi|$ of
$|\xi|\leq 2.4\times 10^{-15},$ (11)
almost five orders of magnitude better than the previous best solid limit
derived using polarimetric observations of the Crab Nebula in the hard X-ray
energy range [19].
From equation 9, the result given in equation (11) implies a constraint on the
isotropic dimension 5 SME parameter of
$k^{(5)}_{(V)00}\leq 4.2\times 10^{-34}\ {\rm GeV}^{-1}.$ (12)
Finally, it should be noted that with the redshift dependence obtained from
Equations (7) and (8), any reasonable redshift for a GRB similar to GRB041219a
and showing detectable polarization will give a constraint on $|\xi|$ below
$\sim 5\times 10^{-15}$ corresponding to a constraint on $k^{(5)}_{(V)00}$
below $\sim 10^{-33}$ GeV-1. This can be seen from Figure 1.
Much better tests of birefringence can be performed by polarization
measurements at higher $\gamma$-ray energies. The technology for measuring
polarization in the 5 to 100 MeV energy range using gas filled detectors is
now being developed and tested [25]. Studies of cosmological sources such as a
GRBs at such energies can probe values of $|\xi|$ several orders of magnitude
smaller than is presently possible.
## 5 Frame Independent Constraint
The vector $n$ in the EFT model given by equation (1) leads to strictly
isotropic physics only in one special frame, usually taken to be the frame in
which the cosmic microwave background is isotropic. In other frames the
dispersion relation will have anisotropic components. This can be taken into
account by using the general SME formalism [17]. There are then 16 independent
$k^{(5)}_{(jm)}$ parameters that are weighted by spherical harmonic
coefficients according to their spin weight with respect to the line of sight
unit vector ${\bf n}$. For GRB041219a this leads to the frame-independent
constraint
$|\sum_{jm}{Y_{jm}}(37^{\circ},0^{\circ})k^{(5)}_{(V)jm}|\leq 1.2\times
10^{-34}\ {\rm GeV^{-1}}.$ (13)
## 6 Other constraints and Implications
The Lorentz violating dispersion relation (2) implies that the group velocity
of photons, $v_{g}=1\pm\xi p/M_{Pl}$, is energy dependent. This leads to an
energy dependent dispersion in the arrival time at Earth for photons spread
over a finite energy range originating in a distant source. The result
obtained from observations of the $\gamma$-ray energy-time profile by the
Fermi satellite for the burst GRB090510 gives a limit of $\xi<0.82$ [26].
Thus, the time of flight constraint from Fermi, while still significant
because it gives $\xi<1$, remains many orders of magnitude weaker than the
birefringence constraint. However, the Fermi constraint is independent of the
EFT assumption of helicity dependence of the group velocity. Perhaps the best
constraint on LIV in general comes from a study of the highest energy cosmic
rays, giving a limit of $4.5\times 10^{-23}$ in the hadronic sector [5].
Thus, all of the present astrophysical data point to the conclusion that LIV
does not occur at the level $\xi(E/M_{Pl})$ with $\xi$ = $\cal{O}$(1). In
fact, in appears that $\xi\ll 1$. What this is telling us about the natures of
space-time and gravity at the Planck scale is still an open question.
## Acknowledgements
I would like to thank Neil Gehrels, Stanley Hunter, Sean Scully, Takanori
Sakamoto, Tonia Venters, and an anonymous referee for helpful discussions.
## References
* [1] S. R. Coleman and S. L. Glashow, Phys. Rev. D 59, 116008 (1999).
* [2] F. .W. Stecker and S. L. Glashow, Astropart. Phys 16, 97 (2001).
* [3] M. Planck, Mitt. Thermodynamik, Folg. 5 (1899)
* [4] T. Jacobson, S. Liberati and D. Mattingly, Annals of Physics 321, 150 (2006).
* [5] F. W. Stecker and S. T. Scully, New J. Phys. 11 085003 (2009).
* [6] R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003).
* [7] D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998).
* [8] J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, arXiv0912.3428 (2009) and references therein.
* [9] D. Larson et al., Astrophys. J. Suppl. 192, 16 (2011)
* [10] R. J. Gleiser and C. N. Kozameh, Phys. Rev. D 64, 083007 (2001).
* [11] Y.-Z. Fan, D.-M. Wei and X. Dong, Mon. Not. Roy. Astr. Soc. 376, 1857 (2007).
* [12] T. Jacobson, S. Liberati, D. Mattingly and F. Stecker, Phys. Rev. Lett. 93, 021101 (2004).
* [13] W. Coburn and S. E. Boggs, Nature 423, 415 (2003).
* [14] http://hesperia.gsfc.nasa.gov/hessi/
* [15] R. E. Rutledge and D. B. Fox, Mon. Not. Roy. Astr. Soc. 350, 1288 (2004).
* [16] C. Wigger et al., Astrophys. J. 613, 1088 (2004).
* [17] V. A. Kostelecký and M. Mewes, Phys. Rev. D 80, 015020 (2009).
* [18] V. A. Kostelecký and N. Russell, Rev. Mod. Phys. 83, 11 (2011).
* [19] L. Maccione et al., Phys. Rev. D 78, 103003 (2008).
* [20] S. Mc Glynn et al., Astron. and Astrophys. 466 895 (2007).
* [21] D. Band et al., Astrophys. J. 413, 281 (1993).
* [22] F.-Y. Wang, S. Qi and Z.-G. Dai, Mon. Not. Roy. Astr. Soc., in press (2011), arXiv:1105.0046.
* [23] L. Xiao and B. E. Schaefer, Astrophys. J. 707, 387 (2009).
* [24] S. McBreen et al. Astron. and Astrophys. 455 433 (2006).
* [25] S. D. Hunter et al., in Space Telescopes and Instrumentation 1020: Ultraviolet to Gamma Rays, Proc. SPIE 7732, ed. et al., 773221 (2010).
* [26] A. Abdo et al., Nature 462, 331 (2009).
|
arxiv-papers
| 2011-02-14T14:36:41 |
2024-09-04T02:49:16.987398
|
{
"license": "Public Domain",
"authors": "Floyd W. Stecker",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/1102.2784"
}
|
1102.2856
|
# Spatially Coupled Codes over the Multiple Access Channel
Shrinivas Kudekar1 and Kenta Kasai2
1 New Mexico Consortium and CNLS, Los Alamos National Laboratory, New Mexico,
USA
Email: skudekar@lanl.gov 2 Dept. of Communications and Integrated Systems,
Tokyo Institute of Technology, 152-8550 Tokyo, Japan.
Email: kenta@comm.ss.titech.ac.jp
###### Abstract
We consider spatially coupled code ensembles over a multiple access channel.
Convolutional LDPC ensembles are one instance of spatially coupled codes. It
was shown recently that, for transmission over the binary erasure channel,
this coupling of individual code ensembles has the effect of increasing the
belief propagation threshold of the coupled ensembles to the maximum
a-posteriori threshold of the underlying ensemble. In this sense, spatially
coupled codes were shown to be capacity achieving. It was observed,
empirically, that these codes are universal in the sense that they achieve
performance close to the Shannon threshold for any general binary-input
memoryless symmetric channels.
In this work we provide further evidence of the threshold saturation phenomena
when transmitting over a class of multiple access channel. We show, by density
evolution analysis and EXIT curves, that the belief propagation threshold of
the coupled ensembles is very close to the ultimate Shannon limit.
## I Introduction
It has long been known that convolutional LDPC (or spatially coupled)
ensembles, introduced by Felström and Zigangirov [1], have excellent
thresholds when transmitting over general binary-input memoryless symmetric-
output (BMS) channels. The fundamental reason underlying this good performance
was recently discussed in detail in [2] for the case when transmission takes
place over the binary erasure channel (BEC). In the limit of large $L$ and
$w$, the spatially-coupled LDPC code ensemble $(\mathtt{l},\mathtt{r},L,w)$
[2] was shown to achieve the MAP threshold of $(\mathtt{l},\mathtt{r})$ code
ensemble (see last paragraph in this section for the definition of the
$(\mathtt{l},\mathtt{r},L,w)$ ensemble). This is the reason why they call this
phenomena threshold saturation via spatial coupling. In a recent paper [3],
Lentmaier and Fettweis independently formulated the same statement as
conjecture.
The phenomena of threshold saturation seems not to be restricted to the BEC.
By computing EBP GEXIT curves [4], it was observed in [5] that threshold
saturation also occurs for general BMS channels. In other words, in the limit
of large $\mathtt{l}$ (keeping $\frac{\mathtt{l}}{\mathtt{r}}$ constant), $L$
and $w$, the coupled ensemble $(\mathtt{l},\mathtt{r},L,w)$ achieves
universally the capacity of the BMS channels under belief propagation (BP)
decoding. Such universality is not a characteristic feature of polar codes [6]
and the irregular LDPC codes [7]. According to the channel, polar codes need
selection of frozen bits [8] and irregular LDPC codes need optimization of
degree distributions.
The principle which underlies the good performance of spatially coupled
ensembles has been shown to apply to many other problems in communications,
and more generally computer science. To mention just a few, the threshold
saturation effect (dynamical threshold of the system being equal to the static
or condensation threshold) of coupled graphical models has recently been shown
to occur for compressed sensing [9], and a variety of graphical models in
statistical physics and computer science like the random $K$-SAT problem,
random graph coloring, and the Curie-Weiss model [10]. Other communication
scenarios where the spatially coupled codes have found immediate application
is to achieve the whole rate-equivocation region of the BEC wiretap channel
[11], and to achieve the symmetric information rate for a class of channels
with memory [12].
It is tempting to conjecture that the same phenomenon occurs for transmission
over general multi-user channels. We provide some empirical evidence via
density evolution (DE) analysis that this is indeed the case. In particular,
we compute EXIT curves for transmission over a multiple access channel (MAC)
with erasures. We show that these curves behave in an identical fashion to the
curves when transmitting over the BEC. We compute fixed points (FPs) of the
coupled DE and show that these FPs have properties identical to the BEC case.
For a review on the literature on convolutional LDPC ensembles, we refer the
reader to [2] and the references therein. As discussed in [2], there are many
basic variants of coupled ensembles. For the sake of convenience of the
reader, we quickly review the ensemble $(\mathtt{l},\mathtt{r},L,w)$. This is
the ensemble we use throughout the paper as it is the simplest to analyze.
### I-A $(\mathtt{l},\mathtt{r},L,w)$ Ensemble [2]
We assume that the variable nodes are at sections $[-L,L]$, $L\in\mathbb{N}$.
At each section there are $M$ variable nodes, $M\in\mathbb{N}$. Conceptually
we think of the check nodes to be located at all integer positions from
$[-\infty,\infty]$. Only some of these positions actually interact with the
variable nodes. At each position there are $\frac{\mathtt{l}}{\mathtt{r}}M$
check nodes. It remains to describe how the connections are chosen. We assume
that each of the $\mathtt{l}$ connections of a variable node at position $i$
is uniformly and independently chosen from the range $[i,\dots,i+w-1]$, where
$w$ is a “smoothing” parameter. In the same way, we assume that each of the
$\mathtt{r}$ connections of a check node at position $i$ is independently
chosen from the range $[i-w+1,\dots,i]$. The design rate of the ensemble
$(\mathtt{l},\mathtt{r},L,w)$, with $w\leq 2L$, is given by
$R(\mathtt{l},\mathtt{r},L,w)=(1-\frac{\mathtt{l}}{\mathtt{r}})-\frac{\mathtt{l}}{\mathtt{r}}\frac{w+1-2\sum_{i=0}^{w}\bigl{(}\frac{i}{w}\bigr{)}^{\mathtt{r}}}{2L+1}.$
A discussion on the above ensemble can be found in [2].
## II Channel Model, Achievable Rate Region, Iterative Decoding and Factor
Graph
### II-A Binary Adder Channel with Erasures
We consider the simplest synchronous 2-user multiple access channel, the
binary adder channel (BAC) with erasure. More precisely, the inputs to the MAC
are binary $X_{1},X_{2}\in\\{0,1\\}$. The users take on the values $0,1$ with
equal probability. The subscripts 1,2 denote the two users. The output
$Y\in\\{0,1,2,\text{?}\\}$ is given by
$\displaystyle Y$
$\displaystyle=\left\\{\begin{array}[]{ll}Z=X_{1}+X_{2}&\text{ with
probability }1-\epsilon\\\ \text{?}&\text{ with probability
}\epsilon,\end{array}\right.$
where $\epsilon$ is the fraction of erasures.
### II-B Achievable Rate Region
We assume that the two users do not coordinate their transmission. This
implies that the joint input distribution has a product form. Let $R_{1}$ and
$R_{2}$ denote the transmission rates of the two users. The achievable rate
region is given as follows.
$\displaystyle R_{1}\leq$ $\displaystyle I(X_{1};Y|X_{2}),$ $\displaystyle
R_{2}\leq$ $\displaystyle I(X_{2};Y|X_{1}),$ $\displaystyle R_{1}+R_{2}\leq$
$\displaystyle I(X_{1},X_{2};Y).$
The mutual information values above can be computed as
$\displaystyle I(X_{1};Y|X_{2})$ $\displaystyle=I(X_{2};Y|X_{1})=1-\epsilon,$
$\displaystyle I(X_{1},X_{2};Y)$ $\displaystyle=\frac{3(1-\epsilon)}{2},$
$\displaystyle I(X_{1};Y)$ $\displaystyle=I(X_{2};Y)=\frac{1-\epsilon}{2}.$
The Shannon limit is defined as the ultimate erasure threshold below which
both users can successfully decode using any decoder. Thus, the Shannon
threshold is given by,
$\displaystyle\epsilon_{\mathrm{Sh}}=\min(1-R_{1},1-R_{2},1-\frac{2}{3}(R_{1}+R_{2})).$
(1)
### II-C Factor Graph and Iterative Decoding
Figure 1 shows the factor graph representation used in the BP decoder
analysis. The channel output is the vector $\underline{y}$ and the user inputs
are $\underline{x}_{1}$ and $\underline{x}_{2}$. Each user has its own code
and there is a function node which connects the two factor graphs (dark
squares in Figure 1). This function node represents the channel factor node
$p(y_{i}|x_{1,i},x_{2,i})$ and we call it the MAC function node (see [13, 14]
for details). Figure 1 shows the spatially coupled ensemble used by each user.
For the ease of illustration, we show the protograph-based variant of
spatially coupled codes. If we do not use coupled codes for transmission, then
the two protographs above will be replaced by the usual LDPC codes.
$\text{-}L$$\cdots$(60,-10)(9.2,0)[b]$\text{-}4$,$\text{-}3$,$\text{-}2$,$\text{-}1$,$0$,$1$,$2$,$3$,$4$$\cdots$$L$
Figure 1: The figure shows two protograph-based spatially coupled codes (each
belonging to one user) in light gray. The two protographs are connected by the
MAC function node shown in dark. Note that in the actual code the MAC function
node connects each variable node of one user to the corresponding variable
node of the other user. For the ease of illustration, we just show connections
across one-half of the variable nodes.
The BP decoder passes messages between the various nodes in the factor graph.
The message passing schedule involves first passing the channel observations
from the MAC function nodes to the variable nodes of both of the users, then
performing one round of BP for both the users (in parallel) and then sending
the extrinsic information back to the MAC function node (from both the users).
## III Uncoupled System: Density Evolution, Exit-like Curves
### III-A Density Evolution
Before we proceed to the analysis of coupled codes, it is instructive to
consider the DE analysis for the uncoupled $(\mathtt{l},\mathtt{r})$-regular
ensemble. More precisely, users 1 and 2 pick a code from the ensemble
$(\mathtt{l}_{1},\mathtt{r}_{1})$-regular and
$(\mathtt{l}_{2},\mathtt{r}_{2})$-regular respectively. From the schedule
given above it is not hard to see that for finite number of iterations and
large blocklengths, the local neighborhood around any node is a tree with high
probability. See [13, 14] for more details on the DE setup. Also, the BAC with
erasures can be thought of as a BEC (for either user) with erasure probability
equal to $\epsilon+(1-\epsilon)\mu/2$, where $\mu$ is the erasure message
flowing into the MAC function node. Indeed, the channel output is either
erased (wp $\epsilon$) or it is not erased (wp $1-\epsilon$) and we are still
uncertain of the transmitted symbol if the output is equal to 1 (occurs wp
1/2) and the other symbol is uncertain (wp $\mu$). The FPs of the DE are then
given by,
$\displaystyle y_{1}$ $\displaystyle=1-(1-x_{1})^{\mathtt{r}_{1}-1},$
$\displaystyle x_{1}$
$\displaystyle=(\epsilon+\frac{1-\epsilon}{2}y_{2}^{\mathtt{l}_{2}})y_{1}^{\mathtt{l}_{1}-1},$
$\displaystyle y_{2}$ $\displaystyle=1-(1-x_{2})^{\mathtt{r}_{2}-1},$
$\displaystyle x_{2}$
$\displaystyle=(\epsilon+\frac{1-\epsilon}{2}y_{1}^{\mathtt{l}_{1}})y_{2}^{\mathtt{l}_{2}-1},$
where $x_{1}(y_{1})$ and $x_{2}(y_{2})$ are variable-to-check (check-to-
variable) erasure messages of user 1 and 2 respectively. Note that if
$\mathtt{l}_{1}=\mathtt{l}_{2}=\mathtt{l}$ and
$\mathtt{r}_{1}=\mathtt{r}_{2}=\mathtt{r}$, then the above equations reduce to
a single parameter equation and is given by
$\displaystyle
x=(\epsilon+\frac{(1-\epsilon)}{2}y^{\mathtt{l}})y^{\mathtt{l}-1},$
$\displaystyle y=1-(1-x)^{\mathtt{r}-1}.$
### III-B Exit-like Curves
We define the BP EXIT-like111The reason we call this function EXIT-like is
because we do not provide any operational interpretation of these curves like
the Area theorem [13]. The curves are drawn only to illustrate that the BP
performance of coupled codes is close to the Shannon threshold, which is the
main result of the paper. function as follows.
$\displaystyle
h^{\mathrm{BP}}(\epsilon)=\frac{3}{2}y_{1}^{\mathtt{l}_{1}}y_{2}^{\mathtt{l}_{2}}+y_{1}^{\mathtt{l}_{1}}(1-y_{2}^{\mathtt{l}_{2}})+(1-y_{1}^{\mathtt{l}_{1}})y_{2}^{\mathtt{l}_{2}}.$
(2)
An intuitive reason as to why we define the BP EXIT function as above is since
the entropy of $Z_{i}=X_{1,i}+X_{2,i}$ is $H(1/4,1/4,1/2)=3/2$ when a priori
messages from both LDPC codes are erased and since the entropy of $Z_{i}$ is 1
when either of them is erased and the other is not.
Assume that all the FPs are parametrized with $x_{1}$ such as
$(x_{1},y_{1}(x_{1}),x_{2}(x_{1}),y_{2}(x_{1}),\epsilon(x_{1})).$ This
assumption is true if
$(\mathtt{l}_{1},\mathtt{r}_{1})=(\mathtt{l}_{2},\mathtt{r}_{2})=(\mathtt{l},\mathtt{r})$
with
$\displaystyle y_{1}(x_{1})$
$\displaystyle=y_{2}(x_{1})=1-(1-x_{1})^{\mathtt{r}-1},$ $\displaystyle
x_{2}(x_{1})$ $\displaystyle=x_{1},$ $\displaystyle\epsilon(x_{1})$
$\displaystyle=\frac{\frac{x_{1}}{y_{1}(x_{1})^{\mathtt{l}-1}}-\frac{y_{2}(x_{1})^{\mathtt{l}}}{2}}{1-\frac{y_{2}(x_{1})^{\mathtt{l}}}{2}}.$
We then have BP EXIT function as follows
$\displaystyle h^{\mathrm{BP}}(x_{1})=$
$\displaystyle\frac{3}{2}y_{1}(x_{1})^{\mathtt{l}_{1}}y_{2}(x_{1})^{\mathtt{l}_{2}}$
$\displaystyle+y_{1}(x_{1})^{\mathtt{l}_{1}}(1-y_{2}(x_{1})^{\mathtt{l}_{2}})$
$\displaystyle+(1-y_{1}(x_{1})^{\mathtt{l}_{1}})y_{2}(x_{1})^{\mathtt{l}_{2}}.$
We also consider the extended BP EXIT (EBP EXIT) curve which is the plot of
all the fixed points of DE. In the case of codes (of each user) being picked
from the same ensemble, the EBP EXIT-like curve is given by the parametric
curve $(h^{\text{\tiny BP}}(x),\epsilon(x))$, where $x$ is the variable-to-
check node message of either code.
###### Example 1
Figure 2 shows the plot of the EBP EXIT-like curve for
$\mathtt{l}_{1}=\mathtt{l}_{2}=3,\mathtt{r}_{1}=\mathtt{r}_{2}=6$. We choose
this particular example since as seen from above it is easier to evaluate the
value of $\epsilon$ given a fixed value of $x$ (the variable-to-check node
message in either of the code). The BP threshold is $\approx 0.12256$ which is
much less than the Shannon threshold of $1/3$.
(50,-8)(20,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$(36,0)(0,30)[l]
,$0.3$,$0.6$,$0.9$,$1.2$$0.0$$\epsilon$
$h^{\text{\scriptsize BP}}$
$\epsilon^{\text{\tiny BP}}\approx 0.12256$
Figure 2: EBP EXIT curve for the case when both users pick a code from the
$(3,6)$ and $(3,6)$. The BP threshold is $\approx 0.12256$ and the Shannon
threshold is $1/3$.
We observe that if we increase the degrees to $(4,8)$ for both the codes, the
BP threshold dramatically drops to zero. Also note the C shape of the EXIT
curve, indicating that there are exactly 3 FPs including a trivial FP plotted
at $(0,\epsilon)$ for each channel value, similar to the BEC case.
## IV Main Results
In this section, we analyze the performance of coupled codes over BAC with
erasures. We use the $(\mathtt{l}_{1},\mathtt{r}_{1},L,w)$ coupled ensemble
for user 1 and $(\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ ensemble for user 2. As a
shorthand notation we use
$(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ to denote
both the ensembles. Our main result is that, via DE analysis, the BP threshold
of the coupled ensemble is very close to the Shannon threshold given by (1).
Furthermore, by increasing the degrees, the BP threshold of the coupled
ensemble goes to the Shannon threshold.
Next, we develop the DE equation when transmitting using the coupled codes.
### IV-A Density Evolution for the
$(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ ensemble
We develop the DE equations assuming that the two users use ensembles of
different degrees. Consider the
$(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ ensemble.
To perform the DE analysis, we already take the limit $M\to\infty$ (the number
of variable nodes in each section).
Let $x_{1,i}$, $i\in\mathbb{Z}$, denote the average erasure probability which
is emitted by variable nodes at position $i$ to check nodes at position $i$
for user 1. Similarly define $x_{2,i}$ for the user 2. For $i\not\in[-L,L]$,
we set $x_{1,i}=x_{2,i}=0$. For $i\in[-L,L]$ the DE is given by
$\displaystyle y_{1,i}$
$\displaystyle=1-(1-\frac{1}{w}\sum_{k=0}^{w-1}x_{1,i-k})^{\mathtt{r}_{1}-1},$
$\displaystyle x_{1,i}$
$\displaystyle=\big{(}\epsilon+\frac{1-\epsilon}{2}\big{(}\frac{1}{w}\sum_{j=0}^{w-1}y_{2,i+j}\big{)}^{\mathtt{l}_{2}}\big{)}\big{(}\frac{1}{w}\sum_{j=0}^{w-1}y_{1,i+j}\big{)}^{\mathtt{l}_{1}-1},$
$\displaystyle y_{2,i}$
$\displaystyle=1-(1-\frac{1}{w}\sum_{k=0}^{w-1}x_{2,i-k})^{\mathtt{r}_{2}-1},$
$\displaystyle x_{2,i}$
$\displaystyle=\big{(}\epsilon+\frac{1-\epsilon}{2}\big{(}\frac{1}{w}\sum_{j=0}^{w-1}y_{1,i+j}\big{)}^{\mathtt{l}_{1}}\big{)}\big{(}\frac{1}{w}\sum_{j=0}^{w-1}y_{2,i+j}\big{)}^{\mathtt{l}_{2}-1}.$
(3)
We will use the notation $\epsilon^{\text{\tiny
BP}}(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ to
denote the threshold of the BP decoder when we use coupled codes for
transmission. Also, we use $\epsilon^{\text{\tiny
BP}}(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2})$ to denote
the BP threshold of the underlying uncoupled ensemble.
As a shorthand we use $g_{1}(x^{1,2}_{i-w+1},\dots,x^{1,2}_{i+w-1})$ to denote
$(\epsilon+\frac{1-\epsilon}{2}(\frac{1}{w}\sum_{j=0}^{w-1}y_{2,i+j})^{\mathtt{l}_{2}})(\frac{1}{w}\sum_{j=0}^{w-1}y_{1,i+j})^{\mathtt{l}_{1}-1}$
and also $g_{2}(x^{1,2}_{i-w+1},\dots,x^{1,2}_{i+w-1})$ to denote
$(\epsilon+\frac{1-\epsilon}{2}(\frac{1}{w}\sum_{j=0}^{w-1}y_{1,i+j})^{\mathtt{l}_{1}})(\frac{1}{w}\sum_{j=0}^{w-1}y_{2,i+j})^{\mathtt{l}_{2}-1}$.
###### Definition 2 (FPs of Density Evolution)
Consider DE for the
$(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ ensemble.
Let $\underline{x}_{1}=(x_{1,-L},\dots,{x}_{1,L})$ and
$\underline{x}_{2}=(x_{2,-L},\dots,{x}_{2,L})$ denote the vector of variable-
to-check erasure messages for user 1 and 2 respectively. We call
$\underline{x}_{1}$ and $\underline{x}_{2}$ the constellation of user 1 and 2
respectively. We say that $(\underline{x}_{1},\underline{x}_{2})$ forms a FP
of DE with channel $\epsilon$ if $(\underline{x}_{1},\underline{x}_{2})$
fulfills (IV-A) for $i\in[-L,L]$. As a shorthand we then say that
$(\epsilon,\underline{x}_{1},\underline{x}_{2})$ is a FP. We say that
$(\epsilon,\underline{x}_{1},\underline{x}_{2})$ is a non-trivial FP if either
$\underline{x}_{1}$ or $\underline{x}_{2}$ is not identically equal to
$0\,\,\forall\,i$. Again, for $i\notin[-L,L]$, $x_{1,i}=x_{2,i}=0$. ∎
###### Definition 3 (Forward DE and Admissible Schedules)
Consider forward DE for the
$(\mathtt{l}_{1},\mathtt{r}_{1},\mathtt{l}_{2},\mathtt{r}_{2},L,w)$ ensemble.
More precisely, pick a channel $\epsilon$ and initialize
$\underline{x}^{(0)}_{1}=\underline{x}^{(0)}_{2}=(1,\dots,1)$. Let
$\underline{x}^{(\ell)}_{1}$ and $\underline{x}^{(\ell)}_{2}$ be the result of
$\ell$ rounds of DE for user 1 and 2 respectively. More precisely,
$\underline{x}^{(\ell+1)}_{1}$ and $\underline{x}^{(\ell+1)}_{2}$ are
generated from $\underline{x}^{(\ell)}_{1}$ and $\underline{x}^{(\ell)}_{2}$
by applying the DE equation (IV-A) to each section $i\in[-L,L]$,
$\displaystyle x_{1,i}^{(\ell+1)}$
$\displaystyle=g_{1}(x_{i-w+1}^{1,2,(\ell)},\dots,x_{i+w-1}^{1,2,(\ell)}),$
$\displaystyle x_{2,i}^{(\ell+1)}$
$\displaystyle=g_{2}(x_{i-w+1}^{1,2,(\ell)},\dots,x_{i+w-1}^{1,2,(\ell)}),$
where we use the notation $x_{i}^{1,2,(\ell)}$ to denote
$(x_{1,i}^{(\ell)},x_{2,i}^{(\ell)})$. We call this the parallel schedule.
More generally, consider a schedule in which in each step $\ell$ an arbitrary
subset of the sections is updated, constrained only by the fact that every
section is updated in infinitely many steps. We call such a schedule
admissible. Again, we call $\underline{x}^{(\ell)}_{1}$ and
$\underline{x}^{(\ell)}_{2}$ the resulting sequence of constellations. ∎
One can show that if we perform forward DE under any admissible schedule, then
the constellations $\underline{x}^{(\ell)}_{1}$ and
$\underline{x}_{2}^{(\ell)}$ converge to a FP of DE and this FP is independent
of schedule. This statement can be proved similar to the one in [2, 13].
For the case when $\mathtt{l}_{1}=\mathtt{l}_{2}$ and
$\mathtt{r}_{1}=\mathtt{r}_{2}$ we have that for any FP, $x_{1,i}=x_{2,i}$ and
$y_{1,i}=y_{2,i}$ for all $i$.
### IV-B Forward DE – Simulation Results
In the examples below, the Shannon threshold is computed using equation (1).
###### Example 4 (Equal Degrees – BP goes to Shannon)
We consider forward DE for the coupled ensembles. More precisely, we fix an
$\epsilon$ and initialize all $x_{1,i}$ and $x_{2,i}$ to 1, for $i\in[-L,L]$.
Then we run the DE given by (IV-A) till we reach a fixed-point. We fix
$L=200$. For $\mathtt{l}_{1}=\mathtt{l}_{2}=3$ and
$\mathtt{r}_{1}=\mathtt{r}_{2}=6$, we have that $\epsilon^{\text{\tiny
BP}}(3,6,3,6,200,3)\approx 0.332287$. If we increase the degrees we get
$\epsilon^{\text{\tiny BP}}(4,8,4,8,200,4)\approx 0.333195$,
$\epsilon^{\text{\tiny BP}}(5,10,5,10,200,5)\approx 0.333286$. We observe that
by increasing the degrees the BP threshold approaches the Shannon threshold of
$1/3$. On the other hand for the uncoupled codes, $\epsilon^{\text{\tiny
BP}}(3,6,3,6)\approx 0.12256$ and for larger degrees the BP threshold is zero.
###### Example 5 (Unequal Degrees – BP goes to Shannon)
We also consider the more general case when the degrees are not equal. For
$\mathtt{l}_{1}=5,\mathtt{r}_{1}=10$ and $\mathtt{l}_{2}=6,\mathtt{r}_{2}=13$
we get $\epsilon^{\text{\tiny BP}}(5,10,6,13,500,10)\approx 0.307647$. The
Shannon threshold in this case is equal to $\approx 0.307692$. For
$\mathtt{l}_{1}=9,\mathtt{r}_{1}=10$ and $\mathtt{l}_{2}=6,\mathtt{r}_{2}=10$
we get $\epsilon^{\text{\tiny BP}}(9,10,6,10,500,10)\approx 0.59992$ and the
Shannon threshold is $=0.6$.
### IV-C EXIT curve plots
We also show via EXIT analysis that the coupling of regular LDPC codes pushes
the BP threshold (of the coupled systems) to the Shannon threshold. For the
purpose of illustration of the threshold saturation phenomena we focus only on
the case when $\mathtt{l}_{1}=\mathtt{l}_{2}$ and
$\mathtt{r}_{1}=\mathtt{r}_{2}$. Thus, the variable-to-check node messages,
for any FP of DE, for both the users are equal (cf. (IV-A)). Now, to plot the
EBP EXIT curve, which is essentially the plot of all the fixed-points of DE,
we define the entropy of a constellation as
$\displaystyle\chi=\frac{1}{2L+1}\sum_{i=-L}^{L}x_{1,i}.$
To plot all the FPs of DE, we first fix a value of $\chi\in[0,1]$ and then run
the reverse DE process given in [4]. Briefly, we start with an initial
variable-to-check message and run it through the check node. Then the
appropriate channel value is found such that the resulting constellation has
entropy equal to $\chi$. This process is run till we get an FP. Figure 3 shows
the plot of the EBP EXIT curve for the $(3,6,3,6,L,3)$ ensemble with
$L=2,4,8,16,32,64,128,256$. We observe that the plot looks very similar to the
case of single user transmission over a BEC. For small values of $L$ there is
a large rateloss and the EBP EXIT curve is to the right. As $L$ increases, the
rateloss diminishes and the curves move to the left. The limiting BP EXIT
curve of the coupled system looks very similar to when we are transmitting
over the BEC. It traces the BP EXIT function of the underlying uncoupled codes
until the channel erasure value is very close to the Shannon threshold and
then drops vertically to almost zero entropy.
(50,-8)(20,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$(36,0)(0,30)[l]
,$0.3$,$0.6$,$0.9$,$1.2$$0.0$$\epsilon$
$h^{\text{\scriptsize BP}}$
Figure 3: The EBP EXIT curve for $(3,6,3,6,L,3)$ with
$L=2,4,8,16,32,64,128,256$. The curve with light gray background is the BP
EXIT curve for the uncoupled $(3,6,3,6)$ ensemble. We see that as $L$
increases the EBP EXIT curves of the coupled system moves to the left. The BP
threshold of the coupled system is $\approx 0.3323$ which is very close to the
Shannon threshold.
### IV-D Shape of the Constellation
Figure 4 shows the constellation of an unstable FP (which cannot be reached by
BP). This FP is obtained via the reverse DE process. This special FP was the
key ingredient in proving threshold saturation over the BEC [2]. Let us
describe the (empirically observed) crucial properties of this constellation.
* (i)
The constellation is symmetric around $i=0$ and is unimodal. The constellation
has $\epsilon\approx 0.3323$, which is close to the Shannon threshold of
$1/3$.
* (ii)
The value in the flat part in the middle is $\approx 0.6548$ which is very
close to the stable FP of DE for the underlying uncoupled $(3,6)$-regular
ensemble at $\epsilon\approx 0.3323$.
* (iii)
The transition from values close to zero to values close to $0.6548$ is very
quick.
(6,0)(14.4,0)[b]$\text{-}16$,$\text{-}14$,$\text{-}12$,$\text{-}10$,$\text{-}8$,$\text{-}6$,$\text{-}4$,$\text{-}2$,0,2,4,6,8,10,12,14,16
Figure 4: The unstable FP shown above has an entropy of $0.28$ and is
obtained via reverse DE. The constellation is symmetric around $0$ and is
unimodal. The flat middle part has value close to $0.6548$ which is the value
of stable FP for the uncoupled system at $\epsilon\approx 0.3323$. Both the
users have identical FP constellation.
## V Discussion
In this paper we show that, by using coupled regular LDPC codes when
transmitting over the 2 user BAC with erasures, the BP threshold can be made
very close to the Shannon threshold. In this sense, the coupled codes are
threshold saturating. We demonstrate this by plotting EXIT-like curves. The
behavior of these curves is very similar to when transmitting over the BEC.
Even the shape of the constellation of an unstable FP of DE is same as the BEC
case. Thus we believe one should be able to provide a proof of this phenomena
on the lines of the BEC proof [2].
Another interesting question is to determine area theorems which will also
further show that the BP threshold of the coupled system goes to the MAP
threshold of the underlying uncoupled codes, when we consider finite degrees.
To do this we would need to define an appropriate EXIT function.
Lastly, it would be interesting to see if we can demonstrate the threshold
saturation phenomena to more general MAC channels, like the 2 user BAC with
additive Gaussian noise.
## VI Acknowledgments
SK acknowledges support of NMC via the NSF collaborative grant CCF-0829945 on
“Harnessing Statistical Physics for Computing and Communications.” SK would
also like to thank Rüdiger Urbanke for his encouragement.
## References
* [1] A. J. Felström and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” _IEEE Trans. Inform. Theory_ , vol. 45, no. 5, pp. 2181–2190, Sept. 1999.
* [2] S. Kudekar, T. Richardson, and R. Urbanke, “Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC,” 2010, e-print: http://arxiv.org/abs/1001.1826.
* [3] M. Lentmaier and G. P. Fettweis, “On the thresholds of generalized LDPC convolutional codes based on protographs,” in _Proc. of the IEEE Int. Symposium on Inform. Theory_ , Austing, TX, USA, June 2010, pp. 709–713.
* [4] C. Méasson, A. Montanari, T. Richardson, and R. Urbanke, “The generalized area theorem and some of its consequences,” _IEEE Trans. Inform. Theory_ , vol. 55, no. 11, pp. 4793–4821, Nov. 2009.
* [5] S. Kudekar, C. Méasson, T. Richardson, and R. Urbanke, “Threshold saturation on BMS channels via spatial coupling,” Apr. 2010, e-print: http://arxiv.org/abs/1004.3742.
* [6] E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” _IEEE Trans. Inform. Theory_ , vol. 55, no. 7, pp. 3051–3073, 2009.
* [7] T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” _IEEE Trans. Inform. Theory_ , vol. 47, no. 2, pp. 619–637, Feb. 2001.
* [8] R. Mori and T. Tanaka, “Performance and construction of polar codes on symmetric binary-input memoryless channels,” Jan. 2009, http://arxiv.org/abs/0901.2207.
* [9] S. Kudekar and H. D. Pfister, “The effect of spatial coupling on compressive sensing,” in _Proc. of the Allerton Conf. on Commun., Control, and Computing_ , Monticello, IL, USA, 2010.
* [10] S. H. Hassani, N. Macris, and R. Urbanke, “Coupled graphical models and their thresholds,” in _Proc. of the IEEE Inform. Theory Workshop_ , Dublin, Ireland, Sept. 2010.
* [11] V. Rathi, R. Urbanke, M. Andersson, and M. Skoglund, “Rate-equivocation optimally spatially coupled LDPC codes for the BEC wiretap channel,” 2010, e-print: http://arxiv.org/abs/1010.1669.
* [12] S. Kudekar and K. Kasai, “Threshold Saturation on Channels with Memory via Spatial Coupling,” 2011, e-print: http://arxiv.org/abs/0211.1669.
* [13] T. Richardson and R. Urbanke, _Modern Coding Theory_. Cambridge University Press, 2008.
* [14] A. Amraoui, S. Dusad, and R. Urbanke, “Achieving general points in the 2-user Gaussian MAC without time-sharing or rate-splitting by means of iterative coding,” in _Proc. of the IEEE Int. Symposium on Inform. Theory_ , Lausanne, Switzerland, June 2002, conference, p. 334.
|
arxiv-papers
| 2011-02-14T19:00:57 |
2024-09-04T02:49:16.993481
|
{
"license": "Public Domain",
"authors": "Shrinivas Kudekar and Kenta Kasai",
"submitter": "Kenta Kasai",
"url": "https://arxiv.org/abs/1102.2856"
}
|
1102.3069
|
# MuMax: a new high-performance micromagnetic simulation tool
A. Vansteenkiste Arne.Vansteenkiste@ugent.be B. Van de Wiele Department of
Solid State Sciences, Ghent University, Krijgslaan 281-S1, B9000 Gent,
Belgium. Department of Electrical Energy, Systems and Automation, Ghent
University, Sint Pietersnieuwstraat 41, B-9000 Ghent, Belgium.
###### Abstract
We present MuMax, a general-purpose micromagnetic simulation tool running on
Graphical Processing Units (GPUs). MuMax is designed for high performance
computations and specifically targets large simulations. In that case speedups
of over a factor 100$\times$ can easily be obtained compared to the CPU-based
OOMMF program developed at NIST. MuMax aims to be general and broadly
applicable. It solves the classical Landau-Lifshitz equation taking into
account the magnetostatic, exchange and anisotropy interactions, thermal
effects and spin-transfer torque. Periodic boundary conditions can optionally
be imposed. A spatial discretization using finite differences in 2 or 3
dimensions can be employed. MuMax is publicly available as open source
software. It can thus be freely used and extended by community. Due to its
high computational performance, MuMax should open up the possibility of
running extensive simulations that would be nearly inaccessible with typical
CPU-based simulators.
###### keywords:
micromagnetism , simulation , GPU
###### PACS:
75.78.Cd , 02.70.Bf
## 1 Introduction
Micromagnetic simulations are indispensable tools in the field of magnetism
research. Hence, micromagnetic simulators like, e.g., OOMMF [1], magpar [2]
and Nmag [3] are widely used. These tools solve the Landau-Lifshitz equation
on regular CPU hardware. Due to the required fine spatial and temporal
discretizations, such simulations can be very time consuming. Limited
computational resources therefore often limit the full capabilities of the
otherwise successful micromagnetic approach.
There is currently a growing interest in running numerical calculations on
graphical processing units (GPUs) instead of CPUs. Although originally
intended for purely graphical purposes, GPUs turn out to be well-suited for
high-performance, general-purpose calculations. Even relatively cheap GPUs can
perform an enormous amount of calculations in parallel. E.g., the nVIDIA
GTX580 GPU used for this work costs less than $ 500 and delivers 1.5 trillion
floating-point operations (Flops) per second, about 2 orders of magnitude more
than a typical CPU.
However, in order to employ this huge numerical power programs need to be
written specifically for GPU hardware, using the programming languages and
tools provided by the GPU manufacturer, and the code also needs to handle many
hardware-specific technicalities. Additionally, the used algorithms need to be
expressed in a highly parallel manner, which is not always easily possible.
Other groups have already implemented micromagnetic simulations on GPU
hardware and report considerable speedups compared to a CPU-only
implementation [4, 5, 6]. At the time of writing, however, none of these
implementations is freely available. MuMax, on the other hand, is available as
open source software and can be readily used by anyone. Its performance also
compares favorably to these other implementations.
## 2 Methods
Since the micromagnetic theory describes the magnetization as a continuum
field $\mathbf{M}(\mathbf{r},t)$, the considered magnetic sample is
discretized in cuboidal finite difference (FD) cells with a uniform
magnetization. The time evolution of the magnetization in each cell is given
by the Landau-Lifshitz equation
$\begin{split}\frac{\partial\mathbf{M}(\mathbf{r},t)}{\partial
t}=&-\frac{\gamma}{1+\alpha^{2}}\mathbf{M}(\mathbf{r},t)\times\mathbf{H}_{eff}(\mathbf{r},t)\\\
&-\frac{\alpha\gamma}{M_{s}(1+\alpha^{2})}\mathbf{M}(\mathbf{r},t)\times\left(\mathbf{M}(\mathbf{r},t)\times\mathbf{H}_{eff}(\mathbf{r},t)\right).\end{split}$
(1)
Here, $M_{s}$ is the saturation magnetization, $\gamma$ the gyromagnetic ratio
and $\alpha$ the damping parameter. The continuum effective field
$\mathbf{H}_{eff}$ has several contributions that depend on the magnetization,
the externally applied field and the material parameters of the considered
sample. When timestepping equation (1) the effective field is evaluated
several times per time step. Hence, the efficiency of micromagnetic software
depends on the efficient evaluation of the different effective field terms at
the one hand and the application of efficient time stepping schemes on the
other hand. MuMax combines both with the huge computational power of GPU
hardware.
### 2.1 Effective field terms
In the present version of MuMax, the effective field can have 5 different
contributions: the magnetostatic field, the exchange field, the applied field,
the anisotropy field and the thermal field. In what follows we present these
terms and comment on their optimized implementation.
#### 2.1.1 Magnetostatic field
The magnetostatic field $\mathbf{H}_{ms}$ accounts for the long-range
interaction throughout the complete sample
$\mathbf{H}_{ms}(\mathbf{r})=-\frac{1}{4\pi}\int_{V}\nabla\nabla\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}\cdot\mathbf{M}(\mathbf{r}^{\prime})\,\mathrm{d}\mathbf{r}^{\prime}.$
(2)
Since the magnetostatic field in one FD cell depends on the magnetization in
all other FD cells, the calculation of $\mathbf{H}_{ms}$ is the most time-
consuming part of a micromagnetic simulation. The chosen method for this
calculation is thus decisive for the performance of the simulator. Therefore,
we opted for a fast Fourier transform (FFT) based method. In this case, the
convolution structure of (2) is exploited. By applying the convolution
theorem, the convolution is accelerated by first Fourier transforming the
magnetization, then multiplying this result with the Fourier-transform of the
convolution kernel and finally inverse transforming this product to obtain the
magnetostatic field. The overall complexity of this method is
$\mathcal{O}(N\log N)$, as it is dominated by the FFTs.
Methods with even lower complexity exist as well. The fast multipole method,
e.g., only has complexity $\mathcal{O}(N)$, but with such a large pre-factor
that in most cases the FFT method remains much faster [7].
A consequence of the FFT method is that the magnetic moments must lie on a
regular grid. This means that a finite difference (FD) spatial discretization
has to be used: space is divided into equal cuboid cells. This method is thus
most suited for rectangular geometries. Other shapes have to be approximated
in a staircase-like fashion. However, thanks to the speedup offered by
MuMax’s, smaller cells may be used to improve this without excessive
performance penalties.
The possibility of adding periodic boundary conditions in one or more
directions is also included in the software. This is done by adding a
sufficiently large number of periodic images to the convolution kernel. The
application of periodic boundary conditions has a positive influence on the
computational time since the magnetization data does not need to be zero
padded in the periodic directions, which roughly halves the time spend on FFTs
for every periodic direction.
#### 2.1.2 Exchange field
The exchange interaction contributes to the effective field in the form of a
laplacian of the magnetization
$\mathbf{H}_{exch}=\frac{2A}{\mu_{0}M_{s}}\nabla^{2}\mathbf{m},$ (3)
with $A$ the exchange stiffness. In discretized form, this can be expressed as
a linear combination of the magnetization of a cell and a number of its
neighbors. MuMax uses a 6-neighbor scheme, similar to [8]. In the case of 2D
simulations (only one FD cell in the z-direction), this method automatically
reduces to a 4-neighbor scheme.
The exchange field calculation is included in the magnetostatic field routines
by simply adding the kernel describing the exchange interaction to the
magnetostatic kernel. In this way, the exchange calculation is essentially
free, as only one joint convolution product is needed to simultaneously
evaluate both the magnetostatic and exchange fields. Moreover, by introducing
the exchange contribution in the magnetostatic field kernel periodic boundary
conditions are directly accounted for if applicable.
#### 2.1.3 Other effective field terms
Next to the above mentioned interaction terms and the applied field
contribution, MuMax provides the ability to include magnetocrystalline
anisotropy. Currently, uniaxial and cubic anisotropy are available. The
considered anisotropy energies are
$\phi_{ani}=K_{u}\sin^{2}\theta$ (4)
and
$\begin{split}\phi_{ani}(\mathbf{r})&=K_{1}\left[\alpha_{1}^{2}(\mathbf{r})\alpha_{2}^{2}(\mathbf{r})+\alpha_{2}^{2}(\mathbf{r})\alpha_{3}^{2}(\mathbf{r})+\alpha_{1}^{2}(\mathbf{r})\alpha_{3}^{2}(\mathbf{r})\right]\\\
&+K_{2}\left[\alpha_{1}^{2}(\mathbf{r})\alpha_{2}^{2}(\mathbf{r})\alpha_{3}^{2}(\mathbf{r})\right]\end{split}$
(5)
for uniaxial and cubical anisotropy respectively. Here, $K_{u}$ and
$(K_{1},K_{2})$ are the uniaxial and cubical anisotropy constants, $\theta$ is
the angle between the local magnetization and uniaxial anisotropy axis and
$\alpha_{i}$ ($i=1,2,3$) are the direction cosines between the local
magnetization and the cubic easy magnetization axes.
Furthermore, thermal effects are included by means of a fluctuating thermal
field
$\mathbf{H}_{th}=\boldsymbol{\eta}(\mathbf{r},t)\sqrt{\frac{2\alpha
k_{B}T}{\gamma\mu_{0}M_{s}V\delta t}}$ (6)
which is added to the effective field $\mathbf{H}_{eff}$ according to [9]. In
(6), $k_{B}$ is the Boltzmann constant, $V$ is the volume of a FD cell,
$\delta t$ is the used time step and $\boldsymbol{\eta}(\mathbf{r},t)$ is a
stochastic vector whose components are Gaussian random numbers, uncorrelated
in space and time with zero mean value and dispersion 1.
#### 2.1.4 Spin-transfer torque
The spin-transfer torque interaction describes the influence of electrical
currents on the local magnetization. Possible applications are spin-transfer
torque random access memory [10] and racetrack memory [11]. MuMax incorporates
the spin-transfer torque description developed by Berger [12], refined by
Zhang and Li [13]
$\begin{split}\frac{\partial\mathbf{M}}{\partial
t}=&-\frac{\gamma}{1+\alpha^{2}}\mathbf{M}\times\mathbf{H}_{eff}\\\
&-\frac{\alpha\gamma}{M_{s}(1+\alpha^{2})}\mathbf{M}\times(\mathbf{M}\times\mathbf{H}_{eff})\\\
&-\frac{b_{j}}{M_{s}^{2}(1+\alpha^{2})}\mathbf{M}\times\left(\mathbf{M}\times(\mathbf{j}\cdot\nabla)\mathbf{M}\right)\\\
&-\frac{b_{j}}{M_{s}(1+\alpha^{2})}(\xi-\alpha)\mathbf{M}\times(\mathbf{j}\cdot\nabla)\mathbf{M}.\end{split}$
(7)
Here, $\xi$ is the degree of non-adiabicity and $b_{j}$ is the coupling
constant between the current density $\mathbf{j}$ and the magnetization
$b_{j}=\frac{P\mu_{B}}{eM_{s}(1+\xi^{2})},$ (8)
with $P$ the polarization of the current density, $\mu_{B}$ the Bohr magneton
and $e$ the electron charge.
### 2.2 Time integration schemes
MuMax provides a range of Runge-Kutta (RK) methods to integrate the Landau-
Lifshitz equation. Currently the user can select between the following
options:
* 1.
RK1: Euler’s method
* 2.
RK2: Heun’s method
* 3.
RK12: Heun-Euler (adaptive step)
* 4.
RK3: Kutta’s method
* 5.
RK23: Bogacki–Shampine (adaptive step)
* 6.
RK4: Classical Runge-Kutta method
* 7.
RKCK: Cash-Karp (adaptive step)
* 8.
RKDP: Dormand–Prince (adaptive step)
The adaptive step methods adjust the time step based on a maximum tolerable
error per integration step that can be set by the user. The other methods can
use either a fixed time step or a fixed maximum variation of m per step.
Depending on the needs of the simulation, a very accurate but relatively slow
high-order solver (e.g. RKDP) or a less accurate but fast solver (e.g. RK12)
can be chosen. Additionally, MuMax incorporates the semi-analytical methods
described in [14]. These methods are specifically tailored to the Landau-
Lifshitz equation.
## 3 GPU-optimized implementation
Since various CPU based micromagnetic tools —well suited for relatively small
micromagnetic problems— are already available, we mainly concentrated on
optimizing MuMax for running very large simulations on GPUs. Nevertheless, the
code can also run in CPU-mode, with multi-threading modalities enabled. In
this way one can get familiar with the capabilities of MuMax before a high-end
GPU has to be purchased.
The GPU-specific parts of MuMax have been developed using nVIDIA’s CUDA
platform. The low-level, performance-critical functions that have to interact
directly with the GPU are written in C/C++. Counterparts of these functions
for the CPU are implemented as well and use FFTW [15] and multi threading. The
high-level parts of MuMax are implemented in ”safe” languages including Java,
Go and Python. This part is independent of the underlying GPU/CPU hardware. In
what follows we will only elaborate on the GPU-optimized implementation of the
low-level functions.
### 3.1 General precautions
A high-end GPU has its own dedicated memory with a high bandwidth (typically a
few hundred GB/s) which enables fast reads and writes on the GPU itself.
Communication with the CPU on the other hand is much slower since this takes
place over a PCI express bus with a much lower bandwidth (typically a few
GB/s). Therefore, our implementation keeps as much data as possible in the
dedicated GPU memory, avoiding CPU-GPU communication. The only large data
transfers occure at initialization time and when output is saved to disk. The
CPU thus only instructs the GPU which subroutines to launch. Hence, the GPU
handles all the major computational jobs.
On the GPU, an enormous number of threads can run in parallel, each performing
a small part of the computations. E.g., the GTX580 GPU used for this work has
512 computational cores grouped in 16 multiprocessors, resulting in total
number of 16384 available parallel threads. However, this huge parallel power
is only optimally exploited when the code is adapted to the specific GPU
architecture. E.g.: threads on the same multiprocessor (”thread _warps_ ”)
should only access the GPU memory in a coalesced way and should ideally
perform the same instructions. When coalesced memory access in not possible,
the so-called _shared memory_ should be used instead of the global GPU memory.
This memory is faster and has better random-access properties but is very
scarce. Our implementation takes into account all these technicalities,
resulting in a very high performance.
### 3.2 GPU-optimization of the convolution product
Generally, most computational time goes to the evaluation of the convolution
product defined by the magnetostatic field. When using fast Fourier transforms
(FFTs), the computations enhance three different stages: (i) forward Fourier
transforming the magnetization data that is zero padded in the non-periodic
directions, (ii) point-by-point multiplying the obtained data with the Fourier
transformed magnetostatic field kernel, (iii) inverse Fourier transforming the
resulting magnetostatic field data. The carefull implementation of these three
stages determines the efficiency of the convolution product and, more general,
of the micromagnetic code.
In the first place, the efficiency of this convolution process is safeguarded
by ensuring that the matrices defined by the magnetostatic field kernel are
completely symmetrical. Consequently, the Fourier transformed kernel data is
purely real. The absence of the imaginary part leads to smaller memory
requirements as well as a much faster evaluation of the point-by-point
multiplications – step (ii).
Furthermore, our GPU implementation of the fast Fourier transforms, which
internally uses the CUDA ”CUFFT” library, is specifically optimized for
micromagnetic applications. The general 3D real-to-complex Fourier transform
(and its inverse) available in the CUFFT library is replaced by a more
efficient implementation in which the set of 1D transforms in the different
directions are performed separately. This way, Fourier transforms on arrays
containing only zeros resulting from the zero padding are avoided. In each
dimension, the set of 1D Fourier transforms are performed on contiguous data
points resulting in the coalesced reading and writing of the data. As a
drawback, the transposition of the data between a set of Fourier transforms in
one and another dimension is needed.
In a straight forward implementation of the required matrix transposes, the
read and write instructions can not be both performed in a coalesced way since
either the input data or the transposed data is not contiguous in global
memory. Therefore, the input data is divided in blocks and copied to shared
memory assigned to a predefined number of GPU threads. There, the data block
is transposed and copied in a coalesced way back to the global GPU memory
space. By inventively using the large number of zero arrays –in the non-
periodic case– this transpose process can be done without (for 2D) or with
only limited (3D) extra memory requirements. The different sets of Fourier
transforms in this approach are performed using the 1D FFT routines available
in the CUFFT library. This implementation outperforms the general 3D real-to-
complex Fourier transform available in the CUFFT library while the built-in 2D
real-to-complex is only faster for small dimensions (for square geometries:
smaller than 512x512 FD cells). This approach ensures the efficient evaluation
of steps (i) and (iii) of the convolution.
### 3.3 Floating point precision
GPUs are in general better suited for single-precision than double-precision
arithmetic. Double-precision performance is not only much slower due to the
smaller number of arithmetic units, but also requires twice the amount of
memory. Since GPU’s typically have limited memory and FFT methods are
relatively memory-intensive, we opted to uses single-precision exclusively.
While, e.g., the finite element method used by Kakay et al. relies heavily on
double precision to obtain an accurate solution [4], our implementation is
designed to remain accurate even at single precission. First, all quantities
are internally stored in units that are well adapted to the problem. More
specifically, we choose units so that $\mu_{0}=\gamma_{0}=M_{s}=A=1$. This
avoids that any other quantity in the simulation becomes exceptionally large
or small —which could cause a loss of precision due to saturation errors. The
conversion to and from internal units is performed transparently to the user.
Secondly, we avoid numerically instable operations like, e.g., subtracting
nearly equal numbers. This avoids that small rounding of errors get amplified.
Finally, and most importantly, we restrict the size of the FFTs to numbers
where the CUFFT implementation is most accurate: $2^{n}\times\\{1,3,5\mathrm{\
or\ }7\\}$. Hence sometimes a slightly larger number of FD cells than strictly
necessary is used to meet this requirement. Fortunately this has no adverse
effect on the performance since CUFFT FFTs with these sizes also happen to be
exceptionally fast (see below). In this way, the combined error introduced by
the forward+inverse FFT was found to be only of the order of
$\mathcal{O}(10^{-6})$, as opposed to a typical error of
$\mathcal{O}(10^{-4})$ for other FFT sizes (estimated from the error on
transforming random data back and forth). Thanks to these precautions we
believe that our implementation should be sufficiently accurate for most
practical applications. Indeed, the uncertainty on material parameters alone
is usually much larger than the FFT error of $10^{-6}$.
## 4 Validation
In order to validate our software, we tested the reliability of the code by
simulating several standard problems. These standard problems are constructed
such that all different contributions in the considered test case influence
the magnetization processes significantly. A correct simulation of standard
problems can be considered as the best possible indication of the validity of
the developed software. In what follows, we consider standard problems
constructed for testing static simulations, dynamic simulations and dynamic
simulations incorporating spin-transfer torque.
### 4.1 static standard problem
The $\mu$MAG standard problem #2 [16] aims at testing quasi static
simulations. A cuboid with dimensions $5d\times d\times 0.1d$ is considered.
Since only magnetostatic and exchange interactions are included, the resulting
static properties only depend on the scaled parameter $d/l_{ex}$, with
$l_{ex}$ the exchange length. The starting configuration is saturation along
the $[1,1,1]$ axis, which is relaxed to the remanent state. This was done by
solving the Landau-Lifshitz equation with a high damping parameter $\alpha=1$.
Figure 1: Standard problem #2. Remanent magnetization along the $x$-axis (left
axis) and along the $y$-axis (right axis) as a function of the scaling
parameter $d$. The full line represents the simulation results from MuMax,
while the circles represent simulation points obtained from McMichael et al.
[17] and from Donahue et al.[18]
The number of FD cells was chosen depending on the size of nanostructure,
making sure the cell size remained below the exchange length. For
$d/l_{ex}\leq 10$, single-domain states with nearly full saturation along the
long axis were found, while for large geometries an S-state occured. The MuMax
simulations considered 200 values for $d/l_{ex}$. On the GPU a total
simulation time of 3’21” was needed to complete the 200 individual
simulations, compared to 34’30” on the CPU. The GPU speedup is here limited by
the relatively small simulation sizes (cfr. Fig. 7).
Figure 1 shows the remanent magnetization in function of the ratio $d/l_{ex}$.
The values obtained with MuMax coincide well with those of other authors [17,
18], validating MuMax for static micromagnetic problems.
### 4.2 dynamic standard problem
The $\mu$MAG standard problem #4 [16] aims at testing the description of the
dynamic magnetization processes by considering the magnetization reversal in a
thin film with dimensions 500 nm $\times$ 125 nm $\times$ 3 nm. Starting from
an initial equilibrium S-state, two different fields are applied. In this
problem only the exchange and magnetostatic interactions are considered
(exchange stiffness $A=1.3\times 10^{-11}$ J/m, saturation magnetization
$M_{s}=8.0\times 10^{5}$ Am-1). When relaxing to the initial equilibrium
S-state, a damping constant equal to 1 is used while during the reversal
itself a damping constant of 0.02 is applied, according to the problem
definition. As proposed in the standard problem, we show in Figs. 2 and 3 the
evolution of the average magnetization components together with the reference
magnetization configuration at the time point when $<M_{x}>$ crosses zero for
the first time, for field 1 ($\mu_{0}H_{x}$=-24.6 mT, $\mu_{0}H_{y}$= 4.3 mT,
$\mu_{0}H_{z}$= 0.0 mT) and field 2 ($\mu_{0}H_{x}$=-35.5 mT, $\mu_{0}H_{y}$=
-6.3 mT, $\mu_{0}H_{z}$= 0.0 mT). A discretization using 128 $\times$ 32
$\times\,$1 FD cells and the RK23 time stepping scheme with a time step around
600 fs (dynamically adapted during the simulation) was used. The relatively
large time steps used to solve this standard problem demonstrate that MuMax
incorporates robust time stepping schemes with accurate adaptive step
mechanisms. The adaptive step algorithms ensure optimal time step lengths and
thus reduce the number of field evaluations, speeding up the simulation. Here,
a total time of only 2.5 seconds was needed to finish this simulation on the
GPU compared to 16 seconds on the CPU. In this case the speedup on GPU is
limited due to the small number of FD cells. When the simulation is repeated
with a finer discretization of 256 $\times$64 $\times$ 2 cells, on the other
hand, the GPU speedup already becomes more pronounced: the simulation finishes
in 46 seconds on the GPU but takes 20’32” on the CPU.
Figure 2: (top) Time evolution of the average magnetization during the
reversal considered in ${\mu}$Mag standard problem #4, field1. The results
obtained with MuMax (black) lie well within the spread of the reference
solutions (grey), taken from [16]. (bottom) Magnetization configuration when
$<M_{x}>$ crosses the zero magnetization for the first time.
Figure 3: (top) Time evolution of the average magnetization during the
reversal considered in $\mu$Mag standard problem #4, field2. This field was
chosen to cause a bifurcation point to make the different solutions diverge.
(bottom) Magnetization configuration when $<M_{x}>$ crosses the zero
magnetization point for the first time.
From Figs. 2 and 3 it is clear that the results obtained with MuMax are well
within the spread of the curves obtained by other authors. Also the
magnetization plots are in close agreement with those available at the
$\mu$Mag website [16].
### 4.3 Spin-transfer torque standard problem [19]
$\mu$Mag does not propose any standard problems that include spin-transfer
torque. Therefore we rely on a standard problem proposed by M. Najafi et al.
[19] to check the validity of the spin-transfer torque description implemented
in MuMax. The standard problem considers a permalloy sample ($A=1.3\times
10^{-11}$ J/m, $M_{s}=8.0\times 10^{5}$ Am-1) with dimensions 100 nm $\times$
100 nm $\times$ 10 nm. The initial equilibrium magnetization state is a
predefined vortex, positioned in the center of the sample and relaxed without
any spin-transfer torque interaction ($\alpha=1.0$). Once relaxed, an
homogeneous spin-polarized dc current $\mathbf{j}=10^{12}$ Am-2 along the
$x$-axis is applied on the sample. Now, $\alpha$ is $0.1$ and the degree of
non-adiabicity $\xi$ is 0.05, see expression (7). Under these circumstances,
the vortex center moves towards a new equilibrium position. The time evolution
of the average in plane magnetization and the magnetization configuration at
$t$=0.73 ns are shown respectively in Fig. 4 and Fig. 5. The results are in
good agreement with those presented in reference [19]. With a discretization
of 128 $\times$ 128 FD cells, 10 minutes of simulation time were needed to
obtain the presented data.
Figure 4: Time evolution of the average in plane magnetization during the
first 8 ns of the spin-transfer torque standard problem. To facilitate the
visual comparison of our results with [19], the average magnetization is
expressed in [A/m] and the same axes ratios are chosen. Figure 5:
Magnetization configuration at t=0.73 ns as found during the simulation of a
standard problem incorporating spin-transfer torque [19]. The vortex core
evolves towards a new equilibrium state under influence of a spin-polarized dc
current allong the horizontal direction.This figure is rendered with the
built-in graphics features present in MuMax.
## 5 Performance
The performance of MuMax on the CPU is roughly comparable to OOMMF. The CPU
performance is thus good, but our main focus is optimizing the GPU code.
Special attention went to fully exploiting the numerical power of the GPU
while focussing on the time- and memory-efficient simulation of large
micromagnetic problems. Figure 6 shows the time required to take one time step
with the Euler method (i.e. effective field evaluation, evaluation of the LL-
equation and magnetization update) on CPU (1 core) and on GPU for 2D and 3D
simulations.
Figure 6: Time required to perform one time step using the Euler method for
(top) 2D geometries with varying dimensions N$\times$N and (bottom) 3D
geometries with varying dimensions N$\times$N$\times$N. The CPU computations
are performed on a 2.8 GHz intel core i7-930 processor, while the GPU
computations are performed on nVIDIA GTX580 GPU hardware.
In both the 2D and 3D case, speedups of up two orders of magnitude are
obtained for large dimensions. For smaller geometries, the speedups decrease
but remain significant. This can be understood by the fact that in these
simulations not enough FD cells are considered to have all
$\mathcal{O}(10^{4})$ available threads at work at the same time. Hence, the
computational power is not fully exploited. Furthermore, Fig. 6 shows that the
CPU performance as well as the GPU performance does not follow a smooth curve.
This is a consequence of the FFTs which are most efficient for powers of two,
possibly multiplied with one small prime (in the benchmarks shown in Fig. 6,
the default rounding to these optimal sizes is not performed). In the GPU
implementation this is even more the case than in the CPU implementation.
E.g., the 2D simulation with dimensions 992 $\times$ 992 is five times slower
than the 2D simulation with dimensions 1024$\times$1024\. This shows that not
only for accuracy reasons, but also for time efficiency reasons, it is most
advantageous to restrict the simulations domain to the optimal dimensions
defined by $2^{n}\times\\{1,3,5\mathrm{\ or\ }7\\}$. Because of the extreme
impact on the performance of MuMax, we opted to standardly rescale the size of
the FD cells such that the dimensions are rounded of to one of these optimal
sizes.
Table 1: Time needed to take one time step with the Euler method on CPU and GPU for 2D geometries (top) and 3D geometries (bottom). size | CPU time (ms) | GPU time (ms) | speedup
---|---|---|---
$32^{2}$ | 0.633 | 0.59 | $\times$ 1.07
$64^{2}$ | 3.092 | 0.60 | $\times$ 5.1
$128^{2}$ | 6.739 | 0.69 | $\times$ 9.7
$256^{2}$ | 59.90 | 1.11 | $\times$ 18
$512^{2}$ | 266.8 | 2.75 | $\times$ 47
$1024^{2}$ | 1166 | 9.07 | $\times$ 128
$2048^{2}$ | 5233 | 35.78 | $\times$ 146
$8^{3}$ | 0.8492 | 0.79 | $\times$ 1.07
$16^{3}$ | 4.066 | 1.03 | $\times$ 3.9
$32^{3}$ | 36.14 | 1.70 | $\times$ 21
$64^{3}$ | 489.6 | 5.52 | $\times$ 88
$128^{3}$ | 4487 | 35.42 | $\times$ 126
Due to the typical architecture of GPUs and the nature of the FFT algorithm,
simulations of geometries with power of two sizes run extremely fast on GPU.
Table 1 gives an overview of the speedups for these sizes for the 2D and 3D
case and Fig. 7 shows the speedup obtained for these power of two sizes by
MuMax compared to the OOMMF code. Both comparisons result in speedups larger
than a factor 100. This means that simulations that used to take several hours
can now be performed in minutes. The comparison between the speedups shown in
Table 1 and Fig. 7 further show that our CPU implementation has indeed a
comparable efficiency regarding to OOMMF. The immense speedups evidence the
fact that MuMax can indeed open completely new research opportunities in
micromagnetic modelling.
Figure 7: Speedup obtained with MuMax running on a GTX580 GPU compared to
OOMMF on a 2.8GHz core i7-930 CPU. The 2D and 3D geometries have sizes
N$\times$N and N$\times$N$\times$N respectively. The lowest speedup for the 16
x 16 x 16 case – an unusually small simulation– is still a factor 4.
## 6 How to use MuMax
MuMax is released as open source software under the GNU General Public License
(GPL) v.3 and can thus be freely used by the community. In addition to the
terms of the GPL, we kindly ask to acknowledge the authors in any publication
or derivative software that uses MuMax, by citing this paper. The MuMax source
code can be obtained via http://dynamat.ugent.be/mumax. To use the software, a
PC with a ”CUDA capable” nVIDIA GPU and a recent 64-bit Linux installation is
required.
A MuMax simulation is entirely specified by an input file passed via the
command-line. I.e., once the input file is written, no further user
interaction is necessary to complete the simulation. This allows, for
instance, to run large batches of simulations unattended. Nevertheless, the
progress of a simulation can easily be checked: the number of time steps
taken, total simulated time, etc. is reported in the terminal, PNG images of
the magnetization state can be output on-the-fly, graphs of the average
magnetization can easily be obtained while the simulation is running, etc.
Furthermore, MuMax’s output format is compatible with OOMMF, enabling the use
of existing post-processing tools to visualize and analyze the output. Built-
in tools for output processing are available as well. The 3D vector field in
Fig. 5, e.g., is rendered by MuMax’s tools.
MuMax input files can be written in Python. This offers powerful control over
the simulation flow and output. As an example, the code snippet below
simulates an MRAM element as in standard problem 2 (see section 4.1). Starting
form a uniform state in the $+x$ direction, it scans the field $B$ in small
steps until the point of coercivity. This illustrates how easily complex
simulations can be defined.
from mumax import *
msat(800e3)
aexch(1.3e-11)
partsize(500e-9, 50e-9, 5e-9)
uniform(1, 0, 0)
B = 0
while avg_m(’x’) > 0:
staticfield(-B, 0, 0)
relax()
B += 1e-4
# B now holds the coercitive field
save(’m’, ’binary’)
Figure 8: MuMax input file snippet illustrating a simulation specification in
Python. After initialization, a field in the $-x$ direction is stepped until
the average magnetization along $x$ reaches zero. The possibility of writing
conditional statements and loops, and obtaining information like the
magnetization state allows to construct arbitrarily complex simulation flows.
## 7 Conclusions and Outlook
MuMax is the first GPU-based micromagnetic solver that is publicly available
as open-source software. Due to the large number of considered interaction
terms and the versatile geometrical options (e.g. periodic boundary
conditions) the software covers many of the classical micromagnetic research
topics. The code is extensively validated by considering several standard
problems and is shown to be reliable. The time gains are extremely large
compared to CPU simulations: for large simulations a speedup with a factor 100
is easily obtained. These enormous speedups will open up new opportunities in
micromagnetic modelling and boost fundamental magnetic research.
In the future, MuMax will be extended towards a yet more multipurpose software
package incorporating other interactions in the Landau-Lifshitz equation:
other expressions for the exchange contribution, exchange bias, magnetoelastic
coupling, etc. Boundary correction methods to help in approximation non-square
geometries better are currently also being considered. Furthermore, the
description of more complex, non uniform microstructures will be made possible
as e.g. nanocrystalline materials. Modules for hysteresis research and
magnetic domain studies will be developed following the presented 2D and 3D
approach as well as the so-called 2.5D approach (infinitely thick geometries).
Furthermore, the efficient simulation of yet larger problems is planned by
introducing multiple GPUs –in one or more machines– for one single simulation.
This way, the up to now limited memory available on the GPU hardware can be
circumvented and the computational power will be further increased. Apart from
requiring efficient communication between the different GPUs, this should be
possible without drastic changes to our code as most of the implementation is
already hardware-independent.
## Acknowledgements
Financial support from the Flanders Research Foundation (FWO) is gratefully
acknowledged. We cordially thank Bartel Van Waeyenberge, Luc Dupré, and Daniël
De Zutter for supporting this research. Furthermore we would also like to
thank André Drews, Claas Abert, Gunnar Selke and Theo Gerhardt from Hamburg
University for the fruitful discussions and feedback.
## References
## References
* [1] MJ Donahue and DG Porter. OOMMF user’s guide, version 1.0. interagency report NISTIR 6376, national institute of standards and technology, gaithersburg, MD, 1999.
* [2] W. Scholz, J. Fidler, T. Schrefl, D. Suess, R. Dittrich, H. Forster, and V. Tsiantos. Scalable parallel micromagnetic solvers for magnetic nanostructures. Comput. Mater. Sci., 28:366–383, 2003.
* [3] T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr. A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag. Ieee Transactions On Magnetics, 43(6):2896–2898, 2007.
* [4] A. Kakay, E. Westphal, and R. Hertel. Speedup of fem micromagnetic simulations with graphical processing units. Ieee Transactions On Magnetics, 46(6):2303–2306, 2010.
* [5] S. J. Li, B. Livshitz, and V. Lomakin. Graphics processing unit accelerated o(n) micromagnetic solver. Ieee Transactions On Magnetics, 46(6):2373–2375, 2010.
* [6] G. Selke, A. Drews, and D.P.F. Möller. Highly efficient micromagnetic simulations using graphics processing units. submitted to IEEE Trans. Magn., 2011.
* [7] B. Van de Wiele, F. Olyslager, and L. Dupre. Application of the fast multipole method for the evaluation of magnetostatic fields in micromagnetic computations. Journal of Computational Physics, 227(23):9913–9932, 2008.
* [8] M. J. Donahue and D. G. Porter. Exchange energy formulations for 3d micromagnetics. Physica B-condensed Matter, 343(1-4):177–183, 2004.
* [9] W. F. Brown Jr. Micromagnetics. Interscience Publishers, New York, NY, 1963.
* [10] S. Bohlens, B. Kruger, A. Drews, M. Bolte, G. Meier, and D. Pfannkuche. Current controlled random-access memory based on magnetic vortex handedness. Applied Physics Letters, 93(14):142508, October 2008.
* [11] S. S. P. Parkin, M. Hayashi, and L. Thomas. Magnetic domain-wall racetrack memory. Science, 320(5873):190–194, April 2008.
* [12] L. Berger. Emission of spin waves by a magnetic multilayer traversed by a current. Physical Review B, 54(13):9353–9358, October 1996.
* [13] S. Zhang and Z. Li. Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets. Physical Review Letters, 93(12):127204, September 2004.
* [14] B. Van de Wiele, F. Olyslager, and L. Dupre. Fast semianalytical time integration schemes for the landau-lifshitz equation. Ieee Transactions On Magnetics, 43(6):2917–2919, 2007.
* [15] M. Frigo and S. G. Johnson. The design and implementation of FFTW3. Proceedings of the Ieee, 93(2):216–231, 2005.
* [16] muMAG Micromagnetic Modeling Activity Group http://www.ctcms.nist.gov/ rdm/mumag.org.html.
* [17] R. D. McMichael, M. J. Donahue, D. G. Porter, and J. Eicke. Comparison of magnetostatic field calculation methods on two-dimensional square grids as applied to a micromagnetic standard problem. Journal of Applied Physics, 85(8):5816–5818, 1999.
* [18] M. J. Donahue, D. G. Porter, R. D. McMichael, and J. Eicke. Behavior of mu mag standard problem no. 2 in the small particle limit. Journal of Applied Physics, 87(9):5520–5522, 2000.
* [19] M. Najafi, B. Kruger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche, D. P. F. Moller, and G. Meier. Proposal for a standard problem for micromagnetic simulations including spin-transfer torque. Journal of Applied Physics, 105(11), 2009.
|
arxiv-papers
| 2011-02-15T13:47:53 |
2024-09-04T02:49:17.005364
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arne Vansteenkiste and Ben Van de Wiele",
"submitter": "Arne Vansteenkiste",
"url": "https://arxiv.org/abs/1102.3069"
}
|
1102.3129
|
1em1em
# Automated Complexity Analysis Based on the Dependency Pair Method††thanks:
This research is partly supported by FWF (Austrian Science Fund) project
P20133, the Grant-in-Aid for Young Scientists Nos. 20800022 and 22700009 of
the Japan Society for the Promotion of Science, and Leading Project e-Society
(MEXT of Japan), and STARC.
Nao Hirokawa
School of Information Science
Japan Advanced Institute of Science and Technology Japan
hirokawa@jaist.ac.jp Georg Moser
Institute of Computer Science
University of Innsbruck Austria
georg.moser@uibk.ac.at
(June 2011)
###### Abstract
This article is concerned with automated complexity analysis of term rewrite
systems. Since these systems underlie much of declarative programming, time
complexity of functions defined by rewrite systems is of particular interest.
Among other results, we present a variant of the dependency pair method for
analysing runtime complexities of term rewrite systems automatically. The
established results significantly extent previously known techniques: we give
examples of rewrite systems subject to our methods that could previously not
been analysed automatically. Furthermore, the techniques have been implemented
in the Tyrolean Complexity Tool. We provide ample numerical data for assessing
the viability of the method.
_Key words_ : Term rewriting, Termination, Complexity Analysis, Automation,
Dependency Pair Method
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Rewriting
2. 2.2 Matrix Interpretations
3. 3 Runtime Complexity
4. 4 Usable Replacement Maps
5. 5 Weak Dependency Pairs
6. 6 The Weight Gap Principle
7. 7 Weak Dependency Graphs
8. 8 Experiments
9. 9 Conclusion
## 1 Introduction
This article is concerned with automated complexity analysis of term rewrite
systems (TRSs for short). Since these systems underlie much of declarative
programming, time complexity of functions defined by TRSs is of particular
interest.
Several notions to assess the complexity of a terminating TRS have been
proposed in the literature, compare [1, 2, 3, 4]. The conceptually simplest
one was suggested by Hofbauer and Lautemann in [2]: the complexity of a given
TRS is measured as the maximal length of derivation sequences. More precisely,
the _derivational complexity function_ with respect to a terminating TRS
relates the maximal derivation height to the size of the initial term.
However, when analysing complexity of a function, it is natural to refine
derivational complexity so that only terms whose arguments are constructor
terms are employed. Conclusively the _runtime complexity function_ with
respect to a TRS relates the length of the longest derivation sequence to the
size of the initial term, where the arguments are supposed to be in normal
form. This terminology was suggested in [4]. A related notion has been studied
in [1], where it is augmented by an _average case_ analysis. Finally [3]
studies the complexity of the functions _computed_ by a given TRS. This latter
notion is extensively studied within _implicit computational complexity
theory_ (_ICC_ for short), see [5] for an overview. A conceptual difference
from runtime complexity is that polynomial computability addresses the number
of steps by means of (deterministic) Turing machines, while runtime complexity
measures the number of rewrite steps which is closely related to operational
semantics of programs. For instance, a statement like a quadratic complexity
of sort algorithm is in the latter sense.
This article presents methods for (over-)estimating runtime complexity
automatically. We establish the following results:
1. 1)
We extend the applicability of direct techniques for complexity results by
showing how the monotonicity constraints can be significantly weakened through
the employ of _usable replacement maps_.
2. 2)
We revisit the _dependency pair method_ in the context of complexity analysis.
The dependency pair method is originally developed for proving termination
[6], and known as one of the most successful methods in automated termination
analysis.
3. 3)
We introduce the _weight gap principle_ which allows the estimation of the
complexity of a TRS in a modular way.
4. 4)
We revisit the dependency graph analysis of the dependency pair method in the
context of complexity analysis. For that we introduce a suitable notion of
_path analysis_ that allows to modularise complexity analysis further.
Note that while we have taken seminal ideas from termination analysis as
starting points, often the underlying principles are crucially different from
those used in termination analysis.
A preliminary version of this article appeared in [4, 7]. Apart from the
correction of some shortcomings, we extend our earlier work in the following
way: First, all results on usable replacement maps are new (see Section 4).
Second, the side condition for the weight gap principle [4, Theorem 24] is
corrected in Section 6. Thirdly, the weight gap principle is extended by
exploiting the initial term conditions and is generalised by means of matrix
interpretations (see Section 6). Finally, the applicability of the path
analysis is strengthened in comparison to the conference version [7] (see
Section 7).
The remainder of this article is organised as follows. In the next section we
recall basic notions. We define runtime complexity and a subclass of matrix
interpretations for its analysis in Section 3. In Section 4 we relate context-
sensitive rewriting to runtime complexity. In the next sections several
ingredients in the dependency pair method are recapitulated for complexity
analysis: dependency pairs and usable rules (Section 5), reduction pairs via
the weight gap principle (Section 6), and dependency graphs (Section 7). In
order to access viability of the presented techniques all techniques have been
implemented in the _Tyrolean Complexity Tool_ 111http://cl-
informatik.uibk.ac.at/software/tct/. (TCT for short) and its empirical data
is provided in Section 8. Finally we conclude the article by mentioning
related works in Section 9.
## 2 Preliminaries
We assume familiarity with term rewriting [8, 9] but briefly review basic
concepts and notations from term rewriting, relative rewriting, and context-
sensitive rewriting. Moreover, we recall matrix interpretations.
### 2.1 Rewriting
Let $\mathcal{V}$ denote a countably infinite set of variables and
$\mathcal{F}$ a signature, such that $\mathcal{F}$ contains at least one
constant. The set of terms over $\mathcal{F}$ and $\mathcal{V}$ is denoted by
$\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V})$. The _root symbol_ of a
term $t$, denoted as $\mathrm{root}(t)$, is either $t$ itself, if
$t\in\mathcal{V}$, or the symbol $f$, if $t=f({t_{1}},\dots,{t_{n}})$. The
_set of position_ $\mathcal{P}\mathsf{os}(t)$ of a term $t$ is defined as
usual. We write
$\mathcal{P}\mathsf{os}_{\mathcal{G}}(t)\subseteq\mathcal{P}\mathsf{os}(t)$
for the set of positions of subterms, whose root symbol is contained in
$\mathcal{G}\subseteq\mathcal{F}$. The subterm of $t$ at position $p$ is
denoted as ${{t}\\!\\!\mid_{p}}$, and $t[u]_{p}$ denotes the term that is
obtained from $t$ by replacing the subterm at $p$ by $u$. The subterm relation
is denoted as $\mathrel{{\trianglelefteq}}$. $\mathcal{V}\mathsf{ar}(t)$
denotes the set of variables occurring in a term $t$. The _size_ $\lvert
t\rvert$ of a term is defined as the number of symbols in $t$:
$\lvert t\rvert\mathrel{:=}\begin{cases}1&\text{if $t$ is a variable}\hbox
to0.0pt{$\;$,\hss}\\\ 1+\sum_{1\leqslant i\leqslant n}\lvert
t_{i}\rvert&\text{if $t=f(t_{1},\dots,t_{n})$}\hbox
to0.0pt{$\;$.\hss}\end{cases}$
A _term rewrite system_ (_TRS_) $\mathcal{R}$ over
$\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V})$ is a _finite_ set of
rewrite rules $l\to r$, such that $l\notin\mathcal{V}$ and
$\mathcal{V}\mathsf{ar}(l)\supseteq\mathcal{V}\mathsf{ar}(r)$. The smallest
rewrite relation that contains $\mathcal{R}$ is denoted by
$\to_{\mathcal{R}}$. The transitive closure of $\to_{\mathcal{R}}$ is denoted
by $\mathrel{\mathrel{\to}_{\mathcal{R}}^{+}}$, and its transitive and
reflexive closure by $\mathrel{\mathrel{\to}_{\mathcal{R}}^{\ast}}$. We simply
write $\to$ for $\to_{\mathcal{R}}$ if $\mathcal{R}$ is clear from context.
Let $s$ and $t$ be terms. If exactly $n$ steps are performed to rewrite $s$ to
$t$ we write $s\to^{n}t$. Sometimes a derivation $s=s_{0}\to s_{1}\to\cdots\to
s_{n}=t$ is denoted as $A\colon s\mathrel{\to}^{\ast}t$ and its length $n$ is
referred to as $\lvert A\rvert$. A term
$s\in\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V})$ is called a _normal
form_ if there is no $t\in\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V})$
such that $s\to t$. With $\mathsf{NF}(\mathcal{R})$ we denote the set of all
normal forms of a term rewrite system $\mathcal{R}$. The _innermost rewrite
relation_
$\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}}$
of a TRS $\mathcal{R}$ is defined on terms as follows:
$s\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}}t$
if there exist a rewrite rule $l\to r\in\mathcal{R}$, a context $C$, and a
substitution $\sigma$ such that $s=C[l\sigma]$, $t=C[r\sigma]$, and all proper
subterms of $l\sigma$ are normal forms of $\mathcal{R}$. _Defined symbols_ of
$\mathcal{R}$ are symbols appearing at root in left-hand sides of
$\mathcal{R}$. The set of defined function symbols is denoted as
$\mathcal{D}$, while the _constructor symbols_
$\mathcal{F}\setminus\mathcal{D}$ are collected in $\mathcal{C}$. We call a
term $t=f({t_{1}},\dots,{t_{n}})$ _basic_ or _constructor based_ if
$f\in\mathcal{D}$ and
$t_{i}\in\operatorname{\mathcal{T}}(\mathcal{C},\mathcal{V})$ for all
$1\leqslant i\leqslant n$. The set of all basic terms are denoted by
$\operatorname{\mathcal{T}_{\mathsf{b}}}$. A TRS $\mathcal{R}$ is called
_duplicating_ if there exists a rule $l\to r\in\mathcal{R}$ such that a
variable occurs more often in $r$ than in $l$. We call a TRS _(innermost)
terminating_ if no infinite (innermost) rewrite sequence exists.
We recall the notion of _relative rewriting_ , cf. [10, 9]. Let $\mathcal{R}$
and $\SS$ be TRSs. The relative TRS $\mathcal{R}/\SS$ is the pair
$(\mathcal{R},\SS)$. We define
${s\mathrel{\mathrel{\to}_{\mathcal{R}/\mathcal{S}}}t}\mathrel{:=}{s\mathrel{\mathrel{\to}_{\mathcal{S}}^{\ast}}\cdot\mathrel{\mathrel{\to}_{\mathcal{R}}}\cdot\mathrel{\mathrel{\to}_{\mathcal{S}}^{\ast}}t}$
and we call $\mathrel{\mathrel{\to}_{\mathcal{R}/\mathcal{S}}}$ the _relative
rewrite relation_ of $\mathcal{R}$ over $\mathcal{S}$. Note that
${\mathrel{\mathrel{\to}_{\mathcal{R}/\mathcal{S}}}}={\mathrel{\mathrel{\to}_{\mathcal{R}}}}$,
if $\SS=\varnothing$. $\mathcal{R}/\mathcal{S}$ is called _terminating_ if
$\mathrel{\mathrel{\to}_{\mathcal{R}/\mathcal{S}}}$ is well-founded. In order
to generalise the innermost rewriting relation to relative rewriting, we
introduce the slightly technical construction of the _restricted_ rewrite
relation, compare [11]. The _restricted rewrite relation
$\mathrel{\smash{\xrightarrow{\mathcal{Q}}}}_{\mathcal{R}}$_ is the
restriction of $\mathrel{\mathrel{\to}_{\mathcal{R}}}$ where all arguments of
the redex are in normal form with respect to the TRS $\mathcal{Q}$. We define
the _innermost relative rewriting relation_ (denoted as
$\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}/\mathcal{S}}}$)
as follows:
${\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}/\mathcal{S}}}}\mathrel{:=}{{\mathrel{\smash{\xrightarrow{\mathcal{R}\cup\mathcal{S}}}}_{\mathcal{S}}^{\ast}}\cdot{\mathrel{\smash{\xrightarrow{\mathcal{R}\cup\mathcal{S}}}}_{\mathcal{R}}}\cdot{\mathrel{\smash{\xrightarrow{\mathcal{R}\cup\mathcal{S}}}}_{\mathcal{S}}^{\ast}}}\hbox
to0.0pt{$\;$,\hss}$
We briefly recall context-sensitive rewriting. A replacement map $\mu$ is a
function with $\mu(f)\subseteq\\{1,\ldots,n\\}$ for all $n$-ary functions with
$n\geqslant 1$. The set $\mathcal{P}\mathsf{os}_{\mu}(t)$ of _$\mu$ -replacing
positions_ in $t$ is defined as follows:
$\mathcal{P}\mathsf{os}_{\mu}(t)=\begin{cases}\\{\epsilon\\}&\text{if $t$ is a
variable}\hbox to0.0pt{$\;$,\hss}\\\
\\{\epsilon\\}\cup\\{ip\mid\text{$i\in\mu(f)$ and
$p\in\mathcal{P}\mathsf{os}_{\mu}(t_{i})$}\\}&\text{if
$t=f({t_{1}},\dots,{t_{n}})$}\hbox to0.0pt{$\;$.\hss}\end{cases}$
A _$\mu$ -step_
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${\mu}$}}}}t$ is a
rewrite step $s\to t$ whose rewrite position is in
$\mathcal{P}\mathsf{os}_{\mu}(s)$. The set of all non-$\mu$-replacing
positions in $t$ is denoted by $\overline{\mathcal{P}\mathsf{os}}_{\mu}(t)$;
namely,
$\overline{\mathcal{P}\mathsf{os}}_{\mu}(t)\mathrel{:=}\mathcal{P}\mathsf{os}(t)\setminus\mathcal{P}\mathsf{os}_{\mu}(t)$.
### 2.2 Matrix Interpretations
One of the most powerful and popular techniques for analysing derivational
complexities is use of orders induced from matrix interpretations [12]. In
order to define it first we define (weakly) monotone algebras.
A _proper order_ is a transitive and irreflexive relation and a _preorder_ (or
_quasi-order_) is a transitive and reflexive relation. A proper order $\succ$
is _well-founded_ if there is no infinite decreasing sequence $t_{1}\succ
t_{2}\succ t_{3}\cdots$. We say a proper order $\succ$ and a TRS $\mathcal{R}$
are _compatible_ if $\mathcal{R}\subseteq{\succ}$.
An $\mathcal{F}$-_algebra_ $\mathcal{A}$ consists of a carrier set $A$ and a
collection of interpretations $f_{\mathcal{A}}$ for each function symbol in
$\mathcal{F}$. By $[\alpha]_{\mathcal{A}}(\cdot)$ we denote the usual
evaluation function of $\mathcal{A}$ according to an assignment $\alpha$ which
maps variables to values in $A$. A _monotone $\mathcal{F}$-algebra_ is a pair
$(\mathcal{A},\succ)$ where $\mathcal{A}$ is an $\mathcal{F}$-algebra and
$\succ$ is a proper order such that for every function symbol
$f\in\mathcal{F}$, $f_{\mathcal{A}}$ is strictly monotone in all coordinates
with respect to $\succ$. A _weakly monotone $\mathcal{F}$-algebra_
$(\mathcal{A},\succcurlyeq)$ is defined similarly, but for every function
symbol $f\in\mathcal{F}$, it suffices that $f_{\mathcal{A}}$ is weakly
monotone in all coordinates (with respect to the quasi-order $\succcurlyeq$).
A monotone $\mathcal{F}$-algebra $(\mathcal{A},\succ)$ is called _well-
founded_ if $\succ$ is well-founded. We write _WMA_ instead of well-founded
monotone algebra.
Any (weakly) monotone $\mathcal{F}$-algebra $(\mathcal{A},\mathrel{R})$
induces a binary relation $\mathrel{R}_{\mathcal{A}}$ on terms: define
$s\mathrel{R}_{\mathcal{A}}t$ if
$[\alpha]_{\mathcal{A}}(s)\mathrel{R}[\alpha]_{\mathcal{A}}(t)$ for all
assignments $\alpha$. Clearly if $\mathrel{R}$ is a proper order (quasi-
order), then $\mathrel{R}_{\mathcal{A}}$ is a proper order (quasi-order) on
terms and if $\mathrel{R}$ is a well-founded, then $\mathrel{R}_{\mathcal{A}}$
is well-founded on terms. We say $\mathcal{A}$ is _compatible_ with a TRS
$\mathcal{R}$ if ${\mathcal{R}}\subseteq{\mathrel{R}_{\mathcal{A}}}$. Let
$\mathrel{{\succcurlyeq}_{\mathcal{A}}}$ denote the quasi-order induced by a
weakly monotone algebra $(\mathcal{A},\succcurlyeq)$, then
$\mathrel{=_{\mathcal{A}}}$ denotes the equivalence (on terms) induced by
$\mathrel{{\succcurlyeq}_{\mathcal{A}}}$. Let $\mu$ denote a replacement map.
Then we call a well-founded algebra $(\mathcal{A},\succ)$ _$\mu$ -monotone_ if
for every function symbol $f\in\mathcal{F}$, $f_{\mathcal{A}}$ is strictly
monotone _on_ $\mu(f)$, i.e., $f_{\mathcal{A}}$ is strictly monotone with
respect to every argument position in $\mu(f)$. Similarly a (strict) relation
$\mathrel{R}$ is called $\mu$-monotone if (strictly) monotone on $\mu(f)$ for
all $f\in\mathcal{F}$. Let $\mathcal{R}$ be a TRS compatible with a
$\mu$-monotone relation $\mathrel{R}$. Then clearly any $\mu$-step
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${\mu}$}}}}t$
implies $s\mathrel{R}t$.
We recall the concept of _matrix interpretations_ on natural numbers (see [12]
but compare also [13]). Let $\mathcal{F}$ denote a signature. We fix a
dimension $d\in\mathbb{N}$ and use the set $\mathbb{N}^{d}$ as the carrier of
an algebra $\mathcal{A}$, together with the following extension of the natural
order $>$ on $\mathbb{N}$:
$(x_{1},x_{2},\ldots,x_{d})>(y_{1},y_{2},\ldots,y_{d})\mathrel{:\Longleftrightarrow}x_{1}>y_{1}\wedge
x_{2}\geqslant y_{2}\wedge\ldots\wedge x_{d}\geqslant y_{d}\hbox
to0.0pt{$\;$.\hss}$
Let $\mu$ be a replacement map. For each $n$-ary function symbol $f$, we
choose as an interpretation a linear function of the following shape:
$f_{\mathcal{A}}\colon(\vec{v}_{1},\ldots,\vec{v}_{n})\mapsto
F_{1}\vec{v}_{1}+\cdots+F_{n}\vec{v}_{n}+\vec{f}\hbox to0.0pt{$\;$,\hss}$
where $\vec{v}_{1},\ldots,\vec{v}_{n}$ are (column) vectors of variables,
$F_{1},\ldots,F_{n}$ are matrices (each of size $d\times d$), and $\vec{f}$ is
a vector over $\mathbb{N}$. Moreover, suppose for any $i\in\mu(f)$ the top
left entry $(F_{i})_{1,1}$ is positive. Then it is easy to see that the
algebra $\mathcal{A}$ forms a $\mu$-monotone WMA. Let $\mathcal{A}$ be a
matrix interpretation, let $\alpha_{0}$ denotes the assignment mapping any
variable to $\vec{0}$, i.e., $\alpha_{0}(x)=\vec{0}$ for all
$x\in\mathcal{V}$, and let $t$ be a term. In the following we write $[t]$,
$[t]_{j}$ as an abbreviation for $[\alpha_{0}]_{\mathcal{A}}(t)$, or
$\left([\alpha_{0}]_{\mathcal{A}}(t)\right)_{j}$ ($1\leqslant j\leqslant d$),
respectively, if the algebra $\mathcal{A}$ is clear from the context.
## 3 Runtime Complexity
In this section we formalise runtime complexity and then define a subclass of
matrix interpretations that give polynomial upper-bounds.
The _derivation height_ of a term $s$ with respect to a well-founded, finitely
branching relation $\to$ is defined as:
${\mathsf{dh}}(s,\to)=\max\\{n\mid\exists t\;s\to^{n}t\\}$. Let $\mathcal{R}$
be a TRS and $T$ be a set of terms. The _complexity function with respect to a
relation $\to$ on $T$_ is defined as follows:
$\operatorname{\mathsf{comp}}(n,T,\mathrel{\to})=\max\\{{\mathsf{dh}}(t,\mathrel{\to})\mid\text{$t\in
T$ and $\lvert t\rvert\leqslant n$}\\}\hbox to0.0pt{$\;$.\hss}$
In particular we are interested in the (innermost) complexity with respect to
$\mathrel{\mathrel{\to}_{\mathcal{R}}}$
($\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}}$)
on the set $\operatorname{\mathcal{T}_{\mathsf{b}}}$ of all _basic_ terms.
###### Definition 3.1.
Let $\mathcal{R}$ be a TRS. We define the _runtime complexity function_
$\mathsf{rc}_{\mathcal{R}}(n)$, the _innermost runtime complexity function_
$\mathsf{rc}_{\mathcal{R}}^{\mathrm{i}}(n)$, and the _derivational complexity
function_ $\mathsf{dc}_{\mathcal{R}}(n)$ as
$\operatorname{\mathsf{comp}}(n,{\operatorname{\mathcal{T}_{\mathsf{b}}}},\mathrel{\mathrel{\to}_{\mathcal{R}}})$,
$\operatorname{\mathsf{comp}}(n,{\operatorname{\mathcal{T}_{\mathsf{b}}}},\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}})$,
and
$\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V}),\mathrel{\mathrel{\to}_{\mathcal{R}}})$,
respectively.
Note that the above complexity functions need not be defined, as the rewrite
relation $\mathrel{\mathrel{\to}_{\mathcal{R}}}$ is not always well-founded
_and_ finitely branching. We sometimes say the (innermost) runtime complexity
of $\mathcal{R}$ is _linear_ , _quadratic_ , or _polynomial_ if there exists a
(linear, quadratic) polynomial $p(n)$ such that
$\mathsf{rc}_{\mathcal{R}}^{(\mathrm{i})}(n)\leqslant p(n)$ for sufficiently
large $n$. The (innermost) runtime complexity of $\mathcal{R}$ is called
_exponential_ if there exist constants $c$, $d$ with $c,d\geqslant 2$ such
that $c^{n}\leqslant\mathsf{rc}_{\mathcal{R}}^{(\mathrm{i})}(n)\leqslant
d^{n}$ for sufficiently large $n$.
The next example illustrates a difference between derivational complexity and
runtime complexity.
###### Example 3.2.
Consider the following TRS $\mathcal{R}_{\mathsf{div}}$222This is Example 3.1
in Arts and Giesl’s collection of TRSs [14].
$\displaystyle 1\colon$ $\displaystyle x-\mathsf{0}$ $\displaystyle\to x$
$\displaystyle\qquad 3\colon$ $\displaystyle\mathsf{0}\div\mathsf{s}(y)$
$\displaystyle\to\mathsf{0}$ $\displaystyle 2\colon$
$\displaystyle\mathsf{s}(x)-\mathsf{s}(y)$ $\displaystyle\to x-y$
$\displaystyle\qquad 4\colon$ $\displaystyle\mathsf{s}(x)\div\mathsf{s}(y)$
$\displaystyle\to\mathsf{s}((x-y)\div\mathsf{s}(y))\hbox to0.0pt{$\;$.\hss}$
Although the functions _computed_ by $\mathcal{R}_{\mathsf{div}}$ are
obviously feasible this is not reflected in the derivational complexity of
$\mathcal{R}_{\mathsf{div}}$. Consider rule 4, which we abbreviate as $C[x]\to
D[x,x]$. Since the maximal derivation height starting with $C^{n}[x]$ equals
$2^{n-1}$ for all $n>0$, $\mathcal{R}_{\mathsf{div}}$ admits (at least)
exponential derivational complexity. In general any duplicating TRS admits (at
least) exponential derivational complexity.
In general it is not possible to bound $\mathsf{dc}_{\mathcal{R}}$
polynomially in $\mathsf{rc}_{\mathcal{R}}$, as witnessed by Example 3.2 and
the observation that the runtime complexity of $\mathcal{R}$ is linear (see
Example 4.10, below). We will use Example 3.2 as our running example.
Below we define classes of orders whose compatibility with a TRS $\mathcal{R}$
bounds its runtime complexity from the above. Note that
${\mathsf{dh}}(t,{\succ})$ is undefined, if the relation $\succ$ is not well-
founded or not finitely branching. In fact compatibility of a constructor TRS
with the polynomial path order $>_{\mathsf{pop*}}$ ([15]) induces polynomial
innermost runtime complexity, whereas
$\mathsf{f}(x)>_{\mathsf{pop*}}\cdots>_{\mathsf{pop*}}\cdots>_{\mathsf{pop*}}\mathsf{g}^{2}(x)>_{\mathsf{pop*}}\mathsf{g}(x)>_{\mathsf{pop*}}x$
holds when precedence $\mathsf{f}>\mathsf{g}$ is used. Hence
${\mathsf{dh}}(t,{>_{\mathsf{pop*}}})$ is undefined, while the order
$>_{\mathsf{pop*}}$ can be employed in complexity analysis.
###### Definition 3.3.
Let $\mathrel{R}$ be a binary relation over terms, let $\succ$ be a proper
order on terms, and let $\operatorname{\mathsf{G}}$ denote a mapping
associating a term with a natural number. Then $\succ$ is
_$\operatorname{\mathsf{G}}$ -collapsible on $\mathrel{R}$_ if
$\operatorname{\mathsf{G}}(s)>\operatorname{\mathsf{G}}(t)$, whenever
${s}\mathrel{R}{t}$ and ${s}\succ{t}$ holds. An order $\succ$ is _collapsible
(on $\mathrel{R}$)_, if there is a mapping $\operatorname{\mathsf{G}}$ such
that $\succ$ is $\operatorname{\mathsf{G}}$-collapsible (on $\mathrel{R}$).
###### Lemma 3.4.
Let $\mathrel{R}$ be a finitely branching and well-founded relation. Further,
let $\succ$ be a $\operatorname{\mathsf{G}}$-collapsible order with
${\mathrel{R}}\subseteq{\succ}$. Then
${\mathsf{dh}}(t,{\mathrel{R}})\leqslant\operatorname{\mathsf{G}}(t)$ holds
for all terms $t$.
The alert reader will have noticed that any proper order $\succ$ is
collapsible on a finitely branching and well-founded relation $\mathrel{R}$:
simply set
$\operatorname{\mathsf{G}}(t)\mathrel{:=}{\mathsf{dh}}(t,{\mathrel{R}})$.
However, this observation is of limited use if we wish to bound the derivation
height of $t$ in independence of $\mathrel{R}$.
If a TRS $\mathcal{R}$ and a $\mu$-monotone matrix interpretation
$\mathcal{A}$ are compatible, $\operatorname{\mathsf{G}}(t)$ can be given by
$[t]_{1}$. In order to estimate derivational or runtime complexity, one needs
to associate $[t]_{1}$ to $|t|$. For this sake we define degrees of matrix
interpretations.
###### Definition 3.5.
A matrix interpretation is of _(basic) degree_ $d$ if there is a constant $c$
such that $[t]_{i}\leqslant c\cdot|t|^{d}$ for all (basic) terms $t$ and $i$,
respectively.
An _upper triangular complexity matrix_ is a matrix $M$ in
$\mathbb{N}^{d\times d}$ such that we have $M_{j,k}=0$ for all $1\leqslant
k<j\leqslant d$, and $M_{j,j}\leqslant 1$ for all $1\leqslant j\leqslant d$.
We say that a WMA $\mathcal{A}$ is a _triangular matrix interpretation_ (_TMI_
for short) if $\mathcal{A}$ is a matrix interpretation (over $\mathbb{N}$) and
all matrices employed are of upper triangular complexity form. It is easy to
define triangular matrix interpretations, such that an algebra $\mathcal{A}$
based on such an interpretation, forms a well-founded _weakly_ monotone
algebra. To simplify notation we will also refer to $\mathcal{A}$ as a TMI, if
no confusion can arise from this. A TMI $\mathcal{A}$ of dimension 1, that is
a linear polynomial, is called a _strongly linear interpretation_ (_SLI_ for
short) if all interpretation functions $f_{\mathcal{A}}$ are strongly linear.
Here a polynomial $P(x_{1},\dots,x_{n})$ is strong linear if
$P(x_{1},\dots,x_{n})=x_{1}+\cdots+x_{n}+c$.
###### Lemma 3.6.
Let $\mathcal{A}$ be a TMI and let $M$ denote the component-wise maximum of
all matrices occurring in $\mathcal{A}$. Further, let $d$ denote the number of
ones occurring along the diagonal of $M$. Then for all $1\leqslant
i,j\leqslant d$ we have $(M^{n})_{i,j}=\operatorname{\mathsf{O}}(n^{d-1})$.
###### Proof.
The lemma is a direct consequence of Lemma 4 in [16] together with the
observation that for any triangular complexity matrix, the diagonal entries
denote the multiset of eigenvalues. ∎
###### Lemma 3.7.
Let $\mathcal{A}$ and $d$ be defined as in Lemma 3.6. Then $\mathcal{A}$ is of
degree $d$.
###### Proof.
For any (triangular) matrix interpretation $\mathcal{A}$, there exist vectors
$\vec{v}_{i}$ and a vector $\vec{w}$ such that the evaluation $[t]$ of $t$ can
be written as follows:
$[t]=\sum_{i=1}^{\ell}\vec{v}_{i}+\vec{w}\hbox to0.0pt{$\;$,\hss}$
where each vector $\vec{v}_{i}$ is the product of those matrices employed in
the interpretation of function symbols in $\mathcal{A}$ and a vector
representing the constant part of a function interpretation. It is not
difficult to see that there is a one-to-one correspondence between the number
of vectors $\vec{v}_{1},\dots,\vec{v}_{\ell}$ and the number of subterms of
$t$ and thus $\ell=\lvert t\rvert$. Moreover for each $\vec{v}_{i}$ the number
of products is less than the depth of $t$ and thus bounded by $\lvert
t\rvert$. In addition, due to Lemma 3.6 the entries of the vectors
$\vec{v}_{i}$ and $\vec{w}$ are bounded by a polynomial of degree at most
$d-1$. Thus for all $1\leqslant j\leqslant d$, there exists $k\leqslant d$
such that $([t])_{j}=\operatorname{\mathsf{O}}(\lvert t\rvert^{k})$. ∎
###### Theorem 3.8.
[16, Theorem 9],[17] Let $\mathcal{A}$ and $d$ be defined as in Lemma 3.6.
Then, $\mathrel{{\succ}_{\mathcal{A}}}$ is
$\operatorname{\mathsf{O}}(n^{d})$-collapsible.
###### Proof.
The theorem is a direct consequence of Lemmas 3.6 and 3.7. ∎
In order to cope with runtime complexity, a similar idea to restricted
polynomial interpretations (see [18]) can be integrated to triangle matrix
interpretations. We call $\mathcal{A}$ a _restricted matrix interpretation_
(_RMI_ for short) if $\mathcal{A}$ is a matrix interpretation, but for each
constructor symbol $f\in\mathcal{F}$, the interpretation $f_{\mathcal{A}}$ of
$f$ employs upper triangular complexity matrices, only. The next theorem is a
direct consequence of the definitions in conjunction with Lemma 3.7.
###### Theorem 3.9.
Let $\mathcal{A}$ be an RMI and let $t$ be a basic term. Further, let $M$
denote the component-wise maximum of all matrices used for the interpretation
of constructor symbol, and let $d$ denote the number of ones occurring along
the diagonal of $M$. Then $\mathcal{A}$ is of basic degree $d$. Furthermore,
if $M$ is the unit matrix then $\mathcal{A}$ is of basic degree $1$.
## 4 Usable Replacement Maps
Unfortunately, there is no RMI compatible with the TRS of our running example.
The reason is that the monotonicity requirement of TMI is too severe for
complexity analysis. Inspired by the idea of Fernández [19], we show how
context-sensitive rewriting is used in complexity analysis. Here we briefly
explain our idea. Let $\mathbf{n}$ denote the numeral $s^{n}(\mathsf{0})$.
Consider the derivation from $\mathbf{4}\div\mathbf{2}$:
$\underline{\mathbf{4}\div\mathbf{2}}\to\mathsf{s}(\underline{(\mathbf{3}-\mathbf{1})}\div\mathbf{2})\to\mathsf{s}((\underline{\mathbf{2}-\mathsf{0}})\div\mathbf{2})\to\mathsf{s}(\underline{\mathbf{2}\div\mathbf{2}})\to\cdots$
where redexes are underlined. Observe that e.g. any second argument of $\div$
is never rewritten. More precisely, any derivation from a basic term consists
of only $\mu$-steps with the replacement map $\mu$:
$\mu(\mathsf{s})=\mu({\div})=\\{1\\}$ and $\mu({-})=\varnothing$.
We present a simple method based on a variant of $\mathsf{ICAP}$ in [20] to
estimate a suitable replacement map. Let $\mu$ be a replacement map. Clearly
the function $\mu$ is representable as set of ordered pairs $(f,i)$. Below we
often confuse the notation of $\mu$ as a function or as a set. Recall that
$\mathcal{P}\mathsf{os}_{\mu}(t)$ denotes the set of _$\mu$ -replacing
positions_ in $t$ and
$\overline{\mathcal{P}\mathsf{os}}_{\mu}(t)=\mathcal{P}\mathsf{os}(t)\setminus\mathcal{P}\mathsf{os}_{\mu}(t)$.
Further, a term $t$ is a _$\mu$ -replacing term_ with respect to a TRS
$\mathcal{R}$ if ${{{t}\\!\\!\mid_{p}}}\not\in{\mathsf{NF}(\mathcal{R})}$
implies that $p\in Pos_{\mu}(t)$. The set of all $\mu$-replacing terms is
denoted by $\mathcal{T}(\mu)$. In the following $\mathcal{R}$ will always
denote a TRS.
###### Definition 4.1.
Let $\mathcal{R}$ be a TRS and let $\mu$ be a replacement map. We defined the
operator $\Upsilon^{\mathcal{R}}$ as follows:
$\Upsilon^{\mathcal{R}}(\mu)\mathrel{:=}\\{(f,i)\mid\text{$l\to
C[f({r_{1}},\dots,{r_{n}})]\in\mathcal{R}$ and
$\mathsf{CAP}_{\mu}^{l}(r_{i})\neq r_{i}$}\\}\hbox to0.0pt{$\;$.\hss}$
Here $\mathsf{CAP}_{\mu}^{s}(t)$ is inductively defined on $t$ as follows:
$\mathsf{CAP}_{\mu}^{s}(t)=\begin{cases}t&\text{$t={{s}\\!\\!\mid_{p}}$ for
some $p\in\overline{\mathcal{P}\mathsf{os}}_{\mu}(s)$}\hbox
to0.0pt{$\;$,\hss}\\\ u&\text{if $t=f({t_{1}},\dots,{t_{n}})$ and $u$ and $l$
unify for no $l\to r\in\mathcal{R}$}\hbox to0.0pt{$\;$,\hss}\\\
y&\text{otherwise}\hbox to0.0pt{$\;$,\hss}\end{cases}$
where,
$u=f(\mathsf{CAP}_{\mu}^{s}(t_{1}),\ldots,\mathsf{CAP}_{\mu}^{s}(t_{n}))$, $y$
is a fresh variable, and
$\mathcal{V}\mathsf{ar}(l)\cap\mathcal{V}\mathsf{ar}(u)=\varnothing$ is
assumed.
We define the _innermost usable replacement map_
${\mu}^{\mathcal{R}}_{\mathsf{i}}$ as follows
${\mu}^{\mathcal{R}}_{\mathsf{i}}\mathrel{:=}\Upsilon^{\mathcal{R}}(\varnothing)$
and let the _usable replacement map_ ${\mu}^{\mathcal{R}}_{\mathsf{f}}$ denote
the least fixed point of $\Upsilon^{\mathcal{R}}$. The existence of
$\Upsilon^{\mathcal{R}}$ follows from the monotonicity of
$\Upsilon^{\mathcal{R}}$. If $\mathcal{R}$ is clear from context, we simple
write ${\mu_{\mathsf{i}}}$, ${\mu_{\mathsf{f}}}$, and $\Upsilon$,
respectively. Usable replacement maps satisfy a desired property for runtime
complexity analysis. In order to see it several preliminary lemmas are
necessary.
First we take a look at $\mathsf{CAP}_{\mu}^{s}(t)$. Suppose
$s\in\mathcal{T}(\mu)$: observe that the function $\mathsf{CAP}_{\mu}^{s}(t)$
replaces a subterm $u$ of $t$ by a fresh variable if $u\sigma$ is a redex for
some $s\sigma\in\mathcal{T}(\mu)$. This is exemplified below.
###### Example 4.2.
Consider the TRS $\mathcal{R}_{\mathsf{div}}$. Let $l\to r$ be rule 4, namely,
$l=\mathsf{s}(x)\div\mathsf{s}(y)$ and $r=\mathsf{s}((x-y)\div\mathsf{s}(y))$.
Suppose $\mu(f)=\varnothing$ for all functions $f$ and let $w$ and $z$ be
fresh variables. The next table summarises $\mathsf{CAP}_{\mu}^{l}(t)$ for
each proper subterm $t$ in $r$. To see the computation process, we also
indicate the term $u$ in Definition 4.1.
$t$ | $x$ | $y$ | $x-y$ | $\mathsf{s}(y)$ | $(x-y)\div\mathsf{s}(y)$
---|---|---|---|---|---
$u$ | – | – | $x-y$ | $\mathsf{s}(y)$ | $w\div\mathsf{s}(y)$
$\mathsf{CAP}_{\mu}^{l}(t)$ | $x$ | $y$ | $w$ | $\mathsf{s}(y)$ | $z$
By underlining proper subterms $t$ in $r$ such that
$\mathsf{CAP}_{\mu}^{l}(t)\neq t$, we have
$\mathsf{s}(\underline{\underline{(x-y)}\div\mathsf{s}(y)})$
which indicates $(\mathsf{s},1),({\div},1)\in\Upsilon(\mu)$.
The next lemma states a role of $\mathsf{CAP}_{\mu}^{s}(t)$.
###### Lemma 4.3.
If $s\sigma\in\mathcal{T}(\mu)$ and $\mathsf{CAP}_{\mu}^{s}(t)=t$ then
$t\sigma\in\mathsf{NF}(\mathcal{R})$.
###### Proof.
We use induction on $t$. Suppose $s\sigma\in\mathcal{T}(\mu)$ and
$\mathsf{CAP}_{\mu}^{s}(t)=t$. If $t={{s}\\!\\!\mid_{p}}$ for some
$p\in\overline{\mathcal{P}\mathsf{os}}_{\mu}(s)$ then
$t\sigma={{(s\sigma)}\\!\\!\mid_{p}}\in\mathsf{NF}$ follows by definition of
$\mathcal{T}(\mu)$.
We can assume that $t=f({t_{1}},\dots,{t_{n}})$. Assume otherwise that
$t=x\in\mathcal{V}$, then $\mathsf{CAP}_{\mu}^{s}(x)=x$ entails that $x\sigma$
occurs at a non-$\mu$-replacing position in $s\sigma$. Hence
$x\sigma\in\mathsf{NF}$ follows from $s\sigma\in\mathcal{T}(\mu)$. Moreover,
by assumption we have:
1. 1)
$\mathsf{CAP}_{\mu}^{s}(t_{i})=t_{i}$ for each $i$, and
2. 2)
there is no rule $l\to r\in\mathcal{R}$ such that $t$ and $l$ unify.
Due to 2) $l\sigma$ is not reducible at the root, and the induction hypothesis
yields $t_{i}\sigma\in\mathsf{NF}$ because of 1). Therefore, we obtain
$t\sigma\in\mathsf{NF}$. ∎
For a smooth inductive proof of our key lemma we prepare a characterisation of
the set of $\mu$-replacing terms $\mathcal{T}(\mu)$.
###### Definition 4.4.
The set
$\\{(f,i)\mid\text{$f({t_{1}},\dots,{t_{n}})\mathrel{{\trianglelefteq}}t$ and
$t_{i}\not\in\mathsf{NF}(\mathcal{R})$}\\}$ is denoted by $\upsilon(t)$.
###### Lemma 4.5.
$\mathcal{T}(\mu)=\\{t\mid\upsilon(t)\subseteq\mu\\}$.
###### Proof.
The inclusion from left to right essentially follows from the definitions. Let
$t\in\mathcal{T}(\mu)$ and let $(f,i)\in\upsilon(t)$. We show $(f,i)\in\mu$.
By Definition 4.4 there is a position $p\in\mathcal{P}\mathsf{os}(t)$ with
${{t}\\!\\!\mid_{p}}=f({t_{1}},\dots,{t_{n}})$ and
${{{t}\\!\\!\mid_{pi}}}\not\in{\mathsf{NF}}$. Thus
$pi\in\mathcal{P}\mathsf{os}_{\mu}(t)$ and
$i\in\mathcal{P}\mathsf{os}_{\mu}({{t}\\!\\!\mid_{p}})$. Hence $(f,i)\in\mu$
is concluded.
Next we consider the reverse direction
${\\{t\mid\upsilon(t)\subseteq\mu\\}}\subseteq{\mathcal{T}(\mu)}$. Let $t$ be
a minimal term such that $\upsilon(t)\subseteq\mu$ and
$t\not\in\mathcal{T}(\mu)$. One can write $t=f({t_{1}},\dots,{t_{n}})$. Then,
there exists a position $p\in\overline{\mathcal{P}\mathsf{os}}_{\mu}(t)$ such
that ${{t}\\!\\!\mid_{p}}\not\in\mathsf{NF}$. Because
$\epsilon\not\in\overline{\mathcal{P}\mathsf{os}}_{\mu}(t)$ holds in general,
$p$ is of the form $iq$ with $i\in\mathbb{N}$. As
$iq\in\overline{\mathcal{P}\mathsf{os}}_{\mu}(t)$ one of $(f,i)\not\in\mu$ or
$q\in\overline{\mathcal{P}\mathsf{os}}_{\mu}({{t}\\!\\!\mid_{i}})$ must hold.
As $t$ is minimal and ${{{t}\\!\\!\mid_{iq}}}\not\in{\mathsf{NF}}$ implies
that ${{{t}\\!\\!\mid_{i}}}\not\in{\mathsf{NF}}$, we have $(f,i)\not\in\mu$.
However, by Definition 4.4, $(f,i)\in\upsilon(t)\subseteq\mu$. Contradiction.
∎
The next lemma about the operator $\Upsilon$ is a key for the main theorem.
Note that every subterm of a $\mu$-replacing term is a $\mu$-replacing term.
###### Lemma 4.6.
If $l\to r\in\mathcal{R}$ and $l\sigma\in\mathcal{T}(\mu)$ then
$r\sigma\in\mathcal{T}(\mu\cup\Upsilon(\mu))$.
###### Proof.
Let $l\to r\in\mathcal{R}$ and suppose $l\sigma\in\mathcal{T}(\mu)$. By Lemma
4.5 we have
$\mathcal{T}(\mu)=\\{t\mid\upsilon(t)\subseteq\mu\\}\qquad\mathcal{T}(\mu\cup\Upsilon(\mu))=\\{t\mid{\upsilon(t)}\subseteq{\mu\cup\Upsilon(\mu)}\\}\hbox
to0.0pt{$\;$.\hss}$
Hence it is sufficient to show
$\upsilon(r\sigma)\subseteq\mu\cup\Upsilon(\mu)$. Let
$(f,i)\in\upsilon(r\sigma)$. There is $p\in\mathcal{P}\mathsf{os}(r\sigma)$
with ${{{r\sigma}\\!\\!\mid_{p}}}={f({t_{1}},\dots,{t_{n}})}$ and
$t_{i}\not\in\mathsf{NF}$. If $p$ is below some variable position of $r$,
${{{r\sigma}\\!\\!\mid_{p}}}$ is a subterm of $l\sigma$, and thus
$\upsilon({{r\sigma}\\!\\!\mid_{p}})\subseteq\upsilon(l\sigma)\subseteq\mu$.
Otherwise, $p$ is a non-variable position of $r$. We may write
${{r}\\!\\!\mid_{p}}=f({r_{1}},\dots,{r_{n}})$ and
$r_{i}\sigma=t_{i}\not\in\mathsf{NF}$. Due to Lemma 4.3 we obtain
$\mathsf{CAP}_{\mu}^{l}(r_{i})\neq r_{i}$. Therefore, $(f,i)\in\Upsilon(\mu)$.
∎
Remark that if $s,t\in\mathcal{T}(\mu)$ and
$p\in\mathcal{P}\mathsf{os}_{\mu}(s)$ then $s[t]_{p}\in\mathcal{T}(\mu)$.
###### Lemma 4.7.
The following implications hold.
1. 1)
If $s\in\mathcal{T}({\mu_{\mathsf{i}}})$ and
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}t$
then $t\in\mathcal{T}({\mu_{\mathsf{i}}})$.
2. 2)
If $s\in\mathcal{T}({\mu_{\mathsf{f}}})$ and $s\to t$ then
$t\in\mathcal{T}({\mu_{\mathsf{f}}})$.
###### Proof.
We show property 1). Suppose $s\in\mathcal{T}({\mu_{\mathsf{i}}})$ and
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}t$
is a rewrite step at $p$. Due to the definition of innermost rewriting, we
have ${{s}\\!\\!\mid_{p}}\in\mathcal{T}(\varnothing)$. Hence,
${{t}\\!\\!\mid_{p}}\in\mathcal{T}({\mu_{\mathsf{i}}})$ is obtained by Lemma
4.6. Because $s\in\mathcal{T}({\mu_{\mathsf{i}}})$ we have
$p\in\mathcal{P}\mathsf{os}_{\mu_{\mathsf{i}}}(s)$. Hence due to
${{t}\\!\\!\mid_{p}}\in\mathcal{T}({\mu_{\mathsf{i}}})$ we conclude
$t=s[{{t}\\!\\!\mid_{p}}]_{p}\in\mathcal{T}({\mu_{\mathsf{i}}})$ due to the
above remark. The proof of 2) proceeds along the same pattern and is left to
the reader. ∎
We arrive at the main result of this section.
###### Theorem 4.8.
Let $\mathcal{R}$ be a TRS, and let $\operatorname{\to^{\ast}}(L)$ denote the
descendants of the set of terms $L$. Then
$\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}^{\ast}_{\mathcal{R}}}(\mathcal{T}(\varnothing))\subseteq\mathcal{T}({\mu_{\mathsf{i}}})$
and
$\mathrel{\mathrel{\to}_{\mathcal{R}}^{\ast}}(\mathcal{T}(\varnothing))\subseteq\mathcal{T}({\mu_{\mathsf{f}}})$.
###### Proof.
Recall that $\operatorname{\to^{\ast}}(L)\mathrel{:=}\\{t\mid\text{$\exists
s\in L$ such that $s\to^{\ast}t$}\\}$. We focus on the second part of the
theorem, where we have to prove that $t\in\mathcal{T}({\mu_{\mathsf{f}}})$,
whenever there exists $s\in\mathcal{T}(\varnothing)$ such that
$s\mathrel{\mathrel{\to}_{\mathcal{R}}^{\ast}}t$. As
$\mathcal{T}(\varnothing)\subseteq\mathcal{T}({\mu_{\mathsf{f}}})$ this
follows directly from Lemma 4.7. ∎
Note that $\mathcal{T}(\varnothing)$ is the set of all argument normalised
terms. Therefore,
${\operatorname{\mathcal{T}_{\mathsf{b}}}}\subseteq{\mathcal{T}(\varnothing)}$.
The following corollary to Theorem 4.8 is immediate.
###### Corollary 4.9.
Let $\mathcal{R}$ be a TRS and let
$\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}$,
$\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{f}}}}$}}}}$
denote the ${\mu_{\mathsf{i}}}$-step and ${\mu_{\mathsf{f}}}$-step relation,
respectively. Then for all terminating terms
$t\in\operatorname{\mathcal{T}_{\mathsf{b}}}$ we have
${\mathsf{dh}}(t,{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}}})\leqslant{\mathsf{dh}}(t,{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}})$
and
${\mathsf{dh}}(t,{\mathrel{\mathrel{\to}_{\mathcal{R}}}})\leqslant{\mathsf{dh}}(t,{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{f}}}}$}}}}})$.
An advantage of the use of context-sensitive rewriting is that the
compatibility requirement of monotone algebra in termination or complexity
analysis is relaxed to $\mu$-monotone algebra. We illustrate its use in the
next example.
###### Example 4.10.
Recall the TRS $\mathcal{R}_{\mathsf{div}}$ given in Example 3.2 above. The
usable argument positions are as follows:
${\mu_{\mathsf{i}}}(\mathsf{-})=\varnothing\quad{\mu_{\mathsf{i}}}(\mathsf{s})={\mu_{\mathsf{i}}}(\mathsf{\div})=\\{1\\}\qquad{\mu_{\mathsf{f}}}(\mathsf{s})={\mu_{\mathsf{f}}}(\mathsf{-})={\mu_{\mathsf{f}}}(\mathsf{\div})=\\{1\\}\hbox
to0.0pt{$\;$.\hss}$
Consider the $1$-dimensional RMI $\mathcal{A}$ (i.e., linear polynomial
interpretations) with
$\displaystyle\mathsf{0}_{\mathcal{A}}$ $\displaystyle=1$
$\displaystyle\mathsf{s}_{\mathcal{A}}(x)$ $\displaystyle=x+2$
$\displaystyle{-_{\mathcal{A}}}(x,y)$ $\displaystyle=x+1$
$\displaystyle{\div_{\mathcal{A}}}(x,y)$ $\displaystyle=3x\hbox
to0.0pt{$\;$.\hss}$
which is strictly ${\mu_{\mathsf{i}}}$-monotone and
${\mu_{\mathsf{f}}}$-monotone. The rules in $\mathcal{R}_{\mathsf{div}}$ are
interpreted and ordered as follows.
$\displaystyle 1\colon\quad$ $\displaystyle x+1$ $\displaystyle>x$
$\displaystyle 3\colon\quad$ $\displaystyle 3$ $\displaystyle>1$
$\displaystyle 2\colon\quad$ $\displaystyle x+3$ $\displaystyle>x+2$
$\displaystyle 4\colon\quad$ $\displaystyle 3x+6$ $\displaystyle>3x+5\hbox
to0.0pt{$\;$.\hss}$
Therefore, $\mathcal{R}_{\mathsf{div}}\subseteq{>_{\mathcal{A}}}$ holds. By an
application of Theorem 3.9 we conclude that the (innermost) runtime complexity
is _linear_ , which is optimal.
We cast the observations in the example into another corollary to Theorem 4.8.
###### Corollary 4.11.
Let $\mathcal{R}$ be a TRS and let $\mathcal{A}$ be a $d$-degree
${\mu_{\mathsf{i}}}$-monotone (or ${\mu_{\mathsf{f}}}$-monotone) RMI
compatible with $\mathcal{R}$. Then the (innermost) runtime complexity
function $\mathsf{rc}_{\mathcal{R}}^{(\mathrm{i})}$ with respect to
$\mathcal{R}$ is bounded by a $d$-degree polynomial.
###### Proof.
It suffices to consider the case for full rewriting. Let $s$, $t$ be terms
such that $s\mathrel{\mathrel{\to}_{\mathcal{R}}}t$. By the theorem, we have
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{f}}}}$}}}}t$.
Furthermore, by assumption
${\mathcal{R}}\subseteq{\mathrel{{\succ}_{\mathcal{A}}}}$ and for any
$f\in\mathcal{F}$, $f_{\mathcal{A}}$ is strictly monotone on all
${\mu_{\mathsf{f}}}(f)$. Thus $s\mathrel{{\succ}_{\mathcal{A}}}t$ follows.
Finally, the corollary follows by application of Theorem 3.9. ∎
We link Theorem 4.8 to related work by Fernández [19]. In [19] it is shown how
context-sensitive rewriting is used for proving innermost termination.
###### Proposition 4.12 ([19]).
A TRS $\mathcal{R}$ is innermost terminating if
$\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}$
is terminating.
###### Proof.
We show the contraposition. If $\mathcal{R}$ is not innermost terminating,
there is an infinite sequence
$t_{0}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}t_{1}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}t_{2}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}\cdots$,
where $t_{0}\in\mathcal{T}(\varnothing)$. From Theorem 4.8 and Lemma 4.7 we
obtain
$t_{0}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}t_{1}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}t_{2}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}\cdots$.
Hence,
$\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny${{\mu_{\mathsf{i}}}}$}}}}$
is not terminating. ∎
One might think that a similar claim holds for full termination if one uses
${\mu_{\mathsf{f}}}$. The next examples clarifies that this is not the case.
###### Example 4.13.
Consider the famous Toyama’s example $\mathcal{R}$
$\displaystyle\mathsf{f}(\mathsf{a},\mathsf{b},x)$
$\displaystyle\to\mathsf{f}(x,x,x)$ $\displaystyle\mathsf{g}(x,y)$
$\displaystyle\to x$ $\displaystyle\mathsf{g}(x,y)$ $\displaystyle\to y\hbox
to0.0pt{$\;$.\hss}$
The replacement map ${\mu_{\mathsf{f}}}$ is empty. Thus, the algebra
$\mathcal{A}$ over $\mathbb{N}$
$\displaystyle\mathsf{f}_{\mathcal{A}}(x,y,z)$ $\displaystyle=\max\\{x-y,0\\}$
$\displaystyle\mathsf{g}_{\mathcal{A}}(x,y)$ $\displaystyle=x+y+1$
$\displaystyle\mathsf{a}_{\mathcal{A}}$ $\displaystyle=1$
$\displaystyle\mathsf{b}_{\mathcal{A}}$ $\displaystyle=0\hbox
to0.0pt{$\;$.\hss}$
is ${\mu_{\mathsf{f}}}$-monotone and we have
$\mathcal{R}\subseteq{>_{\mathcal{A}}}$. However, we should not conclude
termination of $\mathcal{R}$, because
$\mathsf{f}(\mathsf{a},\mathsf{b},\mathsf{g}(\mathsf{a},\mathsf{b}))$ is non-
terminating.
## 5 Weak Dependency Pairs
In Section 4 we investigated argument positions of rewrite steps. This section
is concerned about contexts surrounding rewrite steps. Recall the derivation:
$\displaystyle\boxed{\mathbf{4}\div\mathbf{2}}$
$\displaystyle\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\mathcal{R}_{\mathsf{div}}}}\mathsf{s}(\,\boxed{(\mathbf{3}-\mathbf{1})\div\mathbf{2}}\,)$
$\displaystyle\leavevmode\nobreak\
\mathrel{\to^{2}_{\mathcal{R}_{\mathsf{div}}}}\mathsf{s}(\,\boxed{\mathbf{2}\div\mathbf{2}}\,)$
$\displaystyle\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\mathcal{R}_{\mathsf{div}}}}\mathsf{s}(\mathsf{s}(\,\boxed{(\mathbf{1}-\mathbf{1})\div\mathbf{2}}\,))$
$\displaystyle\leavevmode\nobreak\
\mathrel{\to^{2}_{\mathcal{R}_{\mathsf{div}}}}\mathsf{s}(\mathsf{s}(\,\boxed{\mathsf{0}\div\mathbf{2}}\,))$
$\displaystyle\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\mathcal{R}_{\mathsf{div}}}}\mathsf{s}(\mathsf{s}(\mathsf{0}))\hbox
to0.0pt{$\;$,\hss}$
where we boxed outermost occurrences of defined symbols. Obviously, their
surrounding contexts are not rewritten. Here an idea is to simulate rewrite
steps from basic terms with new rewrite rules, obtained by dropping
unnecessary contexts. In termination analysis this method is known as the
dependency pair method [6]. We recast its main ingredient called dependency
pairs.
Let $X$ be a set of symbols. We write
${C\langle{t_{1},\ldots,t_{n}}\rangle}_{X}$ to denote $C[t_{1},\ldots,t_{n}]$,
whenever $\mathrm{root}(t_{i})\in X$ for all $1\leqslant i\leqslant n$ and $C$
is an $n$-hole context containing no $X$-symbols. (Note that the context $C$
may be degenerate and doesn’t contain a hole $\Box$ or it may be that $C$ is a
hole.) Then, every term $t$ can be uniquely written in the form
${C\langle{t_{1},\ldots,t_{n}}\rangle}_{X}$.
###### Lemma 5.1.
Let $t$ be a terminating term, and let $\sigma$ be a substitution. Then
${\mathsf{dh}}(t\sigma,\to_{\mathcal{R}})=\sum_{1\leqslant i\leqslant
n}{\mathsf{dh}}(t_{i}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})$, whenever
$t={C\langle{t_{1},\ldots,t_{n}}\rangle}_{\mathcal{D}\cup\mathcal{V}}$.
The idea is to replace such a $n$-hole context with a fresh $n$-ary function
symbol. We define the function com as a mapping from tuples of terms to terms
as follows: $\textsc{com}({t_{1}},\dots,{t_{n}})$ is $t_{1}$ if $n=1$, and
$c(t_{1},\ldots,t_{n})$ otherwise. Here $c$ is a fresh $n$-ary function symbol
called _compound symbol_. The above lemma motivates the next definition of
_weak dependency pairs_.
###### Definition 5.2.
Let $t$ be a term. We set $t^{\sharp}\mathrel{:=}t$ if $t\in\mathcal{V}$, and
$t^{\sharp}\mathrel{:=}f^{\sharp}(t_{1},\dots,t_{n})$ if
$t=f({t_{1}},\dots,{t_{n}})$. Here $f^{\sharp}$ is a new $n$-ary function
symbol called _dependency pair symbol_. For a signature $\mathcal{F}$, we
define $\mathcal{F}^{\sharp}=\mathcal{F}\cup\\{f^{\sharp}\mid
f\in\mathcal{F}\\}$. Let $\mathcal{R}$ be a TRS. If
$l\mathrel{\to}r\in\mathcal{R}$ and
$r={C\langle{{u_{1}},\dots,{u_{n}}}\rangle}_{\mathcal{D}\cup\mathcal{V}}$ then
the rewrite rule
$l^{\sharp}\to\textsc{com}(u_{1}^{\sharp},\ldots,u_{n}^{\sharp})$ is called a
_weak dependency pair_ of $\mathcal{R}$. The set of all weak dependency pairs
is denoted by $\operatorname{\mathsf{WDP}}(\mathcal{R})$.
While dependency pair symbols are defined with respect to
$\operatorname{\mathsf{WDP}}(\mathcal{R})$, these symbols are not defined with
respect to the original system $\mathcal{R}$. In the sequel defined symbols
refer to the defined function symbols of $\mathcal{R}$.
###### Example 5.3 (continued from Example 3.2).
The set $\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})$ consists of
the next four weak dependency pairs:
$\displaystyle 5\colon$ $\displaystyle x-^{\sharp}\mathsf{0}$
$\displaystyle\to x$ $\displaystyle\qquad 7\colon$
$\displaystyle\mathsf{0}\div^{\sharp}\mathsf{s}(y)$
$\displaystyle\to\mathsf{c}$ $\displaystyle 6\colon$
$\displaystyle\mathsf{s}(x)-^{\sharp}\mathsf{s}(y)$ $\displaystyle\to
x-^{\sharp}y$ $\displaystyle\qquad 8\colon$
$\displaystyle\mathsf{s}(x)\div^{\sharp}\mathsf{s}(y)$
$\displaystyle\to(x-y)\div^{\sharp}\mathsf{s}(y)\hbox to0.0pt{$\;$.\hss}$
Here $\mathsf{c}$ denotes a fresh compound symbols of arity $0$.
The derivation on page 5 corresponds to the derivation of
$\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})\cup\mathcal{R}_{\mathsf{div}}$:
$\displaystyle\mathbf{4}\div^{\sharp}\mathbf{2}$
$\displaystyle\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})}}\leavevmode\nobreak\
(\mathbf{3}-\mathbf{1})\div^{\sharp}\mathbf{2}$
$\displaystyle\leavevmode\nobreak\
\mathrel{\to^{2}_{\mathcal{R}_{\mathsf{div}}}}\mathbf{2}\div^{\sharp}\mathbf{2}$
$\displaystyle\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})}}\leavevmode\nobreak\
(\mathbf{1}-\mathbf{1})\div^{\sharp}\mathbf{2}$
$\displaystyle\leavevmode\nobreak\
\mathrel{\to^{2}_{\mathcal{R}_{\mathsf{div}}}}\mathsf{0}\div^{\sharp}\mathbf{2}$
$\displaystyle\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})}}\leavevmode\nobreak\
\mathsf{c}\hbox to0.0pt{$\;$,\hss}$
which preserves the length. The next lemma states that this is generally true.
###### Lemma 5.4.
Let $t\in\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V})$ be a terminating
term with defined root. Then we obtain:
${\mathsf{dh}}(t,\mathrel{\mathrel{\to}_{\mathcal{R}}})={\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})$.
###### Proof.
We show
${\mathsf{dh}}(t,\mathrel{\mathrel{\to}_{\mathcal{R}}})\leqslant{\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})$
by induction on ${\mathsf{dh}}(t,{\mathrel{\mathrel{\to}_{\mathcal{R}}}})$.
Let $\ell={\mathsf{dh}}(t,{\mathrel{\mathrel{\to}_{\mathcal{R}}}})$. If
$\ell=0$, the inequality is trivial. Suppose $\ell>0$. Then there exists a
term $u$ such that $t\mathrel{\mathrel{\to}_{\mathcal{R}}}u$ and
${\mathsf{dh}}(u,\mathrel{\mathrel{\to}_{\mathcal{R}}})=\ell-1$. We
distinguish two cases depending on the rewrite position $p$.
1. 1)
If $p$ is a position below the root, then clearly
$\mathrm{root}(u)=\mathrm{root}(t)\in\mathcal{D}$ and
$t^{\sharp}\mathrel{\mathrel{\to}_{\mathcal{R}}}u^{\sharp}$. Induction
hypothesis yields
${\mathsf{dh}}(u,{\mathrel{\mathrel{\to}_{\mathcal{R}}}})\leqslant{\mathsf{dh}}(u^{\sharp},{\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}}})$,
and we obtain
$\ell\leqslant{\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})$.
2. 2)
If $p$ is a root position, then there exist a rewrite rule $l\to
r\in\mathcal{R}$ and a substitution $\sigma$ such that $t=l\sigma$ and
$u=r\sigma$. There exists a context $C$ such that
$r={C\langle{{u_{1}},\dots,{u_{n}}}\rangle}_{\mathcal{D}\cup\mathcal{V}}$ and
thus by definition
$l^{\sharp}\to\textsc{com}(u_{1}^{\sharp},\ldots,u_{n}^{\sharp})\in\operatorname{\mathsf{WDP}}(\mathcal{R})$
such that $t^{\sharp}=l^{\sharp}\sigma$. Now, either $u_{i}\in\mathcal{V}$ or
$\mathrm{root}(u_{i})\in\mathcal{D}$ for every $1\leqslant i\leqslant n$.
Suppose $u_{i}\in\mathcal{V}$. Then $u_{i}^{\sharp}\sigma=u_{i}\sigma$ and
clearly no dependency pair symbol can occur and thus,
${\mathsf{dh}}(u_{i}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})={\mathsf{dh}}(u_{i}^{\sharp}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})={\mathsf{dh}}(u_{i}^{\sharp}\sigma,\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})\hbox
to0.0pt{$\;$.\hss}$
Otherwise, if $\mathrm{root}(u_{i})\in\mathcal{D}$ then
$u_{i}^{\sharp}\sigma=(u_{i}\sigma)^{\sharp}$. Hence
${\mathsf{dh}}(u_{i}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})\leqslant{\mathsf{dh}}(u,\mathrel{\mathrel{\to}_{\mathcal{R}}})<\ell$,
and we conclude
${\mathsf{dh}}(u_{i}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})\leqslant{\mathsf{dh}}(u_{i}^{\sharp}\sigma,\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})$
from the induction hypothesis. Therefore,
$\displaystyle\ell$
$\displaystyle={\mathsf{dh}}(u,\mathrel{\mathrel{\to}_{\mathcal{R}}})+1$
$\displaystyle=\sum_{1\leqslant i\leqslant
n}{\mathsf{dh}}(u_{i}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})+1\leqslant\sum_{1\leqslant
i\leqslant
n}{\mathsf{dh}}(u_{i}^{\sharp}\sigma,\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})+1$
$\displaystyle={\mathsf{dh}}(\textsc{com}(u_{1}^{\sharp},\ldots,u_{n}^{\sharp})\sigma,\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})+1\leqslant{\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})\hbox
to0.0pt{$\;$.\hss}$
Here we used Lemma 5.1 for the second equality.
Note that $t$ is $\mathcal{R}$-reducible if and only if $t^{\sharp}$ is
$\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}$-reducible. Hence as
$t$ is terminating, $t^{\sharp}$ is terminating on
$\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}}$.
Thus, similarly,
${\mathsf{dh}}(t,\mathrel{\mathrel{\to}_{\mathcal{R}}})\geqslant{\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})$
is shown by induction on
${\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\operatorname{\mathsf{WDP}}(\mathcal{R})\cup\mathcal{R}}})$.
∎
In the case of innermost rewriting we need not include collapsing dependency
pairs as in Definition 5.2. This is guaranteed by the next lemma.
###### Lemma 5.5.
Let $t$ be a terminating term and $\sigma$ a substitution such that $x\sigma$
is a normal form of $\mathcal{R}$ for all $x\in\mathcal{V}\mathsf{ar}(t)$.
Then
${\mathsf{dh}}(t\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})=\sum_{1\leqslant
i\leqslant
n}{\mathsf{dh}}(t_{i}\sigma,\mathrel{\mathrel{\to}_{\mathcal{R}}})$, whenever
$t={C\langle{t_{1},\ldots,t_{n}}\rangle}_{\mathcal{D}}$.
###### Definition 5.6.
Let $\mathcal{R}$ be a TRS. If $l\mathrel{\to}r\in\mathcal{R}$ and
$r={C\langle{{u_{1}},\dots,{u_{n}}}\rangle}_{\mathcal{D}}$ then the rewrite
rule $l^{\sharp}\to\textsc{com}(u_{1}^{\sharp},\ldots,u_{n}^{\sharp})$ is
called a _weak innermost dependency pair_ of $\mathcal{R}$. The set of all
weak innermost dependency pairs is denoted by
$\operatorname{\mathsf{WIDP}}(\mathcal{R})$.
###### Example 5.7 (continued from Example 3.2).
The set $\operatorname{\mathsf{WIDP}}(\mathcal{R}_{\mathsf{div}})$ consists of
the next three weak innermost dependency pairs (with respect to
$\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}$):
$\displaystyle\mathsf{s}(x)-^{\sharp}\mathsf{s}(y)$ $\displaystyle\to
x-^{\sharp}y$ $\displaystyle\mathsf{0}\div^{\sharp}\mathsf{s}(y)$
$\displaystyle\to\mathsf{c}$
$\displaystyle\mathsf{s}(x)\div^{\sharp}\mathsf{s}(y)$
$\displaystyle\to(x-y)\div^{\sharp}\mathsf{s}(y)\hbox to0.0pt{$\;$.\hss}$
The next lemma adapts Lemma 5.4 to innermost rewriting.
###### Lemma 5.8.
Let $t$ be an innermost terminating term in
$\operatorname{\mathcal{T}}(\mathcal{F},\mathcal{V})$ with
$\mathrm{root}(t)\in\mathcal{D}$. We have
${\mathsf{dh}}(t,\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}})={\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\operatorname{\mathsf{WIDP}}(\mathcal{R})\cup\mathcal{R}}})$.
Looking at the simulated version of the derivation on page 5, rules 1 and 2
are used, but neither rule 3 nor 4 is used in the $\mathcal{R}$-steps. In
general we can approximate a subsystem of a TRS that can be used in
derivations from basic terms, by employing the notion of usable rules in the
dependency pair method (cf. [6, 21, 22]).
###### Definition 5.9.
We write ${f}\mathrel{\rhd_{\mathsf{d}}}{g}$ if there exists a rewrite rule
$l\to r\in\mathcal{R}$ such that $f=\mathrm{root}(l)$ and $g$ is a defined
function symbol in $\mathcal{F}\mathsf{un}(r)$. For a set $\mathcal{G}$ of
defined function symbols we denote by $\mathcal{R}{\restriction}\mathcal{G}$
the set of rewrite rules $l\to r\in\mathcal{R}$ with
$\mathrm{root}(l)\in\mathcal{G}$. The set $\operatorname{\mathcal{U}}(t)$ of
_usable rules_ of a term $t$ is defined as
$\mathcal{R}{\restriction}\\{g\mid\text{${f}\mathrel{\rhd_{\mathsf{d}}}^{*}{g}$
for some $f\in\mathcal{F}\mathsf{un}(t)$}\\}$. Finally, if $\mathcal{P}$ is a
set of (weak) dependency pairs then
$\operatorname{\mathcal{U}}(\mathcal{P})=\bigcup_{l\to
r\in\mathcal{P}}\operatorname{\mathcal{U}}(r)$.
###### Example 5.10 (continued from Examples 5.3 and 5.7).
The set
$\operatorname{\mathcal{U}}(\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}}))$
of usable rules for the weak dependency pairs consists of the two rules:
$\displaystyle 1\colon$ $\displaystyle x-\mathsf{0}$ $\displaystyle\to x$
$\displaystyle\qquad 2\colon$ $\displaystyle\mathsf{s}(x)-\mathsf{s}(y)$
$\displaystyle\to x-y\hbox to0.0pt{$\;$.\hss}$
Note that we have that
$\operatorname{\mathcal{U}}(\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}}))=\operatorname{\mathcal{U}}(\operatorname{\mathsf{WIDP}}(\mathcal{R}_{\mathsf{div}}))$.
We show a usable rule criterion for complexity analysis by exploiting the
property that the starting terms are basic. Recall that
$\operatorname{\mathcal{T}_{\mathsf{b}}}$ denotes the set of basic terms; we
set $\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}=\\{t^{\sharp}\mid
t\in\operatorname{\mathcal{T}_{\mathsf{b}}}\\}$.
###### Lemma 5.11.
Let $\mathcal{P}$ be a set of weak dependency pairs and let
$(t_{i})_{i=0,1,\ldots}$ be a (finite or infinite) derivation of
$\mathcal{P}\cup\mathcal{R}$. If
$t_{0}\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$ then
$(t_{i})_{i=0,1,\ldots}$ is a derivation of
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$.
###### Proof.
Let $\mathcal{G}$ be the set of all non-usable symbols with respect to
$\mathcal{P}$. We write $P(t)$ if
${{{t}\\!\\!\mid_{q}}}\in{\mathsf{NF}(\mathcal{R})}$ for all
$q\in\mathcal{P}\mathsf{os}_{\mathcal{G}}(t)$. First we prove by induction on
$i$ that $P(t_{i})$ holds for all $i$.
1. 1)
Assume $i=0$. Since
$t_{0}\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$, we have
$t_{0}\in\mathsf{NF}(\mathcal{R})$ and thus
${{{t}\\!\\!\mid_{p}}}\in{\mathsf{NF}(\mathcal{R})}$ for all positions $p$.
The assertion $P$ follows trivially.
2. 2)
Suppose $i>0$. By induction hypothesis, $P(t_{i-1})$ holds, i.e., there exist
$p\in\mathcal{P}\mathsf{os}(t_{i-1})$, a substitution $\sigma$, and
$l\mathrel{\to}r\in\operatorname{\mathcal{U}}(\mathcal{P})\cup\mathcal{P}$,
such that ${{{t_{i-1}}\\!\\!\mid_{p}}}=l\sigma$ and
${{t_{i}}\\!\\!\mid_{p}}=r\sigma$. In order to show property $P$ for $t_{i}$,
we fix a position $q\in\mathcal{P}\mathsf{os}_{\mathcal{G}}(t)$. We have to
show ${{t_{i}}\\!\\!\mid_{q}}\in\mathsf{NF}(\mathcal{R})$. We distinguish
three subcases:
* •
Suppose that $q$ is above $p$. Then ${{t_{i-1}}\\!\\!\mid_{q}}$ is reducible,
but this contradicts the induction hypothesis $P(t_{i-1})$.
* •
Suppose $p$ and $q$ are parallel but distinct. Since
${{t_{i-1}}\\!\\!\mid_{q}}={{t_{i}}\\!\\!\mid_{q}}\in\mathsf{NF}(\mathcal{R})$
holds, we obtain $P(t_{i})$.
* •
Otherwise, $q$ is below $p$. Then, ${{t_{i}}\\!\\!\mid_{q}}$ is a subterm of
$r\sigma$. Because $r$ contains no $\mathcal{G}$-symbols by the definition of
usable symbols, ${{t_{i}}\\!\\!\mid_{q}}$ is a subterm of $x\sigma$ for some
$x\in\mathcal{V}\mathsf{ar}(r)\subseteq\mathcal{V}\mathsf{ar}(l)$. Therefore,
${{t_{i}}\\!\\!\mid_{q}}$ is also a subterm of ${{t_{i-1}}\\!\\!\mid_{q}}$,
from which ${{t_{i}}\\!\\!\mid_{q}}\in\mathsf{NF}(\mathcal{R})$ follows. We
obtain $P(t_{i})$.
Hence property $P$ holds for all $t_{i}$ in the assumed derivation. Thus any
reduction step
$t_{i}\mathrel{\mathrel{\to}_{\mathcal{R}\cup\mathcal{P}}}t_{i+1}$ can be
simulated by a step
$t_{i}\mathrel{\mathrel{\to}_{\operatorname{\mathcal{U}}(\mathcal{P})\cup\mathcal{P}}}t_{i+1}$.
From this the lemma follows. ∎
Note that the proof technique adopted for termination analysis [21, 22] cannot
be directly used in this context. The technique transforms terms in a
derivation to exclude non-usable rules. However, since the size of the initial
term increases, this technique does not suit to our use. On the other hand,
the transformation employed in [22] is adaptable to a complexity analysis in
the large, cf. [23].
The next theorem follows from Lemmas 5.4 and 5.8 in conjunction with the above
Lemma 5.11. It adapts the usable rule criteria to complexity analysis.
###### Theorem 5.12.
Let $\mathcal{R}$ be a TRS and let
$t\in\operatorname{\mathcal{T}_{\mathsf{b}}}$. If $t$ is terminating with
respect to $\mathrel{\to}$ then
${\mathsf{dh}}(t,\mathrel{\to})={\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}})$,
where $\mathrel{\to}$ denotes $\mathrel{\mathrel{\to}_{\mathcal{R}}}$ or
$\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}}$
depending on whether $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$ or
$\mathcal{P}=\operatorname{\mathsf{WIDP}}(\mathcal{R})$.
To clarify the applicability of the theorem in complexity analysis, we
instantiate the theorem by considering RMIs.
###### Corollary 5.13.
Let $\mathcal{R}$ be a TRS, let $\mu$ be the (innermost) usable replacement
map and let $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$ (or
$\mathcal{P}=\operatorname{\mathsf{WIDP}}(\mathcal{R})$). If
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ is compatible with a
$d$-degree $\mu$-monotone RMI $\mathcal{A}$, then the (innermost) runtime
complexity function ${\mathrm{rc}}^{(\mathsf{i})}_{\mathcal{R}}$ with respect
to $\mathcal{R}$ is bounded by a $d$-degree polynomial.
###### Proof.
For simplicity we suppose
$\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$ and let $\mathcal{A}$
be a $\mu$-monotone RMI of degree $d$. Compatibility of $\mathcal{A}$ with
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ implies the well-
foundedness of the relation
$\mathrel{\mathrel{\to}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}}$
on the set of terms $\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$, cf.
Theorem 4.8. This in turn implies the well-foundedness of
$\mathrel{\mathrel{\to}_{\mathcal{R}}}$, cf. Lemma 5.11. Hence Theorem 5.12 is
applicable and we conclude
${\mathsf{dh}}(t,\mathrel{\mathrel{\to}_{\mathcal{R}}})={\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}})$.
On the other hand, due to Theorem 3.9 compatibility with $\mathcal{A}$ implies
that
${\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}})=\operatorname{\mathsf{O}}(\lvert
t^{\sharp}\rvert^{d})$. As $\lvert t^{\sharp}\rvert=\lvert t\rvert$, we can
combine these equalities to conclude polynomial runtime complexity of
$\mathcal{R}$. ∎
The below given example applies Corollary 5.13 to the motivating Example 3.2
introduced in Section 1.
###### Example 5.14 (continued from Example 5.10).
Consider the TRS $\mathcal{R}_{\mathsf{div}}$ for division used as running
example; the weak dependency pairs
$\mathcal{P}\mathrel{:=}\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})$
are given in Example 5.3. We have
$\operatorname{\mathcal{U}}(\mathcal{P})=\\{1,2\\}$ and let
$\SS=\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$. The usable
replacement map $\mu\mathrel{:=}{\mu}^{\SS}_{\mathsf{f}}$ is defined as
follows:
$\displaystyle\mu(\mathsf{s})$
$\displaystyle=\mu(\mathsf{-})=\mu(\mathsf{-}^{\sharp})=\varnothing$
$\displaystyle\mu(\div^{\sharp})$ $\displaystyle=\\{1\\}\hbox
to0.0pt{$\;$.\hss}$
Note that ${\mu}^{\SS}_{\mathsf{f}}$ is smaller than
${\mu}^{\mathcal{R}}_{\mathsf{f}}$ on $\mathcal{F}$ (see Example 4.10).
Consider the $1$-dimensional RMI $\mathcal{A}$ with
$\mathsf{0}_{\mathcal{A}}=\mathsf{c}_{\mathcal{A}}=\mathsf{d}_{\mathcal{A}}=0$,
$\mathsf{s}_{\mathcal{A}}(x)=x+2$,
$\mathsf{-}_{\mathcal{A}}(x,y)=\mathsf{-}^{\sharp}_{\mathcal{A}}(x,y)=x+1$,
and $\div^{\sharp}_{\mathcal{A}}(x,y)=x+1$. The algebra $\mathcal{A}$ is
strictly monotone on all usable argument positions and the rules in $\SS$ are
interpreted and ordered as follows:
$\displaystyle 1\colon\quad$ $\displaystyle x+1$ $\displaystyle>x$
$\displaystyle 5\colon\quad$ $\displaystyle 1$ $\displaystyle>0$
$\displaystyle 7\colon\quad$ $\displaystyle 1$ $\displaystyle>0$
$\displaystyle 2\colon\quad$ $\displaystyle x+3$ $\displaystyle>x+1$
$\displaystyle 6\colon\quad$ $\displaystyle x+3$ $\displaystyle>x+1$
$\displaystyle 8\colon\quad$ $\displaystyle x+3$ $\displaystyle>x+2\hbox
to0.0pt{$\;$.\hss}$
Therefore, $\SS$ is compatible with $\mathcal{A}$ and the runtime complexity
function $\mathsf{rc}_{\mathcal{R}}$ is linear. Remark that by looking at the
coefficients of the interpretations more precise bound can be inferred. Since
all coefficients are at most one, we obtain
$\mathsf{rc}_{\mathcal{R}}(n)\leqslant n+c$ for some $c\in\mathbb{N}$.
It is worth stressing that it is (often) easier to analyse the complexity of
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ than the complexity
of $\mathcal{R}$. This is exemplified by the next example.
###### Example 5.15.
Consider the TRS $\mathcal{R}_{\mathsf{D}}$
$\displaystyle\mathsf{D}(\mathsf{c})$ $\displaystyle\to\mathsf{0}$
$\displaystyle\mathsf{D}(x+y)$ $\displaystyle\to\mathsf{D}(x)+\mathsf{D}(y)$
$\displaystyle\mathsf{D}(x\times y)$
$\displaystyle\to(y\times\mathsf{D}(x))+(x\times\mathsf{D}(y))$
$\displaystyle\mathsf{D}(\mathsf{t})$ $\displaystyle\to\mathsf{1}$
$\displaystyle\mathsf{D}(x-y)$
$\displaystyle\to\mathsf{D}(x)-\mathsf{D}(y)\hbox to0.0pt{$\;$.\hss}$
There is no $1$-dimensional ${\mu_{\mathsf{f}}}$-monotone RMI compatible with
$\mathcal{R}_{\mathsf{D}}$. On the other hand
$\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{D}})$ consists of the five
pairs
$\displaystyle\mathsf{D}^{\sharp}(\mathsf{c})$
$\displaystyle\to\mathsf{c_{1}}$ $\displaystyle\mathsf{D}^{\sharp}(x+y)$
$\displaystyle\to\mathsf{c_{3}}(\mathsf{D}^{\sharp}(x),\mathsf{D}^{\sharp}(y))$
$\displaystyle\mathsf{D}^{\sharp}(x\times y)$
$\displaystyle\to\mathsf{c_{5}}(y,\mathsf{D}^{\sharp}(x),x,\mathsf{D}^{\sharp}(y))$
$\displaystyle\mathsf{D}^{\sharp}(\mathsf{t})$
$\displaystyle\to\mathsf{c_{2}}$ $\displaystyle\mathsf{D}^{\sharp}(x-y)$
$\displaystyle\to\mathsf{c_{4}}(\mathsf{D}^{\sharp}(x),\mathsf{D}^{\sharp}(y))\hbox
to0.0pt{$\;$,\hss}$
and
$\operatorname{\mathcal{U}}(\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{D}}))=\varnothing$.
The usable replacement map ${\mu_{\mathsf{f}}}$ for
$\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{D}})\cup\operatorname{\mathcal{U}}(\mathcal{R}_{\mathsf{D}})$
is defined as
${\mu_{\mathsf{f}}}(\mathsf{c_{3}})={\mu_{\mathsf{f}}}(\mathsf{c_{4}})=\\{1,2\\}$,
${\mu_{\mathsf{f}}}(\mathsf{c_{5}})=\\{2,4\\}$, and
${\mu_{\mathsf{f}}}(f)=\varnothing$ for all other symbols $f$. Since the
$1$-dimensional ${\mu_{\mathsf{f}}}$-monotone RMI $\mathcal{A}$ with
$\displaystyle\mathsf{D}^{\sharp}_{\mathcal{A}}(x)=2x\qquad\mathsf{c}_{\mathcal{A}}=\mathsf{t}_{\mathcal{A}}=1\qquad{+}_{\mathcal{A}}(x,y)={-}_{\mathcal{A}}(x,y)={\times}_{\mathcal{A}}(x,y)=x+y+1$
$\displaystyle\mathsf{c_{1}}_{\mathcal{A}}=\mathsf{c_{2}}_{\mathcal{A}}=0\qquad\mathsf{c_{3}}_{\mathcal{A}}(x,y)=\mathsf{c_{4}}_{\mathcal{A}}(x,y)=x+y\qquad\mathsf{c_{5}}_{\mathcal{A}}(x,y,z,w)=y+w\hbox
to0.0pt{$\;$,\hss}$
is compatible with $\mathcal{R}_{\mathsf{D}}$, linear runtime complexity of
$\mathcal{R}_{\mathsf{D}}$ is concluded. Remark that this bound is optimal.
We conclude this section by discussing the (in-)applicability of standard
dependency pairs (see [6]) in complexity analysis. For that we recall the
definition of standard dependency pairs.
###### Definition 5.16 ([6]).
The set $\mathsf{DP}(\mathcal{R})$ of (standard) _dependency pairs_ of a TRS
$\mathcal{R}$ is defined as $\\{l^{\sharp}\to u^{\sharp}\mid l\to
r\in\mathcal{R},\text{$u\mathrel{{\trianglelefteq}}r$, $\mathrm{root}(u)$ is
defined, and $u\not\mathrel{{\lhd}}l$}\\}$.
The next example shows that Lemma 5.4 (Lemma 5.8) does not hold if we replace
weak (innermost) dependency pairs with standard dependency pairs.
###### Example 5.17.
Consider the one-rule TRS $\mathcal{R}$:
$\mathsf{f}(\mathsf{s}(x))\to\mathsf{g}(\mathsf{f}(x),\mathsf{f}(x))$.
$\mathsf{DP}(\mathcal{R})$ is the singleton of
$\mathsf{f}^{\sharp}(\mathsf{s}(x))\to\mathsf{f}^{\sharp}(x)$. Let
$t_{n}=\mathsf{f}(\mathsf{s}^{n}(x))$ for each $n\geqslant 0$. Since
$t_{n+1}\mathrel{\mathrel{\to}_{\mathcal{R}}}\mathsf{g}(t_{n},t_{n})$ holds
for all $n\geqslant 0$, it is easy to see
${\mathsf{dh}}(t_{n+1},\mathrel{\mathrel{\to}_{\mathcal{R}}})\geqslant 2^{n}$,
while
${\mathsf{dh}}(t_{n+1}^{\sharp},\mathrel{\mathrel{\to}_{\mathsf{DP}(\mathcal{R})\cup\mathcal{R}}})=n$.
## 6 The Weight Gap Principle
Let $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})$ and
recall the derivation over $\mathcal{P}\cup\mathcal{R}_{\mathsf{div}}$ on page
5.3. This derivation can be represented as derivation of $\mathcal{P}$ modulo
$\operatorname{\mathcal{U}}(\mathcal{P})$:
$\mathbf{4}\div^{\sharp}\mathbf{2}\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}\leavevmode\nobreak\
\mathbf{2}\div^{\sharp}\mathbf{2}\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}\leavevmode\nobreak\
\mathsf{0}\div^{\sharp}\mathbf{2}\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}\leavevmode\nobreak\
\mathsf{c}\hbox to0.0pt{$\;$.\hss}$
As we see later linear runtime complexity of
$\operatorname{\mathcal{U}}(\mathcal{P})$ and
$\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})$ can be easily obtained.
If linear runtime complexity of
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ would follow from
them, linear runtime complexity of $\mathcal{R}$ could be established in a
modular way.
In order to bound complexity of relative TRSs we define a variant of a
reduction pair [6]. Note that $\operatorname{\mathsf{G}}$ is associated to a
given collapsible order.
###### Definition 6.1.
A $\mu$-_complexity pair_ for a relative TRS $\mathcal{R}/\mathcal{S}$ is a
pair $({\gtrsim},{\succ})$ such that $\gtrsim$ is a $\mu$-monotone proper
order and $\succ$ is a strict order. Moreover ${\gtrsim}$ and ${\succ}$ are
compatible, that is, ${\gtrsim\cdot\succ}\subseteq{\succ}$ or
${\succ\cdot\gtrsim}\subseteq{\succ}$. Finally $\succ$ is collapsible on
$\mathrel{\mathrel{\to}_{\mathcal{R}/\mathcal{S}}}$ and all compound symbols
are $\mu$-monotone with respect to $\succ$.
###### Lemma 6.2.
Let $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$ and
$({\gtrsim},{\succ})$ a
${\mu_{\mathsf{f}}}^{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}$-complexity
pair for $\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})$. If
$\mathcal{P}\subseteq{\succ}$ and
$\operatorname{\mathcal{U}}(\mathcal{P})\subseteq{\gtrsim}$ then
${\mathsf{dh}}(t,\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}})\leqslant\operatorname{\mathsf{G}}(t)$
for any $t\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$.
###### Example 6.3 (continued from Example 5.14).
Consider the $1$-dimensional RMI $\mathcal{A}$ with
$\displaystyle\mathsf{0}_{\mathcal{A}}$
$\displaystyle=\mathsf{c}_{\mathcal{A}}=\mathsf{d}_{\mathcal{A}}=0$
$\displaystyle\mathsf{s}_{\mathcal{A}}(x)$ $\displaystyle=x+1$
$\displaystyle{-}_{\mathcal{A}}(x,y)$
$\displaystyle={-}^{\sharp}_{\mathcal{A}}(x,y)=\div^{\sharp}_{\mathcal{A}}(x,y)=x\hbox
to0.0pt{$\;$,\hss}$
which yields the complexity pair
$({\mathrel{{\geqslant}_{\mathcal{A}}}},{\mathrel{{>}_{\mathcal{A}}}})$ for
$\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})$. Since
${\mathcal{P}}\subseteq{\mathrel{{>}_{\mathcal{A}}}}$ and
${\operatorname{\mathcal{U}}(\mathcal{P})}\subseteq{\mathrel{{\geqslant}_{\mathcal{A}}}}$
hold,
$\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}},\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}})=\operatorname{\mathsf{O}}(n)$.
First we show the main theorem of this section.
###### Definition 6.4.
Let $\mathcal{A}$ be a matrix interpretation and let $\mathcal{R}/\SS$ be a
relative TRS. A _weight gap_ on a set $T$ of terms is a number
$\Delta\in\mathbb{N}$ such that $s\in{\to^{*}_{\mathcal{R}\cup\SS}}(T)$ and
$s\to_{\mathcal{R}}t$ implies $[t]_{1}-[s]_{1}\leqslant\Delta$.
Let $T$ be a set of terms and let $\mathcal{R}/\SS$ be a relative TRS.
###### Theorem 6.5.
If $\mathcal{R}/\SS$ is terminating, $\mathcal{A}$ admits a _weight gap_
$\Delta$ on $T$, and $\mathcal{A}$ is a matrix interpretation of degree $d$
such that $\SS$ is compatible with $\mathcal{A}$, then there exists
$c\in\mathbb{N}$ such that
${\mathsf{dh}}(t,{\to_{\mathcal{R}\cup\SS}})\leqslant(1+\Delta)\cdot{\mathsf{dh}}(t,{\to_{\mathcal{R}/\SS}})+c\cdot|t|^{d}$
for all $t\in T$. Consequently,
$\operatorname{\mathsf{comp}}(n,T,{\mathrel{\mathrel{\to}_{\mathcal{R}\cup\SS}}})=\operatorname{\mathsf{O}}(\operatorname{\mathsf{comp}}(n,T,{\mathrel{\mathrel{\to}_{\mathcal{R}/\SS}}})+n^{d})$
holds.
###### Proof.
Let $m={\mathsf{dh}}(s,{\mathrel{\mathrel{\to}_{\mathcal{R}/\SS}}})$ and
$n=\lvert s\rvert$. Any derivation of
$\mathrel{\mathrel{\to}_{\mathcal{R}\cup\mathcal{S}}}$ is representable as
follows:
$s=s_{0}\to_{\mathcal{S}}^{k_{0}}t_{0}\to_{\mathcal{R}}s_{1}\to_{\mathcal{S}}^{k_{1}}t_{1}\to_{\mathcal{R}}\cdots\to_{\mathcal{S}}^{k_{m}}t_{m}\hbox
to0.0pt{$\;$.\hss}$
Without loss of generality we may assume that the derivation is maximal and
ground. We observe:
1. 1)
$k_{i}\leqslant[s_{i}]_{1}-[t_{i}]_{1}$ holds for all $0\leqslant i\leqslant
m$. This is because $[s]_{1}>[t]_{1}$, whenever
$s\mathrel{\mathrel{\to}_{\mathcal{S}}}t$ by the assumption $\SS$ is
compatible with $\mathcal{A}$. By definition of $>$, we conclude
$[s]_{1}\geqslant[t]_{1}+1$ whenever
$s\mathrel{\mathrel{\to}_{\mathcal{S}}}t$. From the fact that
$s_{i}\to_{\mathcal{S}}^{k_{i}}t_{i}$ we thus obtain
$k_{i}\leqslant[s_{i}]_{1}-[t_{i}]_{1}$.
2. 2)
$([s_{i+1}])_{1}\leqslant([t_{i}])_{1}+\Delta$ holds for all $0\leqslant i<m$
by the assumption.
3. 3)
There exists a number $c$ such that for any term $s\in T$, $[s]_{1}\leqslant
c\cdot\lvert s\rvert^{d}$. This follows by the degree of $\mathcal{A}$.
We obtain the following inequalities:
$\displaystyle{\mathsf{dh}}(s_{0},\mathrel{\mathrel{\to}_{\mathcal{R}\cup\SS}})$
$\displaystyle=m+k_{0}+\dots+k_{m}$ $\displaystyle\leqslant
m+([s_{0}]_{1}-[t_{0}]_{1})+\dots+([s_{m}]_{1}-[t_{m}]_{1})$
$\displaystyle=m+[s_{0}]_{1}+([s_{1}]_{1}-[t_{0}]_{1})+\dots+([s_{m}]_{1}-[t_{m-1}]_{1})-[t_{m}]_{1}$
$\displaystyle\leqslant
m+[s_{0}]_{1}+([t_{0}]_{1}+\Delta-[t_{0}]_{1})+\dots-[t_{m}]_{1}$
$\displaystyle\leqslant m+[s_{0}]_{1}+m\Delta-[t_{m}]_{1}$
$\displaystyle\leqslant m+[s_{0}]_{1}+m\Delta$
$\displaystyle\leqslant(1+\Delta)m+c\cdot\lvert s_{0}\rvert^{d}\hbox
to0.0pt{$\;$.\hss}$
Here we use property 1) $m$-times in the second line. We used property 2) in
the third line and property 3) in the last line. ∎
A question is when a weight gap is admitted. We present two conditions. We
start with a simple version for derivational complexity, and then we adapt it
for runtime complexity.
We employ a very restrictive form of TMIs. Every $f\in\mathcal{F}$ is
interpreted by the following restricted linear function:
$f_{\mathcal{A}}\colon(\vec{v}_{1},\ldots,\vec{v}_{n})\mapsto\mathbf{1}\vec{v}_{1}+\ldots+\mathbf{1}\vec{v}_{n}+\vec{f}\hbox
to0.0pt{$\;$.\hss}$
I.e., the only matrix employed in this interpretation is the unit matrix
$\mathbf{1}$. Such a matrix interpretation is called _strongly linear_ (_SLMI_
for short).
###### Lemma 6.6.
If $\mathcal{R}$ is non-duplicating and $\mathcal{A}$ is an SLMI, then
$\mathcal{R}/\SS$ and $\mathcal{A}$ admit a weight gap on all terms.
###### Proof.
Let
$\Delta\mathrel{:=}\max\\{[r]_{1}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}[l]_{1}\mid
l\to r\in\mathcal{R}\\}$. We show that $\Delta$ gives a weight gap. In proof,
we first show the following equality.
$\Delta=\max\\{([\alpha]_{\mathcal{A}}(r))_{1}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}([\alpha]_{\mathcal{A}}(l))_{1}\mid
l\to r\in\mathcal{R},\alpha\colon\mathcal{V}\to\mathcal{A}\\}\hbox
to0.0pt{$\;$.\hss}$ (1)
Although the proof is not difficult, we give the full account in order to
utilise it later. Observe that for any matrix interpretation $\mathcal{A}$ and
rule ${l\to r}\in{\mathcal{R}}$, there exist matrices (over $\mathbb{N}$)
$L_{1},\dots,L_{k}$, $R_{1},\dots,R_{k}$ and vectors $\vec{l}$, $\vec{r}$ such
that:
$[\alpha]_{\mathcal{A}}(l)=\sum_{i=1}^{k}L_{i}\cdot\alpha(x_{i})+\vec{l}\hskip
43.05542pt[\alpha]_{\mathcal{A}}(r)=\sum_{i=1}^{k}R_{i}\cdot\alpha(x_{i})+\vec{r}\hbox
to0.0pt{$\;$,\hss}$
where $k$ denotes the cardinality of
$\mathcal{V}\mathsf{ar}(l)\supseteq\mathcal{V}\mathsf{ar}(r)$. Conclusively,
we obtain:
$[\alpha]_{\mathcal{A}}(r)\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}[\alpha]_{\mathcal{A}}(l)=\sum_{i=1}^{k}(R_{i}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}L_{i})\alpha(x_{i})+(\vec{r}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}\vec{l})\hbox
to0.0pt{$\;$.\hss}$ (2)
Here
$\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}$
denotes the natural component-wise extension of the modified minus to vectors.
As $\mathcal{A}$ is an SLMI the matrices $L_{i}$, $R_{i}$ are obtained by
multiplying or adding unit matrices, where the latter case can only happen if
(at least one) of the variables $x_{i}$ occurs multiple times in $l$ or $r$.
Due to the fact that $l\to r$ is non-duplicating, this effect is canceled out.
Thus the right-hand side of (2) is independent on the assignment $\alpha$ and
we conclude:
$[r]_{1}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}[l]_{1}=([\alpha]_{\mathcal{A}}(r)\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}[\alpha]_{\mathcal{A}}(l))_{1}=(\vec{r}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}\vec{l})_{1}\hbox
to0.0pt{$\;$.\hss}$
By definition
$\Delta=\max\\{[r]_{1}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}[l]_{1}\mid
l\to r\in\mathcal{R}\\}$ and thus (1) follows.
Let $C[\Box]$ denote a (possible empty) context such that
$s=C[l\sigma]\mathrel{\mathrel{\to}_{\mathcal{R}}}C[r\sigma]=t$, where
${l\mathrel{\to}r}\in{\mathcal{R}}$ and $\sigma$ a substitution. We prove the
lemma by induction on $C$.
1. 1)
Suppose $C[\Box]=\Box$, that is, $s=l\sigma$ and $t=r\sigma$. There exists an
assignment $\alpha_{1}$ such that $[l\sigma]=[\alpha_{1}]_{\mathcal{A}}(l)$
and $[r\sigma]=[\alpha_{1}]_{\mathcal{A}}(r)$. By (1) we conclude for the
assignment $\alpha_{1}$:
$([\alpha_{1}]_{\mathcal{A}}(l))_{1}+\Delta\geqslant([\alpha_{1}]_{\mathcal{A}}(r))_{1}$.
Therefore in sum we obtain $[s]_{1}+\Delta\geqslant[t]_{1}$.
2. 2)
Suppose $C[\Box]=f(t_{1},\dots,t_{i-1},C^{\prime}[\Box],t_{i+1},\dots,t_{n})$.
Hence, we obtain:
$\displaystyle[f(t_{1},\dots,C^{\prime}[l\sigma],\dots,t_{n})]_{1}+\Delta$
$\displaystyle={}$
$\displaystyle[t_{1}]_{1}+\dots+([C^{\prime}[l\sigma]]_{1}+\Delta)+\dots+[t_{n}]_{1}+(\vec{f})_{1}$
$\displaystyle\geqslant{}$
$\displaystyle[t_{1}]_{1}+\dots+[C^{\prime}[r\sigma]]_{1}+\dots+[t_{n}]_{1}+(\vec{f})_{1}$
$\displaystyle={}$
$\displaystyle[f(t_{1},\dots,C^{\prime}[r\sigma],\dots,t_{n})]_{1}\hbox
to0.0pt{$\;$,\hss}$
for some vector $\vec{f}\in\mathbb{N}^{d}$. In the first and last line, we
employ the fact that $\mathcal{A}$ is strongly linear. In the second line the
induction hypothesis is applied together with the (trivial) fact that
$\mathcal{A}$ is strictly monotone on all arguments of $f$ by definition.
∎
Note that the combination of Theorem 6.5 and Lemma 6.6 corresponds to (the
corrected version of) Theorem 24 in [4]. In [4] 1-dimensional SLMIs are called
_strongly linear interpretations_ (_SLIs_ for short).
###### Example 6.7.
Consider the TRS $\mathcal{R}$
$\displaystyle 1\colon\leavevmode\nobreak\ \mathsf{f}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{f}(x-\mathsf{s}(\mathsf{0}))$ $\displaystyle
2\colon\leavevmode\nobreak\ x-\mathsf{0}$ $\displaystyle\to x$ $\displaystyle
3\colon\leavevmode\nobreak\ \mathsf{s}(x)-\mathsf{s}(y)$ $\displaystyle\to
x-y\hbox to0.0pt{$\;$.\hss}$
$\mathcal{P}\mathrel{:=}\operatorname{\mathsf{WDP}}(\mathcal{R})$ consists of
the three pairs
$\displaystyle\mathsf{f}^{\sharp}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{f}^{\sharp}(x-\mathsf{s}(\mathsf{0}))$ $\displaystyle
x-^{\sharp}\mathsf{0}$ $\displaystyle\to x$
$\displaystyle\mathsf{s}(x)-^{\sharp}\mathsf{s}(y)$ $\displaystyle\to
x-^{\sharp}y\hbox to0.0pt{$\;$,\hss}$
and $\operatorname{\mathcal{U}}(\mathcal{P})=\\{2,3\\}$. Obviously
$\mathcal{P}$ is non-duplicating and there exists an SLI $\mathcal{A}$ with
$\operatorname{\mathcal{U}}(\mathcal{P})\subseteq{\mathrel{{\succ}_{\mathcal{A}}}}$.
Thus, Lemma 6.6 yields a weight gap for
$\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})$. By taking the
$1$-dimensional RMI $\mathcal{B}$ with
$\displaystyle\mathsf{s}_{\mathcal{B}}(x)$ $\displaystyle=x+1$
$\displaystyle{-}_{\mathcal{B}}(x,y)$ $\displaystyle=x$
$\displaystyle\mathsf{f}_{\mathcal{B}}(x)$
$\displaystyle=\mathsf{f}^{\sharp}_{\mathcal{B}}(x)=x$
$\displaystyle\mathsf{0}_{\mathcal{B}}$ $\displaystyle=0$
$\displaystyle{-^{\sharp}}_{\mathcal{B}}(x,y)$ $\displaystyle=x+1\hbox
to0.0pt{$\;$,\hss}$
we obtain $\mathcal{P}\subseteq{\mathrel{{\succ}_{\mathcal{B}}}}$ and
$\operatorname{\mathcal{U}}(\mathcal{P})\subseteq{\mathrel{{\succcurlyeq}_{\mathcal{B}}}}$.
Therefore,
$\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}},{\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}})=\operatorname{\mathsf{O}}(n)$.
Hence,
$\mathsf{rc}_{\mathcal{R}}(n)=\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}},{\mathrel{\mathrel{\to}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}}})=\operatorname{\mathsf{O}}(n)$
is concluded by Theorem 6.5.
The next lemma shows that there is no advantage to consider SLMIs of dimension
$k\geqslant 2$.
###### Lemma 6.8.
If $\mathcal{S}$ is compatible with some SLMI $\mathcal{A}$ then $\mathcal{S}$
is compatible with some SLI $\mathcal{B}$.
###### Proof.
Let $\mathcal{A}$ be an SLMI of dimension $k$. Further, let
$\alpha:\mathcal{V}\to\mathbb{N}$ denote an arbitrary assignment. We define
$\widehat{\alpha}\colon\mathcal{V}\to\mathbb{N}^{k}$ as
$\widehat{\alpha}(x)=(\alpha(x),0,\dots,0)^{\top}$ for each variable $x$. We
define the SLI $\mathcal{B}$ by
$f_{\mathcal{B}}(x_{1},\dots,x_{n})=x_{1}+\cdots+x_{n}+\vec{f}_{1}$. Then,
$\displaystyle f_{\mathcal{B}}(x_{1},\dots,x_{n})$
$\displaystyle=\left((x_{1},0,\dots,0)^{\top}+\cdots+(x_{n},0,\dots,0)^{\top}+\vec{f}\right)_{1}$
$\displaystyle=\left(f_{\mathcal{A}}((x_{1},0,\dots,0)^{\top},\dots,(x_{n},0,\dots,0)^{\top}))\right)_{1}$
Therefore, easy structural induction shows that
$[\alpha]_{\mathcal{B}}(t)=([\widehat{\alpha}]_{\mathcal{A}}(t))_{1}$ for all
terms $t$. Hence, $\mathcal{S}\subseteq{\mathrel{{\succ}_{\mathcal{B}}}}$
whenever $\mathcal{S}\subseteq{\mathrel{{\succ}_{\mathcal{A}}}}$. ∎
The next example shows that in Lemma 6.6 SLMIs cannot be simply replaced by
RMIs.
###### Example 6.9.
Consider the TRSs $\mathcal{R}_{\mathsf{exp}}$
$\displaystyle\mathsf{exp}(\mathsf{0})$
$\displaystyle\to\mathsf{s}(\mathsf{0})$ $\displaystyle\mathsf{d}(\mathsf{0})$
$\displaystyle\to\mathsf{0}$ $\displaystyle\mathsf{exp}(\mathsf{r}(x))$
$\displaystyle\to\mathsf{d}(\mathsf{exp}(x))$
$\displaystyle\mathsf{d}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{s}(\mathsf{s}(\mathsf{d}(x)))\hbox
to0.0pt{$\;$.\hss}$
This TRS formalises the exponentiation function. Setting
$t_{n}=\mathsf{exp}(\mathsf{r}^{n}(\mathsf{0}))$ we obtain
${\mathsf{dh}}(t_{n},\mathrel{\mathrel{\to}_{\mathcal{R}_{\mathsf{exp}}}})\geqslant
2^{n}$ for each $n\geqslant 0$. Thus the runtime complexity of
$\mathcal{R}_{\mathsf{exp}}$ is exponential.
In order to show the claim, we split $\mathcal{R}_{\mathsf{exp}}$ into two
TRSs
$\mathcal{R}=\\{\mathsf{exp}(\mathsf{0})\to\mathsf{s}(0),\mathsf{exp}(\mathsf{r}(x))\to\mathsf{d}(\mathsf{exp}(x))\\}$
and
$\mathcal{S}=\\{\mathsf{d}(\mathsf{0})\to\mathsf{0},\mathsf{d}(\mathsf{s}(x))\to\mathsf{s}(\mathsf{s}(\mathsf{d}(x)))\\}$.
Then it is easy to verify that the next $1$-dimensional RMI $\mathcal{A}$ is
compatible with $\mathcal{S}$:
$\mathsf{0}_{\mathcal{A}}=0\qquad\mathsf{d}_{\mathcal{A}}(x)=3x\qquad\mathsf{s}_{\mathcal{A}}(x)=x+1\hbox
to0.0pt{$\;$.\hss}$
Moreover an upper-bound of
${\mathsf{dh}}(t_{n},{\mathrel{\mathrel{\to}_{\mathcal{R}/\mathcal{S}}}})$ can
be estimated by using the following $1$-dimensional TMI $\mathcal{B}$:
$\mathsf{0}_{\mathcal{B}}=0\qquad\mathsf{d}_{\mathcal{B}}(x)=\mathsf{s}_{\mathcal{B}}(x)=x\qquad\mathsf{exp}_{\mathcal{B}}(x)=\mathsf{r}_{\mathcal{B}}(x)=x+1\hbox
to0.0pt{$\;$.\hss}$
Since
${\mathrel{\mathrel{\to}_{\mathcal{R}}}}\subseteq{\mathrel{{>}_{\mathcal{B}}}}$
and
${\mathrel{\mathrel{\to}_{\mathcal{S}}^{\ast}}}\subseteq{\mathrel{{\geqslant}_{\mathcal{B}}}}$
hold, we have
${\mathrel{\mathrel{\to}_{{\mathcal{R}}/{\mathcal{S}}}}}\subseteq{\mathrel{{>}_{\mathcal{B}}}}$.
Hence
${\mathsf{dh}}(t_{n},\mathrel{\mathrel{\to}_{{\mathcal{R}}/{\mathcal{S}}}})\leqslant[\alpha_{0}]_{\mathcal{B}}(t_{n})=n+2$.
But clearly from this we cannot conclude a polynomial bound on the derivation
length of $\mathcal{R}\cup\mathcal{S}=\mathcal{R}_{\mathsf{exp}}$, as the
runtime complexity of $\mathcal{R}_{\mathsf{exp}}$ is exponential.
Furthermore, non-duplication of $\mathcal{R}$ is also essential for Lemma
6.6.333This example is due to Dieter Hofbauer and Andreas Schnabl.
###### Example 6.10.
Consider the following $\mathcal{R}\cup\SS$
$\displaystyle 1\colon$ $\displaystyle\mathsf{f}(\mathsf{s}(x),y)$
$\displaystyle\to\mathsf{f}(x,\mathsf{d}(y,y,y))$ $\displaystyle\qquad
2\colon$ $\displaystyle\mathsf{d}(\mathsf{0},\mathsf{0},x)$ $\displaystyle\to
x$ $\displaystyle 3\colon$
$\displaystyle\mathsf{d}(\mathsf{s}(x),\mathsf{s}(y),z)$
$\displaystyle\to\mathsf{d}(x,y,\mathsf{s}(z))\hbox to0.0pt{$\;$.\hss}$
Let $\mathcal{R}=\\{1\\}$ and let $\SS=\\{2,3\\}$. The following SLI
$\mathcal{A}$ is compatible with $\SS$:
$\mathsf{d}_{\mathcal{A}}(x,y,z)=x+y+z+1\qquad\mathsf{s}_{\mathcal{A}}(x)=x+1\qquad\mathsf{0}_{\mathcal{A}}=0\hbox
to0.0pt{$\;$.\hss}$
Furthermore, the following ${\mu}^{\mathcal{R}\cup\SS}_{\mathsf{f}}$-monotone
1-dimensional RMI $\mathcal{B}$ orients the rule in $\mathcal{R}$ strictly,
while the rules in $\SS$ are weakly oriented.
$\mathsf{f}_{\mathcal{B}}(x,y)=x\qquad\mathsf{d}_{\mathcal{B}}(x,y,z)=x+y+z\qquad\mathsf{s}_{\mathcal{B}}(x)=x+1\qquad\mathsf{0}_{\mathcal{B}}=0\hbox
to0.0pt{$\;$.\hss}$
Thus,
$\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}_{\mathsf{b}}},{\to_{\mathcal{R}/\SS}})=\operatorname{\mathsf{O}}(n)$
is obtained. If the restriction that $\mathcal{R}$ is non-duplicating could be
dropped from Lemma 6.6, we would conclude
$\mathsf{rc}_{\mathcal{R}\cup\SS}(n)=\operatorname{\mathsf{O}}(n)$. However,
it is easy to see that $\mathsf{rc}_{\mathcal{R}\cup\SS}$ is at least
exponential. Setting
$t_{n}\mathrel{:=}\mathsf{f}(\mathsf{s}^{n}(\mathsf{0}),\mathsf{s}(\mathsf{0}))$,
we obtain
${\mathsf{dh}}(t_{n},\mathrel{\mathrel{\to}_{\mathcal{R}\cup\mathcal{S}}})\geqslant
2^{n}$ for any $n\geqslant 1$.
We present a weight gap condition for runtime complexity analysis. When
considering the derivation in the beginning of this section (on page 6), every
step by a weak dependency pair only takes place as an outermost step.
Exploiting this fact we can relax the restriction that was imposed in the
above examples. To this end, we introduce a generalised notion of non-
duplicating TRSs.
Below
$\max\,\\{\,([\alpha]_{\mathcal{A}}(r))_{1}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}([\alpha]_{\mathcal{A}}(l))_{1}\mid\text{$l\to
r\in\mathcal{P}$ and $\alpha:\mathcal{V}\to\mathcal{A}$}\,\\}$ is referred to
as $\operatorname{\Delta}(\mathcal{A},\mathcal{P})$. We say that a
$\mu$-monotone RMI is _adequate_ if all compound symbols are interpreted as
$\mu$-monotone SLMI.
###### Lemma 6.11.
Let $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$ and let
$\mathcal{A}$ be an adequate
${\mu_{\mathsf{f}}}^{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}$-monotone
RMI. Suppose $\operatorname{\Delta}(\mathcal{A},\mathcal{P})$ is well-defined
on $\mathbb{N}$. Then, $\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})$
and $\mathcal{A}$ admit a weight gap on
$\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$.
###### Proof.
The proof follows the proof of Lemma 6.6. We set
$\Delta=\operatorname{\Delta}(\mathcal{A},\mathcal{P})$. Let
$s\mathrel{\mathrel{\to}_{\mathcal{P}}}t$ with
$s\in{\to_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}}(\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}})$.
One may write $s=C[l\sigma]$ and $t=C[r\sigma]$ with
$l\mathrel{\to}r\in\mathcal{P}$, where $C$ denotes a context. Note that due to
$s\in{\to_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}}(\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}})$
all function symbols above the hole in $C$ are compound symbols. We perform
induction on $C$.
1. 1)
If $C=\Box$ then $[t]_{1}-[s]_{1}\leqslant\Delta$ by the definition of
$\operatorname{\Delta}(\mathcal{A},\mathcal{P})$.
2. 2)
For inductive step, $C$ must be of the form
$c(u_{1},\ldots,u_{i-1},C^{\prime},u_{i+1},\ldots,u_{n})$ with $i\in\mu(c)$.
Since $\mathcal{A}$ is adequate, $c_{\mathcal{A}}$ is a SLMI. The rest of
reasoning is same with 2) in the proof of Lemma 6.6.
∎
###### Example 6.12 (continued from Example 6.3).
Consider the following adequate
${\mu}^{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}_{\mathsf{f}}$-monotone
$1$-dimensional RMI $\mathcal{B}$:
$\displaystyle\mathsf{0}_{\mathcal{B}}$
$\displaystyle=\mathsf{c}_{\mathcal{B}}=\mathsf{d}_{\mathcal{B}}=0$
$\displaystyle\mathsf{s}_{\mathcal{B}}(x)$ $\displaystyle=x+2$
$\displaystyle\mathsf{-}_{\mathcal{B}}(x,y)$
$\displaystyle=\mathsf{-}^{\sharp}_{\mathcal{B}}(x,y)={\div}^{\sharp}_{\mathcal{B}}(x,y)=x+1$
Since $\Delta(\mathcal{B},\mathcal{P})$ is well-defined (indeed $1$),
$\mathcal{B}$ admits the weight gap of Lemma 6.11. Moreover,
$\operatorname{\mathcal{U}}(\mathcal{P})$ is compatible with
${\mathrel{{\succ}_{\mathcal{B}}}}$. As
$\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}},{\mathrel{\mathrel{\to}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}})=\operatorname{\mathsf{O}}(n)$
was shown in Example 6.3, Theorem 6.5 deduces linear runtime complexity for
$\mathcal{R}_{\mathsf{div}}$.
In Lemma 6.11 $\operatorname{\Delta}(\mathcal{A},\mathcal{P})$ must be well-
defined.
###### Example 6.13.
Consider the following TRS $\mathcal{R}$
$\displaystyle 1\colon\leavevmode\nobreak\ $ $\displaystyle\mathsf{f}([\,])$
$\displaystyle\to[\,]$ $\displaystyle 3\colon\leavevmode\nobreak\ $
$\displaystyle\mathsf{g}([\,],z)$ $\displaystyle\to z$ $\displaystyle
2\colon\leavevmode\nobreak\ $ $\displaystyle\mathsf{f}(x:y)$ $\displaystyle\to
x:\mathsf{f}(\mathsf{g}(y,[\,]))\qquad$ $\displaystyle
4\colon\leavevmode\nobreak\ $ $\displaystyle\mathsf{g}(x:y,z)$
$\displaystyle\to\mathsf{g}(y,x:z)$
whose optimal innermost runtime complexity is quadratic. The weak innermost
dependency pairs
$\mathcal{P}\mathrel{:=}\operatorname{\mathsf{WIDP}}(\mathcal{R})$ are
$\displaystyle 5\colon\leavevmode\nobreak\ $
$\displaystyle\mathsf{f}^{\sharp}([\,])$ $\displaystyle\to\mathsf{c}$
$\displaystyle 7\colon\leavevmode\nobreak\ $
$\displaystyle\mathsf{g}^{\sharp}([\,],z)$ $\displaystyle\to\mathsf{d}$
$\displaystyle 6\colon\leavevmode\nobreak\ $
$\displaystyle\mathsf{f}^{\sharp}(x:y)$
$\displaystyle\to\mathsf{f}^{\sharp}(\mathsf{g}(y,[\,]))\qquad$ $\displaystyle
8\colon\leavevmode\nobreak\ $ $\displaystyle\mathsf{g}^{\sharp}(x:y,z)$
$\displaystyle\to\mathsf{g}^{\sharp}(y,x:z)$
and $\operatorname{\mathcal{U}}(\mathcal{P})=\\{3,4\\}$. It is not difficult
to show
$\operatorname{\mathsf{comp}}(n,\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}},{\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}})=\operatorname{\mathsf{O}}(n)$
with a $1$-dimensional RMI. Moreover, the
${\mu}^{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}_{\mathsf{i}}$-monotone
$1$-dimensional RMI $\mathcal{A}$ with
$\displaystyle[\,]_{\mathcal{A}}$ $\displaystyle=0$
$\displaystyle{:}_{\mathcal{A}}(x,y)$ $\displaystyle=y+1$
$\displaystyle\mathsf{g}_{\mathcal{A}}(x,y)$ $\displaystyle=2x+y+1$
$\displaystyle\mathsf{f}_{\mathcal{A}}(x)$
$\displaystyle=\mathsf{f}^{\sharp}_{\mathcal{A}}(x)=x$
$\displaystyle\mathsf{g}^{\sharp}_{\mathcal{A}}(x,y)$ $\displaystyle=0$
$\displaystyle\mathsf{c}_{\mathcal{A}}$
$\displaystyle=\mathsf{d}_{\mathcal{A}}=0$
is compatible with $\operatorname{\mathcal{U}}(\mathcal{P})$. If Lemma 6.11
would be applicable without its well-definedness, linear innermost runtime
complexity of $\mathcal{R}$ would be concluded falsely. Note that
$\operatorname{\Delta}(\mathcal{A},\mathcal{P})$ is _not_ well-defined on
$\mathbb{N}$ due to pair 6.
###### Corollary 6.14.
Let $\mathcal{R}$ be a TRS, $\mathcal{P}$ the set of weak (innermost)
dependency pairs, and $\mu$ be the (innermost) usable replacement map. Suppose
$\mathcal{B}$ is a RMI such that
$(\mathrel{{\succcurlyeq}_{\mathcal{B}}},\mathrel{{\succ}_{\mathcal{B}}})$
forms a $\mu$-complexity pair with
$\operatorname{\mathcal{U}}(\mathcal{P})\subseteq{\mathrel{{\succcurlyeq}_{\mathcal{B}}}}$
and $\mathcal{P}\subseteq{\mathrel{{\succ}_{\mathcal{B}}}}$. Further, suppose
$\mathcal{A}$ is an adequate $\mu$-monotone RMI such that
$\operatorname{\Delta}(\mathcal{A},\mathcal{P})$ is well-defined on
$\mathbb{N}$ and $\mathcal{P}$ is compatible with
$\operatorname{\mathcal{U}}(\mathcal{P})$.
Then the (innermost) runtime complexity function
${\mathrm{rc}}^{(\mathsf{i})}_{\mathcal{R}}$ with respect to $\mathcal{R}$ is
polynomial. Here the degree of the polynomial is given by the maximum of the
degrees of the used RMIs.
Let $\mathcal{A}$ be an RMI as in the corollary. In order to verify that
$\operatorname{\Delta}(\mathcal{A},\mathcal{P})$ is well-defined, we use the
following simple trick in the implementation. Let $l\to r\in\mathcal{P}$ and
let $k$ denotes the cardinality of
$\mathcal{V}\mathsf{ar}(l)\supseteq\mathcal{V}\mathsf{ar}(r)$. Recall the
existence of matrices (over $\mathbb{N}$) $L_{1},\dots,L_{k}$,
$R_{1},\dots,R_{k}$ and vectors $\vec{l}$, $\vec{r}$ such that
$[\alpha]_{\mathcal{A}}(l)\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}[\alpha]_{\mathcal{A}}(r)=\sum_{i=1}^{k}(R_{i}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}L_{i})\alpha(x_{i})+(\vec{r}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}\vec{l})$.
Then $\operatorname{\Delta}(\mathcal{A},\mathcal{P})$ is well-defined if
$(R_{i}\mathbin{\mathchoice{\stackrel{{\scriptstyle\displaystyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\textstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptstyle\cdot}}{{\relbar}}}{\stackrel{{\scriptstyle\scriptscriptstyle\cdot}}{{\relbar}}}}L_{i})\leqslant\mathbf{0}$.
## 7 Weak Dependency Graphs
In this section we extend the above refinements by revisiting dependency
graphs in the context of complexity analysis. Let
$\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{div}})$ and
recall the derivation over
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ on page 6.3. Looking
more closely at this derivation we observe that we do not make use of all weak
dependency pairs in $\mathcal{P}$, but we only employ the pairs $7$ and $8$:
$\mathbf{4}\div^{\sharp}\mathbf{2}\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\\{8\\}/\operatorname{\mathcal{U}}(\mathcal{P})}}\leavevmode\nobreak\
\mathbf{2}\div^{\sharp}\mathbf{2}\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\\{8\\}/\operatorname{\mathcal{U}}(\mathcal{P})}}\leavevmode\nobreak\
\mathsf{0}\div^{\sharp}\mathbf{2}\leavevmode\nobreak\
\mathrel{\mathrel{\to}_{\\{7\\}/\operatorname{\mathcal{U}}(\mathcal{P})}}\leavevmode\nobreak\
\mathsf{c}\hbox to0.0pt{$\;$.\hss}$
Therefore it is a natural idea to modularise our complexity analysis and apply
the previously obtained techniques only to those pairs that are relevant.
Dependencies among weak dependency pairs are formulated by the notion of weak
dependency graphs, which is an easy variant of _dependency graphs_ [6].
###### Definition 7.1.
Let $\mathcal{R}$ be a TRS over a signature $\mathcal{F}$ and let
$\mathcal{P}$ be the set of weak, weak innermost, or (standard) dependency
pairs. The nodes of the _weak dependency graph_
$\operatorname{\mathsf{WDG}}(\mathcal{R})$, _weak innermost dependency graph_
$\operatorname{\mathsf{WIDG}}(\mathcal{R})$, or _dependency graph_
$\operatorname{\mathsf{DG}}(\mathcal{R})$ are the elements of $\mathcal{P}$
and there is an arrow from $s\to t$ to $u\to v$ if and only if there exist a
context $C$ and substitutions
$\sigma,\tau\colon\mathcal{V}\to\mathcal{T}(\mathcal{F},\mathcal{V})$ such
that $t\sigma\mathrel{\to}^{*}C[u\tau]$, where $\mathrel{\to}$ denotes
$\mathrel{\mathrel{\to}_{\mathcal{R}}}$ or
$\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}}$
depending on whether $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$,
$\mathcal{P}=\mathsf{DP}(\mathcal{R})$, or
$\mathcal{P}=\operatorname{\mathsf{WIDP}}(\mathcal{R})$, respectively.
###### Example 7.2 (continued from Example 5.3).
The weak dependency graph
$\operatorname{\mathsf{WDG}}(\mathcal{R}_{\mathsf{div}})$ has the following
form.
6587
Since weak dependency graphs represent call graphs of functions, grouping
mutual parts helps analysis. A graph is called _strongly connected_ if any
node is connected with every other node by a (possibly empty) path. A
_strongly connected component_ (_SCC_ for short) is a maximal strongly
connected subgraph.444We use SCCs in the standard graph theoretic sense, while
in the literature SCCs are sometimes defined as _maximal cycles_ (e.g. [24,
25, 11]). This alternative definition is of limited use in our context.
###### Definition 7.3.
Let $\mathcal{G}$ be a graph, let $\equiv$ denote the equivalence relation
induced by SCCs, and let $\mathcal{P}$ be a SCC in $\mathcal{G}$. Consider the
_congruence graph_ ${\mathcal{G}}_{\equiv}$ induced by the equivalence
relation $\equiv$. The set of all source nodes in ${\mathcal{G}}_{\equiv}$ is
denoted by $\mathsf{Src}({\mathcal{G}}_{\equiv})$. Let
$\mathcal{K}\in{\mathcal{G}}_{\equiv}$ and let $\mathcal{C}$ denote the SCC
represented by $\mathcal{K}$. Then we write $l\to r\in\mathcal{K}$ if $l\to
r\in\mathcal{C}$. For nodes $\mathcal{K}$ and $\mathcal{L}$ in
${\mathcal{G}}_{\equiv}$ we write $\mathcal{K}\mathrel{\leadsto}\mathcal{L}$,
if $\mathcal{K}$ and $\mathcal{L}$ are connected by an edge. The reflexive
(transitive, reflexive-transitive) closure of $\mathrel{\leadsto}$ is denoted
as $\mathrel{\leadsto^{=}}$ ($\mathrel{\leadsto^{+}}$,
$\mathrel{\leadsto^{\ast}}$).
###### Example 7.4 (continued from Example 7.2).
Let $\mathcal{G}$ denote
$\operatorname{\mathsf{WDG}}(\mathcal{R}_{\mathsf{div}})$. There are 4 SCCs in
$\mathcal{G}$: $\\{5\\}$, $\\{6\\}$, $\\{7\\}$, and $\\{8\\}$. Thus the
congruence graph ${\mathcal{G}}_{\equiv}$ has the following form:
6587
Here $\mathsf{Src}({\mathcal{G}}_{\equiv})=\\{\\{6\\},\\{8\\}\\}$.
###### Example 7.5.
Consider the TRS $\mathcal{R}_{\mathsf{gcd}}$ which computes the greatest
common divisor.555This is Example 3.6a in Arts and Giesl’s collection of TRSs
[14].
$\displaystyle 1\colon$ $\displaystyle\mathsf{0}\leqslant y$
$\displaystyle\to\mathsf{true}$ $\displaystyle 6\colon$
$\displaystyle\mathsf{gcd}(\mathsf{0},y)$ $\displaystyle\to y$ $\displaystyle
2\colon$ $\displaystyle\mathsf{s}(x)\leqslant\mathsf{0}$
$\displaystyle\to\mathsf{false}$ $\displaystyle 7\colon$
$\displaystyle\mathsf{gcd}(\mathsf{s}(x),\mathsf{0})$
$\displaystyle\to\mathsf{s}(x)$ $\displaystyle 3\colon$
$\displaystyle\mathsf{s}(x)\leqslant\mathsf{s}(y)$ $\displaystyle\to
x\leqslant y$ $\displaystyle\hskip 12.91663pt8\colon$
$\displaystyle\mathsf{gcd}(\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle\to\mathsf{if_{gcd}}(y\leqslant x,\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle 4\colon$ $\displaystyle x-\mathsf{0}$ $\displaystyle\to x$
$\displaystyle 9\colon$
$\displaystyle\mathsf{if_{gcd}}(\mathsf{true},\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle\to\mathsf{gcd}(x-y,\mathsf{s}(y))$ $\displaystyle 5\colon$
$\displaystyle\mathsf{s}(x)-\mathsf{s}(y)$ $\displaystyle\to x-y$
$\displaystyle 10\colon$
$\displaystyle\mathsf{if_{gcd}}(\mathsf{false},\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle\to\mathsf{gcd}(y-x,\mathsf{s}(x))\hbox to0.0pt{$\;$.\hss}$ The
set $\operatorname{\mathsf{WDP}}(\mathcal{R}_{\mathsf{gcd}})$ consists of the
next ten weak dependency pairs: $\displaystyle 11\colon$
$\displaystyle\mathsf{0}\leqslant^{\sharp}y$ $\displaystyle\to\mathsf{c_{1}}$
$\displaystyle\hskip 12.91663pt16\colon$
$\displaystyle\mathsf{gcd}^{\sharp}(\mathsf{0},y)$ $\displaystyle\to y$
$\displaystyle 12\colon$
$\displaystyle\mathsf{s}(x)\leqslant^{\sharp}\mathsf{0}$
$\displaystyle\to\mathsf{c_{2}}$ $\displaystyle 17\colon$
$\displaystyle\mathsf{gcd}^{\sharp}(\mathsf{s}(x),\mathsf{0})$
$\displaystyle\to x$ $\displaystyle 13\colon$
$\displaystyle\mathsf{s}(x)\leqslant^{\sharp}\mathsf{s}(y)$ $\displaystyle\to
x\leqslant^{\sharp}y$ $\displaystyle 18\colon$
$\displaystyle\mathsf{gcd}^{\sharp}(\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle\to\mathsf{if_{gcd}}^{\sharp}(y\leqslant
x,\mathsf{s}(x),\mathsf{s}(y))$ $\displaystyle 14\colon$
$\displaystyle\mathsf{s}(x)-^{\sharp}\mathsf{0}$ $\displaystyle\to x$
$\displaystyle 19\colon$
$\displaystyle\mathsf{if_{gcd}}^{\sharp}(\mathsf{true},\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle\to\mathsf{gcd}^{\sharp}(x-y,\mathsf{s}(y))$ $\displaystyle
15\colon$ $\displaystyle\mathsf{s}(x)-^{\sharp}\mathsf{s}(y)$
$\displaystyle\to x-^{\sharp}y$ $\displaystyle 20\colon$
$\displaystyle\mathsf{if_{gcd}}^{\sharp}(\mathsf{false},\mathsf{s}(x),\mathsf{s}(y))$
$\displaystyle\to\mathsf{gcd}^{\sharp}(y-x,\mathsf{s}(x))\hbox
to0.0pt{$\;$.\hss}$
The congruence graph ${\mathcal{G}}_{\equiv}$ of
$\mathcal{G}\mathrel{:=}\operatorname{\mathsf{WDG}}(\mathcal{R}_{\mathsf{gcd}})$
has the following form:
1113121514{18,19,20}1617
Here
$\mathsf{Src}({\mathcal{G}}_{\equiv})=\\{\\{13\\},\\{15\\},\\{17\\},\\{18,19,20\\}\\}$.
The main result in this section is stated as follows: Let $\mathcal{R}$ be a
TRS, $\mathcal{P}=\operatorname{\mathsf{WDP}}(\mathcal{R})$,
$\mathcal{G}=\operatorname{\mathsf{WDG}}(\mathcal{R})$, and furthermore
$\operatorname{\mathsf{L}}(t)\mathrel{:=}\max\\{{\mathsf{dh}}(t,\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}})\mid\text{$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$
is a path in ${\mathcal{G}}_{\equiv}$ and
$\mathcal{P}_{1}\in\mathsf{Src}({\mathcal{G}}_{\equiv})$}\\}\hbox
to0.0pt{$\;$,\hss}$
where $\mathcal{Q}=\bigcup_{i=1}^{k}\mathcal{P}_{i}$. Then,
${\mathsf{dh}}(t,{\mathrel{\mathrel{\to}_{\mathcal{R}}}})=\operatorname{\mathsf{O}}(\operatorname{\mathsf{L}}(t))$
holds for all basic term $t$. This means that one may decompose
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ into several smaller
fragments and analyse these fragments separately.
Reconsider the derivation on page 7. The only dependency pairs are from the
set $\\{7,8\\}$. Observe that the order these pairs are applied is
representable by the path $(\\{8\\},\\{7\\})$ in the congruence graph. This
observation is cast into the following definition.
###### Definition 7.6.
Let $\mathcal{P}$ be the set of weak (innermost) dependency pairs and let
$\mathcal{G}$ denote the weak (innermost) dependency graph. Suppose
$A\colon{s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}$
denote a derivation, such that
$s\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$. If $A$ can be written
in the following form:
${s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{1}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{k}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}\hbox
to0.0pt{$\;$,\hss}$
then $A$ is _based on the sequence of nodes
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ (in ${\mathcal{G}}_{\equiv}$)_.
The next lemma is an easy generalisation of the above example.
###### Lemma 7.7.
Let $\mathcal{R}$ be a TRS, let $\mathcal{P}$ be the set of weak (innermost)
dependency pairs and let $\mathcal{G}$ denote the weak (innermost) dependency
graph. Suppose that all compound symbols are nullary. Then any derivation
$A\colon{s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}$
such that $s\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$ is based on a
path in ${\mathcal{G}}_{\equiv}$.
From Lemma 7.7 we see that the above mentioned modularity result easily
follows as long as the arity of the compound symbols is restricted. We lift
the assumption that all compound symbols are nullary. Perhaps surprisingly
this generalisation complicates the matter. As exemplified by the next
example, Lemma 7.7 fails if there exist non-nullary compound symbols.
###### Example 7.8.
Consider the TRS
$\mathcal{R}=\\{\mathsf{f}(\mathsf{0})\to\mathsf{a},\mathsf{f}(\mathsf{s}(x))\to\mathsf{b}(\mathsf{f}(x),\mathsf{f}(x))\\}$.
The set $\operatorname{\mathsf{WDP}}(\mathcal{R})$ consists of the two weak
dependency pairs: $1\colon\mathsf{f}^{\sharp}(\mathsf{0})\to\mathsf{c}$ and
$2\colon\mathsf{f}^{\sharp}(\mathsf{s}(x))\to\mathsf{d}(\mathsf{f}^{\sharp}(x),\mathsf{f}^{\sharp}(x))$.
The corresponding congruence graph only contains the single edge from
$\\{2\\}$ to $\\{1\\}$. Writing $t_{n}$ for
$\mathsf{f}^{\sharp}(\mathsf{s}^{n}(\mathsf{0}))$, we have the sequence
$\displaystyle t_{2}$
$\displaystyle\to_{\\{2\\}}^{2}\mathsf{d}(\mathsf{d}(t_{0},t_{0}),t_{1})\mathrel{\mathrel{\to}_{\\{1\\}}}\mathsf{d}(\mathsf{d}(\mathsf{c},t_{0}),t_{1})$
$\displaystyle\mathrel{\mathrel{\to}_{\\{2\\}}}\mathsf{d}(\mathsf{c}(\mathsf{c},t_{0}),\mathsf{d}(t_{0},t_{0}))\to_{\\{1\\}}^{3}\mathsf{d}(\mathsf{d}(\mathsf{c},\mathsf{c}),\mathsf{d}(\mathsf{c},\mathsf{c}))\hbox
to0.0pt{$\;$.\hss}$
whereas $(\\{2\\},\\{1\\},\\{2\\},\\{1\\})$ is not a path in the graph.
Note that the derivation in Example 7.8 can be reordered (without affecting
its length) such that the derivation becomes based on the path
$(\\{2\\},\\{1\\})$. More generally, we observe that a weak (innermost)
dependency pair containing an $m$-ary ($m>1$) compound symbol can induce $m$
_independent_ derivations. This allows us to reorder (sub-)derivations. We
show this via the following sequence of lemmas.
Let $\mathcal{R}$ be a TRS, let $\mathcal{P}$ denote the set of weak
(innermost) dependency pairs, and let $\mathcal{G}$ denote the weak
(innermost) dependency graph. The set
$\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$ is inductively defined as
follows (i)
$\mathcal{T}^{\sharp}\cup\mathcal{T}\subseteq\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$,
where $\mathcal{T}^{\sharp}=\\{t^{\sharp}\mid t\in\mathcal{T}\\}$ and (ii)
$c(t_{1},\ldots,t_{n})\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$,
whenever
$t_{1},\ldots,t_{n}\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$ and
$c$ a compound symbol. The next lemma formalises an easy observation.
###### Lemma 7.9.
Let $\mathcal{C}$ be a set of nodes in $\mathcal{G}$ and let
$A\colon{t=t_{0}}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{C}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t_{n}}$
denote a derivation based on $\mathcal{C}$ with
$t\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$. Then $A$ has the
following form:
$t=t_{0}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{C}/\operatorname{\mathcal{U}}(\mathcal{P})}}t_{1}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{C}/\operatorname{\mathcal{U}}(\mathcal{P})}}\dots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{C}/\operatorname{\mathcal{U}}(\mathcal{P})}}t_{n}$
where each $t_{i}\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$.
A key is that consecutive two weak dependency pairs may be swappable.
###### Lemma 7.10.
Let $\mathcal{K}$ and $\mathcal{L}$ denote two different nodes in
${\mathcal{G}}_{\equiv}$ such that there is no edge from $\mathcal{K}$ to
$\mathcal{L}$. Let $s\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$ and
suppose the existence of a derivation $A$ of the following form:
${s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{K}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdot\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{L}/\operatorname{\mathcal{U}}(\mathcal{P})}}t\hbox
to0.0pt{$\;$.\hss}$
Then there exists a derivation $B$
${s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{L}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdot\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{K}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}\hbox
to0.0pt{$\;$,\hss}$
such that $\lvert A\rvert=\lvert B\rvert$.
###### Proof.
We only show the full rewriting case since the innermost case is analogous.
According to Lemma 7.9 an arbitrary terms $u$ reachable from $s$ belongs to
$\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$. Writing
${C\langle{u_{1},\ldots,u_{i},\ldots,u_{m}}\rangle}_{\mathcal{F}\cup\mathcal{F}^{\sharp}}$
for $u$, the $m$-hole context $C$ consists of compound symbols and variables,
$u_{1},\ldots,u_{m}\in\mathcal{T}\cup\mathcal{T}^{\sharp}$. Therefore, $A$ can
be written in the following form:
$\displaystyle s$ $\displaystyle\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{n_{1}}\leavevmode\nobreak\ $
$\displaystyle{C\langle{u_{1},\ldots,u_{i},\ldots,u_{m}}\rangle}_{\mathcal{F}\cup\mathcal{F}^{\sharp}}$
$\displaystyle=:u$ $\displaystyle\leavevmode\nobreak\
\to_{\mathcal{L}}\leavevmode\nobreak\ $ $\displaystyle
C[u_{1},\ldots,u_{i}^{\prime},\ldots,u_{m}]$
$\displaystyle\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{n_{2}}\leavevmode\nobreak\ $
$\displaystyle C[v_{1},\ldots,v_{i},\ldots,v_{j},\ldots,v_{m}]$
$\displaystyle\leavevmode\nobreak\ \to_{\mathcal{K}}\leavevmode\nobreak\ $
$\displaystyle C[v_{1},\ldots,v_{i},\ldots,v_{j}^{\prime},\ldots,v_{m}]$
$\displaystyle\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{n_{3}}\leavevmode\nobreak\
t\hbox to0.0pt{$\;$,\hss}$
with $u_{i}^{\prime}\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{k}v_{i}$.
Here $i\neq j$ holds, because $i=j$ induces $\mathcal{L}\leadsto\mathcal{K}$.
Easy induction on $n_{2}$ shows
$\displaystyle s$ $\displaystyle\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{n_{1}}\leavevmode\nobreak\
u\leavevmode\nobreak\ =\leavevmode\nobreak\ $ $\displaystyle
C[u_{1},\ldots,u_{i},\ldots,u_{j},\ldots,u_{m}]$
$\displaystyle\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{n_{2}-k}\leavevmode\nobreak\ $
$\displaystyle C[v_{1},\ldots,u_{i},\ldots,v_{j},\ldots,v_{m}]$
$\displaystyle\leavevmode\nobreak\ \to_{\mathcal{K}}\leavevmode\nobreak\ $
$\displaystyle C[v_{1},\ldots,u_{i},\ldots,v_{j}^{\prime},\ldots,v_{m}]$
$\displaystyle\leavevmode\nobreak\ \to_{\mathcal{L}}\leavevmode\nobreak\ $
$\displaystyle
C[v_{1},\ldots,u_{i}^{\prime},\ldots,v_{j}^{\prime},\ldots,v_{m}]$
$\displaystyle\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{k}\leavevmode\nobreak\ $
$\displaystyle
C[v_{1},\ldots,v_{i},\ldots,v_{j}^{\prime},\ldots,v_{m}]\leavevmode\nobreak\
\to_{\operatorname{\mathcal{U}}(\mathcal{P})}^{n_{3}}t\leavevmode\nobreak\
\hbox to0.0pt{$\;$,\hss}$
which is the desired derivation $B$. ∎
The next lemma states that reordering is partly possible.
###### Lemma 7.11.
Let $s\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$, and let
$A\colon{s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}$
be a derivation based on a sequence of nodes
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ such that
$\mathcal{P}_{1}\in\mathsf{Src}({\mathcal{G}}_{\equiv})$, and let
$(\mathcal{Q}_{1},\ldots,\mathcal{Q}_{\ell})$ be a path in
${\mathcal{G}}_{\equiv}$ with
$\\{\mathcal{P}_{1},\dots,\mathcal{P}_{k}\\}=\\{\mathcal{Q}_{1},\dots,\mathcal{Q}_{\ell}\\}$.
Then there exists a derivation
$B\colon{s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}$
based on $(\mathcal{Q}_{1},\ldots,\mathcal{Q}_{\ell})$ such that $\lvert
A\rvert=\lvert B\rvert$ and $\mathcal{P}_{1}=\mathcal{Q}_{1}$.
###### Proof.
According to Lemma 7.9, for any derivation $A$
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{1}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{n}/\operatorname{\mathcal{U}}(\mathcal{P})}}t\hbox
to0.0pt{$\;$,\hss}$
if $\mathcal{P}_{i}\mathrel{\leadsto}\mathcal{P}_{i+1}$ does not hold, there
is a derivation $B$
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{1}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{i+1}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdot\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{i}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}_{n}/\operatorname{\mathcal{U}}(\mathcal{P})}}t\hbox
to0.0pt{$\;$,\hss}$
with $\lvert A\rvert=\lvert B\rvert$. By assumption
$(\mathcal{Q}_{1},\ldots,\mathcal{Q}_{\ell})$ is a path, whence we obtain
$\mathcal{Q}_{1}\mathrel{\leadsto}\cdots\mathrel{\leadsto}\mathcal{Q}_{\ell}$.
By performing bubble sort with respect to $\mathrel{\leadsto^{+}}$, $A$ is
transformed into the derivation $B$:
$s\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{Q}_{1}/\operatorname{\mathcal{U}}(\mathcal{P})}}\cdots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{Q}_{m}/\operatorname{\mathcal{U}}(\mathcal{P})}}t\hbox
to0.0pt{$\;$,\hss}$
such that $\lvert A\rvert=\lvert B\rvert$. ∎
The next example shows that there is a derivation that cannot be transformed
into a derivation based on a path.
###### Example 7.12.
Consider the TRS
$\mathcal{R}=\\{\mathsf{f}\to\mathsf{b}(\mathsf{g},\mathsf{h}),\mathsf{g}\to\mathsf{a},\mathsf{h}\to\mathsf{a}\\}$.
Thus $\operatorname{\mathsf{WDP}}(\mathcal{R})$ consists of three dependency
pairs:
$1\colon\mathsf{f}^{\sharp}\to\mathsf{c}(\mathsf{g}^{\sharp},\mathsf{h}^{\sharp})$,
$2\colon\mathsf{g}^{\sharp}\to\mathsf{d}$, and
$3\colon\mathsf{h}^{\sharp}\to\mathsf{e}$. Let
$\mathcal{P}\mathrel{:=}\operatorname{\mathsf{WDP}}(\mathcal{R})$ and let
$\mathcal{G}\mathrel{:=}\operatorname{\mathsf{WDG}}(\mathcal{R})$. Note that
${\mathcal{G}}_{\equiv}$ are identical to $\mathcal{G}$. We witness that the
derivation
$\mathsf{f}^{\sharp}\mathrel{\mathrel{\to}_{\mathcal{P}}}\mathsf{c}(\mathsf{g}^{\sharp},\mathsf{h}^{\sharp})\mathrel{\mathrel{\to}_{\mathcal{P}}}\mathsf{c}(\mathsf{d},\mathsf{h}^{\sharp})\mathrel{\mathrel{\to}_{\mathcal{P}}}\mathsf{c}(\mathsf{d},\mathsf{e})\hbox
to0.0pt{$\;$,\hss}$
is based neither on the path $(\\{1\\},\\{2\\})$, nor on the path
$(\\{1\\},\\{3\\})$.
Lemma 7.11 shows that we can reorder a given derivation $A$ that is based on a
sequence of nodes that would in principle form a path in the congruence graph
${\mathcal{G}}_{\equiv}$. The next lemma shows that we can guarantee that any
derivation is based on sequence of different paths.
###### Lemma 7.13.
Let $s\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$ and let
$A\colon{s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}$
be a derivation based on
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k},\mathcal{Q}_{1},\ldots,\mathcal{Q}_{\ell})$,
such that $(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ and
$(\mathcal{Q}_{1},\ldots,\mathcal{Q}_{\ell})$ form two disjoint paths in
$\mathcal{G}$. Then there exists a derivation
$B\colon{s}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{P}/\operatorname{\mathcal{U}}(\mathcal{P})}}{t}$
based on the sequence of nodes
$(\mathcal{Q}_{1},\ldots,\mathcal{Q}_{\ell},\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$
such that $\lvert A\rvert=\lvert B\rvert$.
###### Proof.
The lemma follows by an adaptation of the technique in the proof of Lemma
7.11. ∎
Lemma 7.13 shows that the maximal length of any derivation only differs from
the maximal length of any derivation based on a path by a linear factor,
depending on the size of the congruence graph ${\mathcal{G}}_{\equiv}$. We
arrive at the main result of this section. Recall the definition of
$\operatorname{\mathsf{L}}(\cdot)$ on page 7.5.
###### Theorem 7.14.
Let $\mathcal{R}$ be a TRS and $\mathcal{P}$ the set of weak (innermost)
dependency pairs. Then,
${\mathsf{dh}}(t,\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{R}}})=\operatorname{\mathsf{O}}(\operatorname{\mathsf{L}}(t))$
holds for all $t\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$.
###### Proof.
Let $a$ denotes the maximum arity of compound symbols and $K$ denotes the
number of SCCs in the weak (innermost) dependency graph $\mathcal{G}$. We show
${\mathsf{dh}}(s,\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{R}}})\leqslant
a^{K}\cdot\operatorname{\mathsf{L}}(s)$ holds for all
$s\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$. Theorem 5.12 yields
that
${{\mathsf{dh}}(s,\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{R}}})}={{\mathsf{dh}}(s,\mathrel{\to})}$,
where $\mathrel{\to}$ either denotes
$\mathrel{\mathrel{\to}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}}$
or
$\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}}$.
Let $A\colon{s}\mathrel{\to}^{\ast}{t}$ be a derivation over
$\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})$ such that
$s\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$. Then $A$ is based on a
sequence of nodes in the congruence graph ${\mathcal{G}}_{\equiv}$ such that
there exists a maximal (with respect to subset inclusion) components of
${\mathcal{G}}_{\equiv}$ that includes all these nodes. Let $T$ denote this
maximal component. $T$ forms a directed acyclic graph. In order to
(over-)estimate the number of nodes in this graph we can assume without loss
of generality that $T$ is a tree with root in
$\mathsf{Src}({\mathcal{G}}_{\equiv})$. Note that $K$ bounds the height of
this tree. Thus the number of nodes in the component $T$ is less than
$\frac{a^{K}-1}{a-1}\leqslant a^{K}\hbox to0.0pt{$\;$.\hss}$
Due to Lemma 7.13 the derivation $A$ is conceivable as a sequence of
subderivations based on paths in ${\mathcal{G}}_{\equiv}$. As the number of
nodes in $T$ is bounded from above by $a^{K}$, there exist at most be $a^{K}$
different paths through $T$.
Hence in order to estimate $\lvert A\rvert$, it suffices to estimate the
length of any subderivation $B$ of $A$, based on a specific path. Let
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ be a path in
${\mathcal{P}}_{\equiv}$ such that
$\mathcal{P}_{1}\in\mathsf{Src}({\mathcal{G}}_{\equiv})$ and let $B\colon
u\mathrel{\to}^{n}v$, denote a derivation based on this path. Let
$\mathcal{Q}\mathrel{:=}\bigcup_{i=1}^{k}\mathcal{P}_{i}$. By Definition 7.6
and the definition of usable rules, the derivation $B$ can be written as:
$u=u_{0}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{P}_{1}/\operatorname{\mathcal{U}}(\mathcal{Q})}}u_{n_{1}}\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{P}_{2}/{\operatorname{\mathcal{U}}(\mathcal{Q})}}}\cdots\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}_{\mathcal{P}_{k}/{\operatorname{\mathcal{U}}(\mathcal{Q})}}}u_{n}=v\hbox
to0.0pt{$\;$,\hss}$
where $u\in\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$ each
$u_{i}\in\operatorname{\mathcal{T}^{\sharp}_{\mathsf{c}}}$. Hence $B$ is
contained in
$u\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$(\mathrm{i})$}}}}^{\ast}_{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}}v$
and thus $\lvert B\rvert\leqslant\operatorname{\mathsf{L}}(u)$ by definition.
As the length of a derivation $B$ based on a specific path can be estimated by
$\operatorname{\mathsf{L}}(s)$, we obtain that the length of an arbitrary
derivation is less than $a^{K}\cdot\operatorname{\mathsf{L}}(s)$. This
completes the proof of the theorem. ∎
###### Corollary 7.15.
Let $\mathcal{R}$ be a TRS and let $\mathcal{G}$ denote the weak (innermost)
dependency graph. For every path
$\bar{P}\mathrel{:=}(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ in
${\mathcal{G}}_{\equiv}$ such that
$\mathcal{P}_{1}\in\mathsf{Src}({\mathcal{G}}_{\equiv})$, we set
$\mathcal{Q}\mathrel{:=}\bigcup_{i=1}^{k}\mathcal{P}_{i}$ and suppose
1. 1)
there exist a
${\mu}^{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}_{\mathsf{f}}$-monotone
(${\mu}^{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}_{\mathsf{i}}$-monotone)
and adequate RMI $\mathcal{A}_{\bar{P}}$ that admits the weight gap
$\operatorname{\operatorname{\Delta}}(\mathcal{A}_{\bar{P}},\mathcal{Q})$ on
$\operatorname{\mathcal{T}_{\mathsf{b}}^{\sharp}}$ and $\mathcal{A}_{\bar{P}}$
is compatible with the usable rules $\operatorname{\mathcal{U}}(\mathcal{Q})$,
2. 2)
there exists a
${\mu}^{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}_{\mathsf{f}}$-monotone
(${\mu}^{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}_{\mathsf{i}}$-monotone)
RMI $\mathcal{B}_{\bar{P}}$ such that
$(\mathrel{{\succcurlyeq}_{\mathcal{B}_{\bar{P}}}},\mathrel{{\succ}_{\mathcal{B}_{\bar{P}}}})$
forms a complexity pair for
$\mathcal{P}_{k}/{\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{k-1}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}$,
and
Then the (innermost) runtime complexity of a TRS $\mathcal{R}$ is polynomial.
Here the degree of the polynomial is given by the maximum of the degrees of
the used RMIs.
###### Proof.
We restrict our attention to weak dependency pairs and full rewriting. First
observe that the assumptions imply that any basic term
$t\in\operatorname{\mathcal{T}_{\mathsf{b}}}$ is terminating with respect to
$\mathcal{R}$. Let $\mathcal{P}$ be the set of weak dependency pairs. (Note
that $\mathcal{P}\supseteq\mathcal{Q}$.) By Lemma 5.11 any infinite derivation
with respect to $\mathcal{R}$ starting in $t$ can be translated into an
infinite derivation with respect to
$\operatorname{\mathcal{U}}(\mathcal{P})\cup\mathcal{P}$. Moreover, as the
number of paths in ${\mathcal{G}}_{\equiv}$ is finite, there exist a path
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ in ${\mathcal{G}}_{\equiv}$ and an
infinite rewrite sequence based on this path. This is a contradiction. Hence
we can employ Theorem 6.5 in the following.
Let $(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ be an arbitrary, but fixed path
in the congruence graph ${\mathcal{G}}_{\equiv}$, let
$\mathcal{Q}=\bigcup_{i=1}^{k}\mathcal{P}_{i}$, and let $d$ denote the maximum
of the degrees of the used RMIs. Due to Theorem 6.5 there exists
$c\in\mathbb{N}$ such that:
${\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}})\leqslant(1+\operatorname{\operatorname{\Delta}}(\mathcal{A}_{\bar{P}},\mathcal{Q}))\cdot{\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\mathcal{Q}/\operatorname{\mathcal{U}}(\mathcal{Q})}})+c\cdot\lvert
t\rvert^{d}\hbox to0.0pt{$\;$.\hss}$
Due to Theorem 7.14 it suffices to consider a derivation $A$ based on the path
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$. Suppose $A\colon
s\mathrel{\to^{n}_{\mathcal{Q}/\operatorname{\mathcal{U}}(\mathcal{Q})}}t$.
Then $A$ can be represented as follows:
$s=s_{0}\mathrel{\to^{n_{1}}_{\mathcal{P}_{1}/\operatorname{\mathcal{U}}(\mathcal{P}_{1})}}s_{n_{1}}\mathrel{\to^{n_{2}}_{\mathcal{P}_{2}/{\operatorname{\mathcal{U}}(\mathcal{P}_{1})\cup\operatorname{\mathcal{U}}(\mathcal{P}_{2})}}}\cdots\mathrel{\to^{n_{k}}_{\mathcal{P}_{k}/{\operatorname{\mathcal{U}}(\mathcal{P}_{1})\cup\cdots\cup\operatorname{\mathcal{U}}(\mathcal{P}_{k})}}}s_{n}=t\hbox
to0.0pt{$\;$,\hss}$
such that $n=\sum_{i=1}^{k}n_{i}$. It is sufficient to bound each $n_{i}$ from
the above. Fix $i\in\\{1,\dots,k\\}$. Consider the subderivation
$A^{\prime}\colon
s=s_{0}\mathrel{\to^{n_{1}}_{\mathcal{P}_{1}/\operatorname{\mathcal{U}}(\mathcal{P}_{1})}}s_{n_{1}}\cdots\mathrel{\to^{n_{i}}_{\mathcal{P}_{k}/{\operatorname{\mathcal{U}}(\mathcal{P}_{1})\cup\cdots\cup\operatorname{\mathcal{U}}(\mathcal{P}_{i})}}}s_{n_{i}}\hbox
to0.0pt{$\;$.\hss}$
Then $A^{\prime}$ is contained in $A^{\prime\prime}\colon
s\mathrel{\mathrel{\to}_{\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{i-1}\cup\operatorname{\mathcal{U}}(\mathcal{P}_{1})\cup\cdots\operatorname{\mathcal{U}}(\mathcal{P}_{i})}^{\ast}}\cdot\mathrel{\to^{n_{i}}_{\mathcal{P}_{k}/{\operatorname{\mathcal{U}}(\mathcal{P}_{1})\cup\cdots\cup\operatorname{\mathcal{U}}(\mathcal{P}_{i})}}}s_{n_{i}}$.
Let $\hat{P_{i}}\mathrel{:=}(\mathcal{P}_{1},\ldots,\mathcal{P}_{i})$. By
assumption there exists a $\mu$-monotone complexity pair
$(\mathrel{{\succcurlyeq}_{\mathcal{B}_{\hat{P_{i}}}}},\mathrel{{\succ}_{\mathcal{B}_{\hat{P_{i}}}}})$
such that
$\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{i-1}\cup\operatorname{\mathcal{U}}(\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{i})\subseteq{\mathrel{{\succcurlyeq}_{\mathcal{B}_{\hat{P_{i}}}}}}$
and $\mathcal{P}_{i}\subseteq{\mathrel{{\succ}_{\mathcal{B}_{\hat{P_{i}}}}}}$.
Hence, we obtain
$n_{i}\leqslant([\alpha_{0}]_{\mathcal{B}_{\hat{P_{i}}}}(s))_{1}$ and in sum
${n}\leqslant{k\cdot\lvert s\rvert^{d}}$. Finally, defining the polynomial $p$
as follows:
$p(x)\mathrel{:=}(1+\operatorname{\operatorname{\Delta}}(\mathcal{A}_{\bar{P}},\mathcal{Q}))\cdot
k\cdot x^{d}+c\cdot x^{d}\hbox to0.0pt{$\;$,\hss}$
we conclude
${\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\to}_{\mathcal{Q}\cup\operatorname{\mathcal{U}}(\mathcal{Q})}})\leqslant
p(\lvert t\rvert)$. Note that the polynomial $p$ depends only on the algebras
$\mathcal{A}_{\bar{P}}$ and $\mathcal{B}_{\hat{P_{1}}}$, …,
$\mathcal{B}_{\bar{P_{k}}}$.
As the path $(\mathcal{P}_{1},\ldots,\mathcal{P}_{k})$ was chosen arbitrarily,
there exists a polynomial $q$, depending only on the employed RMIs such that
$\operatorname{\mathsf{L}}(t)\leqslant q(\lvert t\rvert)$. Thus the corollary
follows due to Theorem 7.14. ∎
Let $t$ be an arbitrary term. By definition the set in
$\operatorname{\mathsf{L}}(t)$ may consider
$2^{\operatorname{\mathsf{O}}(n)}$-many paths, where $n$ denotes the number of
nodes in ${\mathcal{G}}_{\equiv}$. However, it suffices to restrict the
definition on page 7.5 to _maximal_ paths. For this refinement
$\operatorname{\mathsf{L}}(t)$ contains at most $n^{2}$ paths. This fact we
employ in implementing the WDG method.
###### Example 7.16 (continued from Example 7.5).
For ${\operatorname{\mathsf{WDG}}(\mathcal{R}_{\mathsf{gcd}})}_{\equiv}$ the
above set consists of 8 paths: $(\\{13\\})$, $(\\{13\\},\\{11\\})$,
$(\\{13\\},\\{12\\})$, $(\\{15\\})$, $(\\{15\\},\\{14\\})$, $(\\{17\\})$,
$(\\{18,19,20\\})$, and $(\\{18,19,20\\},\\{16\\})$. In the following we only
consider the last three paths, since all other paths are similarly handled.
* •
Consider $(\\{17\\})$. Note
$\operatorname{\mathcal{U}}(\\{17\\})=\varnothing$. By taking an arbitrary SLI
$\mathcal{A}$ and the linear restricted interpretation $\mathcal{B}$ with
$\mathsf{gcd}^{\sharp}_{\mathcal{B}}(x,y)=x$ and
$\mathsf{s}_{\mathcal{B}}(x)=x+1$, we have
$\varnothing\subseteq{>_{\mathcal{A}}}$,
$\varnothing\subseteq{\geqslant_{\mathcal{B}}}$, and
$\\{17\\}\subseteq{>_{\mathcal{B}}}$.
* •
Consider $(\\{18,19,20\\})$. Note
$\operatorname{\mathcal{U}}(\\{18,19,20\\})=\\{1,\ldots,5\\}$. The following
RMI $\mathcal{A}$ is adequate for $(\\{18,19,20\\})$ and strictly monotone on
${\mu}^{\mathcal{P}\cup\operatorname{\mathcal{U}}(\mathcal{P})}_{\mathsf{f}}$.
The presentation of $\mathcal{A}$ is succinct as only the signature of the
usable rules $\\{1,\ldots,5\\}$ is of interest.
$\displaystyle\mathsf{true}_{\mathcal{A}}$
$\displaystyle=\mathsf{false}_{\mathcal{A}}=\mathsf{0}_{\mathcal{A}}=\vec{0}$
$\displaystyle\mathsf{s}_{\mathcal{A}}(\vec{x})$
$\displaystyle=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}\vec{x}+\begin{pmatrix}3\\\ 1\end{pmatrix}$
$\displaystyle{\leqslant}_{\mathcal{A}}(\vec{x},\vec{y})$
$\displaystyle=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\vec{y}+\begin{pmatrix}1\\\ 3\end{pmatrix}$
$\displaystyle{-}_{\mathcal{A}}(\vec{x},\vec{y})$
$\displaystyle=\vec{x}+\begin{pmatrix}2\\\ 3\end{pmatrix}\hbox
to0.0pt{$\;$.\hss}$
Further, consider the RMI $\mathcal{B}$ giving rise to the complexity pair
$({\mathrel{{\succcurlyeq}_{\mathcal{B}}}},{\mathrel{{\succ}_{\mathcal{B}}}})$.
$\displaystyle\mathsf{0}_{\mathcal{B}}$
$\displaystyle=\makebox[0.0pt][l]{$\mathsf{true}_{\mathcal{B}}=\mathsf{false}_{\mathcal{B}}=\mathsf{\leqslant}_{\mathcal{B}}(\vec{x},\vec{y})=\vec{0}$}$
$\displaystyle\mathsf{s}_{\mathcal{B}}(\vec{x})$
$\displaystyle=\begin{pmatrix}1&3\\\
0&0\end{pmatrix}\vec{x}+\begin{pmatrix}3\\\ 0\end{pmatrix}$
$\displaystyle{-}_{\mathcal{B}}(\vec{x},\vec{y})$
$\displaystyle=\begin{pmatrix}1&0\\\
2&2\end{pmatrix}\vec{x}+\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$
$\displaystyle\mathsf{if_{gcd}}^{\sharp}_{\mathcal{B}}(x,y,z)$
$\displaystyle=\begin{pmatrix}3&0\\\
0&0\end{pmatrix}\vec{y}+\begin{pmatrix}3&0\\\ 0&0\end{pmatrix}\vec{z}$
$\displaystyle\mathsf{gcd}^{\sharp}_{\mathcal{B}}(x,y)$
$\displaystyle=\makebox[0.0pt][l]{$\begin{pmatrix}3&0\\\
0&0\end{pmatrix}\vec{x}+\begin{pmatrix}3&0\\\
0&0\end{pmatrix}\vec{y}+\begin{pmatrix}2\\\ 0\end{pmatrix}$ \hbox
to0.0pt{$\;$.\hss}}$
We obtain $\\{1,\ldots,5\\}\subseteq{\mathrel{{\succ}_{\mathcal{A}}}}$,
$\\{1,\ldots,5\\}\subseteq{\mathrel{{\succcurlyeq}_{\mathcal{B}}}}$, and
$\\{18,19,20\\}\subseteq{\mathrel{{\succ}_{\mathcal{B}}}}$.
* •
Consider $(\\{18,19,20\\},\\{16\\})$. Note
$\operatorname{\mathcal{U}}(\\{16\\})=\varnothing$. By taking the same
$\mathcal{A}$ and also $\mathcal{B}$ as above, we have
$\\{1,\ldots,5\\}\subseteq{\mathrel{{\succ}_{\mathcal{A}}}}$,
$\\{1,\ldots,5,18,19,20\\}\subseteq{\mathrel{{\succcurlyeq}_{\mathcal{B}}}}$,
and $\\{16\\}\subseteq{\mathrel{{\succ}_{\mathcal{B}}}}$.
Thus, all path constraints are handled by suitably defined RMIs of dimension
2. Hence, the runtime complexity function of $\mathcal{R}_{\mathsf{gcd}}$ is
at most quadratic, which is unfortunately not optimal, as
$\mathsf{rc}_{\mathcal{R}_{\mathsf{gcd}}}$ is linear.
Corollary 7.15 is more powerful than Corollary 6.14. We illustrate it with a
small example.
###### Example 7.17.
Consider the TRS $\mathcal{R}$
$\displaystyle\mathsf{f}(\mathsf{a},\mathsf{s}(x),y)$
$\displaystyle\to\mathsf{f}(\mathsf{a},x,\mathsf{s}(y))$
$\displaystyle\mathsf{f}(\mathsf{b},x,\mathsf{s}(y))$
$\displaystyle\to\mathsf{f}(\mathsf{b},\mathsf{s}(x),y)\hbox
to0.0pt{$\;$.\hss}$
Its weak dependency pairs $\operatorname{\mathsf{WDP}}(\mathcal{R})$ are
$\displaystyle 1\colon\leavevmode\nobreak\
\mathsf{f}^{\sharp}(\mathsf{a},\mathsf{s}(x),y)$
$\displaystyle\to\mathsf{f}^{\sharp}(\mathsf{a},x,\mathsf{s}(y))$
$\displaystyle 2\colon\leavevmode\nobreak\
\mathsf{f}^{\sharp}(\mathsf{b},x,\mathsf{s}(y))$
$\displaystyle\to\mathsf{f}^{\sharp}(\mathsf{b},\mathsf{s}(x),y)\hbox
to0.0pt{$\;$.\hss}$
The corresponding congruence graph consists of the two isolated nodes
$\\{1\\}$ and $\\{2\\}$. It is not difficult to find suitable $1$-dimensional
RMIs for the nodes, and therefore
$\mathsf{rc}_{\mathcal{R}}(n)=\operatorname{\mathsf{O}}(n)$ is concluded. On
the other hand, it can be verified that the linear runtime complexity cannot
be obtained by Corollary 6.14 with a $1$-dimensional RMI.
We conclude this section with a brief comparison of the path analysis
developed here and the use of the dependency graph refinement in termination
analysis. First we recall a theorem on the dependency graph refinement in
conjunction with usable rules and innermost rewriting (see [24], but also
[25]). Similar results hold in the context of full rewriting, see [21, 22].
###### Theorem 7.18 ([24]).
A TRS $\mathcal{R}$ is innermost terminating if for every maximal cycle
$\mathcal{C}$ in the dependency graph
$\operatorname{\mathsf{DG}}(\mathcal{R})$ there exists a reduction pair
$(\gtrsim,\succ)$ such that
${\operatorname{\mathcal{U}}(\mathcal{C})}\subseteq{\gtrsim}$ and
${\mathcal{C}}\subseteq{\succ}$.
The following example shows that in the context of complexity analysis it is
_not_ sufficient to consider each cycle individually.
###### Example 7.19 (continued from Example 6.9).
Consider the TRS $\mathcal{R}_{\mathsf{exp}}$ introduced in Example 6.9.
$\displaystyle\mathsf{exp}(\mathsf{0})$
$\displaystyle\to\mathsf{s}(\mathsf{0})$ $\displaystyle\mathsf{d}(\mathsf{0})$
$\displaystyle\to\mathsf{0}$ $\displaystyle\mathsf{exp}(\mathsf{r}(x))$
$\displaystyle\to\mathsf{d}(\mathsf{exp}(x))$
$\displaystyle\mathsf{d}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{s}(\mathsf{s}(\mathsf{d}(x)))\hbox
to0.0pt{$\;$.\hss}$
Recall that the (innermost) runtime complexity of $\mathcal{R}_{\mathsf{exp}}$
is exponential. Let $\mathcal{P}$ denote the (standard) dependency pairs with
respect to $\mathcal{R}_{\mathsf{exp}}$. Then $\mathcal{P}$ consists of three
pairs:
$1\colon\mathsf{exp}^{\sharp}(\mathsf{r}(x))\to\mathsf{d}^{\sharp}(\mathsf{exp}(x))$,
$2\colon\mathsf{exp}^{\sharp}(\mathsf{r}(x))\to\mathsf{exp}^{\sharp}(x)$, and
$3\colon\mathsf{d}^{\sharp}(\mathsf{s}(x))\to\mathsf{d}^{\sharp}(x)$. Hence
the dependency graph $\operatorname{\mathsf{DG}}(\mathcal{R}_{\mathsf{exp}})$
contains two maximal cycles: $\\{2\\}$ and $\\{3\\}$.
We define two reduction pairs
$(\mathrel{{\succcurlyeq}_{\mathcal{A}}},\mathrel{{\succ}_{\mathcal{A}}})$ and
$(\mathrel{{\succcurlyeq}_{\mathcal{B}}},\mathrel{{\succ}_{\mathcal{B}}})$
such that the conditions of the theorem are fulfilled. Let $\mathcal{A}$ and
$\mathcal{B}$ be SLIs such that $\mathsf{exp}^{\sharp}_{\mathcal{A}}(x)=x$,
$\mathsf{r}_{\mathcal{A}}(x)=x+1$ and
$\mathsf{d}^{\sharp}_{\mathcal{B}}(x)=x$, $\mathsf{s}_{\mathcal{A}}(x)=x+1$.
Hence for any term $t\in\operatorname{\mathcal{T}_{\mathsf{b}}}$, we have that
the derivation heights
${\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\\{2\\}/\operatorname{\mathcal{U}}(\mathcal{P})}})$
and
${\mathsf{dh}}(t^{\sharp},\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\\{3\\}/\operatorname{\mathcal{U}}(\mathcal{P})}})$
are linear in $\lvert t\rvert$, while
${\mathsf{dh}}(t,\mathrel{\mathrel{\smash{\xrightarrow{\raisebox{-2.84526pt}{\tiny{$\mathrm{i}$}}}}}_{\mathcal{R}}})$
is (at least) exponential in $\lvert t\rvert$.
Observe that the problem exemplified by Example 7.19 cannot be circumvented by
replacing the dependency graph employed in Theorem 7.18 with weak (innermost)
dependency graphs. The exponential derivation height of terms $t_{n}$ in
Example 7.19 is not controlled by the cycles $\\{2\\}$ or $\\{3\\}$, but
achieved through the non-cyclic pair $1$ and its usable rules.
Example 7.19 shows an exponential speed-up between the maximal number of
dependency pair steps within a cycle in the dependency graph and the runtime
complexity of the initial TRS. In the context of derivational complexity this
speed-up may even increase to a primitive recursive function, cf. [23].
While Example 7.19 shows that the usable rules need to be taken into account
fully for any complexity analysis, it is perhaps tempting to think that it
should suffice to demand that at least one weak (innermost) dependency pair in
each cycle decreases strictly. However this intuition is deceiving as shown by
the next example.
###### Example 7.20.
Consider the TRS $\mathcal{R}$ of
$\mathsf{f}(\mathsf{s}(x),\mathsf{0})\to\mathsf{f}(x,\mathsf{s}(0))$ and
$\mathsf{f}(x,\mathsf{s}(y))\to\mathsf{f}(x,y)$.
$\operatorname{\mathsf{WDP}}(\mathcal{R})$ consists of
$1\colon\mathsf{f}^{\sharp}(\mathsf{s}(x),\mathsf{0})\to\mathsf{f}^{\sharp}(x,\mathsf{s}(x))$
and $2\colon\mathsf{f}^{\sharp}(x,\mathsf{s}(y))\to\mathsf{f}^{\sharp}(x,y)$,
and the weak dependency graph $\operatorname{\mathsf{WDG}}(\mathcal{R})$
contains two cycles $\\{1,2\\}$ and $\\{2\\}$. There are two linear restricted
interpretations $\mathcal{A}$ and $\mathcal{B}$ such that
$\\{1,2\\}\subseteq{\geqslant_{\mathcal{A}}}\cup{>_{\mathcal{A}}}$,
$\\{1\\}\subseteq{>_{\mathcal{A}}}$, and $\\{2\\}\subseteq{>_{\mathcal{B}}}$.
Here, however, we must not conclude linear runtime complexity, because the
runtime complexity of $\mathcal{R}$ is at least quadratic.
## 8 Experiments
All described techniques have been incorporated into the _Tyrolean Complexity
Tool_ TCT, an open source complexity analyser666Available at http://cl-
informatik.uibk.ac.at/software/tct.. The testbed is based on version 8.0.2 of
the _Termination Problems Database_ (_TPDB_ for short). We consider TRSs
without theory annotation, where the runtime complexity analysis is non-
trivial, that is the set of basic terms is infinite. This testbed comprises
1695 TRSs. All experiments were conducted on a machine that is identical to
the official competition server ($8$ AMD Opteron${}^{\text{\textregistered}}$
885 dual-core processors with 2.8GHz, $8\text{x}8$ GB memory). As timeout we
use 60 seconds. The complete experimental data can be found at http://cl-
informatik.uibk.ac.at/software/tct/experiments, where also the testbed
employed is detailed.
Table 1 summarises the experimental results of the here presented techniques
for full runtime complexity analysis in a restricted setting. The tests are
based on the use of one- and two-dimensional RMIs with coefficients over
$\\{0,1,\ldots,7\\}$ as direct technique (compare Theorem 3.9) as well as in
combination with the WDP method (compare Corollaries 5.13 and 6.14) and the
WDG method (compare Corollary 7.15). Weak dependency graphs are estimated by
the $\mathsf{TCAP}$-based technique ([20]). The tests indicate the power of
the transformation techniques introduced. Note that for linear and quadratic
runtime complexity the latter techniques are more powerful than the direct
approach. Furthermore note that the WDG method provides overall better bounds
than the WDP method.
| full
---|---
result | direct (1) | direct (2) | WDP (1) | WDP (2) | WDG (1) | WDG (2)
$\mathsf{O}(1)$ | 16 | 18 | 0 | 0 | 10 | 10
$\mathsf{O}(n)$ | 106 | 113 | 123 | 70 | 130 | 67
$\mathsf{O}(n^{2})$ | 106 | 148 | 123 | 157 | 130 | 158
timeout (60s) | 20 | 88 | 55 | 127 | 103 | 261
Table 1: Experiment results I (one- and two-dimensional RMIs separated)
However if we consider RMIs upto dimension 3 the picture becomes less clear,
cf. Table 2. Again we compare the direct approach, the WDP and WDG method and
restrict to coefficients over $\\{0,1,\ldots,7\\}$. Consider for example the
test results for cubic runtime complexity with respect to full rewriting.
While the transformation techniques are still more powerful than the direct
approach, the difference is less significant than in Table 1. On one hand this
is due to the fact that RMIs employing matrices of dimension $k$ may have a
degree strictly smaller than $k$, compare Theorem 3.9 and on the other hand
note the increase in timeouts for the more advanced techniques.
Moreover note the seemingly strange behaviour of the WDG method for innermost
rewriting: already for quadratic runtime the WDP method performs better, if we
only consider the number of yes-instances. This seems to contradict the fact
that the WDG method is in theory more powerful than the WDP method. However,
the explanation is simple: first the sets of yes-instances are incomparable
and second the more advanced technique requires more computation power. If we
would use (much) longer timeout the set of yes-instances for WDP would become
a _proper_ subset of the set of yes-instances for WDG. For example the WDG
method can prove cubic runtime complexity of the TRS AProVE_04/Liveness 6.2
from the TPDB, while the WDP method fails to give its bound.
| full | innermost
---|---|---
result | direct | WDP | WDG | direct | WDP | WDG
$\mathsf{O}(1)$ | 18 | 0 | 10 | 20 | 0 | 10
$\mathsf{O}(n)$ | 135 | 141 | 140 | 135 | 142 | 145
$\mathsf{O}(n^{2})$ | 161 | 163 | 162 | 173 | 181 | 172
$\mathsf{O}(n^{3})$ | 163 | 167 | 169 | 179 | 185 | 178
timeout (60s) | 310 | 459 | 715 | 311 | 458 | 718
Table 2: Experiment results II ($1\text{--}3$-dimensional RMIs combined)
In order to assess the advances of this paper in contrast to the conference
versions (see [4, 7]), we present in Table 3 a comparison between RMIs
with/without the use of usable arguments and a comparison of the WDP or WDG
method with/without the use of the extended weight gap principle. Again we
restrict our attention to full rewriting, as the case for innermost rewriting
provides a similar picture (see http://cl-
informatik.uibk.ac.at/software/tct/experiments for the full data).
| full
---|---
result | direct ($-$) | direct ($+$) | WDP ($-$) | WDP ($+$) | WDG ($-$) | WDG ($+$)
$\mathsf{O}(1)$ | 4 | 18 | 5 | 0 | 10 | 10
$\mathsf{O}(n)$ | 105 | 135 | 102 | 141 | 105 | 140
$\mathsf{O}(n^{2})$ | 127 | 161 | 118 | 163 | 119 | 162
$\mathsf{O}(n^{3})$ | 130 | 163 | 120 | 167 | 122 | 169
timeout (60s) | 306 | 310 | 505 | 459 | 655 | 715
Table 3: Experiment results III ($1\text{--}3$-dimensional RMIs combined)
Finally, in Table 4 we present the overall power obtained for the automated
runtime complexity analysis. Here we test the version of TCT that run for
the international annual termination competition
(TERMCOMP)777http://termcomp.uibk.ac.at/termcomp/. in 2010 in comparison to
the most recent version of TCT incorporating all techniques developed in
this paper. In addition we compare with a recent version of CaT.888http://cl-
informatik.uibk.ac.at/software/cat/.
| full | innermost
---|---|---
result | TCT (old) | TCT (new) | CaT | TCT (old) | TCT (new) | CaT
$\mathsf{O}(1)$ | 10 | 3 | 0 | 10 | 3 | 0
$\mathsf{O}(n)$ | 393 | 486 | 439 | 401 | 488 | 439
$\mathsf{O}(n^{2})$ | 394 | 493 | 452 | 403 | 502 | 452
$\mathsf{O}(n^{3})$ | 397 | 495 | 453 | 407 | 505 | 453
$\mathsf{O}(n^{4})$ | 397 | 495 | 454 | 407 | 505 | 454
Table 4: Experiment results IV ($1\text{--}3$-dimensional RMIs combined)
The results in Table 4 clearly show the increase in power in TCT, which is
due to the fact that the techniques developed in this paper have been
incorporated.
## 9 Conclusion
In this article we are concerned with automated complexity analysis of TRSs.
More precisely, we establish new and powerful results that allow the
assessment of polynomial runtime complexity of TRSs fully automatically. We
established the following results: Adapting techniques from context-sensitive
rewriting, we introduced _usable replacement maps_ that allow to increase the
applicability of direct methods. Furthermore we established the _weak
dependency pair method_ as a suitable analog of the dependency pair method in
the context of (runtime) complexity analysis. Refinements of this method have
been presented by the use of the _weight gap principle_ and _weak dependency
graphs_. In the experiments of Section 8 we assessed the viability of these
techniques. It is perhaps worthy of note to mention that our motivating
examples (Examples 3.2, 5.15, and 7.5) could not be handled by any known
technique prior to our results.
To conclude, we briefly mention related work. Based on earlier work by Arai
and the second author (see [26]) Avanzini and the second author introduced
$\text{POP}^{\ast}$ a restriction of the recursive path order (RPO) that
induces polynomial innermost runtime complexity (see [27, 15]). With respect
to derivational complexity, Zankl and Korp generalised a simple variant of our
weight gap principle to achieve a modular derivational complexity analysis
(see [28, 29]). Neurauter et al. refined in [16] matrix interpretations in the
context of derivational complexity derivational complexity (see also [30]).
Furthermore, Waldmann studied in [17] the use of weighted automata in this
setting. Based on [4, 7] Noschinski et al. incorporated a variant of weak
dependency pairs (not yet published) into the termination prover
AProVE.999This novel version of AProVE (see http://aprove.informatik.rwth-
aachen.de/) for (innermost) runtime complexity took part in TERMCOMP in 2010.
Currently this method is restricted to innermost runtime complexity, but
allows for a complexity analysis in the spirit of the dependency pair
framework. Preliminary evidence suggests that this technique is orthogonal to
the methods presented here. While all mentioned results are concerned with
_polynomial_ upper bounds on the derivational or runtime complexity of a
rewrite system, Schnabl and the second author provided in [31, 23, 32] an
analysis of the dependency pair method and its framework from a complexity
point of view. The upshot of this work is that the dependency pair framework
may induce multiple recursive derivational complexity, even if only simple
processors are considered.
Investigations into the complexity of TRSs are strongly influenced by research
in the field of ICC, which contributed the use of restricted forms of
polynomial interpretations to estimate the complexity, cf. [18]. Related
results have also been provided in the study of term rewriting
characterisations of complexity classes (compare [33]). Inspired by Bellantoni
and Cook’s recursion theoretic characterisation of the class of all polynomial
time computable functions in [34], Marion [35] defined LMPO, a variant of RPO
whose compatibility with a TRS implies that the functions computed by the TRS
is polytime computable (compare [3]). A remarkable milestone on this line is
the quasi-interpretation method by Bonfante et al. [36]. The method makes use
of standard termination methods in conjunction with special polynomial
interpretation to characterise the class of polytime computable functions. In
conjunction with _sup-interpretations_ this method is even capable of making
use of _standard_ dependency pairs (see [37]).
In principle we cannot directly compare our result on _polynomial_ runtime
complexity of TRSs with the results provided in the setting of ICC: the notion
of complexity studied is different. However, due to a recent result by
Avanzini and the second author (see [38], but compare also [39, 40]) we know
that the runtime complexity of a TRS is an _invariant_ cost model. Whenever we
have polynomial runtime complexity of a TRS $\mathcal{R}$, the functions
computed by this $\mathcal{R}$ can be implemented on a Turing machine that
runs in polynomial time. In this context, our results provide automated
techniques that can be (almost directly) employed in the context of ICC. The
qualification only refers to the fact that our results are presented for an
abstract form of programs, viz. rewrite systems.
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|
arxiv-papers
| 2011-02-15T17:16:23 |
2024-09-04T02:49:17.014688
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nao Hirokawa, Georg Moser",
"submitter": "Georg Moser",
"url": "https://arxiv.org/abs/1102.3129"
}
|
1102.3155
|
# First observation of the exchange of transverse and longitudinal emittances
J. Ruan ruanjh@fnal.gov Fermi National Accelerator Laboratory, Batavia, IL
60510, USA A.S. Johnson Fermi National Accelerator Laboratory, Batavia, IL
60510, USA A.H. Lumpkin Fermi National Accelerator Laboratory, Batavia, IL
60510, USA R. Thurman-Keup Fermi National Accelerator Laboratory, Batavia,
IL 60510, USA H. Edwards Fermi National Accelerator Laboratory, Batavia, IL
60510, USA R.P. Fliller Current address: Photon Sciences Directorate,
Brookhaven National Laboratory, Upton, NY 11973 T. Koeth Current address:
Institute for Research in Electronics and Applied Physics, University of
Maryland, College Park, MD 20742 Y.-E Sun Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA
###### Abstract
An experimental program to demonstrate a novel phase space manipulation in
which the horizontal and longitudinal emittances of a particle beam are
exchanged has been completed at the Fermilab A0 Photoinjector. A new beamline,
consisting of a $TM_{110}$ deflecting mode cavity flanked by two horizontally
dispersive doglegs has been installed. We report on the first direct
observation of transverse and longitudinal emittance exchange:
{$\varepsilon_{\text{x}}^{n}$, $\varepsilon_{\text{y}}^{n}$,
$\varepsilon_{\text{z}}^{n}$}={$2.9\pm{0.1}$, $2.4\pm{0.1}$,
$13.1\pm{1.3}$}$\Rightarrow${$11.3\pm{1.1}$, $2.9\pm{0.5}$, $3.1\pm{0.3}$} mm-
mrad.
###### pacs:
29.27.-a, 41.85.-p, 41.75.Fr
The next generation of advanced accelerators will benefit from the
optimization of the phase-space volume by beam manipulations. Such
applications include high brightness light sources and improved luminosity for
a linear $e^{+}$ / $e^{-}$ collider. The advent of synchrotron radiation light
sources and free electron lasers (FEL) has been a boon to a wide range of
disciplines, resulting in a constantly increasing demand for brighter sources
and better resolution bib:lcls . This demand translates to requirements on the
properties of the underlying electron beams which produce the light. In
particular, one is driven to find ways to precisely manipulate the phase space
volume of the beam to optimize it for the desired application bib:marie ;
bib:yine . It had been pointed out by Courant that while the total emittance
(i.e. the phase space volume occupied by the beam) of a particle beam is
conserved by a symplectic process, it does allow for the exchange of
emittances between the 3 spatial dimensions bib:courant . Motivated by the FEL
requirement for a small transverse emittance, Cornacchia and Emma developed a
transverse / longitudinal emittance exchange (EEX) concept using a deflecting
mode rf cavity located in the dispersive section of a magnetic chicane
bib:emma . This method however, contained residual couplings between the two
dimensions. Other solutions exist that allow for complete exchange, such as
the proposal by Kim to place a deflecting mode cavity between two magnetic
doglegs bib:kim ; bib:piot .
In this Letter, we present the first experimental results of a near ideal,
one-to-one exchange of transverse and longitudinal normalized emittances
bib:emittance at the Fermilab A0 Photoinjector (A0PI) using the latter
scheme. Unlike the original motivation which was to exchange a large incoming
transverse emittance with a small incoming longitudinal one, this experiment
exchanges a large longitudinal with a small transverse emittance. There is
however, no reason to expect that the opposite would not work as well.
The transfer matrix of the EEX beamline using thin lens elements for the
dipoles and drifts and a thick lens cavity (symplectic) matrix for the 5-cell
structure with the TESLA shape approximated by half-wavelength pillboxes is
$M_{\text{EEX}}=$
$\\!\\!\left(\begin{array}[]{cccc}0&\frac{17\lambda}{40}&-\frac{1}{\alpha}-\frac{33\lambda}{40D}-\frac{L}{D}&-\frac{33\alpha\lambda}{40}-\alpha
L\\\ 0&0&-\frac{1}{D}&-\alpha\\\ -\alpha&-\frac{33\alpha\lambda}{40}-\alpha
L&\frac{17\alpha\lambda}{40D}&\frac{17\alpha^{2}\lambda}{40}\\\
-\frac{1}{D}&-\frac{1}{\alpha}-\frac{33\lambda}{40D}-\frac{L}{D}&\frac{17\lambda}{40D^{2}}&\frac{17\alpha\lambda}{40D}\end{array}\right),$
(1)
where $\alpha$ is the bend angle of a dogleg, $L$ is the length of the drift,
$\lambda$ is the wavelength and the cavity strength is set to $-1/D$, with D
being the dispersion of a single dogleg bib:edwards . In order to relate the
final beam emittances to the initial, uncoupled emittances, we write the
$4\times 4$ beam covariance matrix $\Sigma_{0}$ whose elements are the average
of the second central moments of phase-space variables
$(x,x^{\prime},z,\delta\equiv\frac{p_{z}}{\langle p_{z}\rangle}-1)$,
$\left(\begin{array}[]{cccc}\langle x^{2}\rangle&\langle
xx^{\prime}\rangle&0&0\\\ \langle xx^{\prime}\rangle&\langle x^{\prime
2}\rangle&0&0\\\ 0&0&\langle z^{2}\rangle&\langle z\delta\rangle\\\
0&0&\langle z\delta\rangle&\langle\delta^{2}\rangle\end{array}\right),$ (2)
The beam matrix after traversing the EEX beamline is
$\Sigma_{out}=M_{EEX}\Sigma_{0}M_{EEX}^{T}$. The final rms emittances are
found by taking the determinant of the $2\times 2$ on diagonal sub-blocks of
$\Sigma_{out}$ and can be written in terms of the incoming emittances as,
$\\!\\!\\!\begin{array}[]{c}\varepsilon_{x,\it
out}^{2}=\varepsilon_{z}^{2}+(\frac{17\lambda^{2}}{40D})^{2}\langle x^{\prime
2}\rangle\left[\langle
z^{2}\rangle+\alpha^{2}D^{2}\langle\delta^{2}\rangle+2\alpha D\langle
z\delta\rangle\right]\\\ \\\ \varepsilon_{z,\it
out}^{2}=\varepsilon_{x}^{2}+(\frac{17\lambda^{2}}{40D})^{2}\langle x^{\prime
2}\rangle\left[\langle
z^{2}\rangle+\alpha^{2}D^{2}\langle\delta^{2}\rangle+2\alpha D\langle
z\delta\rangle\right]\\\ \end{array}\\!\\!\\!\\!\\!\\!$ (3)
As can be seen, the non-zero cavity length causes an imperfect exchange which
can, however, be reduced by proper selection of longitudinal or transverse
input parameters bib:fliller ; bib:ray .
The A0PI facility includes an 1.5-cell normal-conducting L-band rf
photocathode gun using a Cs2Te photocathode irradiated by the frequency
quadrupled, UV component of a Nd:Glass drive laser bib:carneiro . The drive
laser can be configured to provide a train of electron beam pulses separated
by 1 $\mu$s with charges up to 1 nC. Two emittance compensation solenoidal
coils are installed as well as a bucking coil which is used to ensure zero
magnetic field at the photocathode. The rf gun is followed by a 9-cell L-band
superconducting cavity, and both a straight ahead and emittance exchange beam
lines as schematically shown in Figure 1.
Figure 1: Top view of the A0 Photoinjector showing elements pertinent to
performing emittance exchange. Elements labeled “X” are diagnostics stations
(beam viewers and/or multi-slit mask locations), “S” are solenoid lenses, “Q”
are quadrupole magnets and “D” are dipole magnets.
The emittance exchange beamline at the A0PI consists of a 3.9 GHz $TM_{110}$
deflecting mode 5 cell cavity located between two horizontal dogleg magnetic
channels. The cavity is a liquid nitrogen cooled, normal conducting variant of
a superconducting version previously developed at Fermilab bib:mcashan ;
bib:PAC07 . The time varying longitudinal electric field gradient,
$dE_{z}/dx$, of the $TM_{110}$ mode provides a linearly sloped field about the
cavity axis. The dispersion introduced by the first magnetic dogleg
horizontally positions off-momentum electrons ($\delta\neq 0$) in the
$TM_{110}$ cavity causing them to receive a negative longitudinal kick
proportional to their $\delta$. As a result, the $TM_{110}$ cavity reduces the
momentum spread. The time varying vertical magnetic field is $90\,^{\circ}$
advanced of the electric field. The synchronous particle is timed to cross the
cavity center at the peak of the electric field when the magnetic field is
zero, and as a consequence, the cavity produces a time dependent positive
(negative) horizontal kick with respect to early (late) particles.
Accurate measurements of the beam parameters are critical to the evaluation of
the EEX process, thus the beamline is equipped with various diagnostic
instruments. Transverse beam profiles are measured by optical transition
radiation (OTR) viewing screens oriented at $45\,^{\circ}$. Both ingoing and
outgoing transverse divergences are measured with the interceptive method of
tungsten slits bib:wang . Downstream slit images are generated by single
crystal YAG:Ce scintillator screens oriented orthogonal to the incident beam
direction. A $45\,^{\circ}$ mirror directs the radiation to the optical
system. This configuration eliminates depth of focus issues from the field of
view and improves resolution bib:lumpkin-FEL .
Example incoming beam and slit images are shown in Figure 2. The beam image is
taken from the OTR screen located at X3. Horizontal and vertical slits of 50
$\mu$m width separated by 1 mm are inserted into the beamline at X3, and the
beamlets are allowed to drift 1.29 m to the YAG:Ce screen located at X6. Image
profiles are projected along the axis and fit with Gaussians. Sample outgoing
emittance measurements are shown in Figure 3. At X23 the horizontal slits are
separated by 2 mm while the vertical slits are spaced at 1 mm. A summary of
input and output data is listed in Table 1. Prior to image analysis, the dark
current contributions have been subtracted by acquiring a background image
with the beam shutter closed. The uncertainty in the emittance includes the
statistical fit uncertainty, pulse to pulse variation, and an estimate of the
uncertainty in the optical resolution based on the differences between
modulation contrast and edge blurring measurements using a calibration target.
A matlab-based program calculates the emittances and the Courant-Snyder
parameters ($\alpha$,$\beta$,$\gamma$) based on the X3-X6 and X23-X24 spot and
slit image pairs. Transverse beam position is monitored by $10$ button beam
position monitors.
Figure 2: Example incoming transverse emittance measurement data. Figure (a)
shows an OTR image of the beam spot at X3 with Gaussian fits to the projected
x and y profiles. Figures (b) and (c) are slit images taken at X6 YAG screen
for x and y divergence measurements, respectively. Gaussian fits to the
projected profiles are shown. Figure 3: Example outgoing transverse emittance
measurement data. Figure (a) shows an YAG:Ce screen image of the beam spot at
X23 with Gaussian fits to the projected x and y profiles. Figures (b) and (c)
are slit images taken at X24 YAG screen for x and y divergence measurements,
respectively. Gaussian fits to the projected profiles are shown.
Projected longitudinal emittance measurements are made by combining energy
spread and bunch length measurements. EEX input and output central momenta and
momentum spreads are measured by two spectrometer magnets and down-stream
viewing screens. Figure 4 shows the energy spread with Gaussian fits as
measured at XS3 and after EEX at XS4. We conservatively report the output
longitudinal emittance by only taking the energy-spread bunch-length product,
$\varepsilon_{\text{z,out}}$ = $\sigma_{\delta}\sigma_{z}$. The bunch length
is then determined at the X9 OTR screen using a Hamamatsu C5680 streak camera
operating with a low jitter synchroscan vertical plug-in unit phase locked to
81.25 MHz as described previously bib:lumpkin . The outgoing energy-spread is
measured at the XS4 screen following the vertical spectrometer magnet. The
bunch length measurement at X24 is made with OTR transported to the streak
camera and with the far infrared coherent transition radiation transported to
a Martin-Puplett interferometer bib:keup . As a graphic example of the effects
on bunch length in the exchange process, Figure 5 shows the effective
compression by about a factor 3 with 5-cell cavity on (blue) compared to off
(red).
Figure 4: Energy spread measurements before and after EEX. The triangles show typical incoming minimum energy spread as measured at XS3 with a Gaussian fit to the projection. After EEX, the energy spread measured at XS4 is shown with dots and a Gaussian fit to the projection. Table 1: Summary of measured input and output rms beam parameters at 14.3 MeV with charge of 250 pC per bunch. Parameter | In | Out | Unit
---|---|---|---
$\sigma_{x}$ | 0.905$\,\pm\,$ | 0.013 | 4.014$\,\pm\,$ | 0.059 | mm
$\sigma_{x^{\prime}}$ | 0.110$\,\pm\,$ | 0.002 | 0.098$\,\pm\,$ | 0.010 | mrad
$\sigma_{z}$ | 2.3$\,\pm\,$ | 0.2 | 0.8$\,\pm\,$ | 0.2 | ps
$\sigma_{\delta}$ | 9.2$\,\pm\,$ | 0.9 | 6.1$\,\pm\,$ | 0.6 | keV
Figure 5: Effect of deflecting mode cavity on bunch length. The dots represent
the bunch length as measured with the streak camera at X24 with the deflecting
mode cavity off. The triangles show a reduction in bunch length when measured
with the deflecting mode cavity on. Each measurement was made over 25 shots.
The direct measurement of the emittance exchange has been performed at
$\approx{14.3}$ MeV with a bunch charge of 250 pC, the latter chosen as a
compromise between diagnostic requirements and space-charge effects. To set up
the incoming longitudinal phase space, the fractional momentum spread was
minimized by operating the booster cavity off crest. Separate experiments have
shown the coherent synchrotron radiation (CSR) production at D3 is minimal at
the selected 9-cell phase setting so we anticipate the emittance growth due to
CSR is also low bib:charles . Input transverse parameters were tuned by
adjusting Q1, Q2 and Q3 for a minimum EEX beamline output bunch-length energy-
spread product, $\sigma_{\delta}\sigma_{z}$. Since the intensity of the
coherent transition radiation is strongly dependent on the bunch length, the
interferometer’s pyroelectric sensors are used to make quick, but
uncalibrated, relative bunch-length measurements. This is very useful in
mapping the effects of the input quadrupole fields on output longitudinal
parameters. A normalized 1/$\sigma_{\delta}\sigma_{z}$ product map is shown in
Figure 6. Complete measurements of the initial and final emittances were
collected with these conditions.
Figure 6: A relative output 1/$\sigma_{\delta}\sigma_{z}$ product map against
input quadrupole currents.
For comparison, a linear transfer matrix model of the EEX beamline has been
assembled in matlab in an effort to explore the behavior of the EEX line. It
includes thick quadrupole and dipole magnets, and uses a thick lens model of
the deflecting mode cavity composed of five zerolength $TM_{110}$ cavities
each separated by a 3.9 GHz freespace halfwavelength drift, which agrees well
with the realistic elliptical cavity transfer function bib:koeth . The
measured emittance exchange transport matrix shows good agreement with the
calculated transport matrix bib:PAC09 .
Results of the measurements are shown in Table 2 and summarized as follows.
The A0PI input beam’s measured horizontal emittance is
$\varepsilon_{\text{x}}^{n}$=$2.9\pm{0.1}$ mm-mrad and the EEX output
longitudinal emittance measured $\varepsilon_{\text{z}}^{n}$=$3.1\pm{0.3}$ mm-
mrad demonstrating a 1:1 transfer of $\varepsilon_{\text{x,in}}^{n}$ to
$\varepsilon_{\text{z,out}}^{n}$. Similarly the input longitudinal emittance,
$\varepsilon_{\text{z,in}}^{n}$=13.1$\pm{1.3}$ mm-mrad and the EEX output
horizontal emittance measured $\varepsilon_{\text{x,out}}^{n}$=$11.3\pm{1.1}$
mm-mrad also show agreement between $\varepsilon_{\text{z,in}}^{n}$ and
$\varepsilon_{\text{x,out}}^{n}$. The vertical emittance was left unaffected,
$\varepsilon_{\text{y,in}}^{n}$=$2.4\pm{0.1}$ mm-mrad $\Rightarrow$
$\varepsilon_{\text{y,in}}^{n}$=$2.9\pm{0.5}$ mm-mrad. The combined results
show the successful exchange of emittance between two planes while conserving
the full 6D phase space volume.
Table 2: Comparison of direct measurements of horizontal transverse ($x$) to longitudinal ($z$) emittance exchange to simulation. Emittance measurements are in units of mm-mrad and are normalized. | Simulated | Measured
---|---|---
| In | Out | In | Out
$\varepsilon_{\text{x}}^{n}$ | 2.9 | 13.2 | 2.9$\,\pm\,$ | 0.1 | 11.3$\,\pm\,$ | 1.1
$\varepsilon_{\text{y}}^{n}$ | 2.4 | 2.4 | 2.4$\,\pm\,$ | 0.1 | 2.9$\,\pm\,$ | 0.5
$\varepsilon_{\text{z}}^{n}$ | 13.1 | 3.2 | 13.1$\,\pm\,$ | 1.3 | 3.1$\,\pm\,$ | 0.3
In summary, a proof-of-principle transverse and longitudinal emittance
exchange has been completed at the Fermilab A0 Photoinjector, demonstrating a
novel fundamental phase space manipulation technique. Further studies are
planned at higher charge values to investigate the possible effects of space
charge and CSR.
###### Acknowledgements.
We are grateful for the technical support of J. Santucci, R. Montiel, W.
Muranyi, B. Tennis, E. Lopez, C. Tan, M. Davidsaver, R. Andrews, B. Popper, G.
Cancelo, B. Chase, J. Branlard and P. Prieto. We greatly appreciate the
discussions and comments from P. Piot (NIU), D. Edwards, M. Cooke, M. Stauffer
and M. Cornacchia (UMD). We thank M. Church, M. Wendt and E. Harms for their
interest and encouragement. This work was supported by Fermi Research
Alliance, LLC under contract No. DE-AC02-06CH11359 with the U.S. Department of
Energy.
## References
* (1) P. Emma _et al._ , Nat. Photon. 4, 6417 (2010).
* (2) N. Yampolsky _et al._ , arXiv:1010.1558v2 [physics.acc-ph].
* (3) Y.-E Sun _et al._ , Phys. Rev. Lett. 105, 234801 (2010).
* (4) E. Courant, in _Perspectives in Modern Physics, Essays in Honor of Hans A. Bethe_ , edited by R. E. Marshak (Interscience Publishers, New York, 1966), pp. 257-260.
* (5) M. Cornacchia and P. Emma, Phys. Rev. ST Accel. Beams 5, 084001 (2002).
* (6) K.-J. Kim and A. Sessler, _Proceedings of the 2006 Electron Cooling Workshop_ , Galena IL (ECOOL06), AIP 821, 115 (2006).
* (7) P. Emma, Z. Huang, and K.-J. Kim, and Ph. Piot,_Phys. Rev. ST Accel. Beams_ 9, 100702 (2006).
* (8) The normalized emittance $\varepsilon^{n}=\gamma\beta\varepsilon$, where $\gamma$ is the Lorentz factor and $\beta=\sqrt{1-\gamma^{-2}}$ relates beams of different energies.
* (9) D. A. Edwards, private communication.
* (10) R. P. Fliller III and T. W. Koeth, _Proceedings of the 2009 Particle Accelerator Conference_ , Vancouver BC, (PAC09), TU4PBI01, (2009).
* (11) R. P. Fliller III _et al., Proceedings of the 2007 Particle Accelerator Conference_ , Albuquerque NM, (PAC07), THPAS094, (2007).
* (12) J.-P. Carneiro _et al._ , Phys. Rev. ST Accel. Beams 8, 040101 (2005).
* (13) M. McAshan and R. Wanzenberg, FNAL TM-2144, 2001.
* (14) T. W. Koeth _et al., Proceedings of the 2007 Particle Accelerator Conference_ , Albuquerque NM, (PAC07), THPAS079, (2007).
* (15) C. H. Wang _et al., International Conference on Accelerator and Large Experimental Physics Control Systems_ , Trieste Italy, (ICALEPCS), 284, (1999).
* (16) A. H. Lumpkin _et al., Proceedings of the 2010 Free Electron Conference_ , Malmö City Sweden, (FEL10), (2010) (to be published).
* (17) A. H. Lumpkin _et al., Proceedings of the 2008 Beam Instrumentation Workshop_ , Lake Tahoe CA, (BIW08), 258, (2008).
* (18) R. M. Thurman-Keup _et al., Proceedings of the 2008 Beam Instrumentation Workshop_ , Lake Tahoe CA, (BIW08), 153, (2008).
* (19) J. C. T. Thangaraj _et al., Advanced Accelerator Workshop 2010_ , Annapolis MD, (AAC10), 643, (2010).
* (20) T. W. Koeth, Ph. D. Dissertation, Rutgers University, Piscataway NJ, (2009).
* (21) T. W. Koeth, _et al., Proceedings of the 2009 Particle Accelerator Conference_ (PAC09), Vancouver BC, FR5PFP020, (2009).
|
arxiv-papers
| 2011-02-15T18:53:59 |
2024-09-04T02:49:17.026635
|
{
"license": "Public Domain",
"authors": "J. Ruan, A. S. Johnson, A. H. Lumpkin, R. Thurman-Keup, H. Edwards, R.\n P. Fliller, T. Koeth, Y. -E Sun",
"submitter": "Amber Johnson",
"url": "https://arxiv.org/abs/1102.3155"
}
|
1102.3263
|
# Roles of axial anomaly on neutral quark matter with color superconducting
phase
Zhao Zhang zhaozhang@pku.org.cn School of Mathematics and Physics, North
China Electric Power University, Beijing 102206, China Teiji Kunihiro
kunihiro@ruby.scphys.kyoto-u.ac.jp Department of Physics, Kyoto University,
Kyoto 606-8502, Japan
###### Abstract
We investigate effects of the axial anomaly term with a chiral-diquark
coupling on the phase diagram within a two-plus-one-flavor Nambu-Jona-Lasinio
(NJL) model under the charge-neutrality and $\beta$-equilibrium constraints.
We find that when such constraints are imposed, the new anomaly term plays a
quite similar role as the vector interaction does on the phase diagram, which
the present authors clarified in a previous work. Thus, there appear several
types of phase structures with multiple critical points at low temperature
$T$, although the phase diagrams with intermediate-$T$ critical point(s) are
never realized without these constraints even within the same model
Lagrangian. This drastic change is attributed to an enhanced interplay between
the chiral and diquark condensates due to the anomaly term at finite
temperature; the u-d diquark coupling is strengthened by the relatively large
chiral condensate of the strange quark through the anomaly term, which in turn
definitely leads to the abnormal behavior of the diquark condensate at finite
$T$, inherent to the asymmetric quark matter. We note that the critical point
from which the crossover region extends to zero temperature appears only when
the strength of the vector interaction is larger than a critical value. We
also show that the chromomagnetic instability of the neutral asymmetric
homogenous two-flavor color superconducting(2CSC) phase is suppressed and can
be even completely cured by the enhanced diquark coupling due to the anomaly
term and/or by the vector interaction.
###### pacs:
12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc
## I INTRODUCTION
It is generally believed that the strongly interacting matter exhibits a rich
phase structure in extreme environment such as at high temperature and high
baryon chemical potential. Experimentally, RHIC (Relativistic Heavy Ion
Collider) and LHC (Large Hadron Collider) may provide more information on this
topics. Theoretically, some results have been already obtained on a sound
basis: First, the lattice simulations of quantum chromodynamics (QCD) indicate
that, for physical quark masses, the transition from the hadronic phase to the
quark gluon plasma (QGP) is a smooth crossover at finite temperature and
vanishing baryon chemical potential Cheng:2009be ; Borsanyi:2010bp , whereas
in the low temperature and very high density region, the techniques of
perturbation QCD can be used and the color flavor locking (CFL) Alford:1998mk
phase is proved to be the ground state of QCD Son:1998uk ; Schafer:1999jg ;
Shovkovy:1999mr ; Schafer:1999fe .
However, the above methods based on the first principle fail at the low
temperature and moderate density region, due to the sign problem or the non-
perturbative effect. Phenomenologically, such a region in the $T$-$\mu$ plane
is more relevant to reality and hence interesting since it is directly related
to the physics of compact stars. On that account, chiral models of QCD such as
the NJL model Nambu:1961tp ; Vogl:1991qt ; Klevansky:1992 ; Hatsuda:1994pi
that embody the basic low-energy characteristics of QCD such as symmetry
properties have been extensively used to explore the $T$-$\mu$ phase diagram
of strongly interacting matter. In particular, such model calculations suggest
that CSC phase may occur at low temperature and large chemical potential (for
reviews, see Rajagopal:2000wf ; Rischke:2003mt ; Buballa:2003qv ;
Alford:2007xm ). In addition, a popular result from the model studies is that
the chiral phase transition always keeps first order at the low-temperature
region Asakawa:1989bq ; Barducci:1989 ; Kunihiro:1991hp ; Berges:1998rc ;
Ruester:2005jc ; Abuki:2005ms . Combined with the crossover transition
confirmed by lattice QCD at zero baryon chemical potential, usually, a
schematic $T$-$\mu$ phase diagram with one chiral critical point(CP) is widely
adopted in the literature Stephanov:2007fk . Such a CP may be located at
relatively high temperature and low baryon chemical potential, which has
attracted considerable attention as it is potentially detectable in heavy-ion
experiments Stephanov:1998dy ; Minami:2009hn .
Generally, there is no reason to rule out the possibility that the QCD phase
diagram may contain more than one chiral CP, especially when the chiral and
diquark condensates are considered simultaneously; some rich structures with
multiple CP’s may be expected for the phase diagram owing to somehow enhanced
interplay between the two types of condensates111Note that multiple critical
points had also been found in two-flavor models of QCD without considering
diquark paringBowman:2008kc ; Ferroni:2010ct .. It has been shown on the basis
of the NJL model that it is indeed the case Kitazawa:2002bc ; Addenda ;
Zhang:2008wx ; Zhang:2009mk ; the QCD phase diagram can admit multiple CP’s
when the repulsive vector interaction Kitazawa:2002bc ; Addenda or the
charge-neutrality and $\beta$-equilibrium Zhang:2008wx or both of these two
ingredients Zhang:2009mk are included 222 Note that the renormalization group
theory for deducing low-energy effective vertex favors the presence of the
vector-type interaction ref:EHS ; ref:SW-reno . . This is because the two
ingredients act so as to enhance the competition between the chiral and
diquark condensates and thus the would-be first-order boundary line in the
low-temperature region of the $T$-$\mu$ plane can be turned into a smooth
crossover or multiply-cut crossover lines with new CP(’s). Indeed it has been
found Zhang:2009mk that the number of the chiral CP’s may vary from zero to
four with the joint effect of these two ingredients. Moreover, the present
authors have shown Zhang:2009mk that the vector interaction can effectively
suppress the chromomagnetic instability Huang:2004bg in the asymmetric
homogeneous CSC phase.
It is noteworthy that a direct coupling term between the chiral and diquark
condensates can be supplied by the axial anomaly Alford:1998mk ; Rapp:1999qa ;
Steiner:2005jm , which thus might lead to a new CP in the low-temperature
region, as first conjectured in Hatsuda:2006ps on the basis of an analysis
using the Ginzburg-Landau (GL) theory in the chiral limit; see also subsequent
detailed analyses Yamamoto:2007ah ; Baym:2008me though still in the chiral
limit. It is, however, to be noted that the GL theory assumes that both the
diquark and chiral condensates are sufficiently small around the phase
boundaries provided that the phase transitions are of second-order. In
addition, the GL theory itself can not determine the coefficients in the
action, and some microscopic model or theory is necessary for such a
determination.
Recently, a microscopic calculation has been done Abuki:2010jq with use of a
three-flavor NJL model for massive quarks incorporating the axial anomaly term
with a form of a six-quark interaction Alford:1998mk ; Rapp:1999qa ;
Steiner:2005jm ; Yamamoto:2007ah , the coupling constant of which is denoted
by $K^{\prime}$: It was claimed in Abuki:2010jq that the low-temperature CP
can exist owing to the axial anomaly for an appropriate range of the model
parameters even off the chiral limit but still with a flavor symmetry as in
the GL approach in the chiral limit. It should be noticed, however, that the
SU(3)-flavor symmetry may lead to a special type of CSC phase, i.e., the CFL
phase, as is taken for granted in Hatsuda:2006ps ; Yamamoto:2007ah ;
Baym:2008me ; Abuki:2010jq , which automatically satisfies the charge-
neutrality and $\beta$-equilibrium constraints.
Then one may suspect that the possible emergence of the new CP might be an
artifact of such an ideal situation with the three-flavor symmetry.
Nevertheless, it is a very interesting possibility that the axial anomaly-
induced interplay between the chiral and diquark condensates would lead to a
new CP in the low-temperature region. Thus it is worth exploring to see
whether a new low-temperature CP is induced by the axial anomaly in a
dynamical model of QCD by considering the realistic situation with the broken
flavor symmetry by the hierarchical current quark masses.We note that once the
quark mass difference of different flavors is taken into account, it becomes a
complicated dynamical problem to make the charge-neutrality and
$\beta$-equilibrium constraints satisfied.
More recently, such a realistic calculation in the framework of a two-plus-
one-flavor NJL model has been done by Basler et al. Basler:2010xy ; they have
shown that such a new low-temperature CP is not found in such a model even
with the axial anomaly term, because an unusual interplay between the chiral
and diquark condensates induced by the anomaly term actually leads favorably
to the 2CSC phase Alford:1997zt ; Rapp:1997zu rather than the CFL phase near
the chiral phase boundary, even in the case with the equal quark mass limit
Basler:2010xy .
It is worth emphasizing here that the constraints by the charge-neutrality and
$\beta$-equilibrium are not taken into consideration in Abuki:2010jq nor
Basler:2010xy in contrast to Zhang:2008wx ; Zhang:2009mk where various types
of multiple-CP structures are found in the phase diagram. Thus the following
two questions arise naturally: Will the results found in Basler:2010xy be
altered or not when the charge-neutrality and $\beta$-equilibrium constraints
and/or the vector interaction are taken into account? Or will the phase
structure with multiple CP’s found in Zhang:2009mk rather persist when taking
into account the coupling between the chiral and diquark condensates induced
by such a six-quark interaction? The main purpose of this paper is to answer
these questions by incorporating the anomaly term that breaks the $U_{A}(1)$
symmetry as well as the vector interaction under the constraints of the
charge-neutrality and $\beta$-equilibrium in the two-plus-one-flavor NJL
model. The present work may be regarded as either an extension of the paper
Basler:2010xy by incorporating the charge-neutrality, $\beta$-equilibrium and
the vector interaction, or an extension of the paper Zhang:2009mk by
including the $K^{\prime}$-interaction.
The main conclusion we reach is that the key results on the phase structure
obtained in Ref. Zhang:2008wx ; Zhang:2009mk persist even when the attractive
$K^{\prime}$-interaction is incorporated. That is, there appear new CP(’s) at
the intermediate temperature owing to charge-neutrality constraint and then
the transition in the low temperature region extending to zero $T$ becomes a
crossover when the strength of the vector interaction becomes larger than a
critical value: Thus the number of the CP’s can be even more than two,
depending on the values of some related coupling constants. Strikingly enough,
we find that the interplay between the chiral and the diquark condensates
induced by the anomaly term even acts toward realizing the multi-CP structure
of the phase diagram under the neutrality and $\beta$-equilibrium constraints
even without the help of the vector interaction. Accordingly, the results in
Ref. Basler:2010xy are modified by considering these constraints. Even though
the chiral boundary in the low-$T$ region extending to zero $T$ also remains
first order in our case in the absence of the vector interaction, which
shrinks and vanishes eventually as $K^{\prime}$ becomes large and exceeds a
critical value.
We shall also examine the chromomagnetic (in)stability under the influence of
the axial anomaly, as was done in Zhang:2009mk . It is well known that the
asymmetric homogenous 2CSC phase suffers from the chromomagnetic instability.
At zero temperature, the calculation based on the hard-dense-loop (HDL) method
Huang:2004bg suggests that the Meissner mass squared of the 8th gluon becomes
negative for $\frac{\delta\mu}{\Delta}>1$ while the 4th-7th gluons acquire
negative Meissner masses squared for $\frac{\delta\mu}{\Delta}>1/\sqrt{2}$;
here $\delta\mu$ and $\Delta$ denote the difference of the chemical potentials
of u and d quarks and the gap, respectively. Note that $\delta\mu$ is just
equal to a half of the electron chemical potential $\mu_{e}$ when the vector
interaction is absent, and this quantity is to be replaced by an effective
chemical potential $\delta\tilde{\mu}$ (see below) when the vector interaction
is present, as shown in Zhang:2009mk . The instability of the asymmetric
homogenous CSC phase should imply the existence of a yet unknown but stable
phase in this region of the $T$-$\mu$ plane. Candidates of such a stable phase
include the Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) phase Giannakis:2004pf and
gluonic phase Gorbar:2005rx . Besides developing the possible new phases, the
instability problem may also be totally or partially gotten rid of by some
other mechanisms. For instance, the instability problem becomes less severe
simply at finite temperature because the smeared Fermi surface relaxes the
mismatch of the Fermi spheres of the asymmetric quark matter Kiriyama:2006jp ;
He:2007cn ; Fukushima:2005cm . Furthermore, it is known that the larger the
quark mass and the stronger the diquark coupling, more suppressed the
instability even at zero temperatureKitazawa:2006zp . Recently, the present
authorsZhang:2009mk have shown that the repulsive vector interaction can also
resolve the instability problem totally or partially. The stability by the
vector interaction is realized due to the following two ingredients: (1) the
density difference between the u and d quarks reduces the mismatch in the
effective chemical potentials; (2) the nonzero vector interaction suppresses
the formation of high density and hence larger quark masses than those
obtained without the interaction are realized. We shall show that the new
anomaly term play a quite similar role as the vector interaction and the
interplay between the chiral and diquark condensates induced by the anomaly
term acts toward suppressing the unstable region of the homogeneous 2CSC phase
in the $T$-$\mu$ plane; the neutral 2CSC phase can become even free from the
chromomagnetic instability if $K^{\prime}$ is larger than a critical value
$K^{\prime}_{c}$, which can be reduced significantly when the vector
interaction is incorporated.
This paper is organized as follows. In Sec.II, the two-plus-one-flavor NJL
model with the extended flavor-mixing six-quark interaction is introduced
under the constraints of the charge-neutrality and $\beta$-equilibrium. The
phase diagram of the neutral strongly interacting matter with the influence of
the axial anomaly is presented in Sec.III. Sec.IV focuses on the role of the
axial anomaly on the chromomagnetic (in)stability. The conclusion and outlook
are given in Sec.V.
## II NJL Model With Axial Anomaly and Vector Interaction
### II.1 The model Lagrangian
We start from the following two-plus-one-flavor NJL model with the vector
interaction Klimt:1989pm ; Zhang:2009mk and two types of six-quark anomaly
terms,
$\mathcal{L}=\bar{\psi}\,(i\partial\hbox
to0.0pt{\hss$\diagup$\kern-2.0pt}-\hat{m}\,)\psi+\mathcal{L}_{\chi}^{(4)}+\mathcal{L}_{d}^{(4)}+\mathcal{L}_{\chi}^{(6)}+\mathcal{L}_{\chi{d}}^{(6)},$
(1)
where $\hat{m}=\text{diag}_{f}(m_{u},m_{d},m_{s})$ denotes the current-quark
mass matrix and
$\displaystyle\mathcal{L}_{\chi}^{(4)}=G_{S}\sum_{i=0}^{8}\left[\left(\bar{\psi}\lambda_{i}^{f}\psi\right)^{2}+\left(\bar{\psi}i\gamma_{5}\lambda_{i}^{f}\psi\right)^{2}\right]-G_{V}\sum_{i=0}^{8}\left[\left(\bar{\psi}\gamma^{\mu}\lambda_{i}^{f}\psi\right)^{2}+\left(\bar{\psi}\gamma^{\mu}\gamma_{5}\lambda_{i}^{f}\psi\right)^{2}\right],$
(2)
$\displaystyle\mathcal{L}_{d}^{(4)}=G_{D}\sum_{i,j=1}^{3}\left[(\bar{\psi}i\gamma_{5}t_{i}^{f}t^{c}_{j}\psi_{C})(\bar{\psi}_{C}i\gamma_{5}t_{i}^{f}t^{c}_{j}\psi)+(\bar{\psi}t_{i}^{f}t^{c}_{j}\psi_{C})(\bar{\psi}_{C}t_{i}^{f}t^{c}_{j}\psi)\right],$
(3)
$\displaystyle\mathcal{L}_{\chi}^{(6)}=-K\left\\{\det_{f}\left[\bar{\psi}\left(1+\gamma_{5}\right)\psi\right]+\det_{f}\left[\bar{\psi}\left(1-\gamma_{5}\right)\psi\right]\right\\},$
(4)
$\displaystyle\mathcal{L}_{\chi{d}}^{(6)}=\frac{K^{\prime}}{8}\sum_{i,j,k=1}^{3}\sum_{\pm}\left[({\psi}t_{i}^{f}t^{c}_{k}(1\pm\gamma_{5}){\psi}_{C})(\bar{\psi}t_{j}^{f}t^{c}_{k}(1\pm\gamma_{5})\bar{\psi}_{C})(\bar{\psi}_{i}(1\pm\gamma_{5})\psi_{j})\right].$
(5)
Here the four-fermion interactions are all invariant under the
$U(3)_{R}\times{U(3)_{L}}$-transformation in the flavor space. In our
notations, the Gell-Mann matrices in flavor (color) space are
$\lambda_{i}^{f(c)}$ with $i=1,\ldots,8$, and
$\lambda_{0}^{f(c)}\equiv\sqrt{2/3}\,\openone_{f(c)}$, and the antisymmetric
one is denoted by $t_{i}^{f(c)}$ with $i=1,2,3$ :
$\displaystyle t_{1}^{f(c)}=\lambda_{7}^{f(c)},\quad
t_{2}^{f(c)}=\lambda_{5}^{f(c)},\quad t_{3}^{f(c)}=\lambda_{2}^{f(c)}.$ (6)
The scalar interaction in $\mathcal{L}_{\chi}^{(4)}$ is responsible for the
dynamical chiral symmetry breaking in the vacuum with the formation of the
chiral condensate, while the vector interaction can be used to investigate the
effect of density-density interaction on the chiral phase transition
Zhang:2009mk 333We remark that the effects of the vector interaction on the
baryon-number susceptibility and the chiral transition in the two-flavor case
are examined in Kunihiro:1991qu and Asakawa:1989bq ; Kitazawa:2002bc ,
respectively.. In Eqs.(3) and (5), $\psi_{C}$ stands for $C\bar{\psi}^{T}$ and
$C=i\gamma_{0}\gamma_{2}$ is the Dirac charge conjugation matrix. We remark
that the suffix $3$ in $t_{3}^{f}$ denotes the channel for the u$-$d pairing,
for example. For lower temperature $T$ and large enough baryon chemical
potential $\mu$, $\mathcal{L}_{d}^{(4)}$ leads to the formation of diquark
condensate in the color-anti-triplet channel Alford:1997zt ; Rapp:1997zu ;
Alford:1998mk . Besides the four-fermion interactions, Lagrangian (1) also
contains two types of six-quark interactions, $\mathcal{L}_{\chi}^{(6)}$ and
$\mathcal{L}_{\chi d}^{(6)}$: the former is the traditional Kobayashi-
Maskawa-’tHooft (KMT) interaction Kobayashi:1970ji ; 't Hooft:1976fv and its
effect on the phase diagram in $T$-$\mu$ plane is fully
examinedKunihiro:1989my ; Kunihiro:1991hp ; Hatsuda:1994pi ; Buballa:2003qv ;
Fu:2007xc ; Kunihiro:2009ds , whereas the latter could be obtained by a Fierz
transformation of the former and induces the coupling between the chiral and
diquark condensates Alford:1998mk ; Rapp:1999qa ; Steiner:2005jm ;
Yamamoto:2007ah ; Abuki:2010jq . We remark that both interactions respect the
flavor symmetry of $SU(3)_{R}\times SU(3)_{L}\times U(1)$ while violating the
$U_{A}(1)$ symmetry as mentioned above. The former is responsible for
accounting for the abnormally large mass of $\eta^{\prime}$ beyond the
Weinberg inequalityWeinberg:1975ui (in contrast to other pseudo Nambu-
Goldstone bosons ) in the effective chiral model and can be identified as an
induced quark interaction from instantons't Hooft:1976fv ; ref:SS . The
introduction of the latter to the Lagrangian expands the study of CSC to the
six-fermion level Steiner:2005jm .
### II.2 The model parameters
The numerical values of some model parameters are given in Table 1. In
contrast to Abuki:2010jq , we only consider the case with realistic quark
masses. The choice of the model parameters is the same as that in
Ruester:2005jc ; Zhang:2009mk ; Basler:2010xy (all following Ref.
Rehberg:1995kh ), where $G_{S}$, the coupling constant for the scalar meson
channel, and $K$, the coupling constant of the KMT term, are fixed by the
vacuum physical observables (meson masses and decay constants). We shall work
in the isospin symmetric limit in two-flavor space with
$m_{u}=m_{d}=5.5\;\text{MeV}$, and a sharp three-momentum cut-off $\Lambda$ is
adopted.
$m_{u,d}$[MeV] | $m_{s}$[MeV] | $G_{S}\Lambda^{2}$ | $K\Lambda^{5}$ | $\Lambda$ [MeV] | $M_{u,d}$ [MeV]
---|---|---|---|---|---
5.5 | 140.7 | 1.835 | 12.36 | 602.3 | 367.7
$f_{\pi}$[MeV] | $m_{\pi}$[MeV] | $m_{K}$ [MeV] | $m_{\eta^{,}}$[MeV] | $m_{\eta}$[MeV] | $M_{s}$ [MeV]
92.4 | 135 | 497.7 | 957.8 | 514.8 | 549.5
Table 1: Model parametrization of two-plus-one-flavor NJL.
In contrast to $G_{S}$ and $K$, no definite observables in the vacuum are
available for determining the coupling constants $G_{V},G_{D}$ and
$K^{\prime}$ in such a quark model, although we could read off their values
from a Fierz transformation of known vertices: the coupling constant
$K^{\prime}$, for instance, can be related to $K$ through the Fierz
transformation of the instanton vertex, and $K^{\prime}$ is found to be
identical to $K$ Abuki:2010jq . Since we are mainly interested in the roles of
$K^{\prime}$ and $G_{V}$ on the chiral phase transition and the chromomagnetic
instability, both these coupling constants are treated as free parameters in
the present work. Following Abuki:2010jq ; Basler:2010xy , we only consider
the attractive interaction between the chiral condensate and the diquark
condensate. Namely, the coupling $K^{\prime}$ is kept positive. As for the
ratio of $G_{D}/G_{S}$, we adopt the standard value from Fierz transformation
in this paper. Due to the contribution from the KMT interaction, the ratio
$G_{D}/G_{S}$ from Fierz transformation should be 0.95 rather than 0.75
obtained by only considering the four-quark interaction Buballa:2003qv . Such
a choice of the coupling has also been used in Refs. Zhang:2009mk and
Basler:2010xy . In the literature, the diquark-diquark interaction near the
standard value from Fierz transformation is usually called the intermediate
coupling.
### II.3 Thermodynamic potential with the constraints of charge-neutrality
and $\beta$-equilibrium
The grand partition function of the NJL model is given by
$Z\equiv{e^{-{\Omega}V/T}}=\int{D\bar{\psi}D\psi}e^{i\int{dx^{4}}(\cal{L}+{\psi^{\dagger}}\hat{\mu}\psi)},$
(7)
where $\Omega$ is the thermodynamic potential density and $\hat{\mu}$ is the
quark chemical potential matrix. In general, the quark chemical potential
matrix $\hat{\mu}$ takes the form Alford:2002kj
$\hat{\mu}=\mu-\mu_{e}Q+\mu_{3}T_{3}+\mu_{8}T_{8},$ (8)
where $\mu$ is the quark chemical potential (i.e. one third of the baryon
chemical potential), $\mu_{e}$ the chemical potential associated with the
(negative) electric charge, and $\mu_{3}$ and $\mu_{8}$ represent the color
chemical potentials corresponding to the Cartan subalgebra in the SU(3)-color
space. The explicit form of the electric charge matrix is
$Q=\text{diag}(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}))$ in flavor space, and
the color charge matrices are $T_{3}=\text{diag}(\frac{1}{2},-\frac{1}{2},0)$
and $T_{8}=\text{diag}(\frac{1}{3},\frac{1}{3},-\frac{2}{3})$ in the color
space. The chemical potentials for the quarks with respective flavor and color
charges are listed below:
$\begin{split}&\mu_{ru}=\mu-\tfrac{2}{3}\mu_{e}+\tfrac{1}{2}\mu_{3}+\tfrac{1}{3}\mu_{8}\,,\quad\mu_{rd}=\mu+\tfrac{1}{3}\mu_{e}+\tfrac{1}{2}\mu_{3}+\tfrac{1}{3}\mu_{8}\,,\quad\mu_{rs}=\mu+\tfrac{1}{3}\mu_{e}+\tfrac{1}{2}\mu_{3}+\tfrac{1}{3}\mu_{8}\,,\\\
&\mu_{gu}=\mu-\tfrac{2}{3}\mu_{e}-\tfrac{1}{2}\mu_{3}+\tfrac{1}{3}\mu_{8}\,,\quad\mu_{gd}=\mu+\tfrac{1}{3}\mu_{e}-\tfrac{1}{2}\mu_{3}+\tfrac{1}{3}\mu_{8}\,,\quad\mu_{gs}=\mu+\tfrac{1}{3}\mu_{e}-\tfrac{1}{2}\mu_{3}+\tfrac{1}{3}\mu_{8}\,,\\\
&\mu_{bu}=\mu-\tfrac{2}{3}\mu_{e}-\tfrac{2}{3}\mu_{8}\,,\qquad\qquad\mu_{bd}=\mu+\tfrac{1}{3}\mu_{e}-\tfrac{2}{3}\mu_{8}\,,\qquad\qquad\mu_{bs}=\mu+\tfrac{1}{3}\mu_{e}-\tfrac{2}{3}\mu_{8}\,.\end{split}$
(9)
Corresponding to the chiral and diquark interactions in Eq. (1), we assume
that the following condensates are formed in the system, namely, the scalar
quark-antiquark condensate
$\sigma_{i}=\langle\bar{\psi}_{i}\psi_{i}\rangle,$ (10)
and the scalar diquark condensate
$s_{i}=\langle\bar{\psi}_{C}i\gamma_{5}t_{i}^{f}t_{i}^{c}\psi\rangle.$ (11)
In addition, we remark that the quark-number (or baryon-number) density
$\rho_{i}=\langle\bar{\psi}_{i}\gamma^{0}\psi_{i}\rangle,$ (12)
has a finite value for finite $\mu$. Note that the indices 1,2 and 3 in Eqs.
(10) and (12) represent u, d and s quarks, respectively, whereas in Eq. (11),
the indices 1, 2 and 3 stand for the diquark condensate in d-s, s-u and u-d
pairing channels, respectively. Here we have assumed the condensates and the
density are all homogeneous; the study of the phase structure with
inhomogeneous condensates and/or baryon-number density Nakano:2004cd ;
Nickel:2009ke ; Nickel:2009wj ; Carignano:2010ac ; Giannakis:2004pf is surely
intriguing but beyond the scope of the present work.
The constituent quark masses and the dynamical Majarona masses are expressed
in terms of these condensates as follows:
$M_{i}=m_{i}-4G_{S}\sigma_{i}+K|\varepsilon_{ijk}|\sigma_{j}\sigma_{k}+\frac{K^{\prime}}{4}|s_{i}|^{2}\,,$
(13)
and
$\Delta_{i}=2(G_{D}-\frac{K^{\prime}}{4}\sigma_{i})s_{i}\,.$ (14)
Similarly, it is convenient to define the dynamical quark chemical potential
for flavor $i$ by
${\tilde{\mu}}_{i}=\mu_{i}-4G_{V}{\rho_{i}}\,.$ (15)
A few remarks are in order here:
1. 1.
Both types of the anomaly terms ${\mathcal{L}_{\chi}}^{(6)}$ and
$\mathcal{L}_{\chi{d}}^{(6)}$ contribute to the constituent quark masses in
Eq. (13), and thus if $K^{\prime}$ and the diquark condensate $s_{i}$ are
finite, chiral symmetry is dynamically broken even when the usual chiral
condensates are absent.
2. 2.
The new anomaly term $\mathcal{L}_{\chi{d}}^{(6)}$ also modifies the formula
for the Majarona mass for the CSC phase so that the chiral condensates affects
the Majorana mass, and hence induce an interplay between the two condensates:
Indeed the ‘bare’ diquark-diquark coupling $G_{D}$ is replaced by an effective
one, ${G_{D}^{\prime}}_{i}\equiv G_{D}-\frac{K^{\prime}}{4}\sigma_{i}$, as
shown in Eq. (14), which is dependent on the chiral condensates. Thus the
flavor-dependent effective coupling ${G_{D}^{\prime}}_{i}$ is now dependent on
$T$ and $\mu$ through $\sigma_{i}$.
3. 3.
Equations. (13) and (14) clearly show that the flavor-mixing occurs not only
in the usual chiral condensates due to ${\mathcal{L}}_{\chi}^{(6)}$ but also
in the diquark condensates owing to ${\mathcal{L}}_{\chi d}^{(6)}$, which
would lead to interesting physical consequences.
4. 4.
It is also to be noted that the dynamical quark chemical potential
$\tilde{\mu}_{i}$ for u and d quarks are different from each other because of
the constraint of electric charge-neutrality ($\mu_{d}>\mu_{u}$) in 2CSC;
notice also, however, that they are dependent only on the respective density
$\rho_{u,d}$ and hence the dynamical chemical potentials $\tilde{\mu}_{u,d}$
tend to come closer because $\rho_{d}>\rho_{u}$ with the common coupling
constant $G_{V}$ Zhang:2009mk .
In the mean field level, the thermodynamic potential for the two-plus-one-
flavor NJL with the charge-neutrality constraints reads
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\Omega_{l}+2G_{S}\sum_{i=1}^{3}\sigma_{i}^{2}-2G_{V}\sum_{i=1}^{3}\rho_{i}^{2}+\sum_{i=1}^{3}(G_{D}-\frac{K^{\prime}}{2}\sigma_{i})\left|s_{i}\right|^{2}$
(16) $\displaystyle-$ $\displaystyle
4K\sigma_{1}\sigma_{2}\sigma_{3}-\frac{T}{2V}\sum_{P}\ln\det\frac{S^{-1}_{MF}}{T}\;.$
Notice the presence of the new cubic-mixing terms among the chiral and diquark
condensates. In Eq. (16), $\Omega_{l}$ denotes the contribution from free
leptons. Note that $\Omega_{l}$ should include the contributions from both
electrons and muons for completeness. Since $M_{\mu}>>M_{e}$ and $M_{e}\approx
0$, ignoring the contribution of muons has little effect on the phase
structure. Therefore, only electrons are considered in our calculation and the
corresponding $\Omega_{l}$ reads
$\Omega_{l}=-\frac{1}{12\pi^{2}}\left(\mu_{e}^{4}+2\pi^{2}T^{2}\mu_{e}^{2}+\frac{7\pi^{4}}{15}T^{4}\right)\,.$
(17)
Due to the large mass difference between s and u [d] quarks, the most favored
phase at low temperature and moderate density tends to be the 2CSC rather than
CFL phase, as demonstrated in the two-plus-one-flavor NJL model Ruester:2005jc
; Abuki:2005ms . Surprisingly enough, if the anomaly term
${{\mathcal{L}}_{\chi d}}^{(6)}$ is incorporated, the 2CSC phase turns to be
still favored in the intermediate density region even when the three flavors
have the equal mass Basler:2010xy . Needless to say, the dominance of the 2CSC
phase over the CFL one is more robust when the realistic mass hierarchy for
the three flavors is adopted. Moreover, it is worth mentioning here that the
mass disparity favors the 2CSC phase with the u-d pairing also through the
inequality of the effective diquark coupling
${G_{D}^{\prime}}_{3}>{G_{D}^{\prime}}_{1,2}$ when the anomaly coupling
$K^{\prime}$ is present; see Eq. (14). Since the main purpose of the present
work is to explore how the axial anomaly term ${\mathcal{L}}_{\chi d}^{(6)}$
affect the phase boundary involving the chiral transition at moderate
densities, under the constraints of the charge-neutrality and
$\beta$-equilibrium, we only consider the 2CSC phase in the following. Note
that $\mu_{3}$ in (8) vanishes in the 2CSC phase because the color SU(2)
symmetry for the red and green quarks are left unbroken.
(a) $K^{\prime}/K=2.0$
(b) $K^{\prime}/K=2.25$
(c) $K^{\prime}/K=2.4$
(d) $K^{\prime}/K=2.8$
Figure 1: The phase diagrams in the $T$-$\mu$ plane for various values of
$K^{\prime}$ in the two-plus-one-flavor NJL model with the charge-neutrality
and $\beta$-equilibrium being kept. The vector interaction is not taken into
account. The thick solid line, thin solid line and dashed line denote the
first order transition, second order transition and chiral crossover,
respectively.
The inverse quark-propagator matrix in the Nambu-Gor’kov formalism takes the
following form in the mean-field approximation,
$S^{-1}_{\mathrm{MF}}(i\omega_{n},\vec{p})=\bigg{(}\begin{array}[]{cc}[{G_{0}^{+}}]^{-1}&\Delta\gamma_{5}t_{3}^{f}t_{3}^{c}\\\
-\Delta^{*}\gamma_{5}t_{3}^{f}t_{3}^{c}&[{G_{0}^{-}}]^{-1}\end{array}\bigg{)}\,,$
(18)
with
$[{G_{0}^{\pm}}]^{-1}=\gamma_{0}(i\omega_{n}\pm\hat{\tilde{\mu}})-\vec{\gamma}\cdot\vec{p}-\hat{M}\,,$
(19)
where $\hat{M}={\rm diag}_{f}(M_{u},M_{d},M_{s})$, $\hat{\tilde{\mu}}={\rm
diag}_{f}(\tilde{\mu}_{u},\tilde{\mu}_{d},\tilde{\mu}_{s})$ and
$\omega_{n}=(2n+1)\pi{T}$ is the Matsubara frequency. Taking the Matsubara
sum, the last part of the thermodynamic potential (16) is expressed as
$-\frac{T}{2V}\sum_{P}\ln\det\frac{S^{-1}_{MF}}{T}=-\sum_{i=1}^{18}\int\frac{d^{3}p}{(2\pi)^{3}}\\{(E_{i}-E_{i}^{0})+2T\ln(1+e^{-E_{i}/T})\\},$
(20)
with the dispersion relations for nine quasi-particles (that is, three flavors
$\times$ three colors; the spin degeneracy is already taken into account in
Eq. (20)) and nine quasi-antiparticles. In Eq. (20), $E_{i}^{0}$ represents
$E_{i}(M=m,\Delta=0,\rho=0)$. The s quark and unpaired blue u and d quarks
have twelve energy dispersion relations with a similar form. For example, the
dispersion relations for the blue u quark and anti blue u quark are
$E_{bu}=E-\tilde{\mu}_{bu}\quad\text{and}\quad\bar{E}_{bu}=E+\tilde{\mu}_{bu}\,,$
(21)
respectively, with $E=\sqrt{\vec{p}^{2}+{M_{u}^{2}}}$. In the $rd$-$gu$ quark
sector with pairing we can find the four dispersion relations,
$\begin{split}E_{\text{$rd$-$gu$}}^{\pm}=E_{\Delta}\pm\tfrac{1}{2}(\tilde{\mu}_{rd}-\tilde{\mu}_{gu})=E_{\Delta}\pm\delta\tilde{\mu}\,,\\\
\bar{E}_{\text{$rd$-$gu$}}^{\pm}=\bar{E}_{\Delta}\pm\tfrac{1}{2}(\tilde{\mu}_{rd}-\tilde{\mu}_{gu})=\bar{E}_{\Delta}\pm\delta\tilde{\mu}\,,\end{split}$
(22)
and the $ru$-$gd$ sector has another four as
$\begin{split}E_{\text{$ru$-$gd$}}^{\pm}=E_{\Delta}\pm\tfrac{1}{2}(\tilde{\mu}_{ru}-\tilde{\mu}_{gd})=E_{\Delta}\mp\delta\tilde{\mu}\,,\\\
\bar{E}_{\text{$ru$-$gd$}}^{\pm}=\bar{E}_{\Delta}\pm\tfrac{1}{2}(\tilde{\mu}_{ru}-\tilde{\mu}_{gd})=\bar{E}_{\Delta}\mp\delta\tilde{\mu}\,,\end{split}$
(23)
where $E_{\Delta}=\sqrt{(E-\bar{\tilde{\mu}})^{2}+\Delta^{2}}$ and
$\bar{E}_{\Delta}=\sqrt{(E+\bar{\tilde{\mu}})^{2}+\Delta^{2}}$; from now on
$\Delta$ stands for $\Delta_{3}$. The average chemical potential is defined by
$\bar{\tilde{\mu}}=\frac{\tilde{\mu}_{rd}+\tilde{\mu}_{gu}}{2}=\frac{\tilde{\mu}_{ru}+\tilde{\mu}_{gd}}{2}=\mu-\frac{\mu_{e}}{6}-2G_{V}(\rho_{1}+\rho_{2})+\frac{\mu_{8}}{3}\,,$
(24)
and the effective mismatch between the chemical potentials of u and d quarks
takes the form
$\delta\tilde{\mu}=\tfrac{1}{2}(\mu_{e}-4G_{V}(\rho_{2}-\rho_{1})).$ (25)
Ignoring the the mass difference between u and d quarks, the determinantal
term in Eq. (16) has an analytical form which greatly simplifies the numerical
calculation. Adopting the variational method, we get the eight non-linear
coupling equations
$\frac{\partial\Omega}{\partial\sigma_{1}}=\frac{\partial\Omega}{\partial\sigma_{3}}=\frac{\partial\Omega}{\partial{s_{3}}}=\frac{\partial\Omega}{\partial\rho_{1}}=\frac{\partial\Omega}{\partial\rho_{2}}=\frac{\partial\Omega}{\partial\rho_{3}}=\frac{\partial\Omega}{\partial\mu_{e}}=\frac{\partial\Omega}{\partial{\mu_{8}}}=0\,.$
(26)
Since $\mu_{8}$ is tiny around the chiral transition region Ruester:2005jc ;
Abuki:2005ms , we shall set it zero with little effect in the numerical
results Zhang:2009mk . Thus, Eq. (26) is then simplified to a system of seven
coupled equations.
Figure 2: The temperature dependence of $M_{u},M_{s},\Delta$ and
$\delta\tilde{\mu}$ for fixed $K^{\prime}/K=2.25$ and three different chemical
potentials $\mu=312$ MeV, 320 MeV and 330 MeV. The constraints of electric
charge-neutrality and $\beta$-equilibrium are imposed while the vector
interaction is not incorporated.
## III Phase Structure With The Axial Anomaly
In this section, we show numerical results of the effects of the new six-quark
interaction (5) on the chiral phase transition under the charge-neutrality and
$\beta$-equilibrium constraints with or without the vector interaction. Since
we are mainly interested in the phase diagram involving chiral transition at
low temperatures, all the phase diagrams will be plotted for the range
$250\text{MeV}<\mu<400\text{MeV}$ where the chiral transition is expected to
be relevant. As for the type of the CSC phase, Ref. Basler:2010xy indicates
that the CFL phase is only realized for $\mu>460\text{MeV}$ even when
$K^{\prime}=0$ with the same model parameters as ours, and an increase of
$K^{\prime}$ pushes the CFL phase to even higher $\mu$ region. Therefore, we
exclusively consider the 2CSC phase near the chiral boundary without a loss of
generality.
In the following, we use the same notations as in Ref. Hatsuda:2006ps ;
Zhang:2008wx ; Zhang:2009mk to distinguish the different regions in the
$T$-$\mu$ phase diagram. Namely, NG, CSC, COE, and NOR refer to the hadronic
(Nambu-Goldstone) phase with $\sigma\neq 0$ and $\Delta=0$, color-
superconducting phase with $\Delta\neq 0$ and $\sigma=0$, coexisting phase
with $\sigma\neq 0$ and $\Delta\neq 0$, and normal phase with
$\sigma=\Delta=0$, respectively, though they have exact meanings only in the
chiral limit.
### III.1 The case without vector interaction
We first show the numerical results in the case without the vector
interaction. The phase diagrams with varying coupling constant $K^{\prime}$
are displayed in Fig. 1. In contrast to Fig. 8 in Ref. Basler:2010xy , the
multi-CP structure can still appear with a choice of $K^{\prime}$ in the phase
diagram when the charge-neutrality, $\beta$-equilibrium and the new axial
anomaly term are simultaneously taken into account. For $K^{\prime}/K=2.0$,
Fig. 1a shows that there exists only one usual chiral CP even though the COE
emerges: We remark that the COE region does not exist when $K^{\prime}/K=0$,
which is not displayed in Fig. 1. Figure 1b shows that when $K^{\prime}/K$ is
increased to 2.25, the chiral transition turns to a crossover at relatively
lower temperatures, and hence there appear two new chiral CP’s. With a further
increase of $K^{\prime}/K$, the boundary line for first-order transition at
higher temperature shrinks and thus the two crossover boundary lines in Fig.
1b join with each other, and eventually only one CP is left in the phase
diagram, as shown in Fig. 1c. When $K^{\prime}/K$ is large enough, Fig. 1d
indicates that the first order boundary vanishes completely and there is no
chiral CP in the phase diagram.
We note that the emergence of the three CP’s in Fig. 1b comes from a joint
effect of the interplay between the chiral and diquark condensates and the
electric charge-neutrality constraint. First of all, we recall that the
abnormal thermal behavior of the diquark condensate that it has a maximum at a
finite temperature in the COE is responsible for the emergence of the multiple
chiral CP structure Kitazawa:2002bc ; Addenda ; Zhang:2008wx ; Zhang:2009mk .
Such a behavior is also observed in the present case, as displayed in Fig. 1b.
As first indicated in Ref. Zhang:2008wx , when $\mu_{e}=\mu_{d}-\mu_{u}$ is
positive, the boundary of the chiral transition is shifted towards higher
$\mu$ region, and leads to the formation of the COE at low-temperature region,
in which the chiral phase transition is significantly weakened by the smearing
of the Fermi surface inherent in the CSC phase. In this regard, $\mu_{e}$
plays a role of an effective vector interaction Kitazawa:2002bc ; Addenda ;
Zhang:2009mk . On the other hand, the chiral anomaly term with positive
$K^{\prime}$ intensifies the competition between the chiral and diquark
condensates due to the enhanced effective diquark-diquark interaction. Thus
when $K^{\prime}$ is increased, the CSC region expands towards lower $\mu$
region in the $T$-$\mu$ plane. Consequently, the COE region tends to be more
easily formed when both $\mu_{e}$ and $K^{\prime}$ take effects. Therefore the
chiral transition is significantly weakened and the smooth crossover gets to
appear with new CP’s in the intermediate temperature owing to the abnormal
thermal behavior of the diquark condensate.
The $T$-dependence of $M_{u},M_{s},\Delta$ and $\delta\tilde{\mu}$ for fixed
$K^{\prime}/K=2.25$ and several values of $\mu$ is shown in Fig. 2. One can
see that, with increasing $T$, the constituent quark masses decrease
persistently while the Majarona mass for CSC first increases, has the maximum
value and then decreases in the COE region and nearby. Let us see the details
for each value of $\mu$. For a small $\mu=312$ MeV, the diquark pairing is
weak and the gap $\Delta$ does not appear at lower temperature region. Thus
the chiral phase transition keeps the nature of the first order at
$T_{C1}\approx{75}$ MeV. At a lager $\mu=320$ MeV, the diquark pairing becomes
significant and the $\Delta$ shows the abnormal thermal behavior with a
maximum value around $T_{C2}\approx 60$ MeV, and hence the chiral phase
transition turns to a smooth crossover owing to the competition with the
diquark condensate in the COE region. For even larger $\mu=330$ MeV, the
diquark pairing becomes more significant and the $\Delta$ still shows the
abnormal thermal behavior. However, the competition between the two
condensates is not strong enough to qualitatively change the nature of the
chiral restoration and a first order transition happens at $T_{C3}\approx 40$
MeV. The reason why the crossover does not occur at $\mu=330$ MeV but happens
at $\mu=320$ MeV can be understood as follows: Starting from the same point
($T=T_{C3}$, $\mu=320$ MeV) in the COE region, an increase of $T$ affects the
nature of the chiral transition more significantly than that of $\mu$ does,
since $T_{C3}<T_{C2}$.
We have seen that the abnormal thermal behavior of the gap $\Delta$ plays an
essential role in realizing the multi-CP structure of the phase diagram. Such
an unusual $T$-dependence of the $\Delta$ can be attributed to the following
two mechanisms: (i) the mismatch between the chemical potentials of u and d
quarks owing to the charge-neutrality and $\beta$-equilibrium constraints and
(ii) the small Fermi spheres of the quarks in the COE region due to the
relatively large quark masses: First, the difference in the chemical
potentials $\delta\tilde{\mu}$ or the mismatch of the Fermi momenta disfavors
the u-d pairing at zero or small temperature. However, as the temperature is
raised in the low-$T$ region, more and more u and d quarks tends to
participate in the pairing due to the smearing of the Fermi surfaces,
especially that of the u quark. Of course, when $T$ is raised too much, the
pairing will be gradually destroyed. Thus the $\Delta$ will have the maximum
value at a finite $T$ and then disappears eventually when $T$ is further
raised. These dual effects of the temperature on the diquark pairing lead to
the abnormal behavior of the $\Delta$. This behavior becomes more prominent in
the 2CSC phase for a weak diquark coupling Shovkovy:2004me or in the COE
region for a moderate or strong diquark coupling Zhang:2008wx ; Zhang:2009mk .
Second, when $T$ is raised, the dynamical quark masses decrease and hence the
Fermi spheres or momenta of u and d quarks grow significantly for a fixed
$\mu$, which means that the density of states at the Fermi surface increases
with $T$, and thus the diquark pairing is enhanced in the COE region. The
increased diquark condensates in turn tend to further suppress the dynamical
quark masses.
Notice that such an increase of the diquark condensate along with increasing
$T$ is expected to be most prominent around the phase boundary of the chiral
transition, including the COE region, where the chiral condensates change most
significantly. For the neutral 2CSC, once the COE is formed, both of these
mechanisms take effects simultaneously and are mutually enhanced, and thus the
formation of the multiple-CP structure is readily made.
This may explain why no intermediate-temperature CP is realized in Ref.
Basler:2010xy where the chiral-diquark interplay is embodied by the anomaly
term but without the charge-neutrality and $\beta$-equilibrium constraints;
the anomaly term solely is insufficient for realizing the abnormal thermal
behavior of the $\Delta$. It should be stressed that the mechanism for the
emergence of the intermediate-temperature CP’s in Fig. 1b is apparently
similar to that in the two-flavor case found in Zhang:2008wx . However, the
strange quark plays an important role in the present case since the chiral
condensate of the strange quark contributes positively to the effective
diquark-diquark coupling for u and d quarks through the axial anomaly. We
should stress that apart from the appearance of intermediate-temperature CP’s,
there is a common feature with and without charge-neutrality constraint: the
chiral transition in the low-$T$ region extending zero temperature keeps first
order provided that $K^{\prime}$ does not exceed a critical value at which the
first-order line completely disappears.
Last but not least, we remark that the $T$-$\mu$ region where the two new low-
temperature CP’s are located in Fig. 1b is free from the chromomagnetic
instability, which is obvious from Fig. 5a; a detailed discussion on this
point will be given in Sec.IV.
### III.2 The case for nonzero vector interaction
In this subsection, we will investigate the phase diagram when both the vector
and the new six-quark interactions are present under the charge-neutrality and
$\beta$-equilibrium constraints.
There are some choices for the value of the vector coupling: The chiral
instanton-anti-instanton molecule model ref:SS gives the ratio
$G_{V}/G_{S}=0.25$, while the Fierz transformation of the vertex given in the
truncated Dyson-Schwinger model ref:RWP gives the ratio 0.5. Thus we rather
treat the $G_{V}/G_{S}$ as a free parameter in the range, $0$-$0.5$.
(a) $K^{\prime}/K=0.55$
(b) $K^{\prime}/K=0.57$
(c) $K^{\prime}/K=0.70$
(d) $K^{\prime}/K=1.0$
Figure 3: The $T$-$\mu$ phase diagrams of the two-plus-one-flavor NJL model
for several values of $K^{\prime}/K$ and fixed $G_{V}/G_{S}=0.25$, where the
charge-neutrality constraint and $\beta$-equilibrium condition are imposed.
With the increase of $K^{\prime}/K$, the number of the critical points changes
and the unstable region characterized by the chromomagnetic instability
(bordered by the dash dotted line) tends to shrink and ultimately vanishes in
the phase diagram. The respective meanings of the various types of lines are
the same as those in Fig. 1.
We first explore the phase diagram in the $T$-$\mu$ plane by varying the ratio
$K^{\prime}/K$ but with $G_{V}/G_{S}$ being fixed as 0.25, the value given in
the instanton-anti-instanton molecule model. When $K^{\prime}/K$ is small and
less than $0.5$, only the usual phase structure with single CP is obtained.
When $K^{\prime}/K$ exceeds $0.5$, other four different types of the CP
structures appear, as displayed in Fig. 3. At $K^{\prime}/K=0.55$, a phase
diagram similar to that in Fig. 1b is obtained, as shown in Fig. 3a, where two
new intermediate-temperature CP’s emerge. When $K^{\prime}/K$ is slightly
increased to 0.57, the chiral transition becomes crossover in the lower-
temperature region which extends to zero temperature; thus the total number of
the CP’s becomes four, which indicates stronger competition between the chiral
and diquark condensates at relatively larger $\mu$. Further increasing
$K^{\prime}/K$, the low-temperature chiral boundary totally turns into a
crossover one and only one first-order transition line with two CP’s attached
remains in the phase diagram, as displayed in Fig. 3c. In this case, the
number of the CP’s is reduced to two accordingly. When $K^{\prime}/K$ is large
enough, Fig. 3d shows that only chiral crossover transition exists in the
phase diagram with no CP.
In comparison with Fig. 1 where the vector interaction is not included, Fig. 3
indicates that the phase structures with multiple CP’s can be realized with
relatively small $K^{\prime}$ owing to the vector interaction. We remark that
all the types of the chiral CP structures displayed in Fig. 3 by varying
$K^{\prime}$ are obtained by varying $G_{V}$ without the anomaly term
Zhang:2009mk . In the present case, the number of the critical points changes
as 1$\rightarrow$ 3 $\rightarrow$ 4 $\rightarrow$ 2 $\rightarrow$ 0 when
$K^{\prime}$ is increased.
As is mentioned before, the Fierz transformation of the instanton vertex leads
to the identity $K^{\prime}=K$, so it is of special interest to investigate
the phase diagram in the case of $K^{\prime}=K$. A series of phase diagram
with fixed $K^{\prime}/K=1$ but varied $G_{V}$ are shown in Fig. 4. One finds
that all the chiral CP structures in Fig. 3 still appear in the phase
diagrams, and moreover, even as large as five CP’s can exist in the phase
diagram, as shown in Fig. 4c. This suggests that the interplay between the
chiral and the diquark condensates in the COE region becomes complicated once
the charge-neutrality, the vector interaction and the axial anomaly are all
taken into account. A comparison with the case of vanishing $K^{\prime}$,
which is given in Fig. 7 in Zhang:2009mk , shows that the parameter range of
$G_{V}$ for realizing the low-temperature CP’s moves towards lower $G_{V}$,
which is actually natural because $K^{\prime}$ gives the same effect as
$G_{V}$ on the chiral transition. It is noteworthy that such lower values of
$G_{V}$ are also close to the standard value of $G_{V}/G_{S}$ derived from the
instanton model. The number of the CP’s changes as 1 $\rightarrow$ 3
$\rightarrow$ 5 $\rightarrow$ 4 $\rightarrow$ 2 $\rightarrow$ 0 with
increasing $G_{V}$.
(a) $G_{V}/G_{S}=0$
(b) $G_{V}/G_{S}=0.193$
(c) $G_{V}/G_{S}=0.195$
(d) $G_{V}/G_{S}=0.197$
(e) $G_{V}/G_{S}=0.23$
(f) $G_{V}/G_{S}=0.3$
Figure 4: The phase diagrams in the two-plus-one-flavor NJL model for fixed
$K^{\prime}/K=1.0$ with $G_{V}/G_{S}$ being varied, where the charge-
neutrality constraint and $\beta$-equilibrium condition are taken into
account. The meanings of the different line types are the same as those in
Fig. 1. The number of the critical points changes along with an increase of
$G_{V}/G_{S}$. All the phase diagrams are free from the chromomagnetic
instability.
The anomaly terms in Eq.(1) are supposed to originate from the instantons,
which are to be screened at finite chemical potential and temperatureref:SS .
Accordingly, both the coupling constants $K$ and $K^{\prime}$ are expected to
diminish around the phase boundary. However, we emphasize that the main effect
of $K^{\prime}$ is to enhance the chiral condensate of the strange quark and
the u-d diquark condensate by each other through the cubic coupling among them
for the realistic quark masses, and Figs.3 and 4 tell us that even smaller
values of $K^{\prime}$ expected at low temperature and moderate density can
still lead to a quite different phase structure with multiple CP’s when the
vector interaction is present under the charge-neutrality constraint.
## IV The Influence On The Chromomagnetic Instability
In this section, we investigate effect of the new axial-anomaly term on the
chromomagnetic instability of the asymmetric homogeneous 2CSC phase by varying
$K^{\prime}$. We shall show that the anomaly-induced interplay between the
chiral and diquark condensates acts toward suppressing the unstable region of
the homogeneous 2CSC phase in the $T$-$\mu$ plane. Thus the 2CSC phase can
become even free from the chromomagnetic instability provided that
$K^{\prime}$ is larger than a critical value $K^{\prime}_{c}$, which can be
reduced significantly when the vector interaction is incorporated.
The magnetic instability region in the $T$-$\mu$ plane is determined by
calculating the Meissner masses squared which can be negative when the charge-
neutrality constraint is imposed. Here we adopt the same method as that in
Kiriyama:2006jp to evaluate the Meissner mass squared
$m_{M}^{2}=\frac{\partial^{2}}{\partial{B^{2}}}[\Omega(\Delta)-\Omega(\Delta=0)]_{B=0},$
(27)
where $B$ has the same meaning as that in Kiriyama:2006jp . Since the strange
quark does not take part in the diquark pairing in the present case, we can
directly use the formula for two-flavor NJL model to calculate the Meissner
mass squared.
The effect of the coupling constant $K^{\prime}$ on the chromomagnetic
instability is shown in Fig.5. We have adopted the model parameters in Table 1
to calculate the Meissner mass squared. Figure 5a displays the change of the
unstable region of the chromomagnetic instability with varying $K^{\prime}$
when the vector interaction is not included. One can see that the instability
region tends to shrink with increasing $K^{\prime}$ and eventually vanishes
for $K^{\prime}/K>0.8$. This suggests that the neutral homogenous 2CSC phase
will be totally free from the chromomagnetic instability if $K^{\prime}=K$
that is derived by the Fierz transformation from the usual instanton vertex.
When taking the vector interaction with $G_{V}/G_{S}=0.5$, the unstable region
shrinks more significantly with increasing $K^{\prime}$ and eventually
disappears in the $T$-$\mu$ plane for $K^{\prime}/K>0.55$, as shown in Fig.5b.
This could be an expected result because of the effect of the vector
interaction on the instability problem found in Zhang:2009mk .
The reason for the suppression of the chromomagnetic instability by
$K^{\prime}$ is understood as follows. First of all, Eq. (14) tells us that
the u-d diquark coupling is enhanced by the presence of the s quark chiral
condensate due to the coupling between the chiral and diquark condensates
induced by the $K^{\prime}$ term. On the other hand, as was first shown in
Kitazawa:2006zp through changing the diquark coupling by hand, the
chromomagnetic instability tends to be suppressed in the strong coupling
region and can be completely gotten rid of when the diquark coupling is strong
enough. Thus we see that the coupling between the u-d diquark and the chiral
s-quark condensates by the $K^{\prime}$ term leads to the suppression of the
chromomagnetic instability. This is a new mechanism of the stabilization of
the gapless 2CSC phase, found in the present work. We here emphasize the
important role of the strange quark and the anomaly term in suppressing the
instability: In contrast to the pure two-flavor case, the rather large chiral
condensate of the strange quark enhances the diquark coupling between the u
and d quarks owing to the axial anomaly in the two-plus-one flavor case, and
this enhancement of the diquark coupling causes the stabilization.
Due to their common effects on the chromomagnetic (in)stability, Fig.5
suggests that the instability may be totally cured in the asymmetric
homogeneous 2CSC phase when the coupling constants of the vector interaction
and the extended six-quark interaction are in an appropriate range.
Admittedly, the present work has only dealt with the case of the so called
intermediate diquark coupling. Nevertheless, for a weaker diquark coupling, it
is expected that the system can be still free from the chromomagnetic
instability only with larger couplings for both the vector interaction and the
anomaly $K^{\prime}$-term.
(a)
(b)
Figure 5: The boundary between the stable and unstable homogenous 2CSC
regions with (right figure) and without (left figure) the vector interaction
in two-plus-one-flavor NJL model. With the increase of the ratio
$K^{\prime}/K\equiv R$, the unstable region with the chromomagnetic
instability in the $T$-$\mu$ plane shrinks and eventually vanishes.
## V CONCLUSIONS AND OUTLOOK
We have explored the phase structure and the chromomagnetic instability of the
strongly interacting matter under the charge-neutrality constraint within a
two-plus-one-flavor NJL model by incorporating a new anomaly term as well as
the conventional KMT interaction. The anomaly terms have the forms of six-
quark interactions and violate the $U_{A}(1)$ symmetry as a reflection of the
axial anomaly of QCD. Similarly to the KMT term, the new anomaly interaction
with the coupling constant $K^{\prime}$ also induces a flavor-mixing which
leads to a direct coupling between the chiral and diquark condensates.
We first investigated the role of the axial anomaly on the emergence of the
low or intermediate-temperature CP(’s) without the vector interaction. Owing
to the large strange quark mass, the favored CSC phase near the chiral
boundary is 2CSC rather than CFL, where the electric chemical potential
$\mu_{e}$ required by the charge-neutrality plays an important role on the
chiral phase transition Zhang:2008wx . The once-declared low-temperature CP in
the symmetric three-flavor limit Abuki:2010jq was ruled out in Ref.
Basler:2010xy due to the actual dominance of the 2CSC over the CFL. We have
shown that this is true under charge-neutrality constraint without the vector
interaction; the chiral transition in the low-$T$ region extending zero
temperature keeps first order provided that $K^{\prime}$ does not exceed a
critical value at which the first-order line completely disappears. However,
the new chiral anomaly term enhances the competition between the chiral and
diquark condensates under charge-neutrality constraint, and gives rise to the
intermediate-temperature CP’s for an appropriate range of $K^{\prime}$. No
such intermediate-temperature CP’s had been found in the same model when only
either the charge-neutrality constraint or the axial anomaly is exclusively
included, as shown in Ruester:2005jc and Basler:2010xy ; both of which did
not take into account the vector interaction, either.
We then investigated the $T$-$\mu$ phase diagram by incorporating the
repulsive vector interaction as well: We remark that this task may be viewed
as an extension of the work Zhang:2009mk , in which the effect of the vector
interaction on the phase diagram is fully explored under the charge-neutrality
constraint, to incorporate the anomaly term. We have found that the cubic
coupling between the chiral and diquark condensates induced by the axial
anomaly does not affect the qualitative results obtained in Zhang:2009mk .
Rather, the vector interaction and the anomaly term jointly act so that the
multiple CP’s are realized. Indeed, by varying $K^{\prime}$ with fixed vector
coupling or vise verse, we have shown that all the types of multiple-CP
structures obtained in Zhang:2009mk can be reproduced. In particular, the
phase transition in the low-$T$ region extending zero temperature becomes a
crossover only when the vector interaction is present with a strength larger
than a critical value. Furthermore, the maximum number of the CP’s can reach
as large as five when both the interactions are put on. In this case, the low-
and intermediate-temperature CP’s can appear even with small values of
$K^{\prime}$ owing to the help by the vector interaction. This is very welcome
because $K^{\prime}$ in the realistic situation at moderate and high density
should be weaker than that in the vacuum, since the anomaly term is supposed
to originate from the instanton configuration which is expected to be
suppressed at finite density.
Besides the influence on the chiral phase transition, we have shown that the
axial anomaly also plays an important role on the suppression of the
chromomagnetic instability for the asymmetric homogenous 2CSC phase, which is
first disclosed in the present work: With an increase of the extended six-
quark interaction, the $T$-$\mu$ region with the chromomagnetic instability
shrinks and eventually vanishes when the coupling $K^{\prime}$ is sufficiently
large. In particular, when taking into account the vector interaction
simultaneously, the chromomagnetic instability is suppressed more
significantly and can be completely gotten rid of by the axial anomaly.
A general and remarkable message obtained from the present investigation is
that the strange quark can significantly affect the properties of the neutral
strongly interacting matter in which the 2CSC phase with u-d pairing is
realized: Even though the strange quark does not directly participate in the
Cooper pairing in the 2CSC, the interplay between the u-d diquark condensate
and the strange chiral condensate induced by the anomaly term can lead to
drastically different phase structure in the $T$-$\mu$ plane under charge-
neutrality constraint.
It should be remarked here that the contribution of other possible cubic
flavor-mixing terms composed of different condensates, such as
$\sigma\rho^{2}=\epsilon^{ijk}\sigma_{i}\rho_{j}\rho_{k},$ (28)
which arise from another type of six-quark interaction
$\mathcal{L}_{\chi{\rho}}^{(6)}\sim\epsilon^{ijk}\epsilon^{lmn}(\bar{\psi}_{i}\gamma^{\mu}(1\pm\gamma_{5})\psi_{l})(\bar{\psi}_{j}\gamma_{\mu}(1\pm\gamma_{5})\psi_{m})(\bar{\psi}_{k}(1\pm\gamma_{5})\psi_{n}),$
(29)
are all neglected in Eq. (16) for simplicity. The interaction (29) can be
derived from the KMT interaction, which may or may not affect the phase
structure. Beside their direct contribution to the thermodynamic potential,
these flavor-mixing terms also modify the dispersion relations of the quasi-
quarks: For example, the dynamical quark mass becomes dependent on the quark-
number density through the term $\sigma\rho^{2}$. It is certainly an
interesting problem to explore the possible effects of these cubic coupling
terms on the phase diagram, we leave such a task to a future work.
Even apart from the neglect of the above vertex (29), there are some caveats
with the present study based on a chiral model that does not embody the
confinement effect, and is relied on the mean-field approximation. The results
obtained in the current study are largely parameter dependent and bears the
shortcomings inherent in the mean-field approximation. For instance, the
result that there can be multiple CP’s associated with the chiral transition
and the CSC actually may merely mean that the QCD matter is very soft for a
simultaneous formation of the diquark and chiral condensates coupled with the
baryonic density along the phase boundary. Of course, a study which
incorporates these fluctuations should be performed, say, by means of the
nonperturbative/functional renormalization group methodAoki:2000wm ;
Berges:2000ew with the present model used as a bare model. More profoundly,
the effect of the confinement should be incorporated even in an effective
model approach, which is a more challenging problem since the mechanism of
confinement is still unclear. Anyway, further studies based on different
models and/or methods are needed to determine whether the low-temperature
CP(’s) exists. One of the tasks of future is exploring whether the low-
temperature CP(’s) persists or not when the inhomogeneous phases are taken
into consideration such as the chiral crystalline phase Nakano:2004cd ;
Nickel:2009ke ; Nickel:2009wj ; Carignano:2010ac or the LOFF phase
Giannakis:2004pf .
###### Acknowledgements.
Z.Z. was supported by the Fundamental Research Funds for the Central
Universities of China. T.K. was partially supported by a Grant-in-Aid for
Scientific Research by the Ministry of Education, Culture, Sports, Science and
Technology (MEXT) of Japan (No.20540265), by Yukawa International Program for
Quark-Hadron Sciences, and by the Grant-in-Aid for the global COE program “
The Next Generation of Physics, Spun from Universality and Emergence ” from
MEXT.
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|
arxiv-papers
| 2011-02-16T08:46:29 |
2024-09-04T02:49:17.033108
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao Zhang and Teiji Kunihiro",
"submitter": "Zhao Zhang",
"url": "https://arxiv.org/abs/1102.3263"
}
|
1102.3285
|
We study novel simulation-like preorders for quotienting nondeterministic Büchi automata.
We define fixed-word delayed simulation,
a new preorder coarser than delayed simulation.
We argue that fixed-word simulation is the coarsest forward simulation-like preorder which can be used for quotienting Büchi automata,
thus improving our understanding of the limits of quotienting.
Also, we show that computing fixed-word simulation is PSPACE-complete.
On the practical side,
we introduce proxy simulations,
which are novel polynomial-time computable preorders sound for quotienting.
In particular, delayed proxy simulation induce quotients that can be smaller by an arbitrarily large factor than direct backward simulation.
We derive proxy simulations as the product of a theory of refinement transformers:
A refinement transformer maps preorders nondecreasingly,
preserving certain properties.
We study under which general conditions refinement transformers are sound for quotienting.
We thank Richard Mayr and Patrick Totzke for helpful discussions,
and two anonymous reviewers for their valuable feedback.
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|
arxiv-papers
| 2011-02-16T10:16:26 |
2024-09-04T02:49:17.042437
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lorenzo Clemente",
"submitter": "Lorenzo Clemente Lorenzo Clemente",
"url": "https://arxiv.org/abs/1102.3285"
}
|
1102.3294
|
# Causal Rate Distortion Function on Abstract Alphabets and Optimal
Reconstruction Kernel
Charalambos D. Charalambous, Photios A. Stavrou and Christos K. Kourtellaris
C. D. Charalambous (chadcha@ucy.ac.cy). P. A. Stavrou
(stavrou.fotios@ucy.ac.cy). C. K. Kourtellaris
(kourtellaris.christos@ucy.ac.cy). The authors are with the Department of
Electrical and Computer Engineering, University of Cyprus, Nicosia, CYPRUS
###### Abstract
A Causal rate distortion function with a general fidelity criterion is
formulated on abstract alphabets and the optimal reconstruction kernel is
derived, which consists of a product of causal kernels. In the process,
general abstract spaces are introduced to show existence of the minimizing
kernel using weak∗-convergence. Certain properties of the causal rate
distortion function are presented.
## I INTRODUCTION
This paper is concerned with lossy data compression subject to distortion or
fidelity criterion and causal decoding on abstract alphabets. Its information
theoretic interpretation is the causal rate distortion function formulated via
the directed information between the source sequence
$X^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{0},X_{1},\ldots,X_{n}\\}$
and its reproduction sequence
$Y^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{Y_{0},Y_{1},\ldots,Y_{n}\\}$
defined by
$\displaystyle I(X^{n}{\rightarrow}Y^{n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i=0}^{n}I(X^{i};Y_{i}|Y^{i-1})$
(1)
The average distortion constraint is
$\displaystyle
E\\{d_{0,n}(X^{n},Y^{n})\\}\leq{D},\>d_{0,n}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\sum^{n}_{i=0}\rho_{0,i}(x^{i},y^{i})$
(2)
where $D\geq 0$, $d_{0,n}(\cdot,\cdot)$ a non-negative distortion function.
Define the causal product of conditional distributions by
$\displaystyle{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\otimes^{n}_{i=0}P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})$
(3)
where $P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})$ denotes the conditional
distribution of $Y_{i}$ given $(Y^{i-1},X^{i}),~{}i=0,1,\ldots,n.$
Since causal codes as defined in [4] satisfy
$P_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|x^{i-1},y^{i-1})=P_{X_{i}|X^{i-1}}(dx_{i}|x^{i-1})$.
$P-a.s$ (see also Lemma II.4), in the analysis it is convenient to express
$I(X^{n}\rightarrow{Y^{n}})$ as a functional of
${\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})$ as follows.
$\displaystyle I(X^{n}\rightarrow{Y^{n}})$
$\displaystyle=\int\log(\frac{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{{P}_{Y^{n}}(dy^{n})})$
$\displaystyle\times{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})P_{X^{n}}(dx^{n})$
(4) $\displaystyle={\mathbb{I}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})$
(5)
where ${\mathbb{I}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})$ indicates
the functional dependence of $I(X^{n}\rightarrow{Y^{n}})$ on
$\\{P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}}\\}$.
The causal information rate distortion function investigated is
$\displaystyle\inf_{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n}):E\big{\\{}d_{0,n}(X^{n},Y^{n})\big{\\}}\leq
D}I(X^{n}\rightarrow{Y^{n}})$ (6)
Under appropriate assumptions on $d_{0,n}(\cdot,\cdot)$ it is shown that the
optimal causal product (reproduction channel)
${\overrightarrow{P}}^{*}_{Y^{n}|X^{n}}$ which achieves the infimum in (6) is
given by
$\displaystyle{\overrightarrow{P}}^{*}_{Y^{n}|X^{n}}(dy^{n}|x^{n})=\otimes^{n}_{i=0}\frac{e^{s\rho_{i}(x^{i},y^{i})}P^{*}_{Y_{i}|Y_{i-1}}(dy_{i}|y^{i-1})}{\int_{{\cal
Y}_{i}}e^{\rho_{i}(x^{i},y^{i})}P^{*}_{Y_{i}|Y_{i-1}}(dy_{i}|y^{i-1})}$ (7)
where $s\leq 0$ is the Lagrange multiplier associated with the fidelity
constraint. The operational meaning of (6) is shown in [5] via coding theorems
(called sequential code), hence this aspect will not be discussed. Rather, the
main emphasis of the paper is the mathematical formulation, the prove of
existence of solution to (6), the derivation of (7), the derivation of a
closed form expression for the causal rate distortion function, and some of
its properties.
The Shannon source code consists of an encoder-decoder pair. The encoder
observes a source sequence
$X^{\infty}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{0},X_{1},\ldots\\}$
and generates a compressed representation $\\{Z_{0},Z_{1},\ldots\\}$. The
decoder upon observing the representation sequence $\\{Z_{0},Z_{1},\ldots\\}$
generates a reproduction sequence $Y_{i}=f_{i}(X^{\infty})$ of $X_{i}$, for
every time step $i$. The dependence of the reproduction sequence on the future
source symbols, in addition to its past and present symbols makes such a
decoder non-causal. In Neuhoff and Gilbert [4], a source code is defined as
causal if the reproduction sequence is such that
$f_{i}(X^{\infty})=f_{i}(\tilde{X}^{\infty})$ whenever
$X^{i}={\tilde{X}}^{i},~{}\forall i=0,1,\ldots$. The definition of a causal
code necessitates that any information theoretic causal rate distortion
function should lead to an optimal reconstruction conditional distribution
which is causally dependent on the source symbols, and (7) has this property.
The classical rate distortion function is defined via the mutual information
between $X^{n}$ and $Y^{n}$, namely, $I(X^{n};Y^{n})$ with average distortion
(2), and the code is assumed non-causal, leading to the well known optimal
reconstruction [1, 3]
$\displaystyle
P_{Y^{n}|X^{n}}^{*}(dy^{n}|x^{n})=\frac{e^{s\sum_{i=0}^{n}\rho_{0,i}(x^{i},y^{i})}P_{Y^{n}}^{*}(dy^{n})}{\int_{{\cal
Y}_{0,n}}e^{s\sum_{i=0}^{n}\rho_{0,i}(x^{i},y^{i})}P_{Y^{n}}^{*}(dy^{n})}$ (8)
Since by chain rule
$P_{Y^{n}|X^{n}}(dy^{n}|X^{n}=x^{n})=\otimes_{i=0}^{n}P_{Y_{i}|Y^{i-1},X^{n}=x^{n}}(dy_{i}|y^{i-1}=y^{i-1},X^{n}=x^{n})$,
the classical rate distortion theory gives a reconstruction $Y_{i}=y_{i}$
which depends on future values of the source symbols,
$(X_{i+1}=x_{i+1},\ldots,X_{n}=x_{n})$ in addition to its past reconstructions
$Y^{i-1}=y^{i-1}$, and past and present source symbols $X^{i}=x^{i}$. The
point to be made here is that, in general, aside from some special examples,
such as the i.i.d source and single letter distortion
$d_{0,n}=\sum^{n}_{i=0}\rho_{i}(x_{i},y_{i})$ [2] the reconstruction
conditional distribution and hence the decoder of the classical rate
distortion function is non-causal. On the other hand, a code is causal if the
reconstruction distribution is causal.
## II PROBLEM FORMULATION
In this section, we introduce the set up of the problem on discrete time sets
$\mathbb{N}^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{0,1,\ldots,n\\}$,
$n\in\mathbb{N}\stackrel{{\scriptstyle\triangle}}{{=}}\\{0,1,2,\ldots\\}$.
Assume all processes are defined on a complete probability space
$(\Omega,{\cal F}(\Omega),\mathbb{P})$ with filtration $\\{{\cal
F}_{t}\\}_{t\geq 0}$. The source and reconstruction alphabets are sequences of
Polish spaces [11] $\\{{\cal X}_{t}:t\in\mathbb{N}\\}$ and $\\{{\cal
Y}_{t}:t\in\mathbb{N}\\}$, respectively, (e.g., ${\cal Y}_{t},{\cal X}_{t}$
are complete separable metric spaces), associated with their corresponding
measurable spaces $({\cal X}_{t},{\cal B}({\cal X}_{t}))$ and $({\cal
Y}_{t},{\cal B}({\cal Y}_{t}))$ (e.g., ${\cal B}({\cal X}_{t})$ is a Borel
$\sigma-$algebra of subsets of the set ${\cal X}_{t}$ generated by closed
sets), $t\in\mathbb{N}$. Sequences of alphabets are identified with the
product spaces $({\cal X}_{0,n},{\cal B}({\cal
X}_{0,n}))\stackrel{{\scriptstyle\triangle}}{{=}}\times_{k=0}^{n}({\cal
X}_{k},{\cal B}({\cal X}_{k}))$, and $({\cal Y}_{0,n},{\cal B}({\cal
Y}_{0,n}))\stackrel{{\scriptstyle\triangle}}{{=}}\times_{k=0}^{n}({\cal
Y}_{k},{\cal B}({\cal Y}_{k}))$. The source and reconstruction are processes
denoted by
$X^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{t}:t\in\mathbb{N}^{n}\\}$,
$X:\mathbb{N}^{n}\times\Omega\mapsto{\cal X}_{t}$, and by
$Y^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{Y_{t}:t\in\mathbb{N}^{n}\\}$,
$Y:\mathbb{N}^{n}\times\Omega\mapsto{\cal Y}_{t}$, respectively. Probability
measures on any measurable space $({\cal Z},{\cal B}({\cal Z}))$ are denoted
by ${\cal M}_{1}({\cal Z})$. It is assumed that the $\sigma$-algebras
$\sigma\\{X^{-1}\\}=\sigma\\{Y^{-1}\\}=\\{\emptyset,\Omega\\}$.
###### Definition II.1
Let $({\cal X},{\cal B}({\cal X})),({\cal Y},{\cal B}({\cal Y}))$ be
measurable spaces in which $\cal Y$ is a Polish Space.
A stochastic Kernel on $\cal Y$ given $\cal X$ is a mapping $q:{\cal B}({\cal
Y})\times{\cal X}\rightarrow[0,1]$ satisfying the following two properties:
1) For every $x\in{\cal X}$, the set function $q(\cdot;x)$ is a probability
measure (possibly finitely additive) on ${\cal B}({\cal Y}).$
2) For every $F\in{\cal B}({\cal Y})$, the function $q(F;\cdot)$ is ${\cal
B}({\cal X})$-measurable.
The set of all such stochastic Kernels is denoted by ${\cal Q}({\cal Y};{\cal
X})$.
An important notion is conditional independence. The Random Variable (R.V.)
${Z}$ is called conditional independent of R.V. $X$ given the R.V. $Y$ if and
only if $X\leftrightarrow Y\leftrightarrow Z$ forms a Markov chain in both
directions.
Stochastic kernels can be used to define non-causal and causal product
reconstruction kernels and associated rate distortion functions.
###### Definition II.2
Given measurable spaces $({\cal X}_{0,n},{\cal B}({\cal X}_{0,n}))$, $({\cal
Y}_{0,n},{\cal B}({\cal Y}_{0,n}))$, and their product spaces, data
compression channels are defined as follows.
1. 1.
A Non-Causal Data Compression Channel is a stochastic kernel
$q_{0,n}(dy^{n};x^{n})\in{\cal Q}({\cal Y}_{0,n};{\cal
X}_{0,n}),n\in\mathbb{N}$.
2. 2.
A Causal Product Data Compression Channel is a product of a sequence of causal
stochastic kernels defined by
$\displaystyle{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})$
$\displaystyle=\otimes_{i=0}^{n}q_{i}(dy_{i};y^{i-1},x^{i})$
where $q_{i}\in{\cal Q}({\cal Y}_{i};{\cal Y}_{0,i-1}\times{\cal
X}_{0,i}),i=0,\ldots,n,~{}n\in\mathbb{N}$.
Note that classical rate distortion theory is concerned with finding the
optimal $P_{Y^{n}|X^{n}}(dy^{n}|X^{n}=x^{n})$, which is generally non-causal,
while in this paper the interest is to find the optimal causal product kernel.
### II-A Causal and Classical Rate Distortion Functions
In this section the classical rate distortion function which has a non-causal
structure is reviewed, and then the causal rate distortion function is
defined.
Given a source probability measure ${\cal\mu}_{0,n}\in{\cal M}_{1}({\cal
X}_{0,n})$ (possibly finite additive) and a reconstruction Kernel
$q_{0,n}\in{\cal Q}({\cal Y}_{0,n};{\cal X}_{0,n})$, one can define three
probability measures as follows.
(P1): The joint measure $P_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n}\times{\cal
X}_{0,n})$:
$\displaystyle P_{0,n}(G_{0,n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}(\mu_{0,n}\otimes
q_{0,n})(G_{0,n}),\>G_{0,n}\in{\cal B}({\cal X}_{0,n})\times{\cal B}({\cal
Y}_{0,n})$ $\displaystyle=\int_{{\cal
X}_{0,n}}q_{0,n}(G_{0,n,x^{n}};x^{n})\mu_{0,n}(d{x^{n}})$
where $G_{0,n,x^{n}}$ is the $x^{n}-$section of $G_{0,n}$ at point ${x^{n}}$
defined by
$G_{0,n,x^{n}}\stackrel{{\scriptstyle\triangle}}{{=}}\\{y^{n}\in{\cal
Y}_{0,n}:(x^{n},y^{n})\in G_{0,n}\\}$ and $\otimes$ denotes the convolution.
(P2): The marginal measure $\nu_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n})$:
$\displaystyle\nu_{0,n}(F_{0,n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}P_{0,n}({\cal
X}_{0,n}\times F_{0,n}),~{}F_{0,n}\in{\cal B}({\cal Y}_{0,n})$
$\displaystyle=\int_{{\cal X}_{0,n}}q_{0,n}(({\cal X}_{0,n}\times
F_{0,n})_{{x}^{n}};{x}^{n})\mu_{0,n}(d{x^{n}})$ $\displaystyle=\int_{{\cal
X}_{0,n}}q_{0,n}(F_{0,n};x^{n})\mu_{0,n}(dx^{n})$
(P3): The product measure $\pi_{0,n}:{\cal B}({\cal X}_{0,n})\times{\cal
B}({\cal Y}_{0,n})\mapsto[0,1]$ of $\mu_{0,n}\in{\cal M}_{1}({\cal X}_{0,n})$
and $\nu_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n})$:
$\displaystyle\pi_{0,n}(G_{0,n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}(\mu_{0,n}\times\nu_{0,n})(G_{0,n}),~{}G_{0,n}\in{\cal
B}({\cal X}_{0,n})\times{\cal B}({\cal Y}_{0,n})$ $\displaystyle=\int_{{\cal
X}_{0,n}}\nu_{0,n}(G_{0,n,x^{n}})\mu_{0,n}(dx^{n})$
The precise definition of mutual information between two sequences of Random
Variables $X^{n}$ and $Y^{n}$, denoted $I(X^{n};Y^{n})$ is defined via the
Kullback-Leibler distance (or relative entropy) between the joint probability
distribution of $(X^{n},Y^{n})$ and the product of its marginal probability
distributions of $X^{n}$ and $Y^{n}$, using the Radon-Nikodym derivative.
Hence, by the construction of probability measures (P1)-(P3), and the chain
rule of relative entropy [11]:
$\displaystyle
I(X^{n};Y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{D}(P_{0,n}||\pi_{0,n})$
(9) $\displaystyle=\int_{{\cal X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{d(\mu_{0,n}\otimes
q_{0,n})}{d(\mu_{0,n}\times\nu_{0,n})}\Big{)}d(\mu_{0,n}\otimes q_{0,n})$
$\displaystyle=\int_{{\cal X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{q_{0,n}(dy^{n};x^{n})}{\nu_{0,n}(dy^{n})}\Big{)}$
$\displaystyle q_{0,n}(dy^{n};dx^{n})\mu_{0,n}(dx^{n})$
$\displaystyle=\int_{{\cal
X}_{0,n}}\mathbb{D}(q_{0,n}(\cdot;x^{n})||\nu_{0,n}(\cdot))\mu_{0,n}(dx^{n})$
$\displaystyle\equiv\mathbb{I}(\mu_{0,n};q_{0,n})$ (10)
Note that $(\ref{re3})$ states that mutual information is expressed as a
functional of $\\{\mu_{0,n},q_{0,n}\\}$ and it is denoted by
$\mathbb{I}(\mu_{0,n};q_{0,n})$. Note that necessary and sufficient conditions
for existence of a Radon-Nikodym derivative for finitely additive measures can
be found in [13]. Moreover, $I(X^{n};Y^{n})$ is also expressed by the sum of
two directed information as follows
$\displaystyle I(X^{n};Y^{n})$
$\displaystyle=I(X^{n}{\rightarrow}Y^{n})+I(X^{n}{\leftarrow}Y^{n})$ (11)
where
$\displaystyle I(X^{n}{\rightarrow}Y^{n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i=0}^{n}I(X^{i};Y_{i}|Y^{i-1})$
(12) $\displaystyle I(X^{n}{\leftarrow}Y^{n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i=0}^{n}I(Y^{i-1};X_{i}|X^{i-1})$
(13)
###### Definition II.3
(Classical Rate Distortion Function) Let $d_{0,n}:{\cal X}_{0,n}\times{\cal
Y}_{0,n}\rightarrow[0,\infty)$, be an ${\cal B}({\cal X}_{0,n})\times{\cal
B}({\cal Y}_{0,n})$-measurable distortion function, and let
$Q_{0,n}(D)\subset{\cal Q}({\cal Y}_{0,n};{\cal X}_{0,n})$ (assuming is non-
empty) denotes the average distortion or fidelity constraint defined by
$\displaystyle
Q_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}q_{0,n}\in{\cal
Q}({\cal Y}_{0,n};{\cal X}_{0,n}):$ $\displaystyle\frac{1}{n+1}\int_{{\cal
X}_{0,n}}\int_{{\cal Y}_{0,n}}d_{0,n}(x^{n},y^{n})q_{0,n}(dy^{n};x^{n})$
$\displaystyle\mu_{0,n}(dx^{n})\leq D\Big{\\}},~{}D\geq 0$ (14)
The classical rate distortion function associated with the non-causal kernel
$q_{0,n}\in{\cal Q}({\cal Y}_{0,n};{\cal X}_{0,n})$ is defined by
$\displaystyle
R_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\inf_{q_{0,n}\in
Q_{0,n}(D)}\frac{1}{n+1}\mathbb{I}(\mu_{0,n};q_{0,n})$ (15)
while its operational meaning can be established via
${\lim}{\sup_{n\rightarrow\infty}}{R_{0,n}}$.
Existence in (15) is shown assuming $d_{0,n}(x^{n};\cdot)$ is bounded
continuous on ${\cal Y}_{0,n}$ and ${\cal Y}_{0,n}$ is compact, using weak-
convergence of probability measures in [3], and for more general
$d_{0,n}(x^{n};\cdot)$ which is only continuous in ${\cal Y}_{0,n}$ using
weak*-convergence of measures [14] on Polish spaces.
A version of the optimal reconstruction kernel which attains the infimum in
(15), [3] is
$\displaystyle q_{0,n}^{*}(dy^{n};x^{n})=\frac{\
e^{sd_{0,n}(x^{n},y^{n})}\nu_{0,n}^{*}(dy^{n})}{\int_{{\cal
Y}_{0,n}}e^{sd_{0,n}(x^{n},y^{n})}\nu_{0,n}^{*}(dy^{n})},\quad s\leq 0$ (16)
where $\nu_{0,n}^{*}\in{\cal M}_{1}({\cal Y}_{0,n})$ is the marginal of
$P_{0,n}^{*}=\mu_{0,n}\otimes q_{0,n}^{*}\in{\cal M}_{1}({\cal
X}_{0,n}\times{\cal Y}_{0,n})$ and $s\leq 0$ is the Lagrange multiplier
associated with the fidelity constraint $(\ref{dc1})$. Unfortunately, for
general sources and distortion function $d_{0,n}$, the optimal reconstruction
$q^{*}_{0,n}(dy^{n};x^{n})=\otimes^{n}_{i=0}q^{*}_{i}(dy_{i};y^{i-1},x^{n})$
is non-causal and introduces delay in the reconstruction processes. On the
other hand, if the solution (16) gives a reconstruction such that
$q^{*}_{0,n}(dy^{n};x^{n})={\overrightarrow{q}}^{*}_{0,n}(dy^{n};x^{n})=\otimes^{n}_{i=0}q^{*}_{i}(dy_{i};y^{i-1},x^{i})$
it will be causal. However, there are only limited examples in which
$(\ref{f6a})$ is causal on the source sequence. For single letter distortion
function
$d_{0,n}(x^{n},y^{n})=\frac{1}{n+1}\sum^{n}_{i=0}\rho_{i}(x_{i},y_{i})$ and
independent sources $\mu_{0,n}(dx^{n})=\otimes^{n}_{i=0}\mu_{i}(dx^{i})$
(e.g., $\\{X_{i}:i\in\mathbb{N}\\}$ are independent) the optimal
reconstruction $q^{*}_{0,n}(dy^{n};x^{n})$ factors into a product of causal
kernels $q^{*}_{0,n}(dy^{n};x^{n})=\otimes^{n}_{i=0}q_{i}(dy_{i},x_{i})$ [2].
This raises the question whether the classical rate distortion function can be
reformulated using the causal product
${\overrightarrow{q}}_{0,n}(dy^{n};x^{n})$.
The next lemma relates causal product reconstruction kernels, mutual
information, directed information, and conditional independence.
###### Lemma II.4
The following are equivalent for each $n\in\mathbb{N}$.
1. 1.
$q_{0,n}(dy^{n};x^{n})={\overrightarrow{q}}_{0,n}(dy^{n};x^{n})$, as defined
in Definition II.2-2)
2. 2.
For each $i=0,1,\ldots,n-1$,
$Y_{i}\leftrightarrow(X^{i},Y^{i-1})\leftrightarrow(X_{i+1},X_{i+2},\ldots,X_{n})$,
forms a Markov chain
3. 3.
$I(X^{n};Y^{n})=I(X^{n}\rightarrow Y^{n})$
4. 4.
$I(X^{n}\leftarrow Y^{n})=0$
5. 5.
For each $i=0,1,\ldots,n-1$, $Y^{i}\leftrightarrow X^{i}\leftrightarrow
X_{i+1}$ forms a Markov chain
Proof. Omitted due to space limitation.
According to Lemma II.4 any source with a satisfying conditional distribution
$P_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|X^{i-1}=x^{i-1},Y^{i-1}=y^{i-1})=P_{X_{i}|X^{i-1}}(dx_{i}|X^{i-1}=x^{i-1}),~{}P-a.s.,$
$\forall{i}\in\mathbb{N}$ is equivalent to any of the equivalent statements of
Lemma II.4. Therefore, for such a source the mutual information becomes
$\displaystyle I(X^{n};Y^{n})=I(X^{n}{\rightarrow}Y^{n})$
$\displaystyle=\int_{{\cal X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{\overrightarrow{q}_{0,n}(dy^{n};x^{n})}{\nu_{0,n}(dy^{n})}\Big{)}$
$\displaystyle\overrightarrow{q}_{0,n}(dy^{n};dx^{n})\mu_{0,n}(dx^{n})$ (17)
$\displaystyle\equiv{\mathbb{I}}(\mu_{0,n};\overrightarrow{q}_{0,n})$ (18)
where (18) states that $I(X^{n};Y^{n})$ is a functional of
$\\{\mu_{0,n},{\overrightarrow{q}}_{0,n}\\}$. Hence, causal rate distortion is
defined by optimizing ${\mathbb{I}}(\mu_{0,n};\overrightarrow{q}_{0,n})$ over
${\overrightarrow{q}}_{0,n}$ which satisfies a distortion constraint.
###### Definition II.5
(Causal Rate Distortion Function) Suppose
$d_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\sum^{n}_{i=0}\rho_{0,i}(x^{i},y^{i})$,
where $\rho_{0,i}:{\cal X}_{0,i}\times{\cal Y}_{0,i}\rightarrow[0,\infty)$, is
a sequence of ${\cal B}({\cal X}_{0,i})\times{\cal B}({\cal
Y}_{0,i})$-measurable distortion functions, and let
$\overrightarrow{Q}_{0,n}(D)$ (assuming is non-empty) denotes the average
distortion or fidelity constraint defined by
$\displaystyle\overrightarrow{Q}_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}\overrightarrow{q}_{0,i}\in{\cal
M}_{1}({\cal Y}_{0,i}),0\leq{i}\leq{n}:$
$\displaystyle\frac{1}{n+1}\sum_{i=0}^{n}\int_{{\cal X}_{0,i}}\int_{{{\cal
Y}}_{0,i}}\rho_{0,i}({x^{i}},{y^{i}})\overrightarrow{q}_{0,i}(d{y}^{i};{x}^{i})$
$\displaystyle\mu_{0,i}(d{x}^{i})\leq D\Big{\\}},~{}D\geq 0$ (19)
The causal rate distortion function associated with the causal product kernel
${\overrightarrow{q}}_{0,n}\in{\overrightarrow{Q}}_{0,n}(D)$ is defined by
$\displaystyle{\overrightarrow{R}}_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\inf_{{\overrightarrow{q}_{0,n}\in\overrightarrow{Q}_{0,n}(D)}}\frac{1}{n+1}{\mathbb{I}}(\mu_{0,n};\overrightarrow{q}_{0,n})$
(20)
while its operational meaning can be established via
$\lim\sup_{n\rightarrow{\infty}}{\overrightarrow{R}}_{0,n}$.
Clearly, ${\overrightarrow{R}}_{0,n}(D)$ is characterized by minimizing
directed information or equivalently
$\mathbb{I}(\mu_{0,n};\overrightarrow{q}_{0,n})$ over the causal product
measure ${\overrightarrow{q}}_{0,n}\in{\overrightarrow{Q}}_{0,n}(D)$.
###### Lemma II.6
$\overrightarrow{q}_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n})$ is uniquely
determined by $\\{q_{i}\in{\cal Q}_{i}({\cal Y}_{i};{\cal
Y}_{0,i-1}\times{\cal X}_{0,i})\\}_{i=0}^{n}$ and vice-versa, $P-a.s$.
Proof. For densities this result is derived in [15].
## III EXISTENCE OF OPTIMAL CAUSAL PRODUCT RECONSTRUCTION KERNEL
In this section, appropriate topologies and function spaces are employed to
show existence of the minimizing causal product kernel in $(\ref{ex12})$. In
the process we also show existence for $R_{0,n}(D)$.
### III-A Abstract Spaces
Let $BC({\cal Y}_{0,n})$ denote the vector space of bounded continuous real
valued functions defined on the Polish space ${\cal Y}_{0,n}$. Furnished with
the sup norm topology, this is a Banach space. The topological dual of
$BC({\cal Y}_{0,n})$ denoted by $\Big{(}BC({\cal Y}_{0,n})\Big{)}^{*}$ is
isometrically isomorphic to the Banach space of finitely additive regular
bounded signed measures on ${\cal Y}_{0,n}$ [7], denoted by $M_{rba}({\cal
Y}_{0,n})$. Let $\Pi_{rba}({\cal Y}_{0,n})\subset M_{rba}({\cal Y}_{0,n})$
denote the set of regular bounded finitely additive probability measures on
${\cal Y}_{0,n}$. Clearly if ${\cal Y}_{0,n}$ is compact, then
$\Big{(}BC({\cal Y}_{0,n})\Big{)}^{*}$ will be isometrically isomorphic to the
space of countably additive signed measures, as in [3]. Denote by
$L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$ the space of all $\mu_{0,n}$-integrable
functions defined on ${\cal X}_{0,n}$ with values in $BC({\cal Y}_{0,n}),$ so
that for each $\phi\in L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$ its norm is
defined by
$\displaystyle\parallel\phi\parallel_{\mu_{0,n}}\stackrel{{\scriptstyle\triangle}}{{=}}\int_{{\cal
X}_{0,n}}||\phi(x^{n})(\cdot)||_{BC({\cal Y}_{0,n})}\mu_{0,n}(dx^{n})<\infty$
The norm topology $\parallel{\phi}\parallel_{\mu_{0,n}}$, makes
$L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$ a Banach space, and it follows from the
theory of “lifting” [10] that the dual of this space is
$L_{\infty}^{w}(\mu_{0,n},M_{rba}({\cal Y}_{0,n}))$, denoting the space of all
$M_{rba}({\cal Y}_{0,n})$ valued functions $\\{q\\}$ which are
weak∗-measurable in the sense that for each $\phi\in BC({\cal Y}_{0,n}),$
$x^{n}\longrightarrow
q_{x^{n}}(\phi)\stackrel{{\scriptstyle\triangle}}{{=}}\int_{{\cal
Y}_{0,n}}\phi(y^{n})q(dy^{n};x^{n})$ is $\mu_{0,n}$-measurable and
$\mu_{0,n}$-essentially bounded.
### III-B Weak∗-Compactness and Existence
Define an admissible set of stochastic kernels associated with classical rate
distortion function by
$\displaystyle
Q_{ad}\stackrel{{\scriptstyle\triangle}}{{=}}L_{\infty}^{w}(\mu_{0,n},\Pi_{rba}({\cal
Y}_{0,n}))\subset L_{\infty}^{w}(\mu_{0,n},M_{rba}({\cal Y}_{0,n}))$
Clearly, $Q_{ad}$ is a unit sphere in $L_{\infty}^{w}(\mu_{0,n},M_{rba}({\cal
Y}_{0,n}))$. For each $\phi{\in}L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$ we can
define a linear functional on $L_{\infty}^{w}(\mu_{0,n},M_{rba}({\cal
Y}_{0,n}))$ by
$\displaystyle\ell_{\phi}(q_{0,n})\stackrel{{\scriptstyle\triangle}}{{=}}\frac{1}{n+1}\int_{{\cal
X}_{0,n}}\Big{(}\int_{{\cal Y}_{0,n}}\phi(x^{n},y^{n})$ $\displaystyle
q_{0,n}(dy^{n};x^{n})\Big{)}\mu_{0,n}(dx^{n})$
This is a bounded, linear and weak∗-continuous functional on
$L_{\infty}^{w}(\mu_{0,n},M_{rba}({\cal Y}_{0,n}))$. For $d_{0,n}:{\cal
X}_{0,n}\times{\cal Y}_{0,n}\rightarrow[0,\infty)$ measurable and
$d_{0,n}{\in}L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$ the distortion constraint
set of the classical rate distortion function is
$\displaystyle
Q_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\\{q{\in}Q_{ad}:\frac{1}{n+1}\ell_{d_{0,n}}(q_{0,n}){\leq}D\\}$
It can be shown that $Q_{0,n}(D)$ is bounded and weak∗-closed subset of
$Q_{ad}$ and hence weak∗-compact (Compactness of $Q_{ad}$ follows from
Alaoglu’s Theorem [7],[12]).
Next, we define the set of causal product kernels as follows.
$\displaystyle{\overrightarrow{\Pi}}_{rba}({\cal Y}_{0,n})$
$\displaystyle=\Big{\\{}{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\otimes_{i=1}^{n}{q}_{i}(dy_{i};y^{i-1},x^{i}):$
$\displaystyle{q}_{i}(dy_{i};y^{i-1},x^{i})\in{\Pi}_{rba}({\cal
Y}_{i}),\>i\in\mathbb{N}^{n}\Big{\\}}$
where $L_{\infty}^{w}(\mu_{0,n},{\overrightarrow{\Pi}}_{rba}({\cal Y}_{0,n}))$
denotes the space of all ${\overrightarrow{\Pi}}_{rba}({\cal Y}_{0,n})$ valued
functions $\\{\overrightarrow{q}\\}$ which are weak∗-measurable in the sense
that for each $\phi\in BC({\cal Y}_{0,n}),$
$x^{n}\rightarrow{\overrightarrow{q}}_{x^{n}}(\phi)\stackrel{{\scriptstyle\triangle}}{{=}}\int_{{\cal
Y}_{0,n}}\phi(y^{n}){\overrightarrow{q}}(dy^{n};x^{n})$ is
$\mu_{0,n}$-measurable and $\mu_{0,n}$-essentially bounded.
Define the admissible set of causal product stochastic kernels associated with
the causal rate distortion function by
$\displaystyle{\overrightarrow{Q}}_{ad}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}L_{\infty}^{w}(\mu_{0,n},{\overrightarrow{\Pi}}_{rba}({\cal
Y}_{0,n}))$
Clearly, ${\overrightarrow{Q}}_{ad}=\\{q_{0,n}\in
Q_{ad}:q_{0,n}(dy^{n};x^{n})=\overrightarrow{q}_{0,n}(dy^{n};x^{n})\\}$. For
$d_{0,n}:{\cal X}_{0,n}\times{\cal Y}_{0,n}\rightarrow[0,\infty)$ which is
measurable and $d_{0,n}{\in}L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$ the
distortion constraint of causal rate distortion function is
$\displaystyle{\overrightarrow{Q}_{{0,n}}(D)}\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}{\overrightarrow{q}}_{0,n}\in{\overrightarrow{Q}}_{ad}:$
$\displaystyle\frac{1}{n+1}\ell_{d_{0,n}}({\overrightarrow{q}}_{0,n})\stackrel{{\scriptstyle\triangle}}{{=}}\int_{{\cal
X}_{0,n}}\biggr{(}\int_{{\cal Y}_{0,n}}d_{0,n}(x^{n},y^{n})$
$\displaystyle{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})\biggr{)}\mu_{0,n}(dx^{n})\leq
D\Big{\\}}$
###### Assumptions III.1
We make the following assumptions.
1. 1.
The set $\overrightarrow{Q}_{ad}$ is weak∗-closed.
2. 2.
The set ${\overrightarrow{Q}_{0,n}(D)}$ is non-empty.
###### Lemma III.2
Suppose Assumptions III.1 hold. Let ${\cal X}_{0,n},{\cal Y}_{0,n}$ be two
Polish spaces and $d_{0,n}:{\cal X}_{0,n}\times{\cal
Y}_{0,n}\rightarrow[0,\infty],$ a measurable, non-negative, extended real
valued function, such that $d_{0,n}{\in}L_{1}(\mu_{0,n},BC({\cal Y}_{0,n}))$.
For any $D\in[0,\infty)$, the set ${\overrightarrow{Q}_{0,n}(D)}$ is
weak∗-compact.
Proof. By Assumptions III.1, $\overrightarrow{Q}_{ad}$ is a weak∗-closed,
hence as a subset of a weak∗-compact set $Q_{ad}$ it is weak∗-compact. Also,
under assumptions III.1, ${\overrightarrow{Q}_{0,n}(D)}$ is bounded and
weak∗-closed and hence it is weak∗-compact (as a weak∗-closed subset of the
weak∗-compact set ${\overrightarrow{Q}}_{ad}$) $\bullet$
###### Theorem III.3
Under Assumptions III.1, ${\overrightarrow{R}}_{0,n}(D)$ has a minimum.
Proof. Follows from Lemma III.2 and the lower semi-continuity of
${\mathbb{I}}(\mu_{0,n};\cdot)$ on ${\overrightarrow{Q}}_{ad}$ $\bullet$
## IV NECESSARY CONDITIONS OF OPTIMALITY OF CAUSAL PRODUCT RATE DISTORTION
FUNCTION
In this section the form of the optimal causal product reconstruction kernels
is derived. The method is based on calculus of variations on the space of
measures [9].
###### Theorem IV.1
Suppose
${\mathbb{I}}_{\mu_{0,n}}({\overrightarrow{q}}_{0,n})\stackrel{{\scriptstyle\triangle}}{{=}}{\mathbb{I}}(\mu_{0,n};\overrightarrow{q}_{0,n})$
is well defined for every ${\overrightarrow{q}}_{0,n}\in
L_{\infty}^{w}({\mu_{0,n},\overrightarrow{\Pi}}_{rba}({\cal Y}_{0,n}))$
possibly taking values from the set $[0,\infty].$ Then
${\overrightarrow{q}}_{0,n}\rightarrow{\mathbb{I}}_{\mu_{0,n}}({\overrightarrow{q}}_{0,n})$
is Gateaux differentiable at every point in
$L_{\infty}^{w}({\mu_{0,n},\overrightarrow{\Pi}}_{rba}({\cal Y}_{0,n})),$ and
the Gateaux derivative at the point ${\overrightarrow{q}}_{0,n}^{0}$ in the
direction ${\overrightarrow{q}}_{0,n}-{\overrightarrow{q}}_{0,n}^{0}$ is given
by
$\displaystyle\delta{\mathbb{I}}_{\mu_{0,n}}({\overrightarrow{q}}_{0,n}^{0};{\overrightarrow{q}}_{0,n}-{\overrightarrow{q}}_{0,n}^{0})$
$\displaystyle=\int_{{\cal X}_{0,n}}\int_{{\cal
Y}_{0,n}}\log\Bigg{(}\frac{{\overrightarrow{q}}_{0,n}^{0}(dy^{n};x^{n})}{\nu_{0,n}^{0}(dy^{n})}\Bigg{)}$
$\displaystyle({\overrightarrow{q}}_{0,n}-{\overrightarrow{q}}_{0,n}^{0})(dy^{n};x^{n})\mu_{0,n}(dx^{n})$
where $\nu_{0,n}^{0}\in{\cal M}_{1}({\cal Y}_{0,n})$ is the marginal measure
corresponding to
${\overrightarrow{q}}_{0,n}^{0}\otimes\mu_{0,n}(dx^{n})\in{\cal M}_{1}({\cal
Y}_{0,n}\times{\cal X}_{0,n})$.
Proof. The proof is based on the fact that the causal product stochastic
kernel ${\overrightarrow{q}}_{0,n}$ is used to show the existence of Gateaux
Differential [9] rather than for individual causal stochastic kernel
$q_{i}(dy_{i};y^{i-1},x^{i})$, $i\in\mathbb{N}^{n}$ $\bullet$
The constrained problem defined by (20) can be reformulated using Lagrange
multipliers as follows (equivalence of constrained and unconstrained problems
follows from [9]).
$\displaystyle{\overrightarrow{R}}_{0,n}(D)=\inf_{{\overrightarrow{q}}_{0,n}\in{\overrightarrow{Q}}_{ad}}\Big{\\{}\frac{1}{n+1}{{\mathbb{I}}}(\mu_{0,n};{\overrightarrow{q}}_{0,n})$
$\displaystyle-s(\ell_{{d}_{0,n}}({\overrightarrow{q}}_{0,n})-D)\Big{\\}}$
(21)
and $s\in(-\infty,0]$ is the Lagrange multiplier.
###### Theorem IV.2
Suppose $d_{0,n}(x^{n},y^{n})=\sum_{i=0}^{n}\rho_{0,i}(x^{i},y^{i})$ and the
assumptions of Lemma III.2 hold. The infimum in $(\ref{ex13})$ is attained at
$\overrightarrow{q}^{*}_{0,n}\in
L_{\infty}^{w}(\mu_{0,n},{\overrightarrow{\Pi}}_{rba}({\cal Y}_{0,n}))$ given
by
$\displaystyle\overrightarrow{q}^{*}_{0,n}(dy^{n};x^{n})=\otimes_{i=0}^{n}\frac{e^{s\rho_{i}(x^{i},y^{i})}\nu^{*}_{i}(dy^{i};y^{i-1})}{\int_{{\cal
Y}_{i}}e^{s\rho_{i}(x^{i},y^{i})}\nu^{*}_{i}(dy_{i};y^{i-1})}$ (22)
and $\nu^{*}_{i}(dy_{i};y^{i-1})\in{\cal Q}({\cal Y}_{i};{\cal Y}_{0,{i-1}})$.
The causal rate distortion function is given by
$\displaystyle{\overrightarrow{R}}_{0,n}(D)=sD-\frac{1}{n+1}\sum_{i=0}^{n}\int_{{{\cal
X}_{0,i}}\times{{\cal Y}_{0,i-1}}}$ $\displaystyle\log\Big{(}\int_{{\cal
Y}_{i}}e^{s\rho_{i}(x^{i},y^{i})}\nu^{*}_{i}(dy_{i};y^{i-1})\Big{)}$
$\displaystyle{{\overrightarrow{q}}^{*}_{0,i-1}}(dy^{i-1};x^{i-1})\otimes\mu_{0,i}(dx^{i})$
(23)
If ${\overrightarrow{R}}_{0,n}(D)>0$ then $s<0$ and
$\displaystyle\frac{1}{n+1}\sum_{i=0}^{n}\int_{{\cal X}_{0,i}}\int_{{\cal
Y}_{0,i}}\rho_{0,i}(x^{i},y^{i}){\overrightarrow{q}}^{*}_{0,i}(dy^{i};x^{i})\mu_{0,i}(dx^{i})=D$
Proof. The fully unconstraint problem of (21) is obtained by introducing
another Lagrange multiplier. Using this and Theorem IV.1 we obtain (22) and
(23) $\bullet$
## V PROPERTIES OF CAUSAL RATE DISTORTION FUNCTION
In this section, we present some important properties of the causal rate
distortion function as it is defined in (20).
###### Theorem V.1
1. 1.
${\overrightarrow{R}}_{0,n}(D)$ is a convex, non-increasing function of $D$
2. 2.
If $\rho_{0,i}\in L^{1}(\pi_{0,i})$ then
a)
${\overrightarrow{R}}_{0,n}(\frac{1}{n+1}\sum_{i=0}^{n}E_{\pi_{0,i}}(\rho_{0,i}))=0$;
b) ${\overrightarrow{R}}_{0,n}(D)$ is non-increasing for $D\in[0,D_{max}]$
where $D_{max}=\frac{1}{n+1}\sum_{i=0}^{n}E_{\pi_{0,i}}(\rho_{0,i})$ and
${\overrightarrow{R}}_{0,n}(D)=0$ for any $D\geq D_{max}$
3. 3.
${\overrightarrow{R}}_{0,n}(D)>0$ for all $D<D_{max}$ and
${\overrightarrow{R}}_{0,n}(D)=0$ for all $D\geq D_{max}$, where
$\displaystyle D_{max}=\min_{\\{y^{n}\\}\in{\cal
Y}_{0,n}}\frac{1}{n+1}\sum_{i=0}^{n}\int_{{\cal
X}_{0,i}}\rho_{0,i}(x^{i},y^{i})\mu_{0,i}(dx^{i})$
if such a minimum exists.
Proof. Omitted due to space limitation.
## VI CONCLUSION AND FUTURE WORK
### VI-A Conclusion
The solution of the causal rate distortion function subject to a reproduction
kernel which is a product of causal kernels is presented, on abstract
alphabets. Some of its properties are also presented. It is believed that the
optimal reconstruction kernel as a product of causal kernels has several
implications in applications where causality of the decoder as a function of
the source is of concern.
### VI-B Future Work
Examples are currently under investigation, and will be presented at the final
version of the paper.
## VII APPENDIX
## References
* [1] T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression. Prentice Hall, Englewood Cliffs, NJ, 1971.
* [2] T. Cover and J. Thomas, Elements of Information Theory. John Wiley & Sons, 1991.
* [3] I. Csiszár, “On an extremum problem of information theory”, Studia Scientiarum Mathematicarum Hungarica, vol. 9, pp. 57–71, 1974.
* [4] D. L. Neuhoff and R. Kent Gilbert, “Causal Source Codes”, IEEE Transactions on Information Theory, vol. IT-28, No.5, pp. 701–713, 1982.
* [5] S. Tatikonda, “Control Over Communication Constraints”, PhD Dissertation, M.I.T., Cambridge, MA, 2000.
* [6] J. Massey, “Causality, Feedback and Directed Information”, in the IEEE International Symposium on Information Theory and its Applications, pp. 303–305, Nov. 27–30, Hawaii, U.S.A.1990.
* [7] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory. Interscience Publishers, Inc., New York, 1958.
* [8] R. M. Gray, Entropy and Information Theory. Springer-Verlag, 1990.
* [9] D. G. Luenberger, Optimization by Vector Space Methods. John Wiley & Sons, 1969.
* [10] A. Ionescu Tulcea & C. Ionescu Tulcea, Topics in the Theory of Lifting, Springer Verlag, Berlin, Heidelberg, New York, 1969.
* [11] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the theory of Large Deviations. John Wiley & Sons, 1997.
* [12] W. Rudin, Functional analysis. McGraw-Hill, 1991.
* [13] H.B.Maynard, A Radon-Nikodym Theorem for Finitely Additive Bounded Measures, Pacific Journal of Mathematics, 83(2), 1979, pp. 401-413.
* [14] F. Rezaei, N. U. Ahmed and C. D. Charalambous, Rate Distortion Theory for General Sources With Potential Application to Image Compression, International Journal of Applied Mathematical Sciences, vol. 3 No. 2, 2006, pp. 141-165.
* [15] H. H. Permuter, T. Weissman, A. Goldsmith, “Finite State Channels with Time-Invariant Deterministic Feedback”, IEEE Transactions on Information Theory, vol.IT-55, No. 2, pp. 644-662, February 2009.
|
arxiv-papers
| 2011-02-16T10:50:44 |
2024-09-04T02:49:17.046622
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Charalambos D. Charalambous, Photios A. Stavrou, Christos K.\n Kourtellaris",
"submitter": "Photios Stavrou Antreas",
"url": "https://arxiv.org/abs/1102.3294"
}
|
1102.3485
|
# The Sun’s small-scale magnetic elements in Solar Cycle 23
C. L. Jin, J. X. Wang, and Q. Song Key Laboratory of Solar Activity of
Chinese Academy of Sciences
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China; cljin@nao.cas.cn; wangjx@nao.cas.cn H. Zhao National Tsing
Hua University, Hsinchu, Taiwan; berserker0715@hotmail.com
###### Abstract
With the unique database from Michelson Doppler Imager aboard the Solar and
Heliospheric Observatory in an interval embodying solar cycle 23, the cyclic
behavior of solar small-scale magnetic elements is studied. More than 13
million small-scale magnetic elements are selected, and the following results
are unclosed. (1) The quiet regions dominated the Sun’s magnetic flux for
about 8 years in the 12.25 year duration of Cycle 23. They contributed (0.94 –
1.44) $\times 10^{23}$ Mx flux to the Sun from the solar minimum to maximum.
The monthly average magnetic flux of the quiet regions is 1.12 times that of
active regions in the cycle. (2) The ratio of quiet region flux to that of the
total Sun equally characterizes the course of a solar cycle. The 6-month
running-average flux ratio of quiet region had been larger than 90.0% for 28
continuous months from July 2007 to October 2009, which characterizes very
well the grand solar minima of Cycles 23-24. (3) From the small to large end
of the flux spectrum, the variations of numbers and total flux of the network
elements show no-correlation, anti-correlation, and correlation with sunspots,
respectively. The anti-correlated elements, covering the flux of (2.9 -
32.0)$\times 10^{18}$ Mx, occupies 77.2% of total element number and 37.4% of
quiet Sun flux. These results provide insight into reason for anti-correlated
variations of small-scale magnetic activity during the solar cycle.
Sun: magnetic fields — Sun: photosphere — Sun: sunspots
## 1 Introduction
No any other astrophysical process but solar cycle leaves massive footprints
on human’s living environment. This eleven-year cycle was discovered by a
Germany pharmacist Schwabe (1843) from the number changes of solar sunspots. A
primary understanding on the solar cycle has been established based on the
theories and simulations of a mean-field magnetohydrodynamic (MHD) dynamo
(Charbonneau 2005). However, new observations are continuously challenging our
understanding by the myriad of new and seemingly conflicting observations. A
more severe challenge comes from observations of small-scale magnetic elements
(see de Wijn et al. 2009). Therefore, it is still a difficult task to explore
the physics of solar cycle.
Since the 1960s it has been observed that small-scale magnetic fields outside
of sunspots are everywhere on the Sun (Sheeley 1966, 1967; Harvey 1971). The
stronger magnetic elements at the boundaries of supergranulation cells are
network elements, while the smaller and weaker elements within the
supergranulation cells are intra-network (IN) elements (Livingston & Harvey
1975; Smithson 1975). Similar to emerging flux regions (EFRs) in sunspot
groups (or active regions) (Zirin 1972), small-scale emerging bipoles named
ephemeral (active) regions (ERs) were described by Harvey and Martin (1973).
They account for the formation of network elements in addition to the debris
from decaying sunspots. It was noticed that the flux emerging rate in ER
exceeds that in sunspots by two orders of magnitude (Zirin 1987). Moreover,
flux generation rate of IN elements exceeds that of ERs by another two orders
of magnitude. Further smaller magnetic fibrils are believed to be mostly
unresolved by present telescopes, yet their aggregation is the dominant
mechanism by which IN and network elements appear (Lamb et al. 2008). A
substantial amount of solar magnetic flux is probably still hidden (Trujillo
et al. 2004).
As soon as the small-scale magnetic elements were identified, great efforts
have been made to understand how they change during a solar cycle and if they
are correlated with sunspots. Diverse observations are reported, igniting
discussions and debates in the literature. The observations are made either
directly from the magnetic measurements, or indirectly from proxies of small-
scale magnetic flux, e.g., the G-band and CaII K bright points and coronal
X-ray bright points. Key revelations are listed below.
1. (1)
No cyclic variations: CaII K emission in solar quiet regions (White &
Livingston 1981); modern X-ray bright points observations (Sattarov et al.
2002; Hara & Nakakubo 2003); magnetic flux of networks (Labonte & Howard
1982); flux spectrum and total flux of network elements with flux $\leq
2.0\times 10^{19}$ Mx (Hagenaar et al. 2003); Stokes $\frac{Q}{I}$ profile
(Trujillo et al. 2004).
2. (2)
Anti-correlation of small-scale fields with sunspot cycle: number of network
bright points in very quiet regions (Muller & Roudier 1984, 1994); HeI 10830 Å
dark points in the higher chromosphere (Harvey 1985); early X-ray bright
points observations (Davis et al. 1977; Davis 1983; Golub et al. 1979); Weak
changes of emergence frequency of ERs with flux less than (3-5)$\times
10^{19}$ Mx (Hagenaar et al. 2003).
3. (3)
Correlation with sunspot cycle: more ERs appeared during active solar
condition (Harvey & Harvey 1974; Harvey 1989); the number (or magnetic flux)
of network structures (Foukal et al. 1991; Meunier 2003); flux distribution
and total flux of network concentrations with flux $(2.0-3.3)\times 10^{19}$
Mx (Hagenaar et al. 2003).
The observations listed above are related to some fundamental, but not yet
resolved questions in solar physics: the origin, dynamics and active role in
Sun’s global processes of solar small-scale magnetic elements, as well as the
controlling physics of solar activity cycle. However, discrepancy among
different authors is not yet understood, implying problems either in the
observations or on the physics used to interpret the observations. A few
aspects make things even more complicated.
First, for the observations of the proxies of small-scale magnetic elements,
the connections between the magnetic elements and their proxies are not well
quantified, and the underlined physics is not known exactly. There seems to be
not a one-to-one correspondence between network elements and network bright
points (Zhao et al. 2009). In other words, the widely-adopted paradigm of
“magnetic bright points” is still questionable. Moreover, the early revelation
about the magnetic properties of coronal X-ray bright points (Golub et al.
1977), needs to be revisited and updated with state-of-the-art observations.
Secondly, quite many reports listed above went back to early solar
observations, which makes us difficult to evaluate the quality of the
observations. We are confused by rather poor resolution, calibration and
consistency in sensitivity in early magnetic measurements. As an example, the
early Mont. Wilson magnetograph observations (Labonte & Howard 1982) were with
a resolution of $\geq$ 12.5-17.5 arcsec, and the calibration was not
consistent time to time. We are simply not able to say anything confidently
about their conclusions. Additionally, early X-ray bright point measurements,
which suffered from low cadence, purported to show a decrease in the number of
X-ray bright points with the solar cycle. More recent higher cadence
observations have called into question whether this effect is real. It reminds
to be seen whether other observations of variation with the solar cycle also
need to be reinterpreted. New observations with careful and thorough data
reduction and interpretation are crucially required.
Thirdly, even for recent observations, sometimes, the different algorithm and
logic in data analysis make us hard to judge the results too. An interesting
example comes from the analysis of full-disk magnetograms of the Michelson
Doppler Imager aboard the Solar and Heliospheric Observatory (MDI/SOHO)
(Scherrer et al. 1995). By adopting the different detection algorithm and
approaches, Meunier (2003) revealed correlation of the network element number
(or flux) with sunspots; in contrast, Hagenaar et al. (2003) declared some
weak anti-correlated emergence rate of ERs and an independence of the total
absolute flux for smaller network concentrations. This discrepancy should be
clarified with new analysis.
To clarify the problem and to close the debates are an essential task in
understanding the solar cycle phenomena. Fortunately, now MDI/SOHO is
providing a unique database - the full-disk magnetograms over more than 13
years, covering the complete 23rd Solar Cycle. The 13.5 year 5-min average
full disk magnetograms are used in the current study. However, the poor
temporal resolution makes the identity of ERs questionable and the sensitivity
of the full-disk magnetograms rules out the possibility to resolve the IN
elements. Therefore, what we have identified in this study is basically the
network magnetic elements.
In this paper, we aim at learning the cyclic variations of quiet Sun’s
magnetic flux and small-scale magnetic elements. To use the full-disc MDI
magnetograms with the temporal coverage of entire Cycle 23 comes from an
awareness of the intermittency of solar cyclic behavior in both the temporal
and spatial domains. By the intermittency to select the magnetograms of a
short interval, e.g., 10-30 hours, in a month for each year, at the ‘supposed’
different cycle phases. would not guarantee a grape of the key characteristics
of a solar cycle. From our understanding, to choose the database that cover
the entire cycle 23 is of overwhelming importance. The database for the
current study is unique in the sense that it is the only space-borne magnetic
measurements of the full Sun, for which the consistency in sensitivity and
resolution persisted for a cycle-long interval. As we are interested in the
global behavior of small-scale magnetic elements, sampling network elements in
a cycle-long temporal domain and in all different flux ranges (or strengths)
are more important than selecting a few high cadence sequences interruptedly.
Moreover, the magnetic elements with different flux (or size) may have
different origins and characteristics, therefore we group all the network
magnetic elements into different categories in accordance with their magnetic
flux.
In section 2, we describe the observations, the technique of calibration, the
evaluation of noise level of the magnetograms, the separation of active
regions and the quiet Sun, and the selection of network elements. In section
3, we present the results of cyclic behavior of quiet region magnetic flux and
small-scale magnetic elements. In section 4, we make the comparison with
previous studies, and consider the possibilities on how to understand the
anti-correlated network magnetic elements with sunspots. In section 5, we draw
the conclusions.
## 2 Observations and methods
The MDI instrument aboard SOHO spacecraft provides the full-disk magnetogram
with a pixel size of 2”. In order to obtain a low noise level, only those
5-min average magnetograms are selected in the study. We extract one observed
full-disk magnetogram per day, and thusly we totally select 3764 magnetograms
from 1996 September to 2010 February, which include the complete 23rd solar
cycle. In order to further reduce the noise level, we apply a boxcar smoothing
function to each magnetogram by a width of 6”$\times$6”. There are two groups
of authors who first pointed out the under-estimation of magnetic flux by
earlier MDI full-disk magnetogram calibration (Berger et al. 2003; Wang et al.
2003). All the magnetograms used in this study are that retrieved after
recalibration of December 2008. For a better understanding about the cyclic
behavior of solar minima of Cycles 22 and 23, we extend the MDI data base by
adding Kitt Peak full-disk magnetograms from August 1996 back to January 1994.
The data merging is made based on a least-square fitting of the mean flux
density of Kitt Peak magnetograms to that of MDI magnetograms for the common
interval of 1996.
We estimate the noise level of these smoothed 5-min average magnetograms
according to the method described by Hagenaar (2001) and Hagenaar et al.
(2003). Based on these magnetograms, we analyze their histograms of magnetic
flux density. The core of the distribution function is fitted by a Gaussian
function $F(x)=F_{max}exp(-x^{2}/2\sigma^{2})$, where the width $\sigma$ of
the Gaussian function, about 6 Mx/cm2, is defined as the noise level.
We assume that the observed line of sight magnetic flux density is a
projection of the intrinsic flux density normal to the solar surface, so the
magnetic flux density for each pixel is corrected(see Hagenaar 2001 and
Hagenaar et al. 2003) as $B_{cal}=B_{obs}(\alpha)/\cos(\alpha)$. The angle
$\alpha$ of each pixel is defined by $\sin(\alpha)=\sqrt{x^{2}+y^{2}}/R$ Where
x and Y are the pixel position referring to the disk center, at which x and y
is equal to 0, and the R is the solar disk radius. After the correlation, the
magnetogram shows the magnetic flux density normal to solar surface.
After the angle is greater 60 degrees, there are less and less magnetic
signals due to the lower magnetic sensitivity and spatial resolution of
MDI/SOHO magnetograms, and the magnetic noise level would increase according
to the magnetic correction 1/cos($\alpha$). Therefore, we only analyze these
pixels with angle $\alpha$ less than 60 degree, i.e., the region included by
the black circle in the left panels of Fig. 1. The flux density of the pixel
with 60o $\leq$ $\alpha$ $\leq$ 90o is set to zero.
For each smoothed and corrected full-disk magnetogram of MDI/SOHO, we apply a
magnetic flux density of 15 Mx/cm2 as a threshold to define the active regions
and their surroundings, and then create a mask for each magnetogram. These
masks include many small clusters and isolated pixels, so only the islands
with area larger 9$\times$9 pixels are defined as the active regions (Hagenaar
et al. 2003). Considering the active regions close to the edge of 60 degree,
in order to avoid missing them in the automatic procedure, we always search
the active regions in the solar disk with angle $\alpha$ less than 70 degree
first, as that shown in the left panels of Fig. 1. Thusly, the islands with
area less than 81 pixels within 60 degree disk are still defined as the active
regions if they have more than 81 pixels searched within 70 degree disk.
Two magnetograms within the 70$\deg$ from disk center at approximately the
solar maximum and minimum phases, respectively, are displayed in the left
panels of Fig. 1. On these retrieved magnetograms the selected ARs are masked
by red curves. The criterion of selecting ARs appears to work well from a
visually examination for the given cases. In the right panels two selected
sub-windows of the magnetograms are shown with contours outlining the network
elements which are selected by a procedure of automatic feature selection. The
yellow and green contours outline the selected network elements that are
belong to the components of correlated and anti-correlated with sunspots in
the solar cycle, respectively (see Section 3.2).
## 3 Results
### 3.1 Cyclic variations of magnetic flux of solar quiet regions
In order to compare the cyclic variations of magnetic flux of active regions
with that of quiet regions, we calculate their magnetic flux, respectively,
which is shown in the left panel of Fig. 2. At the same time, the area ratio
of quiet regions is also computed, which is shown by purple ‘+’ symbols in the
right panel of Fig. 2. It is found that the quiet Sun contributed
$(0.94-1.44)\times 10^{23}$ Mx flux from approximately the solar minimum to
maximum in Cycle 23. The fractional area of quiet regions always exceeds 80%
in the entire solar cycle 23, and decreased from the cycle minimum to maximum
by a factor of 1.2, although their total flux increased by a factor of 1.53;
as a comparison, the active region flux increased by several orders of
magnitude. The measurements confirm the global behavior of the quiet Sun
fields (see Meunier 2003 and Hagenaar et al. 2003). During the 12.25 years of
Cycle 23, from October 1996 to December 2008 (see http://www.ips.gov.au), the
quiet Sun dominated the Sun’s magnetic flux for 7.92 years. The monthly
average magnetic flux of quiet Sun is 1.12 times that of active regions. The
magnetic fields on the quiet Sun, indeed, are a fundamental component of the
Sun’s activity cycle which maintains the Sun’s magnetic energy and Poynting
flux at a certain level.
It is interesting to notice that the ratio of the quiet Sun’s magnetic flux to
solar total flux (referring to as the flux occupation by the quiet Sun)
equally characterizes the course of a solar cycle, like sunspots. The
occupation of 6-month running-average magnetic flux by the quiet Sun is shown
by purple cross symbols in the right panel of Fig. 2. The active region flux
shown in the left panel answers for the variation of sunspot cycle very well.
However, for the quiet regions, the maximum occupation of magnetic flux marks
the minima of solar cycles. For instance, in our data set, the maximum flux
occupation of quiet Sun, which was 96.0%, first happened in October of 1996 at
the beginning of Cycle 23\. The later maxima happened from July 2008 to August
2009. In December of 2008, the beginning of Cycle 24, the maximum occupation
reached 99.3%. The 6-month running-average fractional flux of quiet Sun had
been larger than 90.0% for 28 continuous months (from July 2007 to October
2009), which characterizes the grand solar minima of Cycles 23-24. Staying at
such a low activity level there were 25 months, for which the total AR flux
was less than $10^{22}$ Mx. However, during the minima of Cycles 22-23 for
only intermittent 7 months, i.e., from December 1995 to April 1996 and from
December 1996 to January 1997, we had witnessed the fraction larger than
90.0%. The distinction between two solar minima are so severe, which can be
seen very clearly in Fig. 2.
### 3.2 Cyclic variations of network magnetic elements
After excluding the active regions, we apply the magnetic noise, i.e., 6
Mx/cm2 as a threshold to create a mask for each quiet magnetogram, and define
these magnetic concentrations with more than 10 pixels in size as network
magnetic elements (Hagenaar et al. 2003). More than 13 million network
elements have been identified for the interval from September 1996 to February
2010. The probability distribution function (PDF) of these magnetic elements
in the studied interval is shown in Fig. 3 as the average flux distribution.
From the figure, it can be found that the distribution of magnetic flux of
network magnetic elements mainly concentrate at the flux of 1019 Mx. This peak
distribution is consistent with that found for multiple MDI full-disk datasets
by parnel et al. (2009) (see their Fig. 5)
For an exclusive examination, we divide all the magnetic elements into 96 sub-
groups according to the flux per element. In this way, a statistical sample is
created, covering the range of magnetic flux per element from the smallest
observable network flux of $1.5\times 10^{18}$ Mx for the current data set to
an upper limit of $3.8\times 10^{20}$ Mx. The monthly variation of magnetic
elements for each sub-group is calculated and examined in term of number
density and absolute total flux in the interval from October 1996 to February
2010, embodying the entire Cycle 23. The influence of the area changes of the
quiet Sun on both quantities has been removed. There are 0.3% of network
elements (or clusters) with flux larger than the upper limit, which were
fragments of decayed sunspots and not included in the sample. By following
Hagenaar et al. (2003) tiny flux pieces with less than 10 pixels are not
considered in the study. As a whole the total flux of these tiny flux pieces
showed a small variation in the scope of $(3.5-4.0)\times 10^{22}$ Mx during
the cycle.
The correlation coefficients between the cyclic variation of numbers of
network elements and sunspots are calculated for each sub-group of network
elements and shown in Fig. 4. They are the linear Pearson correlation
coefficients of two vectors for each sub-group elements. Denote the element
number in sub-group $i$ as $N_{i}$ and the sunspot number $N_{s}$, then the
correlation coefficient between $N_{i}$ and $N_{s}$ will be
$\rho(N_{i},N_{s})=Covariance(N_{i},N_{s})/(Variance{N_{i}}\times
Variance{N_{s}})^{\frac{1}{2}}$ (1)
The confidence level about the correlation can be found in some basic
statistics handbook by taking account of how big was of the sample. As the
sample size for each sub-group elements is 162, which is quite large. If the
coefficient is higher than 0.256, then the failure probability of the linear
correlation would already be $<0.001$.
From the small to the large flux spectrum, there appears a remarkable 3-fold
correlation scheme between the network elements and the sunspots: basically
no-correlation, anti-correlation and correlation. This behavior is held for
both the element number and total flux. Either the anti-correlation or the
correlation has been observed at very high confidence level. The majority of
the correlations show a failure probability $\leq 0.001$. Between the anti-
correlation and correlation, there is a narrow range of magnetic flux per
element of (3.2 - 4.3)$\times 10^{19}$ Mx. Network elements falling in this
flux range show a transition from anti-correlation to correlation with sunspot
cycle (see the narrow shaded column in the middle of Fig. 4.)
The dependence of the correlation coefficient on the element flux hints the
possibility that network elements at different segments of the flux spectrum
may present different physical origins and different cyclic behavior
accordingly. For an detailed examination of cyclic behavior of network
elements, we group all the network elements into 4 categories which show,
respectively, no-correlation, anti-correlation, transition from anti-
correlation to correlation, and correlation with sunspot cycle. For each
category, its flux range, percentage in number and in total flux, as well as
the correlation coefficient with sunspots are listed in Table 1. We, then,
discriminate the cyclic variation of magnetic elements in accordance to the
flux range listed in the table. The detailed cyclic variations of each
category network elements are shown in Fig. 5.
Approximate 77.2% of the magnetic elements, covering the flux range of
(2.9-32.0)$\times 10^{18}$ Mx show anti-correlation with the sunspot cycle.
This anti-correlated component contributes 37.4% of network flux during Cycle
23. Transition from anti-correlation to correlation takes place between (3.20
- 4.27) $\times 10^{19}$ Mx. The correlated component elements have magnetic
flux larger than 4.27$\times 10^{19}$ Mx. They occupy approximately 15.7% in
number but 53.5% of total flux of network elements. In the flux range of
(1.5-2.9)$\times 10^{18}$ Mx, network elements show randomly independent
variation with the sunspot cycle. From this data set, they occupy less than
0.6% of network elements and have neglectable total flux. With the poor
sensitivity in flux measurements at the smallest end of the flux spectrum, it
could not be excluded that the non-correlation component manifested some
random noises in flux measurement. More serious efforts with higher resolution
and sensitivity data are necessary to clarify the cyclic behavior of smallest
observable magnetic elements.
The number changes of the network elements in the flux range of (2.9 \-
32.0)$\times 10^{18}$ Mx show obviously anti-phase correlation with sunspot
cycle, so do the changes of their total unsigned flux. However, the cyclic
minimum of this anti-correlation component is not exactly coincided with the
reversed profile of the maximum of the sunspot cycle, implying complexity in
causing the anti-correlation. Meanwhile, the flux changes of the magnetic
elements with flux larger than 4.3$\times 10^{19}$ Mx show remarkable in-phase
correlation with sunspot cycle. The same is true that the profiles of the
maximum of network elements and that of sunspots are not corresponding one
another exactly. There is a 5-7 month delay of their cyclic maximum related to
that of the sunspot cycle. This seems to be related to the characteristics
dispersal time of active region fields.
To further explore the cyclic variation of magnetic elements, we obtain the
PDFs of yearly network magnetic elements according to the magnetic flux, and
compute the differential PDFs, i.e., the difference between the yearly PDFs
and average PDF (see Fig.3). Here, the differential probability distribution
function is abbreviated as DPDF. We plot the DPDFs, and show the variation for
magnetic flux spectrum from 1996 to 2010 in Fig. 6. From the figure, we
confirm the 3-fold scenario of cyclic variations of network elements.
From the solar minimum to solar maximum (see the first column of the figure),
the distribution of magnetic elements in the flux range about (3-30)$\times
10^{18}$ Mx gradually decrease, which shows the anti-correlation variation
with the sunspot cycle; while the distribution of magnetic elements of flux
larger than about 4$\times 10^{19}$ shows the correlation variation with the
sunspot cycle and reaches the peak in the years 2000, 2001 and 2002.
Furthermore, the distribution of magnetic elements with flux of $\sim$
3$\times 10^{19}$ and less than 3$\times 10^{18}$ Mx shows almost no
variation. The distribution of magnetic elements correlated with the sunspot
cycle reaches the smallest values in the years 2007, 2008 and 2009, which are
the solar minima of cycle 23-24; while the distribution of magnetic elements
anti-correlated with the sunspot cycle shows outstanding peak during this long
interval (see the third column of the figure). The distribution characterizes
the long duration of the solar minima of Cycles 23-24. It is noticed that the
distributions in some of the ascending and declining phases (see that in 1998
and 2005) are, more or less, represent the average distribution of small-scale
magnetic elements shown in Fig.3.
## 4 Discussion
With the unique space-borne observations which comprised a complete solar
cycle, we have revealed a 3-fold correlation scheme of the Sun’s small-scale
magnetic elements with sunspot cycle, and identified an anti-correlation
component of network elements that dominates the element population. Before
coming up with conclusion and discussion on the physics, a comparison with
previous studies that adopted the similar approaches and with that same space-
borne MDI observations (see Section 2) is necessary.
Hagenaar et al.(2003) selected high cadence magnetograms of 6 time-sequences,
each of which covered 10-30 hours in a month from 1996 to 2000. These authors
found that the component of network elements with flux $\geq 30\times 10^{18}$
Mx varied in phase with the sunspot cycle. The magnetogram calibration they
adopted had under-estimated the flux density by a factor of about 1.6 (Bergers
et al. 2003; Wang et al. 2003). With the renewed calibration, this component
would consist of magnetic elements with flux $\geq 4.8\times 10^{19}$. This is
a component in our analysis that changes in phase with sunspots. Meunier
(2003) chose strong magnetic elements with threshold flux density of 25 G and
40 G, respectively, and found naturally a correlation of number and flux of
network elements with sunspots.
When the element flux $\leq 20\times 10^{18}$ Mx (i.e., 32$\times 10^{18}$ Mx
in renewed calibration), Hagenaar et al.(2003) declared that both the flux
spectrum of quiet network elements and the total flux changed a little with
the cycle phase. These authors used the interrupted data in the 6 years of the
ascending cycle phase, they would not be able to guarantee a grasp of the real
trend of the cyclic modulation. We tested their results by using the same
6-month 5-m magnetograms, and found a weak change, but anti-phased with the
cycle phase, in both numbers and flux for network elements in this flux range.
In fact, Hagenaar et al.(2003) reported that the number density of network
concentrations on the quiet Sun decreased by less than 20% from 1997 to 2000,
consistent with our approaches. They also suggested an even anti-correlated
changes in flux emergence rate in this low flux range. The revelation of a
remarkable anti-correlation component of network elements with a broad flux
range from several times of $10^{18}$ Mx to 3 times of $10^{19}Mx$ is likely
to be the true nature of small-scale solar magnetism and inspiring new
considerations of the Sun’s magnetism.
Exploration of the magnetic nature of the Sun’s small-scale activity went back
to earlier solar studies. A few pioneer studies stand still as reliable
references in solar physics. Mehltretter (1974) identified that the network
bright points represented magnetic flux concentration with field strength of
1000-2000 G, each of them had a mean flux of 4.7$\times 10^{17}$ Mx. Later,
the work was extended by Muller & Roudier (1984, 1994). They deduced an
average flux of 2.5$\times 10^{17}$ Mx for an network bright points. The flux
range suggested by these authors for the network bright points changing anti-
phased with sunspot cycle, is out of reach by current data base. Muller &
Roudier (1984) also identified the correlation between the network bright
points in the photosphere and coronal XBPs. For the latter, Golub et al.(1977)
carefully studied their magnetic properties, and found the average total flux
associated with a typical XBP was 2.0$\times 10^{19}$ Mx. The magnetic
measurements were obtained at Kitt Peak with a fine scan of 2.5 arcsec
resolution element and $\sim$2G noise level. They are reasonably reliable to
quantify the magnetic flux of an XBP. This typical flux, even with some
uncertainty, e.g., 50% or larger, is still falling in the flux range of the
anti-correlated component of network elements discovered by this study. We
tentatively suggest that the anti-correlated component of magnetic elements
are responsible for the small-scale activity, e.g., the coronal X-ray bright
points. Updated efforts to quantify the magnetic properties of the so-called
magnetic bright points are crucial to final resolve the long-lasting puzzle of
the anti-phase behavior of the Sun’s small-scale activity in a solar cycle.
Observationally, small-scale network elements come from several sources:
fragmentation of active regions, flux emergence in the form of ephemeral
regions, coalescence of intranetwork flux, and products of dynamic interaction
among different sources of magnetic flux. The 3-fold relationship between
network elements and sunspot cycle has immediate implication on the Sun’s
magnetism. As demonstrated by state-of-the-art simulations (see Vögler &
Schüssler 2007), the magnetic elements at the smallest end of the flux
spectrum, either resolved or un-resolved, manifest a local turbulent dynamo
which operates in the near-photosphere and is independent to the sunspot
cycle. On the other hand, at the larger flux end, the magnetic elements are
likely to be the debris of decayed sunspots. They follow, of course, the solar
cycle.
The key issue here is how to understand the majority of magnetic elements
which are anti-correlated with sunspots in the solar cycle. They are not
likely the debris of decayed sunspots, but probably created by turbulent local
dynamo action that, however, is globally affected or controlled by the sunspot
field from the mean-field MHD dynamo. A few possibilities now are being
considered.
First, during the more active times of the Sun, the smaller magnetic elements
created by the turbulent dynamo have more opportunity to encounter sunspots
and their fragments. The same-polarity encountering results in a merging of
those elements to the flux related to sunspots. Whereas, the opposite polarity
encountering causes flux cancelations with the net results of lost smaller
elements and a diffusion of sunspot flux. What accompanied the sunspot flux
diffusion is the reduced smaller elements with the turbulent origin. This
accounts for the anti-correlated magnetic component possibly. By this kind of
interaction magnetic flux from turbulent dynamo actively takes part in the
operation of the solar cycle, helping with more efficient magnetic diffusion.
To quantify this mechanism, studies of dynamic interaction between small-scale
magnetic elements and active regions fields are crucially required.
Secondly, it is also possible that at the solar maximum, the stronger magnetic
field from sunspots tends to suppress the Sun’s global convection in some
measure. As a result, the local dynamo has been abated somehow, and the
network elements created by turbulence are reduced in number and total flux.
This seems to suggest that the turbulent dynamo is, in fact, global but not
local. Unfortunately, so far there have been no definite observations about
the changes in the global solar convection during the sunspot cycle.
Another possibility is that the anti-correlated component represents the
recycling of parts of the previously diffused or submerged magnetic flux from
the mean-field dynamo (Parker 1987). The diffusion of magnetic flux from
sunspots to the deep convection zone requires 5-7 years (Jiang et al. 2007).
Parts of the diffused or submerged flux serves as the seed field for the
globally turbulent dynamo. Its production is naturally out of phase with
sunspots in the solar cycle, and brings up the magnetic elements that anti-
phased with sunspots.
In a recent literature, Thomas and Weiss (2008) proposed a picture of the
solar Dynamo on three scales (one large and two small), which, according to
the above authors, were only loosely coupled to each other. It is not clear if
some unknown interplay of different scale dynamos may result in the
complicated behavior of the Sun’s small-scale fields. If we adopt the common
vision that the smaller magnetic elements are created by a local turbulent
dynamo, then the local turbulent dynamo on a certain scale must have closely
correlated to the global mean-field dynamo. The global dynamo either provides
seed flux or modifies the condition for this ‘global’ turbulent dynamo. At the
smallest end, the dynamo is likely to be more ‘local’. The turbulent dynamo,
either global or purely local, brings a tremendous amount of turbulent flux to
the Sun that continuously interacts with the products of the mean-field
dynamo. The interaction seems to not only help with the operation of the
global dynamo, but also power the ceaseless small-scale magnetic activity and
maintain the Sun’s Poynting flux to Earth and interplanetary space.
## 5 Conclusions
With the unique database from MDI/SOHO in the interval from September 1996 to
February 2010, which embodies the entire Solar Cycle 23, we analyze the cyclic
variations of quiet Sun’s magnetic flux and Sun’s small-scale magnetic
elements.
The quiet regions contributed $(0.94-1.44)\times 10^{23}$ Mx flux from
approximately the solar minimum to maximum in Cycle 23. The fractional area of
quiet regions decreased from the cycle minimum to maximum by a factor of 1.2,
but their total flux increased by a factor of 1.53. The quiet regions dominate
Sun’s magnetic flux over 60% duration of the cycle. Furthermore, the ratio of
the quiet region magnetic flux to the Sun’s total flux can be used to describe
the course of solar cycle, just as sunspots. The maximum flux occupation of
quiet regions marks the minima of solar cycle. The flux occupation on the
quiet Sun had been larger than 90% for 28 continuous months from July 2007 to
October 2009, which seems to equally characterize the grand minima of Cycles
23 and 24.
With increasing magnetic flux per element the number and total flux of the
Sun’s small-scale magnetic elements follow no-correlation, anti-correlation
and correlation changes with sunspots. The anti-correlated component, covering
the flux range of (2.9 - 32.0)$\times 10^{18}$ Mx, occupies 77.2% of total
elements and 37.4% of flux on the quiet Sun. However, the stronger magnetic
elements with flux larger than 4.3$\times 10^{19}$ Mx dominate the quiet Sun
magnetic flux and follow closely the sunspot cycle.
The definitively identified anti-correlated component of the small-scale
magnetic elements seems to offer an interpretation on the puzzling
observations of anti-correlation variation of many types of small-scale
activity with the solar cycle, e.g., the network bright points, HeI 10830 Å
dark points and coronal X-ray bright points.
It is speculated that the anti-correlated small-scale magnetic elements are
products of some local turbulent dynamo or dynamos that is modulated to be
anti-phased with the global mean-field dynamo.
The authors are grateful to Dean-Yi Chou, Sara Martin and Jie Jiang for their
valuable suggestions and discussions. We appreciate the instructive advice and
valuable suggestions of the anonymous referee, by which the paper has been
significantly improved. The work is supported by the National Natural Science
Foundation of China (10873020, 11003024, 40974112, 40731056, 10973019,
40890161, 10921303, 11025315), and the National Basic Research Program of
China (G2011CB811403).
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Table 1: Cyclic variation of the NT elements with different flux range Category | Flux (in Mx) | Number ratio | Flux (ratio) | Cor.
---|---|---|---|---
No-correlation | (1.5–2.9)$\times 10^{18}$ | 0.58% | 6.48$\times 10^{21}$ (0.05%) | -0.04
Anti-correlation | (2.9–32.0)$\times 10^{18}$ | 77.19% | 4.72$\times 10^{24}$ (37.40%) | -0.45
Transition | (3.20–4.27)$\times 10^{19}$ | 6.59% | 1.15$\times 10^{24}$ (9.08%) | -0.03
correlation | (4.27–38.01)$\times 10^{19}$ | 15.65% | 6.74$\times 10^{24}$ (53.46%) | 0.82
Figure 1: Left panels: Two retrieved MDI 5-minute full-disk magnetograms
within 70 $\deg$ from disk center, at approximately the solar maximum and
minimum, respectively. Using the threshold on 15 Mxcm-2 to define the edge of
active region, the islands with area larger 9$\times$9 pixels, i.e., the
regions contoured by red line, is defined as active regions. The purple circle
displays the location $\alpha$=70 $\deg$, and the black circle displays the
location $\alpha$=60 $\deg$. The gray scale saturates at $\pm$50 Mxcm-2. Right
panels: enlarged images for the windows framed in the MDI magnetograms, on
which network elements falling in the flux ranges of (2.9-32.0)$\times$ 1018
Mx and (4.3-38.0)$\times$ 1019 Mx are outlines by green and yellow curves,
respectively (see Section 3.2). Figure 2: The left panel is the flux
variations of ARs (cross symbols in black) and quiet Sun (‘+’ symbols in
purple) in an interval including the entire 23rd Solar Cycle. The red curve
represents the sunspot number changes in the cycle. The shaded columns are the
statistical results based on the Kitt Peak full-disk magnetograms. The
magnetic flux for quiet regions rises from 0.94$\times 10^{23}$ Mx in December
1995 to 1.44$\times 10^{23}$ in May 2002, increases by a factor of 1.53. The
fractional quiet Sun area is shown by purple ‘+’ symbols, in the right panel.
It decreases by a factor of 1.2 from the solar minima to maximum. The ratio of
quiet Sun flux to the total Sun’s flux, the flux occupation of the quiet Sun,
is shown by purple cross symbols in the right panel. The quiet Sun flux has
dominated the Sun’s magnetic flux for 7.92 years in the 12.5 year Cycle 23.
Figure 3: The probability distribution function of element magnetic flux for
all the selected network elements during the interval from September 1996 to
February 2010, i.e., average PDF of the quiet Sun’s small-scale magnetic
elements. The peak distribution at $10^{19}$ Mx is consistent with that found
for multiple MDI full-disk datasets by Parnell et al. (2009). Figure 4:
Correlation coefficients between the sunspot number and network element number
of each of the 96 sub-group elements which are reconstructed according to the
flux per element. There appears a 3-fold correlation scheme between the
network elements and the sunspot cycle: basically no-correlation, anti-
correlation and correlation. At the low end of flux spectrum, there are very
small correction coefficients. With the increasing flux per element, the
correlation coefficients reach approximately to -0.58, then they become
positive and reach as high as 0.92 after a very narrow transition in the flux
range of (3.20-4.27)$\times 10^{19}$ Mx. The color bar represents the
confidence level. Figure 5: Cyclic variations of network element number (right
panel) and flux (left panel) of 4 categories of network elements shown in
Table 1, which represents the 3-fold correlation scheme of network elements
with the sunspot cycle. The green ‘+’ is referring anti-correlation component
elements, while the purple ‘+’ is for the in-phase correlation component
elements. Black and blue dotted lines are elements which have no correlation
or shown transition from anti-correlation to correlation with the solar cycle.
Figure 6: The differential probability distribution function (DPDF), i.e., the
difference between the PDFs of yearly network magnetic elements and average
PDF shown in Fig.3.
|
arxiv-papers
| 2011-02-17T03:18:07 |
2024-09-04T02:49:17.057811
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. L. Jin, J. X. Wang, Q. Song and H. Zhao",
"submitter": "Chunlan Jin",
"url": "https://arxiv.org/abs/1102.3485"
}
|
1102.3487
|
# Baryonium Study in Heavy Baryon Chiral Perturbation Theory
Yue-De Chena111Email: chenyuede-b07@mails.gucas.ac.cn and Cong-Feng
Qiaoa,b222Email: qiaocf@gucas.ac.cn
$a)$ Department of Physics, Graduate University, the Chinese Academy of
Sciences
YuQuan Road 19A, 100049, Beijing, China
$b)$ Theoretical Physics Center for Science Facilities (TPCSF), CAS
YuQuan Road 19B, 100049, Beijing, China
To see whether heavy baryon and anti-baryon can form a bound state, the heavy
baryonium, we study the two-pion exchange interaction potential between them
within the heavy baryon chiral perturbation theory. The obtained potential is
applied to calculate the heavy baryonium masses by solving the Schrödinger
equation. We find it is true that the heavy baryonium may exist in a
reasonable choice of input parameters. The uncertainties remaining in the
potential and their influences on the heavy baryonium mass spectrum are
discussed.
## 1 Introduction
Quark model has achieved great success in describing the experimentally
observed hadronic structures to a large extent. And the quark potential in
between quark and anti-quark deduced from Chromodynamics (QCD) can explain the
meson spectrum quite well. Many of predicted states by potential model were
discovered in experiment and the theoretical predictions are in good agreement
with experimental data, especially in charmonium and bottomonium sectors [1,
2, 3], where the masses of charm and bottom quarks are heavy enough to be
treated non-relativistically. However, things became confused after the
discovery of $X(3872)$ in 2003 at $\mathrm{Belle}$ [4], which was later
confirmed by $\mathrm{BaBar}$ [5]. In recent years, a series of unusual states
in charmonium sector, such as $Y(4260)$, $Y(4360)$, $Y(4660)$, and
$Z^{\pm}(4430)$, were observed in experiment [6]. Due to their extraordinary
decay nature, it is hard to embed them into the conventional charmonium
spectrum, which leads people to treat them as exotic rather than quark-quark
bound states. The typical scenarios in explaining these newly found states
include treating $Y(4260)$ as a hybrid charmonium [7], a $\chi_{c}\rho^{0}$
molecular state [8], a conventional $\Psi(4S)$ [9], an $\omega\chi_{c1}$
molecular state [10], a $\Lambda_{c}\bar{\Lambda}_{c}$ baryonium state [11], a
$D_{1}D$ or $D_{0}D^{*}$ hadronic molecule [12], and a $P$-wave tetraquark
$[cs][\bar{c}\bar{s}]$ state [13]; $Y(4360)$ is interpreted as the candidate
of the charmonium hybrid or an excited D-wave charmonium state, the
$3^{3}D_{1}$ [14] and an excited state of baryonium [16]; $Y(4660)$ is
suggested to be the excited S-wave charmonium states, the $5^{3}S_{1}$ [14]
and $6^{3}S_{1}$ [15], a baryonium state [16, 17], a $f_{0}(980)\Psi^{\prime}$
bound state [18, 19], a $5^{3}S_{1}$-$4^{3}D_{1}$ mixing state [20], and also
a tetraquark state [21, 22]. There have been recently many research works on
”exotic” heavy quarkonium study in experiment and theory. To know more of
recent progress in this respect and to have a more complete list of references
one can see e.g. recent reviews [23, 24] and references therein.
In the baryonium picture, the tri-quark clusters are baryon-like, but not
necessarily colorless. In the pioneer works of heavy baryonium for the
interpretation of newly observed “exotic” structures [11, 16], there were only
phenomenological and kinematic analysis, but without dynamics. In this work we
attempt to study the heavy baryonium interaction potential arising from two-
pion exchanges in the framework of Heavy Baryon Chiral Perturbation Theory
(HBCPT) [25]. The paper is organized as follows. In Section 2, we present the
formalism for the heavy baryon-baryon interaction study; in Section 3 we
perform the numerical study for the mass spectrum of the possible baryonium
with the obtained potential in preceding section; the Section 4 is devoted to
the summary and conclusions. For the sake of reader’s convenience some of the
used formulae are given in the Appendix.
## 2 Formalism
To obtain the heavy baryonium mass spectrum, we first start from extracting
the baryon-baryon interaction potential in the same procedure as for quark-
quark interaction [1].
### 2.1 Heavy Baryonium
In the heavy baryonium picture [16], $\Lambda_{c}$ and $\Sigma^{0}_{c}$ are
taken as basis vectors in two-dimensional space. The baryonia are loosely
bound states of heavy baryon and anti-baryon, namely
$\displaystyle B^{+}_{1}$ $\displaystyle\equiv$
$\displaystyle|\Lambda_{c}^{+}\;\bar{\Sigma}_{c}^{0}>~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle{\rm Triplet:}\;\;\;\;\;B^{0}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}(|\Lambda_{c}^{+}\;\bar{\Lambda}_{c}^{+}>\;-\;|{\Sigma}_{c}^{0}\bar{\Sigma}_{c}^{0}>)$
(1) $\displaystyle B^{-}_{1}$ $\displaystyle\equiv$
$\displaystyle|\bar{\Lambda}^{+}_{c}\;{\Sigma}_{c}^{0}>~{}~{}~{}~{}~{}~{}~{}~{}~{}$
and
$\displaystyle{\rm
Singlet:}\;\;\;\;\;B^{0}_{0}\equiv\frac{1}{\sqrt{2}}(|\Lambda_{c}^{+}\;\bar{\Lambda}_{c}^{+}>\;+\;|{\Sigma}_{c}^{0}\bar{\Sigma}_{c}^{0}>)\
.$ (2)
Here, approximately the transformation in this two-dimensional ”C-spin” space
is invariant, which is in analog to the invariance of isospin transformation
in proton and neutron system.
### 2.2 Effective Chiral Lagrangian
Heavy baryon contains both light and heavy quarks, of which the light
component exhibits the chiral property and the heavy component exhibits heavy
symmetry. Therefore, it is plausible to tackle the problem of heavy baryon
interaction through the heavy chiral perturbation theory. Following we briefly
review the gists of the HBCPT for later use.
In usual chiral perturbation theory, the nonlinear chiral symmetry is realized
by making use of the unitary matrix
$\Sigma=e^{\frac{2iM}{f_{\pi}}}\;,$ (3)
where $M$ is a $3\times 3$ matrix composed of eight Goldstone-boson fields,
i.e.,
$M=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\phantom{+}\pi^{+}&\phantom{+}K^{+}\\\
\phantom{+}\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\phantom{+}K^{0}\\\
\phantom{+}K^{-}&\phantom{+}\bar{K}^{0}&\phantom{+}-\frac{2}{\sqrt{6}}\eta\end{array}\right)\;.$
(4)
Here, $f_{\pi}$ is the $pion$ decay constant.
After the chiral symmetry spontaneously broken, the Goldstone boson
interaction with hadron is introduced through a new matrix [26, 27]
$\xi=\Sigma^{\frac{1}{2}}=e^{\frac{iM}{f_{\pi}}}\;.$ (5)
From $\xi$ one can construct a vector field $V_{\mu}$ and an axial vector
field $A_{\mu}$ with simple chiral transformation properties, i.e.,
$V_{\mu}=\frac{1}{2}(\xi^{{\dagger}}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{{\dagger}})\;,$
(6)
$A_{\mu}=\frac{i}{2}(\xi^{{\dagger}}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{{\dagger}})\;.$
(7)
For our aim, we work only on the leading order vector and axial vector fields
in the expansion of $\xi$ in terms of $f_{\pi}$, they are
$V_{\mu}=\frac{1}{f_{\pi}^{2}}M\partial_{\mu}M\;,$ (8)
$A_{\mu}=-\frac{1}{f_{\pi}}\partial_{\mu}M\;.$ (9)
For heavy baryon, each of the two light quarks is in a triplet of flavor
SU(3), and hence the baryons can be grouped in two different SU(3) multiplets,
the sixtet and antitriplet. The symmetric sixtet and antisymmetric triplet can
be constructed out in $3\times 3$ matrices [27], they are
$B_{6}=\left(\begin{array}[]{ccc}\Sigma_{c}^{++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\frac{1}{\sqrt{2}}\Xi_{c}^{{}^{\prime}+}\\\
\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\Sigma_{c}^{0}&\frac{1}{\sqrt{2}}\Xi_{c}^{{}^{\prime}0}\\\
\frac{1}{\sqrt{2}}\Xi_{c}^{{}^{\prime}+}&\frac{1}{\sqrt{2}}\Xi_{c}^{{}^{\prime}0}&\Omega_{c}^{0}\end{array}\right)\;,$
(10)
and
$B_{\bar{3}}=\left(\begin{array}[]{ccc}0&\Lambda_{c}&\Xi_{c}^{+}\\\
-\Lambda_{c}&0&\Xi_{c}^{-}\\\
-\Xi_{c}^{+}&-\Xi_{c}^{-}&0\end{array}\right)\;,$ (11)
respectively.
Introducing six coupling constant $g_{i}$, $i=1,6$, the general chiral-
invariant Lagrangian then reads [25]
$\displaystyle\mathcal{L_{G}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}tr[\bar{B}_{\bar{3}}(iD\\!\\!\\!/-M_{\bar{3}})B_{\bar{3}}]+tr[\bar{B}_{6}(iD\\!\\!\\!/-M_{6})B_{6}]$
(12) $\displaystyle+$ $\displaystyle
tr[\bar{B}_{6}^{*\mu}[-g_{\mu\nu}(iD\\!\\!\\!/-M_{6}^{*})+i(\gamma_{\mu}D_{\nu}+\gamma_{\nu}D_{\mu})-\gamma_{\mu}(iD\\!\\!\\!/+M_{6}^{*})\gamma_{\nu}]B_{6}^{*\nu}]$
$\displaystyle+$ $\displaystyle
g_{1}tr(\bar{B}_{6}\gamma_{\mu}\gamma_{5}A^{\mu}B_{6})+g_{2}tr(\bar{B}_{6}\gamma_{\mu}\gamma_{5}A^{\mu}B_{\bar{3}})+h.c.$
$\displaystyle+$ $\displaystyle
g_{3}tr(\bar{B}_{6{\mu}}^{*}A^{\mu}B_{6})+h.c.+g_{4}tr(\bar{B}_{6{\mu}}^{*}A^{\mu}B_{\bar{3}})+h.c.$
$\displaystyle+$ $\displaystyle
g_{5}tr(\bar{B}_{6}^{\nu*}\gamma_{\mu}\gamma_{5}A^{\mu}B_{6\nu}^{*})+g_{6}tr(\bar{B}_{\bar{3}}\gamma_{\mu}\gamma_{5}A^{\mu}B_{\bar{3}})\;.$
Here, $B_{6\nu}^{*}$ is a Rarita-Schwinger vector-spinor field for
spin-$\frac{3}{2}$ particle; $M_{\bar{3}}$, $M_{6}$, $M_{6}^{*}$ represent for
heavy baryon mass matrices of corresponding fields; With the help of vector
current $V_{\mu}$ defined in Eq. (8), we may construct the covariant
derivative $D_{\mu}$, which acts on baryon field, as
$D_{\mu}B_{6}=\partial_{\mu}B_{6}+V_{\mu}B_{6}+B_{6}V_{\mu}^{T}\;,$ (13)
$D_{\mu}B_{\bar{3}}=\partial_{\mu}B_{\bar{3}}+V_{\mu}B_{\bar{3}}+B_{\bar{3}}V_{\mu}^{T}\;,$
(14)
where $V_{\mu}^{T}$ stands for the transpose of $V_{\mu}$. Thus, the couplings
of vector current to heavy baryons relevant to our task take the following
form
$\displaystyle\mathcal{L}_{{\mathcal{E}_{1}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}tr(\bar{B}_{\bar{3}}i\gamma^{\mu}V_{\mu}B_{\bar{3}})$
(15) $\displaystyle=$
$\displaystyle\frac{1}{2f_{\pi}^{2}}\bar{\Lambda}_{c}i\gamma^{\mu}(\pi^{0}\partial_{\mu}\pi^{0}+\pi^{-}\partial_{\mu}\pi^{+}+\pi^{+}\partial_{\mu}\pi^{-})\Lambda_{c}\;,$
and
$\displaystyle\mathcal{L}_{{\mathcal{E}_{2}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}tr(\bar{B}_{\bar{3}}B_{\bar{3}}i\gamma^{\mu}V_{\mu}^{T})$
(16) $\displaystyle=$
$\displaystyle\frac{1}{2f_{\pi}^{2}}\bar{\Lambda}_{c}\Lambda_{c}i\gamma^{\mu}(\pi^{0}\partial_{\mu}\pi^{0}+\pi^{-}\partial_{\mu}\pi^{+}+\pi^{+}\partial_{\mu}\pi^{-})\;.$
According to the heavy quark symmetry, there are four constraint relations
among those six coupling constants of the Lagrangian of Eq. (12), i.e.,
$\displaystyle
g_{6}=0\;,\;g_{3}=\frac{\sqrt{3}}{2}g_{1}\;,\;g_{5}=-\frac{3}{2}g_{1}\;,\;g_{4}=-\sqrt{3}g_{2}\;,$
(17)
which means the number of independent couplings are then reduced to two. In
this work, we employ $g_{1}$ and $g_{2}$ for the numerical evaluation as did
in Ref. [25].
Here, to get the dominant interaction potential we restrict our effort only on
the $pion$ exchange processes as usual. Notice that the couplings of $pion$ to
spin-$\frac{3}{2}$ and -$\frac{1}{2}$ baryons, and $pion$ to two
spin-$\frac{1}{2}$ baryons take a similar form, in the following we merely
present the spin-$\frac{3}{2}$ and -$\frac{1}{2}$ baryon-$pion$ coupling for
illustration, i.e.,
$\mathcal{L}_{1}=\frac{g_{3}}{\sqrt{2}f_{\pi}}\bar{\Sigma_{c}}\\!^{0*\mu}\partial_{\mu}\pi^{0}\Sigma_{c}^{0}+h.c.\;,$
(18)
$\mathcal{L}_{2}=-\frac{g_{3}}{\sqrt{2}f_{\pi}}\bar{\Sigma_{c}}\\!^{+*\mu}\partial_{\mu}\pi^{+}\Sigma_{c}^{0}+h.c.\;,$
(19)
$\mathcal{L}_{3}=\frac{g_{4}}{f_{\pi}}\bar{\Sigma_{c}}\\!^{++*\mu}\partial_{\mu}\pi^{+}\Lambda_{c}^{+}+h.c.\;,$
(20)
$\mathcal{L}_{4}=-\frac{g_{4}}{f_{\pi}}\bar{\Sigma_{c}}\\!^{0*\mu}\partial_{\mu}\pi^{-}\Lambda_{c}^{+}+h.c.\;,$
(21)
$\mathcal{L}_{5}=-\frac{g_{4}}{f_{\pi}}\bar{\Sigma_{c}}\\!^{+*\mu}\partial_{\mu}\pi^{0}\Lambda_{c}^{+}+h.c.\;.$
(22)
To get the $pion$ and two spin-$\frac{1}{2}$ baryon couplings one only needs
to replace the $\Sigma_{c}^{*\mu}$ by $\Sigma_{c}$, $g_{3}$ by $g_{1}$,
$g_{4}$ by $g_{2}$, and insert $\gamma^{\mu}\gamma_{5}$ in between the two
baryon fields in Eqs.(18)-(22).
Figure 1: Schematic Diagrams which contribute to the baryonium potential.
### 2.3 Baryonium Potential from Two-pion Exchange
To obtain heavy baryon-baryon interaction potential in configuration space, we
start from writing down the two-body scattering amplitude in the center-of-
mass frame(CMS), i.e. taking $\textbf{p}_{a}=-\textbf{p}_{b}$ and
$\textbf{p}_{a}^{\prime}=-\textbf{p}_{b}^{\prime}$. In CMS the total and
relative four momenta are defined as
$\displaystyle P$ $\displaystyle=$
$\displaystyle(p_{a}\;+\;p_{b})\;=\;(p_{a}^{\prime}\;+\;p_{b}^{\prime})=(E,\;0)\;,$
(23) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p_{a}\;-\;p_{b})\;=\;(0,\;\textbf{p})\;,$ (24)
$\displaystyle p^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p_{a}^{\prime}\;-\;p_{b}^{\prime})\;=\;(0,\;\textbf{p}^{\prime})\;.$
(25)
To perform the calculation, it is convenient to introduce some new variables
as functions of p and $\textbf{p}^{\prime}$, i.e.,
$\displaystyle\mathcal{W}(\textbf{p})$
$\displaystyle=E_{a}(\textbf{p})+E_{b}(\textbf{p})\;,$ (26)
$\displaystyle\mathcal{W}(\textbf{p}^{\prime})$
$\displaystyle=E_{a}(\textbf{p}^{\prime})+E_{b}(\textbf{p}^{\prime})\;,$ (27)
$\displaystyle F_{E}(\textbf{p},\;p_{0})$
$\displaystyle=\frac{1}{2}E+p_{0}-E(\textbf{p})+i\delta\;,$ (28)
where $\delta$ is an infinitesimal quantity introduced in the so-called
$i\delta$ prescription. Following the same procedure as in Refs. [28, 29], it
is straightforward to write down the baryon-baryon scattering kernels, shown
as box and crossed diagrams in Figure 1,
$\displaystyle K_{box}=$ $\displaystyle-$
$\displaystyle\frac{1}{(2\pi)^{2}}(E-\mathcal{W}(\textbf{p}^{\prime}))(E-\mathcal{W}(\textbf{p}))\int
dp_{0}^{\prime}dp_{0}dk_{20}dk_{10}d^{3}\textbf{k}_{2}d^{3}\textbf{k}_{1}$
(29) $\displaystyle\times$
$\displaystyle\frac{i}{(2\pi)^{4}}\delta^{4}(p-p^{\prime}-k_{1}-k_{2})\frac{1}{k_{2}^{2}-m^{2}+i\delta}\frac{1}{F_{E}(\textbf{p}^{\prime},p_{0}^{\prime})F_{E}(-\textbf{p}^{\prime},-p_{0}^{\prime})}$
$\displaystyle\times$
$\displaystyle\frac{\Gamma_{j}\Gamma_{i}\Gamma_{i}\Gamma_{j}}{F_{E}(\textbf{p}-\textbf{k},p_{0}-k_{10})F_{E}(-\textbf{p}+\textbf{k},-p_{0}+k_{10})}\frac{1}{F_{E}(\textbf{p},p_{0})F_{E}(\textbf{p},-p_{0}))}$
$\displaystyle\times$ $\displaystyle\frac{1}{k_{1}^{2}-m^{2}+i\delta}\;,$
$\displaystyle K_{cross}=$ $\displaystyle-$
$\displaystyle\frac{1}{(2\pi)^{2}}(E-\mathcal{W}(\textbf{p}^{\prime}))(E-\mathcal{W}(\textbf{p}))\int
dp_{0}^{\prime}dp_{0}dk_{20}dk_{10}d^{3}\textbf{k}_{2}d^{3}\textbf{k}_{1}$
(30) $\displaystyle\times$
$\displaystyle\frac{i}{(2\pi)^{4}}\delta^{4}(p-p^{\prime}-k_{1}-k_{2})\frac{1}{k_{2}^{2}-m^{2}+i\delta}\frac{1}{F_{E}(\textbf{p}^{\prime},p_{0}^{\prime})F_{E}(-\textbf{p}^{\prime},-p_{0}^{\prime})}$
$\displaystyle\times$
$\displaystyle\frac{\Gamma_{j}\Gamma_{i}\Gamma_{j}\Gamma_{i}}{F_{E}(\textbf{p}-\textbf{k},p_{0}-k_{10})F_{E}(-\textbf{p}^{\prime}-\textbf{k},-p_{0}^{\prime}-k_{10})}\frac{1}{F_{E}(\textbf{p},p_{0})F_{E}(-\textbf{p},-p_{0})}$
$\displaystyle\times$ $\displaystyle\frac{1}{k_{1}^{2}-m^{2}+i\delta}\;.$
Here, $m$ corresponds to the $pion$ mass and $\Gamma_{i,j}$ are heavy
baryon-$pion$ interaction vertices that can be read out from the Lagrangian in
Eqs.(18)-(22). In case of spin-$\frac{3}{2}$ intermediate,
$\displaystyle\Gamma_{j}\Gamma_{i}\Gamma_{i}\Gamma_{j}$ $\displaystyle=$
$\displaystyle\left(\frac{g_{4}}{f_{\pi}}\right)^{4}\bar{u}(-p)k_{2}^{\mu}u_{\mu}(p-k_{1})\bar{u}_{\nu}(p-k_{1})k_{1}^{\nu}u(p)$
(31) $\displaystyle\times$
$\displaystyle\bar{v}(p)(-k_{1}^{\alpha})v_{\alpha}(-p+k_{1})\bar{v}_{\beta}(-p+k_{1})k_{2}^{\beta}v(-p)\;,$
and in case of spin-$\frac{1}{2}$ intermediate
$\displaystyle\Gamma_{j}\Gamma_{i}\Gamma_{i}\Gamma_{j}$ $\displaystyle=$
$\displaystyle\left(\frac{g_{2}}{f_{\pi}}\right)^{4}\bar{u}(-p)\gamma_{\mu}\gamma_{5}k_{2}^{\mu}u(p-k_{1})\bar{u}(p-k_{1})\gamma_{\nu}\gamma_{5}k_{1}^{\nu}u(p)$
(32) $\displaystyle\times$
$\displaystyle\bar{v}(p)\gamma_{\alpha}\gamma_{5}(-k_{1}^{\alpha})v(-p+k_{1})\bar{v}(-p+k_{1})\gamma_{\beta}\gamma_{5}k_{2}^{\beta}v(-p)\;.$
Integrating over $p^{\prime}_{0}$, $p_{0}$, $k_{10}$, and $k_{20}$ in Eq.(29)
one obtains the interaction kernel of box diagram at order
$\mathcal{O}(\frac{1}{M_{H}})$,
$\displaystyle K_{box}=$ $\displaystyle-$
$\displaystyle\frac{1}{(2\pi)^{3}}\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{4E_{\textbf{k}_{1}}E_{\textbf{k}_{2}}}\frac{\Gamma_{j}\Gamma_{i}}{E_{\textbf{p}-\textbf{k}_{1}}+E_{\textbf{p}}-W+E_{\textbf{k}_{1}}}$
(33) $\displaystyle\times$
$\displaystyle\frac{\Gamma_{i}\Gamma_{j}}{E_{\textbf{p}}^{\prime}+E_{\textbf{p}-\textbf{k}_{1}}-W+E_{\textbf{k}_{2}}}\frac{1}{E_{\textbf{p}}+E_{\textbf{p}^{\prime}}-W+E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}}}\;,$
where $M_{H}$ represents one of the heavy baryon mass, $M_{\Lambda_{c}^{+}}$,
$M_{\Sigma^{0}_{c}}$ or $M_{\Sigma_{c}^{*}}$;
$E_{\textbf{p}-\textbf{k}_{1}}=\sqrt{(\textbf{p}-\textbf{k}_{1})^{2}+M_{\Sigma_{c}^{*}}^{2}}$
is the intermediate state energy;
$E_{\textbf{k}_{1}}=\sqrt{\textbf{k}_{1}^{2}+m^{2}}$ and
$E_{\textbf{k}_{2}}=\sqrt{\textbf{k}_{2}^{2}+m^{2}}$ are two $\it{pion}$s’
energies; and $W=2E(\textbf{p})$. With the same procedure, we can get the
interaction kernel of crossed diagram, i.e.
$\displaystyle K_{cross}=$ $\displaystyle-$
$\displaystyle\frac{1}{(2\pi)^{3}}\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{4E_{\textbf{k}_{1}}E_{\textbf{k}_{2}}}\frac{\Gamma_{j}\Gamma_{i}}{E_{\textbf{p}-\textbf{k}_{1}}+E_{\textbf{p}}-W+E_{\textbf{k}_{1}}}$
(34) $\displaystyle\times$
$\displaystyle\frac{\Gamma_{j}\Gamma_{i}}{E_{\textbf{p}}^{\prime}+E_{\textbf{p}^{\prime}+\textbf{k}_{1}}-W+E_{\textbf{k}_{1}}}\frac{1}{E_{\textbf{p}}+E_{\textbf{p}^{\prime}}-W+E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}}}\;.$
Next, since what we are interested in is the heavy baryons, we can further
implement the non-relativistic reduction on spinors with the help of vertices
given in Eqs.(18)-(22). In the end, the non-relativistic reduction for
$\Lambda_{c}^{+}\Sigma_{c}^{+*}\pi^{0}$ and
$\Lambda_{c}^{+}\Sigma_{c}^{+}\pi^{0}$ couplings gives
$i\left(\frac{g_{4}}{f_{\pi}}\right)\bar{u}(p_{2})u_{\mu}(p_{1})(p_{2}-p_{1})^{\mu}=-i\left(\frac{g_{4}}{f_{\pi}}\right)\textbf{S}^{{\dagger}}\cdot\textbf{q}\;,$
(35)
and
$i\left(\frac{g_{2}}{f_{\pi}}\right)\bar{u}(p_{2})\gamma_{\mu}\gamma_{5}u(p_{1})(p_{2}-p_{1})^{\mu}=i\left(\frac{g_{2}}{f_{\pi}}\right)\boldsymbol{\sigma}_{1}\cdot\textbf{q}\;,$
(36)
respectively. Here, $\textbf{q}=\textbf{p}_{2}-\textbf{p}_{1}$ and
$\textbf{S}^{{\dagger}}$ is the spin-$\frac{1}{2}$ to spin-$\frac{3}{2}$
transition operator.
In the process of deriving $\Lambda_{c}^{+}-\bar{\Lambda}_{c}^{+}$ potential,
the $\Sigma_{c}^{+}$ and $\Sigma_{c}^{+*}$ are taken into account as
intermediate states. Using Eqs. (35)-(36) and the explicit forms of spinors
given in the appendix, we can readily obtain the reduction forms for the
$\Sigma_{c}^{+}$ intermediate
$\displaystyle\bar{u}(-p)\gamma_{\mu}\gamma_{5}k_{2}^{\mu}u(p-k_{1})\bar{u}(p-k_{1})\gamma_{\nu}\gamma_{5}k_{1}^{\nu}u(p)\times$
(37)
$\displaystyle\bar{v}(p)\gamma_{\alpha}\gamma_{5}(-k_{1}^{\alpha})v(-p+k_{1})\bar{v}(-p+k_{1})\gamma_{\beta}\gamma_{5}k_{2}^{\beta}v(-p)$
$\displaystyle=$
$\displaystyle(\textbf{k}_{1}\cdot\textbf{k}_{2})^{2}+(\boldsymbol{\sigma}_{1}\cdot\textbf{k}_{1}\times\textbf{k}_{2})(\boldsymbol{\sigma}_{2}\cdot\textbf{k}_{1}\times\textbf{k}_{2})\;,$
the $\Sigma_{c}^{+*}$ intermediate in the box diagram
$\displaystyle\bar{u}(-p)k_{2}^{\mu}u_{\mu}(p-k_{1})\bar{u}_{\nu}(p-k_{1})k_{1}^{\nu}u(p)\times$
(38)
$\displaystyle\bar{v}(p)(-k_{1}^{\alpha})v_{\alpha}(-p+k_{1})\bar{v}_{\beta}(-p+k_{1})k_{2}^{\beta}v(-p)$
$\displaystyle=$
$\displaystyle\frac{4}{9}(\textbf{k}_{1}\cdot\textbf{k}_{2})^{2}-\frac{1}{9}(\boldsymbol{\sigma}_{1}\cdot\textbf{k}_{1}\times\textbf{k}_{2})(\boldsymbol{\sigma}_{2}\cdot\textbf{k}_{1}\times\textbf{k}_{2})\;,$
and the crossed diagram
$\displaystyle\bar{u}(-p)k_{2}^{\mu}u_{\mu}(p-k_{1})\bar{u}_{\nu}(p-k_{1})k_{1}^{\nu}u(p)\times$
(39)
$\displaystyle\bar{v}(p)(-k_{1}^{\alpha})v_{\alpha}(-p+k_{1})\bar{v}_{\beta}(-p+k_{1})k_{2}^{\beta}v(-p)$
$\displaystyle=$
$\displaystyle\frac{4}{9}(\textbf{k}_{1}\cdot\textbf{k}_{2})^{2}+\frac{1}{9}(\boldsymbol{\sigma}_{1}\cdot\textbf{k}_{1}\times\textbf{k}_{2})(\boldsymbol{\sigma}_{2}\cdot\textbf{k}_{1}\times\textbf{k}_{2})\;,$
respectively. Thus, the spinor reduction finally leads to an operator
$\mathcal{O}_{1}(\textbf{k}_{1},\;\textbf{k}_{2})$, of which the variables
$\textbf{k}_{1}$ and $\textbf{k}_{2}$ can be replaced by gradient operators
$\boldsymbol{\nabla}_{1}$ and $\boldsymbol{\nabla}_{2}$ in configuration space
and acting on $\textbf{r}_{1}$ and $\textbf{r}_{2}$, respectively. This
operator is expressed as
$\displaystyle\mathcal{O}_{1}(\textbf{k}_{1},\;\textbf{k}_{2})$
$\displaystyle=$ $\displaystyle
c_{1}O_{1}(\textbf{k}_{1},\;\textbf{k}_{2})+c_{2}O_{2}(\textbf{k}_{1},\;\textbf{k}_{2})$
(40) $\displaystyle=$ $\displaystyle
c_{1}(\textbf{k}_{1}\cdot\textbf{k}_{2})^{2}+c_{2}(\boldsymbol{\sigma}_{1}\cdot\textbf{k}_{1}\times\textbf{k}_{2})(\boldsymbol{\sigma}_{2}\cdot\textbf{k}_{1}\times\textbf{k}_{2})\;.$
Here, the decomposition coefficients $c_{1}$ and $c_{2}$ are given in Table 1.
The first part of Eq. (40) may generate the central potential and the second
part may generate the spin-spin coupling and the tensor potentials, which are
explicitly shown in the Appendix.
Table 1: The values of coefficients $c_{1}$ and $c_{2}$ in the decomposition
of operator $O(\textbf{k}_{1},\;\textbf{k}_{2})$ in Eq. (40). The left one is
for the spin-$\frac{1}{2}$ intermediate state case and the right one is for
the spin-$\frac{3}{2}$ case.
spin-1/2 | $~{}c_{1}$ | $~{}c_{2}$
---|---|---
box | 1 | 1
cross | 1 | 1
spin-3/2 | $c_{1}$ | $c_{2}$
---|---|---
box | $4/9$ | $-1/9$
cross | $4/9$ | $~{}~{}1/9$
To get the leading order central potential, e.g. for
$\Lambda_{c}$-$\bar{\Lambda}_{c}$ system, we first expand the energy in powers
of $\frac{1}{M_{H}}$, but keep only the leading term, like
$\displaystyle\frac{1}{E_{\textbf{p}-\textbf{k}_{1}}+E_{\textbf{p}}-W+E_{\textbf{k}_{1}}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{M_{\Sigma_{c}^{*}}+M_{\Lambda_{c}}-2M_{\Lambda_{c}}+E_{\textbf{k}_{1}}}=\frac{1}{E_{\textbf{k}_{1}}+\Delta_{1}},$
(41)
where $\Delta_{1}=M_{\Sigma_{c}^{*}}-M_{\Lambda_{c}}$ represents the mass
splitting. By virtue of the factorization in integrals given in the Appendix,
we can then make a double Fourier transformation, i.e.,
$\displaystyle
V_{C}^{B}(r_{1},\;r_{2})=-\left(\frac{g_{4}^{4}}{f_{\pi}^{4}}\right)\int\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{(2\pi)^{6}}\frac{\mathcal{O}_{1}(\textbf{k}_{1},\textbf{k}_{2})e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})}{2E_{\textbf{k}_{1}}E_{\textbf{k}_{2}}(E_{\textbf{k}_{1}}+\Delta_{1})(E_{\textbf{k}_{2}}+\Delta_{1})(E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}})}\;,$
(42)
where the superscript $B$ denotes the box diagram and the subscript $C$ means
central potential. Similarly, one can get the central potential from the
crossed diagram contribution
$V_{C}^{C}(r_{1},\;r_{2})=-\left(\frac{g_{4}^{4}}{f_{\pi}^{4}}\right)\int\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{(2\pi)^{6}}\mathcal{O}_{1}(\textbf{k}_{1},\textbf{k}_{2})e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})\
D\;,$ (43)
where the superscript $C$ denote crossed diagram and the subscript $C$ means
central potential, and
$\displaystyle D$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{4E_{\textbf{k}_{1}}E_{\textbf{k}_{2}}}\left[\left(\frac{1}{(E_{\textbf{k}_{1}}+\Delta_{1})^{2}}+\frac{1}{(E_{\textbf{k}_{2}}+\Delta_{1})^{2}}\right)\frac{1}{E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}}}\right.$
(44) $\displaystyle+$
$\displaystyle\\!\\!\\!\left(\frac{1}{(E_{\textbf{k}_{1}}+\Delta_{1})^{2}}\left.+\frac{1}{(E_{\textbf{k}_{2}}+\Delta_{1})^{2}}+\frac{2}{(E_{\textbf{k}_{1}}+\Delta_{1})(E_{\textbf{k}_{2}}+\Delta_{1})}\right)\frac{1}{E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}}+2\Delta_{1}}\right].$
In order to regulate the potentials we have introduced form factors at each
baryon-pion vertex. The resulting $f(\bf k^{2})$ form factors appearing in
Eqs. (42) and (43) will be given in Section 3.
Taking a similar approach as given in above one can readily get the central
potential in other interaction channels and also the tensor potential. Notice
that although there exists the one-pion exchange contribution in
$\Sigma_{c}$-$\Sigma_{c}$ system, due to the $\gamma_{\mu}\gamma_{5}$ nature
in interaction vertex, it only contributes to
$\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$ term, which is out of
our concern in this work. Here we just focus on the central potential.
Figure 2: The triangle and two-pion loop diagrams.
Besides box and crossed diagrams, there are also contributions from triangle
and two-pion loop diagrams as shown in Fig. 2. As in the box and crossed
diagrams, after integrating over energy component, we get the pion-pair
contribution, as shown in the left diagram of Figure 2, as [33]
$V_{triangle}(r_{1},r_{2})=\frac{g_{4}^{2}}{2f_{\pi}^{4}}\int\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{(2\pi)^{6}}\frac{\mathcal{O}_{2}(\textbf{k}_{1},\textbf{k}_{2})(E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}})e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})}{E_{\textbf{k}_{1}}E_{\textbf{k}_{2}}(E_{\textbf{k}_{1}}+\Delta_{1})(E_{\textbf{k}_{2}}+\Delta_{1})}\;,$
(45)
where the
$\mathcal{O}_{2}(\textbf{k}_{1},\textbf{k}_{2})=(\textbf{k}_{1}\cdot\textbf{k}_{2})$
from spinor reduction can be replaced in configuration space by the gradient
operator $(\boldsymbol{\nabla}_{1}\cdot\boldsymbol{\nabla}_{2})$. Similarly,
the two-pion loop contribution, as shown in the right diagram of Figure 2
reads
$V_{2\pi-
loop}(r_{1},r_{2})=\frac{1}{16f_{\pi}^{4}}\int\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{(2\pi)^{6}}e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})A\;.$
(46)
Here,
$A=-\frac{1}{2E_{\textbf{k}_{1}}}-\frac{1}{2E_{\textbf{k}_{2}}}+\frac{2}{E_{\textbf{k}_{1}}+E_{\textbf{k}_{2}}}~{}$.
Expressing Eps. (45) and (46) in the integral representation of
$E_{\textbf{k}_{1}}$, and making the Fourier transformation, one can then
obtain the corresponding potentials.
## 3 Numerical Analysis
With the central potentials obtained in preceding section, one can calculate
the heavy baryonium spectrum by solving the Schrödinger equation. In our
numerical evaluation, the Matlab based package Matslise [31] is employed. The
following inputs from Particle Data Book [32] are used in the numerical
calculation:
$M_{\Lambda_{c}^{+}}=2.286\mathrm{GeV}\;,\;M_{\Sigma_{c}^{0}}=2.454\mathrm{GeV}\;,\;M_{\Sigma_{c}^{*}}=2.518\mathrm{GeV}\;,\;f_{\pi}=0.132\mathrm{GeV}\;,\;m=0.135\mathrm{GeV}\;,$
(47)
and both spin-$\frac{1}{2}$ and -$\frac{3}{2}$ fermion intermediates are taken
into account.
It is obvious that the main uncertainties in the evaluation of heavy baryonium
remain in the couplings of Eq. (17). The magnitudes of the two independent
couplings $g_{1}$ and $g_{2}$ were phenomenologically analyzed in Ref. [25],
and two choices for them were suggested, i.e.,
$g_{1}=\frac{1}{3}\;,\;g_{2}=-\sqrt{\frac{2}{3}}$ (48)
and
$g_{1}=\frac{1}{3}\times 0.75\;,\;g_{2}=-\sqrt{\frac{2}{3}}\times 0.75\;,$
(49)
which implies the $g_{4}$ lies in the scope of 1 to 1.4, similar as estimated
by Ref. [30] in the chiral limit.
### 3.1 Gaussian form factor case
The central potential from two-pion exchange box which can be regularized by
widely used Gaussian form factor
$f(\textbf{k}^{2})=e^{-\textbf{k}^{2}/\Lambda^{2}}$ reads
$\displaystyle V_{CG}^{B}(r_{1},\;r_{2})$ $\displaystyle=$
$\displaystyle-\left(\frac{g_{4}^{4}}{f_{\pi}^{4}}\right)\left[\frac{1}{\pi}\int_{0}^{\infty}\frac{d\lambda}{\Delta_{1}^{2}+\lambda^{2}}O_{1}(\textbf{k}_{1},\textbf{k}_{2})F(\lambda,r_{1})F(\lambda,r_{2})\right.$
(50)
$\displaystyle\left.-\frac{2\Delta_{1}}{\pi^{2}}O_{1}(\textbf{k}_{1},\textbf{k}_{2})\int_{0}^{\infty}\frac{d\lambda}{\Delta_{1}^{2}+\lambda^{2}}F({\lambda,r_{1}})\int_{0}^{\infty}\frac{d\lambda}{\Delta_{1}^{2}+\lambda^{2}}F({\lambda,r_{2}})\right]$
$\displaystyle=$ $\displaystyle\sum_{i}V_{CGi}^{B}+\cdots\;.$
Details of the derivation of Eq. (50) from Eq. (42) can be found in the
Appendix. There, the function $F(\lambda,r)$ is defined by Eq. (76). And,
similarly the central potential from two-pion exchange crossed diagram gives
$\displaystyle V_{CG}^{C}(r_{1},\;r_{2})$ $\displaystyle=$
$\displaystyle-\left(\frac{g_{4}^{4}}{f_{\pi}^{4}}\right)\left[\frac{1}{\pi}\int_{0}^{\infty}\frac{d\lambda(\Delta_{1}^{2}-\lambda^{2})}{(\Delta_{1}^{2}+\lambda^{2})^{2}}O_{1}(\textbf{k}_{1},\textbf{k}_{2})F(\lambda,r_{1})F(\lambda,r_{2})\right]$
(51) $\displaystyle=$ $\displaystyle\sum_{i}V_{CGi}^{C}+\cdots\;.$
Here, the ellipsis represents the high singular terms in $r_{2}\rightarrow
r_{1}=r$ limit, which behave as higher order corrections to the potential and
will not be taken into account in this work, but will be discussed elsewhere.
The central potential of Eq. (50) is obtained in the case of
spin-$\frac{3}{2}$ intermediate state, and the explicit forms of $V_{CGi}$
from box diagram are
$V_{CG1}^{B}=-\frac{g_{4}^{4}\Lambda^{7}}{128\sqrt{2}\pi^{7/2}f_{\pi}^{4}\Delta_{1}^{2}}e^{-\frac{\Lambda^{2}r^{2}}{2}}\;,$
(52)
$V_{CG2}^{B}=-\frac{g_{4}^{4}\Lambda^{5}}{16\sqrt{2}\pi^{7/2}f_{\pi}^{4}\Delta_{1}^{2}r^{2}}e^{-\frac{\Lambda^{2}r^{2}}{2}}\;,$
(53)
$V_{CG3}^{B}=\frac{g_{4}^{4}\Lambda^{3}m^{5/2}e^{m^{2}/\Lambda^{2}}}{32\sqrt{2}\pi^{3}f_{\pi}^{4}\Delta_{1}^{2}r^{3/2}}e^{-\frac{\Lambda^{2}r^{2}}{4}-mr}\;,$
(54)
$V_{CG4}^{B}=\frac{g_{4}^{4}\Lambda^{3}m^{3/2}e^{m^{2}/\Lambda^{2}}}{16\sqrt{2}\pi^{3}f_{\pi}^{4}\Delta_{1}^{2}r^{5/2}}e^{-\frac{\Lambda^{2}r^{2}}{4}-mr}-\frac{g_{4}^{4}m^{9/2}e^{2m^{2}/\Lambda^{2}}}{128\pi^{5/2}f_{\pi}^{4}\Delta_{1}^{2}r^{5/2}}e^{-2mr}\;.$
(55)
With Gaussian form factors it is seen from Eq. (76) in the Appendix that for a
given $\Lambda$ the function $F(\lambda,r)$ is suppressed for large $\lambda$
values, that is the dominant contribution to potential comes from the small
$\lambda$ region. So, in obtaining the analytic expressions of above
potentials and hereafter, we expand the corresponding functions, as defined in
the Appendix, in $\lambda$ and keep only the leading term. In this approach,
the crossed diagram contributes to the potential the same as the box diagram
at the leading order in $\lambda$ expansion, and hence is not presented here.
Similarly, we obtain the potentials from triangle and two-pion loop diagrams,
i.e.,
$\displaystyle V_{CG5}^{T}$ $\displaystyle=$
$\displaystyle\frac{g_{4}^{2}m\Lambda^{3}}{32\sqrt{2}\pi^{7/2}f_{\pi}^{4}\Delta_{1}r^{2}}e^{-\frac{\Lambda^{2}r^{2}}{2}}-\frac{g_{4}^{2}m^{5/2}\Lambda
e^{m^{2}/\Lambda^{2}}}{16\sqrt{2}\pi^{3}f_{\pi}^{4}\Delta_{1}r^{5/2}}e^{-\frac{\Lambda^{2}r^{2}}{4}-mr}$
(56) $\displaystyle+$
$\displaystyle\frac{g_{4}^{2}m^{7/2}e^{2m^{2}/\Lambda^{2}}}{128\pi^{5/2}f_{\pi}^{4}\Delta_{1}r^{5/2}}e^{-2mr}\;,$
and
$V_{CG6}^{L}=-\frac{m^{1/2}\Lambda^{3}}{32\sqrt{2}\pi^{2}f_{\pi}^{4}r^{3/2}}e^{-\frac{1}{4}\Lambda^{2}r^{2}-mr}\;.$
(57)
To get the central potential for the case of spin-$\frac{1}{2}$ intermediate
state, one needs only to make the following replacement
$\displaystyle g_{4}\rightarrow
g_{2}\;,\;\Delta_{1}\rightarrow\Delta^{\prime}_{1}=M_{\Sigma_{c}}-M_{\Lambda_{c}}$
(58)
in Eq.(50).
Note that in above asymptotic expressions we keep only those terms up to order
$\frac{1}{r^{5/2}}$, and more singular terms are not taken into accounted in
this work. The dependence of potential with various parameters are shown in
Figure 3. The results indicate that the potential approaches to zero quickly
in long range in every case, while in short range the potential diverges very
much with different parameters, as expected. As a result, the binding energy
heavily depends on input parameters, the coupling constants and cutoff. One
can read from the figure that in the small coupling situation, the potential
becomes too narrow and shallow to bind two heavy baryons. Table 2 presents the
binding energies of $\Lambda_{c}$-$\bar{\Lambda}_{c}$ and
$\Sigma_{c}$-$\bar{\Sigma}_{c}$ systems with different inputs. Schematically,
the radial wave functions for the ground state of
$\Lambda_{c}$-$\bar{\Lambda}_{c}$ system with Gaussian and monopole form
factors are shown in Figure 4 respectively, while the wave functions for
$\Sigma_{c}$-$\bar{\Sigma}_{c}$ system exhibit similar curves.
Figure 3: The $\Lambda_{c}$-$\bar{\Lambda}_{c}$ central potential behavior in
case of Gaussian form factor versus different parameter choices. Table 2:
Binding energies with different inputs with Gaussian form factor. The left
table is for the $\Lambda_{c}$-$\bar{\Lambda}_{c}$ system, and the right one
for $\Sigma_{c}$-$\bar{\Sigma}_{c}$ system.
$|g_{2}|$ | $\Lambda(\mathrm{GeV})$ | Binding | Baryonium
---|---|---|---
| | energy | mass
$<$0.9 | $<$0.6 | No | -
0.9 | 0.6 | -22 MeV | 4.550 GeV
0.95 | 0.6 | -77 MeV | 4.495 GeV
1.0 | 0.6 | -168 MeV | 4.404 GeV
0.95 | 0.7 | -196 MeV | 4.376 GeV
0.95 | 0.8 | -227 MeV | 4.345 GeV
0.95 | 0.9 | -588 MeV | 3.984 GeV
$g_{1}$ | $\Lambda(\mathrm{GeV})$ | Binding | Baryonium
---|---|---|---
| | energy | mass
$<1.0$ | $<0.8$ | No | -
1.0 | 0.8 | -11 MeV | 4.895 GeV
1.05 | 0.8 | -61 MeV | 4.845 GeV
1.1 | 0.8 | -145 MeV | 4.761 GeV
1.05 | 0.85 | -141 MeV | 4.765 GeV
1.05 | 0.9 | -266 MeV | 4.640 GeV
1.05 | 0.95 | -438 MeV | 4.468 GeV
Figure 4: Radial wave function of $\Lambda_{c}$-$\bar{\Lambda}_{c}$ ground
state. The left figure is for case of Gaussian form factor under the condition
of $|g_{2}|=0.95$ and $\Lambda=0.8$, and the right one is for the case of
monopole form factor with $|g_{2}|=0.9$ and $\Lambda=0.95$.
### 3.2 Monopole form factor case
In order to regulate the singularities at the origin in configuration space,
usually people employ three types form factors in the literature, i.e. the
Gaussian, the monopole, and the dipole form factors [34]. For comparison we
also calculate the potential with monopole form factor using the same
factorization technique, and the basic Fourier transformation for monopole
form factor is presented in Appendix for the sake of convenience. Here, in
obtaining the analytic expressions for potentials we also take the measure of
expanding the corresponding functions in parameter $\lambda$ and keeping only
the leading term. Then, what obtained from the box-diagram contribution reads
$\displaystyle V_{CM}^{B}(r)=$ $\displaystyle-$
$\displaystyle\frac{g_{4}^{4}}{8\pi^{5/2}f_{\pi}^{4}\Delta^{2}r^{5/2}}\left(\frac{m^{9/2}}{4}e^{-2mr}+\frac{\Lambda^{4}m^{1/2}}{4}e^{-2\Lambda
r}\right)$ (59) $\displaystyle+$
$\displaystyle\frac{g_{4}^{4}\Lambda^{5/2}m^{5/2}}{8\sqrt{2}\pi^{5/2}f_{\pi}^{4}\sqrt{m+\Lambda}\Delta_{1}^{2}r^{5/2}}e^{-(m+\Lambda)r}\;.$
Contributions from triangle and two-pion loop diagrams are
$\displaystyle V_{CM}^{T}(r)$ $\displaystyle=$
$\displaystyle\frac{g_{4}^{2}m^{7/2}}{16\pi^{5/2}f_{\pi}^{4}\Delta_{1}r^{5/2}}e^{-2mr}+\frac{g_{4}^{2}m\Lambda^{5/2}}{16\pi^{5/2}f_{\pi}^{4}\Delta_{1}r^{5/2}}e^{-2\Lambda
r}$ (60) $\displaystyle-$
$\displaystyle\frac{g_{4}^{2}m^{5/2}\Lambda^{3/2}}{4\sqrt{2}\pi^{5/2}f_{\pi}^{4}\sqrt{m+\Lambda}\Delta_{1}r^{5/2}}e^{-(m+\Lambda)r}\;$
and
$V_{CM}^{L}(r)=-\frac{(\Lambda^{2}-m^{2})m^{1/2}}{32\sqrt{2}\pi^{3/2}f_{\pi}^{4}r^{3/2}}e^{-(m+\Lambda)r}+\frac{(\Lambda^{2}-m^{2})\Lambda^{1/2}}{32\sqrt{2}\pi^{3/2}f_{\pi}^{4}r^{3/2}}e^{-2\Lambda
r}\;$ (61)
respectively, where superscript $B$, $T$, and $L$ stand for box, triangle and
$2\pi$ loop. Note that since there is no heavy baryon intermediate state in
the $2\pi$ loop process, as shown in the right graph of Figure 2, the
potential range of it appears different.
Figure 5: The $\Lambda_{c}$-$\bar{\Lambda}_{c}$ central potential behavior in
case of monopole form factor versus different choices of inputs.
We find that the structure of potential with monopole form factor is much
simpler than the Gaussian case. The dependence of potential with various
parameters are shown in Fig.5. From the figure one can see that in small
coupling case the potential change less, which means the potential tends to be
insensitive to the small coupling, and hence the binding energy. Solving the
Schrödinger equation we then obtain eigenvalues for different input
parameters, given in Table 3. From the table, we notice that the binding
energy is sensitive to and changes greatly with the variation of $g_{1}$,
$|g_{2}|$ and the cutoff $\Lambda$, the same as the case with Gaussian form
factor. Intuitively, the realistic baryonium can only accommodate small ones
of those parameters.
Table 3: Binding energies with different inputs with monopole form factor. The
left table is for the $\Lambda_{c}$-$\bar{\Lambda}_{c}$ system, and the right
one for $\Sigma_{c}$-$\bar{\Sigma}_{c}$ system.
$|g_{2}|$ | $\Lambda(\mathrm{GeV})$ | Binding | Baryonium
---|---|---|---
| | energy | mass
$<$0.7 | $<$0.9 | No | -
0.8 | 0.95 | -117 MeV | 4.455 GeV
0.85 | 0.95 | -420 MeV | 4.152 GeV
0.9 | 0.95 | -521 MeV | 4.051 GeV
0.7 | 0.9 | -5 MeV | 4.567 GeV
0.7 | 0.95 | -67 MeV | 4.505 GeV
0.7 | 1.0 | -252 MeV | 4.320 GeV
$g_{1}$ | $\Lambda(\mathrm{GeV})$ | Binding | Baryonium
---|---|---|---
| | energy | mass
$<0.9$ | $<0.9$ | No | -
0.95 | 0.95 | -438 MeV | 4.468 GeV
1.0 | 0.95 | -830 MeV | 4.076 GeV
1.05 | 0.95 | -1003 MeV | 3.903 GeV
0.9 | 0.9 | -40 MeV | 4.866 GeV
0.9 | 0.95 | -153 MeV | 4.753 GeV
0.9 | 1.0 | -345 MeV | 4.561 GeV
### 3.3 Ground state of $\Lambda_{b}$-$\bar{\Lambda}_{b}$ baryonium
Table 4: Binding energies with the change of parameters for
$\Lambda_{b}$-$\bar{\Lambda}_{b}$ system. The left table is for the Gaussian
form factor, and the right one for the monopole form factor. Here $g_{b}$
corresponds to $g_{2}$ in charmed baryonium sector
$|g_{b}|$ | $\Lambda(\mathrm{GeV})$ | binding | Baryonium
---|---|---|---
| | energy | mass
$<$0.7 | $<$0.7 | No | No
0.7 | 0.75 | -4 MeV | 11.236 GeV
0.8 | 0.75 | -76 MeV | 11.164 GeV
0.9 | 0.75 | -294 MeV | 10.946 GeV
0.8 | 0.8 | -164 MeV | 11.706 GeV
0.8 | 0.9 | -396 MeV | 10.844 GeV
0.8 | 1.0 | -622 MeV | 10.618 GeV
$|g_{b}|$ | $\Lambda(\mathrm{MeV})$ | Binding | Baryonium
---|---|---|---
| | energy | mass
$<1.0$ | $<0.8$ | No | No
1.0 | 0.8 | -11 MeV | 11.229 GeV
1.05 | 0.8 | -56 Mev | 11.184 GeV
1.1 | 0.8 | -143 MeV | 11.097 GeV
1.05 | 0.8 | -103 Mev | 11.137 GeV
1.05 | 0.9 | -164 MeV | 11.076 GeV
1.05 | 1.0 | -321 MeV | 10.919 GeV
We also estimate the ground state of $\Lambda_{b}$-$\bar{\Lambda}_{b}$
baryonium system with Gaussian and monopole form factors. The result are shown
in Table 4, where $g_{b}$ corresponds to $g_{2}$ in charmed baryonium sector.
Note that since the dominant decay mode of $\Sigma_{b}$ is to
$\Lambda_{b}\pi$, by which we may constrain the $\Sigma_{b}\Lambda_{b}\pi$
coupling from the experiment result, and this may shed lights on the further
investigation on the nature of possible baryonium.
## 4 Summary and Conclusions
In the framework of heavy baryon chiral perturbation theory we have studied
the heavy baryon-baryon interaction, and obtained the interaction potential,
the central potential, in the case of two-pion exchange. The Gaussian and
monopole types form factors are employed to regularized the loop integrals in
the calculation. As a leading order analysis, the tensor potential and higher
order contributions in $\frac{1}{M_{H}}$ expansion are neglected. As expected,
we found that the potential is sensitive to the baryon-pion couplings and the
energy cutoff $\Lambda$ used in the form factor.
We apply the obtained potential to the Schrödinger equation in attempting to
see whether the attraction of two-pion-exchange potential is large enough to
constrain two heavy baryons into a baryonium. We find it true for a reasonable
choice of cutoff $\Lambda$ and baryon-pion couplings, which is quite different
from the conclusion of a recent work in the study of $D\bar{D}$ potential
through two-pion exchange [35]. Since usually the cutoff $\Lambda$ is taken to
be less than the nucleon mass, i.e. about 1 GeV in the literature, in our
calculation we adopt a similar value employed in the nucleon-nucleon case. In
Ref. [35] authors took a fixed coupling $g=0.59$ and obtained the binding with
a large cutoff. While in our calculation for the baryonium system with
Gaussian form factor, there will be no binding in case $g_{1}<1.0$ and
$\Lambda<0.8$. The increase of coupling constant will lead to an even smaller
$\Lambda$ for a given binding energy.
Based on our calculation results it is interesting to note that in case there
exists binding in $\Sigma_{c}$-$\bar{\Sigma}_{c}$ system, with both Gaussian
and monopole factors, the coupling $g_{1}$ will be much bigger than what
conjectured in Ref. [25]. However, for $\Lambda_{c}$-$\bar{\Lambda}_{c}$
system, to form a bound state the baryon-Goldstone coupling $g_{2}$ could be
similar in magnitude as what estimated in the literature.
Notice that the potential depends not only on coupling constants and cutoff
$\Lambda$, it also depends on the types of form factors employed. Our
calculation indicates that the Gaussian form factor and Monopole form factor
are similar in regulating the singularities at origin, and lead to similar
results, with only subtle difference, for both $\Lambda_{c}$ and $\Lambda_{b}$
systems. Numerical result tells that the heavy baryon-baryon potentials are
more sensitive to the coupling constants in the case of Monopole form factor,
but more sensitive to the cutoff $\Lambda$ in the case of Gaussian form
factor. From our calculation it is tempting to conjecture that the recently
observed states $Y(4260)$ and $Y(4360)$, but not $Y(4660)$ [6], in charm
sector could be a $\Lambda_{c}$-$\bar{\Lambda}_{c}$ bound state with
reasonable amount of binding energy, which deserves a further investigation.
Our result also tells that the newly observed “exotic” state in bottom sector,
the $Y_{b}(10890)$ [37], could be treated as the
$\Lambda_{b}$-$\bar{\Lambda}_{b}$ bound state, whereas with an extremely large
binding energy.
It is worth emphasizing at this point that although our calculation result
favors the existence of heavy baryonium, it is still hard to make a definite
conclusion yet, especially with only the leading order two-pion-exchange
potential. The potential sensitivity on coupling constants and energy cutoff
also looks unusual and asks for further investigation. To be more closer to
the truth, one needs to go beyond the leading order of accuracy in
$\frac{1}{M_{H}}$ expansion; one should also investigate the potential while
two baryon-like triquark clusters carry colors as proposed in the heavy
baryonium model [11, 16]; last, but not least, the unknown and difficult to
evaluate annihilation channel effect on the heavy baryonium potential should
also be clarified, especially for heavy baryon-antibaryon interaction, which
nevertheless could be phenomenologically parameterized so to reproduce known
widths of some observed states.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of
China(NSFC) and by the CAS Key Projects KJCX2-yw-N29 and H92A0200S2.
Appendix
In this Appendix, we present more detailed formulas and definitions used for
the sake of reader’s convenience.
The $\gamma$ matrices take the following convention
$\gamma^{0}=\left(\begin{array}[]{ll}1&0\\\
0&-1\end{array}\right)\;,\;\gamma^{i}=\left(\begin{array}[]{ll}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right)\;,\;\gamma_{5}=\left(\begin{array}[]{ll}0&1\\\
1&0\end{array}\right)\;.$ (62)
And the Dirac spinors for $\Sigma_{c}$ read as
$\displaystyle
u(p)=\sqrt{\frac{E+M_{\Sigma}}{2M_{\Sigma}}}\left(\begin{array}[]{l}\chi_{a}\\\
\frac{\boldsymbol{\sigma}\cdot\textbf{p}}{E+M_{\Sigma}}\chi_{a}\end{array}\right)\;,$
(65)
where $\chi_{a}$ is two-component Pauli spinor, and
$\displaystyle
v(p)=\sqrt{\frac{E+M_{\Sigma}}{2M_{\Sigma}}}\left(\begin{array}[]{l}\frac{\boldsymbol{\sigma}\cdot\textbf{p}}{E+M_{\Sigma}}\eta_{a}\\\
\eta_{a}\end{array}\right)\;,$ (68)
where $\eta_{a}=-i\sigma^{2}\chi_{a}^{*}$, and $a=1,2$. Spin-$\frac{3}{2}$
field for $\Sigma^{+*}$ is described by Rarita-Schwinger spinor
$u^{\mu}(p\;,\sigma)$, which can be constructed by spin-$1$ vector and
spin-$\frac{1}{2}$ field [36], that is
$u^{\mu}=\sqrt{\frac{E+M_{\Sigma^{+*}}}{2M_{\Sigma^{+*}}}}L^{(1)}(p)^{\mu}_{\nu}\left(\begin{array}[]{l}1\\\
\frac{\boldsymbol{\sigma}\cdot\textbf{p}}{E+M_{\Sigma^{+*}}}\end{array}\right)S^{\dagger\nu}\psi(\sigma)\;,$
(69)
where $\psi(\sigma)$ is four-component Pauli spinor of a spin-$\frac{3}{2}$
particle, and $L^{(1)}(p)^{\mu}_{\nu}$ is the boost operator for spin-$1$
particle,
$L^{(1)}(p)^{\mu}_{\nu}=\left(\begin{array}[]{ll}\frac{E}{M_{\Sigma^{+*}}}&\hskip
42.67912pt\frac{p_{j}}{M_{\Sigma^{+*}}}\\\
\frac{p_{i}}{M_{\Sigma^{+*}}}&\delta^{i}_{j}-\frac{p^{i}p_{j}}{M_{\Sigma^{+*}}(E+M_{\Sigma^{+*}})}\end{array}\right)\;,$
(70)
where $i,j$ are indices of the space components of momentum $p$. The positive-
and negative-energy projection operators for spin-$\frac{1}{2}$ baryon are
$\displaystyle[\Lambda^{+}(p)]_{\alpha\beta}=\sum_{{\pm}s}u_{\alpha}(p,s)\overline{u}_{\beta}(p,s)=\left(\frac{p\\!\\!\\!/+M_{\Sigma_{c}}}{2M_{\Sigma_{c}}}\right)_{\alpha\beta}\;$
(71)
and
$\displaystyle[\Lambda^{-}(p)]_{\alpha\beta}=-\sum_{{\pm}s}v_{\alpha}(p,s)\overline{v}_{\beta}(p,s)=\left(\frac{-p\\!\\!\\!/+M_{\Sigma_{c}}}{2M_{\Sigma_{c}}}\right)_{\alpha\beta}\;,$
(72)
respectively.
The positive- and negative-energy projection operators for spin-$\frac{3}{2}$
baryon are
$\displaystyle\left[\Lambda^{+}_{\mu\nu}(p)\right]_{\alpha\beta}$
$\displaystyle=\sum_{{\pm}s}u_{\mu,\;\alpha}(p,s)\overline{u}_{\nu,\;\beta}(p,s)$
(73)
$\displaystyle=[\frac{p\\!\\!\\!/+M_{\Sigma_{c}^{*}}}{2M_{\Sigma_{c}^{*}}}]_{\alpha\beta}\left(g_{\mu\nu}-\frac{\gamma_{\mu}\gamma_{\nu}}{3}-\frac{2p_{\mu}p_{\nu}}{3M_{\Sigma_{c}^{*}}^{2}}+\frac{p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu}}{3M_{\Sigma_{c}^{*}}}\right)\;,$
and
$\displaystyle\left[\Lambda^{-}_{\mu\nu}(p)\right]_{\alpha\beta}$
$\displaystyle=-\sum_{{\pm}s}v_{\mu,\;\alpha}(p,s)\overline{v}_{\nu,\;\beta}(p,s)$
(74)
$\displaystyle=[\frac{-p\\!\\!\\!/+M_{\Sigma_{c}^{*}}}{2M_{\Sigma_{c}^{*}}}]_{\alpha\beta}\left(g_{\mu\nu}-\frac{\gamma_{\mu}\gamma_{\nu}}{3}-\frac{2p_{\mu}p_{\nu}}{3M_{\Sigma_{c}^{*}}^{2}}+\frac{p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu}}{3M_{\Sigma_{c}^{*}}}\right)\;,$
respectively. Here, $\mu$ and $\nu$ are Lorentz indices; $\alpha$ and $\beta$
are Dirac spinor indices.
The basic Fourier transformation with Gaussian form factor reads
$\displaystyle I_{2}(m,\;r)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d^{3}\textbf{k}}{(2\pi)^{3}}\frac{e^{i\textbf{kr}}e^{-\textbf{k}^{2}/\Lambda^{2}}}{\textbf{k}^{2}+m^{2}}$
(75) $\displaystyle=$ $\displaystyle\frac{1}{8\pi
r}e^{m^{2}/\Lambda^{2}}\left[e^{-mr}erfc\left(-\frac{\Lambda
r}{2}+\frac{m}{\Lambda}\right)-e^{mr}erfc\left(\frac{\Lambda
r}{2}+\frac{m}{\Lambda}\right)\right]\;,$
and hence
$\displaystyle
F(\lambda,\;r)=\int\frac{d^{3}\textbf{k}}{(2\pi)^{3}}\frac{e^{i\textbf{kr}}e^{-\textbf{k}^{2}/\Lambda^{2}}}{\textbf{k}^{2}+m^{2}+\lambda^{2}}=I_{2}(\sqrt{m^{2}+\lambda^{2}},\;r)e^{-\lambda^{2}/\Lambda^{2}}\;.$
(76)
$erfc(x)$ is complementary error function, which is defined as
$erfc(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}dt\;.$ (77)
The factorization in double Fourier transformation goes like
$\displaystyle H_{11}$ $\displaystyle=$
$\displaystyle\int\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{(2\pi)^{6}}\frac{e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})}{\omega_{1}\omega_{2}(\omega_{1}+a)(\omega_{2}+a)(\omega_{1}+\omega_{2})}$
$\displaystyle=$
$\displaystyle\int\int\frac{d^{3}\textbf{k}_{1}d^{3}\textbf{k}_{2}}{(2\pi)^{6}}\frac{1}{a^{2}}[\frac{2}{\pi}\int_{0}^{\infty}\frac{e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})d\lambda}{(\omega_{1}^{2}+\lambda^{2})(\omega_{2}^{2}+\lambda^{2})}$
$\displaystyle-$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}\frac{e^{i\textbf{k}_{1}\textbf{r}_{1}}e^{i\textbf{k}_{2}\textbf{r}_{2}}f(\textbf{k}_{1}^{2})f(\textbf{k}_{2}^{2})\lambda^{2}d\lambda}{(a^{2}+\lambda^{2})(\omega_{1}^{2}+\lambda^{2})(\omega_{2}^{2}+\lambda^{2})}]-\frac{1}{a}G_{11}(\lambda,\;r_{1})G_{11}(\lambda,\;r_{2})$
$\displaystyle=$
$\displaystyle\frac{2}{\pi}\int_{0}^{\infty}\frac{d\lambda}{a^{2}+\lambda^{2}}F(\lambda,\;r_{1})F(\lambda,\;r_{2})-\frac{1}{a}G_{11}(\lambda,\;r_{1})G_{11}(\lambda,\;r_{2})\;.$
(79)
Here,
$\displaystyle G_{11}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\textbf{k}_{1}}{(2\pi)^{3}}\frac{e^{i\textbf{k}_{1}\textbf{r}}e^{-\textbf{k}_{1}^{2}/\Lambda^{2}}}{\omega_{1}(\omega_{1}+a)}=\int\frac{d^{3}\textbf{k}_{1}}{(2\pi)^{3}}\frac{2a}{\pi}\int_{0}^{\infty}\frac{e^{i\textbf{k}_{1}\textbf{r}}e^{-\textbf{k}_{1}^{2}/\Lambda^{2}}d\lambda}{(a^{2}+\lambda^{2})(\omega_{1}^{2}+\lambda^{2})}$
(80) $\displaystyle=$
$\displaystyle\frac{2a}{\pi}\int_{0}^{\infty}\frac{d\lambda}{(a^{2}+\lambda^{2})}F(\lambda,\;r)\;,$
and for simplicity we define $\omega_{1}=\sqrt{\textbf{k}_{1}^{2}+m^{2}}$ and
$\omega_{2}=\sqrt{\textbf{k}_{2}^{2}+m^{2}}$ .
In the case of the monopole form factor, i.e.
$f(\textbf{k}^{2})=\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\textbf{k}^{2}}$, the
corresponding function to $F(\lambda,\;r)$ reads
$\displaystyle R(\lambda,\;r)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\textbf{k}}{(2\pi)^{3}}\frac{e^{i\textbf{kr}}}{\textbf{k}^{2}+m^{2}+\lambda^{2}}\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}+\textbf{k}^{2}+\lambda^{2}}$
(81) $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r}\left(e^{-r\sqrt{m^{2}+\lambda^{2}}}-e^{-r\sqrt{\Lambda^{2}+\lambda^{2}}}\right)\;.$
Operator $O_{1}(\textbf{k}_{1},\;\textbf{k}_{2})$ contains two parts. The
first part of $O_{1}(\textbf{k}_{1},\;\textbf{k}_{2})$ while acting on
functions in configuration space goes like
$\displaystyle
O_{1}(\textbf{k}_{1},\;\textbf{k}_{2})F(\lambda,\;r_{1})F(\lambda,\;r_{2})$
$\displaystyle=$
$\displaystyle(\textbf{k}_{1}\cdot\textbf{k}_{2})^{2}F(\lambda,\;r_{1})F(\lambda,\;r_{2})$
(82) $\displaystyle=$
$\displaystyle(\nabla_{1i}\nabla_{1j})F(\lambda,\;r_{1})(\nabla_{2i}\nabla_{2j})F(\lambda,\;r_{2})$
$\displaystyle=$
$\displaystyle\frac{2}{r^{2}}F^{\prime}(\lambda,\;r)F^{\prime}(\lambda,\;r)+F^{\prime\prime}(\lambda,\;r)F^{\prime\prime}(\lambda,\;r)\;,$
where
$\nabla_{i}\nabla_{j}=\left(\delta_{ij}-\frac{x_{i}x_{j}}{r^{2}}\right)\left(\frac{1}{r}\frac{d}{dr}\right)+\frac{x_{i}x_{j}}{r^{2}}\left(\frac{d^{2}}{dr^{2}}\right)\;,$
(83)
and the limit $r_{2}\rightarrow r_{1}=r$ is taken. The second part of
$O_{2}(\textbf{k}_{1},\;\textbf{k}_{2})$ while acting on functions in
configuration space goes like
$\displaystyle
O_{2}(\textbf{k}_{1},\;\textbf{k}_{2})F(\lambda,\;r_{1})F(\lambda,\;r_{2})$
$\displaystyle=$
$\displaystyle(\boldsymbol{\sigma}_{1}\cdot\textbf{k}_{1}\times\textbf{k}_{2})(\boldsymbol{\sigma}_{2}\cdot\textbf{k}_{1}\times\textbf{k}_{2})F(\lambda,\;r_{1})F(\lambda,\;r_{2})$
(84) $\displaystyle=$
$\displaystyle\sigma_{1i}\sigma_{2j}\varepsilon_{ikl}\varepsilon_{jmn}(\nabla_{1k}\nabla_{1m})F(\lambda,\;r_{1})(\nabla_{2l}\nabla_{2n})F(\lambda,\;r_{2})$
$\displaystyle=$
$\displaystyle\sigma_{1i}\sigma_{2j}(\delta_{ij}\delta_{km}\delta_{ln}+\delta_{im}\delta_{kn}\delta_{lj}+\delta_{in}\delta_{lm}\delta_{kj}$
$\displaystyle-\delta_{lj}\delta_{km}\delta_{in}-\delta_{lm}\delta_{kn}\delta_{ij}-\delta_{ln}\delta_{im}\delta_{kj})\times$
$\displaystyle(\nabla_{1k}\nabla_{1m})F(\lambda,\;r_{1})(\nabla_{2l}\nabla_{2n})F(\lambda,\;r_{2})$
$\displaystyle=$
$\displaystyle\frac{2}{3}\left[\frac{1}{r^{2}}F^{\prime}(\lambda,\;r)F^{\prime}(\lambda,\;r)+\frac{2}{r}F^{\prime}(\lambda,\;r)F^{\prime\prime}(\lambda,\;r)\right](\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2})$
$\displaystyle+\frac{2}{3}\left(\frac{F^{\prime}(\lambda,\;r)}{r}-F^{\prime\prime}(\lambda,\;r)\right)\frac{1}{r}F^{\prime}(\lambda,\;r)S_{12}\;,$
where $\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$ gives spin-spin
potential and
$S_{12}=\frac{3(\boldsymbol{\sigma}_{1}\cdot\textbf{r})(\boldsymbol{\sigma}_{2}\cdot\textbf{r})}{r^{2}}-\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$
gives the tensor potential.
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|
arxiv-papers
| 2011-02-17T03:38:48 |
2024-09-04T02:49:17.064487
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. D. Chen and C. F. Qiao",
"submitter": "Yuede Chen",
"url": "https://arxiv.org/abs/1102.3487"
}
|
1102.3510
|
# Pair Photoproduction in Constant and Homogeneous Electromagnetic Fields
V.M. Katkov
Budker Institute of Nuclear Physics,
Novosibirsk, 630090, Russia
e-mail: katkov@inp.nsk.su
###### Abstract
The process of pair creation by a photon in a constant and homogeneous
electromagnetic field of an arbitrary configuration is investigating. At high
energy the correction to the standard quasiclassical approximation (SQA) has
been calculated. In the region of intermediate photon energies where SQA is
inapplicable the new approximation, developed recently by authors, is used.
The influence of weak electric field on the process in a magnetic field is
considered. In particular, in the presence of electric field the root
divergence in the probability of pair creation on the Landau energy levels is
vanished. For smaller photon energies the low energy approximation is used.
The found probability describes the absorption of soft photon by particles
created by field. At low photon energy the electric field action dominates and
the influence of magnetic field on the process is connected with the
interaction of it and the magnetic moment of creating particles.
## 1 Introduction
The pair photoproduction in an electromagnetic field is the basic QED reaction
which can play the significant role in many processes.
This process was considered first in a magnetic field. The investigation was
started in 1952 independently by Klepikov and Toll [1, 2]. In Klepikov’s paper
[3], which was based on the solution of the Dirac equation, the probability of
photoproduction had been obtained on the mass shell ( $k^{2}=0,k$ is the
4-momentum of photon. We use the system of units with $\hbar=c=1$ and the
metric $ab=a^{\mu}b_{\mu}=a^{0}b^{0}-\boldsymbol{ab}$). In 1971 Adler [4] had
calculated the photon polarization operator in a magnetic field using the
proper-time technique developed by Schwinger [5] and Batalin and Shabad [6]
had calculated this operator in an electromagnetic field using the Green
function found by Schwinger [5]. In 1975 the contribution of charged-particles
loop in an electromagnetic field with $n$ external photon lines had been
calculated in [7]. For $n=2$ the explicit expressions for the contribution of
scalar and spinor particles to the polarization operator of photon were given
in this work. Making use of the imaginary part of this operator for spinor
particles the pair photoproduction probability was analyzed in the pure
magnetic [8] and the pure electric [9] field.
The probability of pair photoproduction in a constant and homogeneous electric
field in the quasiclassical approximation had been found by Narozhny [10]
using the solution of the Dirac equation in the Sauter potential [11].
Nikishov [12] had obtained the differential distribution of this process also
using the solution of the Dirac equation in the indicated field.
In the present paper we consider the integral probability of pair creation in
a constant and homogeneous electromagnetic field of an arbitrary configuration
basing on the polarization operator [7]. In Sec.2 the exact expression for
this probability has been received for the general case $k^{2}\neq 0.$In Sec.3
the standard quasiclassical approximation (SQA) is outlined for the high-
energy photon $\omega\gg m$ ( $m$ is the electron mass). The corrections to
SQA, determined also the applicability region of SQA, have been calculated.
The found expressions, given in the Lorentz invariant form, contain two
invariant parameters. In Sec.4 the new approach has been developed for the
relatively low energies where SQA is not applicable. This approach is based on
the method proposed in [8]. The obtained probability is valid in the wide
interval of photon energy, which is overlapped with SQA. In Sec.5 the case of
the ”nonrelativistic” photon $\omega\ll m$ is analyzed. In particular, in the
energy region $\omega\lesssim$ $eE/m$ where the previous approach is
inapplicable, the low energy approximation has been developed basing on the
analysis in [9]. In tern the found results have an overlapping region of
applicability with the previous approach. So just as in [9] we have three
overlapping approximations which include all photon energies. At the photon
energy $\omega\ll$ $eE/m$ the probability has been found for arbitrary values
of fields $E$ and $H.$
## 2 General expressions for the probability of process
Our analysis is based on the general expression for the contribution of spinor
particles to the polarization operator obtained in a diagonal form in [7] (see
Eqs. (3.19), (3.33)). The imaginary part of the eigenvalue $\kappa_{i}$ of
this operator on the mass shell $(k^{2}=0)$ determines the probability per
unit length $W_{i}$ of $e^{-}e^{+}$ pair creation by the real photon with the
polarization $e_{i}$ directed along the corresponding eigenvector:
$W_{i}=-\frac{\mathrm{\operatorname{Im}}\kappa_{i}}{\omega};\ \
e_{i}^{\mu}=\frac{b_{i}^{\mu}}{\sqrt{-b_{i}^{2}}},~{}\ b_{2}^{\mu}=\
\left(Bk\right)^{\mu}+\frac{2\Omega_{4}}{\Omega}\left(Ck\right)^{\mu},\ $ (1)
$\
b_{3}^{\mu}=\left(Ck\right)^{\mu}-\frac{2\Omega_{4}}{\Omega}\left(Bk\right)^{\mu};$
$\displaystyle-\mathrm{\operatorname{Im}}\kappa_{2}$
$\displaystyle=r\left(\Omega_{2}-\frac{2\Omega_{4}^{2}}{\Omega}\right),\ \ \
-\mathrm{\operatorname{Im}}\
\kappa_{3}=r\left(\Omega_{3}+\frac{2\Omega_{4}^{2}}{\Omega}\right),\ \ $ (2)
$\displaystyle\ \Omega$
$\displaystyle=\Omega_{3}-\Omega_{2}+\sqrt{(\Omega_{3}-\Omega_{2})^{2}+4\Omega_{4}^{2}},\
\ r=\frac{\omega^{2}-k_{3}^{2}}{4m^{2}}.$
The consideration realizes in the frame where electric $\mathbf{E}$ and
magnetic $\mathbf{H}$ fields are parallel and directed along the axis 3\. In
this frame the tensor of electromagnetic field $F_{\mu\nu}$ and tensors
$F_{\mu\nu}^{\ast}$, $B_{\mu\nu}$ and $C_{\mu\nu}$ have a form
$\displaystyle F_{\mu\nu}$ $\displaystyle=C_{\mu\nu}E+B_{\mu\nu}H,\ \
F_{\mu\nu}^{\ast}=C_{\mu\nu}H-B_{\mu\nu}E,\ \
C_{\mu\nu}=g_{\mu}^{0}g_{\nu}^{3}-g_{\mu}^{3}g_{\nu}^{0},\ $ $\displaystyle\
B_{\mu\nu}$ $\displaystyle=g_{\mu}^{2}g_{\nu}^{1}-g_{\mu}^{1}g_{\nu}^{2};\ \
eE/m^{2}=E/E_{0}\equiv\nu,\ \ eH/m^{2}=H/H_{0}\equiv\mu;$ (3)
$\displaystyle\Omega_{i}$ $\displaystyle=\frac{\alpha
m^{2}}{2\pi\text{{i}}}\mu\nu\int\limits_{-1}^{1}\
dv\int\limits_{-\infty-\text{{i}0}}^{\infty-\text{{i}0 }}\
f_{i}(v,x)\exp(\text{{i}}\psi(v,x))xdx.$ (4)
Here
$\displaystyle f_{1}$ $\displaystyle=\frac{\cos(\mu xv)\cosh(\nu xv)}{\sin(\mu
x)\sinh(\nu x)}-\frac{\cos(\mu x)\cosh(\nu x)\sin(\mu xv)\sinh(\nu
xv)}{\sin^{2}(\mu x)\sinh^{2}(\nu x)},$ $\displaystyle f_{2}$
$\displaystyle=2\frac{\cosh(\nu x)(\cos(\mu x)-\cos(\mu xv))}{\sinh(\nu
x)\sin^{3}(\mu x)}+f_{1},\ $ $\displaystyle\ \ f_{3}$
$\displaystyle=2\frac{\cos(\mu x)(\cosh(\nu x)-\cosh(\nu xv))}{\sin(\mu
x)\sinh^{3}(\nu x)}-f_{1},$ $\displaystyle f_{4}$
$\displaystyle=\frac{\cos(\mu x)\cos(\mu xv)-1}{\sin^{2}(\mu
x)}\frac{\cosh(\nu x)\cosh(\nu xv)-1}{\sinh^{2}(\nu x)}$
$\displaystyle+\frac{\sin(\mu xv)\sinh(\nu xv)}{\sin(\mu x)\sinh(\nu x)};$ (5)
$\displaystyle\ \ \psi(v,x)$ $\displaystyle=2r\left(\frac{\cosh(\nu
x)-\cosh(\nu xv)}{\nu\sinh(\nu x)}+\frac{\cos(\mu x)-\cos(\mu xv)}{\mu\sin(\mu
x)}\right)-x.$ (6)
Let us note that the integration contour in Eq.(4) is passing slightly below
the real axis.
After all calculations have been fulfilled we can return to a covariant form
of the process description using the following expressions
$\displaystyle E^{2},H^{2}$
$\displaystyle=\left(\mathcal{F}^{2}+\mathcal{G}^{2}\right)^{1/2}\pm\mathcal{F,\
\ F=}\left(\mathbf{E}^{2}-\mathbf{H}^{2}\right)\diagup 2,\ \
\mathcal{G=}\mathbf{EH},\ $ $\displaystyle\left(C^{2}\right)_{\mu\nu}$
$\displaystyle=\left(F_{\mu\nu}^{2}+H^{2}g_{\mu\nu}\right)\diagup\left(E^{2}+H^{2}\right),\
\ \left(C^{2}\right)_{\mu\nu}-\left(B^{2}\right)_{\mu\nu}=g_{\mu\nu}.$ (7)
## 3 Quasiclassical approximation
The standard quasiclassical approximation (SQA) was developed first for a
magnetic field in [3], [13], [14]. The SQA is valid for ultrarelativistic
created particles ( $r\gg 1$) and can be derived from Eqs.(4)-(6) by expanding
the functions $f_{i}(v,x)$, $\psi(v,x)$ over $x$ powers. To get the correction
to the probability in SQA we shall keep leading to leading powers of $x$. We
have
$~{}\ b_{2}^{\mu}=\
\left(Bk\right)^{\mu}+\frac{\nu}{\mu}\left(Ck\right)^{\mu}\propto
F^{\mu\nu}k_{\nu},\ \
b_{3}^{\mu}=\left(Ck\right)^{\mu}-\frac{\nu}{\mu}\left(Bk\right)^{\mu}\propto
F^{\ast\mu\nu}k_{\nu};$ (8)
$\displaystyle-\mathrm{\operatorname{Im}}\kappa_{i}$
$\displaystyle=\mathrm{i}\frac{\alpha
m^{2}}{12\pi}r(\mu^{2}+\nu^{2})\int\limits_{-1}^{1}dv\left(1-v^{2}\right)\int\limits_{-\infty}^{\infty}h_{i}(v,x)\left[-\mathrm{i}\gamma\left(v,x\right)\right]xdx;$
$\displaystyle\gamma\left(v,x\right)$
$\displaystyle=x+\frac{x^{3}}{12}r\left(1-v^{2}\right)^{2}\left(\nu^{2}+\mu^{2}\right),$
(9) $\displaystyle h_{2}(v,x)$
$\displaystyle=\frac{3+v^{2}}{2}+\frac{1}{30}\left(15-6v^{2}-v^{4}\right)\left(\mu^{2}-\nu^{2}\right)x^{2}$
$\displaystyle-\frac{\mathrm{i}}{720}r(\mu^{2}+\nu^{2})\left(1-v^{2}\right)^{2}\left(9-v^{2}\right)\left(\mu^{2}-\nu^{2}\right)x^{5},$
$\displaystyle h_{3}(v,x)$
$\displaystyle=3-v^{2}+\frac{1}{60}\left(15-2v^{2}+3v^{4}\right)\left(\mu^{2}-\nu^{2}\right)x^{2}$
$\displaystyle-\frac{\mathrm{i}}{360}r(\mu^{2}+\nu^{2})\left(1-v^{2}\right)^{2}\left(3-v^{2}\right)^{2}\left(\mu^{2}-\nu^{2}\right)x^{5}.$
(10)
We are using the known integrals:
$\displaystyle\int\limits_{-\infty}^{\infty}\cos\left(x+\frac{ax^{3}}{3}\right)dx$
$\displaystyle=\frac{2}{\sqrt{3a}}K_{1/3}\left(\frac{2}{3\sqrt{a}}\right),\ \
$
$\displaystyle\int\limits_{-\infty}^{\infty}x\sin\left(x+\frac{ax^{3}}{3}\right)dx$
$\displaystyle=\frac{2}{\sqrt{3}a}K_{2/3}\left(\frac{2}{3\sqrt{a}}\right).$
(11)
Conserving the first (independent on $x)$ terms in Eq.(10) we obtain the
probabilities in SQA
$\displaystyle W_{i}^{(SQA)}$
$\displaystyle=-\frac{\mathrm{\operatorname{Im}}\kappa_{i}}{\omega}=\frac{\alpha
m^{2}}{3\sqrt{3}\pi\omega}\int\limits_{-1}^{1}\frac{s_{i}}{1-v^{2}}K_{2/3}\left(z\right)dv,\
\ z=\frac{8}{3\left(1-v^{2}\right)\kappa},\ $ $\displaystyle s_{2}$
$\displaystyle=2(3-v^{2}),\ \ \ s_{3}=3+v^{2},\ \ \
\kappa^{2}=4r(\mu^{2}+\nu^{2})=-\frac{e^{2}}{m^{6}}\left(F^{\mu\nu}k_{\nu}\right)^{2}.$
(12)
The correction to SQA has a form
$W_{i}^{(1)}=-\frac{\alpha
m^{2}\widetilde{\mathcal{F}}}{15\sqrt{3}\pi\omega\kappa}\int\limits_{-1}^{1}\frac{dv}{1-v^{2}}G_{i}\left(v,z\right),\
\
\widetilde{\mathcal{F}}=\frac{e^{2}\mathcal{F}}{m^{4}}=\frac{\nu^{2}-\mu^{2}}{2},$
(13)
where
$\displaystyle G_{2}\left(v,z\right)$
$\displaystyle=\left(36+4v^{2}-18z^{2}\right)K_{1/3}\left(z\right)+\left(3v^{2}-57\right)zK_{2/3}\left(z\right),$
$\displaystyle G_{3}\left(v,z\right)$
$\displaystyle=-\left(34+2v^{2}+36z^{2}\right)K_{1/3}\left(z\right)+\left(78-6v^{2}\right)zK_{2/3}\left(z\right).$
(14)
The mathematical transformations of integrals can be found in Appendix C [8].
It is seen that in this order of decomposition the correction does not depend
on the invariant parameter $\mathcal{G}$, because $\mathcal{G}$ is the
pseudoscalar. The asimptotic of the integrals incoming in the correction terms
have been given in the mentioned Appendix C. The asymptotic at $\kappa\ll 1$
will become necessary further
$W_{2}^{(1)}=\frac{4\alpha
m^{2}\widetilde{\mathcal{F}}}{5\omega\kappa^{2}}\sqrt{\frac{2}{3}}\exp\left(-\frac{8}{3\kappa}\right),\
\ W_{3}^{(1)}=2W_{2}^{(1)},\ \ \frac{\
W_{i}^{(1)}}{W_{i}^{(SQA)}}=\frac{64\widetilde{\mathcal{F}}}{15\kappa^{3}}.$
(15)
## 4 Region of intermediate photon energies
In the field which is weak comparing with the critical field $E/E_{0}=\nu\ll
1$ ( $E_{0}=1.32\cdot 10^{16}\operatorname{V}/\operatorname{cm}$), $H/H_{0}$ =
$\mu\ll 1$ $(H_{0}=4.41\cdot 10^{13}\operatorname{G})$ and at the relatively
low photon energies $r\lesssim\nu^{-2/3}$ the standard quasiclassical
approximation Eq.(12) is non-applicable. This follows from the last equality
in Eq.(15). For these energies, if the condition $r\gg\nu^{2}$ is fulfilled,
the method of stationary phase can be applied at integration over $x$ in
Eq.(4). In this case the small values of $v$ contribute to the integral over
$v$. So one can expand the phase $\psi(v,x)$ over $v$ and extend the
integration limit to the infinity. We get
$\Omega_{i}=\frac{\alpha
m^{2}}{2\pi\text{{i}}}\mu\nu\int\limits_{-\infty}^{\infty}\
dv\int\limits_{-\infty}^{\infty\text{ }}\
f_{i}(0,x)\exp\left\\{-\text{{i}}\left[\varphi\left(x\right)+v^{2}\chi\left(x\right)\right]\right\\}xdx,$
(16)
where
$\displaystyle\varphi\left(x\right)$
$\displaystyle=2r\left(\frac{1}{\mu}\tan\frac{\mu
x}{2}-\frac{1}{\nu}\tanh\frac{\nu x}{2}\right)+x,\ $ $\displaystyle\
\chi\left(x\right)$ $\displaystyle=rx^{2}\left(\frac{\nu}{\sinh(\nu
x)}-\frac{\mu}{\sin(\mu x)}\right).$ (17)
From the equation $\varphi^{\prime}(x_{0})=0$ we find the saddle point $x_{0}$
$\tan^{2}\frac{\nu s}{2}+\tanh^{2}\frac{\mu s}{2}=\frac{1}{r},\ \
x_{0}=-\mathrm{i}s.$ (18)
Substituting this value of $\rule{7.22743pt}{7.22743pt}_{0}$ in the
expressions determined the integrals in Eq.(16) we have
$\displaystyle\text{{i}}\varphi\left(x_{0}\right)$
$\displaystyle=2r\left(\frac{1}{\mu}\tanh\frac{\mu
s}{2}-\frac{1}{\nu}\tan\frac{\nu s}{2}\right)+s\equiv b(s),\ $ (19)
$\displaystyle\ \text{{i}}\chi\left(x_{0}\right)$
$\displaystyle=rs^{2}\left(\frac{\nu}{\sin(\nu s)}-\frac{\mu}{\sinh(\mu
s)}\right)\equiv\frac{1}{2}rs^{2}A(s),$ (20) $\displaystyle\
\text{{i}}\varphi^{\prime\prime}(x_{0})$
$\displaystyle=r\left[\nu\sin\frac{\nu s}{2}\diagup\cos^{3}\frac{\nu
s}{2}+\mu\sinh\frac{\mu s}{2}\diagup\cosh^{3}\frac{\mu s}{2}\right]\equiv
rD\left(s\right),$ (21)
$\displaystyle f_{2}(0,x_{0})$ $\displaystyle=\frac{1}{\sinh(\mu s)\sin(\nu
s)}\left[\cos(\nu s)\diagup\cosh^{2}\frac{\mu s}{2}-1\right]\equiv-a_{2}(s),\
$ $\displaystyle f_{3}(0,x_{0})$ $\displaystyle=\frac{1}{\sinh(\mu s)\sin(\nu
s)}\left[1-\cosh\mu s\diagup\cos^{2}\frac{\nu s}{2}\right]\equiv-a_{3}(s),\ $
$\displaystyle\ f_{4}(0,x_{0})$ $\displaystyle=-\left(4\cos^{2}\frac{\nu
s}{2}\cosh^{2}\frac{\mu s}{2}\right)^{-1}\equiv-a_{4}(s).$ (22)
Performing the standard procedure of the stationary phase method and using
Eqs.(1)-(2) one obtains the following expressions
$\displaystyle\Omega_{i}$ $\displaystyle=a_{i}\frac{\alpha
m^{2}\mu\nu}{r\sqrt{AB}}\exp\left(-b\right),\ \ W_{i}=\lambda_{i}\frac{\alpha
m^{2}\mu\nu}{\omega\sqrt{AB}}\exp\left(-b\right);$ (23)
$\displaystyle\lambda_{2}$ $\displaystyle=a_{2}-\frac{2a_{4}^{2}}{a},\ \
\lambda_{3}=a_{3}+\frac{2a_{4}^{2}}{a},\ \
a=a_{3}-a_{2}+\sqrt{\left(a_{3}-a_{2}\right)^{2}+4a_{4}^{2}};$ $\displaystyle
b_{2}^{\mu}$ $\displaystyle=\
\left(Bk\right)^{\mu}+\frac{2a_{4}}{a}\left(Ck\right)^{\mu},\ \
b_{3}^{\mu}=\left(Ck\right)^{\mu}-\frac{2a_{4}}{a}\left(Bk\right)^{\mu}$ (24)
These equations is valid at $r\gg 1$ if the condition $b\gg 1$ is fulfilled.
The first two terms of the decomposition of the functions $s\left(r\right)$
Eq.(18)) and $b\left(s\left(r\right)\right)$ Eq.(19) over $1/r$ are
$s(r)\simeq\frac{4}{\kappa}\left(1-\frac{8\widetilde{\mathcal{F}}}{3\kappa^{2}}\right),\
\ b(r)\simeq\frac{8}{3\kappa}-\frac{64\widetilde{\mathcal{F}}}{15\kappa^{3}},\
\ \kappa^{2}=4(\mu^{2}+\nu^{2})r.$ (25)
It is follows from this formula that the applicability of Eq.(23) is limited
by the condition $\kappa\ll 1$. The main values of the rest terms in
Eqs.(23),(24) have a form
$\displaystyle A$ $\displaystyle=\frac{1}{3}\left(\mu^{2}+\nu^{2}\right)s,\
D=\frac{3}{2}A;\ a_{2}=\frac{\mu^{2}+2\nu^{2}}{4\mu\nu},\
a_{3}=\frac{2\mu^{2}+\nu^{2}}{4\mu\nu},$ $\displaystyle\ \ \ a_{4}$
$\displaystyle=\frac{1}{4},\ \ a=\frac{\mu}{2\nu},\ \
\lambda_{2}=\frac{\mu^{2}+\nu^{2}}{4\mu\nu},\ \ \lambda_{3}=2\lambda_{2},\ $
(26)
and the vectors of polarization are given by Eq.(8). Substituting this values
into equation for $W_{i}$ we have
$W_{2}=\frac{\alpha
m^{2}\kappa}{8\omega}\sqrt{\frac{3}{2}}\exp\left(-\frac{8}{3\kappa}+\frac{64\widetilde{\mathcal{F}}}{15\kappa^{3}}\right),\
\ W_{3}=2W_{2}.$ (27)
In the region of the SQA applicability and for $\kappa\ll 1$ this probability
coincides with the results of the previous section and so the overlapping
region of both approximations exists.
It is interesting to consider the photon energy region $|r-1|\ \ll 1$ in the
presence of a weak electric field $(\nu\ll\mu)$ where in the absence of an
electric field the approach under consideration is valid if the condition
$r-1\gg\mu$ is fulfilled [8]. In this case Eq.(18) and its solutions are given
by the following approximate equations
$\displaystyle\frac{\xi^{2}y_{0}^{2}}{16}$
$\displaystyle\simeq\exp(-y_{0})+\frac{1-r}{4},\ \ y_{0}=\mu s,\
\xi=\frac{\nu}{\mu};$ (28) $\displaystyle\ y_{0}$ $\displaystyle\simeq
2\ln\frac{2}{\xi\ln\frac{4}{\xi}}\left(1-\frac{r-1}{2\xi^{2}\ln\frac{2}{\xi}\ln^{3}\frac{4}{\xi}}\right),\
|r-1|\ \lesssim\xi^{2};$ (29) $\displaystyle y_{0}$
$\displaystyle\simeq\ln\frac{4}{r-1}\left(1-\frac{\xi^{2}}{4\left(r-1\right)}\ln\frac{4}{r-1}\right),\
\ r-1\gg\xi^{2};$ (30) $\displaystyle\xi y_{0}$ $\displaystyle=\nu s\simeq
2\sqrt{1-r},\ \ 1-r\gg\xi^{2}.$ (31)
The applicability of the using saddle-point method is connected with the large
value of the coefficient to the second power $(y-y_{0})^{2}$ of the
decomposition in the phase Eq.(17). In the energy region under consideration
we have
$\mathrm{i}\varphi^{\prime\prime}(x_{0})(x-x_{0})^{2}/2\simeq\frac{\xi^{2}}{4\mu}\left[y_{0}+\frac{y_{0}^{2}}{2}+\frac{2\left(r-1\right)}{\xi^{2}}\right](y-y_{0})^{2}.$
(32)
So, we have from the upper equations that in the case $\nu/\mu=\xi\ll 1,\
|r-1|\ \lesssim\xi^{2}$ Eq.(23) is valid if the condition $\xi^{2}/\mu\gg 1$
is fulfilled. In the case $1\gg r-1\ \gg\xi^{2}\ $the condition $r-1\ \gg\mu$
has to be available for that. And in the case $1\gg 1-r$ $\gg\xi^{2}$ the
condition
$\sqrt{1-r}\xi/\mu=\sqrt{\left(\xi^{2}/\mu\right)\left(1-r\right)/\mu}\gg 1$
is necessary for the applicability of the approach under consideration.
At low photon energy $r\ll 1\ $($\nu^{2}\ll r\ll\nu^{2/3})$ we have
$\displaystyle\nu s$
$\displaystyle\simeq\pi-2\sqrt{r}+r^{3/2}\left(\frac{2}{3}-\tanh^{2}\frac{\pi\eta}{2}\right),$
$\displaystyle b$
$\displaystyle\simeq\frac{1}{\nu}\left(\pi-4\sqrt{r}+\frac{2r}{\eta}\tanh\frac{\pi\eta}{2}\right);$
(33) $\displaystyle a_{2}$
$\displaystyle=\frac{1}{\sqrt{r}\sinh(\pi\eta)}\left(1-\frac{1}{2}\tanh^{2}\frac{\eta\pi}{2}+\frac{\mu}{4r}\coth\pi\eta\right),\
$ $\displaystyle\ a_{3}$
$\displaystyle=\frac{\coth(\pi\eta)}{2r^{3/2}}\left(1+\frac{4\eta\sqrt{r}}{\sinh(2\pi\eta)}\right)\simeq
a,\ \ a_{4}=\left(4r\cosh^{2}\frac{\eta\pi}{2}\right)^{-1},$ (34)
$\displaystyle\ \ \lambda_{2}$
$\displaystyle=\frac{1}{\sqrt{r}\sinh(\pi\eta)}\left[1-\left(\frac{1}{2}+\frac{1}{\cosh(\pi\eta)}\right)\tanh^{2}\frac{\eta\pi}{2}+\frac{\mu}{4r}\coth(\pi\eta)\right],$
$\displaystyle\lambda_{3}$ $\displaystyle\simeq a_{3},\ \
A=\frac{\nu}{\sqrt{r}}\left(1-\frac{2\eta\sqrt{r}}{\sinh(\pi\eta)}\right),\ \
D=\frac{\nu}{r^{3/2}},\ \eta=\frac{\mu}{\nu}.$ (35)
Here we have retained the leading and the leading to leading terms of
decomposition. The term $\propto\mu$ in $a_{2}$ has appeared as the
contribution of the second term in $f_{1}$ ($\propto v^{2}$) in Eq.(5).
Substituting these values into Eq.(23) one obtains the following expression
for the probability of the process
$\displaystyle W_{3}$ $\displaystyle=\frac{\alpha
m^{2}\mu}{2\omega\sqrt{r}}\coth\left(\pi\eta\right)\left(1+\frac{\eta\sqrt{r}}{\sinh(\pi\eta)}+\frac{4\eta\sqrt{r}}{\sinh(2\pi\eta)}\right)\exp\left(-b\right),$
$\displaystyle W_{2}$ $\displaystyle=\frac{\alpha
m^{2}\mu\sqrt{r}}{\omega\sinh(\pi\eta)}\left[1-\frac{2+\cosh(\pi\eta)}{2\cosh(\pi\eta)}\tanh^{2}\frac{\eta\pi}{2}+\frac{\mu}{4r}\coth(\pi\eta)\right]\exp\left(-b\right),$
(36)
where $b$ is given by Eq.(33). One can see out of this equation that $W_{2}\ll
W_{3}.$At $\eta\gg 1$ the probability $W_{3}$ has been increased by the factor
$\eta\pi\exp\left(\pi r/\nu\right)$ in comparison with the case of the absence
of magnetic field. The probability $W_{2}$ has been reduced by the additional
factor $(\exp\left(-\pi\mu/\nu\right))$ and becomes non-applicable at
$\mu\gtrsim\sqrt{r}\gg\nu$. In that case for the probability $W_{2}$ one can
use Eq.(40) which will be get below.
## 5 Approximation at low photon energy
At $r\sim\nu^{2}$ the above approximation becomes non-applicable and another
approach has to be. We close the integration over $x$ contour in Eq.(4) in the
lower half-plane and represent this equation in the following form
$\Omega_{i}=\frac{\alpha m^{2}}{2\pi\text{{i}}}\mu\nu\int\limits_{-1}^{1}\
dv\sum\limits_{n=1}^{\infty}\oint f_{i}(v,x)\exp(\text{{i}}\psi(v,x))xdx,$
(37)
where the path of integration is any simple closed contour around the point
$-$i$\pi n\diagup\nu.$ Let us choose the contour near this point in the
following way $\nu x=-$i$\pi n+\xi_{n},\ \ |\xi_{n}|\ \sim\sqrt{r}\sim\nu$ and
expand the function entering in over the variables $\xi_{n}$. In the case
$\nu\ll 1,$ because of appearance of the factor $\exp\left(-\mathrm{i}\pi
n\diagup\nu\right),$ the main contribution to the sum gives the term $n=1$.
Near the point $-$i$\pi\diagup\nu$ the main terms of expansion such as
$(\xi\equiv\xi_{1})$
$\displaystyle f_{3}$
$\displaystyle=\frac{4\text{{i}}}{\xi^{3}}\coth(\pi\eta)\cos^{2}\frac{\pi
v}{2},\ \ f_{2}=-\frac{1}{\xi^{2}}\frac{\coth(\pi\eta)\text{
}}{\sinh(\pi\eta)}\sinh(v\pi\eta)\ \sin(v\pi),\ $ $\displaystyle\ f_{4}$
$\displaystyle=\frac{2}{\xi^{2}}\frac{\cosh(\pi\eta)-\cosh(v\pi\eta)}{\sinh^{2}(\pi\eta)}\cos^{2}\frac{\pi
v}{2},\ \ \psi=\frac{4r}{\xi\nu}\cos^{2}\frac{\pi
v}{2}-\frac{\xi}{\nu}+\frac{\mathrm{i}\pi}{\nu}.$ (38)
Using the integrals Eq.(7.3.1) and Eq.(7.7.1 (11)) in [15] and substituting
the result in Eqs.(1)-(2) we find
$\displaystyle W_{3}$ $\displaystyle=2\frac{\alpha
m^{2}}{\omega}\eta\pi\coth(\pi\eta)\text{
}\exp\left(-\frac{\pi}{\nu}\right)I_{1}^{2}\left(z\right),\ \ \
z=\frac{2\sqrt{r}}{\nu},$ (39) $\displaystyle W_{2}$
$\displaystyle=\frac{\alpha
m^{2}}{\omega}\mu\coth(\pi\eta)\exp\left(-\frac{\pi}{\nu}\right)\left[\frac{\pi\eta}{\sinh(\pi\eta)}\int\limits_{0}^{1}\cosh(v\pi\eta)I_{0}\left(2z\cos\frac{\pi
v}{2}\right)dv-1\right],$ (40)
where $\mathrm{I}_{n}\left(z\right)$ is the Bessel function of imaginary
argument. At calculation $W_{2}$ the integration by parts over $v$ has been
performed. For $\eta\ll 1$ one obtains
$W_{2}=\frac{\alpha
m^{2}}{\omega}\frac{\nu}{\pi}\exp\left(-\frac{\pi}{\nu}\right)\left(I_{0}^{2}\left(z\right)-1\right).$
(41)
The found probability is applicable for $r\ll\nu.$ Here we have kept the main
terms in $W_{i}$ only.
For $r\gg\nu^{2}$ the asymptotic representation
$\mathrm{I}_{n}\left(z\right)\simeq\exp\left(z\right)\diagup\sqrt{2\pi z}$ can
be used. As a result one obtains the probability Eq.(36) where the leading
terms have to be retained. At very low photon energy $r\ll\nu^{2},$ using the
expansion of the Bessel functions for the small value of argument, we have
$W_{3}=2\frac{\alpha
m^{2}r}{\omega\nu^{2}}\eta\pi\coth(\pi\eta)\exp\left(-\frac{\pi}{\nu}\right),\
\ W_{2}=\frac{\nu}{\pi\left(1+\eta^{2}\right)}W_{3}.$ (42)
The probability under consideration is of interest of theoretics for arbitrary
values $\mu$ and $\nu$. For $r\ll\nu^{2}\diagup\left(1+\nu^{2}\right)$ one can
conserve in the phase $\psi(v,x)$ the term $-x$ only. After integrating over
$v$ we get the following equation for the probability averaged over the photon
polarizations
$\displaystyle W$ $\displaystyle=\frac{W_{2}+W_{3}}{2}=\frac{\alpha
m^{2}r}{\mathrm{i}\pi\omega}\sum\limits_{n=1}^{\infty}\oint
F(y_{n})\exp\left(-\mathrm{i}\frac{y_{n}}{\nu}\right)dy_{n},$ $\displaystyle
F(y)$ $\displaystyle=\frac{\cosh(y)\left(\eta y\cos\left(\eta
y\right)-\sin\left(\eta y\right)\right)}{\sinh y\sin^{3}\eta
y}+\frac{\eta\cos(\eta y)\left(y\cosh y-\sinh y\right)}{\sinh^{3}y\sin(\eta
y)}.$ (43)
Summing the residues in the points $y_{n}=-\mathrm{i}n\pi$ one obtains
$\displaystyle W$ $\displaystyle=\frac{\alpha
m^{2}r}{\omega}\sum\limits_{n=1}^{\infty}\exp\left(-\frac{\pi
n}{\nu}\right)\Phi\left(z_{n}\right),\ \ z_{n}=\eta\pi n,$ (44)
$\displaystyle\Phi\left(z_{n}\right)$
$\displaystyle=\frac{z_{n}}{\nu^{2}}\coth
z_{n}+\frac{2}{\sinh^{2}z_{n}}\left[\frac{\eta
z_{n}}{\nu}+\left(1+\eta^{2}\right)z_{n}\coth z_{n}-1\right].$ (45)
In the absence of magnetic field ( $\eta\rightarrow 0,$ $z_{n}\rightarrow 0$ )
we have
$\displaystyle\Phi$ $\displaystyle=\frac{1}{\nu^{2}}+\frac{2}{\nu\pi
n}+\frac{2}{\pi^{2}n^{2}}+\frac{2}{3},$ $\displaystyle W$
$\displaystyle=\frac{\alpha
m^{2}r}{\omega}\left[\left(\frac{1}{\nu^{2}}+\frac{2}{3}\right)\frac{1}{e^{\pi/\nu}-1}-\frac{2}{\pi\nu}\ln\left(1-e^{-\pi/\nu}\right)+\frac{2}{\pi^{2}}\mathrm{Li}_{2}\left(e^{-\pi/\nu}\right)\right],$
(46)
where $\mathrm{Li}_{2}\left(z\right)$ is the Euler dilogarithm. In the
opposite case $\eta\gg 1$ one obtains
$\Phi=\frac{\pi\eta n}{\nu^{2}},\ \ W=\frac{\alpha
m^{2}r}{\omega\nu^{2}}\frac{\pi\eta}{4}\sinh^{-2}\frac{\pi}{2\nu}.$ (47)
## 6 Conclusion
We have considered the process of pair creation in constant and homogeneous
electromagnetic fields with a real photon taking part in. The probability of
the process has been calculated using three different overlapping
approximation. In the region of SQA applicability the created by a photon
particles have ultrarelativistic energies. The role of fields in this case is
to transfer the required transverse momentum and the electric and magnetic
field actions are equivalent. But even in this case it is necessary to note a
special significance of a weak electric field $E=\xi H$ $(\xi\ll 1)$ in the
removal of the root divergence of the probability when the particles of pair
are created on the Landau levels with the electron and positron momentum
$p_{3}=0$ [8]. The frame is used where $k_{3}=0.$
Generally speaking, at $\xi\ll 1$ the formation time $t_{c}$ of the process
under consideration is $1/\mu.$ Here we use units $\hbar=c=m=1.$ At this time
the particles of creating pair gets the momentum $\delta p_{3}\sim\xi.$ If the
value $\xi^{2}$ becomes more larger than the distance apart Landau levels
$2\mu$ $(\nu^{2}\gg\mu^{3})$ all levels have been overlapped. Under this
condition the divergence of the probability is vanished and the new
quasiclassical approach is valid even in the energy region $r-1\lesssim\mu$
where it has been inapplicable in the absence of electric field [8]. In the
opposite case $\nu^{2}\ll\mu^{3}$ for the small value of $p_{3}\ll\sqrt{\mu},$
in the region where the influence of electric field is negligible, the
formation time of the process $t_{f}$ is $1/p_{3}^{2}$ and $\delta
p_{3}\sim\nu/p_{3}^{2}$ $\ll p_{3}$ . It is follows from above that
$\nu^{1/3}\ll p_{3}\ll\sqrt{\mu}$ . At this condition the value of
discontinuity is $\sqrt{t_{f}/t_{c}}\sim\sqrt{\mu}/p_{3}.$For $\nu^{1/3}\gg
p_{3}$ the time $t_{f}$ is determined by the self-consistent equation
$\delta\varepsilon^{2}\sim 1/t_{f}\sim\nu^{2}t_{f}^{2},$ $t_{f}\sim\nu^{-2/3}$
and the value of discontinuity becomes $\sqrt{\mu
t_{f}}\sim(\mu^{3}/\nu^{2})^{1/6}$ instead of $\sqrt{\mu}/p_{3}.$
In the region $\omega\lesssim 2m$ $(r\lesssim 1)$ the energy transfer from
electric field to the created particles becomes appreciable and for
$\omega\ll$ $m$ it determines the probability of the process mainly. At
$\omega\ll eE/m$ the photon assistance in the pair creation comes to the end
and the probability under consideration defines the probability of photon
absorption by the particles created by electromagnetic fields. The influence
of a magnetic field on the process is connected with the interaction of the
magnetic moment of the created particles and magnetic field. This interaction,
in particular, has appeared in the distinction of the pair creation
probability by field for scalar and spinor particles [5].
## References
* [1] N.P.Klepikov, Ph.D. dissertation, Moscow State University, 1952 (unpublished).
* [2] J.S.Toll, Ph.D. dissertation, Prinston University, 1952 (unpublished).
* [3] N.P.Klepikov, Zh. Eksp. Teor. Fiz., 26, 19 (1954).
* [4] S.L.Adler, Ann. Phys. (N.Y.), 67, 599 (1971).
* [5] J.Schwinger, Phys. Rev., 82, 664 (1951).
* [6] I.A.Batalin and A.E.Shabad, Sov. Phys. JETP, 33, 483 (1971).
* [7] V.N.Baier, V.M.Katkov and V.M.Strakhovenko, Sov. Phys. JETP, 41, 198, (1975).
* [8] V.N.Baier and V.M.Katkov, Phys.Rev., D 75, 073009 (2007).
* [9] V.N.Baier and V.M.Katkov, arXiv:0912.5250v1 [hep-ph]; Preprint BINP 2009-38, Novosibirsk, 2009.
* [10] N.B.Narozhny, Zh. Eksp.Teor.Fiz., 54, 676 (1968).
* [11] F.Sauter, Z.Phys. 69, 742 (1931).
* [12] A.I.Nikishov, Zh.Eksp.Teor.Fiz., 59, 1262 (1970).
* [13] V.N.Baier and V.M.Katkov, Sov. Phys. JETP, 26, 854 (1968).
* [14] W.Tsai and T.Erber, Phys. Rev. D 10, 492 (1974)
* [15] H.Bateman, A.Erdélyi, Higher Transcendental Functions, v.II, McGraw-Hill Book Co, New York, 1953.
|
arxiv-papers
| 2011-02-17T07:11:34 |
2024-09-04T02:49:17.071751
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.M. Katkov",
"submitter": "Valery Katkov",
"url": "https://arxiv.org/abs/1102.3510"
}
|
1102.3557
|
# Electroweak Chiral Lagrangian from TC2 Model with nontrivial TC fermion
condensation and walking
Feng-Jun Ge1, Shao-Zhou Jiang2111Email address: jsz@gxu.edu.cn(S.Z.Jiang).,
Qing Wang1,3222Corresponding author at: Department of Physics, Tsinghua
University, Beijing 100084, P.R.China
Email address: wangq@mail.tsinghua.edu.cn(Q.Wang). 1Department of Physics,
Tsinghua University, Beijing 100084, P.R.China
2College of Physics Science and Technology, Guangxi University, Nanning,
Guangxi 530004, P.R.China
3Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.china
(April 29, 2011)
###### Abstract
The electroweak chiral Lagrangian for the topcolor assisted technicolor model
proposed by K. Lane, which uses nontrivial patterns of techniquark
condensation and walking, was investigated in this study. We found that the
features of the model are qualitatively similar to those of Lane’s previous
natural TC2 prototype model, but there is no limit on the upper bound of the
$Z^{\prime}$ mass. We discuss the phase structure and possible walking
behavior of the model. We obtained the values of all coefficients of the
electroweak chiral Lagrangian up to an order of $p^{4}$. We show that although
the walking effect reduces the S parameter to half its original value, it
maintains an order of $2$. Moreover, a special hyper-charge arrangement is
needed to achieve further reductions in its value.
PACS numbers: 12.60.Nz; 11.10.Lm, 11.30.Rd, 12.10.Dm
††preprint: TUHEP-TH-11175
## I Introduction
Modern technicolor (TC) models of dynamical electroweak symmetry breaking
require assistance for top-color interactions that are strong in the TeV
energy region to provide the large mass of the top quark, and a walking
technicolor (WTC) gauge coupling to aid in the avoidance of large flavor-
changing neutral current (FCNC) effects. The first addition consists of a
class of topcolor-assisted technicolor (TC2) models made through the careful
arrangement of TC, topcolor, extended hypercharge groups, and relevant
techniquark and Standard Model (SM) fermion representations. With the help of
extended technicolor (ETC), we expect that technicolor condensates will form
and provide the mass for the weak vector bosons. ETC provides the mass for the
light quarks and leptons and a bottom-quark-sized mass to the top. The largest
contribution to the top-quark mass is from the formation of a top-quark
condensate through the dynamics of the topcolor gauge sector. The second
addition is based on the phase diagram of strongly coupled TC gauge theories
involving fermions in arbitrary representations of the gauge group. With
suitable choices for the TC group and techniquark representations, WTC is a
natural option for situations with asymptotic freedom that are nearly
conformal. In this case, the TC gauge coupling has an approximate infrared-
stable fixed point (the zero of the beta function) $\alpha_{*}$ which is
slightly larger than the critical value $\alpha_{c}$ necessary for techniquark
condensate formation. In such a theory, for values of $\alpha$ above
$\alpha_{*}$, as the energy scale decreases $\alpha$ increases. However, its
rate of increase decreases to zero as $\alpha$ approaches $\alpha_{*}$. Hence,
over an extended energy interval, $\alpha$ is order O(1), and it is slowly
varying which leads a large anomalous dimension $\gamma\simeq 1$ for the
bilinear local techniquark operator. This results in the enhancement of the SM
fermion and those undiscovered pseudo goldstone boson masses, which achieve
realistic scales while maintaining sufficient suppression of FCNC effects.
The typical gauge group of the TC2 models is
$\displaystyle SU(N)_{\mathrm{TC}}\otimes SU(3)_{1}\otimes SU(3)_{2}\otimes
SU(2)_{L}\otimes U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ (1)
in which the topcolor and extended hypercharge groups $SU(3)_{1}\otimes
SU(3)_{2}\otimes U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ spontaneously break into
their diagonal subgroups $SU(3)_{C}\otimes U(1)_{Y}$ at an energy of a few
TeV. The remaining electroweak groups $SU(2)_{L}\otimes U(1)_{Y}$
spontaneously break into their electromagnetic subgroup $U(1)_{\mathrm{em}}$
at electroweak scale because of a combination of a top-quark condensate and
techniquark condensate. In the simplest example of Hill’s TC2 model Hill95 ,
there are separate color and weak hypercharge gauge groups for the heavy third
generation quarks and leptons and for the two lighter generations. The third
generation transforms under a strongly coupled $SU(3)_{1}\otimes U(1)_{1}$ and
maintains its usual charges. However, the light generations transform
conventionally under a weakly coupled $SU(3)_{2}\otimes U(1)_{2}$. Near 1 TeV,
these four groups break into a diagonal subgroup of ordinary color and
hypercharge, $SU(3)_{C}\otimes U(1)_{Y}$. The desired condensation pattern
occurs because the $U(1)_{1}$ couplings are such that the spontaneously broken
$SU(3)_{1}\otimes U(1)_{1}$ interactions are supercritical only for the top
quark.
After Hill’s proposal was made, Chivukula, Dobrescu, and Terning CDT claimed
that the techniquarks required to break the top and bottom quark chiral
symmetries are likely to have custodial-isospin violating couplings to the
strong $U(1)_{1}$. To maintain a $\rho\simeq 1$, the $U(1)_{1}$ interaction
must be so weak that it is necessary to fine-tune the $SU(3)_{1}$ coupling.
This results in the implementation of the theory being unnatural. To remedy
this isospin violation and improve the suitability of the model, K. Lane
proposed a natural prototype TC2 model in Ref.Lane95 . In that model, the
different techniquark isodoublets, $T^{t}$ and $T^{b}$, provide ETC mass to
the top and bottom quarks. These doublets then could have different $U(1)_{1}$
charges, which are isospin conserving for the right and left handed parts of
each doublet. The $U(1)$ symmetries presented in the model automatically avoid
the problem of $B_{d}-\bar{B}_{d}$ mixing raised by KominisKominis . To
achieve the mixing of the magnitude observed between the heavy and light
generations while breaking the strong top-color interactions near 1 TeV, K.
Lane also proposed an alternative model based on the nontrivial patterns of
techniquark condensation and discussed its phenomenologyLane96 . In this new
model, to break the extended hypercharge groups into $U(1)_{Y}$, a set of
electrically neutral $SU(2)$ singlet techniquarks belonging to the
antisymmetric tensor representation of the TC group were added into the model.
This, in combination with other techniquarks, further ensures the technicolor
coupling walks. With so many techniquarks, one may wonder whether the $S$
parameter of the model can be small. Although qualitatively the large number
of techniquarks will increase the value of $S$, walking effects and certain
arrangements of the hypercharges of the techniquarks may compensate for this
increase, and result in a small overall $S$ parameter. One aim of this paper
is to examine this possibility.
In fact, our interests are not limited to the S parameter, which is one of the
low energy constants (LECs) of the bosonic part of the standard electroweak
chiral Lagrangian (EWCL)EWCL . Rather, our interests include all EWCL LECs. In
our previous studies, we compiled a formulation for computing the bosonic part
of the EWCL LECs for orders up to $p^{4}$ for the one-doublet TC model
discussed in Ref.1D , Hill’s schematic TC2 model Hill95 in Ref.HongHao08 , K.
Lane’s natural prototype TC2 model Lane95 in Ref.JunYi09 and a hypercharge-
universal TC2 model Sekhar in Ref.LangPLB . Here, the bosonic part of the
EWCL is the part that only involves SM electroweak gauge fields and
corresponding Goldstone fields. This part describes the electroweak symmetry
breaking effects on the electroweak gauge fields, but the parts of the EWCL
dealing with matter also include SM fermions which describe the electroweak
symmetry breaking effects on the SM fermion fields. In the literature, these
two parts are proposed in Refs.EWCL and EWCLfermion separately because they
have independent characteristics. The reason that we choose to compute the
bosonic part of the EWCL in isolation is that the matter part is more complex
than the bosonic part. Moreover, some of the three-dimensional fermion mass
terms and six-dimensional FCNC terms were already discussed in Lane’s original
paper Lane96 . In this paper, we only discuss the bosonic part of the EWCL for
the first stage of computing the LECs that are generalized from the $S$
parameter, and leave the part dealing with matter for future discussion. The
EWCL is an universal platform which enables us to compare different underlying
models with experimental data and extract the true physical theory that guides
our world. To achieve this comparison, we compute the EWCL coefficients model
by model. This study is the fourth paper in a series, starting with
Ref.HongHao08 , in which we compute these strongly coupled physics models.
Here, we focus on K. Lane’s alternative TC2 model with nontrivial TC fermion
condensation and walkingLane96 , which was mentioned previously. Corresponding
to recent advances in the understanding of the phase diagram of the $SU(N)$
gauge theories and the new possibilities for model buildingNewWalking , this
work offers a modern way to investigate walking effects in a realistic
strongly-coupled theory with complex structures.
In this paper, except for some conventional calculations that are similar to
those in our previous papers, we focus on the effects of walking that have not
been discussed before. We will compare the different situations of walking,
ideal walking, and running; and examine their effects on the $S$ parameter. In
the next section, we first review K. Lane’s alternative TC2 model with
nontrivial condensation and walkingLane96 and discuss its phase structure. In
section III, we apply our formulation developed in Ref.HongHao08 to Lane’s
model Lane96 . We perform these dynamical calculations through several steps:
first we integrate in the Goldstone field, U. Then, we integrate out the
technigluons and techniquarks by solving the Schwinger-Dyson equation (SDE)
for techniquarks. Next, we integrate out the colorons and $Z^{\prime}$,
perform a low energy expansion, and compute the effective action. Finally, we
obtain the EWCL coefficients. For simplicity, some details of the derivation
and computation in this section are placed in the appendices. Section IV.
contains numerical results and discussions. Section V. is a short summary and
discussion.
## II Review of the Model and its phase structure
Consider K. Lane’s TC2 model Lane96 with nontrivial TC fermion condensation
and walking, in which the group is given by (1). Because we are only
interested in the bosonic part of EWCL, which is independent of the SM
fermions, we do not list their representations and $U(1)$ charge arrangements
here. The left gauge charges for the techniquarks are shown in Table I. There
are three sets of techniquarks. The first set includes $T^{1}$ and $T^{2}$.
These are the specific techniquarks of the model and are expected to have
twisted condensates that generate $SU(3)_{1}\otimes SU(3)_{2}\rightarrow
SU(3)_{c}$ and electroweak breaking, and a sufficient level of generation
mixing. The second set includes $T^{l}$, $T^{t}$ and $T^{b}$, which are the
standard TC2 techniquarks from Lane’s first natural prototype TC2 model Lane95
. They supply the ETC mass to the SM fermions, including the top and bottom.
The third set consists of the high-dimensional representation field $\psi$,
which is responsible for generating $U(1)_{1}\otimes U(1)_{2}\rightarrow
U(1)_{Y}$ and ensuring theory walking.
TABLE I. Gauge charge assignments of the techniquarks in Lane’s TC2 model.
field$\setminus$group | $SU(N)_{\mathrm{TC}}$ | $SU(3)_{1}$ | $SU(3)_{2}$ | $SU(2)_{L}$ | $U(1)_{1}$ | $U(1)_{2}$
---|---|---|---|---|---|---
field,coupling | $G^{\alpha}_{\mu},g_{\mathrm{TC}}$ | $A^{A}_{1\mu},h_{1}$ | $A^{A}_{2\mu},h_{2}$ | $W_{\mu}^{a},g_{2}$ | $B_{1\mu},q_{1}$ | $B_{2\mu},q_{2}$
$T_{L}^{1}$ | N | 3 | 1 | 2 | $u_{1}$ | $u_{2}$
$U_{R}^{1}$ | N | 3 | 1 | 1 | $v_{1}$ | $v_{2}+\frac{1}{2}$
$D_{R}^{1}$ | N | 3 | 1 | 1 | $v_{1}$ | $v_{2}-\frac{1}{2}$
$T_{L}^{2}$ | N | 1 | 3 | 2 | $v_{1}$ | $v_{2}$
$U_{R}^{2}$ | N | 1 | 3 | 1 | $u_{1}$ | $u_{2}+\frac{1}{2}$
$D_{R}^{2}$ | N | 1 | 3 | 1 | $u_{1}$ | $u_{2}-\frac{1}{2}$
$T_{L}^{l}$ | N | 1 | 1 | 2 | $x_{1}$ | $x_{2}$
$U_{R}^{l}$ | N | 1 | 1 | 1 | $x_{1}^{\prime}$ | $x_{2}^{\prime}+\frac{1}{2}$
$D_{R}^{l}$ | N | 1 | 1 | 1 | $x_{1}^{\prime}$ | $x_{2}^{\prime}-\frac{1}{2}$
$T_{L}^{t}$ | N | 1 | 1 | 2 | $y_{1}$ | $y_{2}$
$U_{R}^{t}$ | N | 1 | 1 | 1 | $y_{1}^{\prime}$ | $y_{2}^{\prime}+\frac{1}{2}$
$D_{R}^{t}$ | N | 1 | 1 | 1 | $y_{1}^{\prime}$ | $y_{2}^{\prime}-\frac{1}{2}$
$T_{L}^{b}$ | N | 1 | 1 | 2 | $z_{1}$ | $z_{2}$
$U_{R}^{b}$ | N | 1 | 1 | 1 | $z_{1}^{\prime}$ | $z_{2}^{\prime}+\frac{1}{2}$
$D_{R}^{b}$ | N | 1 | 1 | 1 | $z_{1}^{\prime}$ | $z_{2}^{\prime}-\frac{1}{2}$
$\psi_{L}$ | $\frac{1}{2}N(N-1)$ | 1 | 1 | 1 | $\xi$ | $-\xi$
$\psi_{R}$ | $\frac{1}{2}N(N-1)$ | 1 | 1 | 1 | $\xi^{\prime}$ | $-\xi^{\prime}$
The details of the ETC interaction are not specified in Lane’s original
paperLane96 ; this prohibits quantitative computations. The effects on the
EWCL LECs from these ETC operators can be ignored in our calculation because
the relevant operators are small. Unfortunately, although we know from Ref.
JunYi09 that its contribution to the EWCL LECs is small, the effective four-
fermion coupling may become strong enough to change the results of the current
walking theoryETC . When the effective four-fermion coupling exceeds its
critical value, the position of the infrared fixed point changes
significantly. For the first step of the investigation, we ignore this case by
assuming that the four-fermion coupling does not exceed the critical value and
leave discussion of more general effects for future studies.
A number of constraints were given in Lane’s original paperLane96 to limit
and simplify the charges:
* •
To ensure that the techniquark condensates conserve electric charge,
$u_{1}+u_{2}=v_{1}+v_{2}$, $x_{1}+x_{2}=x_{1}^{\prime}+x_{2}^{\prime}$,
$y_{1}+y_{2}=y_{1}^{\prime}+y_{2}^{\prime}$, and
$z_{1}+z_{2}=z_{1}^{\prime}+z_{2}^{\prime}$.
* •
The $U(1)_{1}$ charges of the techniquarks respect custodial isospin.
* •
For the $U(1)_{1}$ charges of $T^{1}$ and $T^{2}$: while $u_{1}\neq v_{1}$,
the broken $U(1)_{1}$ interactions favor the condensation of $T^{1}$ with
$T^{2}$. If this interaction is stronger than the $SU(3)_{1}$ attraction of
$T^{1}$ to itself and we neglect the other vacuum-aligning ETC interactions,
then $\langle\bar{T}^{i}_{L}T^{j}_{R}\rangle\propto(i\tau^{2})_{ij}$ in each
charge sector.
* •
$u_{1}\neq v_{1}$ implies $Y_{1i}\neq Y_{1i}^{\prime}$ for the fermions.
* •
For the $SU(N)_{\mathrm{TC}}$ antisymmetric tensor $\psi$,
$\xi^{\prime}\neq\xi$ guarantees $U(1)_{1}\otimes U(1)_{2}\rightarrow
U(1)_{Y}$ when $\langle\overline{\psi_{L}}\psi_{R}\rangle$ forms.
The Lagrangian of the model is
$S[G,A_{1},A_{2},W,B_{1},B_{2},\bar{T},T,\bar{\psi},\psi]=\int
d^{4}x[\mathcal{L}_{\mathrm{gauge~{}kinetic}}+\mathcal{L}_{\mathrm{techniquark}}+\mathcal{L}_{\mathrm{SM~{}fermion}}]\;,$
(2)
with
$\displaystyle\mathcal{L}_{\mathrm{gauge~{}kinetic}}=-\frac{1}{4}\bigg{[}G_{\mu\nu}^{\alpha}G^{\alpha,\mu\nu}+A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1\mu\nu}B^{1,\mu\nu}+B_{2\mu\nu}B^{2,\mu\nu}\bigg{]}$
and
$\displaystyle\mathcal{L}_{\mathrm{techniquark}}=$
$\displaystyle+\bar{T}^{1}[i\not{\partial}\\!-\\!g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!\\!-\\!h_{1}\frac{\lambda^{A}}{2}\not{A}_{1}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1}P_{L}\\!\\!-\\!q_{2}u_{2}\not{B}_{2}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1}P_{R}\\!\\!-\\!q_{2}(v_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{1}$
$\displaystyle+\bar{T}^{2}[i\not{\partial}\\!-\\!g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!\\!-\\!h_{2}\frac{\lambda^{A}}{2}\not{A}_{2}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1}P_{L}\\!\\!-\\!q_{2}v_{2}\not{B}_{2}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1}P_{R}\\!\\!-\\!q_{2}(u_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{2}$
$\displaystyle+\bar{T}^{l}[i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!-q_{1}x_{1}\not{B}_{1}P_{L}\\!-q_{2}x_{2}\not{B}_{2}P_{L}\\!-q_{1}x_{1}^{\prime}\not{B}_{1}P_{R}\\!-q_{2}(x_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{l}$
$\displaystyle+\bar{T}^{t}[i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!-q_{1}y_{1}\not{B}_{1}P_{L}\\!-q_{2}y_{2}\not{B}_{2}P_{L}\\!-q_{1}y_{1}^{\prime}\not{B}_{1}P_{R}\\!-q_{2}(y_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{t}$
$\displaystyle+\bar{T}^{b}[i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!-q_{1}z_{1}\not{B}_{1}P_{L}\\!-q_{2}z_{2}\not{B}_{2}P_{L}\\!-q_{1}z_{1}^{\prime}\not{B}_{1}P_{R}\\!-q_{2}(z_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{b}$
$\displaystyle+\bar{\psi}[i\not{\partial}-g_{\rm
TC}\tilde{t}^{\alpha}\not{G}^{\alpha}\\!-q_{1}\xi\not{B}_{1}P_{L}\\!+q_{2}\xi\not{B}_{2}P_{L}-q_{1}\xi_{1}^{\prime}\not{B}_{1}P_{R}\\!+q_{2}\xi^{\prime}\not{B}_{2}P_{R}]\psi\;.$
(3)
Where $\lambda^{A}$ is the three-dimensional Gellman matrix for topcolor
interaction, $\tau^{a}$ is the Pauli matrix for the electroweak interaction,
$t^{\alpha}$ is the $SU(N)_{\mathrm{TC}}$ fundamental representation matrix,
$\tilde{t}^{\alpha}$ is the $SU(N)_{\mathrm{TC}}$ antisymmetric tensor
representation matrix. We do not specify $\mathcal{L}_{\mathrm{SM~{}fermion}}$
which is not relevant to our discussions for the present approximation.
Now we will discuss the phase structure of the model. The two-loop $\beta$
function of the $SU(N)_{\mathrm{TC}}$ coupling, $g_{\mathrm{TC}}$, is333The
reason that we chose the two-loop $\beta$ function instead of the one-loop
version is that it can generate the walking effects needed for the model.
Otherwise, the model setting must be rearranged. Physically, we expect that
the most significant contribution should come from the TC interaction. The SM
particle mass does not reach the TC scale, and the masses of the colorons and
$Z^{\prime}$ slightly exceed this scale, all of their contributions are
expected to be smaller than those of the TC interactions. For simplicity in
the first stage approximation, we ignore the possible effects from SM
particles, colorons, and $Z^{\prime}$. We also ignore the high-dimension ETC
interactions. We will investigate the accuracy of this approximation in a
future study of all of these effects.
$\displaystyle\beta(\alpha)$ $\displaystyle=$
$\displaystyle-\beta_{0}\frac{g_{\mathrm{TC}}^{3}}{(4\pi)^{2}}-\beta_{1}\frac{g_{\mathrm{TC}}^{5}}{(4\pi)^{4}}\hskip
56.9055pt\alpha\equiv\frac{g_{\mathrm{TC}}^{2}}{4\pi}\;.$ (4)
In this case, the two coefficients $\beta_{0}$ and $\beta_{1}$444Here we apply
the convention of Ref.ConformalWindow . are
$\displaystyle 2N\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{11}{3}C_{2}(SU(N)_{\mathrm{TC}})-\frac{4}{3}[T(R_{1})+T(R_{2})+T(R_{3})]$
(5) $\displaystyle(2N)^{2}\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{34}{3}C_{2}^{2}(SU(N)_{\mathrm{TC}})-{\displaystyle\sum_{i=1}^{3}}[\frac{20}{3}C_{2}(SU(N)_{\mathrm{TC}})T(R_{i})+4C_{2}(R_{i})T(R_{i})]\;.$
(6)
The representations of the three sets of techniquarks mentioned above are
labeled $R_{1}$, $R_{2}$ and $R_{3}$. Their corresponding parameters are given
in Table II.
TABLE II. The representation parameters of this model. $d(R)$ is the dimension
of the representation, and $d(SU(N)_{\mathrm{TC}})$ is the number of group
generators. $C_{2}(R_{i})$ and $C_{2}(SU(N)_{\mathrm{TC}})$ are the quadratic
Casimir operators of the representation $R_{i}$ and the adjoint
representation, respectively. $N_{f}$ is the number of techniquarks in the
same representation, $N_{f}C_{2}(R)d(R)=T(R)d(G)$
$i$ | $d(R_{i})$ | $C_{2}(R_{i})$ | $C_{2}(SU(N)_{\mathrm{TC}})$ | $T(R_{i})$ | $d(SU(N)_{\mathrm{TC}})$ | $N_{f}$
---|---|---|---|---|---|---
1 | $N$ | $N^{2}-1$ | $2N^{2}$ | $N_{f}N$ | $N^{2}-1$ | 12
2 | $N$ | $N^{2}-1$ | $2N^{2}$ | $N_{f}N$ | $N^{2}-1$ | 6
3 | $N(N-1)/2$ | $2(N+1)(N-2)$ | $2N^{2}$ | $N_{f}N(N-2)$ | $N^{2}-1$ | 1
The reason that we only use the two-loop $\beta$ function is that the three-
loop term of the $\beta$ function is scheme dependent. Usually, it is only
used for error estimates. The behavior of the TC coupling, $\alpha$, is guided
by the renormalization group equation
$\mu\frac{\partial\alpha}{\partial\mu}=\beta$. From the equation, we know that
$\beta_{0}>0$ corresponds to the case in which the TC interaction allows
asymptotic freedom. However, $\beta_{0}<0$ corresponds to the loss of
asymptotic freedom, or non-asymptotic freedom. From (5) and Table II, we find
that the critical value dividing asymptotic freedom and non-asymptotic freedom
is determined by $\beta_{0}=0$ and leads $N=32/9$. If further ($\beta_{0}>0$
and $\beta_{1}<0$), TC interaction creates a Banks-Zaks infrared fixed point
$\alpha_{*}=-\frac{4\pi\beta_{0}}{\beta_{1}}$ BZ , which corresponds to the
zero of the $\beta$ function. In the more general case, an infrared fixed
point may not exist , which often happens in the situation in which the number
of fermions is small. This is the case for QCD. In this model, because there
are already too many technifermions, we have checked that the infrared fixed
point always exists. The existence of an infrared fixed point requires that
the coupling remains nearly constant over a given range of infrared energy
scales, i.e., it walks. This is the modern realization of the walking
mechanism. When an infrared fixed point exists, the two-loop $\beta$ function
dictates the following energy scale dependence of the TC coupling:
$\displaystyle\frac{1}{\alpha(x)}=\frac{\beta_{0}}{2\pi}\ln
x+\frac{1}{\alpha_{*}}\ln\frac{\alpha(x)}{\alpha_{*}-\alpha(x)}\hskip
85.35826ptx=\frac{q^{2}}{\Lambda^{2}_{w}}\;.$ (7)
Where the parameter $\Lambda_{w}$ is roughly the length of the interval of
constant coupling in the infrared region. At this scale, the coupling constant
completes the walk and begins a fast run in which it exhibits typical
asymptotic freedom behavior. In Section IV, we show that in the ideal walking
situation, $\Lambda_{w}$ can be interpreted as the ETC scale. It is often
referred to as $\Lambda_{\mathrm{ETC}}$ in the literatureYamawaki . Moreover,
in the standard running situation, $\Lambda_{w}$ can be treated as the TC
scale (or $\Lambda_{\mathrm{TC}}$). Realistically, in our model, the system is
somewhere between the cases of running and ideal walking, which suggests that
$\Lambda_{\mathrm{TC}}<\Lambda_{w}<\Lambda_{\mathrm{ETC}}$. This change from
$\Lambda_{\mathrm{ETC}}$ to $\Lambda_{w}$ also reflects the fact that
$\alpha(x)$ in the presence of some walking effects does not depend on the
value of $\Lambda_{\mathrm{ETC}}$ too much. However, in the ideal walking
theory they are very much correlated. Furthermore, the existence of both
asymptotic freedom and an infrared fixed point will divide the theory into two
different phases. One phase is the asymptotic freedom phase in which
$\alpha\leq\alpha_{*}$. In this case, the coupling $\alpha$ increases from
zero to $\alpha_{*}$ monotonically while the energy scale decreases from the
ultraviolet region to the infrared region. The other phase is the non-
asymptotic freedom phase, where $\alpha\geq\alpha_{*}$. In this case, the
coupling $\alpha$ decreases from infinity to $\alpha_{*}$ monotonically while
the energy scale decreases from the ultraviolet region to the infrared region.
Furthermore, the ladder approximation Schwinger-Dyson equation (SDE) for
techniquark self-energy predicts a critical coupling:
$\displaystyle\alpha_{c}=\frac{2\pi N}{3C_{2}(R)}$ (8)
for techniquarks that belong to the techni-gauge group representation, $R$.
While the infrared fixed point $\alpha_{*}$ exceeds its critical coupling
$\alpha_{c}$, spontaneous chiral symmetry breaking occurs, and the SDE
automatically develops nonzero techniquark self-energies and condensates.
However, when $\alpha_{*}$ is less than $\alpha_{c}$, there is no spontaneous
chiral symmetry breaking, and the techniquark self-energy vanishes. Later, we
will see that to ensure the correctness of our $\beta$ function, the nonzero
values of the techniquark self-energy and condensate must be small enough
compare to $\Lambda_{w}$. This dictates that $\alpha_{*}$ can only be larger
than $\alpha_{c}$ by a small amount. In practice, $\alpha_{*}$ may not be so
close in value to $\alpha_{c}$, this will cause inaccuracy in our
computations. We will estimate this error in later calculations. For the cases
discussed above for different values of TC coupling and different choices of
$N$, our model may exhibit different behaviors and then form different phases.
We present555Because $N_{f}$ is fixed in the model, we depict the phase
diagram in terms of $N$ and $\alpha$, instead of $N$ and $N_{f}$, which is
more commonly done in the literature. Comparing our Fig.1 to the phase diagram
depicted by Fig.1 in Ref.ConformalWindow , our phase diagram corresponds to a
horizontal line with a fixed $N_{f}$ in their diagram. Their phase diagram
only provides information about $N_{f}$ and $N$. Our phase diagram does not
provide information about $N_{f}$ , but does provide more information about
the running coupling constant. a phase diagram of our model in Fig.1.
Figure 1: Phases of Lane’s alternative TC2 model with nontrivial TC fermion
condensation and walking. The blue solid line represents the infrared fixed
point $\alpha_{*}$. The red dashed line denotes the critical coupling of the
first and second techniquark sets(fundamental representation of
$SU(N)_{\mathrm{TC}})$). The black dashed-dotted line denotes the critical
coupling of the third techniquark set(antisymmetric representation of
$SU(N)_{\mathrm{TC}})$). The magenta dotted line shows the value $N=32/9$ from
$\beta_{0}=0$.
From Fig.1, we can see that the blue line (infrared fixed point) divides the
phase space into two parts: the region above the blue line represents the non-
asymptotic freedom phase and that below the blue line represents the
asymptotic freedom phase.
In the asymptotic freedom phase, $\alpha$ runs from $\alpha_{*}$ (blue line)
to zero, as the energy scale increases. The blue line crosses the red dashed
line (critical coupling of the first and second techniquark sets) and the
black dashed-dotted line (critical coupling of the third techniquark set) at
two points, which divide the blue line into three segments. The trapezoids
(and triangle) under these segments form the three sub-regions of the
asymptotic freedom phase. From left to right, the blue region is the conformal
region, where $\alpha$ is always below its critical value and no techniquark
condensation forms. Therefore, there is no spontaneous chiral symmetry
breaking. The second red region is the intermediate mixture region, where
$\alpha$ is always below the critical value $\alpha_{c,1}=\alpha_{c,2}$, but
will cross $\alpha_{c,3}$ as the energy scale decreases. This means the third
set of techniquarks forms condensates, but the first and second sets do not.
The yellow and green regions are the ones that we mainly focus on in this
paper. In these regions, $\alpha$ will cross all its critical values as the
energy scale decreases. Thus, all techniquarks have nonzero self-energies and
condensates. Therefore, this is the model required for spontaneous chiral
symmetry breaking.
In the yellow region, the unique TC coupling in the infrared energy region
approaches that of the infrared fixed point, critical values $\alpha_{c}$ of
the first and second techniquark sets (within a magnitude of 0.2 ), and that
of the third techniquark set (within a magnitude of 0.4 ), as the energy scale
decreases. This causes a near conformal behavior in which the value of the
techniquark self-energy is very small (corresponding to a tiny mass). For at
least two reasons, this region is the most important to the investigation of
the walking effect. First, the lower the techniquark self-energy, the more
accurate and reliable our estimate of the $\beta$ function over the energy
region will be. This is because we have used the $\overline{\mathrm{MS}}$
scheme, which assumes massless techniquarks, to obtain the coefficients of the
$\beta$ function in (5) and (6). Second, if a techniquark has a significant
mass, it will decouple and not contribute to the $\beta$ function in the low
energy region. Therefore, in the extreme infrared region, because of
spontaneous chiral symmetry breaking, we cannot treat techniquarks as
massless. Therefore, we need to ignore techniquark contributions if they have
mass. The coupling without these techniquark contributions will run (rather
than walk) to a very large value and will not reach its original infrared
fixed point. We show this special running behavior in the infrared energy
region for $N=6$ using a dashed magenta line near the vertical axis in Fig.2.
A techniquark self-energy on the order of $F_{\mathrm{TC}}$ leads to an
infrared interval of the same order size, which is small in comparison to the
typical scale for $\Lambda_{w}$. The smaller the $F_{\mathrm{TC}}$ is, the
more accurately (7) describes the coupling walking behavior. Therefore, we
expect that replacing the running behavior in this region with an infrared
fixed point will only cause errors of order $F_{\mathrm{TC}}/\Lambda_{w}$ in
the solution of the SDE for the techniquark self-energy. In this model,
because our techniquarks belong to different representations of the TC group,
which leads to different critical couplings, there is not a unique point where
the $\alpha_{*}$ is equal to all the critical coupling values. Usually this is
a necessary component of modern walking theory.
Furthermore, the minimum integer $N$ closest to the conformal region is $N=6$,
but the value $N=4$ was chosen in Lane’s original paperLane96 and does not
satisfy the walking requirements of this study. Although we do not have an
unique $\alpha_{*}$ that is equal to all the critical coupling values and
$N=6$ is perhaps too far from the conformal region, our numerical results
given in section IV show that walking effects are present. Therefore, we do
achieve the situation where the infrared fixed point is not enough but
sufficiently close to the critical coupling. In fact, even if we found a
unique infrared fixed point $\alpha_{*}$ meets all the critical couplings and
an integer $N$ very near the conformal region, the walking results would not
be significantly more reliable. This is because of the large number of
assumptions made in our calculations. These assumptions include: ignoring
higher-order loops (error of $1/16\pi^{2}$), SM particles of mass $m$ (error
of $m^{2}/F^{2}_{\mathrm{TC}}$), and gauge fields such as coloron and
$Z^{\prime}$ (error of $F^{2}_{\mathrm{TC}}/M^{2}_{\mathrm{coloron}}$ and
$F^{2}_{\mathrm{TC}}/M^{2}_{Z^{\prime}}$ in the $\beta$ function). The
precision in the critical value is now only at the two-loop level. As we
mentioned before, the ETC effects may also play a role. One known effect from
the ETC interactionETC is that while the coupling of the ETC-induced
effective four-fermion interaction exceeds its critical value, the area of the
conformal window will be substantially reduced. In this sense, we must include
all the above-mentioned corrections before we can quantitatively improve the
precision of the present calculation of the possible walking effects of the
model. In the asymptotic freedom phase, we show the scale dependence of the TC
coupling according to formula (7) for different values of $N$ in Fig.2.
Figure 2: Energy scale dependence of the TC coupling, $\alpha$, determined
using (7).
From Fig.2, it can be seen that in the asymptotic freedom phase, the smaller
the value of $N$, the flatter the curve. In other words, the smaller the slope
of the curve or corresponding value of the $\beta$ , the larger the impact on
the walking effect. From Fig.1, we know that when $N\leq 5$, there is no
overall spontaneous chiral symmetry breaking. Therefore, the minimum value of
$N$ at which spontaneous chiral symmetry breaking occurs and results in the
largest walking effect is $N=6$. Throughout this paper, we will use $N=6$ in
our quantitative computations.
## III Derivation of the EWCL from Lane’s Model
Our goal is to obtain
$\displaystyle\exp\bigg{(}iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]\bigg{)}$
$\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}\mathcal{D}T^{1}\mathcal{D}\bar{T}^{2}\mathcal{D}T^{2}\mathcal{D}\bar{T}^{l}\mathcal{D}T^{l}\mathcal{D}\bar{T}^{t}\mathcal{D}T^{t}\mathcal{D}\bar{T}^{b}\mathcal{D}T^{b}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(9)
$\displaystyle\times\exp\bigg{(}iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T},T,\bar{\psi},\psi]\bigg{)}\bigg{|}_{A^{A}_{\mu}=0}$
$\displaystyle=$
$\displaystyle\mathcal{N}[W_{\mu}^{a},B_{\mu}]\int\mathcal{D}\mu(U)\exp\bigg{(}iS_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]\bigg{)}\;,$
(10)
where $S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]\equiv\int
d^{4}x{\displaystyle\sum_{i}}\mathcal{L}_{i}$ is the action of the EWCL.
$B_{\mu}$ is the gauge field of $U(1)_{Y}$ and $Z^{\prime}_{\mu}$ is the gauge
field of $U(1)^{\prime}\equiv U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}/U(1)_{Y}$. They
are related to $B_{1\mu}$ and $B_{2\mu}$ through the mixing angle $\theta$ by
$\displaystyle\begin{pmatrix}B_{1\mu}&B_{2\mu}\end{pmatrix}=\begin{pmatrix}Z_{\mu}^{\prime}&B_{\mu}\end{pmatrix}\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\hskip 56.9055ptg_{1}\equiv
q_{1}\sin\theta=q_{2}\cos\theta\;.$ (11)
In (9) $A^{A}_{\mu}$ is the gluon field of $SU(3)_{c}$ and $B^{A}_{\mu}$ is
the gauge field of $SU(3)_{1}\otimes SU(3)_{2}/SU(3)_{c}$. They are related to
$A^{A}_{1\mu}$ and $A^{A}_{2\mu}$ through the mixing angle $\theta^{\prime}$
by
$\displaystyle\begin{pmatrix}A_{1\mu}^{A}&A_{2\mu}^{A}\end{pmatrix}=\begin{pmatrix}B_{\mu}^{A}&A_{\mu}^{A}\end{pmatrix}\begin{pmatrix}\cos\theta^{\prime}&-\sin\theta^{\prime}\\\
\sin\theta^{\prime}&\cos\theta^{\prime}\end{pmatrix}\hskip
56.9055ptg_{3}\equiv h_{1}\sin\theta^{\prime}=h_{2}\cos\theta^{\prime}\;.$
(12)
In the next section, we will use Schwinger-Dyson analysis that the $SU(N)_{\rm
TC}$ interaction induces techniquark condensates
$\langle\overline{\psi_{L}}\psi_{R}\rangle\neq 0$ and
$\langle\overline{T_{L}}^{i}T^{j}_{R}\rangle\neq 0$ for $i,j=1,2$. They
trigger the extended hypercharge symmetry breaking, $U(1)_{Y_{1}}\otimes
U(1)_{Y_{2}}\rightarrow U(1)_{Y}$, and the topcolor symmetry breaking,
$SU(3)_{1}\otimes SU(3)_{2}\rightarrow SU(3)_{c}$, at a TeV energy scale.
These processes leave a singlet heavy state $Z_{\mu}^{\prime}$ in broken
$U(1)^{\prime}$ and colorons $B^{A}_{\mu}$ in the broken $SU(3)_{1}\otimes
SU(3)_{2}/SU(3)_{c}$, respectively. Because this work is only concerned with
the EWCL, we ignored the gluon field by taking $A^{A}_{\mu}=0$.
In (10), $U$ is the standard electroweak Goldstone boson, which can be
expressed in terms of a dimensionless unitary unimodular $2\times 2$ matrix
field, $\mathcal{D}\mu$ denotes the normalized functional integration measure
on $U$. The normalization factor $\mathcal{N}[W_{\mu}^{a},B_{\mu}]$ is
determined through the requirement that when the TC interaction is switched
off, $S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]$ must vanish. This fixes it at:
$\displaystyle\mathcal{N}[W_{\mu}^{a},B_{\mu}]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}\mathcal{D}T^{1}\mathcal{D}\bar{T}^{2}\mathcal{D}T^{2}\mathcal{D}\bar{T}^{l}\mathcal{D}T^{l}\mathcal{D}\bar{T}^{t}\mathcal{D}T^{t}\mathcal{D}\bar{T}^{b}\mathcal{D}T^{b}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(13)
$\displaystyle\times\exp\bigg{(}iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T},T,\bar{\psi},\psi]\bigg{)}\bigg{|}_{A^{A}_{\mu}=0,\mbox{\tiny
ignore TC interation}}\;.~{}~{}~{}$
In Ref.EWCL , the EWCL was constructed with building blocks which are
$SU(2)_{L}$ covariant and $U(1)_{Y}$ invariant as $T\equiv
U\tau^{3}U^{\dagger}$, $V_{\mu}\equiv(D_{\mu}U)U^{\dagger}$,
$g_{1}B_{\mu\nu}$, $g_{2}W_{\mu\nu}\equiv
g_{2}\frac{\tau^{a}}{2}W_{\mu\nu}^{a}$. Where $B_{\mu\nu}$ and $W_{\mu\nu}$
are the field strengths of the $U(1)_{Y}$ and $SU(2)_{L}$ gauge fields,
respectively. Alternatively, in Ref.HongHao08 , we reformulated the EWCL
equivalently using $SU(2)_{L}$ invariant and $U(1)_{Y}$ covariant building
blocks as $\tau^{3}$, $X_{\mu}\equiv U^{\dagger}(D_{\mu}U)$,
$g_{1}B_{\mu\nu}$, $\overline{W}_{\mu\nu}\equiv U^{\dagger}g_{2}W_{\mu\nu}U$.
In which, $\tau^{3}$ and $g_{1}B_{\mu\nu}$ are both $SU(2)_{L}$ and $U(1)_{Y}$
invariant, but $X_{\mu}$ and $\overline{W}_{\mu\nu}$ are bilinearly $U(1)_{Y}$
covariant. The second formulation was used throughout this paper. In Table
III, we detail the relationship between the two formalisms.
TABLE III. Symmetry breaking sector of the EWCL
$S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]=\int
d^{4}x{\displaystyle\sum_{i}}\mathcal{L}_{i}$
Formulation in Ref.EWCL Formulation in Ref.HongHao08 ${\cal L}^{(2)}$
$\frac{1}{4}f^{2}{\rm
tr}[(D_{\mu}U^{\dagger})(D^{\mu}U)]=-\frac{1}{4}f^{2}{\rm tr}(V_{\mu}V^{\mu})$
$-\frac{1}{4}f^{2}{\rm tr}(X_{\mu}X^{\mu})$ ${\cal L}^{(2)\prime}$
$\frac{1}{4}\beta_{1}f^{2}[{\rm tr}(TV_{\mu})]^{2}$
$\frac{1}{4}\beta_{1}f^{2}[{\rm tr}(\tau^{3}X_{\mu})]^{2}$ ${\cal L}_{1}$
$\frac{1}{2}\alpha_{1}g_{2}g_{1}B_{\mu\nu}{\rm tr}(TW^{\mu\nu})$
$\frac{1}{2}\alpha_{1}g_{1}B_{\mu\nu}{\rm tr}(\tau^{3}\overline{W}^{\mu\nu})$
${\cal L}_{2}$ $\frac{1}{2}i\alpha_{2}g_{1}B_{\mu\nu}{\rm
tr}(T[V^{\mu},V^{\nu}])$ $i\alpha_{2}g_{1}B_{\mu\nu}{\rm
tr}(\tau^{3}X^{\mu}X^{\nu})$ ${\cal L}_{3}$ $i\alpha_{3}g_{2}{\rm
tr}(W_{\mu\nu}[V^{\mu},V^{\nu}])$ $2i\alpha_{3}{\rm
tr}(\overline{W}_{\mu\nu}X^{\mu}X^{\nu})$ ${\cal L}_{4}$ $\alpha_{4}[{\rm
tr}(V_{\mu}V_{\nu})]^{2}$ $\alpha_{4}[{\rm tr}(X_{\mu}X_{\nu})]^{2}$ ${\cal
L}_{5}$ $\alpha_{5}[{\rm tr}(V_{\mu}V^{\mu})]^{2}$ $\alpha_{5}[{\rm
tr}(X_{\mu}X^{\mu})]^{2}$ ${\cal L}_{6}$ $\alpha_{6}{\rm
tr}(V_{\mu}V_{\nu}){\rm tr}(TV^{\mu}){\rm tr}(TV^{\nu})$ $\alpha_{6}{\rm
tr}(X_{\mu}X_{\nu}){\rm tr}(\tau^{3}X^{\mu}){\rm tr}(\tau^{3}X^{\nu})$ ${\cal
L}_{7}$ $\alpha_{7}{\rm tr}(V_{\mu}V^{\mu}){\rm tr}(TV_{\nu}){\rm
tr}(TV^{\nu})$ $\alpha_{7}{\rm tr}(X_{\mu}X^{\mu}){\rm
tr}(\tau^{3}X_{\nu}){\rm tr}(\tau^{3}X^{\nu})$ ${\cal L}_{8}$
$\frac{1}{4}\alpha_{8}g_{2}^{2}[{\rm tr}(TW_{\mu\nu})]^{2}$
$\frac{1}{4}\alpha_{8}[{\rm tr}(\tau^{3}\overline{W}_{\mu\nu})]^{2}$ ${\cal
L}_{9}$ $\frac{1}{2}i\alpha_{9}g_{2}{\rm tr}(TW_{\mu\nu}){\rm
tr}(T[V^{\mu},V^{\nu}])$ $i\alpha_{9}{\rm
tr}(\tau^{3}\overline{W}_{\mu\nu}){\rm tr}(\tau^{3}X^{\mu}X^{\nu})$ ${\cal
L}_{10}$ $\frac{1}{2}\alpha_{10}[{\rm tr}(TV_{\mu}){\rm tr}(TV_{\nu})]^{2}$
$\frac{1}{2}\alpha_{10}[{\rm tr}(\tau^{3}X_{\mu}){\rm
tr}(\tau^{3}X_{\nu})]^{2}$ ${\cal L}_{11}$
$\alpha_{11}g_{2}\epsilon^{\mu\nu\rho\lambda}{\rm tr}(TV_{\mu}){\rm
tr}(V_{\nu}W_{\rho\lambda})$ $\alpha_{11}\epsilon^{\mu\nu\rho\lambda}{\rm
tr}(\tau^{3}X_{\mu}){\rm tr}(X_{\nu}\overline{W}_{\rho\lambda})$ ${\cal
L}_{12}$ $\alpha_{12}g_{2}{\rm tr}(TV_{\mu}){\rm tr}(V_{\nu}W^{\mu\nu})$
$\alpha_{12}{\rm tr}(\tau^{3}X_{\mu}){\rm tr}(X_{\nu}\overline{W}^{\mu\nu})$
${\cal L}_{13}$
$\alpha_{13}g_{2}g_{1}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}{\rm
tr}(TW_{\rho\sigma})$
$\alpha_{13}\epsilon^{\mu\nu\rho\sigma}g_{1}B_{\mu\nu}{\rm
tr}(\tau^{3}\overline{W}_{\rho\sigma})$ ${\cal L}_{14}$
$\alpha_{14}g_{2}^{2}\epsilon^{\mu\nu\rho\sigma}{\rm tr}(TW_{\mu\nu}){\rm
tr}(TW_{\rho\sigma})$ $\alpha_{14}\epsilon^{\mu\nu\rho\sigma}{\rm
tr}(\tau^{3}\overline{W}_{\mu\nu}){\rm tr}(\tau^{3}\overline{W}_{\rho\sigma})$
From (9) and (10), it can be seen that to obtain the EWCL, we must integrate
in the electroweak Goldstone boson field, $U$. We also need to integrate out
the series of fields which include the three sets of techniquarks, $\psi$,
$T^{1}$, $T^{2}$, $T^{l}$, $T^{t}$, $T^{b}$ and the technigluon
$G_{\mu}^{\alpha}$, and the colorons $B^{A}_{\mu}$ and $Z_{\mu}^{\prime}$. In
the following subsections, we divide this work into five steps.
### III.1 Integrating in the electroweak Goldstone boson field $U$
We introduce a local $2\times 2$ operator
$O(x)\equiv\mathrm{tr}[T^{1}_{L}\bar{T}^{1}_{R}+T^{2}_{L}\bar{T}^{2}_{R}+T^{l}_{L}\bar{T}^{l}_{R}+T^{t}_{L}\bar{T}^{t}_{R}+T^{b}_{L}\bar{T}^{b}_{R}](x)$
(14)
In this case, $\mathrm{tr}$ are the traces with respect to the Lorentz,
$SU(N)_{\mathrm{TC}}$, $SU(3)_{1}$ and $SU(3)_{2}$ indices. The transformation
of $O(x)$ under $SU(2)_{L}\times U(1)_{Y}$ is
$O(x)\rightarrow V_{L}(x)O(x)V_{R}^{\dagger}(x)\hskip
56.9055ptV_{L}(x)=e^{i\frac{\tau^{a}}{2}\theta^{a}(x)}\qquad
V_{R}(x)=e^{-i\frac{\tau^{3}}{2}\theta^{0}(x)}\;.$ (15)
Then we decompose $O(x)$ as
$O(x)=\xi^{\dagger}_{L}(x)\sigma(x)\xi_{R}(x)$ (16)
Where $\sigma(x)$ which is represented using a Hermitian matrix, describes the
modular degree of freedom; and $\xi_{L}(x)$ and $\xi_{R}(x)$, which are
represented using unitary matrices, describe the phase degrees of freedom of
$SU(2)_{L}$ and $U(1)_{Y}$ respectively. Their transformations under
$SU(2)_{L}\otimes U(1)_{Y}$ are
$\displaystyle\sigma(x)\rightarrow h(x)\sigma(x)h^{\dagger}(x)\hskip
28.45274pt\xi_{L}(x)\rightarrow h(x)\xi_{L}(x)V^{\dagger}_{L}(x)\hskip
28.45274pt\xi_{R}(x)\rightarrow
h(x)\xi_{R}(x)V^{\dagger}_{R}(x)~{}~{}~{}~{}~{}~{}$ (17)
where
$h(x)=e^{i\theta_{h}(x)\frac{\tau^{3}}{2}}$ (18)
belongs to an induced hidden local $U(1)$ symmetry group. Next, we define a
new field
$U(x)\equiv\xi_{L}^{\dagger}(x)\xi_{R}(x)\;,$ (19)
which is the nonlinear realization of the Goldstone boson field in the EWCL.
Subtracting the $\sigma(x)$ field, we find that the present decomposition
results in a constraint
$\xi_{L}(x)O(x)\xi_{R}^{\dagger}(x)-\xi_{R}(x)O^{\dagger}(x)\xi^{\dagger}_{L}(x)=0$
and its functional expression is
$\int\mathcal{D}_{\mu}(U)\mathcal{F}[O]\delta(\xi_{L}O\xi^{\dagger}_{R}-\xi_{R}O^{\dagger}\xi^{\dagger}_{L})=\mathrm{const}\;,$
(20)
where $\mathcal{D}_{\mu}(U)$ is an effective invariant integration measure;
and $\mathcal{F}[O]$ only depends on $O$ and is invariant under
$SU(2)_{L}\otimes U(1)_{Y}$ transformations. This causes the value of the
integrated quantity to be a constant. Inserting the above identity into (9),
we have
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}\mathcal{D}T^{1}\mathcal{D}\bar{T}^{2}\mathcal{D}T^{2}\mathcal{D}\bar{T}^{l}\mathcal{D}T^{l}\mathcal{D}\bar{T}^{t}\mathcal{D}T^{t}\mathcal{D}\bar{T}^{b}\mathcal{D}T^{b}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(21)
$\displaystyle\times\int\mathcal{D}_{\mu}(U)\mathcal{F}[O]\delta(\xi_{L}O\xi^{\dagger}_{R}-\xi_{R}O^{\dagger}\xi^{\dagger}_{L})e^{iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T},T,\bar{\psi},\psi]}\bigg{|}_{A^{A}_{\mu}=0}\\!.~{}~{}~{}$
Using a special $SU(2)_{L}\otimes U(1)_{Y}$ rotation for $V_{L}(x)=\xi_{L}(x)$
and $V_{R}(x)=\xi_{R}(x)$ and labeling the fields after rotation with the
subscript, ξ, the above path integral becomes:
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}_{\xi}\mathcal{D}T^{1}_{\xi}\mathcal{D}\bar{T}^{2}_{\xi}\mathcal{D}T^{2}_{\xi}\mathcal{D}\bar{T}^{l}_{\xi}\mathcal{D}T^{l}_{\xi}\mathcal{D}\bar{T}^{t}_{\xi}\mathcal{D}T^{t}_{\xi}\mathcal{D}\bar{T}^{b}_{\xi}\mathcal{D}T^{b}_{\xi}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(22)
$\displaystyle\times\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})e^{iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}\bigg{|}_{A^{A}_{\mu}=0}\\!.~{}~{}~{}$
where we have used the result that the functional integration measure,
$\mathcal{F}[O]$ and the action on the exponential of the integrand are
invariant under $SU(2)_{L}\otimes U(1)_{Y}$ transformations. From Table I, it
can be seen that:
$\displaystyle T^{1}_{\xi
L}=e^{-i(u_{1}+u_{2})\theta_{0}}P_{L}\xi_{L}T^{1}_{L}\hskip
56.9055ptT^{1}_{\xi R}=e^{-i(v_{1}+v_{2})\theta_{0}}P_{R}\xi_{R}T^{1}_{R}$
$\displaystyle T^{2}_{\xi
L}=e^{-i(v_{1}+v_{2})\theta_{0}}P_{L}\xi_{L}T^{2}_{L}\hskip
56.9055ptT^{2}_{\xi R}=e^{-i(u_{1}+u_{2})\theta_{0}}P_{R}\xi_{R}T^{2}_{R}$
$\displaystyle T^{l}_{\xi
L}=e^{-i(x_{1}+x_{2})\theta_{0}}P_{L}\xi_{L}T^{l}_{L}\hskip
56.9055ptT^{l}_{\xi
R}=e^{-i(x^{\prime}_{1}+x^{\prime}_{2})\theta_{0}}P_{R}\xi_{R}T^{l}_{R}$ (23)
$\displaystyle T^{t}_{\xi
L}=e^{-i(y_{1}+y_{2})\theta_{0}}P_{L}\xi_{L}T^{t}_{L}\hskip
56.9055ptT^{t}_{\xi
R}=e^{-i(y^{\prime}_{1}+y^{\prime}_{2})\theta_{0}}P_{R}\xi_{R}T^{t}_{R}$
$\displaystyle T^{b}_{\xi
L}=e^{-i(z_{1}+z_{2})\theta_{0}}P_{L}\xi_{L}T^{b}_{L}\hskip
56.9055ptT^{b}_{\xi
R}=e^{-i(z^{\prime}_{1}+z^{\prime}_{2})\theta_{0}}P_{R}\xi_{R}T^{b}_{R}\;,$
Furthermore,
$\displaystyle
g_{2}\frac{\tau^{a}}{2}W^{a}_{\xi,\mu}=\xi_{L}[g_{2}\frac{\tau^{a}}{2}W^{a}_{\mu}-i\partial_{\mu}]\xi_{L}^{\dagger}$
(24) $\displaystyle
g_{1}\frac{\tau^{3}}{2}B_{\xi,\mu}=\xi_{R}[g_{1}\frac{\tau^{3}}{2}B_{\mu}-i\partial_{\mu}]\xi_{R}^{\dagger}\hskip
28.45274pt\begin{pmatrix}B_{1\xi,\mu}&B_{2\xi,\mu}\end{pmatrix}=\begin{pmatrix}Z_{\mu}^{\prime}&B_{\xi,\mu}\end{pmatrix}\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\;.~{}~{}~{}~{}$ (25)
Note the fields without the subscript ξ in (22) are the fields that are
invariant under $SU(2)_{L}\otimes U(1)_{Y}$ rotation.
### III.2 Integrating out the technigluons
As a second step,we integrate out the technigluon in (22) using:
$\displaystyle\int\mathcal{D}G_{\mu}^{\alpha}e^{iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}=e^{iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]+iS_{\mathrm{TC1}}[A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}\;,~{}~{}~{}$
(26)
where we choose
$\displaystyle
e^{iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}=\int\mathcal{D}G_{\mu}^{\alpha}~{}e^{i\int
d^{4}x(-\frac{1}{4}G_{\mu\nu}^{\alpha}G^{\alpha,\mu\nu}-g_{TC}G_{\mu}^{\alpha}J^{\mu\alpha})}$
(27) $\displaystyle
S_{\mathrm{TC1}}[A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]=S[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]\bigg{|}_{G_{\mu}^{\alpha}=0}~{}~{}$
(28)
and
$\displaystyle J^{\mu\alpha}$ $\displaystyle=$
$\displaystyle\bar{\psi}\tilde{t}^{\alpha}\gamma^{\mu}\psi+\tilde{J}^{\mu\alpha}$
(29) $\displaystyle\tilde{J}^{\mu\alpha}$ $\displaystyle=$
$\displaystyle\bar{T}^{1}_{\xi}t^{\alpha}\gamma^{\mu}T^{1}_{\xi}+\bar{T}^{2}_{\xi}t^{\alpha}\gamma^{\mu}T^{2}_{\xi}+\bar{T}^{l}_{\xi}t^{\alpha}\gamma^{\mu}T^{l}_{\xi}+\bar{T}^{t}_{\xi}t^{\alpha}\gamma^{\mu}T^{t}_{\xi}+\bar{T}^{b}_{\xi}t^{\alpha}\gamma^{\mu}T^{b}_{\xi}\;.$
(30)
Integrating out the technigluon fields in (27), we get
$\displaystyle
iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]=\sum_{n=2}^{\infty}\int
d^{4}x_{1}\ldots
d^{4}x_{n}\frac{(-ig_{\mathrm{TC}})^{n}}{n!}G_{\mu_{1}\ldots\mu_{n}}^{\alpha_{1}\ldots\alpha_{n}}(x_{1},\ldots,x_{n})J_{\alpha_{1}}^{\mu_{1}}(x_{1})\ldots
J_{\alpha_{n}}^{\mu_{n}}(x_{n})\;,~{}~{}~{}$ (31)
where
$G_{\mu_{1}\ldots\mu_{n}}^{\alpha_{1}\ldots\alpha_{n}}(x_{1},\ldots,x_{n})$ is
a n-point Green’s function for the technigluons.
### III.3 Integrating out the techniquarks
Combining (22) and (26), our starting $S_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]$,
after integrating in the electroweak Goldstone boson field $U$ and integrating
out the technigluons, becomes
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}_{\xi}\mathcal{D}T^{1}_{\xi}\mathcal{D}\bar{T}^{2}_{\xi}\mathcal{D}T^{2}_{\xi}\mathcal{D}\bar{T}^{l}_{\xi}\mathcal{D}T^{l}_{\xi}\mathcal{D}\bar{T}^{t}_{\xi}\mathcal{D}T^{t}_{\xi}\mathcal{D}\bar{T}^{b}_{\xi}\mathcal{D}T^{b}_{\xi}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
$\displaystyle\times\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})e^{iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]+iS_{\mathrm{TC1}}[A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}\bigg{|}_{A^{A}_{\mu}=0}\\!.$
After some detailed derivations and approximations which can be found in
Appendix A, we get:
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})\int\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}~{}\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}$ (33)
$\displaystyle+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1,\mu\nu}B^{1,\mu\nu}+B_{2,\mu\nu}B^{2,\mu\nu})]$
$\displaystyle+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{]}_{A^{A}_{\mu}=0}\;,$
The various quantities appearing in (33) are defined at the end of Appendix A.
Furthermore, in Appendix B, we have shown that the techniquark self energies
$\tilde{\Sigma}$, $\hat{\Sigma}$, $\bar{\Sigma}$ and $\bar{\Sigma}_{5}$
satisfy the following SDEs,
$\displaystyle\tilde{\Sigma}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N+1)(N-2)}{4\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\tilde{\Sigma}(q_{E}^{2})}{q_{E}^{2}+\tilde{\Sigma}^{2}(q_{E}^{2})}$
(34) $\displaystyle\hat{\Sigma}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N^{2}-1)}{8\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\hat{\Sigma}(q_{E}^{2})}{q_{E}^{2}+\hat{\Sigma}^{2}(q_{E}^{2})}$
(35) $\displaystyle\bar{\Sigma}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N^{2}-1)}{8\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\bar{\Sigma}(q_{E}^{2})}{q_{E}^{2}+\bar{\Sigma}^{2}(q_{E}^{2})+\bar{\Sigma}_{5}^{2}(q_{E}^{2})}$
(36) $\displaystyle\bar{\Sigma}_{5}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N^{2}-1)}{8\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\bar{\Sigma}_{5}(q_{E}^{2})}{q_{E}^{2}+\bar{\Sigma}^{2}(q_{E}^{2})+\bar{\Sigma}_{5}^{2}(q_{E}^{2})}\;,$
(37)
where the technigluon propagator is parameterized though the TC running
coupling constant $\alpha$ as
$\displaystyle
G_{\mu\nu}^{\alpha\beta}(x,y)=\\!\int\\!\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\frac{-i\delta^{\alpha\beta}}{p^{2}[1\\!+\\!\Pi(-p^{2})]}\bigg{(}g_{\mu\nu}\\!-\frac{p_{\mu}p_{\nu}}{p^{2}}\bigg{)}\hskip
28.45274pt\alpha(p_{E}^{2})\equiv\frac{g^{2}_{\mathrm{TC}}}{4\pi[1\\!+\\!\Pi(p_{E}^{2})]}\;.~{}~{}~{}$
(38)
### III.4 Integrating out the colorons and the low energy expansion
Before integrating out the coloron field, we first discuss its mass which is
determined by the kinetic and mass terms. From the exponential of the
integrand in (33), it can be seen that there is already a standard coloron
kinetic term from
$-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu})$. The
first set of techniquarks contributes to the quantum loop corrections to the
coloron kinetic and mass terms through the term
$\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]$
in (33). Through detailed computations, we find that these corrections are
$\displaystyle\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{|}_{\mbox{\tiny
coloron kinetic and mass terms}}$ $\displaystyle=\frac{i}{4}\int
d^{4}x\bigg{[}Cg_{3}^{2}(\tan\theta^{\prime}\\!+\\!\tan\theta^{\prime})^{2}B^{A}_{\mu}B_{A}^{\mu}-(\partial^{\mu}B^{A}_{\nu}-\partial^{\nu}B^{A}_{\mu})^{2}[\mathcal{K}g_{3}^{2}(\cot^{2}\theta^{\prime}\\!+\\!\tan^{2}\theta^{\prime})$
$\displaystyle+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}g_{3}^{2}(\tan\theta^{\prime}\\!-\\!\cot\theta^{\prime})^{2}+\frac{1}{2}\hat{E}g_{3}^{2}(\tan\theta^{\prime}+\cot\theta^{\prime})^{2}]\bigg{]}\;,$
(39)
In this case, the coefficients are given at the beginning of Appendix C.
Combining the standard coloron kinetic term in (33) and the techniquark
quantum loop correction given by (39), we find the formula for the coloron
mass to be:
$\displaystyle
M_{\mathrm{coloron}}^{2}=\frac{C}{\hat{E}+2(\mathcal{K}+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})+(2/g_{3}^{2}-8\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})/(\cot\theta^{\prime}+\tan\theta^{\prime})^{2}}\;.$ (40)
In Appendix C, we integrate out the coloron fields and perform the low energy
expansion. Finally we obtain,
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle e^{i\int
d^{4}x[-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}]}\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})\int\mathcal{D}Z_{\mu}^{\prime}~{}e^{iS_{0}+iS_{Z^{\prime}}}\;.~{}~{}~{}~{}~{}$
(41)
Where detailed expressions of $S_{0}$ and $S_{Z^{\prime}}$ are given in (112)
and (116) respectively in Appendix C.
### III.5 Integrating out $Z^{\prime}$
We denote the resulting action after the integration over $Z^{\prime}$ as
$\displaystyle\int\mathcal{D}Z^{\prime}_{\mu}~{}e^{iS_{Z^{\prime}}}=e^{i\bar{S}_{Z^{\prime}}}\;.$
(42)
We can use the loop expansion to calculate the above integral:
$\displaystyle\bar{S}_{Z^{\prime}}=S_{Z^{\prime}}\bigg{|}_{Z^{\prime}=Z^{\prime}_{c}}+\mbox{loop
corrections}$ (43)
where the classical field $Z^{\prime}_{c}$ satisfies:
$\displaystyle\frac{\partial}{\partial
Z^{\prime}_{c,\mu}(x)}\bigg{[}S_{Z^{\prime}}+\mbox{loop
corrections}\bigg{]}=0\;.$ (44)
Using this method, we integrate out the $Z^{\prime}$ field in Appendix D and
simplify the result $\bar{S}_{Z^{\prime}}$ given in (140) into the form of
EWCL. Furthermore, combining (42) and (41) together, we find
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle e^{i\int
d^{4}x[-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}]}\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})~{}e^{iS_{0}+i\bar{S}_{Z^{\prime}}}\;.~{}~{}~{}~{}~{}$
(45)
Comparing this with (10) and Table.III, we obtain all the EWCL LECs. Our final
analytical results for the EWCL LECs (up to an order of $p^{4}$) are
$\displaystyle f^{2}=5\hat{F}_{0}^{2}\hskip
28.45274pt\beta_{1}=\frac{10a_{3}^{2}\hat{F}_{0}^{2}}{\bar{M}_{Z^{\prime}}^{2}}\hskip
28.45274pt\alpha_{1}=\frac{5}{2}(1-2\beta_{1})(\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})+\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\gamma\beta_{1}}{2a_{3}}$
$\displaystyle\alpha_{2}=(\beta_{1}-\frac{1}{2})(\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})+\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\gamma\beta_{1}}{2a_{3}}\hskip
28.45274pt\alpha_{3}=(\beta_{1}-\frac{1}{2})(\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0})~{}~{}~{}~{}~{}~{}$
$\displaystyle\alpha_{4}=(2\beta_{1}+\frac{1}{4})(\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})+(\frac{5}{16}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-\frac{5}{32}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})+\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}$
$\displaystyle\alpha_{5}=-\frac{5}{2}(4\beta_{1}+\frac{1}{4})\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{4}(3\beta_{1}+\frac{1}{4})\hat{\mathcal{K}}_{14}^{\Sigma\neq
0}+\frac{5}{32}(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\hat{\mathcal{K}}_{4}^{\Sigma\neq
0})-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}$
$\displaystyle\alpha_{6}=-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\beta_{1}^{2}}{4a_{3}^{2}}[-(2a_{0}^{2}+\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-10a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+5a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}$ $\displaystyle\hskip
14.22636pt+2a_{0}^{2}\hat{D}_{4}]-\frac{\beta_{1}}{2}(\frac{5}{2}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+15\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}-5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})$
$\displaystyle\alpha_{7}=\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\beta_{1}^{2}}{4a_{3}^{2}}[(\frac{5}{2}a_{3}^{2}+a_{0}^{2}+\frac{1}{2}\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+(a_{0}^{2}+\frac{1}{2}\hat{a}_{0}^{2}-\frac{5}{2}a^{2}_{3})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-10a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+5a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}+a_{0}^{2}\hat{D}_{3}]$
$\displaystyle\hskip
14.22636pt-\frac{\beta_{1}}{2}(\frac{5}{4}\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\frac{5}{4}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-15\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}+5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})$
$\displaystyle\alpha_{8}=-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}+10\beta_{1}(\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}\\!-\hat{\mathcal{K}}_{13}^{\Sigma\neq 0})\hskip
28.45274pt\alpha_{9}=-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}\\!+\beta_{1}(5\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}\\!-10\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}\\!+\frac{5}{4}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0})~{}~{}~{}~{}$
$\displaystyle\alpha_{10}=\frac{5\beta_{1}^{2}}{4}(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0})+\frac{\beta_{1}^{4}}{8a_{3}^{4}}g_{4Z}-\frac{\beta_{1}^{3}}{2a_{3}^{3}}[(2a_{3}^{3}+6a_{0}^{2}a_{3}+3\hat{a}_{0}^{2}a_{3})(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq 0})+2a_{0}^{2}a_{3}\hat{D}_{2}]$
$\displaystyle\alpha_{11}=\alpha_{12}=\alpha_{13}=\alpha_{14}=0\;.$ (46)
## IV Numerical results and discussion
We first analyze the general features of the EWCL LECs obtained in the
previous section, which are similar to those in Lane’s first natural prototype
TC2 modelJunYi09 :
* •
The contributions of the $p^{4}$-order coefficients are divided into two
parts: the contribution from the three sets of techniquarks and the
$Z^{\prime}$ contribution
* •
All correction terms from the $Z^{\prime}$ particle to the EWCL LECs are
proportional to powers of $\beta_{1}$ which vanish if the mixing disappear
($\theta=0$). This can be seen from (46) and (128) which show that:
$\beta_{1}=\frac{10g_{1}^{2}\hat{F}_{0}^{2}\tan^{2}\theta}{16\bar{M}_{Z^{\prime}}^{2}}$.
By using the relation $\alpha_{\mathrm{em}}T=2\beta_{1}$, we can express all
LECs in terms of the $T$ parameter. Later in the paper, we show the $T$
dependence of the LECs.
* •
From (46) (for $f^{2}$ and $\beta_{1}$), combined with (128), (121) , the
relation $\alpha_{\mathrm{em}}T=2\beta_{1}$ and the relationships of the
hyper-charges from Ref.Lane96 , we have
$\displaystyle\alpha_{\mathrm{em}}T$ $\displaystyle=$
$\displaystyle\bigg{[}1+\frac{2}{5}[\frac{81\tilde{F}_{0}^{2}}{4\hat{F}_{0}^{2}}+716+4(1-\frac{F_{0}^{\prime
2}}{\hat{F}_{0}^{2}})](u_{1}-v_{1})^{2}(1+\cot^{2}\theta)^{2}\bigg{]}^{-1}\;.~{}~{}~{}$
If we include the numerical result that $F^{\prime 2}_{0}<\hat{F_{0}^{2}}$,
the above result implies that $T$ must be positive and has an upper bound. The
upper bound is:
$\displaystyle\alpha_{\mathrm{em}}T_{\mathrm{Max}}=\frac{1}{1+\frac{2}{5}[\frac{81\tilde{F}_{0}^{2}}{4\hat{F}_{0}^{2}}+716+4(1-\frac{F_{0}^{\prime
2}}{\hat{F}_{0}^{2}})](u_{1}-v_{1})^{2}}\;.~{}~{}$ (47)
* •
Because numerical calculation shows that $\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}\\!-\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}<0$ and $\beta_{1}$ is positive,
$\alpha_{8}$ is negative based on (46). Then $U=-16\pi\alpha_{8}$ which is a
coefficient given in Ref.EWCL , is always positive in the present model.
Combining (121), (122) and (136), we find,
$\displaystyle
2\frac{\tilde{F}_{0}^{2}}{M_{Z^{\prime}}^{2}}g_{1}^{2}(\cot\theta+\tan\theta)^{2}\xi^{2}+4\frac{\hat{F}_{0}^{2}}{M_{Z^{\prime}}^{2}}(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})-8\frac{F^{\prime
2}_{0}}{M_{Z^{\prime}}^{2}}a_{0}^{2}$ (48)
$\displaystyle=1+[4(\cot\theta+\tan\theta)^{2}\xi^{2}+2\tan^{2}\theta+8\hat{v}+3\tan^{2}\theta+\hat{y}]\mathcal{K}g_{1}^{2}+4(\cot\theta+\tan\theta)^{2}\xi^{2}\tilde{\mathcal{K}}_{2}^{\Sigma\neq
0}g_{1}^{2}$
$\displaystyle+8(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+[40a_{3}^{2}+2(\hat{t}+\hat{s})g_{1}^{2}]\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-15\hat{D}_{0}a_{0}^{2}\;.$
We treat the above equation as a constraint on $\mathcal{K}$. This is done as
following: A suitable choice is made for the hypercharges (this will be
discussed later), electroweak gauge coupling, $T$ and $M_{Z^{\prime}}$. We
already know most of the parameters in (48), except $\tilde{F}_{0}$,
$\hat{F}_{0}$, $F^{\prime 2}_{0}$, $\tilde{\mathcal{K}}_{2}^{\Sigma\neq 0}$,
$\hat{\mathcal{K}}_{2}^{\Sigma\neq 0}$, $\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}$ and $\hat{D}_{0}$. By solving the SDEs, (34), (35), (36), (37), we can
obtain the techniquark self-energies, $\tilde{\Sigma}$, $\hat{\Sigma}$,
$\bar{\Sigma}$, $\bar{\Sigma}_{5}$. Furthermore, substituting the resulting
techniquark self-energies into the formulae given in Appendix E and (107), we
can obtain $\tilde{F}_{0}$, $\hat{F}_{0}$, $F^{\prime 2}_{0}$,
$\tilde{\mathcal{K}}_{2}^{\Sigma\neq 0}$, $\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}$, $\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}$ and $\hat{D}_{0}$ from (48). Now,
aside from $\mathcal{K}$ all the parameters in (48) are known. Then we can use
(48) to fix the value of $\mathcal{K}$. Once $\mathcal{K}$ is fixed, with the
help of (96), we can determine the ratio of the infrared cutoff $\kappa$ and
ultraviolet cutoff $\Lambda$. Numerical calculations show that this is unlike
the results in Refs.JunYi09 ; LangPLB , where the condition $\Lambda>\kappa$
occurs through the definitions used for the calculations and offers stringent
constraints on the allowed region for $T$ and the upper bound for
$M_{Z^{\prime}}$. In our model, $\Lambda>\kappa$ is naturally satisfied for
real values of $M_{Z^{\prime}}$. For example, we find that $\ln\kappa/\Lambda$
is about $-7.6$ and $-9.0$ for $M_{Z^{\prime}}$ values of 0.5TeV and 1TeV,
respectively.
With the above qualitative features, we now can generate numerical results.
First, we take $N=6$ which yields an infrared fixed point of
$\alpha_{w}=88\pi/523$. Then, we take $f=250$GeV. This completely fixes the
two-loop value at $\Lambda_{w}=5.5$TeV through the running behavior of (7),
SDE (35), $f^{2}=5\hat{F}_{0}^{2}$ and (142) which sets up the relationship
between $\hat{F}_{0}^{2}$ techniquark self-energy. This value of $\Lambda_{w}$
is smaller than the expected conventional ETC scale. Therefore, we cannot
interpret it as $\Lambda_{\mathrm{ETC}}$. Later, we will see that this is
because the walking effect is not large enough, and more ideal walking can
lead to a larger $\Lambda_{w}$. The current result with
$\Lambda_{w}\ll\Lambda_{\mathrm{ETC}}$ shows that our running coupling
constant cannot always walk from extreme infrared energy regions to the ETC
scale, $\Lambda_{\mathrm{ETC}}$. Instead, it can only walk a shorter distance
to the scale, $\Lambda_{w}$. Beyond $\Lambda_{w}$, it will run and fall
quickly exhibiting conventional asymptotic freedom behavior. Another
theoretical parameter is the coloron mass given by (40), which theoretically
depends on the values $\theta^{\prime}$, introduced in (12) and $\Theta$,
introduced in (91). We find the largest coloron mass occurs for
$\Theta=\pi/2$, i.e., the self-energies for the first set of techniquarks are
completely contributed by the twisted part of the set,
$\bar{\Sigma}_{5}=\hat{\Sigma}\sin\Theta$ and $\bar{\Sigma}=0$. Using this
value of $\Theta=\pi/2$, in Fig.3, we plot the coloron mass in terms of the
$T$ parameter. We used four values of $M_{Z^{\prime}}=0.5,1,2,5$TeV
(corresponding to $\ln\kappa/\Lambda\sim$ -7.6,-9.0,-9.4 and -9.5). We found
that that the coloron mass is not sensitive to $\theta^{\prime}$.
Figure 3: Coloron mass for Lane’s model.
From Fig.3, it can be seen that the coloron mass is roughly half the 1 TeV
expected in Lane’s original paperLane96 . The reason is that we included a
techniquark loop correction in the coloron kinetic term, which appeared in
(40) with the coefficients $\hat{E}$, $\mathcal{K}$ and
$\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}$. If we denote the coloron mass without
this correction as $M_{\mathrm{bare~{}coloron}}$ which was the notation used
in Lane’s original workLane96 , then our numerical calculation shows that:
$M_{\mathrm{bare~{}coloron}}/M_{\mathrm{coloron}}\sim\frac{2}{3}(\tan\theta^{\prime}+\cot\theta^{\prime})$.
This leads to a larger value for $M_{\mathrm{bare~{}coloron}}$. In fact, if we
carefully examine the denominator of (40), the structure of this kinetic term
correction can be divided into three parts: the tree order term
$2/g_{3}^{2}(\cot\theta^{\prime}+\tan\theta^{\prime})^{2}$, the techniquark
self-energy dependent part $\hat{E}+2\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-8\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}/(\cot\theta^{\prime}+\tan\theta^{\prime})^{2}$, and the techniquark self-
energy independent part $2\mathcal{K}$. The numerical calculation shows that
the main contribution comes from the techniquark self-energy dependent part,
which is an order of magnitude larger than the contributions from the other
two parts. Because the coloron mass is small666The small coloron mass forces
us to switch the order of integration over the coloron and Z’, i.e., instead
of integrating out the coloron before the Z’ boson, we need to integrate out
Z’ and then the coloron. We have performed the computation using this new
procedure and found the same result as that of the present paper, i.e.
switching the order of integration yields no correction. We found that the
possible correction from switching this order of integration depends on the
classical field $B_{A,c}^{\mu}$ caused by the coloron integration. These
classical coloron fields are determined by stationary equations. In both
cases, the stationary equations offer the null solution, $B_{A,c}^{\mu}=0$ ,
which was used in our results., we will use $\Theta=\pi/2$ to give the largest
coloron mass for all the following computations.
To provide numerical values for all the EWCL LECs, we need to choose the
various hyper-charges for the model. Note that the arrangement of the hyper-
charges given in Lane’s original paperLane96 is not suitable here because
that result used $N=4$. We showed in Section II that for the modern
interpretation of our two-loop based phase structure model, we use $N=6$, and
recalculate the hyper-charges. According to a series of relations among
different hyper-charges given by K. Lane in Ref.Lane96 , we need to use three
hyper-charges $x_{1}$, $y_{1}$ and $y_{1}+y_{2}$. We use a treatment similar
to the one used by K. Lane in Ref.Lane96 . Namely, we use $x_{1}=y_{1}$,
$y_{1}+y_{2}=0$. Furthermore, this requires that $u=(u_{1}-v_{1})/2\sim 1$.
These fully fix the typical values of all the hyper-charges. By ”typical” we
mean that the value of the hyper-charges must satisfy all 23 constraint
equations given in Ref. Lane96 and two more constraints: $x_{1}=y_{1}$,
$y_{1}+y_{2}=0$. The last two constraints were not explicitly mentioned in
Ref.Lane96 , but the detailed example used them. These typical hyper-charges
are: $a=-39$, $a^{\prime}=-46$, $b=14$, $b^{\prime}=8.2$, $c=-39$,
$c^{\prime}=-46$, $d=-12$, $d^{\prime}=-14$, $\xi=4.6$, $\xi^{\prime}=-4.6$,
$x_{1}=25$, $x^{\prime}_{1}=19$, $x_{2}=-26$, $x^{\prime}_{2}=-19$,
$y_{1}=25$, $y^{\prime}_{1}=23$, $y_{2}=-25$, $y^{\prime}_{2}=-23$,
$z_{1}=-7.7$, $z^{\prime}_{1}=19$, $z_{2}=7.7$, $z^{\prime}_{2}=-19$,
$u_{1}=-4.1$, $v_{1}=-6.1$, $u_{2}=4.2$, $v_{2}=6.2$.Using this set of typical
hyper-charges, combined with the other necessary inputs for the model, which
were discussed in the previous paragraph, (47) yields an upper bound,
$T_{\mathrm{max}}=0.035$. We show $S=-16\pi\alpha_{1}$ in Fig.4, and
$U=-16\pi\alpha_{8}$ in Fig.5.
Figure 4: $S$ parameter for Lane’s model.
Figure 5: $U$ parameter for Lane’s model.
From Fig.4, it can be seen that the value of $S$ is generally larger than 2,
which is not in agreement with experimental data. This value of the $S$
parameter already includes the walking effects in the model, which we will
discuss later. To examine the possibility of reducing the value of the $S$
parameter through the choice of hyper-charges, we found that when the input
hyper-charges $x_{1},y_{1}$ are not constrained by the requirement
$x_{1}=y_{1}$ and are much larger than 1, $S$ may achieve small values. Fig.6
shows the case with: $x_{1}=-50,y_{1}=36,y_{2}=-12$ which leads $a=-19$,
$a^{\prime}=-22$, $b=7$, $b^{\prime}=4$, $c=-19$, $c^{\prime}=-22$, $d=-6$,
$d^{\prime}=-7$, $\xi=2.3$, $\xi^{\prime}=-2.3$, $x_{1}=-50$,
$x^{\prime}_{1}=-53$, $x_{2}=2.7$,
$x^{\prime}_{2}=5.7$,$y_{1}=36$,$y^{\prime}_{1}=35$,$y_{2}=-12$,
$y^{\prime}_{2}=-11$, $z_{1}=20$, $z^{\prime}_{1}=33$, $z_{2}=3.6$,
$z^{\prime}_{2}=-9.4$, $u_{1}=0.41$, $v_{1}=-0.59$, $u_{2}=-0.41$,
$v_{2}=0.59$. The $S$ parameter can achieve negative values with larger values
of $T$. There may be other sets of hyper-charges which can also yield small or
even negative values of $S$, but typically these hyper-charges have large
values.
Figure 6: $S$ parameter for various choices of the hyper-charges:
$x_{1}=-50,y_{1}=36,y_{2}=-12$.
Excluding the $S$ and $U$ parameters, the leftmost eight non-zero parameters
$\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6},\alpha_{7},\alpha_{9},\alpha_{10}$
are shown in Fig.7 to Fig.13. $\alpha_{3}$ and $\alpha_{10}$ are independent
of $M_{Z^{\prime}}$ and are shown in the same figure.
Figure 7: $\alpha_{2}$ parameter for Lane’s model.
Figure 8: $\alpha_{3}$ and $\alpha_{10}$ parameters for Lane’s model.
Figure 9: $\alpha_{4}$ parameter for Lane’s model.
Figure 10: $\alpha_{5}$ parameter for Lane’s model.
Figure 11: $\alpha_{6}$ parameter for Lane’s model.
Figure 12: $\alpha_{7}$ parameter for Lane’s model.
Figure 13: $\alpha_{9}$ parameter for Lane’s model.
We found that $\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},$ are on the order
of $10^{-2}$, $\alpha_{6},\alpha_{7},\alpha_{9}$ are on the order of $10^{-5}$
and $\alpha_{10}$ is on the order of $10^{-10}$.
Previously, we discussed the three other TC2 modelsHill95 ; Lane95 ; Sekhar .
In Table IV., we list the different features and the orders of magnitude for
all the LECs of these TC2 models. In Fig.14, Fig.15, Fig.16,Fig.17 and Fig.18,
we show the ten nonzero LECs from these four TC2 models for comparison. This
comparison may be useful to other researchers as they consider the needs of
future models.
TABLE IV. Features and LECs of the TC2 models Hill95 , Lane95 , Sekhar and
Lane96
Property or LEC | Schematic TC2${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Hill95}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | Natural TC2${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Lane95}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | Hypercharge Universal${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Sekhar}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ | Present${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Lane96}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
---|---|---|---|---
Upper bound of $M_{Z^{\prime}}$ | $\surd$ | $\surd$ | $\surd$ | $\times$
Negative $S$ | $M_{Z^{\prime}}\\!<\\!0.44$TeV or $T\\!>\\!0.17$ | $\times$ | $T\geq 10^{-1}$ | choose hypercharges
Typical $S\\!=\\!-16\pi\alpha_{1}$ | $\sim 0.3$ | $\sim 0.8$ | $\sim 1$ | $\sim 2$
$\alpha_{2}$ | $-10^{-3}$ | $-10^{-3}$ | $-10^{-3}$ | $-10^{-2}$
$\alpha_{3}$ | $-10^{-3}$ | $3\times$ result of Hill95 | $-10^{-3}$ | $-10^{-2}$
$\alpha_{4}$ | $10^{-3}$ | $3\times$ result of Hill95 | $10^{-3}$ | $10^{-2}$
$\alpha_{5}$ | $-10^{-3}$ | $3\times$ result of Hill95 | $-10^{-3}$ | $-10^{-2}$
$\alpha_{6}$ | $\sim-10^{-4}$ | $\sim-10^{-3}$ | $\sim-10^{-4}$ | $\sim-10^{-5}$
$\alpha_{7}$ | $\sim 10^{-4}$ | $\sim 10^{-3}$ | $\sim 10^{-4}$ | $\sim 10^{-5}$
$\alpha_{8}=-\frac{U}{16\pi}$ | $\sim-10^{-4}$ | $3\times$ result of Hill95 | $\sim-10^{-4}$ | $\sim-10^{-5}$
$\alpha_{9}$ | $\sim-10^{-4}$ | $3\times$ result of Hill95 | $\sim-10^{-4}$ | $\sim-10^{-5}$
$\alpha_{10}$ | $\sim-10^{-8}$ | $\sim-10^{-8}$ | $\sim-10^{-7}$ | $\sim 10^{-10}$
Figure 14: $\alpha_{1}$ and $\alpha_{2}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.
Figure 15: $\alpha_{3}$ and $\alpha_{4}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.
Figure 16: $\alpha_{5}$ and $\alpha_{6}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.
Figure 17: $\alpha_{7}$ and $\alpha_{8}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.
Figure 18: $\alpha_{9}$ and $\alpha_{10}$ of the TC2 model Hill95 -Hill,
Lane95 -Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve
are the masses of the $Z^{\prime}$ boson in TeV.
Finally, we estimate the magnitude of the walking effect in the present model.
Because the primary contribution to the walking effect is from the running
coupling constant, which appears in the kernel of the SDE, we can measure the
walking effect by comparing two other running behaviors:
* •
Running $\alpha$: Rather than using a two-loop running coupling constant (7)
which exhibits an approximation of walking behavior in $N=6$ and spontaneous
chiral symmetry breaking, we used the one-loop running coupling constant used
in our previous workHongHao08 ; JunYi09 ; LangPLB as
$\displaystyle\alpha(x)=\frac{4\pi}{\beta_{0}}\times\left\\{\begin{array}[]{lll}7&&\ln
x\leq-2\\\ 7-\frac{4}{5}(2+\ln x)^{2}&&-2\leq\ln x\leq 0.5\\\ \frac{1}{\ln
x}&&\ln x\geq 0.5\end{array}\right.\hskip
56.9055ptx=\frac{p^{2}}{\Lambda^{2}_{\mathrm{TC}}}\;.~{}~{}~{}~{}$ (52)
Eq.(52) was originally introduced in Ref.Runalpha . The general principle of
the technique is to use a plateau in the low energy region to normalize the
possibly infinite value in the infrared region that is predicted using the
perturbative result and smoothly connect this infrared plateau with the
ultraviolet asymptotic freedom running behavior. Note that if we ignore the
two-loop term in the $\beta$ function in this model and normalize the infrared
coupling constant such that it has a finite value, we can qualitatively obtain
the above form of the running coupling constant. Furthermore, this
approximation at the one-loop level suggests that $\Lambda_{w}$ must be
treated as $\Lambda_{\mathrm{TC}}$ in this running situation. The change from
one-loop running to two-loop walking reflects the evolution of our
understanding of the gauge-coupling running behavior in non-abelian gauge
theory. In addition, the decision to use the latter model in this study is
important because it confirms the existence of the infrared fixed pointLattice
which qualitatively supports the modern two-loop-based explanation of walking.
* •
Ideal walking $\alpha$: Rather than using a two-loop running coupling constant
(7) and a value of $\alpha_{*}=88\pi/523$ that is not close in value to the
critical coupling $\alpha_{c}=4\pi/35$ for the first and second set of
techniquarks, we use the same running coupling constant but change the value
of $\alpha_{*}$ in (7) by artificially requiring that
$\alpha_{*}=1.02\alpha_{c}=1.02*4\pi/35$. Although this is not a realistic
case for the model, it is closer to the conformal situation, and therefore,
ideal walking.
The reason we must consider the above cases is because our analytical
estimation using the $\beta$ may cause some error. Therefore, we can use these
two extremes to investigate the effect of changes in the situation on our
results. We show three different behaviors of $\alpha$ in Fig.20. It can be
seen that $\alpha_{r}$ is much bigger than $\alpha_{w}$ only in the extreme
infrared region, and that the running behavior corresponding to
$1.02\alpha_{c}$ is smaller than that corresponding to $\alpha_{w}$ over most
of the energy region. From a comparison of Fig.20 with Fig.2, it can be seen
that the running effect increases the height of the infrared plateau and
narrows its length. To contrast other differences resulting from these
different couplings, in Fig.21, we show the techniquark self-energies,
$\tilde{\Sigma}$ and $\hat{\Sigma}$, which are determined by the SDEs (34) and
(35). We found that the closer the system came to walking, the lower and wider
the techniquark self-energy plateau was. By contrast, during running, the
plateau was higher and narrower. For fixed $f=250$GeV, we found that the
running situation produces a value of $\Lambda_{\mathrm{TC}}=0.21$TeV
($\Lambda_{\mathrm{ETC}}$ in the running case cannot be determined solely by
the running behavior and requires some other physical parameters to be known).
This result is consistent with the estimate of $\Lambda_{\mathrm{TC}}\simeq
2f\sqrt{3/N}$ given in Ref.LambdaTC . Our walking and ideal walking situations
yield:
$\displaystyle\Lambda_{w}=\left\\{\begin{array}[]{lll}5.5\mathrm{TeV}&&\mbox{walking}\\\
958\mathrm{TeV}&&\mbox{ideal walking}\end{array}\right.$ (55)
From this, it can be seen that $\Lambda_{w}$ is very sensitive to the walking
effect. The closer the system is to ideal walking, the bigger the value of
$\Lambda_{w}$. This was further checked by calculating $\Lambda_{w}$ for
several values of
$\alpha_{*}/\alpha_{c}=1.04,1.06,1.08,1.1,1.12,1.14,1.16,1.18,1.2$. These
points were then plotted as a curve in Fig.19 to quantitatively show the
sensitivity of $\Lambda_{w}$ to the degree of walking.
Figure 19: Dependence of the $\Lambda_{w}$ (TeV) on the degree of walking.
The small value of $\Lambda_{w}$ in our walking situation suggests that the
walking effect in the present model is not large enough. In an ideal walking
situation, $\Lambda_{w}$ is large and can be treated as
$\Lambda_{\mathrm{ETC}}$.
Figure 20: Three different couplings. $\alpha_{w}$ is the coupling used in our
calculation. $\alpha_{r}$ is the running coupling, which is given in (52).
Here, we show $\alpha_{r}/5$ to facilitate comparison between the couplings.
$1.02\alpha_{c}$ is the ideal walking coupling, where
$\alpha_{*}=1.02\alpha_{c}$.
Figure 21: Techniquark self-energies for three different couplings:
$\hat{\Sigma}_{w}$ and $\tilde{\Sigma}_{w}$ the self-energies for the second
and third sets of techniquarks for the coupling that we used in our
calculation. $\hat{\Sigma}_{r}$ and $\tilde{\Sigma}_{r}$ are the self-energies
for the second and third sets of techniquarks for the running coupling, which
is given in (52). Here, we show $\hat{\Sigma}_{r}/5$ and
$\tilde{\Sigma}_{r}/5$ to facilitate comparison between the self-energies.
$\hat{\Sigma}_{1.02\alpha_{c}}$ and $\tilde{\Sigma}_{1.02\alpha_{c}}$ are the
self-energies for the second and third sets of techniquarks for the ideal
walking coupling, where $\alpha_{*}=1.02\alpha_{c}$.
To show the effect of walking on the $S$ parameter, in Fig.22, we show the
value of $S$ for couplings corresponding to running and ideal walking. It can
be seen that for ideal walking (the upper bound on $T$ is reduced to 0.012 in
this case), $S$ is only slightly smaller than 2. Therefore, our prediction
that $S$ is about 2 is not significantly altered, even as one approaches the
walking region. However, Fig.22 shows that for running, $S$ is doubled by
reaching a value of 4. This implies that because of the existence of the
infrared fixed point, the walking only reduces the $S$ parameter by a factor
of 2. Furthermore, comparing the values of the $S$ parameters at different
couplings with their perturbative values
$S_{\mathrm{pert}}=N_{D}*N/6\pi=9/\pi$, we found that the perturbative value
of $S$ lies just between our realistic value and that of the running case.
Figure 22: $S$ parameters for the running and walking cases.
For the effect of walking on the other EWCL LECs, our numerical calculation
shows that for $\alpha_{2},\alpha_{3},\alpha_{4}$ walking reduces these LECs
to roughly $65\%$ of their original values in the running case. $\alpha_{5}$,
similar to the $S$ parameter, is reduced by the walking effect to half of its
original value in the running case. $\alpha_{6},\alpha_{7},\alpha_{9}$ are
reduced by one order of magnitude by the walking effect, but their signs are
preserved. $\alpha_{10}$ is reduced by two orders of magnitude and changes in
sign. Using the expression for $\alpha_{10}$ given by (46), the numerical
computation shows that some cancellations occur here. It is these
cancellations that result in $\alpha_{10}$being the smallest among the EWCL
LECs. Because of this cancellation, if the techniquark self-energy is changed,
more sign changes may occur. This cancellation may reduce the reliability of
our estimate of $\alpha_{10}$ and $\alpha_{10}$ may be seen as one of the
limitations of the calculation for the approximations used. We found that not
all LECs are sensitive to how close to ideal walking the theory is. The only
major exception is $\alpha_{10}$. Finally, we found that walking has almost no
effect on the coloron mass. We interpret this to mean that the techniquark
self-energy will change the value of the coloron mass significantly, but
walking, which changes the form of the techniquark self-energy, does not have
a large effect on the coloron mass. In fact, some quantities, such as
$\Lambda_{w}$ are sensitive to this detailed form of the techniquark self-
energy, but some other quantities, such as the coloron mass, are not.
## V Summary
In this paper, we discuss K. Lane’s TC2 Model in the presence of nontrivial TC
fermion condensation and walking. We focus on the walking effects in the
model, which has not been discussed before. We also discuss the phase
structure of the model in terms of the two-loop $\beta$ function of the TC
coupling of the model. We found that to have both an infrared fixed point and
spontaneous chiral symmetry breaking, the minimum $N$ for the TC group $SU(N)$
is $N=6$. This is the optimal choice because it is the value that is the most
conformal that can be used in our model. Although this choice differs from the
critical values, $N^{c}_{1,2}=5.42$ for the first and second sets of
techniquarks and $N^{c}_{3}=4.93$ for the third set of techniquarks (Fig.1),
walking effects occur in the computed EWCL LECs. We can understand this
explicit walking effect qualitatively through the relation, $N-N^{c}_{i}\ll
N^{c}_{i}$ for $i=1,2,3$. For $N=6$, using the technique used in our previous
studiesHongHao08 ; JunYi09 ; LangPLB ,we derive the EWCL from Lane’s model and
calculate the EWCL LECs up to an order of $p^{4}$. We found that the primary
contributions to the $p^{4}$ order coefficients arise from the three sets of
techniquarks and $Z^{\prime}$. There is no limit on the upper bound of the
$Z^{\prime}$ mass which differs from the TC2 modelsHill95 ; Lane95 ; Sekhar
that we discussed previously. Moreover, all corrections from the $Z^{\prime}$
particle are at least proportional to $\beta_{1}$ and vanish for a mixing of
$\theta=0$. It is especially important that the scale parameter,
$\Lambda_{w}$, appears in the solution of the two-loop $\beta$. This signifies
that the scale of walking cannot be assumed to be $\Lambda_{\mathrm{ETC}}$ in
this model because, generally,
$\Lambda_{\mathrm{TC}}\leq\Lambda_{w}\leq\Lambda_{\mathrm{ETC}}$. We found
that $\Lambda_{w}=5.5$TeV. The value of $\Lambda_{w}$ is small because it is
sensitive to the walking effect. However, our choice of $N$ differs from its
critical value, and does not exhibit a sufficient walking effect. We verified
that in a more ideal walking case, $\Lambda_{w}$ can be increased by at least
two orders of magnitude. The ratio
$(\Lambda_{\mathrm{ETC}}-\Lambda_{w})/\Lambda_{\mathrm{ETC}}$ can be used as a
measurement of the deviation of our theory from ideal walking. We also found
that the coloron mass is roughly half of its expected value of 1 TeV and is
independent of the walking effect. The small coloron mass occurs as the result
of including a correction from the coloron kinetic term for which the main
contribution is from the techniquark self-energy. The $T$ and $U$ parameters
are positive, and there is an upper bound for the $T$ parameter. For our
choice of typical hyper-charges, the upper bound of the $T$ parameter is
0.035, which is well below the experimentally measured bound from PDG. The $S$
parameter is about 2 for our choice of typical hyper-charges, which already
exceeds the experimentally verified constraint that it be half of the value
from the running case, but similar to that of the ideal walking case. To
reduce the value of the $S$ parameter, one can change hyper-charges. This can
result in $S$ being negative for slightly larger values of $T$. This allows
for a case in which both $S$ and $T$ are within the bound from PDG. The
leftmost nine nonzero LECs, $\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5}$ are
on the order of $10^{-2}$ which matches the estimate obtained from naive
dimensional analysis. $\alpha_{6},\alpha_{7},\alpha_{9}$ are on the order of
$10^{-5}$ and $\alpha_{10}$ is on order of $10^{-10}$. This is because
$\alpha_{6},\alpha_{7},\alpha_{9}$,and especially $\alpha_{10}$, are sensitive
to walking effects. Comparing these results with the constraints imposed by
the precision dataprecision , we find that the results are consistent with the
constraints from the precision data. However, $\alpha_{3}$ has the correct
order of magnitude, but the wrong sign.
Previously, we investigated bosonic contribution to the EWCL LECs for most of
the TC2 models. In the future, we will focus on calculating the EWCL LECs in
four areas: The first will be to explore new physics models, including the
non-TC2-type models. The second will be to investigate the part of the EWCL
dealing with matter. In particular, we will focus on the top quark. The third
will be to deepen our understanding of the structure of the model we are
currently discussing in areas such as phase diagrams and the infrared behavior
of the gauge coupling constant. The fourth will be to improve the precision of
the computation and reduce the number of approximations necessary. With an
increasing number of models in our EWCL platform, it will be effective for
future investigations of the electroweak symmetry breaking mechanisms.
## Acknowledgments
This work was supported by the National Science Foundation of China (NSFC)
under Grants No. 10875065 and 11075085.
## Appendix A Process of integrating out the techniquarks
To integrate out the techniquarks, which we have done in previous
studiesHongHao08 ; JunYi09 ; LangPLB , we assume only four fermion
interactions in (31), because a naive dimensional analysis indicates that the
contributions from higher dimensional operators are usually suppressed in the
low energy region. Also, this approximation leads to the conventional ladder
approximation, which is often used in discussions of the SDE. This yields:
$\displaystyle
iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]\approx\int
d^{4}x_{1}d^{4}x_{2}\frac{(-ig_{\mathrm{TC}})^{2}}{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})J_{\alpha_{1}}^{\mu_{1}}(x_{1})J_{\alpha_{2}}^{\mu_{2}}(x_{2})$
$\displaystyle=-\frac{g^{2}_{\mathrm{TC}}}{2}\int
d^{4}x_{1}d^{4}x_{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})\bigg{[}\bar{\psi}(x_{1})\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}\psi(x_{1})\bar{\psi}(x_{2})\tilde{t}^{\alpha_{2}}\gamma^{\mu_{2}}\psi(x_{2})$
$\displaystyle+{\displaystyle\sum_{i,j=1,2,l,t,b}}\bar{T}^{i}_{\xi}(x_{1})t^{\alpha_{1}}\gamma^{\mu_{1}}T^{i}_{\xi}(x_{1})\bar{T}^{j}_{\xi}(x_{2})t^{\alpha_{2}}\gamma^{\mu_{2}}T^{j}_{\xi}(x_{2})+2{\displaystyle\sum_{i=1,2,l,t,b}}\bar{\psi}(x_{1})\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}\psi(x_{1})\bar{T}^{i}_{\xi}(x_{2})t^{\alpha_{2}}\gamma^{\mu_{2}}T^{i}_{\xi}(x_{2})\bigg{]}$
$\displaystyle\approx\int
d^{4}x_{1}d^{4}x_{2}\bigg{[}\bar{\psi}^{\sigma}(x_{1})\tilde{\Pi}_{\sigma\rho}(x_{1},x_{2})\psi^{\rho}(x_{2})+{\displaystyle\sum_{i,j=1,2,l,t,b}}\bar{T}^{i\sigma}_{\xi}(x_{1})\Pi^{ij}_{\sigma\rho}(x_{1},x_{2})\bar{T}^{j\rho}_{\xi}(x_{2})\bigg{]}\;,$
(56)
where we have used (29) and (30). And
$\displaystyle\tilde{\Pi}_{\sigma\rho}(x_{1},x_{2})$ $\displaystyle\equiv$
$\displaystyle-g^{2}_{\mathrm{TC}}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}_{\sigma\sigma_{1}}\langle\psi^{\sigma_{1}}(x_{1})\bar{\psi}^{\rho_{2}}(x_{2})\rangle\tilde{t}^{\alpha_{2}}\gamma^{\mu_{2}}_{\rho_{2}\rho}$
(57) $\displaystyle\Pi^{ij}_{\sigma\rho}(x_{1},x_{2})$ $\displaystyle\equiv$
$\displaystyle-g^{2}_{\mathrm{TC}}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})t^{\alpha_{1}}\gamma^{\mu_{1}}_{\sigma\sigma_{1}}\langle
T^{i\sigma_{1}}(x_{1})\bar{T}^{j\rho_{2}}(x_{2}){\rangle}t^{\alpha_{2}}\gamma^{\mu_{2}}_{\rho_{2}\rho}\;.$
(58)
To obtain (56), we have used the average field approximation and approximated
the four-fermion interactions using their vacuum expectation values (VEVs).
Furthermore, we used the result:
$\langle\bar{\psi}(x)\gamma^{\mu}\psi(x)\rangle=\langle\bar{T}^{i}(x)\gamma^{\mu}T^{j}(x)\rangle=0$,
which can be obtained from the Lorentz invariance;
$\langle\bar{\psi}(x)T^{i}(x)\rangle=\langle\bar{T}^{i}(x)\psi(x)\rangle=0$,
which was assumed in Lane’s original paper Lane96 and can be verified as a
solution to the SDE. In fact, one can confirm that the VEVs between the
different sets of techniquarks vanish and VEVs among the different
techniquarks of the second set also vanish. For (56), this yields:
$\displaystyle iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]$
$\displaystyle\approx$ $\displaystyle\int
d^{4}x_{1}d^{4}x_{2}\bigg{[}\bar{\psi}^{\sigma}(x_{1})\tilde{\Pi}_{\sigma\rho}(x_{1},x_{2})\psi^{\rho}(x_{2})+{\displaystyle\sum_{i,j=1,2}}\bar{T}^{i\sigma}_{\xi}(x_{1})\bar{\Pi}^{ij}_{\sigma\rho}(x_{1},x_{2})T^{j\rho}_{\xi}(x_{2})$
(59)
$\displaystyle+{\displaystyle\sum_{i=l,t,b}}\bar{T}^{i\sigma}_{\xi}(x_{1})\hat{\Pi}_{\sigma\rho}(x_{1},x_{2})T^{i\rho}_{\xi}(x_{2})\bigg{]}$
with
$\displaystyle\Pi^{ij}_{\sigma\rho}(x_{1},x_{2})=\left\\{\begin{array}[]{lll}\bar{\Pi}^{ij}_{\sigma\rho}(x_{1},x_{2})&{}{}{}&i,j=1,2\\\
&&\\\ \hat{\Pi}_{\sigma\rho}(x_{1},x_{2})&&i,j=l,t,b\end{array}\right.\;.$
(63)
Therefore $\bar{\Pi}$, $\hat{\Pi}$ and $\tilde{\Pi}$ represent the fermion
self-energies for the first, second, and third sets of techniquarks,
respectively. Following the treatment in our previous studiesHongHao08 ;
JunYi09 ; LangPLB , these techniquark self-energies can be approximated as:
$\displaystyle\hat{\Pi}^{ij}_{\sigma\rho}(x,y)\approx-\delta_{\sigma\rho}[\hat{\Sigma}(\overline{\nabla}_{x}^{2})\delta^{4}(x\\!-\\!y)]_{ij}\hskip
14.22636pt\tilde{\Pi}_{\sigma\rho}(x,y)\approx-\delta_{\sigma\rho}\tilde{\Sigma}(\partial_{x}^{2})\delta^{4}(x\\!-\\!y)\hskip
14.22636pt\overline{\nabla}^{\mu}\\!=\\!\partial^{\mu}\\!-\\!iV_{2\xi}^{\mu}~{}~{}~{}$
(64)
$\displaystyle\bar{\Pi}^{ij}_{\sigma\rho}(x,y)\approx-[\delta_{\sigma\rho}\bar{\Sigma}(\hat{\nabla}_{x}^{2})+i\gamma^{5}_{\sigma\rho}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}_{x}^{2})]_{ij}\delta^{4}(x-y)\hskip
71.13188pt\hat{\nabla}^{\mu}=\partial^{\mu}-iV_{1\xi}^{\mu}\bigg{|}_{v_{1}=0}\;,~{}~{}~{}~{}~{}$
(65)
where $V_{2\xi}^{\mu}$, $V_{1\xi}^{\mu}$ and $v_{1}^{\mu}$ will be discussed
later in the appendices. The above approximation is the lowest order of a
dynamical perturbation originally proposed by Pagels and Stokar in Ref.DPT .
In this perturbation, all source dependent parts are expressed in terms of the
techniquark self-energy and the detailed dependence is determined by including
the minimal contribution that is covariant with the local chiral symmetry. An
important result of this dynamical perturbation is that the lowest order,
which includes the fermion loop terms, yields spontaneous chiral symmetry
breaking and is dominated by the fermion self-energy. In our previous
studiesHongHao08 ; JunYi09 ; LangPLB , the $\Pi$ functions are diagonal in the
spinor space, but in this model, $\bar{\Pi}_{\sigma\rho}(x,y)$ in (65) differs
from the conventional expression. In this case, there is an extra term
($\bar{\Sigma}_{5}$) that is proportional to $\gamma^{5}$ and $\tau^{2}$ (in
isospin space) because of the special model arrangement that generates
nontrivial twisted TC fermion condensation. This condensation will stimulate
topcolor symmetry breaking: $SU(3)_{1}\otimes SU(3)_{2}\rightarrow SU(3)_{c}$
and generate the coloron mass. Later, we will discuss the appearance of this
term and determine the functions corresponding to $\hat{\Sigma}$,
$\tilde{\Sigma}$, $\bar{\Sigma}$ and $\bar{\Sigma}_{5}$ .
With the results from (59)-(65), the techniquark interactions in (III.3)
become bilinear, and we can complete the integration over the techniquarks and
obtain (33), which is given in the text. Where:
$\displaystyle
V_{1\xi}=\begin{pmatrix}v_{1}\\!+v_{2}-g_{3}\frac{\lambda^{A}}{2}B^{A}\cot\theta^{\prime}&0\\\
0&v_{1}\\!+v_{2}+g_{3}\frac{\lambda^{A}}{2}B^{A}\tan\theta^{\prime}\end{pmatrix}\hskip
19.91684ptA_{1\xi}=\begin{pmatrix}a_{1}\\!+a_{2}&\\\
&a_{1}\\!-a_{2}\end{pmatrix}~{}~{}~{}~{}~{}$ (66) $\displaystyle
V_{2\xi}=\begin{pmatrix}v_{l}&0&0\\\ 0&v_{t}&0\\\ 0&0&v_{b}\end{pmatrix}\hskip
219.08612ptA_{2\xi}=\begin{pmatrix}a_{l}&0&0\\\ 0&a_{t}&0\\\
0&0&a_{b}\end{pmatrix}\;.$ (67)
The prime in $\mathrm{Tr^{\prime}}$ denotes the trace of the extra $2\times 2$
space for the first two sets of techniquarks, and the double prime in
$\mathrm{Tr"}$ denotes the trace of the extra $3\times 3$ space for the third
set of techniquarks with:
$\displaystyle
v_{1}=-\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}(B_{\xi}-Z^{\prime}\tan\theta)$
$\displaystyle
v_{2}=-\frac{1}{2}g_{1}(u_{2}+v_{2})(B_{\xi}-Z^{\prime}\tan\theta)-\frac{1}{2}g_{1}(u_{1}+v_{1})(B_{\xi}+Z^{\prime}\cot\theta)$
(68) $\displaystyle
a_{1}=\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}(B_{\xi}-Z^{\prime}\tan\theta)\hskip
56.9055pta_{2}=\frac{1}{2}g_{1}(u_{1}-v_{1})(\cot\theta+\tan\theta)Z^{\prime}~{}~{}~{}~{}$
(69) $\displaystyle
v_{i}=-\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{g_{1}}{2}\frac{\tau^{3}}{2}(B_{\xi}\\!-\\!Z^{\prime}\tan\theta)-\frac{g_{1}}{2}(x^{i}_{2}\\!+x^{i\prime}_{2})(B_{\xi}\\!-\\!Z^{\prime}\tan\theta)-\frac{g_{1}}{2}(x^{i}_{1}\\!+x^{i\prime}_{1})(B_{\xi}\\!+\\!Z^{\prime}\cot\theta)$
$\displaystyle
a_{i}=\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}(B_{\xi}-Z^{\prime}\tan\theta)+\frac{1}{2}g_{1}(x^{i}_{1}-x^{i\prime}_{1})(\cot\theta+\tan\theta)Z^{\prime}\hskip
42.67912pti=l,t,b\;.$ (70)
We have used the relation
$\displaystyle
iq_{1}\xi{B}_{1\xi,\mu}P_{L}-iq_{2}\xi{B}_{2\xi,\mu}P_{L}+iq_{1}\xi_{1}^{\prime}{B}_{1\xi,\mu}P_{R}-iq_{2}\xi^{\prime}{B}_{2\xi,\mu}P_{R}=-ig_{1}(\cot\theta+\tan\theta)\xi
Z^{\prime}_{\mu}\gamma^{5}$ (71)
$\displaystyle-\\!h_{1}\frac{\lambda^{A}}{2}\not{A}_{1}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1\xi}P_{L}\\!\\!-\\!q_{2}u_{2}\not{B}_{2\xi}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1\xi}P_{R}\\!\\!-\\!q_{2}(v_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}$
$\displaystyle=\not{v}_{1}\\!+\not{v}_{2}-g_{3}\frac{\lambda^{A}}{2}\not{B}^{A}\cot\theta^{\prime}+(\not{a}_{1}+\not{a}_{2})\gamma^{5}$
(72)
$\displaystyle-\\!h_{2}\frac{\lambda^{A}}{2}\not{A}_{2}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1\xi}P_{L}\\!\\!-\\!q_{2}v_{2}\not{B}_{2\xi}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1\xi}P_{R}\\!\\!-\\!q_{2}(u_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}$
$\displaystyle=\not{v}_{1}\\!+\not{v}_{2}+g_{3}\frac{\lambda^{A}}{2}\not{B}^{A}\tan\theta^{\prime}+(\not{a}_{1}-\not{a}_{2})\gamma^{5}$
(73) $\displaystyle-
g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!-q_{1}x_{1}\not{B}_{1\xi}P_{L}\\!-q_{2}x_{2}\not{B}_{2\xi}P_{L}\\!-q_{1}x_{1}^{\prime}\not{B}_{1\xi}P_{R}\\!-q_{2}(x_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}=\not{v}_{l}+\not{a}_{l}\gamma^{5}$
(74) $\displaystyle-
g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!-q_{1}y_{1}\not{B}_{1\xi}P_{L}\\!-q_{2}y_{2}\not{B}_{2\xi}P_{L}\\!-q_{1}y_{1}^{\prime}\not{B}_{1\xi}P_{R}\\!-q_{2}(y_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}=\not{v}_{t}+\not{a}_{t}\gamma^{5}$
(75) $\displaystyle-
g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!-q_{1}z_{1}\not{B}_{1\xi}P_{L}\\!-q_{2}z_{2}\not{B}_{2\xi}P_{L}\\!-q_{1}z_{1}^{\prime}\not{B}_{1\xi}P_{R}\\!-q_{2}(z_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}=\not{v}_{t}+\not{a}_{t}\gamma^{5}\;.$
(76)
## Appendix B Derivation of the Schwinger-Dyson equations for the techniquark
self-energies
In this appendix, we derive the SDE for the techniquark self-energies. We
start from the path integral given in (III.3), and fix the functional
integration over the $U$, $B^{A}_{\mu}$ and $Z^{\prime}_{\mu}$ fields. The
total functional derivative of the integrand with respect to $\bar{\psi}$ and
$\bar{T}_{\xi}^{i}$ is zero, which yields:
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\mu(\psi,T)~{}\frac{\delta}{\delta\bar{\psi}^{\sigma}(x)}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}+iS_{\mathrm{source}}}\bigg{|}_{A^{A}_{\mu}=0}$
(77) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\mu(\psi,T)\frac{\delta}{\delta\bar{T}^{i,\sigma}_{\xi}(x)}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}+iS_{\mathrm{source}}}\bigg{|}_{A^{A}_{\mu}=0}$
(78) $\displaystyle\mathcal{D}\mu(\psi,T)$ $\displaystyle\equiv$
$\displaystyle\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}_{\xi}\mathcal{D}T^{1}_{\xi}\mathcal{D}\bar{T}^{2}_{\xi}\mathcal{D}T^{2}_{\xi}\mathcal{D}\bar{T}^{l}_{\xi}\mathcal{D}T^{l}_{\xi}\mathcal{D}\bar{T}^{t}_{\xi}\mathcal{D}T^{t}_{\xi}\mathcal{D}\bar{T}^{b}_{\xi}\mathcal{D}T^{b}_{\xi}\;,$
(79)
In this case, we have introduced source terms with external sources $\bar{I}$
and $\bar{J}$ to help to derive the SDEs:
$\displaystyle iS_{\mathrm{source}}=\int
d^{4}x\bigg{[}\bar{\psi}(x)I(x)+{\displaystyle\sum_{i=1,2,l,t,b}}\bar{T}^{i}(x)J^{i}(x)\bigg{]}\;.$
(80)
We derive $I^{\rho}(y)$ for both sides of (77) and remove all external
sources. We obtain:
$\displaystyle
0=S_{\psi\sigma\rho}^{-1}(x,y)+i[i\not{\partial}_{x}\\!+g_{1}(\cot\theta\\!+\\!\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}]_{\sigma\rho}\delta(x\\!-\\!y)-g_{\mathrm{TC}}^{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x,y)[\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}S(x,y)\tilde{t}^{\alpha_{2}}\gamma^{\mu_{2}}]_{\sigma\rho}$
(81) $\displaystyle
S_{\psi\sigma\rho}(x,y)\equiv\langle\psi^{\sigma}(x)\bar{\psi}^{\rho}(y)\rangle=\frac{\int\mathcal{D}\mu(\psi,T)~{}\psi^{\sigma}(x)\bar{\psi}^{\rho}(y)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}{\int\mathcal{D}\mu(\psi,T)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}\bigg{|}_{A^{A}_{\mu}=0}\;.$
(82)
(81) is the SDE in coordinate space for the third set of techniquarks.
Combining (57) and (81), we find that $S_{\psi\sigma\rho}(x,y)$, which is
determined by the SDE, relates to $\tilde{\Pi}_{\sigma\rho}(x,y)$, introduced
in (57), through:
$\displaystyle
0=S_{\psi\sigma\rho}^{-1}(x,y)+i[i\not{\partial}_{x}\\!+g_{1}(\cot\theta\\!+\\!\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}]_{\sigma\rho}\delta(x\\!-\\!y)+\tilde{\Pi}_{\sigma\rho}(x,y)=0\;.$
(83)
Similarly we derive $J^{j\rho}(y)$ for both sides of (78), and remove all
external sources, We obtain:
$\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{1\xi}\\!+\\!\not{A}_{1\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)-g_{\mathrm{TC}}^{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x,y)[t^{\alpha_{1}}\gamma^{\mu_{1}}S(x,y)t^{\alpha_{2}}\gamma^{\mu_{2}}]^{ij}_{\sigma\rho}$
$\displaystyle\hskip 312.9803pti,j=1,2$ (84) $\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{2\xi}\\!+\\!\not{A}_{2\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)-g_{\mathrm{TC}}^{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x,y)[t^{\alpha_{1}}\gamma^{\mu_{1}}S(x,y)t^{\alpha_{2}}\gamma^{\mu_{2}}]^{ij}_{\sigma\rho}$
$\displaystyle\hskip 312.9803pti,j=l,t,b\;,$ (85)
where
$\displaystyle S^{ij}_{T\sigma\rho}(x,y)\equiv\langle
T^{i\sigma}(x)\bar{T}^{j\rho}(y)\rangle=\frac{\int\mathcal{D}\mu(\psi,T)~{}T^{i\sigma}(x)\bar{T}^{j\rho}(y)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}{\int\mathcal{D}\mu(\psi,T)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}\bigg{|}_{A^{A}_{\mu}=0}\;.$
(86)
(84) and (85) are the SDEs in the coordinate space of the first and second
sets of techniquarks. Combining (58), (63), (84) and (85), we find that
$S^{ij}_{T\sigma\rho}(x,y)$ which is determined by the SDE, relates to
$\bar{\Pi}^{ij}_{\sigma\rho}(x,y)$ and $\hat{\Pi}^{ij}_{\sigma\rho}(x,y)$,
introduced in (58) and (63), through:
$\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{1\xi}\\!+\\!\not{A}_{1\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)+\bar{\Pi}^{ij}_{\sigma\rho}(x,y)\hskip
28.45274pti,j=1,2$ (87) $\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{2\xi}\\!+\\!\not{A}_{2\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)+\hat{\Pi}^{ij}_{\sigma\rho}(x,y)\hskip
28.45274pti,j=l,t,b\;.$ (88)
Following the treatment in our previous works HongHao08 ; JunYi09 ; LangPLB ,
the techniquark self-energies $\hat{\Sigma}$ and $\tilde{\Sigma}$ in (64) and
$\bar{\Sigma}$, $\bar{\Sigma}_{5}$ in (65) are determined by removing the
gauge fields in the SDEs. Using this approximation, we find the three sets of
techniquarks:
$\displaystyle
S_{\psi\sigma\rho}(x,y)=\\!\int\\!\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\bigg{[}\frac{i}{\not{p}\\!-\\!\tilde{\Sigma}(-p^{2})}\bigg{]}_{\sigma\rho}\hskip
28.45274ptS_{T\sigma\rho}^{ij}(x,y)=\\!\int\\!\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\bigg{[}\frac{i\delta_{ij}}{\not{p}\\!-\\!\hat{\Sigma}(-p^{2})}\bigg{]}_{\sigma\rho}$
$\displaystyle\hskip 275.99164pti,j=l,t,b$ (89) $\displaystyle
S_{T\sigma\rho}^{ij}(x,y)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\bigg{[}\frac{i}{\not{p}-\bar{\Sigma}(-p^{2})-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(-p^{2})}\bigg{]}^{ij}_{\sigma\rho}\hskip
56.9055pti,j=1,2\;,$ (90)
In Euclidean space, we obtain(34), (35), (36) and (37)in the main text.
In terms of $\hat{\Sigma}$, comparing(35) with (36) and (37),we can construct
$\bar{\Sigma}$ and $\bar{\Sigma}_{5}$ as follows:
$\displaystyle\bar{\Sigma}(p_{E}^{2})=\hat{\Sigma}(p_{E}^{2})\cos\Theta\hskip
56.9055pt\bar{\Sigma}_{5}(p_{E}^{2})=\hat{\Sigma}(p_{E}^{2})\sin\Theta\;.$
(91)
$\Theta$ at the present stage in the computation is an arbitrary constant, and
we have verified that the vacuum energy generated by $\bar{\Sigma}$ and
$\bar{\Sigma}_{5}$ only depends on
$\bar{\Sigma}^{2}+\bar{\Sigma}^{2}_{5}=\hat{\Sigma}^{2}$, which is independent
of $\Theta$. Later we show that the coloron mass is dependent on $\Theta$ and
the present model gives a relatively small coloron mass (several hundred GeV).
In practice, we use the value of $\Theta$ which offers the largest coloron
mass. Once nonzero techniquark self-energies are present, we will have nonzero
techniquark condensates:
$\displaystyle\langle\bar{T}^{i}_{L}(x)T^{j}_{R}(x)\rangle=-2N\\!\int\\!\frac{d^{4}p_{E}}{(2\pi)^{4}}\bigg{[}\frac{\delta_{ij}\bar{\Sigma}(p_{E}^{2})}{p_{E}^{2}\\!+\\!\bar{\Sigma}^{2}(p_{E}^{2})\\!+\\!\bar{\Sigma}_{5}^{2}(p_{E}^{2})}-\frac{i\tau^{2}_{ij}\bar{\Sigma}_{5}(p_{E}^{2})}{p_{E}^{2}\\!+\\!\bar{\Sigma}^{2}(p_{E}^{2})\\!+\\!\bar{\Sigma}_{5}^{2}(p_{E}^{2})}\frac{p_{E}^{2}\\!-\\!\bar{\Sigma}^{2}(p_{E}^{2})}{p_{E}^{2}\\!+\\!\bar{\Sigma}^{2}(p_{E}^{2})}\bigg{]}$
$\displaystyle\hskip 361.3499pti,j=1,2,$ (92)
$\displaystyle\langle\bar{T}^{i}_{L}(x)T^{j}_{R}(x)\rangle=-2N\delta_{ij}\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{\hat{\Sigma}(p_{E}^{2})}{p_{E}^{2}+\hat{\Sigma}^{2}(p_{E}^{2})}\hskip
142.26378pti,j=l,t,b,$ (93)
$\displaystyle\langle\bar{\psi}_{L}(x)\psi_{R}(x)\rangle=-N(N-1)\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{\tilde{\Sigma}(p_{E}^{2})}{p_{E}^{2}+\tilde{\Sigma}^{2}(p_{E}^{2})}\;.$
(94)
Note that the first techniquark set has a nontrivial twisted condensation:
$\langle\bar{T}^{1}_{L}(x)T^{2}_{R}(x)\rangle=-\langle\bar{T}^{2}_{L}(x)T^{1}_{R}(x)\rangle\neq
0$ resulting from the nonzero self-energies.
## Appendix C Integrating out the colorons and the low energy expansion
The coefficients in (39) are,
$\displaystyle C=\int
d^{4}\tilde{k}[-2\tau+\tau^{2}k_{E}^{2}+16\tau^{2}\bar{\Sigma}^{2}_{5}]$ (95)
$\displaystyle\mathcal{K}=-\frac{1}{48\pi^{2}}[\ln\frac{\kappa^{2}}{\Lambda^{2}}+\gamma]\hskip
56.9055pt\kappa,\Lambda~{}\mbox{: infrared and ultraviolet cutoffs}$ (96)
$\displaystyle\hat{E}=\int
d^{4}\tilde{k}[\tau^{2}+16\tau^{2}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}+4\tau^{2}k_{E}^{2}\bar{\Sigma}^{\prime
2}+8\tau^{2}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime\prime}+4\tau^{2}k_{E}^{2}\bar{\Sigma}_{5}^{\prime
2}-\frac{1}{3}\tau^{3}k_{E}^{2}-\frac{16}{3}\tau^{3}\bar{\Sigma}_{5}^{2}$
$\displaystyle\hskip
28.45274pt-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}-6\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}-\frac{32}{3}\tau^{3}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}-\frac{32}{3}\tau^{3}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}\bar{\Sigma}^{\prime\prime}-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}^{\prime
2}$ $\displaystyle\hskip
28.45274pt-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime\prime}-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}_{5}^{\prime
2}-\frac{32}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}-\frac{16}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime\prime}\bar{\Sigma}_{5}^{2}-\frac{16}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}$ $\displaystyle\hskip
28.45274pt-16\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}^{2}\bar{\Sigma}_{5}^{\prime
2}+\frac{1}{18}\tau^{4}k_{E}^{4}+\frac{4}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}+\frac{2}{9}\tau^{4}k_{E}^{4}\bar{\Sigma}\bar{\Sigma}^{\prime}+\frac{2}{9}\tau^{4}k_{E}^{4}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}$
$\displaystyle\hskip
28.45274pt+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{3}\bar{\Sigma}^{\prime}+\frac{2}{9}\tau^{4}k_{E}^{4}\bar{\Sigma}^{2}\bar{\Sigma}^{\prime
2}+\tau^{4}k_{E}^{4}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}+\tau^{4}k_{E}^{4}\bar{\Sigma}_{5}^{2}\bar{\Sigma}_{5}^{\prime
2}$ $\displaystyle\hskip
28.45274pt+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}+\frac{32}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{4}\bar{\Sigma}_{5}^{\prime
2}]$ (97) $\displaystyle\int
d^{4}\tilde{k}=N\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int\frac{d^{4}k_{E}}{(2\pi)^{4}}e^{-\tau[k_{E}^{2}+\bar{\Sigma}^{2}(k_{E}^{2})]},\hskip
28.45274pt\bar{\Sigma}=\bar{\Sigma}(k_{E}^{2}),\hskip
28.45274pt\bar{\Sigma}_{5}=\bar{\Sigma}_{5}(k_{E}^{2})\;,$ (98)
Where $\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}$ are the coefficients that are
introduced later in (102), $\Lambda$ is a cutoff that is not sensitive to
changes for values between 10 TeV and 100 TeV for our walking theory. In our
practical calculation, we set it to 40 TeV. Combining the standard coloron
kinetic term in (33) and the techniquark quantum loop correction given by
(39), we obtain the formula for the coloron mass (40) given in the text. With
the coloron mass from (40), we can discuss coloron field integration in (40),
we then discuss coloron field integration in (33). This can be achieved using
the standard loop expansion:
$\displaystyle\int\mathcal{D}B_{\mu}^{A}~{}\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1,\mu\nu}B^{1,\mu\nu}+B_{2,\mu\nu}B^{2,\mu\nu})]$
$\displaystyle+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{]}_{A^{A}_{\mu}=0}$
$\displaystyle=\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1,\mu\nu}B^{1,\mu\nu}+B_{2,\mu\nu}B^{2,\mu\nu})]$
$\displaystyle+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]+\mbox{loop
corrections}\bigg{]}_{A^{A}_{\mu}=0,B^{A}_{\mu}=B^{A}_{\mu,c}}\;.$ (99)
And $B^{A}_{\mu,c}$ is determined by requiring that the result reach its
extremum at $B^{A}_{\mu}=B^{A}_{\mu,c}$. One can show that $B^{A}_{\mu,c}=0$
is one solution. Consequently, (33) becomes:
$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle e^{i\int
d^{4}x[-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}]}\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})\int\mathcal{D}Z_{\mu}^{\prime}$
(100) $\displaystyle\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}Z^{\prime}_{\mu\nu}Z^{\prime\mu\nu}]+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]+\mbox{loop
corrections}\bigg{]}_{A^{A}_{\mu}=B^{A}_{\mu}=0}\;.$
Note that we are interested in the bosonic part of the EWCL, those operators
involve explicit top quark fields, which belong to the part of the EWCL
dealing with matter, are beyond the scope of this paper. The top quark loop
term (especially the top quark condensate) is expected to essentially
contribute only to the top quark mass and not to the W and Z masses in TC2
models. This suggests that the contribution from top quark condensation to the
bosonic part of the EWCL may also be small (we will show this in the future in
a separate paper). Consequently, colorons, which are important in the
formation of top-quark condensates and contribute the majority of the top-
quark mass, only play a passive role in our present calculations. From
(12),the requirement, $A^{A}_{\mu}=B^{A}_{\mu}=0$ in (100) is equivalent to
the requirement, $A^{A}_{1\mu}=A^{A}_{2\mu}=0$.
Now, with the help of a technique used in our previous studiesHongHao08 ;
JunYi09 ; LangPLB , we take low energy expansion for the three TrLn terms in
(100):
$\displaystyle\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]\bigg{|}_{\mathrm{normal~{}part}}$
(101) $\displaystyle=i\int
d^{4}x(\cot\theta\\!+\\!\tan\theta)^{2}\bigg{[}\tilde{F}_{0}^{2}g_{1}^{2}\xi^{2}Z^{\prime
2}-(\mathcal{K}+\tilde{\mathcal{K}}_{2}^{\Sigma\neq
0})g_{1}^{2}\xi^{2}{Z}^{\prime}_{\mu\nu}{Z}^{\prime\mu\nu}-\tilde{\mathcal{K}}_{1}^{\Sigma\neq
0}g_{1}^{2}\xi^{2}(\partial^{\mu}Z_{\mu}^{\prime})^{2}$
$\displaystyle+(\tilde{\mathcal{K}}_{3}^{\Sigma\neq
0}+\tilde{\mathcal{K}}_{4}^{\Sigma\neq
0})g_{1}^{4}(\cot\theta+\tan\theta)^{2}\xi^{4}Z^{\prime 4}\bigg{]}+O(p^{6})$
$\displaystyle\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{|}_{\mathrm{normal~{}part}}$
(102) $\displaystyle=i\int
d^{4}x\bigg{\\{}\hat{F}_{0}^{2}A_{1\xi}^{2}-8F^{\prime
2}_{0}g_{1}^{2}u^{2}(\cot\theta+\tan\theta)^{2}Z^{\prime
2}-\frac{1}{2}\mathcal{K}\bigg{[}g_{2}^{2}W^{a}_{\mu\nu}W^{a\mu\nu}+g_{1}^{2}[1+4(u_{1}+u_{2})^{2}$
$\displaystyle+4(v_{1}+v_{2})^{2}]B_{\mu\nu}B^{\mu\nu}+g_{1}^{2}[4(u_{2}\tan\theta-
u_{1}\cot\theta)^{2}+4(v_{2}\tan\theta-v_{1}\cot\theta)^{2}+\tan^{2}\theta$
$\displaystyle+4\hat{D}_{0}u^{2}(\cot\theta+\tan\theta)^{2}]Z^{\prime}_{\mu\nu}{Z^{\prime}}^{\mu\nu}-2g_{1}^{2}[4(u_{1}+u_{2})(u_{2}\tan\theta-
u_{1}\cot\theta)$ $\displaystyle+4(v_{1}+v_{2})(v_{2}\tan\theta-
v_{1}\cot\theta)+\tan\theta]B_{\mu\nu}{Z^{\prime}}^{\mu\nu}\bigg{]}+\mathrm{tr}\bigg{[}-\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}(d_{\mu}A_{1\xi}^{\mu})^{2}+\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}(A_{1\xi}^{2})^{2}$ $\displaystyle-\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}(d_{\mu}A_{1\xi\nu}-d_{\nu}A_{1\xi\mu})^{2}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}(A_{1\xi\mu}A_{1\xi\nu})^{2}-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}V_{1\xi\mu\nu}V^{\mu\nu}_{1\xi}+i\hat{\mathcal{K}}_{14}^{\Sigma\neq
0}V_{1\xi\mu\nu}A_{1\xi}^{\mu}A_{1\xi}^{\nu}\bigg{]}$
$\displaystyle-8[\hat{D}_{1}a_{0}^{4}+\hat{D}_{2}a_{0}^{2}a_{3}^{2}]Z^{\prime
4}+\hat{D}_{3}a_{0}^{2}Z^{\prime
2}\mathrm{tr}(X^{\mu}X_{\mu})+2\hat{D}_{4}a_{0}^{2}Z^{\prime}_{\mu}Z^{\prime}_{\nu}\mathrm{tr}(X^{\mu}X^{\nu})$
$\displaystyle+4i\hat{D}_{2}a_{0}^{2}a_{3}Z^{\prime
2}Z^{\prime}_{\mu}\mathrm{tr}(X^{\mu}\tau^{3})\bigg{\\}}+O(p^{6})$
$\displaystyle\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]\bigg{|}_{\mathrm{normal~{}part}}$
(103) $\displaystyle=i\int
d^{4}x\sum_{\eta=l,t,b}\mathrm{tr}_{f}\bigg{[}\hat{F}_{0}^{2}a^{\eta
2}-\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}(d_{\mu}a^{\eta\mu})^{2}-\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}(d_{\mu}a_{\nu}^{\eta}-d_{\nu}a_{\mu}^{\eta})^{2}+\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}(a^{\eta 2})^{2}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}(a_{\mu}^{\eta}a_{\nu}^{\eta})^{2}$
$\displaystyle-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}v_{\mu\nu}^{\eta}v^{\eta\mu\nu}+i\hat{\mathcal{K}}_{14}^{\Sigma\neq
0}a_{\mu}^{\eta}a_{\nu}^{\eta}v^{\eta\mu\nu}\bigg{]}+O(p^{6})\;,$
where
$\displaystyle
d_{\mu}A_{1\xi\nu}=\partial_{\mu}A_{1\xi\nu}-i[V_{1\xi\mu},A_{1\xi\nu}]\hskip
56.9055ptV_{1\xi\mu\nu}=\partial_{\mu}V_{1\xi\nu}-\partial_{\nu}V_{1\xi\mu}-i[V_{1\xi\mu},V_{1\xi\nu}]$
(104) $\displaystyle
d_{\mu}a^{\eta}_{\nu}=\partial_{\mu}a^{\eta}_{\nu}-i[v^{\eta}_{\mu},a^{\eta}_{\nu}]\hskip
108.12054ptv^{\eta}_{\mu\nu}=\partial_{\mu}v^{\eta}_{\nu}-\partial_{\nu}v^{\eta}_{\mu}-i[v^{\eta}_{\mu},v^{\eta}_{\nu}]$
(105) $\displaystyle F^{\prime 2}_{0}=\int
d^{4}\tilde{k}~{}2\tau\bar{\Sigma}^{2}_{5}$ (106)
$\displaystyle\hat{D}_{0}=\int
d^{4}\tilde{k}~{}[2\tau^{2}\bar{\Sigma}_{5}\bar{\Sigma}^{\prime}_{5}+\tau^{2}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}^{\prime\prime}_{5}-\frac{2}{3}\tau^{3}\bar{\Sigma}^{2}_{5}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}^{\prime}_{5}-\frac{4}{3}\tau^{3}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}-\frac{4}{3}\tau^{3}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}$
$\displaystyle\hskip
14.22636pt-\frac{4}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime\prime}\bar{\Sigma}_{5}^{2}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime\prime}-\frac{10}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}^{2}\bar{\Sigma}_{5}^{\prime
2}+\frac{1}{6}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}$ $\displaystyle\hskip
14.22636pt+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}+\frac{4}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{4}\bar{\Sigma}_{5}^{\prime
2}]$ (107) $\displaystyle\hat{D}_{1}=\int
d^{4}\tilde{k}~{}[2\tau^{3}\bar{\Sigma}_{5}^{2}-\frac{1}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}-\frac{4}{3}\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}-\frac{2}{3}\tau^{4}\bar{\Sigma}_{5}^{4}]$
(108) $\displaystyle\hat{D}_{2}=\int
d^{4}\tilde{k}~{}[2\tau^{3}\bar{\Sigma}_{5}^{2}+\frac{1}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}-4\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}]$
(109) $\displaystyle\hat{D}_{3}=\int
d^{4}\tilde{k}~{}[\frac{1}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}-\frac{4}{3}\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}]$
(110) $\displaystyle\hat{D}_{4}=\int
d^{4}\tilde{k}~{}[\tau^{3}\bar{\Sigma}_{5}^{2}-\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}-\frac{1}{3}\tau^{4}\bar{\Sigma}_{5}^{4}]\;.$
(111)
$\hat{F}_{0}^{2}$ and $\hat{\mathcal{K}}_{i}^{\Sigma\neq 0}$ are functions of
the techniquark self-energy $\hat{\Sigma}(p_{E}^{2})$ which is determined by
(35). Detailed expressions for these quantities are given in (142) and (143)
of Appendix.E. Similarly, $\tilde{F}_{0}^{2}$ and
$\tilde{\mathcal{K}}_{i}^{\Sigma\neq 0}$ are functions of the techniquark
self-energy $\tilde{\Sigma}(p_{E}^{2})$ , which is determined by (34).
Detailed expressions for these quantities are given in (142) and (143) of
Appendix.E. In this case, the substitution,
$\hat{\Sigma}\rightarrow\tilde{\Sigma}$ is used.
With expansions (101),(102) and (103) and (66)-(70), and by ignoring loop
corrections, we can express (100) as (41) in the text. In this case, $S_{0}$
and $S_{Z^{\prime}}$ are $Z^{\prime}$ independent and dependent parts of the
actions:
$\displaystyle S_{0}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\bigg{\\{}-(\frac{5}{4}\mathcal{K}+\frac{1}{4}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+\frac{5}{8}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+\frac{3}{8}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})g_{2}^{2}W_{\mu\nu}^{a}W^{a,\mu\nu}-[(\frac{5}{4}+2\hat{u}+2\hat{x})\mathcal{K}+\frac{5}{8}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}$ (112)
$\displaystyle+(\frac{5}{8}+2\hat{u}+2\hat{x})\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}]g_{1}^{2}B_{\mu\nu}B^{\mu\nu}+(\frac{5}{8}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}+\frac{5}{32}\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\frac{5}{32}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{16}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})(\mathrm{tr}[X_{\mu}X^{\mu}])^{2}$
$\displaystyle+(\frac{5}{16}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+\frac{5}{8}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{16}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})\mathrm{tr}[X^{\mu}X_{\nu}]\mathrm{tr}[X_{\mu}X^{\nu}]+(\frac{5}{4}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}-\frac{5}{4}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})g_{1}\mathrm{tr}[\overline{W}^{\mu\nu}\tau^{3}]B_{\mu\nu}$
$\displaystyle+(-\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})i\mathrm{tr}[\overline{W}_{\mu\nu}X^{\mu}X^{\nu}]+(-\frac{5}{4}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{16}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ig_{1}B_{\mu\nu}\mathrm{tr}[\tau^{3}X^{\mu}X^{\nu}]$
$\displaystyle+\frac{1}{2}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}\mathrm{tr}[U^{{\dagger}}(D^{\mu}D_{\mu}U)U^{{\dagger}}(D^{\nu}D_{\nu}U)+2U^{{\dagger}}(D^{\mu}D_{\mu}U)(D^{\nu}U^{{\dagger}})(D_{\nu}U)]$
$\displaystyle+\frac{3}{4}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}\mathrm{tr}[U^{{\dagger}}(D^{\mu}D_{\mu}U)U^{{\dagger}}(D^{\nu}D_{\nu}U)+2U^{{\dagger}}(D^{\mu}D_{\mu}U)(D^{\nu}U^{{\dagger}})(D_{\nu}U)]\bigg{\\}}\;,$
where
$\displaystyle U(x)=\xi_{L}^{{\dagger}}(x)\xi_{R}(x)\hskip
71.13188ptX_{\mu}=U^{{\dagger}}(D_{\mu}U)\hskip
71.13188pt\overline{W}_{\mu\nu}=U^{{\dagger}}g_{2}\frac{\tau^{a}}{2}W^{a}_{\mu\nu}U~{}~{}~{}~{}~{}$
(113) $\displaystyle
D_{\mu}U=\partial_{\mu}U+ig_{2}\frac{\tau^{a}}{2}W_{\mu}^{a}U-ig_{1}U\frac{\tau^{3}}{2}B_{\mu}\hskip
28.45274ptD_{\mu}U^{{\dagger}}=\partial_{\mu}U^{{\dagger}}-ig_{2}U^{{\dagger}}\frac{\tau^{a}}{2}W_{\mu}^{a}+ig_{1}\frac{\tau^{3}}{2}B_{\mu}U^{{\dagger}}~{}~{}~{}~{}~{}~{}~{}$
(114)
$\displaystyle\hat{x}=(x_{1}+x_{2})^{2}+(y_{1}+y_{2})^{2}+(z_{1}+z_{2})^{2}\hskip
85.35826pt\hat{u}=(u_{1}+u_{2})^{2}+(v_{1}+v_{2})^{2}\;.$ (115)
While
$\displaystyle S_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x~{}\bigg{[}\frac{1}{2}Z^{\prime}_{\mu}D_{Z}^{-1,\mu\nu}Z^{\prime}_{\nu}+Z^{\prime,\mu}J_{Z,\mu}+Z^{\prime
2}Z_{\mu}^{\prime}J^{\mu}_{3Z}+g_{4Z}Z^{\prime 4}\bigg{]}\;,~{}~{}~{}~{}$
(116)
with
$\displaystyle
D_{Z}^{-1,\mu\nu}=g^{\mu\nu}(c_{Z^{\prime}}^{2}\partial^{2}+\bar{M}^{2}_{Z^{\prime}})-(1+\lambda_{Z})\partial^{\mu}\partial^{\nu}+\Delta^{\mu\nu}_{Z}(X)$
(117) $\displaystyle
J_{Z}^{\mu}=J_{Z0}^{\mu}+g_{1}\gamma\partial^{\nu}B_{\mu\nu}+\tilde{J}_{Z}^{\mu}$
(118) $\displaystyle
g_{4Z}=[10a_{3}^{4}+12a_{3}^{2}(2a_{0}^{2}+\hat{a}_{0}^{2})+4a_{0}^{4}+2\hat{a}_{0}^{4}](\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq 0})$ $\displaystyle\hskip
17.07182pt+g_{1}^{4}(\tan\theta+\cot\theta)^{4}\xi^{4}(\tilde{\mathcal{K}}_{3}^{\Sigma\neq
0}\\!+\\!\tilde{\mathcal{K}}_{4}^{\Sigma\neq
0})-8\hat{D}_{1}a_{0}^{4}-8\hat{D}_{2}a_{0}^{2}a_{3}^{2}~{}~{}~{}$ (119)
$\displaystyle
J_{3Z}^{\mu}=-i[(10a_{3}^{3}+12a_{0}^{2}a_{3}^{2}+6\hat{a}_{0}^{2}a_{3})(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0})+4a_{0}^{2}a_{3}\hat{D}_{2}]\mathrm{tr}[X^{\mu}\tau^{3}]\;,$ (120)
where
$\displaystyle\bar{M}^{2}_{Z^{\prime}}=2\tilde{F}_{0}^{2}g_{1}^{2}(\cot\theta+\tan\theta)^{2}\xi^{2}+4\hat{F}_{0}^{2}(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})-8F^{\prime
2}_{0}a_{0}^{2}$ (121) $\displaystyle
c^{2}_{Z^{\prime}}=1+[4(\cot\theta+\tan\theta)^{2}\xi^{2}+2\tan^{2}\theta+8\hat{v}+3\tan^{2}\theta+\hat{y}]\mathcal{K}g_{1}^{2}+4(\cot\theta+\tan\theta)^{2}\xi^{2}\tilde{\mathcal{K}}_{2}^{\Sigma\neq
0}g_{1}^{2}$ $\displaystyle\hskip
17.07182pt+8(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+[40a_{3}^{2}+2(\hat{t}+\hat{s})g_{1}^{2}]\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-16\hat{D}_{0}a_{0}^{2}$ (122)
$\displaystyle\lambda_{Z}=-2g_{1}^{2}(\tan\theta+\cot\theta)^{2}\tilde{\mathcal{K}}_{1}^{\Sigma\neq
0}-4(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}$ (123)
$\displaystyle\Delta^{\mu\nu}_{Z}(X)=[40a_{3}^{2}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}-(4a_{0}^{2}+2\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-(4a_{0}^{2}+2\hat{a}_{0}^{2}+10a_{3}^{2})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-20a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+10a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}$ $\displaystyle\hskip
42.67912pt+2a_{0}^{2}\hat{D}_{4}]\mathrm{tr}[X^{\mu}X^{\nu}]-(20\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}+5\hat{\mathcal{K}}_{3}^{\Sigma\neq 0}-10\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})a_{3}^{2}\mathrm{tr}[X^{\mu}\tau^{3}]\mathrm{tr}[X^{\nu}\tau^{3}]$
$\displaystyle\hskip
42.67912pt+g^{\mu\nu}[(5a_{3}^{2}+2a_{0}^{2}+\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+(2a_{0}^{2}+2\hat{a}_{0}^{2}-5a_{3}^{2})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-20a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+10a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}$ $\displaystyle\hskip
42.67912pt+a_{0}^{2}\hat{D}_{3}]\mathrm{tr}[X^{\lambda}X_{\lambda}]-g^{\mu\nu}(5\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+10\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}-5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})a_{3}^{2}tr[X_{\lambda}\tau^{3}]\mathrm{tr}[X^{\lambda}\tau^{3}]$ (124)
$\displaystyle
J_{Z0}^{\mu}=-5ia_{3}\hat{F}_{0}^{2}\mathrm{tr}[X^{\mu}\tau^{3}]$ (125)
$\displaystyle\gamma=2[5a_{3}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+(5a_{3}+4g_{1}\hat{w}+2g_{1}\hat{z})\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+(4\hat{w}+\frac{5}{2}\tan\theta+2\hat{z})g_{1}\mathcal{K}]$ (126)
$\displaystyle\tilde{J}_{Z}^{\mu}=10(-\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})a_{3}\partial_{\nu}\mathrm{tr}[\overline{W}^{\mu\nu}\tau^{3}]+10(\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{1}{4}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ia_{3}\partial_{\nu}\mathrm{tr}[X^{\mu}X^{\nu}\tau^{3}]$
$\displaystyle\hskip 14.22636pt+5(\frac{1}{4}\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\frac{1}{4}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{1}{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ia_{3}\mathrm{tr}[X^{\nu}X_{\nu}]tr[X^{\mu}\tau^{3}]$ $\displaystyle\hskip
14.22636pt+5(\frac{1}{2}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{1}{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ia_{3}\mathrm{tr}[X^{\mu}X_{\nu}]\mathrm{tr}[X^{\nu}\tau^{3}]$
$\displaystyle\hskip 14.22636pt+(-5\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{4}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})a_{3}\mathrm{tr}[\overline{W}^{\mu\nu}(X_{\nu}\tau^{3}-\tau^{3}X_{\nu})]$
$\displaystyle\hskip 14.22636pt+5ia_{3}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}\mathrm{tr}\bigg{[}U^{{\dagger}}(D^{\nu}D_{\nu}U)U^{{\dagger}}D^{\mu}U\tau^{3}-U^{{\dagger}}(D^{\nu}D_{\nu}U)\tau^{3}U^{{\dagger}}D^{\mu}U-\partial^{\mu}[U^{{\dagger}}(D^{\nu}D_{\nu}U)\tau^{3}]\bigg{]}$
$\displaystyle\hskip 14.22636pt+i\hat{a}_{0}{\mathcal{K}}_{1}^{\Sigma\neq
0}\partial^{\mu}\mathrm{tr}[X^{\nu}X_{\nu}-U^{\dagger}(D^{\nu}D_{\nu}U)]$
(127)
in which
$\displaystyle
a_{0}=\frac{1}{2}g_{1}(u_{1}-v_{1})(\cot\theta-\tan\theta)\hskip
85.35826pta_{3}=\frac{1}{4}g_{1}\tan\theta$ (128)
$\displaystyle\hat{a}^{2}_{0}=\frac{1}{4}g^{2}_{1}(\tan\theta+\cot\theta)^{2}[(x_{1}-x_{1}^{\prime})^{2}+(y_{1}-y_{1}^{\prime})^{2}+(z_{1}-z_{1}^{\prime})^{2}]$
(129)
$\displaystyle\hat{a}^{4}_{0}=\frac{1}{16}g_{1}^{4}(\tan\theta+\cot\theta)^{4}[(x_{1}-x_{1}^{\prime})^{4}+(y_{1}-y_{1}^{\prime})^{4}+(z_{1}-z_{1}^{\prime})^{4}]$
(130) $\displaystyle\hat{v}=(u_{2}\tan\theta-
u_{1}\cot\theta)^{2}+(v_{2}\tan\theta-v_{1}\cot\theta)^{2}$
$\displaystyle\hat{w}=(u_{1}+u_{2})(u_{2}\tan\theta-
u_{1}\cot\theta)+(v_{1}+v_{2})(v_{2}\tan\theta-v_{1}\cot\theta)$ (131)
$\displaystyle\hat{t}=2[(u_{2}+v_{2})\tan\theta-(u_{1}+v_{1})\cot\theta]^{2}$
(132) $\displaystyle\hat{y}=(x_{2}^{\prime}\tan\theta-
x_{1}^{\prime}\cot\theta)^{2}+(x_{2}\tan\theta-
x_{1}\cot\theta)^{2}+(y_{2}^{\prime}\tan\theta-y_{1}^{\prime}\cot\theta)^{2}$
$\displaystyle\hskip 5.69046pt+(y_{2}\tan\theta-
y_{1}\cot\theta)^{2}+(z_{2}^{\prime}\tan\theta-
z_{1}^{\prime}\cot\theta)^{2}+(z_{2}\tan\theta-z_{1}\cot\theta)^{2}$ (133)
$\displaystyle\hat{z}=(x_{1}+x_{2})[(x_{2}^{\prime}+x_{2})\tan\theta-(x_{1}^{\prime}+x_{1})\cot\theta]+(y_{1}+y_{2})[(y_{2}^{\prime}+y_{2})\tan\theta-(y_{1}^{\prime}+y_{1})\cot\theta]$
$\displaystyle\hskip
5.69046pt+(z_{1}+z_{2})[(z_{2}^{\prime}+z_{2})\tan\theta-(z_{1}^{\prime}+z_{1})\cot\theta]$
(134)
$\displaystyle\hat{s}=[(x_{2}^{\prime}+x_{2})\tan\theta-(x_{1}^{\prime}+x_{1})\cot\theta]^{2}+[(y_{2}^{\prime}+y_{2})\tan\theta-(y_{1}^{\prime}+y_{1})\cot\theta]^{2}$
$\displaystyle\hskip
5.69046pt+[(z_{2}^{\prime}+z_{2})\tan\theta-(z_{1}^{\prime}+z_{1})\cot\theta]^{2}$
(135)
From (116) and (117), it can be seen that the $Z^{\prime}$ mass squared,
$M_{Z^{\prime}}^{2}$, is determined by:
$\displaystyle
M_{Z^{\prime}}^{2}=\frac{\bar{M}_{Z^{\prime}}^{2}}{c_{Z^{\prime}}^{2}}\;.$
(136)
## Appendix D Process of integrating out $Z^{\prime}$
From (116), the solution of Eq.(44) is
$\displaystyle
Z_{c}^{\prime\mu}(x)=-D^{\mu\nu}_{Z}J_{Z,\nu}(x)+O(p^{3})+\mbox{loop
corrections}\;,$ (137)
then
$\displaystyle\bar{S}_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\bigg{[}-\frac{1}{2}J_{Z,\mu}D_{Z}^{\mu\nu}J_{Z,\nu}-J_{3Z,\mu^{\prime}}(D_{Z}^{\mu^{\prime}\nu^{\prime}}J_{Z,\nu^{\prime}})(D_{Z}^{\mu\nu}J_{Z,\nu})^{2}+g_{4Z}(D_{Z}^{\mu\nu}J_{Z,\nu})^{4}\bigg{]}$
(138) $\displaystyle+\mbox{loop corrections}\;,~{}~{}~{}$
where
$\displaystyle
D_{Z}^{-1,\mu\nu}D_{Z,\nu\lambda}=D_{Z}^{\mu\nu}D_{Z,\nu\lambda}^{-1}=g^{\mu}_{\lambda}\;,$
(139)
It can be shown that if our accuracy is on the order of $p^{4}$, then $p^{1}$
order $Z_{c}^{\prime}$ solution is sufficient because all contributions from
$p^{3}$ order $Z^{\prime}_{c}$ are at least on the order of $p^{6}$.
Combining (138), (117) and (118)and ignoring loop corrections, we obtain:
$\displaystyle\bar{S}_{Z^{\prime}}=\\!\int\\!d^{4}x\bigg{[}-\frac{1}{2}J_{Z0,\mu}D_{Z}^{\mu\nu}J_{Z0,\nu}-\frac{1}{\bar{M}_{Z^{\prime}}^{2}}J_{Z0,\mu}(\tilde{J}^{\mu}_{Z}\\!+\\!g_{1}\gamma\partial_{\nu}B^{\mu\nu})-\frac{1}{\bar{M}_{Z^{\prime}}^{6}}J_{3Z,\mu}J_{Z0}^{\mu}J_{Z0}^{2}+\frac{g_{4Z}}{\bar{M}_{Z^{\prime}}^{8}}J_{Z0}^{4}\bigg{]}\;.~{}~{}~{}~{}$
(140)
With the help of the following algebraic relations,
$\displaystyle\partial_{\mu}\mathrm{tr}[\tau^{3}X^{\mu}]=0$
$\displaystyle\mathrm{tr}[\tau^{3}(\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu})]=-2\mathrm{tr}(\tau^{3}X_{\mu}X_{\nu})+i\mathrm{tr}(\tau^{3}\overline{W}_{\mu\nu})-ig_{1}B_{\mu\nu}$
$\displaystyle{\rm tr}(\tau^{3}X_{\mu}X_{\nu}){\rm
tr}(\tau^{3}X^{\mu}X^{\nu})$ (141) $\displaystyle=[{\rm
tr}(X_{\mu}X_{\nu})]^{2}-[{\rm tr}(X_{\mu}X^{\mu})]^{2}-{\rm
tr}(X_{\mu}X_{\nu}){\rm tr}(\tau^{3}X^{\mu}){\rm tr}(\tau^{3}X^{\nu})+{\rm
tr}(X_{\mu}X^{\mu})[{\rm tr}(\tau^{3}X_{\nu})]^{2}$
$\displaystyle\mathrm{tr}(TA)\mathrm{tr}(TBC)+\mathrm{tr}(TB)\mathrm{tr}(TCA)+\mathrm{tr}(TC)\mathrm{tr}(TAB)=2\mathrm{tr}(ABC)\;,$
where $\mathrm{tr}A=\mathrm{tr}B=\mathrm{tr}C=0$ and $T^{2}=1$. We can
simplify (140) into the form of the EWCL.
## Appendix E $\mathcal{K}$ coefficients
In Minkowski space,
$\displaystyle\hat{F}_{0}^{2}$ $\displaystyle=$ $\displaystyle 2\int
d\tilde{p}\bigg{[}(-2\Sigma^{2}_{p}-p^{2}\Sigma_{p}\Sigma^{\prime}_{p})X_{p}^{2}+(2\Sigma^{2}_{p}+p^{2}\Sigma_{p}\Sigma^{\prime}_{p})\frac{X_{p}}{\Lambda^{2}}\bigg{]},$
(142) $\displaystyle{\cal K}_{1}$ $\displaystyle=$ $\displaystyle 2\int
d\tilde{p}\bigg{[}-2A_{p}X_{p}^{3}+2A_{p}\frac{X_{p}^{2}}{\Lambda^{2}}-A_{p}\frac{X_{p}}{\Lambda^{4}}+\frac{p^{2}}{2}\Sigma^{\prime
2}_{p}\frac{X_{p}}{\Lambda^{2}}-\frac{p^{2}}{2}\Sigma^{\prime
2}_{p}X_{p}^{2},\bigg{]},$ $\displaystyle{\cal K}_{2}$ $\displaystyle=$
$\displaystyle\int
d\tilde{p}\bigg{[}-2B_{p}X_{p}^{3}+2B_{p}\frac{X_{p}^{2}}{\Lambda^{2}}-B_{p}\frac{X_{p}}{\Lambda^{4}}+\frac{p^{2}}{2}\Sigma^{\prime
2}_{p}\frac{X_{p}}{\Lambda^{2}},-\frac{p^{2}}{2}\Sigma^{\prime
2}_{p}X_{p}^{2}\bigg{]},$ $\displaystyle{\cal K}_{3}$ $\displaystyle=$
$\displaystyle 2\int
d\tilde{p}\bigg{[}(\frac{4\Sigma^{4}_{p}}{3}-\frac{2p^{2}\Sigma^{2}_{p}}{3}+\frac{p^{4}}{18})(6X_{p}^{4}-\frac{6X_{p}^{3}}{\Lambda^{2}}+\frac{3X_{p}^{2}}{\Lambda^{4}}-\frac{X_{p}}{\Lambda^{6}}),$
$\displaystyle+(-4\Sigma^{2}_{p}+\frac{p^{2}}{2})(-2X_{p}^{3}+\frac{2X_{p}^{2}}{\Lambda^{2}}-\frac{X_{p}}{\Lambda^{4}})-\frac{X_{p}}{\Lambda^{2}}+X_{p}^{2}\bigg{]},$
$\displaystyle{\cal K}_{4}$ $\displaystyle=$ $\displaystyle\int
d\tilde{p}\bigg{[}(\frac{-4\Sigma^{4}_{p}}{3}+\frac{2p^{2}\Sigma^{2}_{p}}{3}+\frac{p^{4}}{18})(6X_{p}^{4}-\frac{6X_{p}^{3}}{\Lambda^{2}}+\frac{3X_{p}^{2}}{\Lambda^{4}}-\frac{X_{p}}{\Lambda^{6}})+4\Sigma^{2}_{p}(-2X_{p}^{3}+\frac{2X_{p}^{2}}{\Lambda^{2}}$
$\displaystyle-\frac{X_{p}}{\Lambda^{4}})+\frac{X_{p}}{\Lambda^{2}}-X_{p}^{2}\bigg{]},$
$\displaystyle{\cal K}_{5}$ $\displaystyle=$ $\displaystyle{\cal K}_{6}=0,$
$\displaystyle{\cal K}_{7}$ $\displaystyle=$ $\displaystyle 2\int
d\tilde{p}\bigg{[}(3\Sigma^{2}_{p}+2p^{2}\Sigma_{p}\Sigma^{\prime}_{p})X_{p}^{2}+[-2\Sigma^{2}_{p}-p^{2}(1+2\Sigma_{p}\Sigma^{\prime}_{p})]\frac{X_{p}}{\Lambda^{2}}\bigg{]},$
$\displaystyle{\cal K}_{8}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\cal K}_{9}$ $\displaystyle=$ $\displaystyle 2\int
d\tilde{p}\bigg{[}(\Sigma^{2}_{p}+2p^{2}\Sigma_{p}\Sigma^{\prime}_{p})X_{p}^{2}-p^{2}(1+2\Sigma_{p}\Sigma^{\prime}_{p})\frac{X_{p}}{\Lambda^{2}}\bigg{]},$
$\displaystyle{\cal K}_{10}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\cal K}_{11}$ $\displaystyle=$ $\displaystyle 4\int
d\tilde{p}\bigg{[}(-4\Sigma^{3}_{p}+p^{2}\Sigma_{p})X_{p}^{3}+(4\Sigma^{3}_{p}-p^{2}\Sigma_{p})\frac{X_{p}}{\Lambda^{2}}-(2\Sigma^{3}_{p}-\frac{1}{2}p^{2}\Sigma_{p})\frac{X_{p}}{\Lambda^{4}}+3\Sigma_{p}\frac{X_{p}}{\Lambda^{2}}$
$\displaystyle-3\Sigma_{p}X_{p}^{2}\bigg{]},$ $\displaystyle{\cal K}_{12}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle{\cal K}_{13}$
$\displaystyle=$ $\displaystyle\int
d\tilde{p}\bigg{[}(\frac{1}{2}p^{2}\Sigma^{\prime}_{p}\Sigma^{\prime\prime}_{p}+\frac{1}{6}p^{2}\Sigma_{p}\Sigma^{\prime\prime\prime}_{p})X_{p}+(C_{p}-D_{p})\frac{X_{p}}{\Lambda^{2}}-(C_{p}-D_{p})X_{p}^{2}-2E_{p}X_{p}^{3}$
$\displaystyle+2E_{p}\frac{X_{p}^{2}}{\Lambda^{2}}-E_{p}\frac{X_{p}^{2}}{\Lambda^{4}}\bigg{]},$
$\displaystyle{\cal K}_{14}$ $\displaystyle=$ $\displaystyle-4\int
d\tilde{p}\bigg{[}-2F_{p}X_{p}^{3}+2F_{p}\frac{X_{p}^{2}}{\Lambda^{2}}-F_{p}\frac{X_{p}}{\Lambda^{4}}+\frac{p^{2}}{2}\Sigma_{p}^{\prime
2}\frac{X_{p}}{\Lambda^{2}}-\frac{p^{2}}{2}\Sigma^{\prime
2}_{p}X_{p}^{2}\bigg{]},$ $\displaystyle{\cal K}_{15}$ $\displaystyle=$
$\displaystyle-4\int
d\tilde{p}\bigg{[}-(\Sigma_{p}+\frac{1}{2}p^{2}\Sigma^{\prime}_{p})\frac{X_{p}}{\Lambda^{2}}+(\Sigma_{p}+\frac{1}{2}p^{2}\Sigma^{\prime}_{p})X_{p}^{2}\bigg{]},$
$\displaystyle{\cal K}^{\Sigma\neq 0}_{i}$ $\displaystyle=$
$\displaystyle{\cal K}_{i}-{\cal K}_{i}\bigg{|}_{\hat{\Sigma}=0}\hskip
56.9055pti=1,2,\ldots,15$ (143)
in which the short notations are
$\displaystyle\int d\tilde{p}\equiv
iN\int\frac{d^{4}p}{(2\pi)^{4}}e^{\frac{p^{2}-\hat{\Sigma}^{2}(p^{2})}{\Lambda^{2}}},$
(144) $\displaystyle\Sigma_{p}$ $\displaystyle\equiv$
$\displaystyle\hat{\Sigma}(p^{2}),$ $\displaystyle X_{p}$
$\displaystyle\equiv$ $\displaystyle\frac{1}{p^{2}-\hat{\Sigma}^{2}(p^{2})},$
$\displaystyle A_{p}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}p^{2}\Sigma_{p}\Sigma^{\prime}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})-\frac{1}{3}\Sigma^{2}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})+\frac{1}{3}p^{2}\Sigma^{2}_{p}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p})$
$\displaystyle-\frac{1}{6}p^{4}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p}),$ $\displaystyle B_{p}$
$\displaystyle=$
$\displaystyle-\frac{2}{3}p^{2}\Sigma_{p}\Sigma^{\prime}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})-\frac{1}{3}\Sigma^{2}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})+\frac{1}{3}p^{2}\Sigma^{2}_{p}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p})$
$\displaystyle-\frac{1}{18}p^{4}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p})-\frac{1}{6}p^{2}(-1-2\Sigma_{p}\Sigma^{\prime}_{p}),$
$\displaystyle C_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{3}-\frac{1}{3}\Sigma_{p}\Sigma^{\prime}_{p}-\frac{1}{2}p^{2}\Sigma^{\prime
2}_{p},$ $\displaystyle D_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p^{2}\Sigma^{\prime
2}_{p}-\frac{1}{3}p^{2}\Sigma_{p}\Sigma^{\prime\prime}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})-\frac{2}{9}p^{4}\Sigma^{\prime}_{p}\Sigma^{\prime\prime}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})]-\frac{2}{9}p^{4}\Sigma^{\prime
2}_{(}p-\Sigma^{\prime 2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p})$
$\displaystyle-\frac{1}{3}p^{2}\Sigma_{p}\Sigma^{\prime}_{p}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p}),$ $\displaystyle E_{p}$
$\displaystyle=$
$\displaystyle-\frac{1}{6}p^{2}\Sigma_{p}\Sigma^{\prime}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})^{2}-\frac{1}{9}kp^{4}\Sigma^{\prime
2}_{p}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})^{2},$ $\displaystyle F_{p}$
$\displaystyle=$
$\displaystyle-\frac{4}{3}p^{2}\Sigma_{p}\Sigma^{\prime}_{p}+\frac{4}{3}p^{2}(\Sigma_{p}\Sigma^{\prime}_{p})^{2}-\frac{2}{3}\Sigma^{2}_{p}+\frac{2}{3}\Sigma_{p}^{3}\Sigma^{\prime}_{p}+\frac{1}{3}p^{2}\Sigma^{2}_{p}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p})$ (145)
$\displaystyle-\frac{1}{9}p^{4}(-\Sigma^{\prime
2}_{p}-\Sigma_{p}\Sigma^{\prime\prime}_{p})-\frac{1}{3}p^{2}(-1-2\Sigma_{p}\Sigma^{\prime}_{p})-\frac{1}{2}p^{2}.$
## References
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* (7) E. Farhi and L. Susskind, Phys. Rep. 74, 277 (1981) and references therein.
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* (9) J.Y.Lang, S.Z.Jiang and Q.Wang, Phys. Rev. D. 79, 015002(2009).
* (10) F.Braam, M.Flossdorf, R.S.Chivukula, S.DiChiara and E.H.Simmons, Phys. Rev. D 77, 055005(2008).
* (11) J.Y.Lang, S.Z.Jiang and Q.Wang, Phys. Lett. B 673, 63(2009).
* (12) E.Bagan, D.Espriu, J.Manzano, Phys. Rev. D 60, 114035(1999).
* (13) F. Sannino, arXiv:0804.0182[hep-ph].
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* (15) D. D. Dietrich and F. Sannino, Phys. Rev. D. 75, 085018(2007).
* (16) T. Banks and A. Zaks, Nucl. Phys. B 196, 189(1982).
* (17) K.Yamawaki, Int. J. Mod. Phys. A 25, 5128(2010).
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* (19) T.Appelquist, G.T.Fleming and E.T.Neil, Phys. Rev. D 79, 076010(2009).
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|
arxiv-papers
| 2011-02-17T10:58:27 |
2024-09-04T02:49:17.080051
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feng-Jun Ge, Shao-Zhou Jiang, Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/1102.3557"
}
|
1102.3607
|
Fairness issues in a chain of IEEE 802.11 stations
Bertrand Ducourthial | Yacine Khaled | Stéphane Mottelet
---|---|---
Heudiasyc lab., UMR-CNRS 6599 | LMAC lab., EA 2222
Université de Technologie de Compiègne
B.P. 20529, F-60205 Compiègne cedex, FRANCE
Email: firstname.name@utc.fr
July 2005
Abstract:
We study a simple general scenario of ad hoc networks based on IEEE 802.11
wireless communications, consisting in a chain of transmitters, each of them
being in the carrier sense area of its neighbors. Each transmitter always
attempts to send some data frames to one receiver in its transmission area,
forming a pair sender-receiver. This scenario includes the three pairs
fairness problem introduced in [1], and allows to study some fairness issues
of the IEEE 802.11 medium access mechanism.
We show by simulation that interesting phenomena appear, depending on the
number $n$ of pairs in the chain and of its parity. We also point out a
notable asymptotic behavior. We introduce a powerful modeling, by simply
considering the probability for a transmitter to send data while its neighbors
are waiting. This model leads to a non-linear system of equations, which
matches very well the simulations, and which allows to study both small and
very large chains. We then analyze the fairness issue in the chain regarding
some parameters, as well as the asymptotic behavior. By studying very long
chains, we notice good asymptotic fairness of the IEEE 802.11 medium sharing
mechanism. As an application, we show how to increase the fairness in a chain
of three pairs.
###### Contents
1. 1 Introduction
1. 1.1 Motivations
2. 1.2 Related work
3. 1.3 Contributions and outlines
2. 2 IEEE 802.11 standard in ad hoc mode
1. 2.1 Physical layer
1. 2.1.1 PMD sublayer
2. 2.1.2 PLCP sublayer
2. 2.2 Medium Access Control layer
1. 2.2.1 Frames
2. 2.2.2 Delays
3. 2.2.3 RTS/CTS
4. 2.2.4 Backoff
3. 3 Fairness issues in a chain of senders
1. 3.1 Transmission ranges considerations
2. 3.2 Fairness in a chain of senders
3. 3.3 The three pairs fairness problem
4. 4 Simulation of a chain of senders
1. 4.1 Configuration and parameters
2. 4.2 Fairness in a chain of four pairs
3. 4.3 Fairness in a chain of five pairs
4. 4.4 Fairness in a chain of six pairs
5. 4.5 Fairness in a chain of one hundred pairs
5. 5 Mathematical modeling
1. 5.1 Modeling with a non-linear system of equations
2. 5.2 Analytical results
3. 5.3 Validation with ns-2 results
6. 6 Analysis of the model
1. 6.1 Proving the existence of a solution
2. 6.2 Asymptotic behavior
3. 6.3 Maximization of fairness with respect to $\alpha$
7. 7 Discussion
1. 7.1 Asymptotic flat area
2. 7.2 Asymptotic optimal alpha
3. 7.3 Asymptotic comparison of modeling and simulation
4. 7.4 Interpretation of the $\alpha$ coefficient
5. 7.5 Obtaining the maximal fairness
8. 8 Conclusion
## 1 Introduction
### 1.1 Motivations
Recently, wireless networks have increasingly received attention from the
networking community. Although several wireless communication standards have
been proposed, the IEEE 802.11 protocol [2, 3, 4] is the most widely used, and
constitutes the de facto solution for practical network connection offering
mobility, flexibility, low cost of deployment and use. This success leads to
many studies of the protocol, in various situations (either ad hoc or with
access point) and by different means (experimentation, simulation, modeling).
It remains that, besides its qualities, the 802.11 protocol, and particularly
its medium access control mechanism, suffers from some imperfections in terms
of global throughput and fairness between nodes. Our work deals with some
fairness issues with 802.11 protocol in ad hoc mode.
We study a simple but general scenario, where some nodes (hereby called
_senders_) try to continuously send some data to one of their neighbors
(hereby called _receiver_), not necessarily always the same. The senders form
a chain, each of which being in the carrier sense area of its neighbors
(Figure 1).
Figure 1: A chain of _senders_.
In [1], the authors study a similar scenario composed of three pairs, and
shows that the central pair obtains a very poor throughput compared to the
border pairs. For instance, with a sending rate of 2 Mbits/s, the central pair
has only a throughput of 0.04 Mbits/s compared to 1.55 Mbits/s for the
external ones (the throughput of a single alone pair is 1.59 Mbits/s in this
situation).
This scenario is a particular case of the chain of senders scenario we study
in this paper. It combines both EIFS delay mechanism and asymmetry of the
chain in terms of number of neighbor senders. We show that interesting
phenomena appear when the number of pairs increases in the chain. These
phenomena depend on the number $n$ of pairs as well as on its parity. Moreover
a notable asymptotic behavior appears when $n$ increases. We provide a
powerful modeling which leads, among others, to interesting conclusions in
terms of fairness both for small and large chains. This analysis allows us to
better understand the DCF properties and to improve the fairness in a chain,
especially in the three pairs case.
### 1.2 Related work
There is a large amount of literature dealing with the performances of the
IEEE 802.11 _Distributed Coordinated Function_ (DCF) responsible of the shared
radio medium sharing.
In [5], a relation between the necessary and real time for sending some data
is given, allowing to estimate the DCF capacity. In [6] the authors make an
analytical study of the rates calculation of the DCF using Markov chain. The
authors prove that the performances of the DCF depends on the minimal
contention window and on the number of stations in the network.
In [7], a modeling of the IEEE 802.11 DCF with stochastic Petri nets is
proposed. Among other results, the authors show that the EIFS delay used when
a collision occurs can be advantageous when the network is not saturated.
In [8], the authors modify the model suggested in [6], and give an estimation
of the throughput as a function of the number of stations in the network and
of the ambient noise. Reusing works of [6, 5], the authors improve their
results in [9], by taking into account the contention window increasing in
case of collision.
Besides throughput evaluation, some studies deal with the DCF fairness.
In [10], the authors present a case where the binary exponential backoff (BEB)
lead to an unfair situation. Indeed, consider a situation where the contention
window of the competing transmitters are large due to collisions. As soon as a
node succeeds in sending a frame, it will reset its contention window. As a
consequence, it will generally wait for smaller backoff than others for its
further transmissions, and then gain more easily access to the shared medium.
To resolve such problems, the authors design the medium access protocol MACAW.
In [11], the authors present the relevance of the EIFS mechanism to the
fairness. They show that the EIFS delay can be too large or too small
according to some scenarios. The authors propose then an adaptive mechanism
for determining the EIFS delay, based on a measurement of the occupation time
of the medium.
In [12] the authors propose an evaluation of the DCF fairness, by means of
maximization of some differentiable concave functions, under a set of
constraints representing the impossibility for two close transmitters to
simultaneously transmit a frame with success. Some unfair situations relying
on asymmetric topologies are studied by simulation. They also study fairness
per packets and fairness per flow: two mobiles with the same probability of
access to the medium do not constitute an equitable scenario when one of both
must retransmit more flow than the other.
In [1, 13], the authors study an unfair scenario called _three pairs problem_
by means of simulations and experimentations. This scenario relies on an
asymmetric topology composed of three pairs of nodes. Pair 2 is placed between
pairs 1 and 3, and is in the carrier sense of its both neighbors. The
emissions of pairs 1 and 3 are not synchronized, and when the pair 2 wants to
emit, it is necessary that the silence periods of the other mobiles overlap.
However the probability of such a covering is weak.
This scenario has been modeled in [14] with a discrete time Markov chain. The
authors obtain results close to the simulations.
### 1.3 Contributions and outlines
In Section 2, we summarize the main characteristics of the IEEE 802.11
standard when used in ad hoc networks with 802.11b devices. We then present in
Section 3 our chain of senders scenario. Numerical values are given assuming a
Lucent Orinoco 802.11b wireless device. Comments of the three pairs fairness
problem introduced in [1] are also given.
In Section 4, we show by simulation using Network Simulator, that interesting
phenomena appear when varying the number $n$ of pairs: i) chance to gain
access to the medium for the $i$th sender-receiver pair depends on the parity
of $i$, ii) the fairness increases with $n$ especially for central pairs and
iii) the system has an asymptotic behavior when $n$ increases.
In Section 5, we introduce a new modeling of such a phenomenon. Although it is
quite simple, it allows to match results of simulations both for small and
large values of $n$, depending on a $\alpha$ coefficient. This coefficient
corresponds to the probability of emission when the neighbor senders are
waiting. For small values of $n$, we give close expressions (depending on
$\alpha$) for the probability of emission of a given pair.
In Section 6, we prove that a stationary state exists for each pair for any
length of the chain. Moreover, this stationary state converge to an asymptotic
stationary state when $n$ increases. This confirms the simulations. We also
show that some values of $\alpha$ allows to maximize the fairness, expressed
as entropy [15].
In Section 7, we comment these results, and we show that when $n$ is large,
the fairness is almost optimal near the center of the chain. We also show that
the simulation results tend to this ideal case when $n$ increases. Finally, we
sketch the relationship between $\alpha$ and the IEEE 802.11 protocol, and we
explain how to optimize the fairness by means of packet size tuning relying on
$n$ and $\alpha$. As an application of our analytical study, we maximize the
fairness in the three pairs scenario.
Concluding remarks end the paper.
## 2 IEEE 802.11 standard in ad hoc mode
The IEEE 802.11 standard implements several types of wireless communications
[2]. We focus on the most widely used for ad hoc networking with 802.11b
compliant devices in order to explain the numerical values of this paper. We
first begin by the physical layer and then we summarize the medium access
layer. Note that the numerical values depend on the physical layer we
describe, but this is not the case for the fairness issues we point out, which
appears also in protocols based on other physical layer (such 802.11a or
802.11g for instance).
### 2.1 Physical layer
In the 802.11 standard, the physical layer (PHY) is divided into two
sublayers: the _Physical Medium Dependent_ (PMD) covered by the _Physical
Layer Convergence Sublayer_.
#### 2.1.1 PMD sublayer
Besides the infra-red communications, the 802.11 PMD has been declined into
two physical layers for radio-communications, based on spread spectrum: FHSS
and DSSS. The spread spectrum techniques uses a wider bandwidth than needed
for sending a message, leading to low power density and redundancy: less
energy is diffused on a given frequency causing less interferences with the
environment, and a given information is present in several frequencies
ensuring better noise robustness. Others physical layers have been introduced
in some addenda: HR-DSSS in [3] and OFDM in [4]. With the _Channel Agility_
option, a PMD can switch from one modulation to another. However, ad hoc
networks based on the IEEE 802.11b standard mainly rely on the DSSS and HR-
DSSS PMD layers, operating in the 2.4-2.485 GHz frequency range included into
the _Industrial, Scientific and Medical_ (ISM) frequencies. We know summarize
these modulations.
For the _Direct Sequence Spread Spectrum_ (DSSS), the 2.4 GHz ISM range is
divided into 14 channels of 22 MHz each, with partial overlapping. A single
frequency is used for transmission. However a chipping technique adds
redundancy to increase the robustness: each bit of data is coded by a sequence
of eleven chips using a Barker code. The modulation technique is the
_Differential Binary (resp. Quadrature) Phase Shift Keying_ (DBPSK, resp.
DQPSK) which offers a sending rate of 1 Mbits/s for the DBPSK and 2 Mbits/s
for the DQPSK. In these techniques, a phase rotation is performed depending on
the symbol to send (either one bit for DBPSK or two for DQPSK). These
modulation techniques admit a better minimum signal to noise ratio of about 12
dB than FHSS. However the transmission is more sensitive to multi-paths, and
to Bluetooth emissions (which uses the same bandwidth range).
The _High Rate DSSS_ (HR-DSSS) uses a more complex modulation technique called
_Complementary Code Keying_ (CCK). A sending rate of 5.5 Mbits/s (resp. 11
Mbits/s) is reached with four (resp. eight) symbols per chips. The different
sending rates are chosen dynamically on the basis of transmission conditions,
for instance the signal to noise ratio (this is not normalized). In practice,
in outdoor environment, the 11Mbits/s is admissible until about 200 m, the 5.5
Mbits/s until about 300 m, the 2 Mbits/s until 400 m and the 1 Mbits/s until
500 m. This of course depends on the devices (power), antenna (gain) and
environment (outdoor/indoor, obstacles, noise…).
#### 2.1.2 PLCP sublayer
This sublayer makes a link between the different PMD layers and the MAC layer
(which should not depend on the physical layer, either current or future). It
prepares the MAC formated packets for the relevant PMD layer. A header and a
preamble are inserted before any sent data in order to synchronize the sender
and the receiver, to choose the modulation technique, and so on. As explained
above, several data rates are available in the IEEE 802.11b standard based on
DSSS modulations: 1 Mbits/s, 2 Mbits/s, 5.5 Mbits/s and 11 Mbits/s. While the
norm admits the optional _short preamble and header_ option (120 bits
partially sent at 2 Mbits/s requiring 96 $\mu$s), both preamble and header are
generally sent at the low sending rate (1Mbits/s using the DBPSK modulation)
in order to be understood by every stations (_long preamble and header_
default option).
The (long) PLCP preamble is composed of 128 bits used for sender and receiver
auto-synchronization (SYNC field) and 16 bits for the _Start Field Delimiter_
(SFD), that indicates the beginning of the frame. This corresponds to 144
$\mu$s. The (long) PLCP header is composed of the SIGNAL field (8 bits) to
indicate the modulation technique which is used (either DBPSK or DQPSK), the
SERVICES field (8 bits, currently unused), the LENGTH field (16 bits) to
indicate the number of microseconds required for transmitting the data of the
MAC layer, and the CRC field (16 bits) used for the cyclic redundancy code
checking. This corresponds to 48 $\mu$s. PLCP preamble and header lead to a
total of 192 $\mu$s at the beginning of any sending.
The PLCD sublayer also implements the Carrier Sense/Clear Channel Assessment
(CS/CCA) procedure, which gives informations on the medium (either idle or
busy). It is used to detect the beginning of a network signal which can be
received (CS), and to determine whether the channel is clear prior to transmit
a packet (CCA). The duration of this procedure depends on the modulation
technique: 27 $\mu$s for FHSS, less than 15 $\mu$s for DSSS and HR-DSSS. It
impacts the value of the _aSlotTime_ constant used by the MAC layer. By adding
other PHY-dependent delays, we found a slot time of 50 $\mu$s for the FHSS and
20 $\mu$s for the DSSS and HR-DSSS modulations.
### 2.2 Medium Access Control layer
The purpose of the MAC layer is to control the access to the shared medium by
the neighborhood nodes. Two methods have been defined: the _Distributed
Coordination Function_ (DCF) and the _Point Coordination Function_ (PCF). The
fundamental access method is the DCF; the PCF is optional. We focus on the DCF
method which is the only used in practice (PCF is rarely implemented). We
first describe frames to explain durations used in the rest of the paper.
#### 2.2.1 Frames
A MAC frame is composed of a _MAC header_ (10 to 30 bytes, depending on the
kind of frame), a body (0 to 2312 bytes) and a _Frame Check Sequence_ (FCS, 4
bytes). The MAC header contains at least a _Frame Control_ field (2 bytes), a
_Duration_ field (2 bytes) and a MAC address (6 bytes) leading to a minimum
frame size of 14 bytes with the FCS field and an empty body. The header of a
frame sent from one mobile to another one in an ad hoc network is 24 bytes
width.
Any frame is acknowledged by the receiver (unicast), implementing a positive
acknowledgment. If the acknowledgment has not been received before a delay
ACK_TIMEOUT, the frame is sent again. An acknowledgment is a 14 bytes length
MAC frame (needing 304 $\mu$s at 1 Mbits/s when adding the PLCP header and
preamble).
#### 2.2.2 Delays
The DCF implements a _Carrier Sense Multiple Access_ protocol with _Collision
Avoidance_ (CSMA/CA). It is designed to reduce the collision probability by
inserting some delays between contiguous frames (_interframe spaces_ , IFS).
The duration of the delay depends on the situation. Any transmission should
begin by a _DCF IFS_ (DIFS) delay. The acknowledgment is sent by the receiver
after a _Short IFS_ (SIFS). The SIFS is smaller than the DIFS to give priority
to the acknowledgement to other transmissions.
If a station $S_{2}$ receives a frame but is not able to understand it
(erroneous frame), it waits during an _Extended IFS_ (EIFS) instead of a DIFS
before sending. This could be a frame sent by $S_{1}$ to $R_{1}$, and these
stations are too far from $S_{2}$ to allow a good reception by this station
(preamble and header are sent using the DBPSK modulation at 1 Mbits/s, and can
be understood while the rest of the frame sent at higher rate with a different
modulation could not be understood). The EIFS delay allows to $R_{1}$ to
acknowledge the frame sent by $S_{1}$. This prevents some cases when $S_{2}$
does not hear the acknowledgment sent by $R_{1}$, and begins a transmission
that could prevent the acknowledgment reception on $S_{1}$. The station
$S_{2}$ will switch from EIFS to DIFS delays after receiving a correct frame.
As for the aSlotTime constant, the duration of the SIFS delay depends on the
PHY layer. It is equal to 10 $\mu$s for DSSS and HR-DSSS. The DIFS delay is
equal to a SIFS delay plus two aSlotTime, leading to 50 $\mu$s for DSSS and
HR-DSSS. The EIFS delay is equal to a SIFS delay plus the duration of an
acknowledgment (sent at the lowest sending rate of 1 Mbits/s) plus the
duration of a preamble and a header of the PLCP sublayer plus a DIFS delay,
leading to 364 $\mu$s for DSSS and HR-DSSS.
#### 2.2.3 RTS/CTS
Both physical and virtual mechanisms are available to sense the carrier. As
already seen, the PLCP sublayer provides a CS/CCA function which is used by
the MAC layer to probe the channel. Moreover, each station maintains a
_Network Allocation Vector_ (NAV) in order to foresee the channel liberation.
The NAV is updated using the duration field included in the received frames. A
station cannot attempt to transmit if its NAV indicates that the medium is
busy. However a station $S_{2}$ which is not in the neighborhood of the sender
$S_{1}$ but is in the neighborhood of the receiver $R_{1}$ could begin to send
data during the current transmission from $S_{1}$ to $R_{1}$, leading to a
congestion on $R_{1}$. To avoid this problem (_hidden station_), the sender
$S_{1}$ can first send a _Request To Send_ (RTS) message to $R_{1}$, which
will then reply by a _Clear To Send_ (CTS). The station $S_{2}$ will receive
the CTS message, and will then update its NAV, preventing it to send data
during the transmission $S_{1}\rightarrow R_{1}$. The frames RTS and CTS are
followed by a SIFS delay.
A RTS frame has the same length than an ACK frame (14 bytes, 304 $\mu$s at 1
Mbits/s). A CTS frame is 20 bytes long (352 $\mu$s at 1 Mbits/s) because the
header contains an additional MAC addresses. These frames are supposed to be
shorter than the data frames, and then less subject to collisions. Depending
on the configuration, this mechanism is i) never used, ii) always used or iii)
used when the frame length is larger than a threshold.
#### 2.2.4 Backoff
Despite the inter-frames delays and the carrier sense before any transmission,
several stations could decide to send simultaneously as soon as the medium is
clear. To minimize such a situation, any station waits for a random delay
called _backoff time_ before beginning a transmission.
After the DIFS or EIFS delay has expired, and if no current backoff time
remains, the station generates a random number $x$ between $0$ and the value
of a _Contention Window_ (CW). The backoff time is then equal to
$x\times$aSlotTime. Each time the channel is idle during aSlotTime
microseconds, the backoff time is decreased of aSlotTime microseconds. The
backoff time does not decrease if the medium is busy. The transmission can
begin if the channel is idle and both the delay (either DIFS or EIFS) and the
backoff time has been expired.
The value of the contention window belongs to the interval CWmin and CWmax,
where CWmin depends on the physical layer (31 for DSSS and HR-DSSS) and CWmax
equals to 1023. At the beginning, CW is equal to CWmin. Every time an attempt
to transmit fails, the contention window is doubled
($\mbox{CW}\leftarrow\mbox{CW}\times 2+1$) until it reaches CWmax. The
contention window is reset to CWmin after a successful transmission (or after
a fixed number of attempts). A successful transmission includes an
acknowledged frame as well as the receiving of a CTS frame in response to a
RTS frame.
## 3 Fairness issues in a chain of senders
The DCF mechanism described in the previous section ensures a fair access to
the shared medium when the competing nodes are able to hear each of them.
However in more complexe multi-hop ad hoc networks, some cases of unfairness
could be caused by asymmetry of the topology, or by the use of the EIFS delay
by some nodes while others use the DIFS [12, 13]. In this section, we present
an unfair case which appears in a chain of senders. This is a more general
case than the already known _three pair problem_ introduced in [1]. We begin
by some considerations on distances between mobiles.
### 3.1 Transmission ranges considerations
In the 802.11 standard, the PHY layer reports the reception of a message only
if the _Signal to Noise Ratio_ (SNR) is larger than a fixed threshold
(SNR_THRESHOLD). A signal sent with a given transmission power will be
received with a smaller reception power because of signal attenuation, fading,
etc. This defines the _transmission range_ (Rtx) which is the maximal distance
to ensure a successful reception if there is no interference. The transmission
range mainly relies on radio propagation properties (attenuation), and on the
modulation technique used, that is on the environment and on the sending rate.
As explained in the previous section, the PHY layer is also asked for carrier
sense detection (CS/CCA procedure). This mainly relies on the antenna
sensitivity. From a given distance called _Carrier Sensing Range_ and denoted
Rcs, the transmission of a far station is no more detected. Generally, the
transmission range Rtx is smaller than the carrier sensing range Rcs. For
instance, for a Lucent Orinoco wireless card, with a sending rate of 2
Mbits/s, Rtx equals 400 m while Rcs equals 670 m [16].
Suppose that a station $S_{1}$ sends a frame and a station $R_{1}$ tries to
receive it. For the reception to be feasible, we should have $d(S_{1},R_{1})<$
Rtx where $d()$ denotes the Euclidean distance (here we admits an outdoor
environment). Now let us consider a third station $S_{2}$ further from $R_{1}$
than $S_{1}$ that also sends some frames. On $R_{1}$, the reception power of
the signal sent by $S_{2}$ (denoted by $P_{r2}$) is smaller than the one of
$S_{1}$ (denoted by $P_{r1}$) and the signal of $S_{2}$ is considered as
noise. By comparing the ratio $P_{r1}/P_{r2}$ to the SNR_THRESHOLD, and by
considering a signal attenuation in $1/d^{4}$ (corresponding to an outdoor
environment modeled by the two-ray ground propagation model outside the
Fresnel zone), [16] determines an _interference range_ Ri, which is equal to
1.78 Rtx. This is the maximal distance until which a station can disrupt a
reception because of concurrent sending.
These considerations lead to the following main cases (depicted on Figure 2),
where the station $S_{1}$ sends some frames to $R_{1}$ while another station
$S_{2}$ could perturb this communication by its own emissions:
Figure 2: Communication ranges for a Lucent Orinoco 802.11b card in outdoor
environment, with a sending rate of 2 Mbits/s [16].
* •
If the station $S_{2}$ is in the area $A$, carrier sensing and backoff allow
to share the medium between $S_{1}$ and $S_{2}$.
* •
If the station $S_{2}$ is in the area $E$, it is commonly called _hidden
station_ [17], and the RTS/CTS mechanism will prevent the collision on
$R_{1}$.
* •
If $S_{2}$ is in the area $I\cup J$, the sending of $S_{1}$ and $S_{2}$ will
lead to some collisions on $R_{1}$ even if the RTS/CTS mechanism is used.
Since $R_{1}$ will not acknowledge frames sent by $S_{1}$, $S_{2}$ will
increase its contention window.
* •
If $S_{2}$ is in the area $B$, then it will receive the frames of $S_{1}$
without understanding them and will presume erroneous frames. As a
consequence, it will wait an EIFS delay instead of a DIFS one, allowing
$R_{1}$ to send the acknowledgment to $S_{1}$.
* •
If $S_{2}$ is in the area $D\cup F$, then it will receive the frames of both
$S_{1}$ and $R_{1}$ without understanding them and wait an EIFS delay.
* •
If $S_{2}$ is in the area $C\cup G$, then it will receive the frames of
$S_{1}$ without understanding them, and will use the EIFS delay. But it may
also perturb the sending of some frames by $R_{1}$ (CTS and ACK), leading to a
contention window increasing on $S_{1}$.
* •
Finally, if $S_{2}$ is in $H$, then its sending will create some collisions on
$S_{1}$ during the reception of the CTS and ACK frames sent by $R_{1}$, and
$S_{1}$ will increase its contention window.
### 3.2 Fairness in a chain of senders
In this paper, we study the fairness in a chain of senders, where each sender
has one or several receivers which are not themselves senders (see Figure 1):
a sender continuously sends some data frames to one of its neighbors, not
necessarily always the same. As explained previously, several kinds of
interaction can appear between neighbor senders and in some case their
receivers. However many studies have already be done on the increase of the
contention window. In this paper, we focus on the impact of the EIFS delay,
which appears when a sender is in the area $B\cup C\cup D\cup F\cup G$ (see
Figure 2) of its neighbors, combined with the chain topology.
For the purpose of our study, we suppose that each sender is in the area $B$.
We noticed that very similar results are obtained when the sender is in the
area $D\cup F$, but the system stabilizes much slowly. Moreover, as our
simulations have been done with network simulator [18] (see Section 4), the
interferences which may appear in the areas $C\cup G$ could not be taken into
account.
This chain of senders scenario could rarely happen in a wireless LAN network
were the mobile nodes share an access point, because in such a situation the
stations are generally in the transmission range of either the sender or the
receiver (i.e.. $A\cup E$ in Figure 2). But it could appear more often in ad
hoc network when the nodes are widely spread in the space, and when they are
moving. More fundamentally, as we will see, this case study allows some
interesting conclusions on the IEEE 802.11 standard.
### 3.3 The three pairs fairness problem
In [1, 14], a specific scenario has been studied, where strong inequity
appears. It is based on asymmetry between some pairs of communicating nodes,
and on the use of the EIFS delay. In this scenario, three pairs of
communicating nodes are considered. In each pair $i$ ($1\leq i\leq 3$), the
sender $S_{i}$ and the receiver $R_{i}$ are close enough to establish a
communication. Moreover, the sender $S_{i}$ has many data to send to the
receiver $R_{i}$ in the same pair so that it always tries to gain access to
the medium. The three pairs are placed in such a way that the senders can
detect an emission in a neighbor pair without understanding the emission.
This is a particular case of our chains of transmitters scenario, as depicted
for instance in Figure 1. Here, there is a single receiver per sender. These
pairs of senders-receivers are not necessarily arranged on a line, but a
sender is in the carrier sense area of its neighbors.
Simulations have been done in [1] as well as real experiments confirming the
simulations. Figure 3 displays simulations results of a chain of three pairs
of senders-receivers, with the parameters we will use in the following
section. As already shown in [1] (with different parameters), we notice a
strong inequity: the two external pairs can reach a throughput larger than
1.55 Mbits/s, whereas the central pair has a throughput which does not exceed
0.04 Mbits/s. Note that in the same conditions, the throughput of a single
pair is equal to 1.59 Mbits/s.
Figure 3: Fairness problem with three pairs.
To explain these results, one can remark that the central pair has to compete
with two neighbors to access the channel, and then a smaller throughput than
the border pairs (which have only to compete with one neighbor) is expected.
Moreover, the EIFS mechanism applies as soon as a neighbor is sending, and
this happens more frequently for the central pair.
## 4 Simulation of a chain of senders
In the previous section, we introduced the chain of senders scenario, which
includes the three pairs fairness problem studied in [1, 14]. In such a
scenario, the central pair has many difficulties to gain access to the channel
compared to its two neighbors. But if those neighbors have more than one
competitors, this could help the central pairs. In the following we study by
simulation the impact of the number of pairs on the fairness in the chain of
senders. This scenario combines both the EIFS mechanism and the border effect
of the chain (some nodes have a single neighbor while some others have two),
which is expected to be less and less important when the minimal distance to a
border pair increases.
### 4.1 Configuration and parameters
Our simulations have been done using Network Simulator v2.28 [18], with
parameters described previously and corresponding to a Lucent Orinoco 802.11b
device (see Section 2 and Figure 2). Without loss of generality, we assume a
single receiver per sender, leading to a chain of senders-receivers pairs.
These pairs are arranged as shown in Figure 4. Similar results should be
obtained with a less regular pattern (Figure 1), provided that the condition
described in the chain of senders scenario introduced in Section 3 are
fulfilled.
Figure 4: Chain of sender-receiver used for the simulations.
The data rate has been fixed to 2 Mbits/s, which corresponds to the Figures 2
and 4. Each sender always tries to send some UDP packets corresponding to a
1500 bytes MAC frame (see Section 2), using the RTS/CTS mechanism. Note that
we did not notice a significant influence of RTS/CTS mechanism. The
propagation model is the _two-ray ground_ , corresponding to an outdoor
environment with a single reflection on the ground. Others parameters are:
transmission power (15 dBm), antenna height (0.9 m), receiving threshold
(-91dBm), carrier sense threshold (-100dBm) [16]. The next sections show some
results when the number of pairs is varying.
### 4.2 Fairness in a chain of four pairs
Figure 5 displays simulation results for a chain of four pairs. We observe a
different behavior than with three pairs (Figure 3). The external pairs have a
throughput around 1.06 Mbits/s, whereas the two central ones reach only 0.53
Mbits/s. As previously said, this difference is explained by the number of
competitors: a single for the border pairs, and two for the central ones.
Figure 5: Fairness problem with four pairs.
Fairness is better than with three pairs because when the pair 1 acquires the
channel, pair 2 is waiting and then pairs 3 and 4 have both a single
competitor. By comparison with the three pairs chain, when the pair 1 acquires
the channel, the other border pair always gains access to the channel. Hence,
with four pairs, the central pairs can have a more frequent access to the
channel than the central pair in a chain of three pairs.
Note however that when the pair 2 gains access to the channel, pairs 1 and 3
are waiting and then pair 4 acquires the channel without difficulties. This
explains the difference between central pairs and border pairs.
### 4.3 Fairness in a chain of five pairs
Figure 6: Fairness problem with five pairs.
Simulation results for five pairs are given in Figure 6. As we can see, pairs
1, 3 and 5 have throughputs close to the maximum, whereas pairs 1 and 2 have
very low throughputs. Indeed, when the pair 1 gains access to the channel, the
pair two is waiting and the pairs 3, 4 and 5 have a similar behavior than a
three pair chain.
We observed a similar phenomenon with 7, 9 and 11 pairs.
### 4.4 Fairness in a chain of six pairs
Simulation results for six pairs are given in Figure 7. They are not so far
than results for four pairs, except that pairs 2 and 5 have less bandwidth
than central pairs 3 and 4, and that central pairs in the chain of four pairs.
Here, even if the border pair 6 acquires the channel, pair 2 could have more
than one competitor, which is not the case in a chain of four pairs. Note that
the pattern can also be seen as two neighbors chains of three pairs.
Figure 7: Fairness problem with six pairs.
We observed some similar behaviors for the chains with a larger even number of
pairs, as seen in Figure 8 with eight pairs.
Figure 8: Fairness problem with eight pairs
### 4.5 Fairness in a chain of one hundred pairs
As explained below, the fairness pattern in a chain of $n$ pairs depends on
the parity of $n$, which is an interesting phenomenon. When $n$ is odd, the
fairness is bad (Figures 3 and 6). When $n$ is even, some more complex
patterns appear with better fairness (Figures 7 and 8).
However we also observed some evolutions of these patterns when $n$ increases.
We then simulated a very large chain, in order to have an idea of the
asymptotic behavior.
Figure 9 displays the simulation results for a chain of one hundred of pairs.
We observed that the same result is obtained with a chain of 101 pairs, which
confirms that the influence of the parity of $n$ tends to decrease when $n$
increases. Moreover, for a chain of 101 pairs, one can see that the closer is
an even pair from the middle, the larger is its throughput. This is explained
by the fact that the influence of the border pairs is less important. As a
consequence, the closer is an even pair from a border, the smaller is its
throughput.
Figure 9: Fairness problem with one hundred pairs.
In this chain, the throughput of external pairs (1.39 Mbits/s) is very close
to the maximum (1.59 Mbits/s), measured in a single pair in the same
conditions. In the central flat area, the throughput of the pairs is close to
0.75 Mbits/s (about half of the throughput of the external pairs). As a
consequence of the existence of this flat area, the insertion of a new pair
has less influence on the throughput of other pairs when $n$ is large, and
when the new pair is inserted near the middle of the chain.
## 5 Mathematical modeling
In the previous section, we have shown that a chain of senders presents some
interesting phenomena, depending on the number $n$ of pairs in the chain, and
on the parity of $n$. The three pairs fairness problem introduced in [1]
appear as a sub-case of the chain of senders scenario presented in Section 3.
In this section, in order to study this phenomena and to improve the fairness,
we propose a simple modeling of such a phenomenon, before comparing the model
with the simulations.
### 5.1 Modeling with a non-linear system of equations
In [14], a mathematical modeling has been proposed for the three pairs
configuration, by means of discrete time Markov chains. Such a modeling gives
numerical results close to the simulations obtained with the ns-2 network
simulator, and not so far from real experiments of [1]. Moreover, it allows to
study the influence of some parameters variations on the fairness. However, it
is not easily generalizable when the number of pairs increases. Indeed, a
state of the Markov chain needs to capture the relative remaining backoff
delays of the pairs, which leads to many states. Moreover, transitions are
more complex when the number of pairs (and then interactions) increases.
We propose a new modeling, based on a non-linear systems of $n$ equations
whose solution gives the probabilities of emission of each pair. It allows an
analytical study both for small and large values of the number $n$ of pairs.
Let us consider a chain of $n$ pairs numbered from $1$ to $n$. For the purpose
of the modeling, we admit that there are two border pairs (pair $0$ and
$n+1$), which never send data.
We consider the random process $y_{i}(t)$ taking value $1$ if the
$i^{\mathrm{th}}$ pair is sending data at time $t$ and 0 if the pair is idle.
In fact for any $t$, the random variable $y_{i}(t)$ follows a Bernouilli’s
law. We now make a simple analysis of the communication mechanism in order to
obtain some relationships between the variables $y_{i}(t)$, for $i=1\dots n$.
Some data can be sent in a given pair $i$ only if its neighbor pairs are idle.
Thus we have the implication
$y_{i}(t)=1\Longrightarrow y_{i-1}(t)=y_{i+1}(t)=0.$ (1)
But before emitting, the sender first waits after delays and CTS frames, so
the converse of (1) is not true. To take this into account, we introduce a new
random process $z_{i}(t)$ such that
$P\left(z_{i}(t)=1\left|y_{i-1}(t)=y_{i+1}(t)=0\right.\right)=\alpha,$
where $0<\alpha<1$ and we consider that data can be sent in pair $i$ at time
$t$ if neighbor pairs are idle and $z_{i}(t)=1$. Thus we can write the
algebraic relationship
$y_{i}(t)=z_{i}(t)\left(1-y_{i-1}(t)\right)\left(1-y_{i+1}(t)\right),~{}i=1\dots
n.$ (2)
Since we want to describe some average behavior, we consider the rate of
emission as the limit when $T\rightarrow\infty$ of the time elapsed in the
emitting state between $t=0$ and $t=T$ divided by $T$
$x_{i}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}y_{i}(t)\,dt.$
In virtue of the Limit Central Theorem we have $x_{i}=E[y_{i}(t)]$, where
$E[.]$ denotes the mathematical expectation, and we have of course
$E[y_{i}(t)]=P(i\mbox{ is emitting at time }t),$
since $y_{i}(t)$ follows a Bernouilli’s law. Hence, we can take the
mathematical expectation on both sides of (2), and we obtain, by neglecting
the correlation between pairs $i-1$ and $i+1$
$x_{i}=\alpha(1-x_{i-1})(1-x_{i+1}),~{}i=1\dots n.$ (3)
### 5.2 Analytical results
The modeling introduced above allows to obtain, by substitution of unknowns
and by using symmetry relationships, a closed form of probabilities of
emission, at least for small values of $n$. For instance, for $n=3$, we have:
$x_{1}=\frac{2\alpha^{2}-1+\sqrt{(1-2\alpha^{2})^{2}-4\alpha^{3}(\alpha-1)}}{2\alpha^{2}}$
For $n=4$, we have:
$x_{1}=\frac{1+\alpha-\sqrt{(1-\alpha)(1+3\alpha)}}{2\alpha}$
Similar expressions can be found for other pairs, but for $n>8$, there is no
analytical formula because using the substitution technique leads to a
polynomial with degree greater or equal than $5$, and the solution of (3) has
to be computed with numerical techniques.
### 5.3 Validation with ns-2 results
In order to compare these results with those given by the ns-2 network
simulator, we normalize both results by the value of the first external pair.
Indeed, during a period of $t$ seconds, the $i^{\mathrm{th}}$ pair can send
data during $t_{i}=x_{i}\times T$ seconds. Let $r_{i}$ be the sending rate of
the $i^{\mathrm{th}}$ pair determined by ns-2, in bits/seconds. We have
$r_{i}\times T=r_{max}\times t_{i}$ where $r_{max}$ represents the maximal
sending rate depending on the configuration and $t_{i}$ the total time during
which the $i^{\mathrm{th}}$ pair has sent data. Thus $r_{i}/t_{i}$ is a
constant equal to $r_{max}/T$, and we have $r_{i}/t_{i}=r_{1}/t_{1}$ and then
$r_{i}/r_{1}=t_{i}/t_{1}=x_{i}/x_{1}$. We then compare the throughputs of each
pair divided by the throughput of the first one ($r_{i}/r_{1}$) with the
probability of emission of each pair divided by the probability of emission of
the first one ($x_{i}/x_{1}$).
We have done a least squares fitting with respect to $\alpha$ to approximate
the ns-2 results. For instance, for $n=3$, $n=5$ and $n=7$, we obtain values
of $\alpha$ respectively equal to $0.862$ and $0.838$ and $0.812$. These
values of $\alpha$ lead to numerical results very close to those obtained with
ns-2 network simulator, as seen in Figure 10 (a discussion of these values is
given in Section 7). The slightly differences are insignificant compared to
the unavoidable approximations of the network simulator. Nevertheless, this
first observation is only a rough validation of our modeling, and a precise
analysis of the model itself is necessary.
Figure 10: Comparison of $ns-2$ results and mathematical modeling for $n=3$,
$5$ and $7$.
## 6 Analysis of the model
Our simple modeling of the chain of senders scenario fits very well with the
simulations results for some given values of $\alpha$ (that we discuss in
Section 7). In this section, we use this modeling to determine the asymptotic
behavior of the chain, as well as to establish the relationship between
$\alpha$ and the fairness.
### 6.1 Proving the existence of a solution
Let us consider the $n$ values $x^{k}_{1}\ldots x^{k}_{n}$ as the components
of the vector $x^{(k)}\in\mathbb{R}^{n}$, and the iterative process by means
of a function $F_{\alpha}$ defined on vectors:
$x^{(k+1)}=F_{\alpha}(x^{(k)}).$ (4)
We have:
$F_{\alpha}(x)=\alpha\left(\begin{array}[]{c}1-x_{2}\\\ (1-x_{1})(1-x_{3})\\\
\vdots\\\ (1-x_{n-2})(1-x_{n})\\\ (1-x_{n-1})\end{array}\right).$ (5)
The algorithm (4) is nothing but the so-called successive approximation method
to determine iteratively a solution of the equation $x=F_{\alpha}(x)$. The
convergence toward a unique solution $\hat{x}\in E$ is guaranteed provided the
application $F_{\alpha}:E\rightarrow E$ is a contraction in some domain
$E\subset\mathbb{R}^{n}$ (this is the well-known ”contraction mapping
theorem”, see [19]). To show that $F_{\alpha}$ is a contraction we can use,
since $F_{\alpha}$ is differentiable, the derivative $F_{\alpha}^{\prime}$
given by the matrix
$F_{\alpha}^{\prime}(x)=\alpha\left(\begin{array}[]{ccccc}0&-1&0&0&0\\\
x_{3}-1&0&x_{1}-1&0&0\\\ &\ddots&\ddots&\ddots&\\\ &&1-x_{n}&0&1-x_{n-2}\\\
0&0&0&-1&0\end{array}\right).$
If we take the supremum norm, i.e. $\|x\|=\max_{1\leq i\leq n}|x_{i}|$, we can
show that $\|F_{\alpha}^{\prime}(x)\|<1$, provided that
$|x_{k}-1|<\frac{1}{2\alpha},\;1\leq k\leq n$, i.e. $F_{\alpha}$ is a
contraction on the subspace $E$ defined by
$E=\\{x\in\mathbb{R}^{n},\;\|x-\mathbf{1}\|<\frac{1}{2\alpha}$, where
$\mathbf{1}=(1,\dots,1)$.
A direct application of this result is that the algorithm (4) converges to the
unique solution of $x=F_{\alpha}(x)$ e.g. by taking $x^{(0)}=(1,\dots,1)$.
### 6.2 Asymptotic behavior
As for the simulations, we observe the convergence to an asymptotic behavior.
And the different behaviors between odd and even values of $n$ tend to
disappear when $n$ increases. Figure 11 shows the probability of emission of
pairs $k=1$ to $8$ for $n=31$ and $n=32$, for $\alpha=0.75$. For much greater
values of $n$, the difference between the rates of the first $n/2$ pairs for
$n$ (even) and $n+1$ pairs is negligible (typically less that $10^{-5}$ for
$n=100$). Thus, without loss of generality, we will continue our study by
considering only even values of $n$ in the simulations.
Figure 11: Simulation of probabilities of emission of pairs $k=1$ to $8$ for
$n=31$ and $n=32$ ($\alpha=0.75$)
### 6.3 Maximization of fairness with respect to $\alpha$
Among other possible criteria (see [20] and [21]), one way of maximizing the
fairness between all pairs is to maximize the entropy (see [15]) of the
distribution of probability of emission $\\{x_{i}\\}_{i=1\dots n}$, i.e. the
function
$E(x)=-\sum_{k=1}^{n}x_{i}\log x_{i}.$
Hence, we consider the function $J(\alpha)=\frac{1}{n}E(x(\alpha))$ where
$x(\alpha)$ is the unique solution of the equation $x=F_{\alpha}(x)$ and the
factor $\frac{1}{n}$ is used to allow some comparisons of results between
different values of $n$.
We search for the value $\hat{\alpha}$ such that
$J(\hat{\alpha})\geq J(\alpha),\;\forall\alpha\in[0,1].$ (6)
The Figure 12 represents $J(\alpha)$ with respect to $\alpha$ for
$n=10,20,100$ and $500$. For these values of $n$ we have respectively
$\hat{\alpha}=0.5536,0.5977,0.6826,0.7309$.
Figure 12: $J(\alpha)$ with respect to $\alpha$ for $n=10,20,100$ and $500$
The derivative of $J(\alpha)$ with respect to $\alpha$ is computed by using
the classical adjoint state method, i.e. we consider the Lagrangian
$L(\alpha,x,\lambda)=\frac{1}{n}E(x)+\lambda^{\top}(x-F_{\alpha}(x)),$
where $\lambda$ is a vector of $\mathbb{R}^{n}$ and $\top$ denotes the
transposition. The function $F_{\alpha}$ has been defined in Equation (5). We
have, of course, $J(\alpha)=L(\alpha,x(\alpha),\lambda)$ for any $\lambda$. We
choose $\lambda=\lambda(\alpha)$ such that
$\frac{\partial L}{\partial x}(\alpha,x(\alpha),\lambda(\alpha))=0,$
which leads to
$\lambda(\alpha)=\frac{1}{n}[F_{\alpha}^{\prime}(\alpha)-I]^{-1}\nabla
E(x(\alpha))$, where $\nabla E$ is the gradient of $E(x)$ with respect to $x$.
We have finally
$\displaystyle J^{\prime}(\alpha)$ $\displaystyle=$
$\displaystyle-\lambda(\alpha)^{\top}\left(\frac{\partial
L}{\partial\alpha}F_{\alpha}(x(\alpha))\right),$ (7) $\displaystyle=$
$\displaystyle-\frac{1}{\alpha}\lambda(\alpha)^{\top}F_{\alpha}(x(\alpha)).$
(8)
The computation of $x(\alpha)$ is done with a Newton type method, much faster
than the simple fixed point method suggested by Equation (4), and the
optimization is performed by the Quasi Newton BFGS method available in Scilab
(see [22]).
## 7 Discussion
In the previous section, the chain of senders scenario has been analyzed on
the basis of the modeling introduced in Section 5. Note that as far as the
mathematical model is concerned, the non-linear systems of equations (3) is
obtained by assuming that the emission states of pairs $i$ and $i+1$ are
independent from a probabilistic point of view. While this assumption (also
assumed in [23]) may be questionable, it is relevant because our modeling
considers the stationary behavior of the chain.
In this section, we discuss the asymptotic values obtained in the analysis
before interpreting $\alpha$ in a practical point of view.
### 7.1 Asymptotic flat area
If we study the asymptotic behavior of results, we see that for large values
of $n$ and the optimal value $\alpha=\hat{\alpha}$, the optimal probabilities
of emission (see Figure 13) exhibit a large flat area with a value very close
to $\frac{1}{3}$ (the $\frac{1}{3}$ value will be discussed below). This flat
area ensures that the insertion of a new pair will not disturb the rate for
close neighbors.
Figure 13: Probabilities of emission for $n=100$ and optimal $\alpha$. The
dotted line is at probability $1/3$.
Moreover, for $n=100,500,1000$ and $2000$ the value of the optimal probability
corresponding to this flat area is respectively equal to $0.3177$, $0.3290$,
$0.3313$ and $0.3325$.
To understand the convergence of this value to $1/3$, we must consider the
idealized situation where there is an infinite number of pairs, or
equivalently, the situation where the number of pairs is large enough to allow
to form a circle, where the last pair numbered $k=n$ has the pairs $k=n-1$ and
$k=1$ as neighbors. Hence, there is no border effect since all pairs have two
neighbor pairs.
So let us consider the $i^{\mathrm{th}}$ pair and its neighbors pairs numbered
$i-1$ an $i+1$, and a very simple model of channel acquirement: each sender of
each pair generates a realization of a random variable $u_{i}$ (uniformly
distributed in the interval $[a,b]$). We consider that the $i^{\mathrm{th}}$
pair will acquire the channel if $u_{i}<u_{i+1}$ and $u_{i}<u_{i-1}$. The
probability of this event can be calculated as follows:
$\displaystyle P(u_{i}<u_{i+1},u_{i}<u_{i-1})$ $\displaystyle=$
$\displaystyle\int_{a}^{b}\int_{a}^{u_{i}}\int_{a}^{u_{i}}\,\frac{d_{u_{i+1}}d_{u_{i-1}}d_{u_{i}}}{(b-a)^{3}},$
$\displaystyle=$
$\displaystyle\frac{1}{(b-a)^{3}}\int_{a}^{b}(u_{i}-a)^{2}\,d_{u_{i}},$
$\displaystyle=$ $\displaystyle\frac{1}{3}.$
Hence, the value $\frac{1}{3}$ can be understood as a limiting value
exhibiting the maximum fairness that can be obtained. This value of
$\frac{1}{3}$ is asymptotically obtained in our model, by maximizing the
entropy of the distribution of probabilities: this is a very interesting
behavior.
### 7.2 Asymptotic optimal alpha
Another interesting phenomenon is the apparent convergence of the optimal
value $\hat{\alpha}$ to $0.75$ when $n$ tends to the infinity, as it can be
seen on Figure 14.
Figure 14: Optimal $\alpha$ with respect to $n$.
This is not so surprising, as we will show it in the following analysis.
Consider the same idealized situation as before, where the pairs are arranged
to form a circle: the probabilities of emission $\\{x_{k}\\}_{k=1\dots n}$ are
necessarily invariant with respect to a shift of indices, since all pairs will
always have two neighbors. Hence we have $x_{k}=x_{1}$, $\forall k$, and the
system of $n$ equations $x=F_{\alpha}(x)$ giving the probabilities is
equivalent to the scalar equation $x_{1}=\alpha(1-x_{1})^{2}$. In this case
the entropy is already maximized since all values are equal. Then, if we are
looking for the value of $\alpha$ giving the maximum probability of emission
in such a configuration, i.e. $x_{1}=\frac{1}{3}$, we obtain
$\alpha=\frac{x_{1}}{(1-x_{1})^{2}}=0.75$. This value is in fact completely
determined by the topology of the neighborhood.
### 7.3 Asymptotic comparison of modeling and simulation
We have compared the normalized rates obtained via ns-2 and via the
mathematical model for $n=100$ pairs (the rates and probabilities are
normalized with respect to the pair exhibiting the maximum value, as explained
in Section 5.3). On Figure 15 we can see that the mathematical model with
$\alpha=0.6825$, corresponding to the maximum entropy, gives an excellent
approximation of ns-2 results.
Figure 15: Probability of emission of the first $50$ pairs of 100 obtained by
ns-2 and mathematical model for optimal $\alpha=0.6825$.
Hence, it appears that the asymptotic behavior of the chain of $n$ IEEE 802.11
senders-receivers (as defined in Section 4) tends to the maximum entropy when
$n$ tends to the infinity. This is a surprising result.
### 7.4 Interpretation of the $\alpha$ coefficient
We defined $\alpha$ as the probability of sending for a given pair when its
neighbors are not sending. Interpreting $\alpha$ implies to determine whether
a pair is sending or not when its neighbors are not sending. This in fact
depends on what is able to hear a neighbor sender, and then on what area it is
on Figure 2. As for previous simulations, we suppose that the neighbors
senders are in the area $B$, and that a sender can only hear transmission of a
neighbor sender, and not of a neighbor receiver. A neighbor pair is then
considering as sending only when the sender (and not the receiver) is sending,
and waiting in other cases.
Before any transmission, a sender has to wait for a delay, and in many cases
this is an EIFS delay instead of a DIFS one. During this delay, chances are
large that its neighbors are sending. This means that this delay is not part
of the time wasting by a pair while it could send because its neighbors are
not sending. To the contrary, neighbor senders are not sending during the
backoff delay.
Figure 16 summarizes a complete transmission of a $s$ bytes MAC frame between
a sender $S_{i}$ and a receiver $R_{i}$ using numerical values given in
Section 2 ($d$ denotes the sending rate, and 0.5 represents the mathematical
expectation of a random variable on $[0,1]$).
sender $S_{i}$ receiver $R_{i}$ DIFS or EIFS 50 or 364 $\mu$s aSlotTime
$\times$ CW $\times 0.5$ 310$\mu$s RTS 304 $\mu$s SIFS 10 $\mu$s CTS 352
$\mu$s SIFS 10 $\mu$s header and preamble (PHY) 192 $\mu$s $s$ data bytes
(MAC) $8\times s/d$ $\mu$s SIFS 10 $\mu$s ACK 304 $\mu$s
Figure 16: Complete transmission of a $s$ bytes MAC data frame at $d$Mbits/s.
We suppose that CW = 31, leading to an average backoff time of 310 $\mu$s (we
indeed rarely observed a contention window larger than 31 in our simulations,
see discussions concerning the areas in Section 3). Based on the previous
considerations, the waiting time $T_{w}$ while the neighbors are waiting
corresponds to the backoff (310 $\mu$s), the SIFS delays ($3\times 10$
$\mu$s), the CTS (352 $\mu$s) and ACK (304 $\mu$s) frames sent by the
receiver: $T_{w}=996$. The sending time $T_{s}$ while the neighbors are
waiting corresponds to the RTS (304 $\mu$s) and data frame (192 + $8s/d$
$\mu$s): $T_{s}=496+8s/d$. Since $T_{s}=\alpha(T_{s}+T_{w})$, we have
$\alpha=\frac{496+\frac{8s}{d}}{1492+\frac{8s}{d}}$
In our simulations, the sending rate has been fixed to 2Mbits/s ($d=2$) and a
data MAC frame is equal to 1500 bytes ($s=1500$). We then find $\alpha=0.867$.
This value is very close to those found in Section 5.3.
### 7.5 Obtaining the maximal fairness
The previous equation shows a relationship between $\alpha$ and the frame size
$s$. We then simulated a three pairs chain while varying the packet size. The
throughput of each pair has been normalized by the reference throughput of a
single pair ($1.59$ Mbits/s in our configuration) in order to compute the
entropy.
Results are displayed in Figure 17. We can show that the maximum entropy is
reached for a packet size of 250 bytes. This corresponds to $\alpha=0.6$,
which is close to the optimal $\hat{\alpha}=0.655$.
Figure 17: Entropy versus packet size.
## 8 Conclusion
In this paper, we developed a scenario for ad hoc networks relying on IEEE
802.11 wireless communications composed of a chain of senders, such that each
of them is in the carrier sense area of its neighbors. This scenario combines
the EIFS mechanism with the asymmetry of a chain, where two nodes have only
one neighbor while the others have two. This scenario includes the three pairs
fairness problem [1].
We show that interesting patterns appear when the number $n$ of sender-
receiver pairs in the chain increases. These phenomena depend on the parity of
$n$. For small values of $n$, the fairness is better if $n$ is even than if
$n$ is odd. We also point out an asymptotic behavior when $n$ increases, with
a large central flat area. By means of a simple modeling, we provide an
analytical study of this scenario, which explains the phenomena observed by
simulation. Moreover, this modeling clearly highlights a link between the
fairness and the packet size.
Besides the curious fairness phenomena we pointed out in the chain of senders,
it is interesting to notice that this simple modeling relying on a single
coefficient $\alpha$ is able to render the complex situation of concurrent
transmissions using the IEEE 802.11 standard. Previous modeling were based on
Markov chains and were not really adapted for $n$ larger than 3. This
coefficient expresses the probability for a sender to transmit a frame while
its neighbors are waiting. Indeed, a sender does not fully use the channel,
even when its neighbors are waiting.
Another interesting contribution is the asymptotic results. When the number of
pairs is large, the probability of emission for a sender near the middle of
the chain is very close to the optimal value (1/3). This optimal probability
corresponds to $\alpha=3/4$. Moreover this value gives also the maximal
fairness (expressed by means of entropy) when $n$ tends to infinity. The
consequence is that, to reach the optimal case, a sender should waste 1/4 of
the time it is granted for sending. We should also notice that when $n$
increases, the chain of IEEE 802.11 senders-receivers tends to this ideal
case.
This ideal value of $\alpha$ is correct for very large values of $n$, which
does not correspond to real cases. However, for a given $n$, the modeling is
able to give the optimal $\alpha$, allowing to deduce the (approximative)
optimal packet size. When applying this method on the chain of three pairs, we
found an ideal MAC frame of 250 bytes. Simulation results with such a frame
size lead to results very close to the optimal fairness.
Among possible further works, we would like to point out other uses of such a
simple modeling, for more complex scenario.
## References
* [1] D. Dhoutaut and I. Guérin-Lassous, “Impact of heavy traffic beyond communication range in multi-hops ad hoc networks,” in _International Network Conference_ , Plymouth, July 2002.
* [2] LAN MAN Standards Comittee of the IEEE Computer Society, “Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications,” The IEEE, Tech. Rep., June 1999, (reaffirmed 12 June 2003 by IEEE-SA Standard Board).
* [3] ——, “Part 11: Wireless LAN medium access control (MAC) and physical layer (phy) specifications: Higher-speed physical layer extension in the 2.4 ghz band,” The IEEE, Tech. Rep., June 1999, (reaffirmed 12 June 2003 by IEEE-SA Standard Board).
* [4] ——, “Part 11: Wireless LAN medium access control (MAC) and physical layer (phy) specifications: Higher-speed physical layer in the 5 ghz band,” The IEEE, Tech. Rep., June 1999, (reaffirmed 12 June 2003 by IEEE-SA Standard Board).
* [5] F. Cali, M. Conti, and E. Gregori, “IEEE 802.11 wireless LAN: Capacity analysis and protocol enhancement,” in _IEEE Infocom_ , San Francisco, March 1998.
* [6] G. Bianci, “Performance analysis of the IEEE 802.11 distributed coordination function,” _IEEE Journal on Selected Areas in Communications_ , vol. 18, no. 3, pp. 535–547, March 2000.
* [7] A. Heindl and R. German, “Performance modeling of IEEE 802.11 wireless LANs with stochastic Petri nets,” _Perform. Eval._ , vol. 44, no. 1-4, pp. 139–164, 2001.
* [8] V. Vishnevsky and A. Lyakhov, “802.11 LANs: Saturation throughput in the presence of noise,” in _IFIP-TC6 Networking Conference_ , Pisa, 2002.
* [9] ——, “IEEE 802.11 wireless LAN: Saturation throughput analysis with seizing effect consideration,” _Cluster Computing_ , vol. 5, no. 2, April 2002.
* [10] V. Bharghavan, A. Demers, S. Shenker, and L. Zhang, “MACAW: a media access protocol for wireless LANs,” in _ACM Sigcomm_ , London, August 1994.
* [11] Z. Li, S. Enandi, and A. Gupta, “Improving MAC performance in wireless ad hoc networks using enhanced carrier sensing (ECS),” in _IFIP-TC6 Networking Conference_ , Athena, May 2004.
* [12] T. Nandagopal, T.-E. Kim, X. Gao, and V. Bharghavan, “Achieving MAC layer fairness in wireless packet networks,” in _ACM Mobicom_ , Massachusets, August 2000.
* [13] C. Chaudet, D. Dhoutaut, and I. Guŕin Lassous, “Experiments of some performance issues with IEEE 802.11b in ad hoc networks,” in _Proc. of WONS_ , St Moritz, January 2005.
* [14] C. Chaudet, I. Guérin-Lassous, E. Thierry, and B. Gaujal, “Study of the impact of asymmetry and carrier sense mechanism in IEEE 802.11 multi-hops networks through a basic case,” in _PE-WASUN_ , Venice, September 2004.
* [15] E. T. Jaynes, “Information theory and statistical mechanics,” _Phys. Rev._ , vol. 106, no. 4, pp. 620–630, 1957.
* [16] K. Xu, M. Gerla, and B. Sang, “How effective is the IEEE 802.11 RTS/CTS handshake in ad hoc networks?” in _Proc. of IEEE Globecom 2002_ , Tapei, Taiwan, R.O.C., November 2002.
* [17] L. Kleinrock and F. Tobagi, “Packet switching in radio channels: Part I – carrier sense multiple-access characteristics,” _IEEE Transactions on Communications_ , vol. 23, no. 12, pp. 1400–1416, 1975.
* [18] “Network simulator 2: http://www.isi.edu/nsnam/ns/.”
* [19] W. Rudin, _Principles of Mathematical Analysis_. New York: McGraw Hill, 1964.
* [20] C. Koksal, H. Kassab, and H. Balakrishnan, “An analysis of short-term fairness in wireless media access protocols,” in _Proceeding of ACM Sigmetrics_ , 2000\.
* [21] T. Bonald and L. Massoulié, “Impact of fairness on Internet performance,” in _Proc. of Sigmetrics Performance_ , Cambridge, USA, 2001.
* [22] C. Bunks, J. Chancelier, F. Delebecque, C. Gomeza, M. Goursat, R. Nikoukhah, and S. Steer, _Engineering and Scientific Computing with SCILAB_. Birkhäuser, 1999.
* [23] C. Wang, B. Li, and L. Li, “A new collision resolution mechanism to enhance the performance of IEEE 802.11 DCF,” _IEEE Transactions on Vehicular Technology_ , vol. 53, no. 4, pp. 1235–1246, July 2004.
|
arxiv-papers
| 2011-02-17T15:33:15 |
2024-09-04T02:49:17.091273
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bertrand Ducourthial and Yacine Khaled and St\\'ephane Mottelet",
"submitter": "St\\'ephane Mottelet",
"url": "https://arxiv.org/abs/1102.3607"
}
|
1102.3655
|
# Symmetry analysis of possible superconducting states in KxFe2Se2
superconductors
I.I. Mazin Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
###### Abstract
A newly discovered family of the Fe-based superconductors is isostructural
with the so-called 122 family of Fe pnictides, but has a qualitatively
different doping state. Early experiments indicate that superconductivity is
nodeless, yet prerequisites for the $s_{\pm}$ nodeless state (generally
believed to be realized in Fe superconductors) are missing. It is tempting to
assign a $d-$ wave symmetry to the new materials, and it does seem at first
glance that such a state may be nodeless. Yet a more careful analysis shows
that it is not possible, given the particular 122 crystallography, and that
the possible choice of admissible symmetries is severly limited: it is either
a conventional single-sign $s_{+}$ state, or another $s_{\pm}$ state,
different from the one believed to be present in other Fe-based
superconductors.
###### pacs:
74.20.Pq,74.25.Jb,74.70.Xa
Recent reports of superconductivity at $T_{c}$ in excess of 35 Kfirstreports
in Se-based iron superconductors (FeBS) isostructural with BaFe2As2 (the so-
called 122 structure) have triggered a new surge of interest among the physics
community. These materials are believed by many to open a new page in Fe-based
superconductivity. Indeed, the formal composition, AFe2Se${}_{2},$ where A is
an alkali metal, corresponds to a formal doping of 0.5 electron off the
standard for FeBS parent compounds (LaFeAsO, BaFe2As${}_{2},$ or FeSe) valence
state of iron, Fe2+. Such a large doping in other materials, such as
Ba(Fe,Co)2As2 leads to a complete suppression of supercondctivity, which has
been generally ascribedWen ; french to disappearence of the hole pockets of
the Fermi surface and formal violation of the quasinesting condition for the
$s_{\pm}$ superconductivity.
Indeed all band structure calculations showbands1 that in AFe2Se2 the hole
bands are well under the Fermi surface (for the reported experimental crystal
structure of KFe2Se${}_{2},$ about 60 meV), and this is confirmed by
preliminary ARPES resultsARPES1 ; ARPES2 ; DingTl . This has led to
speculations that in this subfamily it is not the familiar $s_{\pm}$
superconductivity that is realized, but a $d-$wave superconductivityARPES1 ;
Graser ; Lee ; Balat of the sort discussed in an early paper by Kuroki $et$
$al$Kuroki . Unfortunately, these speculations are entirely based upon the
“unfolded” Brillouin zone description of the electronic structure, a
simplified model that neglects the symmetry lowering due to the As or Se atoms
and the fact that in the real unit cell there are two Fe ions, and not one.
Furthermore they implicitly assume that spin susceptibility corresponding to
the “checkerboard” wave vector, $Q=(\overline{\pi},\overline{\pi}),$ is
substantially enhanced, despite the fact that this vector corresponds to an
electron-electron interband transition that is much less efficient in
enhancing susceptibility (here and below, we used the bar when we work in the
unfolded Brillouin zone). This assumption is supported by model calculations
based on an onsite Hubbard HamiltonianGraser , but its applicability to FeBS
is still an open question.
In this paper, we will critically address these two assumptrions, and will
show that the latter assumption is supported by first principles calculations,
but the former assumpion is actually very misleading. We will present a
general symmetry analysis of possible superconducting symmetries supported by
the Fermi surface topology existing in AFe2Se${}_{2}.$ This analysis is not
limited by a specific density functional calculation, but is based on the
general crystallographic considerations appropriate for this crystal
structure. It appears that it is impossible to fold down a nodeless $d-$wave
state so as to avoid formation of line nodes. Thus, emerging experimental
evidence from ARPES, ARPES1 ; ARPES2 ; DingTl , specific heatWenSH , NMRNMR ,
and opticsopt that superconductivity in AFe2Se2 is nodeless is a strong
argument against $d-$wave. A conventional $s-$state is also unlikely based on
the proximity to magnetism and actual observation of a coexistance of
superconductivity and magnetism. We emphasize that the symmetry of the folded
Fermi surfaces does allow for a nodeless state, which however has an overall
$s$ symmetry and can also be called $s_{\pm}$, as it is strongly sign-
changing. Unlike the $s_{\pm}$ advocated for the “old” FeBS it is not driven
by ($\overline{\pi},0)$ spin fluctuations and cannot be derived from
considering an unfolded Brillouin zone Fermi surface.
Figure 1: A cartoon showing a generic 3D Fermi surface for an AFe2Se2
material in the unfolded (one Fe/cell) Brillouin zone. Different colors show
the signs of the order parameter in a nodeless $d-$wave state, allowed in the
unfolded zone. The $\Gamma$ point is in the center (no Fermi surface pockets
around $\Gamma$), and the electron pockets are around the $\bar{X},\bar{Y}$
points.
The unfolded Fermi surface topology in materials with the 122 structure is
controlled by two factors: ellipticity of individual electron pockets and
their $k_{z}$ dispersion (Fig. 1). The ellipticity in the unfolded zone is
determined by the relative position of the $xy$ and $xz/yz$ levels of Fe, and
the relative dispersion of the bands derived from them. IndeedPALee , the
point on the Fermi surface located between $\overline{\Gamma}$ and
$\overline{X}$ has a purely $xy$ character, while that between
$\overline{\Gamma}$ and M̄ a pure $yz$ character. At the $\overline{X}$ point
the $xy$ state is slightly below the $yz$ state, but has a stronger
dispersion, therefore depending on the system parameters and the Fermi level
the corresponding point of the Fermi surface may be more removed from
$\overline{X},$ or less. In the 1111 compounds, the first to have been
investigated, the dispersion of the $xy$ band is not high enough to reverse
the natural trend, so the Fermi surface remains elongated in the
$\overline{\Gamma}\overline{X}$ (1,0) direction.
For both $xy$ and $xz/yz$ bands the hopping mainly proceeds via As (Se)
$p-$orbitals. The $xy$ states mainly hop through the $p_{z}$ orbital (see Ref.
Ole for more detailed discussions), and $xz$ ($yz)$ via $p_{y}$ ($p_{x})$
orbitals. If there is a considerable interlayer hopping between the $p$
orbitals, whether direct (11 family) or assisted (122 family), the ellipticity
becomes $k_{z}-$dependent. For instance, in FeSe there is noticeable overlap
between the Se $p_{z}$ orbitals, so that they form a dispersive band with the
maximum at $k_{z}=0$ and the minimum at $k_{z}=\pi/c.$ Obviously,
hybridization is stronger when the $p_{z}$ states are higher, therefore the
Fermi surface ellipticity is completely suppressed in the $k_{z}$=0 plane,
while rather strong in the $k_{z}=\pi/c$ plane, which leads to formation of
the characteristic “bellies” in the Fermi surface of FeSe. On the other hand,
$p_{x,y}$ orbitals in FeSe do not overlap in the neighboring layers, so the
$xz$ and $yz$ bands have very little $k_{z}$ dispersion, so that the inner
barrels of the electronic pockets in this compound are practically 2D.
In 122, the interlayer hopping proceeds mainly via the Ba (K) sites, and thus
the $k_{z}$ dispersion is comparable (but opposite in sign!) for the $xy$ and
$xz/yz$ bands. As a result, when going from the $k_{z}=0$ plane to the
$k_{z}=\pi/c$ plane the longer axis of the Fermi pocket shrinks, and the
shorter expands, so that the ellipticity actually changes sign.
Importantly, the symmetry operation that folds down the single-Fe Brillouin
zone when the unit cell is doubled according to the As (Se) site symmetry is
different in the 11 and 1111 structures, as compared to the 122 structure. In
the former case, the operation in question is the translation by
$(\bar{\pi},\bar{\pi},0),$ without any shift in the $k_{z}$ direction, in the
latter by $(\bar{\pi},\bar{\pi},\bar{\pi}).$ Thus the folded Fermi surface in
11 and in 1111 has full fourfold symmetry, while that in the 122 has such
symmetry only for one particular $k_{z,}$ namely $k_{z}=\pi/2c.$ Furthermore,
in 122 the folded bands are not degenerate along the MX (now the labels are
without the bars, that is, corresponding to the folded BZ), as they were in
11/1111. Finally, there is a considerable (at least on the scale of the
superconducting gap) hybridization when the folded bands cross (except for
$k_{z}=0).$
Now we are ready to analyze possible superconducting symmetries in the actual
AFe2Se2 materials. We shall not adhere strictly to the calculated band
structure and the Fermi surfaces, but rather consider several possibilities
allowed by symmetry. Let us start first from a $d-$wave state in the unfolded
BZ, as derived in Refs. Kuroki ; Lee ; Graser . In Fig. 1 we show by the two
colors the signs of the order parameter. Obviously in the unfolded BZ such a
state has no nodes.
Figure 2: A cartoon showing a folded 3D Fermi surface for an AFe2Se2
material, assuming a finite ellipticity, but zero $k_{z}$ dispersion.
Different colors show the signs of the order parameter in a $d-$wave state.
Wherever the two colors meet, turning on hybridization due to the Se potential
creates nodes in the order parameter.
Let us now assume that the $k_{z}$ dispersion is negligible, while the
ellipticity remains finite. After folding, but before turning on the
hybridization, we have the picture shown in Fig. 2. The border between the red
and the blue colored regions now becomes a nodal lineParker . In this case, we
have four such lines for each pair of electron pockets. One can think of an
effective “thickness” of the nodal lines, meaning the distance in the momentum
space over which the sign of the order parameter changes. This is defined by
the ratio of the hybridization gap at the point where the bands cross and
their typical energy separation. Analysis of the first principle calculations
for both As and Se based 122 compounds indicates that this width is varying
between zero (unless spin-orbit interaction is taken into account) and a
number of the order of 1. Thus, the effect of the nodal lines on
thermodynamical properties is comparable to that in one-band $d-$wave
superconductors such as cuprates and therefore should be easily detectable.
Let us now gradually turn on the $k_{z}$ dispersion. Nothing changes for
$k_{z}=\pi/2c,$ that is, there are four equidistant nodes in this plane, which
we can label as 1, 2, 3 and 4. As we move towards $k_{z}=0,$ nodes 1 and 2 get
closer to each other, and so do nodes 3 and 4. As we move towards
$k_{z}=\pi/c,$ the other pairs get closer, nodes 1 and 4, and nodes 2 and 3.
Thus, instead of four vertical node lines we get four wiggly lines, otherwise
similar in properties to the pure 2D case in Fig. 3. Averaged over all
$k_{z},$ they still have the fourfould symmetry and the observable properties
should be very similar to the 2D case. A notable exception is ARPES. That
technique should detect gap nodes along the (0,1) and (1,0) direction when
probing $k_{z}=\pi/2c,$ which should gradually shift away from these
directions when the probed momentum is different.
Figure 3: Same as Fig. 2, but assuming a moderated $k_{z}$ dispersion. The
plane at $k_{z}=\pi/2c$ is shown, and one of the Fermi surfaces is clipped
above this plane to show how the nodal points move away from their high
symmetry positions.
This is actually the case in density functional calculations for the
stoichiometric compounds in the reported crystal structure; the intersection
lines of the two FSs folded on top of each other never close, and a $d-$wave
superconductivity in this system must retain all four vertical node lines.
Suppose however that these calculations underestimate the $k_{z}$ dispersion
(this is somewhat unlikely, as band structure calculations tend to produce too
diffuse orbitals and too much hopping, but let us assume for the sake of
generality that this is possible). In that case, at some finite value of
$\tilde{k}_{z}$ such that $0<\tilde{k}_{z}<\pi/2c$ nodes 1 and 2 will merge
and annihilate, and so will nodes 3 and 4, while at $k_{z}=\pi-\tilde{k}_{z}$
the other two pairs will annihilate. As a result, we will have a $horizontal$
wiggly node line, the less wiggly the stronger is the 3D dispersion (Fig. 4.
Importantly, a full node line remains present in any band structure, whatever
assumption one makes about the 3D dispersion and ellipticity. Thus, the fact
that fully developed node lines are inconsistent with numerous reported
experiments excludes a d-wave pairing as a viable possibility.
Figure 4: Same as Fig. 3, but assuming a very strong $k_{z}$ dispersion.
An interesting alternative presents itself if we look closely at the
calculated ab-initio Fermi surfaces of KFe2Se2. One feature that distinguishes
them from those in As-based materials is a very small ellipticity and,
compared to the As-based 122 family, very little $k_{z}$ dispersionnotebands .
Looking at the constant-$k_{z}$ cuts (Fig. 5) of the Fermi surface, we observe
that we are in a regime where the separation of the two FSs is comparable
with, or smaller than the hybridization. In this case, a reasonable
approximation would be to neglect both ellipticity and $k_{z}-$dispersion, and
analyze the possible superconducting symmetry in this model. First of all in
this approximation the resulting FSs are two concentric cylinders that touch
at $k_{z}=0$ but are split otherwise. The wave functions on these cylinders
are, respectively, the odd and the even combinations of the original and the
downfolded bands.
Figure 5: Cuts of the Fermi surfaces calculated for K0.8Fe2Se2 using LAPW
band structure, and the experimental lattice parameter and atomic positions.
Upper panel: $k_{z}=0$. Lower panel: $k_{z}=\pi/2c$ (half way between $\Gamma$
and $Z$
.
Thus, if the pairing interaction in the unfolded BZ exists only in the
interband (interpocket) channel, as is implicitly or explicitly assumed in
most current theories, it becomes identically zero after downfolding and
hybridization. In fact, in this limit, when hybridization is strong everywhere
in the BZ, the spin susceptibility and the pairing interaction must be
computed from scratch using the 2-Fe unit cell (and the folded BZ).
Importantly, one can easily imagine an interaction that would lead to a
nodeless state in such a system. Indeed, if the interaction is stronger
between the bonding and antibonding band, than between different points in the
same band, the resulting interaction will again be a sign-changing s-wave,
with all inner barrels having one sign of the order parameter, and the other
the opposite sign (A very similar state was unsuccessfully proposed for
bilayer cuprates 15 years agobilayer ).
Naively, one may think that one can construct a d-wave state where the signs
of the order parameter will be swapped as one goes around from one M point in
the BZ to another. Yet this is not allowed by symmetry, for (2$\pi/a,0,\pi/c)$
and (0,2$\pi/b,\pi/c)$ (2 Fe/cell notations) are reciprocal lattice vectors,
so translating by any of these vectors must retain both the amplitude and the
phase of the superconducting order parameter. Incidentally, this symmetry
requirement is not always appreciated, and there have been “$d-$wave”
suggestions ($e.g.,$ Ref.Balat ) that violate it.
Let us now discuss possible magnetic interactions in this system. Both from
the Fermiology point of view and from experimentNMR it is clear that familiar
spin fluctuations with the wave vector ($\pi/a,\pi/b,q_{z})$ are absent in
this system. As discussed above, model calculations based on an unfolded band
structure are much less well justified than in the old pnictides, at least if
one believes the band structure calculations. In principle, one can
controllably calculate the spin resposne using the full density functional
theory Serega , however, there are no codes widely available that are
implementing such capability.
On the other hand, one can gain some insight regarding the DFT spin response
at $q=0$, in particular, on the relative strength of the fluctuations in the
FM and in the AFM (checkerboard) channels, in a different way. To this end,
let us write the full spin suseptibility in the the local density functional
theorynote :
$\chi^{FM}=\frac{\chi_{0}^{FM}}{1-I\chi_{0}^{FM}},\;\chi^{AFM}=\frac{\chi_{0}^{AFM}}{1-I\chi_{0}^{AFM}},$
(1)
where $I=2\delta^{2}E_{xc}/\delta M_{Fe}^{2}$ is the iron Stoner factor, which
we, as the first approximation, will consider independent of the magnetic
pattern. Note that spin-unrestricted calculations for all magnetic patterns,
ferromagnetic, checkerboard, or the stripe phase similar to ferropnictides
converge to large magnetic moment solutions not helpful in analyzing the
linear response of the nonmagnetic phase (Table 1).
Table 1: Calculated energies (the nonmagnetic state is taken as zero) for various stable and metastable magnetic states of KFe2Se2. | $M_{Fe},\mu_{B}$ | $\Delta E,$ meV/Fe
---|---|---
FM (LDA) | 2.8 | $+13$
FM (GGA) | 2.9 | $-140$
AFM-cb (LDA) | 1.8 | -111
AFM-cb (GGA) | 2.1 | $-192$
stripe (LDA) | 2.2 | $-169$
stripe (GGA) | 2.4 | $-290$
To circumvent this problem, we will use a modification of the standard LAPW
package ”WIEN2k”, which allows for a phenomenological account of itinerant
spin fluctuations by tuning the Hund’s rule couplingBlaha . It appears that
the unaltered LDA (and even GGA) functional solution in the nonmagnetic phase
is stable against weak FM perturbations (Fig. 6), even though it is unstable
against formation of a large magnetic momentstripe . It requires scaling $I$
up by 40% to make it unstable, thus $\chi_{0}^{FM}\approx 1/(1.4I)=0.7I.$ at
the same time, scaling $I$ down by $\alpha\approx 0.7,$ we make the
checkerboard pattern also marginally stable, thus $\chi_{0}^{AFM}\approx
1/(0.7I)\approx 2\chi_{0}^{FM}.$ Thus, the Fermiology favors the checkerboard
antiferromagnetic fluctuations about twice more than the ferromagnetic ones.
Figure 6: Fixed spin moment calculations for the uniform (ferromagnetic)
susceptibility in KFe2Se2.
This is in some sense encouraging. If both FM and AFM fluctuations are
present, they can actually provide coupling between the bonding and
antibonding sheets of the folded Fermi surface, even if the hybridization is
very strong (if only AFM fluctuations are present, this coupling vanishes in
the limit of strong hybridization). It may or may not be stronger than the
intraband coupling. Only full calculations of susceptibility in the two Fe
unit cell will give us the answer. Yet, we can firmly conclude that the only
state compatible with two experimental observations, (1) that the
superconducting gap does not have nodes and (2) that superconductivity emerges
in immediate proximity of an ordered magnetic phase, is again an $s_{\pm}$
state, but this time with the order parameter changing sign between the
bonding and antibonding state. It is also worth noting that if a 3D electron
pocket is present at $\Gamma,$ as calculations and several ARPES experiments
suggest, in the proposed $d-$wave symmetryGraser ; Lee ; Balat it would be
cut by four nodal lines which would also have been seen in the experiment. The
concentric $s_{\pm}$ state discussed above does not require any nodes on this
pocket.
Finally, a word of caution is in place. While it is useful, and, arguably,
imperative, at this point of time, to establish the symmetry restrictions on
possible order parameter in AFe2Se2 compounds, the exprerimental situation is
by far not clear. The compositions reported range from $\sim$0.8 hole/Fe doped
(K0.65Fe1.41Se${}_{2},$ Ref. TorchettiNMR ), compared to the stoichiometric
AFe2Se${}_{2},$ to $\sim 0.9$ electron/Fe (Tl0.63K0.37Fe1.78Se${}_{2},$ Ref.
DingTl ). Se-deficient samples have also been reportedSe-def . There have been
credible reports about particular ordering of vacanciesvac . Yet, the
superconducting properties seem to be remarkably similar. Is it fortuitous
that ARPES finds electronic structures remarkably similar to those computed
for stoichiometric compounds, despite large deviations from stoichometry? More
experiments will be needed before we can gain quantitative understanding. Yet
the statements based solely on crystallographic symmetry, and most of the
conclusions of this paper belong to this class, should hold, and have to be
kept in mind.
###### Acknowledgements.
I acknowledge discussions with Andrey Chubukov, Sigfried Graser, Peter
Hirschfeld, and Douglas Sclapino. I am partcularly thankful to Ole Andersen
and Lilia Boeri for helping me figure out the factors that control the
ellipticity and the $k_{z}-$dispersion in the 122 structure.
## References
* (1) Jiangang Guo, Shifeng Jin, Gang Wang, Shunchong Wang, Kaixing Zhu, Tingting Zhou, Meng He, and Xiaolong Chen, Superconductivity in the iron selenide KxFe2Se2 ($0\leq x\leq 1.0$), Phys. Rev. B 82, 180520 (2010)
* (2) Lei Fang, Huiqian Luo, Peng Cheng, Zhaosheng Wang, Ying Jia, Gang Mu, Bing Shen, I. I. Mazin, Lei Shan, Cong Ren, and Hai-Hu Wen, Roles of multiband effects and electron-hole asymmetry in the superconductivity and normal-state properties of Ba(Fe1-xCox)2As2, Phys. Rev. B 80, 140508 (2009).
* (3) V. Brouet, M. Marsi, B. Mansart, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fèvre, F. Bertran, F. Rullier-Albenque, A. Forget, and D. Colson, Nesting between hole and electron pockets in Ba(Fe1-xCox)2As2 (x=0–0.3) observed with angle-resolved photoemission, Phys. Rev. B 80, 165115 (2009) .
* (4) I.R. Shein, A.L. Ivanovskii, Electronic structure and Fermi surface of new K intercalated iron selenide superconductor KxFe2Se2, arXiv:1012.5164; Chao Cao and Jianhui Dai, Electronic Structure of KFe2Se2 from First-Principles Calculations, arXiv:1012.5621; I.A. Nekrasov, M.V. Sadovskii, Electronic Structure, Topological Phase Transitions and Superconductivity in (K,Cs)xFe2Se2, Pisma ZhETF 93, 182 (2011).
* (5) T. Qian, X.-P. Wang, W.-C. Jin, P. Zhang, P. Richard, G. Xu, X. Dai, Z. Fang, J.-G. Guo, X.-L. Chen, H. Ding, Absence of holelike Fermi surface in superconducting K0.8Fe1.7Se2 revealed by ARPES, arXiv:1012.6017
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* (8) T.A. Maier, S. Graser, P.J. Hirschfeld, D.J. Scalapino, d-wave pairing from spin fluctuations in the KxFe2-ySe2 superconductors, arXiv:1101.4988
* (9) Fa Wang, Fan Yang, Miao Gao, Zhong-Yi Lu, Tao Xiang, Dung-Hai Lee, The Electron Pairing of KxFe2-ySe2, arXiv:1101.4390.
* (10) Tanmoy Das, A. V. Balatsky. Stripes, spin resonance and $d_{x^{2}-y^{2}}$-pairing symmetry in FeSe-based layered superconductors, arXiv:1101.6056
* (11) K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki, Phys. Rev. Lett. 101, 087004 (2008)
* (12) Bin Zeng, Bing Shen, Genfu Chen, Jianbao He, Duming Wang, Chunhong Li, Hai-Hu Wen, Nodeless superconductivity in KxFe2-ySe2 single crystals revealed by low temperature specific heat, arXiv:1101.5117.
* (13) L. Ma, J.B. He, D.M. Wang, G.F. Chen, and W. Yu, 77Se NMR Evidence of Strongly Coupled Superconductivity in K0.8Fe2-xSe${}_{2},$ arXiv:11101.3687
* (14) R. H. Yuan, T. Dong, G. F. Chen, J. B. He, D. M. Wang, N. L. Wang, Observation of a small superconducting energy gap in K0.7Fe1.8Se2 by optical spectroscopy, arXiv:1102.1381.
* (15) P.A. Lee and X.-G. Wen, Phys. Rev. B 78, 144517 (2008).
* (16) O.K. Andersen and L. Boeri, On the multi-orbital band structure and itinerant magnetism of iron-based superconductors. Ann. der Phys. 523, 8 (2011).
* (17) Doubling of a unit cell and corresponding downfolding of the BZ does not necessaly lead to formation of nodes. However, it was shown by D. Parker, M. G. Vavilov, A. V. Chubukov, and I. I. Mazin (Coexistence of superconductivity and a spin-density wave in pnictide superconductors: Gap symmetry and nodal lines, Phys. Rev. B80, 100508, 2009), that if the symmetry lowering occurse in the charge channel (charge density wave), hybridization of bands with the opposite signes of the order parameter leads to gap nodes, while if it occurse in the spin channel the nodes are avoided. Symmetry lowering due to the Se ions occurs in the charge channel and thus nodal lines are necessarily formed.
* (18) This is true for my own calculations, which were performed in the experimental crystal structure using the standard WIEN2k band structure package, as well as for other reported calculationsbands1 , although pseudopotential calculations reprted in Ref. Graser have a somewhat stronger $k_{z}$ dispersion.
* (19) I.I. Mazin, and O.K. Andersen, $s-$wave superconductivity from antiferromagnetic spin fluctuation model for YBa2Cu3O7, A.I.Liechtenstein, I.I. Mazin, and O.K. Andersen, Phys. Rev. Lett. 74, 2303 (1995).
* (20) S.Y. Savrasov, Linear Response Calculations of Spin Fluctuations, Phys. Rev. Lett. 81, 2570 (1998).
* (21) There are a number of simplifications in this expression, for instance, we have neglected the so-called local field effects and possible $\mathbf{q-}$dependence of the Stoner factor $J$, but these simplifications are not principal for our qualitative estimate.
* (22) Incidentally, the same is true for the “stripe” phase, the pattern that is the ground state in our calculations and is observed experimentally in pnictides. While the large-moment solution is very stable, the nonmagnetic state is stable against small perturbations of this symmetry, not only is LDA but also in GGA.
* (23) This is implemented by the following recipe: first, at each iteration, the standard LDA or GGA potential for the spin-up and spin-down channel is calculated; then, it is rescaled according to formula $v_{\uparrow}(\mathbf{r})=[v_{\uparrow}(\mathbf{r)+v}_{\downarrow}(\mathbf{r})]/2+\alpha\mathbf{[}v_{\uparrow}(\mathbf{r)-v}_{\downarrow}(\mathbf{r)}]/2=v_{\uparrow}(\mathbf{r})(1+\alpha)/2\mathbf{+v}_{\downarrow}(\mathbf{r)}(1-\alpha)/2,$ $v_{\downarrow}(\mathbf{r})=v_{\uparrow}(\mathbf{r})(1-\alpha)/2\mathbf{+v}_{\downarrow}(\mathbf{r)}(1+\alpha)/2,$ where $0\leq\alpha\leq 1.$
* (24) D. A. Torchetti, M. Fu, D. C. Christensen, K. J. Nelson, T. Imai, H. C. Lei, C. Petrovic, 77Se NMR Investigation of the KxFe2-ySe2 High Tc Superconductor (Tc=33 K), arXiv:1101.4967.
* (25) J. J. Ying, X. F. Wang, X. G. Luo, A. F. Wang, M. Zhang, Y. J.Yan, Z. J. Xiang, R. H. Liu, P. Cheng, G. J. Ye and X. H. Chen, Superconductivity and Magnetic Properties of high-quality single crystals of AxFe2Se2 (A = K and Cs), arXiv:1012.5552
* (26) P. Zavalij, W. Bao, X. F. Wang, J. J. Ying, X. H. Chen, D. M. Wang, J. B. He, X. Q. Wang, G.F Chen, P-Y Hsieh, Q. Huang, M. A. Green, On the Structure of Vacancy Ordered Superconducting Potassium Iron Selenide, arXiv:1101.4882; J. Bacsa, A.Y. Ganin, Y. Takabayashi, K.E. Christensen, K. Prassides, M.J. Rosseinsky, J.B. Claridge, Cation vacancy order in the K0.8+xFe1.6-ySe2 system: five-fold cell expansion accommodates 20% tetrahedral vacancies, arXiv:1102.0488.
|
arxiv-papers
| 2011-02-17T18:37:31 |
2024-09-04T02:49:17.101242
|
{
"license": "Public Domain",
"authors": "I.I. Mazin",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/1102.3655"
}
|
1102.3684
|
# Optimal estimation of entanglement in optical qubit systems
Giorgio Brida INRIM, I-10135, Torino, Italy Ivo P. Degiovanni INRIM,
I-10135, Torino, Italy Angela Florio INRIM, I-10135, Torino, Italy Marco
Genovese m.genovese@inrim.it INRIM, I-10135, Torino, Italy Paolo Giorda
giorda@isi.it ISI Foundation, I-10133, Torino, Italy Alice Meda INRIM,
I-10135, Torino, Italy Matteo G. A. Paris matteo.paris@fisica.unimi.it
Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano,
Italy Alexander P. Shurupov INRIM, I-10135, Torino, Italy
###### Abstract
We address the experimental determination of entanglement for systems made of
a pair of polarization qubits. We exploit quantum estimation theory to derive
optimal estimators, which are then implemented to achieve ultimate bound to
precision. In particular, we present a set of experiments aimed at measuring
the amount of entanglement for states belonging to different families of pure
and mixed two-qubit two-photon states. Our scheme is based on visibility
measurements of quantum correlations and achieves the ultimate precision
allowed by quantum mechanics in the limit of Poissonian distribution of
coincidence counts. Although optimal estimation of entanglement does not
require the full tomography of the states we have also performed state
reconstruction using two different sets of tomographic projectors and
explicitly shown that they provide a less precise determination of
entanglement. The use of optimal estimators also allows us to compare and
statistically assess the different noise models used to describe decoherence
effects occuring in the generation of entanglement.
###### pacs:
03.67.Mn, 03.65.Ta
## I Introduction
The sort of quantum correlations captured by the notion of entanglement
represents a central resource for quantum information processing. Therefore,
the precise characterization of entangled states is a crucial issue for the
development of quantum technologies. In fact, quantification and detection of
entanglement have been extensively investigated, see ren1 ; ren2 ; ren3 for a
review, and different approaches have been developed to extract the amount of
entanglement of a state from a given set of measurement results bayE ; Wun09 ;
Eis07 ; KA06 . Of course, in order to evaluate the entanglement of a quantum
state one may resort to full quantum state tomography LNP that, however,
becomes impractical in higher dimensions and may be affected by large
uncertainty TE94 ; AS2011 . Other methods, requiring a reduced number of
observables, are based on visibility measurements Jae93 , Bell’ tests c2 ; w3
, entanglement witnesses h1 ; t2 ; G3 ; w ; w1 or are related to Schmidt
number pas ; fed ; f . Many of them has been implemented experimentally mg ;
wei ; ser ; nos ; ch ; mat , also in the presence of decoherence effects buc06
; dav07 .
As a matter of fact, any quantitative measure of entanglement corresponds to a
nonlinear function of the density operator and thus it cannot be associated to
a quantum observable. As a consequence, ultimate bounds to the precision of
entanglement measurements cannot be inferred from uncertainty relations. Any
procedure aimed to evaluate the amount of entanglement of a quantum state is
ultimately a parameter estimation problem, where the value of entanglement is
indirectly inferred from the measurement of one or more proper observables
EE08 . An optimization problem thus naturally arises when one looks for the
ultimate bounds to precision, i.e. the smallest value of the entanglement that
can be discriminated according to quantum mechanics, and tries to determine
the optimal measurements achieving those bounds. This optimization problem may
be properly addressed in the framework of quantum estimation theory qet1 ;
qet2 ; qet3 , which provides analytical tools to find the optimal measurement
and to derive ultimate bounds to the precision of entanglement estimation. In
particular, being entanglement an intrinsic property of quantum states, we
adopt local quantum estimation theory and look for optimal estimators
maximizing the Fisher information EE08 ; LQE .
In this paper, we address experimental determination of entanglement for two-
qubit optical systems and apply quantum estimation theory to derive optimal
estimators and ultimate bound to precision. This technique has been
successfully applied in EEE to estimate the entanglement of a pair of
polarization qubit with the ultimate precision allowed by quantum mechanics.
Here we refine and extend the results of EEE in two directions: On the one
hand we present a set of experiments aimed at estimating the amount of
entanglement of a larger class of families of two-qubit mixed photon states.
On the other hand, we have performed full state reconstruction using two
different tomographic sets of projectors in order to show explicitly that the
evaluation of entanglement from the knowledge of the reconstructed density
matrix provides a less precise determination. In our scheme entanglement, is
evaluated through visibility measurements and estimators are built by a
suitable combination of coincidence counts with different settings. Those
estimators turn out to be optimal and to provide estimation with the ultimate
precision in the limit of Poissonian distribution of coincidence counts. In
addition, we demonstrate experimentally that optimality is robust against
deviation from the Poissonian behaviour. Our approach allows entanglement
estimation at the quantum limit, and it is also useful to compare different
noise models using only information extracted from experimental data.
The paper is structured as follows. In the next Section we briefly review the
basic notions of local estimation theory, whereas in Section III we apply them
to estimation of entanglement of states belonging to two relevant families of
mixed states. Section IV describes in details the experimental apparatus used
to demonstrate our theoretical results, which are described in the Section V.
A detailed discussion of the experimental results is given in Section VI,
whereas Section VII closes the paper with some concluding remarks.
## II Local quantum estimation theory
We now give the basis ingredients for the local estimation theory starting
with the classical case. Suppose we have a set of parameters
${\boldsymbol{\lambda}}=(\lambda_{1},\cdots,\lambda_{n})\in\Lambda\subseteq\mathbb{R}^{n}$
labelling different states of the physical system of interest. A statistical
model of our system is a set of probability distributions
$S=(p_{{\boldsymbol{\lambda}}}(x)|{\boldsymbol{\lambda}}\in\Lambda)$ such that
$\Omega$ is the sample space of the random variable $x$. The fundamental
question in estimation theory is how to optimally estimate the unknown true
values of the parameters ${\boldsymbol{\lambda}}$ given a sequence of outcomes
of measurement on the system $\\{x_{1},\cdots,x_{\scriptscriptstyle M}\\}$.
From an geometrical information perspective, this problem was first treated by
Fisher who introduced for the case $N=1$ the now called Fisher information
metric $F({\boldsymbol{\lambda}})$:
$\displaystyle[F({\boldsymbol{\lambda}})]_{ij}$ $\displaystyle=$
$\displaystyle\int_{\Omega}\\!\\!dx\,p_{{\boldsymbol{\lambda}}}(x)\,\partial_{i}\log
p_{{\boldsymbol{\lambda}}}(x)\,\partial_{j}\log
p_{{\boldsymbol{\lambda}}}(x)=$ (1) $\displaystyle=$
$\displaystyle\int_{\Omega}\\!\\!dx\,\frac{\partial_{i}p_{{\boldsymbol{\lambda}}}(x)\partial_{j}p_{{\boldsymbol{\lambda}}}(x)}{p_{{\boldsymbol{\lambda}}}(x)}$
where $\partial_{i}\equiv\partial_{\lambda_{i}}$. $F({\boldsymbol{\lambda}})$
is a positive definite matrix that represents a metric on the parameter space
$\Lambda$ and whose information geometric content is given by the best
resolution with which one can distinguish neighbouring points in the parameter
space. The Fisher information metric is additive, therefore for a sequence of
independent and identically distributed measurements with outcomes
$\\{x_{1},\cdots,x_{\scriptscriptstyle M}\\}$, $F^{\scriptscriptstyle
M}({\boldsymbol{\lambda}})=MF({\boldsymbol{\lambda}})$. The next step in the
estimation theory requires the introduction of the concept of estimator; the
latter is any algorithm or rule of inference, which allows one to extract a
value for the unknown parameters on the basis of the sole knowledge acquired
via the measurement process, i.e. the sequence of outcomes
$\\{x_{1},\cdots,x_{\scriptscriptstyle M}\\}$. We say that the random variable
$\hat{\boldsymbol{\lambda}}:\Omega^{\scriptscriptstyle M}\rightarrow\Lambda$
is an unbiased estimator if
$E[{\hat{\boldsymbol{\lambda}}}]={\boldsymbol{\lambda}}$ i.e., its expected
value coincides with the true value of the parameter(s). The ultimate bound on
the precision with which one can estimate the parameters
${\boldsymbol{\lambda}}$ is given by the Cramer-Rao theorem , which can be
stated in terms of the covariance matrix
$\hbox{Cov}[\hat{\boldsymbol{\lambda}}]_{ij}=E[\hat{\lambda}_{i}\hat{\lambda}_{j}]-E[\hat{\lambda}_{i}]E[\hat{\lambda}_{j}]$
as:
$\hbox{Cov}[\hat{\boldsymbol{\lambda}}]\geq\frac{1}{M}F({\boldsymbol{\lambda}})^{-1}.$
(2)
In particular, for a single parameter the inequality reads
$\hbox{Var}[\hat{\lambda}]\geq\frac{1}{MF(\lambda)}\,,$
i.e. the variance of the estimator, and therefore the precision of any
estimation procedure, cannot be smaller than the inverse of the Fisher
information times the number of repeated measurements. In the general case,
the inequality for the variance of each of the parameters, i.e.
$\hbox{Var}[\hat{\lambda}_{i}]\geq\frac{1}{M}[F({\boldsymbol{\lambda}})^{-1}]_{ii}\,,$
holds only at fixed values of the others parameters.
The previous results can be extended to the quantum realm, also taking into
account all the possible measurements that one can implement on the systems.
The quantum statistical model is given by a set of density operators depending
on the parameters ${\boldsymbol{\lambda}}$:
$S=\\{\rho_{\boldsymbol{\lambda}}|{\boldsymbol{\lambda}}\in\Lambda\\}$. A
measurement corresponds to a Positive Operator Valued Measure (POVM), i.e. a
set of positive operators $\mathcal{E}=\\{E_{i}\\}$ such that
$\sum_{i}E_{i}E_{i}^{\dagger}=\openone$ and such that
$p_{{\boldsymbol{\lambda}}}(i)=\hbox{Tr}[E_{i}\rho_{\boldsymbol{\lambda}}]$ is
the probability of having the $i$-th outcome. The Fisher information matrix
$F_{\mathcal{E}}(\boldsymbol{\lambda})$ in Eq. (1) for a specific measurement
process $\mathcal{E}$ can then be written in terms of the classical
probabilities $p_{{\boldsymbol{\lambda}}}(i)$. What is now specific to the
quantum estimation process is that the optimization over measurement processes
$\mathcal{E}$ may be carried out. The problem has been solved in terms of the
inequality ($A>B$ means that $A-B$ is a positive matrix)
$F_{\mathcal{E}}(\boldsymbol{\lambda})\leq H(\boldsymbol{\lambda})$ (3)
that states that the Fisher information of any measurement process is upper
bounded by the Quantum Fisher information $H(\boldsymbol{\lambda})$ (QFI). The
latter is an $n\times n$ positive definite real matrix which can be expressed
in terms of a set of $n$ positive, zero mean operators called symmetric
logarithmic derivatives (SLD) $L_{i}$, each satisfying the following partial
differential equation
$\partial_{i}\rho_{{\boldsymbol{\lambda}}}=\frac{1}{2}(L_{i}\rho_{\boldsymbol{\lambda}}+\rho_{\boldsymbol{\lambda}}L_{i})$
(4)
In particular, if one expresses the density matrix in its spectral
decomposition
$\rho_{\boldsymbol{\lambda}}=\sum_{i}p_{i}\left|\psi_{i}\rangle\\!\langle\psi_{i}\right|,$
(5)
the SLD pertaining to the $i$-th parameter is
$L_{i}=2\sum_{n,m}\frac{\langle\psi_{n}|\partial_{i}\rho_{\boldsymbol{\lambda}}|\psi_{m}\rangle}{p_{n}+p_{m}}|\psi_{n}\rangle\langle\psi_{m}|,$
(6)
where
$\displaystyle\partial_{i}\rho_{\boldsymbol{\lambda}}=$
$\displaystyle\sum_{n}\partial_{i}p_{n}\left|\psi_{n}\rangle\\!\langle\psi_{n}\right|+$
(7)
$\displaystyle+\sum_{n}p_{n}(\left|\partial_{i}\psi_{n}\rangle\\!\langle\psi_{n}\right|+\left|\psi_{n}\rangle\\!\langle\partial_{i}\psi_{n}\right|)\,,$
accounts for the dependence of both the eigenvalues and the eigenvectors on
the set of parameters ${\boldsymbol{\lambda}}$. In terms of the $L_{i}$’s the
elements of the QFI can be written as:
$[H(\boldsymbol{\lambda})]_{ij}=\hbox{Tr}\left[\rho_{\boldsymbol{\lambda}}\,\frac{L_{i}L_{j}+L_{j}L_{i}}{2}\right].$
(8)
By using the spectral decomposition of $\rho_{\boldsymbol{\lambda}}$, the QFI
can be expressed in terms of the partial derivatives of the eigenvalues and of
the eigenvectors as:
$\displaystyle[H(\boldsymbol{\lambda})]_{ij}=\sum_{n}\frac{(\partial_{i}p_{n})(\partial_{j}p_{n})}{p_{n}}+\sum_{n,m}\frac{(p_{n}-p_{m})^{2}}{p_{n}+p_{m}}\times$
(9)
$\displaystyle\times\Big{(}\langle\psi_{n}|\partial_{i}\psi_{m}\rangle\langle\partial_{j}\psi_{m}|\psi_{n}\rangle+\langle\psi_{n}|\partial_{j}\psi_{m}\rangle\langle\partial_{i}\psi_{m}|\psi_{n}\rangle\Big{)}\,.$
## III Estimation of entanglement for two-qubit systems
We now apply the formalism described in the previous Section to obtain
explicitly the ultimate bound to precision on the estimation of entanglement
for two relevant statistical models, i.e. for two families of two-qubit states
that will be used in the following.
### III.1 The decoherence model
The first statistical model we are going to deal with corresponds to the set
of the states described by the following two-parameter family of density
operators
$\varrho=p\left|\psi\rangle\\!\langle\psi\right|+(1-p)D,$ (10)
where
$|\psi\rangle=\sqrt{q}\,|{\hbox{\small HH}}\rangle+\sqrt{1-q}\,|{\hbox{\small
VV}}\rangle$ (11)
represents a pure polarization two-photon state with horizontal $H$ and
vertical $V$ polarization, and $D=q\,\left|{\hbox{\small
HH}}\rangle\\!\langle{\hbox{\small HH}}\right|+(1-q)\,\left|{\hbox{\small
VV}}\rangle\\!\langle{\hbox{\small VV}}\right|$ describes a mixed contribution
coming from the decoherence of $|\psi\rangle$, $p\in[0,1]$. We will refer to
this set as the decoherence model for $|\psi\rangle$. For the state $\varrho$,
both the two non zero eigenvalues
$\lambda_{\pm}=(1\pm\sqrt{1-4(1-p^{2})q+4(1-p^{2})q^{2}})\,$
and their respective eigenvectors
$\displaystyle{\bf{v}}_{\pm}$
$\displaystyle=\frac{1}{\sqrt{N_{\pm}}}\left\\{-f_{\pm}(p,q),0,0,g(p,q)\right\\}$
(12) $\displaystyle N_{\pm}$ $\displaystyle=\sqrt{g^{2}(p,q)\pm
f^{2}_{\pm}(p,q)}$ $\displaystyle f_{\pm}(p,q)$
$\displaystyle=1-2q\pm\sqrt{1-4(1-p^{2})q+4(1-p^{2})q^{2}}$ $\displaystyle
g(p,q)$ $\displaystyle=2p\sqrt{q(1-q)}\,$
depend on the parameters $p,q$. The straightforward calculations of the
partial derivatives in Eq. (9) show that both the eigenvalues and the
eigenvectors contribute to the diagonal and off-diagonal terms of the QFI.
However, the sum of the different contributions results in a simplified
expression, and the QFI
$H(p,q)=\mbox{diag}\left(\frac{4(1-q)q}{1-p^{2}},\frac{1}{q-q^{2}}\right)$
(13)
is diagonal. From this expression we see that the variance on any estimator
$\hat{q}$ for the parameter $q$ is independent on the mixing parameter $p$ and
is bounded, apart from the statistical scaling, by the inverse of
corresponding element of the QFI matrix
$\hbox{Var}[\hat{q}]\geq\frac{q(1-q)}{M}\,.$
The lower bound is maximal in correspondence of $q=1/2$, i.e. when the state
$|\psi\rangle$ is maximally entangled. We are now interested in estimating the
value of entanglement of the overall state $\varrho$. To this aim we remind
that the negativity of entanglement defined as
$\epsilon=||\varrho^{T_{A}}||_{1}-1$ (14)
is a good measure of entanglement for two qubit systems. In Eq. (14) $T_{A}$
denotes partial transposition with respect to system $A$, and $||...||_{1}$ is
the trace norm. Entanglement negativity for states belonging to the
decoherence model is given by
$\epsilon=2p\sqrt{(1-q)q}\,.$ (15)
In order to reexpress the QFI in terms of the negativity we make the change of
variable $p\rightarrow p,q\rightarrow(p-\sqrt{p^{2}-\epsilon})/2p$; the QFI
changes according to the Jacobian of the transformation and the lower bound to
the covariance matrix of the estimators $\hat{p},\hat{\epsilon}$ now reads:
$\displaystyle\hbox{Cov}[\hat{p},\hat{\epsilon}]$ $\displaystyle\geq
H^{-1}(p,\epsilon)$ (16)
$\displaystyle=\left(\begin{array}[]{cc}p^{2}(1-p^{2})\,\epsilon^{-2}&p(1-p^{2})\,\epsilon^{-1}\\\
p(1-p^{2})\,\epsilon^{-1}&1-\epsilon^{2}\\\ \end{array}\right)$ (19)
From this expression we see that the lower bound for the variance of any
estimator $\hat{\epsilon}$ of the negativity of the state $\varrho$ is
independent on $p$ and is minimal in case of maximal entanglement
$\hbox{Var}[\hat{\epsilon}]\geq\frac{1}{M}(1-\epsilon^{2})\,.$ (20)
### III.2 The Werner model
A second statistical model of interest for our analysis corresponds to the set
of states described by the following two-parameter family of density operator
$\varrho^{\prime}=p\left|\psi\rangle\\!\langle\psi\right|+\frac{1-p}{4}\openone\otimes\openone\,.$
(21)
The states of Eq. (21) are obtained by depolarizing the pure entangled state
$|\psi\rangle$. We will refer to this family as the Werner model for
$|\psi\rangle$. As in the previous example upon varying the parameter $p$ we
may tune the purity of the state, whereas the amount of entanglement depends
on both parameters. The eigenvalues of $\varrho^{\prime}$ depends only on $p$,
whereas the eigenvectors depends only on $q$. The QFI matrix is thus given by
the diagonal form
$H(p,q)=\mbox{diag}\left\\{\frac{3}{1+(2-3p)p},\,\,\frac{p^{2}}{q(1-q)(1+p)}\right\\}\,$
(22)
and the inverses of the diagonal elements correspond to the ultimate bounds to
${\mathrm{Var}}(\hat{p})$ and ${\mathrm{Var}}(\hat{q})$ for any estimator of
$p$ and $q$, either at fixed value of the other parameter or in a joint
estimation procedure. Entanglement of Werner states may be evaluated in terms
of negativity,
$\epsilon=\max\left\\{0,\frac{1}{2}\left[p\left(1+4\sqrt{q(q-1)}\right)-1\right]\right\\}\>,$
(23)
which implies that Werner states are entangled for
$[1+4\sqrt{q(1-q)}]^{-1}<p<1.$
Upon inverting Eq. (23) for $p$ or $q$ we may parametrize the Werner states
using $(p,\epsilon)$ and evaluate the QFI matrix $H(p,\epsilon)$, their
inverses and, in turn, the corresponding bounds to the precision of
entanglement estimation. The main result is that the ultimate bound to the
variance, depend only very slightly on the other free parameter ($q$ or $p$).
In other words, estimation procedures performed at fixed value of $p$ or $q$
respectively show different precision, but the differences are negligible in
the whole range of variations of the parameters. We do not report here the
analytic expression of the inverse QFI at fixed $p$ or $q$, which is quite
cumbersome. However, as it can be easily checked, we note that the bound on
the variance on $\hat{\epsilon}$ that can be derived by the expression of
$H(p,\epsilon)^{-1}$ simply coincides to first order with the bound in Eq.
(20) already evaluated for the decoherence model. We therefore use in the
following, also for the Werner model, the bound given in Eq. (20). It can be
shown that, for the set of values of $p$ that will be relevant for our
experimental analysis, this approximation is negligible with respect to all
the other sources of uncertainty.
## IV Experimental apparatus
The family of entangled states, investigated in our work, is constituted by
polarization entangled states of the field obtained by coherently
superimposing two orthogonally polarized type-I parametric downconversion
emissions (PDC), as schematically depicted in Fig. 1. The linear horizontal
polarization of an argon laser beam, at wavelength $\lambda$ = 351 nm filtered
by dispersion prism and Glan-Thompson prism (GP), is rotated at angle $\phi$
by using half-waveplate (WP0). It is fundamental for our application that only
the laser line $\lambda$ = 351.1 nm is used. For this reason we have
introduced in the setup a prism as wavelength selector for eliminating
wavelengths other than $\lambda$ = 351.1 nm. In particular the closest one at
$\lambda$ = 351.4 nm, which could realize an unwanted phase-matching condition
in our PDC setup. Then, the laser beam is addressed to a pair of non-linear
beta barium borate (BBO) crystals ($l$ = 1 mm), having optical axis in
orthogonal planes, where PDC process occurs, resulting in creation of
biphotons with orthogonal polarization kw ; nos1 . Upon changing the
polarization of the UV pump, we change the amount of PDC light, generated by
each crystal. For example, PDC occurs only in crystal one if the polarization
of the pump beam is horizontal, while for having a balanced PDC process in
both crystals we have set the angle $\phi$ at $45^{\circ}$, having diagonal
polarization of pump beam.
In order to compensate phase shifts, due to ordinary and extraordinary path in
the crystals, we tilt the quartz plates QP, introduced between the halfwave
plate WP0 and BBO crystals, at angle $\varphi$, thus fixing the relative phase
between biphoton components generated in first and second crystal.
Figure 1: (Color online) Experimental setup to generate polarization entangled
two-photon states with variable entanglement and to estimate its value with
the ultimate precision allowed by quantum mechanics. A continuous wave Argon
pump laser beam with wavelength $\lambda$ = 351.1 nm is filtered with a
dispersion prism and then passes through a Glan-Thompson prism and a half-wave
plate WP0 that rotates the polarization by an angle $\phi$. PDC light is
generated by two thin type-I BBO crystals ($l$ = 1 mm). After the crystals the
pump is stopped by a filter (UVF), and the biphoton field is split on a
nonpolarizing 50-50 beam splitter (BS). Then it passes through half-wave
plates (WP1, WP2) and interference filters (IF), centered at the degeneracy
702 nm. Finally the biphotons are focused on commercial single photon
detectors (D1, D2).
In order to maintain stable the phase-matching conditions, BBO crystals and QP
are placed in a closed aluminium box internally covered by polystyrene used as
thermic insulator. The box is equipped with a controlled heating system with a
standard feedback circuit. We have experimentally verified that the
temperature stabilization system ensures appropriate control on the phase
shift. After the box the pump is stopped by an ultraviolet filter (UVF), and
the biphoton field is split on a non-polarizing 50-50 beam splitter (BS). With
the postelection performed by a coincidence count circuit (CC), we can refer
to our state as an optical ququart kulik , which is entangled in two
variables: polarization and spatial mode.
In ideal conditions the output state is described by the pure state
$|\psi_{\phi\varphi}\rangle=\cos\phi|{\hbox{\small HH}}\rangle+\sin\phi
e^{i\Phi(\varphi)}|{\hbox{\small VV}}\rangle$ (24)
where $\phi/2$ is rotation angle of pump halfwaveplate WP0 and $\Phi(\varphi)$
corresponds to phase shift between pair of horizontal photons created in the
first crystal and pair of vertical photons from the second crystal. After
passing the half-waveplates (WP1,WP2) in each spatial mode, the biphoton field
is projected into a linear vertical polarization state by means of Glan-
Thompson polarizers. Phase plates WP1 and WP2 are mounted on precision
rotation stages with high resolution and fully motor controlled, that allow
rotating the polarization of the beams in the course of measurement process.
Spectral selection is performed by interference filters (IF) with central
wavelength $\lambda$ = 702 nm and FWHM = 3 nm. Short focal lenses collimate
resulting biphoton field into single photon avalanche detectors (D1, D2).
Electrical signal from detectors is used by coincidence count scheme (CC) with
time window $\tau$ = 1 ns.
The measurements performed at the output are described as projection of state
into factorized linearly-polarized two-photon state:
$\Pi_{x}(\alpha,\beta)=|\alpha+s\frac{\pi}{2}\rangle\langle\alpha+s\frac{\pi}{2}|\otimes|\beta+s^{\prime}\frac{\pi}{2}\rangle\langle\beta+s^{\prime}\frac{\pi}{2}|$
(25)
where $x=\\{s+2s^{\prime}\\}$, $s,s^{\prime}=0,1$.
Figure 2: Probability of coincidence counts while performing projection
measurement $\Pi_{0}(\frac{\pi}{4},\frac{\pi}{4})$ on state
$|\psi_{\phi\varphi}\rangle$ having $\phi=\frac{\pi}{4}$ as function of quartz
plates tilting angle $\varphi$.
In Fig. 2 we show the dependence of the probability of the coincidence counts
$p_{0}(\varphi)=\langle\psi_{\tfrac{\pi}{4}\varphi}|\Pi_{0}(\frac{\pi}{4},\frac{\pi}{4})|\psi_{\tfrac{\pi}{4}\varphi}\rangle\,,$
as function of quartz plates QP tilting angle $\varphi$. The maximum of this
curve corresponds to phase shift between photon pairs
$\Phi(\varphi_{\scriptscriptstyle M})=0$ and the output state is the Bell
maximally entangled state
$|\Phi^{+}\rangle\equiv|\psi_{\frac{\pi}{4}\varphi_{\scriptscriptstyle
M}}\rangle\propto|{\hbox{\small HH}}\rangle+|{\hbox{\small VV}}\rangle\,,$
while the minimum of that curve corresponds to the maximally entangled state
$|\Phi^{-}\rangle\equiv|\psi_{\frac{\pi}{4}\varphi_{m}}\rangle\propto|{\hbox{\small
HH}}\rangle-|{\hbox{\small VV}}\rangle\,.$
In this work we have fixed the tilting angle of quartz plates to have zero
phase shift, thus, the family of states in Eq. (24) reduce to the one of Eq.
(11) where $q=\cos^{2}(\phi)$.
## V Entanglement estimators
In order to estimate the entanglement content of the states produced by the
experimental set up described in the previous Section, one has to choose an
estimator $\hat{\epsilon}$ to extract the value of entanglement from the
experimental data. We will compare three different approaches: two are based
on full tomography of the polarization two-photon and one is based on
implementing the optimal estimator able to saturate the ultimate bound derived
via the QFI.
Quantum state tomography is an experimental procedure providing full density
matrix reconstruction of a quantum system. This is realized by means of a set
of measurements performed on an ensemble of identical quantum systems LNP .
For a quantum state belonging four-dimensional Hilbert space at least 16
linearly independent measurements are needed to reconstruct full density
matrix and, typically, each measurement corresponds to a local projection of
the input two-qubit state. To be able to perform this set of 16 linearly
independent measurements we added a quarter-waveplate in each measurement arm
just before the half-waveplates (WP1,WP2). The first used tomographic protocol
(J16) KB00 ; Ja01 involves projective measurements performed directly on some
components of the Stokes vector. In particular, the measurement set
corresponds to projection onto polarizations HH, HV, VV, VH, RH, RV, DV, DH,
DR, DD, RD, HD, VD, VL, HL, RL, where H, V, R, L, D, denotes horizontal,
vertical, right and left circular and $45^{\circ}$ diagonal polarizations,
respectively. Here, for example, the measurement setting HR means measuring
horizontal polarization on the first qubit and right circular polarization on
the second qubit. Another approach bog04 ; reh04 involves local projection of
each qubit symmetrically placed on Poincare sphere. Extension of this method
to four-dimensional case (R16) allows obtaining higher fidelity of the
reconstructed states Bur08 ; OurTomo2010 with respect to the previous one.
Once the density matrix of the generated state has been reconstructed, the
negativity of the state can be evaluated inserting the reconstructed matrix
elements in Eq. (14). The precision the tomographic estimation of entanglement
is limited by the uncertainties on the matrix elements. The overall
uncertainty on the estimated value of entanglement may be evaluated by error
propagating. In the following, after describing the implementation of optimal
measurement, we will compare its precision with that of tomographic
estimation.
We first start to briefly describe the estimator for the class of states
defined by Eq. (10). As already described in EEE , an optimal estimator of the
entanglement can be found by noticing that the expressions of the
probabilities
$p_{x}(\epsilon;\alpha,\beta)=\hbox{Tr}[\varrho\>\Pi_{x}(\alpha,\beta)]$
obtained by the projection of the state $\varrho$ on measurement operators in
Eq. (25) with $x=0,1,2,3$, allows writing the following set of unbiased
estimators
$\hat{\epsilon}(\alpha,\beta)=\frac{V(\alpha,\beta)-\cos(2\alpha)\cos(2\beta)}{\sin(2\alpha)\sin(2\beta)},$
(26)
where $V(\alpha,\beta)=p_{0}-p_{1}-p_{2}+p_{3}$ is the expected value of two-
qubit quantum correlations (QC). Furthermore, the estimators corresponding to
the measurement angles $\alpha,\beta=\pm\pi/4$ are optimal, as can be seen by
evaluating the Fisher information
$F_{\epsilon}(\alpha,\beta)=\sum_{x}p_{x}(\epsilon;\alpha,\beta)[\partial_{\epsilon}\log
p_{x}(\epsilon;\alpha,\beta)]^{2}\,,$
which for the chosen angles gives $F_{\epsilon}(\frac{\pi}{4},\frac{\pi}{4})$
equal to QFI. Then we have to express these optimal estimators,
$\hat{\epsilon}=V(\pm\pi/4,\pm\pi/4)$, in terms of the coincidences counts,
which are the results of the measurement process. This can be done by fixing
for example $\alpha=\beta=-\pi/4$ and then, for each measurement run
$j=1,..,M=40$, one records the vector
$\mathbf{k}_{j}=\\{k_{0,j},k_{1,j},k_{2,j},k_{3,j}\\}$, where $k_{x,j}\equiv
k_{x,j}(-\pi/4,-\pi/4)$, is the number of coincidence counts for the projector
$\Pi_{x}$ defined in Eq. (25) as measured by the coincidence circuit during a
single time window of $10$ seconds, and whose expected distribution is given
$p_{x}(\epsilon;\alpha,\beta)=\hbox{Tr}[\varrho\,\Pi_{x}(\alpha,\beta)]\,.$
Finally, we have to derive the probabilities $p_{x}(\epsilon;-\pi/4,-\pi/4)$
in the expression of $V(\alpha,\beta)$ in terms of the relative frequencies
$k_{x,j}(\alpha,\beta)/K_{j}$, where $K_{j}=\sum_{x}k_{x,j}$ is the total
number of coincidences. For large values of $K_{j}$ the coincidence rates
$k_{x,j}(\alpha,\beta)/K_{j}$ converges to the probability
$p_{x}(\epsilon;\alpha,\beta)$. Therefore, the optimal estimator can be
written as desired in terms of the coincidences’ vector:
$\hat{\epsilon}\equiv\hat{\epsilon}(\bf{k}_{j})$.
A second statistical model, which is a possible candidate to represent the
output of our experiment, is the Werner model of Eq. (21). From the physical
point of view it corresponds to incorporate in our our scheme a portion of
“fake” coincidences that results from dark counts of SPADs and from the
influence of the ambient unpolarized luminescence. Since this light is
unpolarized, its density operator can be described by the identity in (21).
The distribution of coincidences is given by
$p_{x}^{\prime}(\epsilon;\alpha,\beta)=\hbox{Tr}[\varrho^{\prime}\,\Pi_{x}(\alpha,\beta)]\,,$
and the unbiased estimators for the mixing parameter and the entanglement
negativity of the state by
$\displaystyle\hat{p}^{\prime}$ $\displaystyle=V(0,0)$
$\displaystyle\hat{\epsilon}^{\prime}$
$\displaystyle=-\frac{1}{2}+\frac{1}{2}V(0,0)+V(-\pi/4,-\pi/4)\,.$ (27)
where $V(0,0)=V(\alpha=0,\beta=0)$ has been defined above. The estimators may
be then written in terms of the coincidence vectors $\mathbf{k}_{j}$, which
was previously defined and that is used for $V(-\pi/4,\pi/4)$, and
$\mathbf{r}_{j}=\\{r_{0,j},r_{1,j},r_{2,j},r_{3,j}\\}$, which is used in an
analogous way to define the probabilities in for $V(0,0)$ and whose elements
are defined as $r_{x,j}\equiv r_{x,j}(0,0)$ i.e., the number of coincidence
counts for the projector $\Pi_{x}$ (25) with $\alpha=0,\beta=0$; in this case
the total number of coincidences is $R_{j}=\sum_{x}r_{x,j}$. The estimators
can then be written as $\hat{p}^{\prime}=\hat{p}^{\prime}(\mathbf{r}_{j})$,
and
$\hat{\epsilon}^{\prime}=\hat{\epsilon}^{\prime}(\mathbf{k}_{j},\mathbf{r}_{j})$.
## VI Results
We first observe that for $\hat{\epsilon}(\bf{k}_{j})$ and finite $K_{j}$s the
uncertainty in the estimation of the entanglement are mostly due to
fluctuations $\delta k_{x}$ in the coincidence counts $k_{x,j}$ around their
average values $\left\langle k_{x}\right\rangle=\sum_{j}k_{x,j}/M$. Thus, if
we want to establish under which conditions on the fluctuations $\delta k_{x}$
the variance of the estimator $\hat{\epsilon}(\bf{k}_{j})$ satisfies the
required bound, we have to implement standard uncertainty propagation with the
derivatives $\partial_{x}\equiv\partial/\partial k_{x}$ evaluated for
$k_{x}\equiv\left\langle k_{x}\right\rangle$, and assuming independence among
fluctuations at different angles, we have
$\displaystyle\hbox{Var}(\hat{\epsilon})$ $\displaystyle=$
$\displaystyle\sum_{x}|\partial_{x}\hat{\epsilon}|^{2}\delta k_{x}^{2}$ (28)
$\displaystyle=$ $\displaystyle\frac{4}{\left\langle
K\right\rangle^{4}}\Big{[}\big{(}\langle k_{0}\rangle+\langle
k_{3}\rangle\big{)}^{2}\big{(}\delta k_{1}^{2}+\delta k_{2}^{2}\big{)}$
$\displaystyle+\big{(}\langle k_{1}\rangle+\langle
k_{2}\rangle\big{)}^{2}\big{(}\delta k_{0}^{2}+\delta
k_{3}^{2}\big{)}\Big{]}\,.$
If we now assume that the counting processes have a Poissonian statistics,
i.e. $\delta k_{x}^{2}=\hbox{Var}(k_{x})=\left\langle k_{x}\right\rangle^{2}$,
then it is straightforward to prove that
$\hbox{Var}(\hat{\epsilon})=\frac{4}{\left\langle
K\right\rangle^{3}}\,(k_{0}+k_{3})(k_{1}+k_{2})=\frac{1}{\left\langle
K\right\rangle}\,(1-\hat{\epsilon}^{2})$
i.e. QC measurements allow for optimal estimation of entanglement with
precision at the quantum limit. Since the inverse of QFI is given by
$[H^{-1}]_{\epsilon\epsilon}=1-\epsilon^{2}$ for a wide range of two-qubit
families of states EE08 , the above calculations suggest that this is a
general result. In particular, following the discussion at the end of section
III, the above result is true also for the Werner state. In other words, given
a source emitting polarization two-qubit states with coincidence counting
statistics satisfying the Poissonian hypothesis, then the experimental setup
of Fig. 1 allows for optimal estimation of entanglement at the quantum limit
by means of a QC estimator. We finally note that in order to test the
Poissonian hypothesis in our experiment we evaluated the Fano factor, which is
defined as $F=\frac{\sigma_{\tau}^{2}}{\mu_{\tau}},$ where $\sigma_{\tau}^{2}$
is the variance and $\mu_{\tau}$ is the mean of a random process in some time
window $\tau$. For a Poissonian process Fano factor should be equal to unity.
In our experiment we had slightly different values EEE , but the method still
allows for optimal estimation, thus showing the robustness of optimal
measurement against deviation from Poissonian behaviour.
### VI.1 Almost pure states
The experimental setup of Fig. 1 allows for the preparation of quantum states
with high value of purity, namely having mixing parameter $p$ close to unity.
In these conditions both family of states in Eq.s (10) and (21) described in
section III are expected to give a reliable estimation of entanglement. In
order to verify this assessment, in the first part of our experiment we have
performed measurements with different values of initial entanglement
corresponding to different values of $q$, i.e. of the angle $\phi$ determined
by WP0. We first consider the decoherence model of Eq. (10). This model can be
considered as a description of the decoherence mechanisms occurring in the
experimental setup due to fluctuations of the relative phase between the two
polarization components, which results in fluctuation of phase shift between
biphoton created in two crystals. Our experimental procedure is based on
$M=40$ repeated acquisitions of coincidence vector
$\boldsymbol{k}_{j}=\\{k_{0j},k_{1j},k_{2j},k_{3j}\\}$. We have randomized the
composition of $\boldsymbol{k}_{j}$ over the sequence of measurements to avoid
spurious correlations, and finally we have estimated entanglement as the
sample mean
$\langle\hat{\epsilon}\rangle=\sum_{j}\hat{\epsilon}(\boldsymbol{k}_{j})/M$.
The corresponding uncertainty has been evaluated by the sample variance
$\hbox{Var}(\hat{\epsilon})=\sum_{j}[\hat{\epsilon}(\boldsymbol{k}_{j})-\langle\hat{\epsilon}\rangle]^{2}/(M-1)$.
In order to verify the compatibility of data with the decoherence model of Eq.
(10) we need to estimate the negativity with a second procedure, namely we
make use of the estimation of the parameter $p$, quantifying the amount of
mixing introduced by decoherence processes. We therefore define an unbiased
estimator $\hat{p}$ by first reversing formula of the negativity i.e.,
$p=\frac{1}{2}\epsilon/\sqrt{q(1-q)}$. We then note that the values of $q$ and
$1-q$ in this model are given by the probabilities relative to the projective
measurements $\Pi_{0}(0,0)$ and $\Pi_{3}(0,0)$ respectively, that can be
expressed in terms of the elements $r_{0,j}$ and $r_{3,j}$. The estimator for
$p$ then reads
$\hat{p}(\boldsymbol{r}_{j},\boldsymbol{k}_{j})=\frac{1}{2}\hat{\epsilon}(\boldsymbol{k}_{j})\frac{R_{j}}{\sqrt{r_{0,j}r_{3,j}}}\,,$
where again $R_{j}=\sum_{x}r_{x,j}$. Rewriting the negativity defined in Eq.
(14) in terms of the pump polarization angle $\phi$ we obtain $\epsilon=p\sin
2\phi$. Thus the reference value $\epsilon_{t}$ of the negativity is then
inferred as $\epsilon_{t}=\langle\hat{p}\rangle\>\sin{2\phi}$, i.e. using the
knowledge of $\phi$ and the estimation $\left\langle p\right\rangle$ of the
mixing parameter. By making use of the relations in Eq. (27) one can apply the
same arguments to the Werner case and derive an appropriate expression for
$\epsilon_{t}$. Upon evaluating the corresponding sample means and variances
we can therefore obtain the first result of our analysis. This is illustrated
on Fig. 3 where we report the estimated value of entanglement as a function of
the reference one assuming, for the description of the output signals, the
families $\varrho$ (left plot) and $\varrho^{\prime}$ (right plot)
respectively. Here the uncertainty bars denote the $3\sigma$ confidence
interval and from this plots it is apparent that the experimental data are
compatible with both models.
Figure 3: (Color online) Estimated value of entanglement as a function of the
reference one assuming, for the description of the output signals, the
families $\varrho$ (left plot) and $\varrho^{\prime}$ (right plot). The
uncertainty bars stays for the $3\sigma$ confidence interval.
Notice that the reference value is built, on the basis of a given model, in
part with informations coming from the experimental settings (the tuning of
the angle $\phi$) and in part from the results of suitably chosen coincidence
measurements. On the other hand, the estimated value of entanglement is
obtained solely with experimental quantities. In principle, we are not
expecting the reference value to be more precise that the estimated one. The
idea here is to use two different estimates of the same quantity
(entanglement) obtained in two different and independent ways in order to to
discriminate and validate the different statistical models. Following our
analysis, a given model is not suitable for the description of our system if
the two different estimates that can be derived by that model, together with
the resulting errors, are not compatible.
It is interesting to compare these results, in particular the ones which refer
to the decoherence model (left plot in Fig. 3), with those obtained for a
different set of measurements data presented in EEE . In that case a less
precise control of the temperature of the PDC generation system made more
relevant the fluctuation of the phase and thus the state more mixed.
Therefore, in that case, a self-consistent statistical analysis of the
acquired data allowed discriminating between the two statistical models
identifying the decoherence model of Eq. (10) as the correct one for the
experimental set up used in EEE . In the present case, which includes that the
already mentioned control in temperature, the states obtained are nearly pure
and thus one cannot expect the different characterization of noise to be
relevant. Furthermore, to experimentally obtain more pure state one should
reduce the collection angle of PDC emission. This obviously reduces the rate
of coincidence counts, thus inducing an increase of the variance of both the
estimators, for negativity and purity parameter respectively.
Figure 4: (Color Online) Estimation of entanglement at the quantum limit. The
plot shows the estimated value of entanglement $\langle\hat{\epsilon}\rangle$
according to the decoherence (left) and Werner (right) models as a function of
the reference one $\epsilon_{t}$. The uncertainty bars on
$\langle\hat{\epsilon}\rangle$ denotes the quantity
$\sqrt{\hbox{Var}(\hat{\epsilon})\times\langle K\rangle}$, i.e. the square
root of the sample variance multiplied by the average number of total
coincidences $\langle K\rangle$. The gray area corresponds to values within
the inverse of the quantum Fisher information $\epsilon_{t}\pm
H_{\epsilon_{t}}^{-1/2}$. Uncertainty bars on the abscissae are due to
fluctuations in the estimation of the mixing parameter.
We now pass to evaluate the optimality of our estimation procedure. In Fig. 4
we show, for the decoherence (left) and Werner (right) model, the estimated
value of entanglement as a function of the reference one obtained for
different values $q=0.97,0.93,0.88,0.78,0.5$ (i.e.
$\phi=10^{\circ},15^{\circ},20^{\circ},28^{\circ},45^{\circ}$). Note that the
corresponding estimated mixing parameter in both model is larger than $0.97$
for all points. The uncertainty bars on $\langle\hat{\epsilon}\rangle$ denotes
the quantity $\sqrt{\hbox{Var}(\hat{\epsilon})\times\langle K\rangle}$, i.e.
the square root of the sample variance multiplied by the average number of
total coincidences $\langle K\rangle$. This is in order to allow a direct
comparison with the Cramer-Rao bound in term of the inverse of the Fisher
information (the gray area). Uncertainty bars on the abscissae correspond to
fluctuations $\delta\epsilon_{t}$ in the determination of $\epsilon_{t}$, due
to fluctuations in the estimation of the mixing parameter with the procedure
outlined above. The plot shows that our procedure allows estimating the
entanglement with a precision at the quantum limit for any value of $q$. From
the figure it is also apparent that, due to the high purity achieved with the
experimental set up that includes the active temperature control, and,
therefore, due to the irrelevance of the decoherence introduced, both the
models give optimal estimation. Notice that this conclusion is robust against
the fact that the statistics is not exactly Poissonian.
### VI.2 Comparison with tomographic estimation
We compared our results with estimation of entanglement from density matrix
elements obtained exploiting two different procedures of quantum state
tomography KB00 ; Ja01 . We found that the reconstructed density matrices are,
for both tomography protocols, statistically compatible within with both the
two models of Eqs. (10) and (21). As an example we present in Fig. 5 real and
imaginary part of reconstructed density matrices of maximally entangled state
corresponding to $q=\frac{1}{2}$ (i.e., $\phi=45^{\circ}$).
Figure 5: (Color online) Real (left) and Imaginary (right) part of the
tomographically reconstructed density matrix for the maximally entangled state
with J16 (top) and R16 (bottom) protocols. All the real elements, except the
four dominant, and the imaginary ones are compatible with zero within the
estimated tomographic uncertainties (not shown in the figure).
In fact, the tomographic procedure also allowed us to estimate entanglement
and the corresponding variance. In order to have a fair comparison of the
uncertainties obtained with different methods we have set measurement time for
the tomographic reconstruction such to have thr total number of registered
coincidences counts equal to $M\langle K\rangle$, i.e. the total number of
coincidence in the optimal measurement. The values of negativity calculated
directly using the reconstructed density matrices and its variance (obtained
by error propagation) for the maximally entangled state are presented in Table
1 together with the determination obtained from the optimal measurement
maximizing the QFI. All three negativity values overlap in their uncertainty
intervals; the three methods are therefore coherent. Furthermore, it is
evident from the presented results that the optimal method devised in this
paper allows, at fixed sample size, for a sensitive reduction of the
uncertainty in entanglement estimation.
Method | $\epsilon$ | | $\delta\epsilon$
---|---|---|---
Optimal | 0.972 | $\pm$ | 0.011
Tomography: J16 | 0.984 | $\pm$ | 0.048
Tomography: R16 | 0.957 | $\pm$ | 0.046
Table 1: Estimated value of entanglement with different methods. The
uncertainty $\delta\epsilon$ is calculated usinv Eq. (28) for the optimal
method and with error propagation for tomographic estimation.
### VI.3 Statistical mixtures
In order to check our method in different working regimes we applied the
estimation procedure to a set of mixed states obtained in a controlled way,
i.e. by adding some portion of unentangled light to pure entangled state. As
we have described in the previous section, our experimental set up allows us
to obtain states with an extremely high purity. In the following we thus
assume that the output state of our apparatus is the pure state as in Eq.
(11). Then, if one is able, for example, to mix in a controlled way the
components $\left|HH\rangle\\!\langle HH\right|$ and
$\left|VV\rangle\\!\langle VV\right|$ to the maximally entangled states one
obtains the states
$\displaystyle\varrho$ $\displaystyle=$ $\displaystyle
p\left|\psi\rangle\\!\langle\psi\right|+(1-p)\,D$ (29) $\displaystyle D$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(\left|HH\rangle\\!\langle
HH\right|+\left|VV\rangle\\!\langle VV\right|)\right)\,,$
which correspond to the model (10) with an adjustable mixing parameter. In
practice, in order to tune the value of the mixing parameter $p$ we have
measured coincidence counts for states $\left|{\hbox{\small
HH}}\rangle\\!\langle{\hbox{\small HH}}\right|$ and $\left|{\hbox{\small
VV}}\rangle\\!\langle{\hbox{\small VV}}\right|$ for different time intervals.
The sample of coincidence counts is then added to experimental data obtained
for the maximally entangled pure state and then analyzed as in the previous
section.
Figure 6: (Color Online) Estimation of entanglement at the quantum limit. The
plot shows the estimated value of entanglement $\langle\hat{\epsilon}\rangle$
as a function of the reference one $\epsilon_{t}$. In the left panel we report
estimated entanglement for mixed states generated according to the decoherence
model (29). In the right panel we report estimated entanglemed for mixed
states generated according to the Werner model (30). The points correspond to
different portions of incoherent addition from both crystals.
In the left panel of Fig. 6 we show the estimated value of entanglement as a
function of the reference one for the originally maximally entangled state
($q=\frac{1}{2}$) and for states prepared with mixing parameter
$p=99.5\%,83\%,74\%,50\%,33\%$.
A similar analysis may carried out for the Werner model. In this case, in
order to tune the value of the mixing parameter $p$ one should supplement the
coincidences vectors $\bf{k}_{j}$ and $\bf{r}_{j}$ with values coming from
unpolarized light. This can be achieved by measuring coincidence counts for
$\left|{\hbox{\small HH}}\rangle\\!\langle{\hbox{\small HH}}\right|$,
$\left|{\hbox{\small HV}}\rangle\\!\langle{\hbox{\small HV}}\right|$,
$\left|{\hbox{\small VH}}\rangle\\!\langle{\hbox{\small VH}}\right|$ and
$\left|{\hbox{\small VV}}\rangle\\!\langle{\hbox{\small VV}}\right|$ for
different time intervals. The measured values are then added to the previously
measured values for pure maximally entangled state. In this way, one can get
data corresponding to
$\displaystyle\varrho^{\prime}=p\left|\psi\rangle\\!\langle\psi\right|+(1-p)\frac{\mathbbm{I}}{4}$
(30)
which correspond to a Werner state with tunable depolarizing parameter. After
performing measurement and analysis set described in previous section we can
estimate entanglement and mixing parameter value in this family of states. In
the right panel Fig. 6 we show the estimated value of entanglement as a
function of the actual one for the originally maximally entangled state and
mixture parameter $p=99.5\%,76\%,62\%,52\%,45\%$. As one can evince from the
presented figure our method provides optimal entanglement estimation also for
mixed states.
## VII Conclusions
In this paper we have addressed in detail the estimation of entanglement for
pairs of polarization qubits. Our scheme is based on visibility measurements
of quantum correlations and allows optimally estimating entanglement of
families of two-photon polarization entangled states without the need of
performing full tomography. Our procedure is self-consistent and allows
estimating the amount of entanglement with the ultimate precision imposed by
quantum mechanics. Although optimal estimation of entanglement does not
require the full tomography of the states we have also performed state
reconstruction using two different sets of projectors and explicitly shown
that they provide a less precise determination of entanglement.
The technique has been demonstrated for nearly pure states as well as for
controlled mixtures in order to confirm its reliability in any working regime.
With a suitable choice of correlation measurements it may be extended to a
generic class of two-photon entangled states. The statistical reliability of
our method suggests a wider use in precise monitoring of external parameters
assisted by entanglement.
## Acknowledgements
This work has been supported by Associazione Sviluppo Piemonte. MGAP and PG
thanks Marco Genoni for several useful discussions. MGAP thanks Simone Cialdi
and Davide Brivio for useful discussions.
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|
arxiv-papers
| 2011-02-17T20:39:39 |
2024-09-04T02:49:17.107480
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Brida, I. P. Degiovanni, A. Florio, M. Genovese, P. Giorda, A.\n Meda, M. G. A. Paris, A. Shurupov",
"submitter": "Matteo G. A. Paris",
"url": "https://arxiv.org/abs/1102.3684"
}
|
1102.3704
|
# A Model for the Sources of the Slow Solar Wind
S. K. Antiochos NASA Goddard Space Flight Center, Greenbelt, MD, 20771
spiro.antiochos@nasa.gov Z. Mikić, V. S. Titov, R. Lionello, J. A. Linker
Predictive Science, Inc., San Diego, CA 92121
###### Abstract
Models for the origin of the slow solar wind must account for two seemingly-
contradictory observations: The slow wind has the composition of the closed-
field corona, implying that it originates from the continuous opening and
closing of flux at the boundary between open and closed field. On the other
hand, the slow wind also has large angular width, up to $\sim 60^{\circ}$,
suggesting that its source extends far from the open-closed boundary. We
propose a model that can explain both observations. The key idea is that the
source of the slow wind at the Sun is a network of narrow (possibly singular)
open-field corridors that map to a web of separatrices and quasi-separatrix
layers in the heliosphere. We compute analytically the topology of an open-
field corridor and show that it produces a quasi-separatrix layer in the
heliosphere that extends to angles far from the heliospheric current sheet. We
then use an MHD code and MDI/SOHO observations of the photospheric magnetic
field to calculate numerically, with high spatial resolution, the quasi-steady
solar wind and magnetic field for a time period preceding the August 1, 2008
total solar eclipse. Our numerical results imply that, at least for this time
period, a web of separatrices (which we term an S-web) forms with sufficient
density and extent in the heliosphere to account for the observed properties
of the slow wind. We discuss the implications of our S-web model for the
structure and dynamics of the corona and heliosphere, and propose further
tests of the model.
Sun: magnetic field — Sun: corona — Sun: solar wind
## 1 Introduction
Decades of in situ measurements of the heliosphere have firmly established
that the Sun’s wind consists of two distinct types: “fast” and “slow”. In
terms of its origins at the Sun, the best understood is the fast wind, which
typically exhibits speeds in excess of 600 km/s at 1 AU and beyond (e.g.,
McComas et al., 2008). The fast wind is measured to be approximately steady,
except for some Alfvénic turbulence (e.g., Bame et al., 1977; Bruno & Carbone,
2005). This wind is known to originate from coronal holes, regions that appear
dark in XUV and X-ray images, due to a plasma density that is substantially
lower ($<50\%$) than in surrounding coronal regions (Zirker, 1977). As implied
by eclipse and coronagraph images, the magnetic field in coronal holes is
open—appearing mainly radial and stretching out without end—whereas the field
in the surrounding regions is closed, looping back down to the photosphere.
Hence, the fast wind corresponds to the steady wind predicted by Parker in his
classic work (Parker, 1958, 1963).
The slow wind, however, is much less understood. In particular, its origin at
the Sun has long been one of the major unsolved problems in solar/heliospheric
physics. This wind has a number of observed features that distinguish it
physically from the fast wind. First, its speeds are typically slower,
$<500\,{\rm km/s}$. More important, the slow wind appears to be inherently
non-steady when compared to the fast wind (e.g., Bame et al., 1977; Schwenn,
1990; Gosling, 1997; McComas et al., 2000). It exhibits strong and continuous
variability in both plasma (for example, speed and composition) and magnetic
field properties; variability that cannot be described as simply Alfvénic
disturbances superimposed on a steady background (Zurbuchen & von Steiger,
2006; Bruno & Carbone, 2005). Finally, its location in the heliosphere is
distinct; it is generally found surrounding the heliospheric current sheet
(HCS) (e.g., Burlaga et al., 2002). A key point is that the HCS is always
embedded inside slow wind, never fast. From the presently available spacecraft
observations, it is not possible to rule out the possibility that slow wind
also occurs in locations unconnected to the HCS, in other words, that there
are pockets of slow wind with no embedded HCS and surrounded completely by
fast wind. However, the present data are certainly consistent with the picture
that, at least, during solar minimum when the corona-wind mapping can be
determined with some accuracy, all slow wind originates from a band that
encompasses the HCS, so that the mapping of the slow wind down to the Sun
appears to connect to or near the helmet streamer belt (e.g., Gosling, 1997;
Zhao et al., 2009).
Another key feature of the slow wind is its latitudinal extent, which
typically ranges from $40^{\circ}$–$60^{\circ}$ near solar minimum, a time
when it is easiest to distinguish the sources of fast and slow wind. Within
this broad region of slow wind the actual HCS, across which the magnetic field
changes direction, is very narrow. As for any current sheet, one can identify
in the heliospheric data a scale over which the field becomes small and the
plasma beta, defined as the ratio of the gas pressure $P_{g}$ to the magnetic
pressure $B^{2}/8\pi$, becomes large. This region is termed the plasma sheet
and is usually of the order of a few degrees in angular width (e.g.,
Winterhalter et al., 1994; Bavassano et al., 1997; Wang et al., 2000; Crooker
et al., 2004). It is important to note that the HCS is often not symmetrically
located within the broad band of slow wind, but is often found nearer to one
edge of the slow wind region (Burlaga et al., 2002). It is also important to
note that the field almost never vanishes at the HCS, as would be expected for
a true steady-state. This observation implies that, at least, the wind near
the HCS must be continuously dynamic.
The final and most critical feature of the slow wind that distinguishes it
from the fast is the plasma composition (Geiss et al., 1995; von Steiger et
al., 1995). It is well-known that in the closed field corona, the ratio of the
abundances of elements with low first ionization potential (FIP), such as Mg
and Fe, to those with high FIP, such as C and Ne, is a factor 4 or so higher
than in the photosphere (e.g., Meyer, 1985; Feldman & Widing, 2003). This so-
called FIP effect is not seen in the fast wind, which has abundances similar
to those of the photosphere; but, it is present in the slow wind, which has
abundances similar to that of the closed corona (Gosling, 1997; Zurbuchen &
von Steiger, 2006; Zurbuchen, 2007).
Along with the difference in elemental abundances, the slow and fast wind also
exhibit clear differences in their ion charge state abundances, for example,
the ratio of ${\rm O}^{7}/{\rm O}^{6}$. This ratio can be used to determine
the “freeze-in” temperature of the ion charge states at the source of the
wind. Close to the Sun where the time scales for ionization and recombination
are much shorter than the plasma’s expansion time-scales, the ion charge
states are approximately in ionization equilibrium with the local electron
temperature. As the solar wind plasma expands outward, however, the electron
density drops rapidly and the recombination time scales become so large that
the ionic charge states stop changing, freezing-in the electron temperature at
this point. The freeze-in radius varies for the different ions, but is
typically 1 - 3 R⊙. The data show that the slow wind has a higher freeze-in
temperature ($\geq 1.5\times 10^{6}\,{\rm K}$) than the fast wind ($\leq
1.2\times 10^{6}\,{\rm K}$) (von Steiger et al., 1997, 2001; Zurbuchen et al.,
1999, 2002). Note, however, that this freeze-in temperature corresponds only
to the electron temperature in the low corona. The proton and ion temperatures
measured in situ and in coronal holes by UVCS, for example, (e.g., Kohl et
al., 2006) show the opposite trend in that the ion temperatures are
substantially higher in the fast wind than in the slow (Marsch, 2006). The
origin of these differences in the ion temperatures between the two winds is
still not clear, but in any case, both the ion and freeze-in temperatures
suggest that the sources of the two winds near the Sun are physically
different.
The elemental abundances track very well the ionic abundances, indicating that
there is a consistent compositional distinction between the two winds.
Furthermore, the two winds have markedly different temporal variability in
elemental and ionic composition. The fast exhibits an approximately constant
composition; whereas the slow exhibits large and continuous variability, so
that its elemental composition varies from coronal to near photospheric. The
composition results suggest that the fast wind has a unique origin, presumably
in coronal holes, but that the slow wind originates from a mixture of sources.
In fact, Zurbuchen and coworkers have argued that the compositional
differences, rather than the speed, are what truly distinguish the two winds,
because it is possible to find solar wind whose composition and constancy
match that of the “fast wind,” but that has relatively slow speed, $<500\,{\rm
km/s}$ (Zhao et al., 2009). Note also that, as determined by the composition
measurements (Zurbuchen et al., 1999), the boundary between the slow and fast
wind in the heliosphere is sharp, of order a few degrees in angular extent,
much smaller than the angular width of the slow wind region, but comparable to
that of the plasma sheet. An important point is that the observed sharpness of
the composition transition is not merely a dynamical effect, because it does
not depend on whether the stream-stream transition is fast to slow or slow to
fast (Geiss et al., 1995; Zurbuchen, 2007). We conclude, therefore, that the
fast and slow winds are far more appropriately described as the steady and
unsteady winds, and that the boundary layer between the two winds is much
narrower than the width of either wind.
Since the differences in plasma composition of the two winds must be due to
differences in their origins at the Sun, the composition data place severe
constraints on the possible sources of the slow wind. In particular, the data
imply that the slow wind originates in the dynamic opening of closed magnetic
flux, which releases closed-corona plasma into the wind. Such a process would
also naturally explain the difference in variability between the fast and slow
wind.
It should be emphasized, however, that this constraint on the slow wind’s
origin is not universally accepted. Several authors have argued that the slow
wind originates from open-field coronal holes, just like the fast wind, but
from the edges of the holes, where the field expands super-radially as it
extends from the photosphere out to the heliosphere (e.g., Kovalenko, 1981;
Wang & Sheeley, 1991; Cranmer & van Ballegooijen, 2005; Cranmer et al., 2007;
Wang et al., 2009). The hypothesis is that a large expansion factor can both
slow down the wind by affecting the location of wave energy deposition in
coronal flux tubes, and change the plasma composition by the FIP mechanism
proposed by Laming (2004). Note that in the expansion factor model, as in all
steady state wind solutions, the properties of the wind in a given flux tube
are determined uniquely, in most cases, by the flux tube geometry and the
forms of the heating and momentum deposition (Cranmer et al., 2007). Of course
the detailed forms of the heating and momentum deposition will depend on the
flux tube geometry, and may depend on other factors, as well, but the
dependence on these other factors cannot be dominant; otherwise the calculated
wind speed would not be well correlated with expansion factor. In other words,
two flux tubes on the Sun with identical geometry should have similar
heating/momentum deposition and end up with the same wind properties.
Therefore, the steady-state models inherently predict a tight correlation
between speed and composition (e.g., Cranmer et al., 2007).
The problem, however, is that observations indicate that wind speed is not
tightly correlated with composition. The wind from small equatorial coronal
holes with a large expansion factor is indeed slow, with speeds $<500\,{\rm
km/s}$, in good agreement with the predictions of the expansion factor models.
But this wind has photospheric FIP ratios, so it is still considered to be
“fast wind” (Zhao et al., 2009). Furthermore, this not-so-fast wind has the
temporal quasi-steadiness of the fast wind, rather than the quasi-chaotic time
variation of the slow wind.
We conclude, therefore, that the most likely source for the true slow wind,
that with FIP-enhanced coronal composition, is the closed-field corona. In
this case, the process that releases the coronal plasma to the wind must be
either the opening of closed flux or interchange reconnection between open and
closed magnetic field lines. This latter process is the underlying mechanism
invoked by Fisk and co-workers (Fisk et al., 1998; Fisk, 2003; Fisk & Zhao,
2009) in their model for the heliospheric field. These authors argue that open
flux can diffuse freely throughout the solar surface, even deep inside the
helmet streamer region. If so, then the interchange reconnection between open
and closed magnetic field lines would naturally account for both the
composition and geometrical properties of the slow wind. The difficulty with
this model is that it has not been demonstrated that such open flux diffusion
can actually occur. In fact, detailed MHD simulations indicate that it is
difficult to bring open fields into closed-field regions without having them
close down (Edmondson et al., 2010; Linker et al., 2010). The simulation
results are in agreement with Antiochos et al. (2007), who argued that, for
the low-beta corona, basic MHD force balance forbids the presence of open flux
deep inside the closed helmet streamer region.
Within the context of MHD models, the most likely location for the release of
closed-field plasma is from the tops of helmet streamers (the Y-point at the
bottom of the HCS), where the balance between gas pressure and magnetic
pressure is most sensitive to perturbations. A number of authors have argued
that streamer tops are unstable and should undergo continual opening and
closing as a result of thermal instability (Suess et al., 1996; Endeve et al.,
2004; Rappazzo et al., 2005). Even if streamer tops are stable, it seems
inevitable that the constant emergence and disappearance of photospheric flux
and the constant motions of the photospheric would force them to be
continuously evolving. Furthermore, coronagraph observations often show the
ejection of “blobs” from the tops of streamers and into the HCS (Sheeley et
al., 1997).
Although this streamer top model seems promising in that it naturally explains
both the composition and variability, it has difficulty in accounting for the
large angular widths of the slow wind. One would expect the instabilities to
be confined to the high-plasma beta region about the current sheet. In fact,
the plasma emanating from the streamer tops, the so-called stalks, is observed
to be only $\sim 3^{\circ}$–$6^{\circ}$ wide, which agrees well with the
plasma sheet width in the heliosphere (Bavassano et al., 1997; Wang et al.,
2000). Even if the plasma sheet width were to be widened by the Kelvin-
Helmholtz instability (e.g., Einaudi et al., 1999), there would not be enough
mass flux from the narrow region at the streamer tops to account for the slow
wind. The streamer-top models can account for a thin band of slow wind around
the HCS, but it seems unlikely that this is the origin of the bulk of the slow
wind, which can extend as far as $30^{\circ}$ in latitude from the HCS.
In order to be compatible with the in situ data, we require some process that
releases closed-field plasma onto open field lines that, in the heliosphere,
can be far from the HCS. This requirement seems impossible to satisfy, because
the plasma release must occur at the boundary between the open and closed
field in the corona, which maps directly to the HCS. We describe below,
however, a magnetic topology that resolves this slow wind paradox: the flux
associated with an open-field corridor can be simultaneously near to and far
from the open-closed boundary!
## 2 The Topology of an Open-Field Corridor
Figure 1 illustrates the magnetic connectivity from the photosphere to the
heliosphere that results from an open-field corridor. The dark yellow inner
sphere in the figure represents the photosphere, while the light yellow, semi-
transparent one represents an arbitrary radial surface in the open-field
heliosphere, say at $5R_{\odot}$ The green line on the photosphere marks the
boundary between open (gray) and closed (yellow) field regions, which is
mapped by the magnetic field (red lines) to the HCS (thick green line) at the
$5R_{\odot}$ surface. The green line at the HCS is also the polarity inversion
line at this surface. Note that the four points, a, b, c, and d, which are
meant to represent the end-points of the corridor at the Sun, map sequentially
to the corresponding points a′, b′, c′, and d′ along the HCS.
The open field pattern at the photosphere of Fig. 1 consists of a large polar
coronal hole and, as is often seen, a smaller low-latitude hole. In recent
work, we argued that if the two holes are in the same photospheric polarity
region, then by our uniqueness conjecture the holes must be connected by an
open field corridor, as illustrated above (Antiochos et al., 2007). It is
evident from the figure that the flux in the corridor maps on the heliospheric
surface to a thin arc (light gray band), bounded at both ends by the HCS. The
flux between the arc and the HCS maps to the low-latitude extension while the
flux outside the arc maps to the main part of the polar coronal hole. The
corridor and its associated arc are the footprints of two quasi-separatrix
layers (QSLs, e.g., Priest & Démoulin, 1995; Démoulin et al., 1996) that
combine into a hyperbolic flux tube, as has been described in detail by Titov
et al. (2002, 2008) for the case of closed magnetic configurations. In
contrast, the HCS is a true separatrix.
The key point for understanding the origin of the slow wind is that, just like
the HCS, the QSL arc in the heliosphere can also be a source region for slow
wind. If the open-field corridor at the Sun is sufficiently narrow, then the
continual evolution of the photosphere, driven by the ever-present
supergranular flow and flux emergence/submergence in particular, will
continually change the exact location of this corridor. But, by the uniqueness
conjecture (Antiochos et al., 2007), the corridor is a topologically robust
feature, similar to a null-point, and must be present on the photosphere as
long as the low-latitude coronal hole extension is present. Its location and
shape, however, will vary in response to local photospheric changes. These
variations require field line opening/closing and interchange reconnection,
thereby releasing closed-field plasma all along the QSL arc in the
heliosphere. Therefore, if the QSL arc extends to high latitudes, this will
naturally produce slow wind with an extent far from the HCS.
To determine whether the QSL resulting from an open field corridor does,
indeed, reach high heliospheric latitudes, we have calculated an example of a
field such as that of Fig. 1 using the source surface model (Altschuler &
Newkirk, 1969; Schatten et al., 1969; Hoeksema, 1991). The field is most
easily determined from the image-dipole formula derived by Antiochos et al.
(2007). For a dipole with moment ${\bf d}$ located at a point ${\bf r}_{d}$
inside the Sun, and a source surface at radius $R_{S}$, the magnetic field
${\bf B}$ is determined from the potential $\Phi$ via ${\bf B}=-\nabla\Phi$,
where $\Phi$ is given by:
$\Phi=\frac{{\bf d}\cdot({\bf r}-{\bf r}_{d})}{|{\bf r}-{\bf
r}_{d}|^{3}}-\frac{R_{S}r_{d}^{3}{\bf d}\cdot(R_{S}^{2}{\bf r}-r^{2}{\bf
r}_{d})}{|r_{d}^{2}{\bf r}-R_{S}^{2}{\bf r}_{d}|^{3}}.$ (1)
This field satisfies the source-surface boundary condition that
$B_{\theta}=B_{\phi}=0$ at $r=R_{S}$, since $\Phi=0$ there. The advantage of
this formulation is that most active regions can be approximated by a
collection of dipoles, and one can build up a field of arbitrary complexity by
simply adding a series of dipoles of the form of Eq. (1). Each dipole is
specified in terms of its position in spherical coordinates ${\bf
r}_{d}=r_{d}{\bf\hat{r}}(\theta_{d},\phi_{d})$, where $r_{d}$, $\theta_{d}$,
and $\phi_{d}$ specify the location of the dipole, and the spherical
components of its dipole moment, ${\bf d}=(d_{r},d_{\theta},d_{\phi})$.
Figure 2 shows the field computed from Eq. (1) for the case of two dipoles: a
sun-centered global dipole with a dipole moment of unit magnitude directed
along the north polar axis, and an equatorial “active region” dipole at ${\bf
r}_{d}=0.8R_{\odot}{\bf\hat{r}}(90^{\circ},0^{\circ})$ with a northward-
pointing dipole moment ${\bf d}=(0,-0.2,0)$. The source surface radius is
chosen as $R_{S}=4R_{\odot}$, though the exact value is not critical for our
argument. Note that for convenience in viewing the magnetic field, we have
selected the dipole parameters so that the system has symmetry across both the
equatorial $(\theta=90^{\circ})$ and meridional $(\phi=0)$ planes. Also, for
ease of viewing, we show in the Fig. 2 only the front hemisphere defined by
the angular region $(15^{\circ}\leq\theta\leq 90^{\circ})$ and
$(-90^{\circ}\leq\phi\leq 90^{\circ})$.
The solar surface, the photosphere, corresponds to the gray grid in Fig. 2.
The colored contours on this surface correspond to contours of radial flux,
indicating the presence of the active region dipole at the equator. We
selected the parameters for the active region dipole so that its structure
would be easily resolved. It is evident from Fig. 2 that the region is large
compared to real active regions, which are generally only a few degrees in
angular extent. On the other hand, the maximum field strength at the dipole
center is only $\sim 20$ times that of the polar region, which is much less
than the corresponding ratio for solar active regions, so the flux ratio
between the active region and global background field is approximately
correct. This ratio is the important parameter to obtain a coronal hole
extension.
The thick black line along the equator is the $B_{r}=0$ contour, i.e., the
polarity inversion line. The thick black line above the solar surface is the
polarity inversion line at the source surface, i.e., the bottom of the HCS.
Red field lines are traced at equal intervals along the HCS down to the solar
surface. These define the boundary between open and closed field lines. As
expected, the effect of the equatorial dipole is to pull the open-closed
boundary down to lower latitudes; in other words, to create a low-latitude
extension of the coronal hole, which can be seen as the gray shaded region in
the Figure. Far from the dipole, the coronal hole boundary is at a latitude of
$\sim 54^{\circ}$, whereas at the meridional symmetry plane the boundary drops
down to $\sim 26^{\circ}$.
For the large spatial scale of our active region dipole, the extension of the
coronal hole down to low latitudes is gradual rather than in the form of a
distinct “elephant trunk”, but the basic effect is clearly present. There is
no open-field corridor in Fig. 2, but let us now add another dipole to the
system, displaced $20^{\circ}$ in both latitude and longitude from the
equatorial one and a factor of five times weaker. This dipole is located at
${\bf r}_{d}=0.8R_{\odot}{\bf\hat{r}}(70^{\circ},20^{\circ})$ with a primarily
southward-pointing dipole moment ${\bf d}=(0,0.05,0)$. In order to maintain
the equatorial and meridional symmetry, as mentioned earlier, we actually add
4 dipoles symmetrically located about the equatorial and meridional planes.
The resulting field is shown in Figure 3. The effect of the additional dipoles
is to add high-latitude polarity inversion lines to the system. These
“squeeze” the open-flux extension of Fig. 2 to form a narrow corridor and a
low-latitude coronal hole. As in Fig. 2, red field lines are traced from
equidistant footpoints along the HCS down to the solar surface. The red
footpoints at the photosphere appear to traverse the boundary of the low-
latitude hole and then jump abruptly to the polar hole boundary, which implies
that the mapping defined by the field develops extreme gradients in the region
connecting the two holes. To clarify this point, we have traced two sets of
field lines, colored in blue, from footpoints that are closely located at the
HCS. The corresponding solar footpoints are much more widely spaced, running
along the corridor. The resulting structure, Fig. 3, looks very similar to the
mapping drawn in Fig. 1, in that the closely spaced pairs of points a′,b′ and
c′,d′ at the HCS map to far-separated points a,b and c,d at the solar surface.
Note also that although the footpoints of the two sets of blue lines approach
each other very closely at the photosphere, they are far separated at the HCS,
by a distance of order $R_{\odot}$. This result indicates that even though the
low-latitude coronal hole has small area, it contains a substantial magnetic
flux. As is evident from the colored contours in Fig. 3, the photospheric
field strength in the low-latitude hole is large due to the presence of the
active region dipole.
The analytic model underlying Fig. 3 has similar topology to the case shown
schematically in Fig. 1. The low-latitude coronal hole extension in Fig. 3 is
connected to the main polar hole by a corridor that becomes very narrow.
Furthermore, this type of topology is not difficult to obtain. It is often
observed in quasi-steady MHD solutions for observed photospheric fields, as
will be shown below. A similar corridor was found for Carrington rotation 1922
(Antiochos et al., 2007).
The question now is whether the open flux in the corridor connects to large
latitudes in the heliosphere. To answer this question, we trace field lines
from a set of photospheric footpoints lying on a latitudinal line segment
spanning the narrowest width of the corridor, which is only of order
$5{,}000\,{\rm km}$ at the photosphere. Fig. 4b shows the footpoints and the
field lines (green) near the photosphere and Fig. 4a shows where they map to
on the source surface. We note that the corridor maps to high latitudes. In
fact, for this analytic case, the corridor mapping defines a QSL arc that
reaches latitudes $>45^{\circ}$, greater than that of the observed slow wind.
This result, that the corridor maps to heliospheric latitudes far above the
HCS, is robust in that it is not sensitive to the exact position of the
secondary dipole. The position and geometry of the corridor, on the other
hand, is very sensitive to the photospheric flux distribution. For example,
its width would change or even become singular (Titov et al., 2011), and its
location would change substantially if the secondary dipoles were moved in
longitude. Based on flux conservation arguments, and the fact that the
heliospheric magnetic field is almost uniform in latitude, we can argue that
the angular extent of the QSL arc, however, would be expected to depend
primarily on the ratio of the flux in the low-latitude coronal hole extension
to that in the polar hole. For example, in the extreme case that the fluxes
were equal, the corridor mapping would be expected to reach the heliospheric
pole ($90^{\circ}$ from the HCS!), irrespective of the geometry of the
corridor or of the coronal holes.
## 3 The S-Web Model
If the width of the corridor at the photosphere is small compared to the scale
of typical motions there, such as the supergranular flow, we expect that the
whole corridor will continuously disrupt and reform at the photosphere and,
consequently, closed-field plasma will be released by reconnection all along
the QSL arc in the heliosphere. Therefore, the topology of Fig. 2 may be able
to resolve the slow wind paradox. The overriding question, however, is whether
there are enough such corridors and corresponding QSL arcs in the heliosphere
to account for the slow wind that is observed. The flux distribution of Fig. 2
produces only one such arc, which would certainly not be sufficient to
reproduce the observed slow wind. There are two issues that must be addressed,
the number of arcs (their density and extent on the Sun and heliosphere), and
the amount of mass and energy that each arc can be expected to release. In
this paper we concentrate on the first issue and only briefly discuss the
second in Section 4 below, because addressing this issue requires fully
dynamic calculations.
In order to address the issue of the number of QSL arcs, we calculated the
quasi-steady model for an observed photospheric flux distribution. Figure 5a
shows the photospheric radial field as derived from MDI observations on SOHO
(Scherrer et al., 1995) for a time period preceding the August 1, 2008 total
solar eclipse. This calculation was used to predict the structure of the
corona prior to the eclipse, using magnetic field data measured during the
period June 25–July 21, 2008\. The prediction compares very favorably with
images of the corona taken during the eclipse in Mongolia (Rušin et al.,
2010). Note that the high resolution of the calculation captures the details
of many small-scale bipoles in the photospheric magnetic field (Harvey, 1985).
This has been incorporated into the idea of the “magnetic carpet” (Schrijver
et al., 1997). We also show the polarity inversion line $B_{r}=0$ slightly
above the photosphere, at $r=1.05R_{\odot}$ to delineate the magnetic polarity
of the large-scale structures. (The polarity inversion line in the photosphere
itself shows an enormous complexity that overshadows its usefulness to discern
the large-scale magnetic polarity.)
The quasi-steady model was calculated by using the 3D MHD code MAS. The MAS
code and its implementation are described in detail by Mikić & Linker (1994),
Mikić et al. (1999), Linker et al. (1999), and Lionello et al. (2009). MAS
solves the time-dependent MHD equations, including a realistic energy equation
with optically thin radiation and thermal conduction parallel to the magnetic
field. Given the magnetic field at the photosphere and an assumption for the
coronal heating source, the MHD equations are advanced until the magnetic
field settles down close to steady state. MHD models are generally considered
to be the most sophisticated implementation of Parker’s solar wind theory
because they incorporate all the essential physics, including the balance
between gas pressure and Lorentz force. An important assumption is the form of
coronal heating, which is prescribed empirically at the present time since the
coronal heating process is still unknown. The parameters of the empirical
heating model are constrained by observations of coronal emission in EUV and
X-rays (e.g., Lionello et al., 2009), as well as by solar wind measurements.
Details on the assumed form for the heating and on the thermodynamics used in
the MAS code can be found in Mikić et al. (2007) and Lionello et al. (2009).
In order to capture as much of the photospheric magnetic structure as
possible, we ran the MAS code with unprecedented resolution. Our calculation
used more than 16 million mesh cells and was run on over 4000 processors of
NSF’s Ranger supercomputer at the Texas Advanced Computing Center, making it
possible to include much of the small-scale structure of the photospheric
field in both the quiet sun and in coronal holes, as shown in Fig. 5a. These
calculations are unique in the degree to which they capture the small-scale
structure of the measured magnetic field.
Figure 5b shows the distribution of open and closed magnetic field regions at
the solar surface as determined by the model. It is evident that there are
many low-latitude coronal hole extensions, similar to that in Fig. 3, but with
much more structure. Several of these extensions appear to be disconnected
from the main polar holes, but this is partly due to the limited resolution of
the figure. A few of these coronal hole extensions are indeed connected by
very thin corridors in the photosphere, though many are only linked to the
polar coronal holes in a singular manner, as described in detail by Titov et
al. (2011), and as discussed further below.
The open field pattern in Fig. 5b is clearly complex, but the important issue
is the degree of complexity of the mapping into the heliosphere and, in
particular, the structure of the separatrices and QSLs there. We determined
the open field mapping in great detail by tracing tens of millions of magnetic
field lines. The topology of this mapping, as evidenced by structures such as
separatrices and QSLs, is most easily seen by analyzing the squashing factor
$Q$ (Titov et al., 2002; Titov, 2007). $Q$ is a measure of the distortion in
the magnetic field mapping, and is directly related to the gradients in the
connectivity. QSLs are regions of very large $Q$; we generally define them as
any region with $Q>10^{3}$. True separatrices such as the HCS have infinite
$Q$, because the mapping is singular there, but when computed numerically they
appear as surfaces with very large (unresolved) values of $Q$. The gray arc at
$r=5R_{\odot}$ in Fig. 1 is a QSL in the open field, and consequently would be
a region of high $Q$. The green HCS would also be a region of high (infinite)
$Q$. As will be seen below, a high-resolution analysis of the $Q$ properties
of our MHD simulation is extremely informative.
Figure 6a shows $Q$ in a meridional plane at a central Carrington longitude of
$23.33^{\circ}$ at the time of the eclipse at 10:21UT, while Figure 6b shows
magnetic field lines traced from the vicinity of the solar limbs at the same
time. We see that $Q$ outlines the boundary between open and closed field,
which is a true separatrix surface, but it is apparent that there is much more
detailed structure in both the closed and open field regions. The complex
structure of $Q$ in the closed-field region is expected; it simply reflects
the fact that the photospheric field consists of many small bipoles; but,
there is also substantial structure in the open field near the open-closed
boundary. Note the presence of a “pseudostreamer” on the NE limb, a feature
that has been discussed by Wang et al. (2007). The relationship of
pseudostreamers to open hole corridors and the S-web is discussed in detail in
Titov et al. (2011)
Figure 7a shows $Q$ in the spherical surface at $r=10R_{\odot}$ using a
logarithmic scale. This is the structure that is expected to map into the
inner heliosphere (appropriately wrapped into a spiral magnetic field by solar
rotation), since the magnetic field has reached its asymptotic structure by
this radius. The thick black line is the heliospheric current sheet (at which
$B_{r}$ reverses sign). Figure 7b shows the magnitude of $B_{r}$ at the same
radial surface $r=10R_{\odot}$. Note that the choice of $10R_{\odot}$ is not
crucial. Any surface in the heliosphere (where the field is all open) yields
similar results.
It is important to emphasize that the apparent structure in $Q$ expresses only
the connectivity of the open field, not its actual magnitude. In spite of the
enormous magnetic complexity at the solar surface, the radial field
distribution in the heliosphere is completely unremarkable, Fig. 7b. There is
a single polarity inversion line denoting a single HCS, as is generally
observed near solar minimum, and this HCS runs more or less equatorial. The
radial field is essentially uniform away from the HCS, as would be expected
from simple pressure balance. (Careful examination of the plot of $B_{r}$
shows that there is a faint semblance of the structure that can be seen in
$Q$, but it is only a small perturbation.)
On the other hand, the $Q$ map at this surface is remarkable, indeed, Fig. 7a.
We see that surrounding the HCS is a broad web of separatrices and QSLs of
enormous complexity. There are at least four striking features of this S-web.
First, it has an angular extent in latitude of approximately $40^{\circ}$,
sufficient to account for the observed extent of the slow wind. Note also that
the angular extent does vary with longitude, but only by a factor of two or
so. Second, the HCS is not necessarily in the center of the S-web, but is
sometimes near its edge. This can explain the frequent observation that the
HCS is usually not centrally located within slow wind streams (e.g., Burlaga
et al., 2002). Third, the boundary between the S-web layer and the featureless
polar hole region is sharp; it is narrow compared to the width of the S-web.
This can explain the observation that the transition from slow to fast wind as
measured by the composition data is narrow compared to the slow wind region
itself (Zurbuchen et al., 1999).
In order to explore the details of how coronal hole extensions connect to the
polar holes, we calculated coronal hole areas at different heights in the
corona. Figure 8 shows the location of a region near longitude $75^{\circ}$
and latitude $15^{\circ}$N in which we explored the connection between the
low-latitude coronal hole extensions (of negative polarity, shown in blue) in
detail. It is evident that the coronal hole extensions in this region appear
disconnected from the north polar hole in the photosphere, but connect with it
low in the corona (at heights approximately between $0.01R_{\odot}$ and
$0.02R_{\odot}$ above the photosphere). Figure 9 shows explicitly how these
coronal holes connect in the low corona. The three-dimensional shape of the
coronal hole boundary is shown as a green semi-transparent surface in the low
corona in the region detailed in Figure 8. This is the boundary between open
and closed field regions. The regions marked by A, B, and C show examples in
which the extensions of coronal holes are not connected in the photosphere, at
least by any measurable open-field corridor, but appear to connect above the
photosphere in the low corona. These regions are also indicated in Figure 8
for ease of cross-reference. Despite the fact that these coronal holes are
“disconnected” in the photosphere, they always remain topologically linked in
a singular manner with the polar coronal hole, as discussed by Titov et al.
(2011).
Finally, note that the connections of the high-$Q$ lines between the
neighborhood of the HCS and the photosphere and low corona that were
postulated by the uniqueness conjecture (Antiochos et al., 2007) are largely
present, even though the insight from these new high-resolution MHD
simulations has led us to generalize the uniqueness conjecture. We have found
that, in general, coronal hole extensions are sometimes connected to the polar
holes in the photosphere via narrow corridors, as originally postulated
(Antiochos et al., 2007), but in other instances they are disconnected in the
photosphere, but remain topologically linked to the polar holes (Titov et al.,
2011). In either case, these connections are responsible for the formation of
the S-web. It should be emphasized that in order to capture the intricate
structure of these connections, very high resolution models are required that
can incorporate some of the complexity of the photospheric magnetic carpet
fields. Given sufficient resolution, the S-web should appear as a generic
feature of all quasi-steady models, including the PFSS. In fact, the PFSS
models should be more effective than the MHD for studying the complex topology
of the S-web, because they allow for much higher spatial resolution than is
possible with an MHD code. On the other hand, for quantitative comparison with
observations, the MHD models should be more effective, because they include
the gas thermal and kinetic pressure forces and Lorentz forces that we know
are present in the real corona.
## 4 Discussion
The major conclusion from our results is that the underlying premise of the
streamer top model is valid. The slow wind is expected to originate from the
release of closed-field plasma due to the dynamic rearrangement of the open-
closed field boundary. The key new addition of our S-web model to this picture
is that the inherent complexity of the photospheric field leads to a network
of narrowly connected and disconnected coronal holes that nevertheless always
remain linked. This produces a separatrix web in the heliosphere that extends
the release of slow wind to regions that significantly depart from the HCS.
Hence, our model accounts for both the observed composition and the broad
extent of the slow wind.
One immediate prediction from the model is that the angular width of the slow
wind is determined primarily by the complexity of the flux distribution in the
photosphere. This complexity produces a very convoluted polarity inversion
line in the low corona and an intricate coronal hole pattern (Figure 5). Our
ability to identify the S-web and its manifestations rests on high-resolution
calculations that are beginning to capture the multitude of small dipoles in
the photospheric magnetic field. If the solar field were a pure dipole,
producing an inversion line that runs straight along the equator, then only
the polar coronal holes would be present and there would be no separatrix web
in the heliosphere. For this “basal” (though idealized) slow wind case, if we
assume that the dynamic broadening of the open-closed boundary at the Sun is
of order the scale of a supergranule, $\sim 30{,}000\,{\rm km}$, the angular
extent of the wind would be only of order $3^{\circ}$–$5^{\circ}$, and would
be centered about the HCS. Of course, the solar field is never a simple
dipole.
At the present time we do not know if the complexity seen in Figures 5–7 is
typical, or whether it is particular to this late declining phase of Cycle 23.
It should be noted that the present minimum appears to be somewhat different
than the previous few minima. In particular, the polar field strength is
significantly weaker (e.g., Luhmann et al., 2009).
The S-web model predicts that for time periods during which extensions of
coronal holes away from the main polar holes are less prevalent than in Cycle
23, the angular extent of the slow wind region would be smaller. In fact,
there is clear evidence from radio scintillation data (Tokumaru et al., 2010)
and recent Ulysses solar wind measurements that the Cycle 23 minimum has a
substantially broader and more structured slow wind region than that of the
previous cycle. Indeed, during the previous minimum (circa 1996), equatorial
coronal hole extensions were less common than during the recent solar minimum.
Further high-resolution numerical calculations will be needed to address this
result.
Another prediction of the model is that the slow wind region is actually a
mixture of winds. It is evident from Fig. 7 that the separatrix web is not
space-filling. There are regions within the broad S-web band where the wind
emanates from the low-latitude coronal hole extensions. These regions are
likely to have large expansion factor, so that the wind will be slow compared
to the fast wind from the polar regions, but its composition will be different
than that of closed-field plasma. Our model, therefore, naturally explains the
observed variability of the slow wind composition.
A key aspect of the S-web model that has yet to be calculated is the dynamic
release of closed-field plasma. Although our quasi-steady calculations allow
us to investigate the topology of the field, and to identify the structure of
the separatrix web in the heliosphere, they do not actually produce a slow
wind with closed-field composition. For this we need fully dynamic simulations
that include the driving due to photospheric motions (e.g., resulting from
differential rotation) and flux emergence. Such simulations are now being
performed in 3D (e.g., Edmondson et al., 2009, 2010; Linker et al., 2010) for
simplified photospheric flux distributions and driving flows. These
simulations do verify the basic idea of the S-web model that open-field
corridors will form and evolve in response to photospheric motions (Edmondson
et al., 2009). Higher resolution simulations will be needed, however, to test
the model in detail. On the other hand, it seems unlikely that dynamic
calculations with the degree of structure present in Fig. 7 will be feasible
in the near future. It is likely that a definitive treatment of the slow wind
will require the development of a statistical theory of the dynamics of the
S-web model.
This work has been supported by the NASA TR&T, SR&T, and HTP Programs. The
work has benefited greatly from the authors’ participation in the NASA TR&T
focused science team on the solar-heliospheric magnetic field. SKA thanks J.
Karpen for invaluable scientific discussions and help with the graphics.
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Figure 1: Magnetic field topology of an open field region consisting of a
large polar coronal hole and a smaller low-latitude hole connected by an open-
field corridor. The inner surface is the photosphere, with the dark gray and
bright yellow regions corresponding to open and closed field respectively. The
outer transparent surface is a radial surface in the heliosphere. The dark
green line is the polarity inversion line and the light gray arc indicates
where the open-field corridor maps to on this outer surface.
Figure 2: (Top) Open-closed magnetic field topology for a photospheric flux
distribution due to a global dipole and an equatorial dipole. The gray shaded
region indicates the polar coronal hole (the open flux region). The contours
on the inner surface indicate radial field magnitude at the photosphere. The
black lines correspond to the polarity inversion line at the photosphere and
source surface. The red lines are magnetic field lines. (Bottom) Close-up near
the solar surface of the magnetic field above.
Figure 3: As in Figure 2, but for a flux distribution that includes
additional high-latitude dipoles. Two additional polarity inversion lines can
be seen at the photosphere. The blue field lines outline an open-field
corridor. Note that the system is symmetric about the meridional plane
$\phi=0$.
Figure 4: (Top) Open field lines (green) traced from photospheric footpoints
along a line segment spanning the narrowest part of the corridor. The lines
clearly extend to high latitude above the HCS. (Bottom) Close-up near the
solar surface showing the photospheric footpoints of the corridor field lines.
Figure 5: (a) Distribution of the radial component of the magnetic field in
the photosphere that was used in the MHD simulation to predict the structure
of the corona for the August 1, 2008 eclipse, as deduced from MDI
measurements. (b) The open and closed field regions in the photosphere as
determined from the MHD solution. The polarity inversion line ($B_{r}=0$) at a
height $r=1.05R_{\odot}$ is superimposed on these images to aid in identifying
the polarity of the large-scale magnetic flux. Figure 6: (a) Plot of the
squashing factor $Q$ on a logarithmic scale in a meridional plane at the time
of the eclipse on August 1, 2008 at 10:21UT. In this view, solar north is
vertically up and the $B_{0}$ angle is zero. [At the time of the eclipse
$B_{0}=5.8^{\circ}$, so this view is slightly different than what would have
been observed.] The Sun’s surface is colored by the value of $B_{r}$ with the
same scaling as that in Fig. 5. (b) Magnetic field lines traced from the
vicinity of the limbs at the same time, showing the structure of the open and
closed field regions. Figure 7: (a) Plot of the squashing factor $Q$ in the
spherical surface $r=10R_{\odot}$ on a logarithmic scale versus longitude and
latitude. (b) Plot of $B_{r}$ in the same spherical surface. The HCS (i.e.,
the location of $B_{r}=0$) is superimposed on these images as a thick black
line. The complex structure in $Q$ in the vicinity of the HCS is produced by
the S-web. Figure 8: The variation of coronal hole shape with height above
the photosphere. The top panel shows coronal holes at the photosphere, as in
Fig. 5b. The black square shows a $100^{\circ}\times 100^{\circ}$ region
centered at longitude $75^{\circ}$ and latitude $15^{\circ}$N that was used to
compute the variation of coronal hole shape with radius in the lower panels.
Note that the extended coronal holes connect to the polar holes low in the
corona. The regions denoted by A, B, and C are cross-referenced with the
corresponding regions in Fig. 9. Figure 9: The three-dimensional shape of
the coronal hole boundary (semi-transparent surface) in the region detailed in
Figure 8, showing that some of the coronal hole extensions (blue areas on the
surface of the sphere) connect with the north polar hole low in the corona.
The top panel shows a view in which the surface is artificially stretched in
radius by a factor of 3$\times$ to show details near the photosphere. The
bottom left panel shows the same view without the radial stretching. The
bottom right panel shows the region detailed in the context of the whole Sun.
The regions denoted by A, B, and C are cross-referenced with the corresponding
regions in Fig. 8.
|
arxiv-papers
| 2011-02-17T21:25:50 |
2024-09-04T02:49:17.114155
|
{
"license": "Public Domain",
"authors": "S. K. Antiochos, Z. Miki\\'c, V. S. Titov, R. Lionello, and J. A.\n Linker",
"submitter": "Spiro K. Antiochos",
"url": "https://arxiv.org/abs/1102.3704"
}
|
1102.3951
|
# Generalized McKay Quivers, Root System and Kac-Moody Algebras
Bo Hou College of Applied Science, Beijing University of Technology, Beijing
100124, People’s Republic China houbo@emails.bjut.edu.cn and Shilin Yang
College of Applied Science, Beijing University of Technology, Beijing 100124,
People’s Republic China slyang@bjut.edu.cn
###### Abstract.
Let $Q$ be a finite quiver and $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ a finite
abelian group. Assume that $\widehat{Q}$ and $\Gamma$ is the generalized Mckay
quiver and the valued graph corresponding to $(Q,G)$ respectively. In this
paper we discuss the relationship between indecomposable
$\widehat{Q}$-representations and the root system of Kac-Moody algebra
$\mathfrak{g}(\Gamma)$. Moreover, we may lift $G$ to
$\overline{G}\subseteq\mbox{Aut}(\mathfrak{g}(\widehat{Q}))$ such that
$\mathfrak{g}(\Gamma)$ embeds into the fixed point algebra
$\mathfrak{g}(\widehat{Q})^{\overline{G}}$ and
$\mathfrak{g}(\widehat{Q})^{\overline{G}}$ as $\mathfrak{g}(\Gamma)$-module is
integrable.
###### Key words and phrases:
Generalized McKay quiver, Representation of quiver, Root system, Kac-Moody
algebra
###### 2000 Mathematics Subject Classification:
Primary 16G10, 16G20, 17B67
The second author was supported by the Science and Technology Program of
Beijing Education Committee (Grant No. KM200710005013) and Foundation of
Selected Excellent Science and Technology Activity for Returned Scholars of
Beijing.
## 1\. Introduction
Thirty years ago, McKay introduced a class of quivers, now called the McKay
quivers, for some finite subgroups of the general linear group [16]. Let
$\mathbb{C}$ denote the complex number field. McKay observed that the McKay
quiver for $G\subseteq\mathrm{SL}(2,\mathbb{C})$ is the double quiver of the
extended Dynkin quiver $\widetilde{A}_{n}$, $\widetilde{D}_{n}$,
$\widetilde{E}_{6}$, $\widetilde{E}_{7}$, $\widetilde{E}_{8}$ respectively.
Furthermore, the corresponding Dynkin diagram is the same as the one occurring
in the minimal resolution of singularities for the quotient surface
$\mathbb{C}/G$ (see [4]). McKay quiver has played an important role in many
mathematical fields such as quantum group, algebraic geometry, mathematics
physics and representation theory (see, for examples [2, 5, 9, 8, 15, 17]).
Let $V$ be a finite vector space over a field $\mathbbm{k}$ of characteristic
0 and $G\subseteq\mathrm{GL}_{\mathbbm{k}}(V)$ a finite group. Assume that
$\mathrm{T}_{\mathbbm{k}}(V)$ is the tensor algeba of $V$ over $\mathbbm{k}$.
It is well-known that the skew group algebra $\mathrm{T}_{\mathbbm{k}}(V)\ast
G$ is Morita equivalent to the path algebra $\mathbbm{k}\widehat{Q}$, where
$\widehat{Q}$ is the McKay quiver of $G$ (see [9]). In other words, the McKay
quiver realizes the Gabriel quiver of $\mathrm{T}_{\mathbbm{k}}(V)\ast G$. It
is natural to ask how to determine the Gabriel quiver of skew group algebra
$\Lambda\ast G$ for any algebra $\Lambda$. Recently, for any path algebra
$\mathbbm{k}Q$ over an algebraically closed field $\mathbbm{k}$ and a finite
group $G$ such that char$\mathbbm{k}\nmid|G|$, if the action of $G$ on
$\mathbbm{k}Q$ permutes the set of primitive idempotents and stabilizing the
vector space spanned by the arrows, Demonet in [6] has constructed a quiver
$\widehat{Q}$ such that the path algebra $\mathbbm{k}\widehat{Q}$ is Morita
equivalent to the skew group algebra $\mathbbm{k}Q\ast G$. The quiver
$\widehat{Q}$ can be viewed as a generalization of McKay quiver, which is
called the generalized McKay quiver of $(Q,G)$ in this paper.
Given a finite quiver $Q$ with an admissible automorphism ${\bf a}$. Hubery in
[11, 12] described the correspondence between dimension vectors of the
isomorphically invariant $Q$-indecomposables and the positive root system of
${\mathfrak{g}}(\Gamma)$, where $\Gamma$ is the valued graph of $(Q,{\bf a})$.
Motivated by Hubery’s work, the aim of this paper is to establish the
correspondence between the indecomposable $\widehat{Q}$-representations and
the positive roots of the symmetrizable Kac-Moody algebra
$\mathfrak{g}(\Gamma)$ of the valued graph $\Gamma$ associated to $(Q,G)$,
where $Q$ is a finite quiver and $G$ is a finite abelian automorphism group of
$\mathbbm{k}Q$. Moreover, we can lift $G$ to an automorphism group
$\overline{G}$ of Kac-Moody algebra $\mathfrak{g}:=\mathfrak{g}(\widehat{Q})$
of $\widehat{Q}$, such that $\mathfrak{g}(\Gamma)$ can be embedded into the
fixed point subalgebra $\mathfrak{g}^{\overline{G}}$. In this case, we also
show that $\mathfrak{g}^{\overline{G}}$ as a $\mathfrak{g}(\Gamma)$-module is
integrable. Compared with Hubery’s work, a more general description is given
by approach of the generalized McKay quiver.
For a finite quiver $Q=(I,E)$ and a finite abelian group
$G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ (the algebra automorphism group of
$\mathbbm{k}Q$). We always assume that the action of $G$ on $Q$ is admissible,
i.e., no arrow connects to vertices in the same orbit. Then we can get a
valued graph $\Gamma$ without loops and a generalized McKay quiver
$\widehat{Q}$ corresponding to $(Q,G)$. By [18], we can define an action of
$G$ on $\mathbbm{k}\widehat{Q}$ due to the Morita equivalence between the skew
group algebra $\mathbbm{k}Q\ast G$ and $\mathbbm{k}\widehat{Q}$. Therefore
this action induces an action on $\widehat{Q}$-representations. Let $G_{X}$ be
a complete set of left coset representatives of $H_{X}=\\{g\in
G\mid{{}^{g}X}\cong X\\}$ in $G$ for any $\widehat{Q}$-representation $X$, let
$\mathbb{Z}I$, $\mathbb{Z}\widehat{I}$ and $\mathbb{Z}\mathcal{I}$ be the root
lattice of $Q$, $\widehat{Q}$ and $\Gamma$, respectively. Applying the
equivalence between representation category of $\widehat{Q}$ and module
category of the skew group algebra $\mathbbm{k}Q\ast G$ and the fact that each
$\mathbbm{k}Q\ast G$ module as a $Q$-representation is $G$-invariant, we
define a map
$h:\quad\mathbb{Z}\widehat{I}\longrightarrow(\mathbb{Z}I)^{G}\longrightarrow\mathbb{Z}\mathcal{I}$
where $(\mathbb{Z}I)^{G}$ is the fixed point set of $\mathbb{Z}I$ under the
action of $G$. The map $h$ builds a bridge between the dimension vectors of
indecomposable $\widehat{Q}$-representations and the root system of Kac-Moody
algebra $\mathfrak{g}(\Gamma)$. The first main result of this paper is
described as follows.
###### Theorem 1.1.
Let $Q$ be a quiver without loops and with an admissible action of a finite
abelian subgroup $G\subseteq{\rm Aut}(\mathbbm{k}Q)$, where $\mathbbm{k}$ is
an algebraically closed field with ${\rm char}\mathbbm{k}\nmid|G|$. Assume
that $\Gamma$ and $\widehat{Q}$ is the valued graph and generalized McKay
quiver associated to $(Q,G)$. Then
(1) the images under $h$ of the dimension vectors of all the indecomposable
$\widehat{Q}$-representations give the positive root system of the
symmetrisable Kac-Moody algebra $\mathfrak{g}(\Gamma)$;
(2) for each positive real root $\alpha$ of $\mathfrak{g}(\Gamma)$, let $X$ be
a $\widehat{Q}$-representation such that $h({\bf dim}X)=\alpha$. Then there
are $|G_{X}|$ indecomposable $\widehat{Q}$-representations (up to isomorphism)
such that their dimension vectors under $h$ are $\alpha$.
The proof of this theorem is based on understanding the relationship among
indecomposable $\widehat{Q}$-representations, indecomposable $\mathbbm{k}Q\ast
G$-modules and indecomposable $G$-invariant $Q$-representations. In the proof,
we also need the dual between $(Q,G)$ and $(\widehat{Q},G)$. This duality is
first discussed in [18] for a finite quiver with an automorphism. Here we give
a general and strict proof by the generalized McKay quiver.
Next we consider the relationship between Kac-Moody algebra
$\mathfrak{g}(\Gamma)$ and the fixed point subalgebra
$\mathfrak{g}^{\overline{G}}$. The action of $G$ on $\widehat{Q}$ naturally
induces an action on the derived algebra $\mathfrak{g}^{\prime}$ of
$\mathfrak{g}$. Let $\Omega=\\{g_{1},g_{2},\cdots,g_{n}\\}$ be a set of
generators of $G$. Following from [14], we lift $G$ to
$\overline{G}\subseteq\mbox{Aut}(\mathfrak{g})$ corresponding to a family of
linear maps
$\\{\psi_{i}:=\psi_{g_{i}}:\mathfrak{h}/\mathfrak{h}^{\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$, where $\mathfrak{c}$ is the center of $\mathfrak{g}$,
$\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ is the Cartan subalgebra of
$\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ respectively. Denote by $C$ the
symmetrisable generalized Cartan matrix of the valued graph $\Gamma$. Then, we
can give a realization
$(\mathcal{H}^{\overline{G}},\\{\epsilon_{i}\\},\\{h_{i}\\})$ of $C$ by the
fixed point set $\mathfrak{h}^{\overline{G}}$ of $\mathfrak{h}$, and we obtain
that
###### Theorem 1.2.
For the lifting $\overline{G}$ of $G$ corresponding to
$\\{\psi_{i}:\mathfrak{h}/\mathfrak{h}^{\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$ such that
$\psi_{i}\big{(}(\mathcal{H}+\mathfrak{h}^{\prime})/\mathfrak{h}^{\prime}\big{)}=0$,
there is a monomorphism
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}.$
Moreover this monomorphism endows $\mathfrak{g}^{\overline{G}}$ with an
integrable $\mathfrak{g}(\Gamma)$-module structure under the adjoint action of
$\mathfrak{g}(\Gamma)$. In particular, if $Q$ is a finite union of Dynkin
quivers, then $\mathfrak{g}(\Gamma)\cong\mathfrak{g}^{\overline{G}}$ as Lie
algebras.
In the end of this paper, two examples are given to elucidate our results.
Throughout this paper, let $\mathbbm{k}$ denote an algebraic closed field and
$\mathbb{Z}$ denote the set of integers. We denote by $G$ the finite group
such that char$\mathbbm{k}\nmid|G|$, denote by ${\bf mod}$-$\Lambda$ the
category of (right) $\Lambda$-modules for any $\mathbbm{k}$-algebra $\Lambda$.
## 2\. Preliminaries
2.1. Recall that a quiver $Q=(I,E)$ is an oriented graph with $I$ the set of
vertices and $E$ the set of arrows. A quiver $Q$ is said to be finite if $I$
and $E$ are all finite set. An arrow in $Q$ is called a loop if its staring
vertex coincides with its terminating vertex. In this paper we only consider a
finite quiver without loops. Therefore we have a path algebra $\mathbbm{k}Q$
for a quiver $Q$ (see [1, 3]).
A representation $X=(X_{i},X_{\alpha})$ of the quiver $Q=(I,E)$ consists of a
family of $\mathbbm{k}$-vector spaces $X_{i}$ for $i\in I$, together with a
family of $\mathbbm{k}$-linear maps $X_{\alpha}:X_{i}\rightarrow X_{j}$ for
$\alpha:i\rightarrow j$ in $E$. Given two representations $X$ and $Y$ of $Q$,
a morphism $\varphi:X\rightarrow Y$ is given by a family of
$\mathbbm{k}$-linear maps $\varphi_{i}:X_{i}\rightarrow Y_{i}~{}(i\in I)$ such
that $\varphi_{j}\circ X_{\alpha}=Y_{\alpha}\circ\varphi_{i}$ for each arrow
$\alpha:i\rightarrow j$. It is well-known that the category of representations
of $Q$ is naturally equivalent to the category of $\mathbbm{k}Q$-modules (see
[1, 3]). Thus we always identify a $\mathbbm{k}Q$-module $X$ with a
$Q$-representation $(X_{i},X_{\alpha})$ in this paper.
2.2. Assume that $\Lambda$ is a $\mathbbm{k}$-algebra and $G$ acts on
$\Lambda$, the skew group algebra of $\Lambda$ under the action of $G$ is by
definition the $\mathbbm{k}$-algebra whose underlying $\mathbbm{k}$-vector
space is $\Lambda\otimes_{\mathbbm{k}}\mathbbm{k}[G]$ and whose multiplication
is defined by
$(\lambda\otimes g)(\lambda^{\prime}\otimes g^{\prime})=\lambda
g(\lambda^{\prime})\otimes gg^{\prime}$
for all $\lambda,\lambda^{\prime}\in\Lambda$ and $g,g^{\prime}\in G$ (see
[18]). For convenience, we denote this algebra by $\Lambda\ast G$, denote the
element $\lambda\otimes g$ in $\Lambda\ast G$ by $\lambda g$. Note that
$\Lambda$ and $\mathbbm{k}[G]$ can be viewed as subalgebras of $\Lambda\ast
G$.
Let $\Lambda=\mathbbm{k}Q$ be the path algebra for a quiver $Q=(I,E)$. Assume
that $G$ acts on $\mathbbm{k}Q$ permuting the set of primitive idempotents
$\\{e_{i}\mid i\in I\\}$ and stabilizing the vector space spanned by the
arrows. Let $\mathcal{I}$ denote a set of representatives of class of $I$
under the action of $G$. For any $i\in I$, let $G_{i}$ denote the subgroup of
$G$ stabilizing $e_{i}$, For each $i\in I$, there exist some $g\in G$ such
that $g^{-1}(i)\in\mathcal{I}$. We fix such a $g$ and denote it by
$\kappa_{i}$. Let $\mathcal{O}_{i}$ be the orbit of $i$ under the action of
$G$. For $(i,j)\in\mathcal{I}^{2}$, $G$ acts on
$\mathcal{O}_{i}\times\mathcal{O}_{j}$ by the diagonal action. A set of
representatives of the classes of this action will be denoted by
${\mathcal{F}}_{ij}$.
For $i,j\in I$, we denote by $E_{ij}\subseteq\mathbbm{k}Q$ the vector space
spanned by the arrows from $i$ to $j$ and regard it as a left and right
$\mathbbm{k}[G_{ij}]:=\mathbbm{k}[G_{i}\cap G_{j}]$-module by restricting the
action of $G$. In [6] Demonet defined the quiver
$\widehat{Q}=(\widehat{I},\widehat{E})$ as follows
$\widehat{I}=\bigcup_{i\in\mathcal{I}}\\{i\\}\times\mbox{irr}G_{i},$
where $\mbox{irr}G_{i}$ is a set of representatives of isomorphism classes of
irreducible representations of $G_{i}$. The set of arrows of $\widehat{Q}$
from $(i,\rho)$ to $(j,\sigma)$ is a basis of
$\bigoplus_{(i^{\prime},j^{\prime})\in{\mathcal{F}}_{ij}}\mbox{Hom}_{\mathbbm{k}[G_{i^{\prime}j^{\prime}}]}\left((\rho\cdot\kappa_{i^{\prime}})|_{G_{i^{\prime}j^{\prime}}},~{}(\sigma\cdot\kappa_{j^{\prime}})|_{G_{i^{\prime}j^{\prime}}}\otimes_{\mathbbm{k}}E_{i^{\prime}j^{\prime}}\right),$
where the representation $\rho\cdot\kappa_{i^{\prime}}$ of $G_{i^{\prime}}$ is
the same as $\rho$ as a $\mathbbm{k}$-vector space, and
$(\rho\cdot\kappa_{i^{\prime}})g=\rho\kappa_{i^{\prime}}g\kappa_{i^{\prime}}^{-1}$
for each $g\in
G_{i^{\prime}}=\kappa_{i^{\prime}}^{-1}G_{i}\kappa_{i^{\prime}}$. Demonet
yielded the following theorem.
###### Theorem 2.1.
(see [6]) The category mod-$\mathbbm{k}\widehat{Q}$ is equivalent to the
category mod-$\mathbbm{k}Q\ast G$.
In particular, if the quiver $Q$ is a singular vertex with $m$ loops, we can
view $G$ as a subgroup of ${\bf GL}_{m}(\mathbbm{k})$. Then the quiver
$\widehat{Q}$ is just the McKay quiver of $G$. Thus, we view $\widehat{Q}$ as
a generalization of McKay quiver in general. Furthormore, for any factor
algebra $\mathbbm{k}Q/J$, the shew group algebra $(\mathbbm{k}Q/J)\ast G$ is
Morita equivalent to a factor algebra of $\mathbbm{k}\widehat{Q}$. This
implies that the generalized McKay quiver can realize the Garbiel quiver of
$\Lambda\ast G$ for any basic algebra $\Lambda$.
2.3. For a quiver $Q=(I,E)$, there is a corresponding symmetric generalized
Cartan matrix $A=(a_{ij})$ indexed by $I$ with entries
$a_{ij}=\left\\{\begin{array}[]{ll}2,&i=j;\\\ -|\\{\mbox{edges between
vertices }i\mbox{ and }j\\}|,&i\neq j.\end{array}\right.$
It is obvious that $A$ is independent of the orientation of $Q$.
Denote by $\mathfrak{g}(Q)$ for the associated symmetric Kac-Moody algebra
corresponding to $A$ with the simple root set $\Pi=\\{\varepsilon_{i}\mid i\in
I\\}$ and root system $\Delta_{Q}$. The root lattice $\mathbb{Z}I$ of $Q$ is
the free abelian group on $\Pi$, with the partially order such that
$\alpha=\sum_{i\in I}\alpha_{i}\varepsilon_{i}\geq 0$ if and only if
$\alpha_{i}\geq 0$ for all $i\in I$. We endow $\mathbb{Z}I$ with a symmetric
bilinear form $(-,-)_{Q}$ via $(\varepsilon_{i},\varepsilon_{j})_{Q}=a_{ij}$.
Then, for each vertex $i\in I$, we have a reflection
$r_{i}:\alpha\mapsto\alpha-(\alpha,\varepsilon_{i})_{Q}\varepsilon_{i}$. These
reflections generate the Weyl group $\mathcal{W}(Q)$ of $Q$. The real roots of
$Q$ are given by the images under $\mathcal{W}(Q)$ of the simple roots
$\varepsilon_{i}$ and the imaginary roots are given by $\pm$ the images under
$\mathcal{W}(Q)$ of the fundamental set
$F_{Q}:=\\{\alpha>0\mid(\alpha,\varepsilon_{i})_{Q}\leq 0\mbox{ for all
}i\mbox{ and the support of }\alpha\mbox{ is connected}\\}.$
Suppose that the action of $G$ on path algebra $\mathbbm{k}Q$ permutes the set
of primitive idempotents. The action of $G$ is said to be admissible if no
arrow connects to vertices in the same $G$-orbit. For any quiver $Q$ with an
admissible action of $G$, we can construct a symmetric matrix $B=(b_{ij})$
indexed by $\mathcal{I}$, where
$b_{ij}=\left\\{\begin{array}[]{ll}2|\mathcal{O}_{i}|,&i=j;\\\
-|\\{\mbox{edges between vertices in }\mathcal{O}_{i}\mbox{ and
}\mathcal{O}_{j}\\}|,&i\neq j.\end{array}\right.$
Let $d_{i}:=b_{ii}/2=|\mathcal{O}_{i}|$ and $D=\mbox{diag}(d_{i})$. Then
$C=(c_{ij})=D^{-1}B$ is a symmetrisable generalized Cartan matrix indexed by
$\mathcal{I}$. It is well-known that there is a unique valued graph $\Gamma$
corresponding to the matrix $C$ by [7]. The valued graph $\Gamma$ has the
vertex set $\mathcal{I}$ and an edge $i$—-$j$ equipped with the ordered pair
$(|c_{ji}|,|c_{ij}|)$ whenever $c_{ij}\neq 0$. Since the action of $G$ is
admissible, $\Gamma$ has no loops. For each connected component
$\Gamma^{\prime}$ of the graph $\Gamma$, we always take the representative set
$\mathcal{I}$ such that the underlying graph of the full subquiver generated
by the vertices in $\Gamma^{\prime}$ is connected.
Denote by $\mathfrak{g}(\Gamma)$ for the associated symmetric Kac-Moody
algebra corresponding to $C$. The simple root set and root system of $\Gamma$
are denoted by $\Pi_{\Gamma}=\\{\overline{\varepsilon}_{i}\mid
i\in\mathcal{I}\\}$ and $\Delta_{\Gamma}$. Let $\mathbb{Z}\mathcal{I}$ denote
the root lattice of $\Gamma$. There is a symmetric bilinear form
$(-,-)_{\Gamma}$ determined by $B$ on $\mathbb{Z}\mathcal{I}$ such that
$(\overline{\varepsilon}_{i},\overline{\varepsilon}_{j})_{\Gamma}=b_{ij}$, and
a reflection $\gamma_{i}$ on $\mathbb{Z}\mathcal{I}$ defined by
$\gamma_{i}:\alpha\mapsto\alpha-\frac{1}{d_{i}}(\alpha,\overline{\varepsilon}_{i})_{\Gamma}\overline{\varepsilon}_{i}$
for each $i\in{\mathcal{I}}$. These reflections generate the Weyl group
$\mathcal{W}(\Gamma)$ of $\Gamma$. Similarly, we have the real roots and the
imaginary roots associated to $\Gamma$ (see [13]).
## 3\. Proof of Theorem 1.1
From now on, unless otherwise stated we fix a finite group
$G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ and assume that the action of $G$ is
admissible. Let $\widehat{Q}$ and $\Gamma$ be the generalized Mckay quiver and
the valued graph corresponding to $(Q,G)$. In this section, we show that the
correspondence between indecomposable representations of $\widehat{Q}$ and the
positive root system of $\Gamma$.
3.1. The group $G$ acts naturally on the root lattice $\mathbb{Z}I$, i.e.,
$g(\varepsilon_{i})=\varepsilon_{g(i)}$ for any $g\in G$. It is easy to check
that this action preserves the partial order $\geq$ and the bilinear form
$(-,-)_{Q}$ is $G$-invariant. Let
$(\mathbb{Z}I)^{G}:=\\{\alpha\in\mathbb{Z}I\mid g(\alpha)=\alpha\mbox{ for any
}g\in G\\}.$
There is a canonical bijection
$f:\quad(\mathbb{Z}I)^{G}\longrightarrow\mathbb{Z}\mathcal{I}$
given by
$f\Big{(}\sum_{i\in
I}\alpha_{i}\varepsilon_{i}\Big{)}=\sum_{i\in\mathcal{I}}\alpha_{i}\overline{\varepsilon}_{i}.$
The admissibility of the action of $G$ implies that the reflections $r_{i}$
and $r_{j}$ commute whenever $i$ and $j$ lie in the same $G$-orbit. Therefore
the element
$S_{i}:=\prod_{i^{\prime}\in\mathcal{O}_{i}}r_{i^{\prime}}\in\mathcal{W}(Q)$
is well-defined for any $i\in\mathcal{I}$. Note that $g\circ
r_{i}=r_{g(i)}\circ g$ for any $g\in G$, we have $S_{i}\in
C_{G}(\mathcal{W}(Q))$, the set of elements in the Weyl group commuting with
the action of $G$. By induction on the length of the element in
$C_{G}(\mathcal{W}(Q))$, it is easy to check that $C_{G}(\mathcal{W}(Q))$ is
generated by $S_{i}$, $i\in\mathcal{I}$.
Similar to [12, Lemma 3], we have
###### Lemma 3.1.
For any $\alpha,\beta\in(\mathbb{Z}I)^{G}$, we have
(1) $(\alpha,\beta)_{Q}=(f(\alpha),f(\beta))_{\Gamma}$;
(2) $f(S_{i}(\alpha))=\gamma_{i}(f(\alpha))\in\mathbb{Z}\mathcal{I}$ for
$i\in\mathcal{I}$.
(3) The map $\gamma_{i}\mapsto S_{i}$ induces a group isomorphism
$\mathcal{W}(\Gamma)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}C_{G}(\mathcal{W}(Q))$.
###### Proof.
(1) Set
$\varepsilon^{i}:=\sum_{i^{\prime}\in\mathcal{O}_{i}}\varepsilon_{i^{\prime}}$.
Then $\\{\varepsilon^{i}\mid i\in\mathcal{I}\\}$ is a basis of
$(\mathbb{Z}I)^{G}$. Since
$(\varepsilon^{i},\varepsilon^{j})_{Q}=\sum_{i^{\prime}\in\mathcal{O}_{i}\atop
j^{\prime}\in\mathcal{O}_{j}}a_{i^{\prime}j^{\prime}}=b_{ij}=(\overline{\varepsilon}_{i},\overline{\varepsilon}_{j})_{\Gamma}$
for any $i,j\in\mathcal{I}$, (1) is obvious.
(2) Since the bilinear form $(-,-)_{Q}$ is $G$-invariant, we have
$S_{i}(\alpha)=\alpha-\sum_{i^{\prime}\in\mathcal{O}_{i}}(\alpha,\varepsilon_{i^{\prime}})_{Q}\varepsilon_{i^{\prime}}=\alpha-\sum_{i^{\prime}\in\mathcal{O}_{i}}\frac{1}{d_{i}}\big{(}\alpha,\sum_{j\in\mathcal{O}_{i}}\varepsilon_{j}\big{)}_{Q}\varepsilon_{i^{\prime}}=\alpha-\frac{1}{d_{i}}(f(\alpha),\overline{\varepsilon}_{i^{\prime}})_{\Gamma}\varepsilon^{i}$
by (1). We obtain that
$f(S_{i}(\alpha))=f(\alpha)-\frac{1}{d_{i}}(f(\alpha),\overline{\varepsilon}_{i^{\prime}})_{\Gamma}\overline{\varepsilon}_{i}=\gamma_{i}(f(\alpha)).$
(3) By the result of (2), it is easy to check that $\gamma_{i}$ and $S_{i}$
satisfy the same relations. Thus $\mathcal{W}(\Gamma)\cong
C_{G}(\mathcal{W}(Q))$. ∎
For a given $\alpha\in\mathbb{Z}I$, let $H_{\alpha}=\\{g\in G\mid
g(\alpha)=\alpha\\}$. Then $H_{\alpha}$ is a subgroup of $G$. We denote by
$G_{\alpha}$ a complete set of left coset representatives of $H_{\alpha}$ in
$G$, and let
$\Sigma(\alpha):=\sum_{g\in G_{\alpha}}g(\alpha).$
Obviously, $\Sigma(\alpha)\in(\mathbb{Z}I)^{G}$ and we have
###### Lemma 3.2.
The map $\alpha\mapsto f(\sigma(\alpha))$ induces a surjection
$\pi:\Delta_{Q}\rightarrow\Delta_{\Gamma}$. Moreover, if $f(\sigma(\alpha))$
is a real root, $\alpha$ has to be real and unique up to $G$-orbit.
###### Proof.
First, for any $\omega\in C_{G}(\mathcal{W}(Q))$, we have
$H_{\alpha}=H_{\omega(\alpha)}$ since the action of $C_{G}(\mathcal{W}(Q))$
and the action of $G$ on $\mathbb{Z}I$ is commutative. Thus we can take
$G_{\alpha}=G_{\omega(\alpha)}$ for any $\omega\in C_{G}(\mathcal{W}(Q))$.
We now consider $\beta:=\omega^{\prime}(f(\Sigma(\alpha)))$ with
$\omega^{\prime}\in\mathcal{W}(\Gamma)$. Let $\omega\in C_{G}(\mathcal{W}(Q))$
be the element corresponding to $\omega^{\prime}$ under the isomorphism in
Lemma 3.1(3). Then $\beta=f(\omega(\Sigma(\alpha)))=f(\Sigma(\omega(\alpha)))$
has connected support since the support of $\alpha$ is connected. It is either
positive or negative since $\Sigma$ preserves the partial order $\geq$. Denote
by $\mathcal{O}_{\beta}$ the orbit of $\beta$ under the action of
$\mathcal{W}(\Gamma)$. Then
* •
if all elements in $\mathcal{O}_{\beta}$ are positive, the element in
$\mathcal{O}_{\beta}$ with minimal height lies in $F_{\Gamma}$;
* •
if all elements in $\mathcal{O}_{\beta}$ are negative, the element in
$\mathcal{O}_{\beta}$ with maximal height lies in $-F_{\Gamma}$;
* •
otherwise, there exists a positive number $m$ and $i\in\mathcal{I}$ such that
$m\overline{\varepsilon}_{i}\in\mathcal{O}_{\beta}$.
In the last case, we have $\omega(\alpha)=m\varepsilon_{i^{\prime}}$ for some
$\omega\in\mathcal{W}(Q)$, $i^{\prime}\in\mathcal{O}_{i}$. But
$\omega(\alpha)\in\Delta_{Q}$, we must have $m=1$ and so that
$\overline{\varepsilon}_{i}\in\mathcal{O}_{\beta}$. Thus $\beta$ is a root of
$\Gamma$ and $\pi:\Delta_{Q}\rightarrow\Delta_{\Gamma}$, $\alpha\mapsto
f(\Sigma(\alpha))$ is well-defined.
Now, we prove that the map $\pi$ is surjective. Here we only need to show that
$F_{\Gamma}$ lies in the image of $\pi$. For any $\beta\in F_{\Gamma}$,
$\gamma:=f^{-1}(\beta)$ satisfies
$0\geq(\beta,\overline{\varepsilon}_{i})_{\Gamma}=(\gamma,\Sigma(\varepsilon_{i^{\prime}}))_{Q}=\sum_{g\in
G_{\varepsilon_{i^{\prime}}}}(\gamma,g(\varepsilon_{i^{\prime}}))_{Q}=d_{i}(\gamma,\varepsilon_{i^{\prime}})_{Q}$
for any $i\in\mathcal{I}$ and $i^{\prime}\in\mathcal{O}_{i}$. Thus any
connected component $\alpha$ of $\gamma$ lies in $F_{Q}$ and
$\Sigma(\alpha)=\gamma$. By Lemma 3.1(3) we get the proof. ∎
For any $g\in G$, we have an additive autoequivalence functor
$\displaystyle F_{g}:\quad\mbox{{\bf mod}-}\mathbbm{k}Q$
$\displaystyle\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}Q$
$\displaystyle M$ $\displaystyle\mapsto\qquad{{}^{g}M}$
where the $\mathbbm{k}Q$-module ${{}^{g}M}$ is defined by taking the same
underlying vector space as $M$ with the action $m\cdot\lambda=mg(\lambda)$ for
$m\in M$ and $\lambda\in\mathbbm{k}Q$, and $F_{g}(\psi)=\psi$ for any
homomorphism $\psi:M\rightarrow N$. Let $(M_{i},~{}M_{\alpha})_{i\in
I,\alpha\in E}$ be the $Q$-representation corresponding to $M$. Then the
$Q$-representation ${}^{g}M$ is $(^{g}X_{i},{{}^{g}X}_{\alpha})_{i\in
I,\alpha\in E}$, where ${}^{g}X_{i}=X_{g(i)}$ and
${}^{g}X_{\alpha}=\sum_{\beta}\zeta_{\beta}X_{\beta}$ if
$g(\alpha)=\sum_{\beta}\zeta_{\beta}\beta$, $\beta\in E$,
$\zeta_{\beta}\in\mathbbm{k}$.
A $\mathbbm{k}Q$-module $M$ is said to be $G$-invariant if $F_{g}(M)\cong M$
for any $g\in G$, a $G$-invariant $\mathbbm{k}Q$-module $M$ is said to be
indecomposable $G$-invariant if $M$ is non-zero and $M$ cannot be written as a
direct sum of two non-zero $G$-invariant $\mathbbm{k}Q$-modules. It is known
that $\mathbbm{k}Q$-module $M$ has a $\mathbbm{k}Q\ast G$-module structure if
and only if $M$ is $G$-invariant, and the full subcategory of $\mbox{{\bf
mod}-}\mathbbm{k}Q$ generated by the $G$-invariant $\mathbbm{k}Q$-module is
also a Krull-Schmidt category (see [10]).
For a given $\mathbbm{k}Q$-module $M$, we let $H_{M}:=\\{g\in G\mid
F_{g}(M)\cong M\\}$ and $G_{M}$ be a complete set of left coset
representatives of $H_{M}$ in $G$. Then for each $\mathbbm{k}Q$-module $M$, we
define a $G$-invariant $\mathbbm{k}Q$-module
$\sum(M):=\bigoplus_{g\in G_{M}}{{}^{g}M}.$
It is easy to see that each $G$-invariant $\mathbbm{k}Q$-module has this form.
For each $\mathbbm{k}Q$-module $M$, we denote the dimension vector of $M$ by
the linear combination ${\bf dim}X:=\sum_{i\in
I}\mbox{dim}X_{i}\,\varepsilon_{i}\in\mathbb{Z}I$. It is easy to see that
${\bf dim}F_{g}(M)=g({\bf dim}M)$ for any $g\in G$ and $M\in\mbox{{\bf
mod}-}\mathbbm{k}Q$.
###### Proposition 3.3.
For any indecomposable $G$-invariant $\mathbbm{k}Q$-module $M$, $f({\bf
dim}M)$ is a root of $\Gamma$. Moreover, for any positive real root $\beta$ of
$\Gamma$, there is a unique (up to isomorphism) indecomposable $G$-invariant
$\mathbbm{k}Q$-module $M$ with $\frac{1}{2}({\bf dim}M,{\bf dim}M)_{Q}$
indecomposable summands $($as $\mathbbm{k}Q$-module$)$ such that $f({\bf
dim}M)=\beta$.
###### Proof.
Let $N$ be an indecomposable $\mathbbm{k}Q$-module and $\alpha:={\bf dim}N$.
Then $\sum(N)$ is an indecomposable $G$-invariant $\mathbbm{k}Q$-module with
dimension vector $\sum_{g\in G_{N}}g(\alpha)$. We claim that
$\sum_{g\in G_{N}}g(\alpha)=m\Sigma(\alpha)$
for some positive integer $m$. Indeed, since $H_{N}\subseteq H_{\alpha}$, we
have $|H_{\alpha}|=m|H_{N}|$ and so that $|G_{N}|=m|G_{\alpha}|$ for some
positive integer $m$. Note that
$\sum_{g\in G_{\alpha}}g(\alpha)=\frac{\sum_{g\in
G}g(\alpha)}{|H_{\alpha}|}\quad\hbox{ and }\quad\sum_{g\in
G_{N}}g(\alpha)=\frac{\sum_{g\in G}g(\alpha)}{|H_{N}|},$
we obtain that
${\bf dim}\sum(N)=\sum_{g\in G_{N}}g(\alpha)=m\sum_{g\in
G_{\alpha}}g(\alpha)=m\Sigma(\alpha).$
In particular, if $\alpha$ is a real root of $Q$, then $H_{N}=H_{\alpha}$ and
so that we take $G_{N}=G_{\alpha}$ in this case. Therefore, $f({\bf
dim}\sum(N))\in\Delta_{\Gamma}$. Note that for every indecomposable
$G$-invariant $\mathbbm{k}Q$-module $M$, there is an indecomposable
$\mathbbm{k}Q$-module $N$ such that $M\cong\sum(N)$, we get $f({\bf
dim}M)\in\Delta_{\Gamma}$.
If $\beta:=f({\bf dim}M)$ is a real root with $f({\bf
dim}M)=\omega^{\prime}(\overline{\varepsilon}_{i})$ for some
$\omega^{\prime}\in\mathcal{W}(\Gamma)$ and $i\in\mathcal{I}$, then ${\bf
dim}M=\omega(\Sigma(\varepsilon_{i^{\prime}}))=\Sigma(\omega(\varepsilon_{i^{\prime}}))$
for any $i^{\prime}\in\mathcal{O}_{i}$, where $\omega\in
C_{G}(\mathcal{W}(Q))$ corresponding to $\omega^{\prime}$, by the proof of
Lemma 3.2. Denote by $N$ the unique indecomposable $\mathbbm{k}Q$-module with
${\bf dim}N=\omega(\varepsilon_{i^{\prime}})$, then $M=\sum(N)$ is the unique
indecomposable $G$-invariant $\mathbbm{k}Q$-module satisfying ${\bf
dim}M=\omega(\Sigma(\varepsilon_{i^{\prime}}))$ and $M$ is independent on the
taking of $i^{\prime}\in\mathcal{O}_{i}$. Finally, note that
$\frac{1}{2}({\bf dim}M,{\bf
dim}M)_{Q}=\frac{1}{2}(\Sigma(\varepsilon_{i^{\prime}}),\Sigma(\varepsilon_{i^{\prime}}))_{Q}=d_{i}=|G_{\varepsilon_{i^{\prime}}}|=|G_{N}|,$
we are done. ∎
We suppose now that $G$ is abelian and let
$e:=\sum_{i\in\mathcal{I}}e_{i}\in\mathbbm{k}Q\subseteq\mathbbm{k}Q\ast G,$
where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to
vertex $i\in I$. By the proof of [6, Theorem 1], we know that
$\mathbbm{k}Q\ast G$ is Morita equivalent to $e\mathbbm{k}Q\ast Ge$ and
$e\mathbbm{k}Q\ast Ge\cong\mathbbm{k}\widehat{Q}$. Thus we view the functor
$\displaystyle E:\quad\mbox{{\bf mod}-}\mathbbm{k}Q\ast G$
$\displaystyle\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}\widehat{Q}$
$\displaystyle M$ $\displaystyle\mapsto\qquad eM$
as the equivalence functor between ${\bf mod}$-$\mathbbm{k}Q\ast G$ and ${\bf
mod}$-$\mathbbm{k}\widehat{Q}$. Denote by
$\begin{array}[]{ll}F:=\mathbbm{k}Q\ast G\otimes_{\mathbbm{k}Q}-:&\mbox{{\bf
mod}-}\mathbbm{k}Q\quad\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}Q\ast
G\\\ H:=\mbox{Res}|_{\mathbbm{k}Q}:&\mbox{{\bf mod}-}\mathbbm{k}Q\ast
G\quad\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}Q\end{array}$
Following from [6, Theorem 1.1], $(H,F)$ and $(F,H)$ are adjoint pairs.
Moreover, for any $\mathbbm{k}\widehat{Q}$-module $X$, $HE^{-1}(X)$ is a
$G$-invariant $\mathbbm{k}Q$-module and there is a $\mathbbm{k}Q$-module $M$
such that $HE^{-1}(X)\cong\sum(M)$, where $E^{-1}$ is the quasi-inverse of
$E$. Identifying $X$ with a $\widehat{Q}$-representation
$(X_{i\rho},X_{\alpha})$, we have
$\sum_{\rho\in{\rm irr}G_{i}}X_{i\rho}\cong
e_{i}HE^{-1}(X)e_{i}\cong\bigoplus_{g\in G_{M}}(^{g}M)_{i}.$
Suppose ${\bf
dim}X:=\sum_{(i\rho)\in\widehat{I}}\alpha_{i\rho}\varepsilon_{i\rho}$, then
$\sum_{\rho\in{\rm irr}G_{i}}\alpha_{i\rho}=\sum_{g\in
G_{M}}\mbox{dim}(^{g}M)_{i}=f\Big{(}{\bf dim}\sum(M)\Big{)}_{i}.$
Therefore, the Moriat equivalence and the restriction functor induce a map
$h:\quad\mathbb{Z}\widehat{I}\longrightarrow\mathbb{Z}\mathcal{I}$
given by $h(\alpha)_{i}=\sum_{\rho\in{\rm
irr}G_{i}}\alpha_{i\rho}\overline{\varepsilon}_{i}$ for any
$\alpha=\sum_{(i\rho)\in\widehat{I}}\alpha_{i\rho}\varepsilon_{i\rho}\in\mathbb{Z}\widehat{I}$.
The restriction of $h$ to the root system $\Delta_{\widehat{Q}}$ is also
denoted by $h$. Then $h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is
well-defined since $X$ is an indecomposable $\mathbbm{k}\widehat{Q}$-module if
and only if $M$ is an indecomposable $\mathbbm{k}Q$-module. By Proposition
3.3, we have
###### Corollary 3.4.
For any indecomposable $\widehat{Q}$-representation $X$, $h({\bf dim}X)$ is a
positive root of $\Gamma$.
Up to now, we have obtained the map
$h:\mathbb{Z}\widehat{I}\rightarrow\mathbb{Z}\mathcal{I}$ and have shown the
half of Theorem 1.1(1). Before completing the proof of Theorem 1.1, we should
define an action of $G$ on $\mathbbm{k}\widehat{Q}$ and give the dual between
$(Q,G)$ and $(\widehat{Q},G)$. In the following subsection, we first describe
the duality of $(Q,G)$.
3.2. We write the abelian group $G$ as the product of some finite cyclic
group, i.e.,
$G=\langle g_{1}\rangle\times\langle g_{2}\rangle\times\cdots\times\langle
g_{n}\rangle,$
where the order of $g_{i}$ is $m_{i}$ for $1\leq i\leq n$. Then
$|G|=m_{1}m_{2}\cdots m_{n}$.
We now define an action of $G$ on $\widehat{Q}$. Since $G$ is abelian, all the
characters $\chi$ of $G$ are linear. The set of all the characters of $G$ is
an abelian group with the multiplication
$\chi\chi^{\prime}(g)=\chi(g)\chi^{\prime}(g),$
for all $g\in G$. We denote this group by $\widetilde{G}$. Setting
$\varphi:G\rightarrow\widetilde{G}$ by
$\varphi(g)=\chi_{g},\quad\chi_{g}(g^{\prime})=\xi_{1}^{t_{1}s_{1}}\xi_{2}^{t_{2}s_{2}}\cdots\xi_{n}^{t_{n}s_{n}}$
if $g=g_{1}^{t_{1}}g_{2}^{t_{2}}\cdots g_{n}^{t_{n}}$ and
$g^{\prime}=g_{1}^{s_{1}}g_{2}^{s_{2}}\cdots g_{n}^{s_{n}}$, where $\xi_{i}$
is a primitive $m_{i}$-th root of unity. It is easy to see that $\varphi$ is a
group isomorphism. By [18], we define a linear action of $G$ on
$\mathbbm{k}Q\ast G$ by setting
$g(\lambda h)=\chi_{g}(h)\lambda h,$
for all $g\in G,\lambda h\in\mathbbm{k}Q\ast G$. Then
$G\subseteq\mbox{Aut}(\mathbbm{k}Q\ast G)$. By [18, Proposition 5.1], we have
###### Proposition 3.5.
The map $\psi:(\mathbbm{k}Q\ast G)\ast G\rightarrow{\rm
End}_{\mathbbm{k}Q}(\mathbbm{k}Q\ast G)$ defined by
$\psi(\lambda gh)(\mu h^{\prime})=\chi_{h}(h^{\prime})\lambda g\mu h^{\prime}$
is an algebra isomorphism. It follows that $(\mathbbm{k}Q\ast G)\ast G$ is
Morita equivalent to $\mathbbm{k}Q$.
Since $e\mathbbm{k}Q\ast Ge\cong\mathbbm{k}\widehat{Q}$ and the action of $G$
on $\mathbbm{k}Q\ast G$ stabilizes $e$, the action of $G$ on $\mathbbm{k}Q\ast
G$ naturally induces an action of $G$ on $\mathbbm{k}\widehat{Q}$ such that
$G\subseteq\mbox{Aut}(\mathbbm{k}\widehat{Q})$. Therefore, we get a skew group
algebra $\mathbbm{k}\widehat{Q}\ast G$ under this action. Let
$\widehat{\widehat{Q}}$ be the generalized Mckay quiver of $(\widehat{Q},G)$.
Then, there is a Morita equivalence between $\mathbbm{k}\widehat{Q}\ast G$ and
$\mathbbm{k}\widehat{\widehat{Q}}$ by Theorem 2.1.
###### Proposition 3.6.
Let $\widehat{Q}$ be the generalized McKay quiver of $(Q,G)$ under the action
of $G$ defined as above. Then the generalized McKay quiver
$\widehat{\widehat{Q}}$ of $(\widehat{Q},G)$ coincides with $Q$.
Thus we get the dual between $(Q,G)$ and $(\widehat{Q},G)$. Now, for the
relationship between quivers $Q$ and $\widehat{Q}$, and the action of $G$ on
$\widehat{Q}$, we give some more description. Note that the stabilizer $G_{i}$
of $i\in I$ has the form
$G_{i}=\langle g_{1}^{d_{i_{1}}}\rangle\times\langle
g_{2}^{d_{i_{2}}}\rangle\times\cdots\times\langle g_{n}^{d_{i_{n}}}\rangle,$
where
$\nu_{i_{j}}:=|\langle
g_{j}^{d_{i_{j}}}\rangle|=\frac{m_{i}}{d_{i_{j}}},\qquad 1\leq j\leq n,$
and so that
$d_{i}=|\mathcal{O}_{i}|=\frac{|G|}{|G_{i}|}=d_{i_{1}}\times
d_{i_{2}}\times\cdots\times d_{i_{n}}.$
We set
$\displaystyle e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}$
$\displaystyle\qquad=\frac{1}{|G_{i}|}\sum_{j_{1}=0}^{\nu_{i_{1}}-1}\sum_{j_{2}=0}^{\nu_{i_{2}}-1}\cdots\sum_{j_{n}=0}^{\nu_{i_{n}}-1}\xi_{1}^{d_{i_{1}}j_{1}s_{i_{1}}}\xi_{2}^{d_{i_{2}}j_{2}s_{i_{2}}}\cdots\xi_{n}^{d_{i_{n}}j_{n}s_{i_{n}}}g_{1}^{d_{i_{1}}j_{1}}g_{2}^{d_{i_{2}}j_{2}}\cdots
g_{n}^{d_{i_{n}}j_{n}}.$
Then one can check that
$\big{\\{}e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}\mid
s_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}\mbox{ for all }1\leq j\leq
n\big{\\}}$ is a complete set of primitive orthogonal idempotents of
$\mathbbm{k}[G_{i}]$.
It is obvious that
$g_{j}(e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})})=e_{(i,s_{i_{1}},\cdots,s_{i_{j-1}},s^{\prime}_{i_{j}},s_{i_{j+1}},\cdots,s_{i_{n}})}$
for any $1\leq j\leq n,$ where
$s^{\prime}_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}$ and
$s^{\prime}_{i_{j}}\equiv s_{i_{j}}+1\mod{\nu_{i_{j}}}$. Since for each
idempotent $e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}$, there is a unique
corresponding one dimensional irreducible representation $\rho$ of $G_{i}$
defined by the group homomorphism $\phi_{\rho}:G_{i}\rightarrow\mathbbm{k}$,
$g_{j}^{d_{i_{j}}}\mapsto\xi^{d_{i_{j}}s_{i_{j}}}$, for $1\leq j\leq n$. Thus
we can index the vertices set $\widehat{I}$ by some sequences
$(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})$, i.e.,
$\widehat{I}=\left\\{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})\mid
i\in\mathcal{I},s_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}\mbox{ for all
}1\leq j\leq n\right\\}.$
Then the action of $G$ on $\widehat{I}$ is clearly and so that the orbit of
$(i,\rho)\in\widehat{I}$ has the form
$\left\\{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})\mid
s_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}\mbox{ for }1\leq j\leq
n\\}=\\{(i,\rho)\mid\rho\in\mbox{irr}G_{i}\right\\}$
for some $i\in\mathcal{I}$. Furthermore, it is easy to see that if the action
of $G$ on $\mathbbm{k}Q$ is admissible, then so is on
$\mathbbm{k}\widehat{Q}$.
For any $i,j\in I$, we consider the group $G_{ij}:=G_{i}\cap G_{j}=\langle
g_{1}^{t_{1}}\rangle\times\langle
g_{2}^{t_{2}}\rangle\times\cdots\times\langle g_{n}^{t_{n}}\rangle$, where
$t_{l}$ is the least common multiple of $d_{i_{l}}$ and $d_{j_{l}}$ for $1\leq
l\leq n$. Note that the vector space $E_{ij}$ spanned by arrows
$\alpha:i\rightarrow j$ in $Q$ is a $\mathbbm{k}[G_{ij}]$-bimodule, we can
find a basis of $E_{ij}$ such that the action of $G_{ij}$ is diagonal. That
is, if $g=g_{1}^{t_{1}}g_{2}^{t_{2}}\cdots
g_{n}^{t_{n}}\in\mathbbm{k}[G_{ij}]$, then for any basis element
$\alpha^{\prime}\in E_{ij}$,
$g(\alpha^{\prime})=\xi_{1}^{t_{1}r_{1}}\xi_{2}^{t_{2}r_{2}}\cdots\xi_{n}^{t_{n}r_{n}}\alpha^{\prime}$
for some $r_{1},r_{2},\cdots,r_{n}\in\mathbb{Z}$. Since $G_{ij}$ is abelian,
the number of the basis elements of $E_{ij}$ is just the number of arrows from
$i$ to $j$ in $Q$. Moreover, it is easy to see that the $t_{1}t_{2}\cdots
t_{n}$ elements
$\alpha^{\prime},~{}g_{1}(\alpha^{\prime}),\cdots,~{}g_{n}(\alpha^{\prime}),~{}g^{2}_{1}(\alpha^{\prime}),~{}g_{1}g_{2}(\alpha^{\prime}),\cdots,~{}g^{2}_{n}(\alpha^{\prime}),\cdots\cdots,~{}g^{t_{1}-1}_{1}g^{t_{2}-1}_{2}\cdots
g^{t_{n}-1}_{n}(\alpha^{\prime})$
are linearly independent. That is, for any arrow $\alpha:i\rightarrow j$ in
$Q$, there are $t_{1}t_{2}\cdots t_{n}$ arrows in its orbit.
On the other hand, we can calculate that
$\displaystyle
e_{(j,s_{j_{1}},s_{j_{2}},\cdots,s_{j_{n}})}\alpha^{\prime}e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}$
$\displaystyle\qquad=\frac{d_{i}d_{j}}{|G|^{2}}\sum_{p_{1}=0}^{\nu_{i_{1}}-1}\cdots\sum_{p_{n}=0}^{\nu_{i_{n}}-1}\sum_{q_{1}=0}^{\nu_{j_{1}}-1}\cdots\sum_{q_{n}=0}^{\nu_{j_{n}}-1}\xi_{1}^{d_{i_{1}}p_{1}s_{i_{1}}+d_{j_{1}}q_{1}s_{j_{1}}}\cdots\xi_{n}^{d_{i_{n}}p_{n}s_{i_{n}}+d_{j_{n}}q_{n}s_{j_{n}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
g_{1}^{d_{j_{1}}q_{1}}\cdots
g_{n}^{d_{j_{n}}q_{n}}(\alpha^{\prime})g_{1}^{d_{i_{1}}p_{1}+d_{j_{1}}q_{1}}\cdots
g_{n}^{d_{i_{n}}p_{n}+d_{j_{n}}q_{n}}.$
We write
$\displaystyle d_{i_{l}}p_{l}$ $\displaystyle=$ $\displaystyle
P_{l}t_{l}+d_{i_{l}}p^{\prime}_{l},\qquad\hbox{ where }0\leq
P_{l}<\frac{m_{l}}{t_{l}},\quad 0\leq p^{\prime}_{l}<\frac{t_{l}}{d_{i_{l}}},$
$\displaystyle d_{j_{l}}q_{l}$ $\displaystyle=$ $\displaystyle
P^{\prime}_{l}t_{l}+d_{j_{l}}q^{\prime}_{l},\qquad\hbox{ where }0\leq
P^{\prime}_{l}<\frac{m_{l}}{t_{l}},\quad 0\leq
q^{\prime}_{l}<\frac{t_{l}}{d_{j_{l}}},$ $\displaystyle d_{i_{l}}k_{l}$
$\displaystyle\equiv$
$\displaystyle(P_{l}+P^{\prime}_{l})t_{l}+d_{i_{l}}p^{\prime}_{l}\mod{m_{i}},\qquad\hbox{
where }0\leq k_{l}<\nu_{i_{l}},$
for all $0\leq l\leq n$. Then the right side of the equation becomes
$\displaystyle\frac{d_{i}d_{j}}{|G|^{2}}$
$\displaystyle\Bigg{(}\sum_{P^{\prime}_{1}=0}^{\frac{m_{1}}{t_{1}}-1}\xi_{1}^{P^{\prime}_{1}t_{1}(r_{1}+s_{j_{1}}-s_{i_{1}})}\Bigg{)}\cdots\Bigg{(}\sum_{P^{\prime}_{n}=0}^{\frac{m_{n}}{t_{n}}-1}\xi_{n}^{P^{\prime}_{n}t_{n}(r_{n}+s_{j_{n}}-s_{i_{n}})}\Bigg{)}$
$\displaystyle\quad\Bigg{(}\sum_{k_{1}=0}^{\nu_{i_{1}}-1}\cdots\sum_{k_{n}=0}^{\nu_{i_{n}}-1}\sum_{q^{\prime}_{1}=0}^{\frac{t_{1}}{d_{j_{1}}}-1}\cdots\sum_{q^{\prime}_{n}=0}^{\frac{t_{n}}{d_{j_{n}}}-1}\xi_{1}^{d_{i_{1}}k_{1}s_{i_{1}}+d_{j_{1}}q^{\prime}_{1}s_{j_{1}}}\cdots\xi_{n}^{d_{i_{n}}k_{n}s_{i_{n}}+d_{j_{n}}q^{\prime}_{n}s_{j_{n}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
g_{1}^{d_{j_{1}}q^{\prime}_{1}}\cdots
g_{n}^{d_{j_{n}}q^{\prime}_{n}}(\alpha^{\prime})g_{1}^{d_{i_{1}}k_{1}+d_{j_{1}}q^{\prime}_{1}}\cdots
g_{n}^{d_{i_{n}}k_{n}+d_{j_{n}}q^{\prime}_{n}}\Bigg{)}.$
Note that
$\displaystyle\Big{\\{}g_{1}^{d_{j_{1}}q^{\prime}_{1}}\cdots
g_{n}^{d_{j_{n}}q^{\prime}_{n}}(\alpha^{\prime})g_{1}^{d_{i_{1}}k_{1}+d_{j_{1}}q^{\prime}_{1}}$
$\displaystyle\cdots g_{n}^{d_{i_{n}}k_{n}+d_{j_{n}}q^{\prime}_{n}}$
$\displaystyle\mid 0\leq k_{l}<\nu_{i_{l}},~{}0\leq
q^{\prime}_{l}<\frac{t_{l}}{d_{j_{l}}}\mbox{ for }1\leq l\leq n\Big{\\}}$
is a linearly independent set. We obtain that
$e_{(j,s_{j_{1}},s_{j_{2}},\cdots,s_{j_{n}})}\alpha^{\prime}e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}\neq
0$ if and only if $s_{i_{l}}\equiv s_{j_{l}}+r_{l}\mod{\frac{m_{l}}{t_{l}}}$
for all $0\leq l\leq n$. It follows that there are $\frac{t_{1}\cdots
t_{n}|G|}{d_{i}d_{j}}$ arrows in $\widehat{Q}$ for each arrow
$\alpha:i\rightarrow j$ in $Q$.
Denote by $\widehat{A}=(a_{(i\rho)(j\sigma)})_{\widehat{I}\times\widehat{I}}$
the Cartan matrix of $\widehat{Q}$, by $\widehat{\Gamma}$ the valued quiver
corresponding to $(\widehat{Q},G)$ and by
$\widehat{C}=(\widehat{c}_{ij})_{\mathcal{I}\times\mathcal{I}}=\widehat{D}^{-1}\widehat{B}$
the generalized Cartan matrix of $\widehat{\Gamma}$, where
$\widehat{B}=(\widehat{b}_{ij})_{\mathcal{I}\times\mathcal{I}}$ is symmetric,
$\widehat{D}=\mbox{diag}(\widehat{d}_{i})$ is diagonal. Then
$\frac{1}{t_{1}\cdots t_{n}}\sum_{i^{\prime}\in\mathcal{O}_{i}\atop
j^{\prime}\in\mathcal{O}_{j}}a_{i^{\prime}j^{\prime}}=\frac{d_{i}d_{j}}{t_{1}\cdots
t_{n}|G|}\sum_{\rho\in{\rm irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}a_{(i\rho)(j\sigma)}.$
It follows that $\widehat{b}_{ij}=\frac{|G|}{d_{i}d_{j}}b_{ij}$,
$\widehat{D}=|G|D^{-1}$, $\widehat{B}=|G|D^{-1}BD^{-1}$ and
$\widehat{C}=(\widehat{D})^{-1}\widehat{B}=BD^{-1}=C^{T}$, the transpose of
$C$. Therefore $\Gamma$ and $\widehat{\Gamma}$ are dual valued graph in the
sense of [13].
###### Remark 3.7.
If $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ is a finite abelian group, we have
given the dual of $(Q,G)$ and $(\widehat{Q},G)$ (see Proposition 3.6).
However, for a non-abelian group $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$, the
conclusion does not hold in general. For example, Let $Q$ be the quiver
It is well-known that the quiver automorphism group of $Q$ is the group
$S_{3}$. Accordingly, we obtain the generalized McKay quiver $\widehat{Q}$ of
$(Q,S_{3})$ as follows
One can check that there does not exist a subgroup $G^{\prime}$ of
$\mbox{Aut}(\mathbbm{k}\widehat{Q})$ such that the generalized McKay quiver of
$(\widehat{Q},G^{\prime})$ is $Q$.
But if the action of $G$ is “good”, there exists the duality still. For
example, we consider the finite non-abelian group
$G=\left\langle a,b\mid a^{3}=b^{2},b^{4}=1,aba=b\right\rangle$
and the quiver $Q$:
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{picture}(50.0,32.0)\put(24.0,1.0){$\bullet$}
\put(24.0,10.0){$\bullet$} \put(37.0,24.0){$\bullet$}
\put(10.0,24.0){$\bullet$} \put(44.0,30.0){$\bullet$}
\put(2.0,30.0){$\bullet$} \put(25.0,9.0){\vector(0,-1){6.0}}
\put(26.0,13.0){\vector(1,1){10.0}} \put(22.5,13.0){\vector(-1,1){10.0}}
\put(13.0,24.0){\vector(1,-1){10.0}} \put(37.0,22.5){\vector(-1,-1){10.0}}
\put(14.5,26.0){\vector(1,0){20.0}} \put(34.5,25.0){\vector(-1,0){20.0}}
\put(9.5,26.0){\vector(-4,3){5.0}} \put(39.0,26.0){\vector(4,3){5.0}}
\put(22.0,26.0){$\alpha^{\ast}$} \put(26.0,23.0){$\alpha$}
\put(18.0,20.0){$\beta$} \put(16.0,14.0){$\beta^{\ast}$}
\put(28.0,17.0){$\gamma$} \put(34.0,17.0){$\gamma^{\ast}$}
\put(39.0,22.0){\small$1$} \put(46.0,28.0){\small$1^{\prime}$}
\put(8.0,22.0){\small$2$} \put(0.0,28.0){\small$2^{\prime}$}
\put(27.0,9.0){\small$3$} \put(27.0,0.0){\small$3^{\prime}$}
\put(21.0,6.0){$\sigma_{3}$} \put(8.0,28.0){$\sigma_{2}$}
\put(38.0,28.0){$\sigma_{1}$} \end{picture}.$
The action of $G$ is given by
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $\alpha$ | $\alpha^{\ast}$
---|---|---|---|---|---|---|---|---
$a$ | $e_{2}$ | $e_{3}$ | $e_{1}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{1^{\prime}}$ | $\beta$ | $\beta^{\ast}$
$b$ | $e_{1}$ | $e_{3}$ | $e_{2}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{2^{\prime}}$ | $-\gamma^{\ast}$ | $\gamma$
| $\beta$ | $\beta^{\ast}$ | $\gamma$ | $\gamma^{\ast}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$
---|---|---|---|---|---|---|---
$a$ | $\gamma$ | $\gamma^{\ast}$ | $\alpha$ | $\alpha^{\ast}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{1}$
$b$ | $-\beta^{\ast}$ | $\beta$ | $-\alpha^{\ast}$ | $\alpha$ | $\sigma_{1}$ | $\sigma_{3}$ | $\sigma_{2}$
where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to
vertex $i$, $i\in\\{1,2,3,1^{\prime},2^{\prime},$ $3^{\prime}\\}$. By direct
calculation, one see that the generalized McKay quiver of $(Q,G)$ is as
follows.
Now, we define an action of $G$ on $\mathbbm{k}\widehat{Q}$ by setting
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$a$ | $e_{3}$ | $e_{4}$ | $e_{1}$ | $e_{2}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{1^{\prime}}$ | $e_{4^{\prime}}$ | $\xi^{2}\alpha_{3}$ | $\xi^{4}\alpha_{4}$ | $\xi^{2}\alpha_{1}$ | $\xi^{4}\alpha_{2}$
$b$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{1}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{1^{\prime}}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{1}$
| $\alpha^{\ast}_{1}$ | $\alpha^{\ast}_{2}$ | $\alpha^{\ast}_{3}$ | $\alpha^{\ast}_{4}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{4}$
---|---|---|---|---|---|---|---|---
$a$ | $\xi^{2}\alpha^{\ast}_{3}$ | $\xi^{4}\alpha^{\ast}_{4}$ | $\xi^{2}\alpha^{\ast}_{1}$ | $\xi^{4}\alpha^{\ast}_{2}$ | $\sigma_{3}$ | $\sigma_{4}$ | $\sigma_{1}$ | $\sigma_{2}$
$b$ | $\alpha^{\ast}_{2}$ | $\alpha^{\ast}_{3}$ | $\alpha^{\ast}_{4}$ | $\alpha^{\ast}_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{4}$ | $\sigma_{1}$
where $\xi$ is a primitive $6$-th root of unity. Then, one can check that
$\widehat{\widehat{Q}}=Q$.
3.3. Consider the admissible action of finite abelian group $G$ on
$\mathbbm{k}\widehat{Q}$ induced from the action of $G$ on $\mathbbm{k}Q$ as
the discussion above, we set
$\begin{array}[]{ll}F^{\prime}:=(\mathbbm{k}Q\ast G)\ast
G\otimes_{\mathbbm{k}Q\ast G}-:&\mbox{{\bf mod}-}\mathbbm{k}Q\ast
G\longrightarrow\mbox{{\bf mod}-}(\mathbbm{k}Q\ast G)\ast G\\\
H^{\prime}:=\mbox{Res}|_{\mathbbm{k}Q\ast G}:&\mbox{{\bf
mod}-}(\mathbbm{k}Q\ast G)\ast G\longrightarrow\mbox{{\bf
mod}-}\mathbbm{k}Q\ast G\end{array}$
Similar to the functors $F$ and $H$, one can check that
$(H^{\prime},F^{\prime})$ and $(F^{\prime},H^{\prime})$ are adjoint pairs.
Note that the Morita equivalence $\mbox{{\bf
mod}-}\mathbbm{k}Q\rightarrow\mbox{{\bf mod}-}(\mathbbm{k}Q\ast G)\ast G$ is
given by $\mathcal{M}:={{}_{(\mathbbm{k}Q\ast G)\ast G}\mathbbm{k}Q\ast
G}\otimes_{\mathbbm{k}Q}-$, we have
###### Lemma 3.8.
There are natural isomorphisms
$F\cong H^{\prime}\mathcal{M}\qquad\mbox{ and }\qquad
F^{\prime}\cong\mathcal{M}H.$
###### Proof.
First, $H^{\prime}\mathcal{M}={{}_{\mathbbm{k}Q\ast G}\mathbbm{k}Q\ast
G}\otimes_{\mathbbm{k}Q}-=F$ is clear. Next, since $(H^{\prime},F^{\prime})$
is an adjoint pair, for any $\mathbbm{k}Q$-module $X$ and $\mathbbm{k}Q\ast
G$-module $Y$, we have
$\displaystyle\mbox{Hom}_{\mathbbm{k}Q}(X,\mathcal{M}^{-1}F^{\prime}(Y))$
$\displaystyle\cong\mbox{Hom}_{(\mathbbm{k}Q\ast G)\ast
G}(\mathcal{M}(X),F^{\prime}(Y))$
$\displaystyle\cong\mbox{Hom}_{\mathbbm{k}Q\ast
G}(H^{\prime}\mathcal{M}(X),Y)\cong\mbox{Hom}_{\mathbbm{k}Q\ast G}(F(X),Y).$
This implies that $(F,\mathcal{M}^{-1}F^{\prime})$ is an adjoint pair and so
that $H\cong\mathcal{M}^{-1}F^{\prime}$, $F^{\prime}\cong\mathcal{M}H$. ∎
By Lemma 3.8 and [18, Proposition 1.8], we have the following proposition
immediately.
###### Proposition 3.9.
Let $X$ and $Y$ be indecomposable $\mathbbm{k}Q\ast G$-modules. Then
(1) $FH(X)\cong H^{\prime}F^{\prime}(X)\cong\bigoplus_{g\in G}{{}^{g}X}$;
(2) $H(X)\cong H(Y)$ if and only if $F^{\prime}(X)\cong F^{\prime}(Y)$, if and
only if $Y\cong{{}^{g}X}$ for some $g\in G$;
(3) $H(X)$ (or $F^{\prime}(X)$) has exactly $|H_{X}|$ indecomposable summands.
###### Remark 3.10.
Consider the action of $G$ on $\mathbbm{k}Q\ast G$, we denote by
$H_{X}:=\\{g\in G\mid F_{g}(X)\cong X\\}$
and by $G_{X}$ a complete set of left coset representatives of $H_{X}$ in $G$,
for any $X\in\mbox{{\bf mod}-}\mathbbm{k}Q\ast G$. In [10], we have shown that
the number of indecomposable summands of $F^{\prime}(X)$ is just $|H_{X}|$
whenever $G$ is abelian (see [10, Theorem 1.2]). This means that $H(X)$ has
$|H_{X}|$ indecomposable summands. Note that $H(X)$ is an indecomposable
$G$-invariant $\mathbbm{k}Q$-module, there exists a unique indecomposable
$\mathbbm{k}Q$-module $M$ such that $H(X)\cong\sum(M)$. Therefore, we have
$|H_{X}|=|G_{M}|$ and $|G_{X}|=|H_{M}|$. Following from Proposition 3.9(2),
for an indecomposable $\mathbbm{k}Q$-module $M$, there are $|G_{X}|=|H_{M}|$
non-isomorphic indecomposable $\mathbbm{k}Q\ast G$-module structures on
$\sum(M)$. This coincides with the result in [10].
For the generalized McKay quiver $\widehat{Q}$, we denote by
$(-,-)_{\widehat{Q}}$ the bilinear form on $\mathbb{Z}\widehat{I}$ determined
by $\widehat{A}$, by $\Delta_{\widehat{Q}}$ the root system of $\widehat{Q}$
with simple roots $\varepsilon_{i\rho}$, $(i,\rho)\in\widehat{I}$, and by
$\mathcal{W}(\widehat{Q})$ the Weyl group of $\widehat{Q}$ with simple
reflections $r_{i\rho}$, $(i,\rho)\in\widehat{I}$. Consider the map
$h:\mathbb{Z}\widehat{I}\rightarrow\mathbb{Z}\mathcal{I}$ defined above, we
have
###### Lemma 3.11.
Let $\widehat{S}_{i}:=\prod_{\rho\in{\rm irr}G_{i}}r_{i\rho}$ for
$i\in\mathcal{I}$. Then, for each $i\in\mathcal{I}$ and
$\beta=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}\in\mathbb{Z}\widehat{I}$,
we have
(1) $(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}=d_{i}\sum_{\rho\in{\rm
irr}G_{i}}(\beta,\varepsilon_{i\rho})_{\widehat{Q}}$;
(2) $h(\widehat{S}_{i}(\beta))=\gamma_{i}(h(\beta))$;
(3) the map $\gamma_{i}\mapsto\widehat{S}_{i}$ induces an isomorphism
$\mathcal{W}(\Gamma)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}C_{G}(\mathcal{W}(\widehat{Q}))$,
the set of elements in $\mathcal{W}(\widehat{Q})$ commuting with the action of
$G$.
###### Proof.
(1) By the dual between $(Q,G)$ and $(\widehat{Q},G)$, we obtain that
$b_{ij}=\sum_{i^{\prime}\in\mathcal{O}_{i}\atop
j^{\prime}\in\mathcal{O}_{j}}a_{i^{\prime}j^{\prime}}=\frac{d_{i}d_{j}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm irr}G_{j}}a_{(i\rho)(j\sigma)},$
and so that
$b_{ij}=d_{i}\sum_{\rho\in{\rm irr}G_{i}}a_{(i\rho)(j\sigma)}$
for any $\sigma\in{\rm irr}G_{j}$. Therefore, we get
$(h(\beta),\overline{\epsilon}_{i})_{\Gamma}=\sum_{i,j\in\mathcal{I}}b_{ij}h(\beta)_{j}=d_{i}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}a_{(i\rho)(j\sigma)}\beta_{j\sigma}=d_{i}\sum_{\rho\in{\rm
irr}G_{i}}(\beta,\varepsilon_{i\rho})_{\widehat{Q}}.$
(2) Firstly, $\widehat{S}_{i}$ is well-defined since the action of $G$ on
$\widehat{Q}$ is admissible. Secondly, it is easy to check that the bilinear
form $(-,-)_{\widehat{Q}}$ is $G$-invariant and $\widehat{S}_{i}$ commutes
with the action of $G$. Thus, we have
$h(\widehat{S}_{i}(\beta))=h(\beta)-\sum_{\rho\in{\rm
irr}G_{i}}(\beta,\varepsilon_{i\rho})_{\widehat{Q}}\overline{\varepsilon}_{i}=h(\beta)-\frac{1}{d_{i}}(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}\overline{\varepsilon}_{i}=\gamma_{i}(h(\beta)).$
(3) By induction on the length, one can check that
$C_{G}(\mathcal{W}(\widehat{Q}))$ is generated by $\widehat{S}_{i}$,
$i\in\mathcal{I}$. Following from (2), we get $\gamma_{i}\mapsto S_{i}$
induces an isomorphism. ∎
We are in a position to complete the proof of Theorem 1.1. We have shown that
for any positive root $\alpha\in\Delta_{\Gamma}$, there exists an
indecomposable $\widehat{Q}$-representation X such that $h({\bf
dim}X)=\alpha$. Moreover, if $\alpha$ is real, the number of $X$ (up to
isomorphism) can be determined. Applying the technique in [12, Proposition
15], we have
###### Proposition 3.12.
The map $h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is a surjection. If
$\alpha\in\Delta_{\Gamma}$ is a positive real root, then there is a unique
$G$-orbit of roots mapping to $\alpha$, and all of which are real.
###### Proof.
Firstly, by Corollary 3.4, the map
$h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is well-defined. To show
the surjectivity, we need to fine the preimages of all the fundamental roots
in $\Delta_{\widehat{Q}}$.
We suppose that $\Gamma$ is connected. Then, for any $\alpha\in F_{\Gamma}$,
we consider the set
${\mathcal{R}}:=\\{\beta\in\Delta_{\widehat{Q}}\mid\beta\mbox{ is positive and
}h(\beta)\leq\alpha\\}.$
Since ${\mathcal{R}}$ is finite and non-empty, we take an element $\beta$ with
maximal height. Suppose that $h(\beta)_{i}<\alpha_{i}$ for all
$i\in\mathcal{I}$, then for any $\rho\in\mbox{irr}G_{i}$,
$h(\beta+\varepsilon_{i\rho})=h(\beta)+\overline{\varepsilon}_{i}\leq\alpha$.
By the maximality of $\beta$, $\beta+\varepsilon_{i\rho}$ is not a root and so
that $(\beta,\varepsilon_{i\rho})_{\widehat{Q}}\geq 0$. Thus
$(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}\geq 0$ for all
$i\in\mathcal{I}$. We conclude that $h(\beta)$ and $\alpha$ have the same
support, for otherwise, we can find such a vertex $(i,\rho)$ adjacent to the
support of $\beta$ such that $(\beta,\varepsilon_{i\rho})_{\widehat{Q}}<0$.
We take $\alpha\in F_{\Gamma}$ such that the support of $\alpha$ is
${\mathcal{I}}$, and set
$\Phi:=\\{i\in\mathcal{I}\mid h(\beta)_{i}=\alpha_{i}\\}.$
If $\Phi$ is the empty set, then $\beta+\varepsilon_{i\rho}$ is not a root for
any vertex $(i,\rho)\in\widehat{Q}$, and so that the connected component of
$\widehat{Q}$ which $\beta$ lies in is Dynkin (see [13, Proposition 4.9]).
Therefore, $\widehat{Q}$ must be a disjoint union of copies of this Dynkin
quiver, all in a single $G$-orbit. Thus $\widehat{Q}$ and $Q$ are
representation finite [18], $\Gamma$ is a connected Dynkin diagram. This
contradicts to that $\alpha$ is a imaginary root.
It follows that $\Phi$ is non-empty. We denote by $\widetilde{\Phi}$ the full
subgraph of $\Gamma$ determined by $\Phi$. Let $T$ be a non-empty connected
component of $\Gamma-\widetilde{\Phi}$, and let $\widetilde{\beta}$ be the
restriction of $h(\beta)$ to $T$. If $T\neq\emptyset$, then for all vertices
$j\in T$, we have
$(\widetilde{\beta},\overline{\varepsilon}_{j})_{T}\geq(h(\beta),\overline{\varepsilon}_{j})_{\Gamma}\geq
0$, where $(-,-)_{T}$ is the restriction of $(-,-)_{\Gamma}$ on $T$. Moreover,
note that there exists a vertex $j\in T$ adjacent to $\widetilde{\Phi}$, we
have $(\widetilde{\beta},\overline{\varepsilon}_{j})_{T}>0$. Therefore, $T$ is
a Dynkin diagram [13, Corollary 4.9]. On the other hand, let
$\widetilde{\beta}^{\prime}$ be the restriction of $\alpha-h(\beta)$ to $T$.
Then $\widetilde{\beta}^{\prime}$ has the support $T$, and for any vertex
$j\in T$,
$(\widetilde{\beta}^{\prime},\overline{\varepsilon}_{j})_{T}=(\alpha-h(\beta),\overline{\varepsilon}_{j})_{\Gamma}=(\alpha,\overline{\varepsilon}_{j})_{\Gamma}-(h(\beta),\overline{\varepsilon}_{j})_{\Gamma}\leq
0.$
Hence $T$ is not Dynkin. This is a contradiction. Therefore, $T$ is empty,
$\widetilde{\Phi}=\Gamma$ and so that $h(\beta)=\alpha$. Thus, we have shown
that $h$ is surjective by Lemma 3.11(3).
In general, assume that $\Gamma$ is non-connected. In this case,
$F_{\Gamma}=\bigcup F_{\Gamma^{\prime}}$, where $\Gamma^{\prime}$ run over all
connected components of $\Gamma$. By the discussion above, we see that any
element $\alpha\in F_{\Gamma}$, there exists an element
$\beta\in\Delta_{\widehat{Q}}$ such that $h(\beta)=\alpha$. Hence, $h$ is also
surjective.
Finally, for any real root $\alpha\in\Delta_{\Gamma}$, we let
$\beta\in\Delta_{\widehat{Q}}$ be the element such that $h(\beta)=\alpha$.
Then, there is an element $\omega^{\prime}\in\mathcal{W}(\Gamma)$ and
$i\in\mathcal{I}$ such that
$\omega^{\prime}(\alpha)=\overline{\varepsilon}_{i}$. Let $\omega$ be the
element in $C_{G}(\mathcal{W}(\widehat{Q}))$ corresponding to
$\omega^{\prime}$. It follows that $\omega(\beta)$ must also be a simple root
$\varepsilon_{i\rho}$ for some $\rho\in\mbox{irr}G_{i}$. Therefore $\beta$ is
real and uniquely determined up to a $G$-orbit. ∎
Consider the action of $G$ on $\mathbbm{k}\widehat{Q}$, any $g\in G$ also
induces an additive autoequivalence functor $F_{g}:\mbox{{\bf
mod}-}\mathbbm{k}\widehat{Q}\rightarrow\mbox{{\bf
mod}-}\mathbbm{k}\widehat{Q}$, $X\mapsto{{}^{g}X}$. Here we also denote by
$G_{X}$ a complete set of left coset representatives of $H_{X}:=\\{g\in G\mid
F_{g}(X)\cong X\\}$ in $G$, for any $X\in\mbox{{\bf
mod}-}\mathbbm{k}\widehat{Q}$. Following from Kac Theorem, for any positive
real root $\beta\in\Delta_{\widehat{Q}}$, there exists a unique
$\widehat{Q}$-representation $X$ such that ${\bf dim}X=\beta$ and
$H_{X}=H_{\beta}$. By Proposition 3.12, there are $|G_{X}|$ indecomposable
$\widehat{Q}$-representations (up to isomorphism) such that the image of their
dimension vector under the map $h$ are $\alpha$, if $h({\bf dim}X)=\alpha$.
Thus the proof of Theorem 1.1 is completed.
## 4\. Proof of Theorem 1.2
Assume that $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ is a finite abelian group. In
this section, we lift $G$ to $\overline{G}\subseteq\mbox{Aut}(\mathfrak{g})$
such that the Kac-Moody algebra $\mathfrak{g}(\Gamma)$ can be embedded into
the fixed point algebra $\mathfrak{g}^{\overline{G}}$. In this case,
$\mathfrak{g}^{\overline{G}}$ is integrable as a
$\mathfrak{g}(\Gamma)$-module.
Firstly, we recall some notations of Kac-Moody algebras. For a symmetricable
generalized Cartan matrix $C=(c_{ij})$ of size $n$ and rank $l$, there exist a
diagonal matrix $D=\mbox{diag}(d_{1},\cdots,d_{n})$ and a symmetric matrix
$B=(b_{ij})$ such that $C=D^{-1}B$. In fact, $d_{i}(1\leq i\leq n)$ may be
chosen to be positive integers. Let $\mathfrak{h}$ be a $2n-l$ dimension
$\mathbbm{k}$-vector space. Choose linearly independent sets
$\left\\{H_{i}\in\mathfrak{h}|1\leq i\leq n\right\\}$ and
$\left\\{\varepsilon_{i}\in\mathfrak{h}^{\ast}|1\leq i\leq n\right\\}$ such
that $\varepsilon_{j}(H_{i})=c_{ij}$. Then the triple
$\left(\mathfrak{h},\\{\varepsilon_{i}\\},\\{H_{i}\\}\right)_{1\leq i\leq n}$
is called a (minimal) realization of $C$. Since any two realizations of $C$
are isomorphic, there is a unique (up to isomorphism) Kac-Moody algebra
$\mathfrak{g}(C)$ generated by $\mathfrak{h}$, $E_{i}$, $F_{i}$, $1\leq i\leq
n$, with relations
$\begin{array}[]{llll}\quad[H,H^{\prime}]=0,&[H,E_{j}]=\varepsilon_{j}(H)E_{j},&(\mbox{ad}E_{i})^{1-c_{ij}}E_{j}=0,\\\
\quad[E_{i},F_{j}]=\delta_{ij}H_{i},&[H,F_{j}]=-\varepsilon_{j}(H)F_{j},&(\mbox{ad}F_{i})^{1-c_{ij}}F_{j}=0.\end{array}$
for any $H,H^{\prime}\in\mathfrak{h}$, where $\delta_{ij}$ is the Kronecker
sign. Moreover, the center $\mathfrak{c}$ of $\mathfrak{g}(C)$ is given by
$\\{H\in\mathfrak{h}\mid\varepsilon_{i}(H)=0\mbox{ for all }1\leq i\leq
n\\}\subseteq[\mathfrak{g}(C),\mathfrak{g}(C)].$
For the details one can see [13].
For the pair $(Q,G)$, we have obtained the valued graph $\Gamma$ with
symmetricable generalized Cartan matrix $C=(c_{ij})$ of size $|\mathcal{I}|$
and the generalized McKay quiver $\widehat{Q}$ with symmetric generalized
Cartan matrix $\widehat{A}=(a_{(i\rho)(j\sigma)})$ of size $|\widehat{I}|$,
see Section 2. Therefore we have Kac-Moody algebras
$\mathfrak{g}(\Gamma):=\mathfrak{g}(C)$ corresponding to the realization
$\big{(}\mathfrak{h}(\Gamma),\\{\overline{\varepsilon}_{i}\\},\\{\overline{H}_{i}\\}\big{)}$
of $C$ and $\mathfrak{g}:=\mathfrak{g}(\widehat{Q})=\mathfrak{g}(\widehat{A})$
corresponding to the realization
$\big{(}\mathfrak{h},\\{\varepsilon_{i\rho}\\},\\{H_{i\rho}\\}\big{)}$ of
$\widehat{A}$. Denote by $r$ and $s$ the coranks of $C$ and $\widehat{A}$,
then $\mbox{dim}_{\mathbbm{k}}\mathfrak{h}(\Gamma)=|\mathcal{I}|+r$ and
$\mbox{dim}_{\mathbbm{k}}\mathfrak{h}=|\widehat{I}|+s$.
We suppose that $\mathfrak{g}(\Gamma)$ generated by $\mathfrak{h}(\Gamma)$ and
$\overline{E}_{i},\overline{F}_{i}$, $i\in\mathcal{I}$. There is a symmetric
bilinear form $(-,-)_{\Gamma}$ on $\mathfrak{h}(\Gamma)$ such that
$(\overline{H}_{i},\overline{H})_{\Gamma}=\frac{1}{d_{i}}\overline{\varepsilon}_{i}(\overline{H})$
for all $\overline{H}\in\mathfrak{h}(\Gamma)$. Then we can extend it uniquely
to an invariant non-degenerate symmetric bilinear form on
$\mathfrak{g}(\Gamma)$ such that
$(\overline{E}_{i},\overline{F}_{i})_{\Gamma}=\frac{1}{d_{i}}.$
Moreover, $(-,-)_{\Gamma}$ determines a bijection
$\nu:\mathfrak{h}(\Gamma)\rightarrow\mathfrak{h}^{\ast}(\Gamma)$ sending
$\overline{H}_{i}$ to $\frac{1}{d_{i}}\overline{\varepsilon}_{i}$, and hence
induces a bilinear form on $\mathfrak{h}^{\ast}(\Gamma)$. We also denote this
bilinear form by $(-,-)_{\Gamma}$. Note that
$(\overline{\varepsilon}_{i},\overline{\varepsilon}_{i})_{\Gamma}=b_{ij}$. It
recovers the bilinear form defined in Section 2.3 for the root lattice
$\mathbb{Z}\mathcal{I}$. Similarly, there is a symmetric bilinear form on
$\mathfrak{h}^{\ast}=\mathfrak{h}^{\ast}(\widehat{Q})$ with
$(\varepsilon_{i\rho},\varepsilon_{j\sigma})_{\widehat{Q}}=a_{(i\rho)(j\sigma)}.$
We now consider the action of $G$ on the quiver $\widehat{Q}$ defined in
Section 3.2. Recall that the derived algebra $\mathfrak{g}^{\prime}$ of
$\mathfrak{g}$ is generated by $H_{i\rho}$, $E_{i\rho}$, $F_{i\rho}$,
$(i,\rho)\in\widehat{I}$ and the action of $G$ on $\widehat{Q}$ satisfies
$a_{(i\rho)(j\sigma)}=a_{(i\rho^{\prime})(j\sigma^{\prime})},\quad\hbox{ if
}(i,\rho^{\prime})=g(i,\rho)\hbox{ and }(j,\sigma^{\prime})=g(j,\sigma)$
for some $g\in G$. Then, there is a natural action of $G$ on
$\mathfrak{g}^{\prime}$ given by
$g(H_{i\rho})=H_{i\rho^{\prime}},\quad g(E_{i\rho})=E_{i\rho^{\prime}},\quad
g(F_{i\rho})=F_{i\rho^{\prime}}$
for any $g\in G$. Denote by $\mathfrak{h}^{\prime}(\Gamma)$ and
$\mathfrak{h}^{\prime}$ the Cartan subalgebra of
$\mathfrak{g}^{\prime}(\Gamma):=[\mathfrak{g}(\Gamma),\mathfrak{g}(\Gamma)]$
and $\mathfrak{g}^{\prime}$ respectively. It is easy to see that the map
$\phi:\quad\mathfrak{h}^{\prime}(\Gamma)\rightarrow(\mathfrak{h}^{\prime})^{G}$
given by $\phi(\overline{H}_{i})=\sum_{\rho\in{\rm irr}G_{i}}H_{i\rho}$ is an
isomorphism and
$(\overline{H},\overline{H}^{\prime})_{\Gamma}=\frac{1}{|G|}(\phi(\overline{H}),\phi(\overline{H}^{\prime}))_{\widehat{Q}}$
for $\overline{H},\overline{H}\in\mathfrak{h}^{\prime}(\Gamma)$. In
particular, the fixed point subalgebra $\mathfrak{c}^{G}$ of the center of
$\mathfrak{g}(\widehat{Q})$ is isomorphic to the center $\mathfrak{c}(\Gamma)$
of $\mathfrak{g}(\Gamma)$.
We wish to extend the action of $G$ on $\mathfrak{g}^{\prime}$ to the whole
Lie algebra $\mathfrak{g}$. Let $\mbox{Aut}(\widehat{A})$ denote the set of
permutations $g$ of $\widehat{I}$ satisfying
$a_{(i\rho)(j\sigma)}=a_{(l\rho^{\prime})(k\sigma^{\prime})}\quad\hbox{ if
}(l,\rho^{\prime})=g(i,\rho)\hbox{ and }(k,\sigma^{\prime})=g(j,\sigma).$
Let $\mbox{DAut}(\mathfrak{g})$ denote the subgroup of
$\mbox{Aut}(\mathfrak{g})$ consisting of the automorphisms preserving each of
the sets $\mathfrak{h}$, $\\{E_{i\rho}\\}$ and $\\{F_{i\rho}\\}$.
###### Proposition 4.1.
(see [14, Section 4.19]) There is a short exact sequence
$0\rightarrow{\rm
Hom}_{\mathbbm{k}}(\mathfrak{h}/\mathfrak{h}^{\prime},\mathfrak{c})\longrightarrow{\rm
DAut}(\mathfrak{g})\longrightarrow{\rm Aut}(\widehat{A})\rightarrow 0.$
###### Proof.
It is easy to see that $\overline{g}(H_{i\rho})=H_{j\sigma}$,
$\overline{g}(E_{i\rho})=E_{j\sigma}$ and
$\overline{g}(F_{i\rho})=F_{j\sigma}$ for any
$\overline{g}\in\mbox{DAut}(\mathfrak{g})$. Thus, there exists a unique
permutation $g\in\mbox{Aut}(\widehat{A})$ corresponding to $\bar{g}$ such that
$(j,\sigma)=g(i,\rho)$. Moreover, each $g\in\mbox{Aut}(\widehat{A})$ can be
obtained in this way.
Let $\Lambda:=\mathbbm{k}\widehat{I}$ be the subspace of $\mathfrak{h}^{\ast}$
spanned by $\\{\varepsilon_{i\rho}\mid(i,\rho)\in\widehat{I}\\}$. Then there
is an natural action of $\mbox{Aut}(\widehat{A})$ on $\Lambda$:
$g(\varepsilon_{i\rho})=\varepsilon_{j\sigma}$, where $(j,\sigma)=g(i,\rho)$,
$g\in G$, and it induces an action of $\mbox{Aut}(\widehat{A})$ on the
quotient space $\mathfrak{h}/\mathfrak{c}$ since $\mathfrak{h}/\mathfrak{c}$
is dual to $\Lambda$. It maps $H_{i\rho}\mod{\mathfrak{c}}$ to
$H_{j\sigma}\mod{\mathfrak{c}}$, and so that
$\mathfrak{h}^{\prime}/\mathfrak{c}$ is $\mbox{Aut}(\widehat{A})$-stable.
Since $\mbox{Aut}(\widehat{A})$ is finite, there exists
$\mathfrak{h}^{\prime\prime}$ such that
$\mathfrak{h}=\mathfrak{h}^{\prime}\oplus\mathfrak{h}^{\prime\prime}$ and
$(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}$ is
$\mbox{Aut}(\widehat{A})$-stable. For any $g\in\mbox{Aut}(\widehat{A})$, we
can define an automorphism $\overline{g}\in\mbox{DAut}(\mathfrak{g})$ by
$\overline{g}(H_{i\rho})=H_{j\sigma},\quad\overline{g}(E_{i\rho})=E_{j\sigma}\quad\hbox{
and }\quad\overline{g}(F_{i\rho})=F_{j\sigma},$
and $\overline{g}|_{\mathfrak{h}^{\prime\prime}}$ is the pull-back of $g$ on
$(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}$.
Obviously, the kernel of the map
$\mbox{DAut}(\mathfrak{g})\rightarrow\mbox{Aut}(\widehat{A})$ is the subgroup
$\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ consisting of all
automorphisms acting trivially on $\mathfrak{g}^{\prime}$. One can check that
an automorphism $\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ if
and only if there exists a map
$\varphi:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}$ such that
$\alpha(H)=H+\varphi(H)$ for all $H\in\mathfrak{h}^{\prime\prime}$. Thus,
there are isomorphisms
$\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})\cong\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}^{\prime\prime},\mathfrak{c})\cong\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}/\mathfrak{h}^{\prime},\mathfrak{c})$.
∎
Therefore, for each $\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$
and $g\in\mbox{Aut}(\widehat{A})$, we have an element
$\overline{g}\in\mbox{DAut}(\mathfrak{g})$ by setting
$\overline{g}|_{\mathfrak{g}^{\prime}}=g$ and
$\overline{g}|_{\mathfrak{h}^{\prime\prime}}=\alpha$. Moreover, for any
$\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ corresponding to
$\varphi:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}$, it is easy to
see that $\alpha^{t}(H)=H+t\varphi(H)$ for any $t\in\mathbb{Z}$ and
$H\in\mathfrak{h}^{\prime\prime}$. That is to say, an automorphism
$\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ has finite order if
and only if the corresponding map
$\varphi:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}$ is zero.
We now fix $\Omega=\\{g_{1},g_{2},\cdots,g_{n}\\}$ a set of generators of $G$.
We can view $G$ as a finite abelian subgroup of $\mbox{Aut}(\widehat{A})$. By
Proposition 4.1, we can lift $G$ to an automorphism group
$\overline{G}=\\{\overline{g}\mid g\in G\\}$ of $\mathfrak{g}$ corresponding
to a set of linear maps
$\\{\varphi_{i}=\varphi_{g_{i}}:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$. It is easy to see that for any $H\in\mathfrak{h}$, we have
$\varepsilon_{i\rho^{\prime}}(\overline{g}(H))=\varepsilon_{i\rho}(H)$ if
$(i,\rho^{\prime})=g(i,\rho)$. Let
$\mathcal{S}:=\mbox{span}\\{\varepsilon_{i\rho}-\varepsilon_{i\rho^{\prime}}\mid
i\in\mathcal{I},~{}\rho,\rho^{\prime}\in\mbox{irr}G_{i}\\}\subseteq\mathfrak{h}^{\ast}$
and
$\mathcal{H}:=\\{H\in\mathfrak{h}\mid\varepsilon_{i\rho}(H)=\varepsilon_{i\rho^{\prime}}(H)\mbox{
for all }\rho,\rho^{\prime}\in\mbox{irr}G_{i},\mbox{ and
}i\in\mathcal{I}\\}=\mbox{ann}_{\mathfrak{h}}\mathcal{S}.$
Then $\mathcal{H}$ contains the center $\mathfrak{c}$,
$\mathcal{H}/\mathfrak{c}=(\mathfrak{h}/\mathfrak{c})^{G}$ and so that, for
any lifting $\overline{G}$ of $G$,
$\mathcal{H}^{\overline{G}}=\mathfrak{h}^{\overline{G}}.$
###### Lemma 4.2.
$\mathcal{H}$ has $\mathbbm{k}$-dimension $|\mathcal{I}|+s$,
$\mathcal{H}\cap\mathfrak{h}^{\prime}$ has $\mathbbm{k}$-dimension
$|\mathcal{I}|+s-r$ and therefore $\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$
has $\mathbbm{k}$-dimension $r$.
###### Proof.
Note that
$\\{\varepsilon_{i\rho}-\varepsilon_{i\rho^{\prime}}\mid
i\in\mathcal{I},~{}\rho^{\prime}\in\mbox{irr}G_{i}\setminus\rho\\}$
is a basis of $\mathcal{S}$, we obtain that
$\mbox{dim}_{\mathbbm{k}}\mathcal{H}=\mbox{dim}_{\mathbbm{k}}\mathfrak{h}-\mbox{dim}_{\mathbbm{k}}\mathcal{S}=|\mathcal{I}|+s$.
Since
$(\mathcal{H}\cap\mathfrak{h}^{\prime})/\mathfrak{c}=(\mathfrak{h}^{\prime}/\mathfrak{c})^{G}$
is isomorphic to $(\mathfrak{h}^{\prime})^{G}/\mathfrak{c}^{G}$,
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{h}^{\prime})^{G}=|\mathcal{I}|$ and
$\mbox{dim}_{\mathbbm{k}}\mathfrak{c}^{G}=\mbox{dim}_{\mathbbm{k}}\mathfrak{c}(\Gamma)=r$,
$\mathcal{H}\cap\mathfrak{h}^{\prime}$ has $\mathbbm{k}$-dimension
$|\mathcal{I}|+s-r$ and so that $\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$
has $\mathbbm{k}$-dimension $r$. ∎
###### Proposition 4.3.
Let $\overline{G}$ be a lifting of $G$ to $\mathfrak{g}$ corresponding to
$\\{\varphi_{i}:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}\mid 1\leq
i\leq n\\}$. Then
$\bigg{(}\mathcal{H}^{\overline{G}},\bigg{\\{}\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}\bigg{\\}},\bigg{\\{}\sum_{\rho\in{\rm
irr}G_{i}}H_{i\rho}\bigg{\\}}\bigg{)}$
is a realization of $C$ if and only if
$\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ for all $1\leq
i\leq n$.
###### Proof.
We denote by
$H_{i}:=\sum_{\rho\in{\rm irr}G_{i}}H_{i\rho}\quad\hbox{ and
}\quad\epsilon_{i}:=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}$
for all $i\in\mathcal{I}$. Since $\\{H_{i}\mid i\in\mathcal{I}\\}$ is a basis
of $(\mathcal{H}\cap\mathfrak{h}^{\prime})^{G}$, $\mathcal{H}^{\overline{G}}$
has dimension $|\mathcal{I}|+r$ if and only if there are
$h^{\prime}_{1},h^{\prime}_{2},\cdots,h^{\prime}_{r}\in\mathcal{H}^{\overline{G}}$
spanning the complementary space of
$(\mathcal{H}\cap\mathfrak{h}^{\prime})^{G}$ in $\mathcal{H}^{\overline{G}}$.
Since $(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}$ is $G$-stable,
$\big{(}(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}\big{)}^{G}$
has $\mathbbm{k}$-dimension $r$ by Lemma 4.2. We can find linearly independent
elements
$h^{\prime\prime}_{1},h^{\prime\prime}_{2},\cdots,h^{\prime\prime}_{r}\in\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$
such that $h^{\prime\prime}_{i}\mod{\mathfrak{c}}$ are fixed by $G$. Since
$\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ for all $i$,
$h^{\prime\prime}_{1},h^{\prime\prime}_{2},\cdots,h^{\prime\prime}_{r}$ are
$G$-stable and form a basis of $\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$.
Therefore, we take $h^{\prime}_{i}=h^{\prime\prime}_{i}$ for all $1\leq i\leq
r$. On the other hand, if we can find such elements
$h^{\prime}_{1},h^{\prime}_{2},\cdots,h^{\prime}_{r}$, then each
$h^{\prime\prime}_{i}$ has the form
$h^{\prime\prime}_{i}=\sum_{j=1}^{s}p_{ij}h^{\prime}_{j}-\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}H_{j\sigma}$
for some $p_{ij},q_{i(j\sigma)}\in\mathbbm{k}$, and
$\displaystyle\varphi_{l}(h^{\prime\prime}_{i})$
$\displaystyle=\overline{g}_{l}\Big{(}\sum_{j=1}^{s}p_{ij}h^{\prime}_{j}-\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}H_{j\sigma}\Big{)}-\sum_{j=1}^{s}p_{ij}h^{\prime}_{j}+\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}H_{j\sigma}$
$\displaystyle=\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}(H_{j\sigma}-H_{j\sigma^{1}}),$
where $(j,\sigma^{1})=g_{l}(j,\sigma)$. It follows that
$t\varphi_{l}(h^{\prime\prime}_{i})=\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}(H_{j\sigma}-H_{j\sigma^{t}})$
for any $t\in\mathbb{Z}$, where $(j,\sigma^{t})=g_{l}^{t}(j,\sigma)$. Note
that $\widehat{I}$ is a finite set, there exist some $t\in\mathbb{Z}$ such
that $g_{l}^{t}(j,\sigma)=(j,\sigma)$ for all $(j,\sigma)\in\widehat{I}$, and
so that $t\varphi_{l}(h^{\prime\prime}_{i})=0$,
$\varphi_{l}(h^{\prime\prime}_{i})=0$ for all $i$ and $l$. Thus
$\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ for any $1\leq
i\leq n$.
Since
$\epsilon_{j}(H_{i})=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}\varepsilon_{j\sigma}(H_{i\rho})=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm irr}G_{j}}a_{(i\rho)(j\sigma)}=c_{ij}$
and $H_{i}$ $(i\in\mathcal{I})$ are linearly independent, it remains to show
$\epsilon_{i}$, $i\in\mathcal{I}$ are linearly independent modulo
$\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}^{\overline{G}})$. Let
$\epsilon:=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}\in\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}^{\overline{G}}),\quad\mu_{j}\in\mathbbm{k}.$
Then
$0=\epsilon(H_{i})=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}(H_{i})=\sum_{j\in\mathcal{I}}c_{ij}\mu_{j}$
for all $i\in\mathcal{I}$, and so that
$\epsilon(H_{i\rho})=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}(H_{i\rho})=\frac{1}{|G|}\sum_{j\in\mathcal{I}}b_{ij}\mu_{j}=\frac{d_{i}}{|G|}\sum_{j\in\mathcal{I}}c_{ij}\mu_{j}=0$
for all $(i,\rho)\in\widehat{I}$. Therefore,
$\epsilon\in\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}^{\overline{G}}+\mathfrak{h}^{\prime})=\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}+\mathfrak{h}^{\prime})\subseteq\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}})=\mathcal{S},$
and
$g(\epsilon)=g\bigg{(}\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}\bigg{)}=g\bigg{(}\sum_{j\in\mathcal{I}}\frac{d_{j}\mu_{j}}{|G|}\sum_{\sigma\in{\rm
irr}G_{j}}\varepsilon_{j\sigma}\bigg{)}=\epsilon$
for any $g\in G$. It concludes that
$\epsilon=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}=0$ by the construction of
$\mathcal{S}$. Therefore, $\mu_{j}=0$ for all $j\in\mathcal{I}$, and so that
$\epsilon_{j}$ are linearly independent in $\mathfrak{h}^{\ast}$.
The proof is completed. ∎
###### Remark 4.4.
Since
$\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}^{\prime\prime},\mathfrak{c})\cong\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}/\mathfrak{h}^{\prime},\mathfrak{c}),$
for any lifting $\overline{G}$ of $G$, there exist a family of maps
$\\{\psi_{i}=\psi_{g_{i}}:\mathfrak{h}/\mathfrak{h}^{\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$ corresponding to it. Moreover, it is easy to see that the
condition $\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ is
equivalent to
$\psi_{i}((\mathcal{H}+\mathfrak{h}^{\prime})/\mathfrak{h}^{\prime})=0$.
Now we can prove the main results of this section.
###### Proposition 4.5.
There is a monomorphism
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$, and for
the lifting $\overline{G}$ of $G$ corresponding to
$\\{\varphi_{i}:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}\mid 1\leq
i\leq n\\}$ with $\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$,
we can extend this monomorphism to the whole Lie algebra such that
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$
is also a monomorphism.
###### Proof.
We set
$H_{i}:=\sum_{\rho\in{\rm irr}G_{i}}H_{i\rho},\quad E_{i}:=\sum_{\rho\in{\rm
irr}G_{i}}E_{i\rho},\quad F_{i}:=\sum_{\rho\in{\rm irr}G_{i}}F_{i\rho}$
for all $i\in\mathcal{I}$. Then
$H_{i},E_{i},F_{i}\in(\mathfrak{g}^{\prime})^{G}$ and
$\displaystyle[H_{i},H_{j}]=0,$ $\displaystyle[E_{i},F_{j}]=\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}[E_{i\rho},F_{j\sigma}]=\delta_{ij}\sum_{\rho\in{\rm
irr}G_{i}}H_{i\rho}=\delta_{ij}H_{i},$
$\displaystyle[H_{i},E_{j}]=\sum_{\rho\in{\rm irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}[H_{i\rho},E_{j\sigma}]=\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}a_{(i\rho)(j\sigma)}E_{j\sigma}=c_{ij}\sum_{\sigma\in{\rm
irr}G_{j}}E_{j\sigma}=c_{ij}E_{j}.$
Similarly, we have $[H_{i},F_{j}]=c_{ij}F_{j}$ for any $i,j\in\mathcal{I}$.
Note that $\mbox{ad}E_{i\rho}$ and $\mbox{ad}E_{i\rho^{\prime}}$ commute for
any $\rho,\rho^{\prime}\in\mbox{irr}G_{i}$, we have
$(\mbox{ad}E_{i})^{n}=\sum_{\lambda}\Phi_{\lambda}^{n}\prod_{\rho\in{\rm
irr}G_{i}}(\mbox{ad}E_{i\rho})^{\lambda_{\rho}}$
for any positive integer $n$, where $\lambda$ takes thought all the sequence
$\lambda=(\lambda_{\rho})_{\rho\in{\rm irr}G_{i}}$ satisfying
$\sum_{\rho\in{\rm irr}G_{i}}\lambda_{\rho}=n,$
and
$\Phi_{\lambda}^{n}=\left({\begin{array}[]{*{20}c}n\\\ \rho_{1}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}n-\rho_{1}\\\ \rho_{2}\\\
\end{array}}\right)\cdots\left({\begin{array}[]{*{20}c}n-\rho_{1}-\cdots-\rho_{|{\rm
irr}G_{i}|-1}\\\ \rho_{|{\rm irr}G_{i}|}\\\ \end{array}}\right)$
for any $\lambda=(\rho_{1},\rho_{2},\cdots,\rho_{|{\rm irr}G_{i}|})$. In
particular, if $n=1-c_{ij}$, then $\lambda_{\rho}>1-a_{(i\rho)(j\sigma)}$ for
some $\rho\in\mbox{irr}G_{i}$ and so that
$(\mbox{ad}E_{i\rho})^{\lambda_{\rho}}E_{j\sigma}=0,\qquad(\mbox{ad}E_{i})^{1-c_{ij}}E_{j}=0.$
Similarly, $(\mbox{ad}F_{i})^{1-c_{ij}}F_{j}=0$ for any $i,j\in\mathcal{I}$.
Therefore, there exists a non-zero homomorphism
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$.
Since
$\big{(}\mathcal{H}^{\overline{G}},\\{\frac{d_{i}}{|G|}\phi(\varepsilon_{i})\\},\\{\phi(H_{i})\\}\big{)}$
is a realization of $C$ by Proposition 4.3, there is an isomorphism
$\mathfrak{h}(\Gamma)\rightarrow\mathcal{H}^{\overline{G}}$,
$\overline{H}_{i}\rightarrow H_{i}$. Therefore we can get a homomorphism
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$ by compositing
the homomorphisms
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$ and
$\mathfrak{h}(\Gamma)\rightarrow\mathcal{H}^{\overline{G}}$. By [13,
Proposition 1.7(b)],
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$ and
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$ are
monomorphisms. ∎
Now, we can identify $\mathfrak{g}(\Gamma)$ with a subalgebra of
$\mathfrak{g}^{\overline{G}}$. Following from Section 3.1, the map
$h:\quad\mathbb{Z}\widehat{I}\rightarrow\mathbb{Z}\mathcal{I},\qquad\beta\mapsto
h(\beta),\quad h(\beta)_{i}=\sum_{\rho\in{\rm irr}G_{i}}\beta_{i\rho},$
satisfies
$d_{i}\bigg{(}\beta,\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}\bigg{)}_{\widehat{Q}}=(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}$
for all
$\beta=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}\in\mathbb{Z}\widehat{I}$
and $h(\Delta_{\widehat{Q}})=\Delta_{\Gamma}$ by Proposition 3.12.
###### Proposition 4.6.
The monomorphism $\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$
endows $\mathfrak{g}^{\overline{G}}$ with an integrable
$\mathfrak{g}(\Gamma)$-module structure under the adjoint action of
$\mathfrak{g}(\Gamma)$.
###### Proof.
Firstly, we identity the realization
$\big{(}\mathfrak{h}(\Gamma),\\{\overline{\varepsilon}_{i}\\},\\{\overline{H}_{i}\\}\big{)}$
with
$\big{(}\mathcal{H}^{\overline{G}},\\{\epsilon_{i}\\},\\{H_{i}\\}\big{)}$. For
any non-zero
$\beta=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}\in\Delta_{\widehat{Q}}$
and $H\in\mathcal{H}^{\overline{G}}$, we have
$\varepsilon_{i\rho}(H)=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}(H)=\overline{\varepsilon}_{i}(H)$
and
$\beta(H)=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}(H)=\sum_{i\in\mathcal{I}}\Big{(}\sum_{\rho\in{\rm
irr}G_{i}}\beta_{i\rho}\Big{)}\overline{\varepsilon}_{i}(H)=\sum_{i\in\mathcal{I}}h(\beta)_{i}\overline{\varepsilon}_{i}(H)=h(\beta)(H).$
Denote by $H_{\beta}=\\{g\in G\mid g(\beta)=\beta\\}$ and $G_{\beta}$ a
complete set of left coset representatives of $H_{\beta}$ in $G$. Then
$H_{\beta}$ acts on the root space $\mathfrak{g}_{\beta}$. Suppose that
$x\in\mathfrak{g}_{\beta}$ satisfies $g(x)=x$ for any $g\in H_{\beta}$. Let
$\Sigma(x):=\sum_{g\in G_{\beta}}g(x).$
It is easy to see that $\Sigma(x)\in\mathfrak{g}^{\overline{G}}$ and
$[H,\Sigma(x)]=\sum_{g\in G_{\beta}}[H,g(x)]=\sum_{g\in
G_{\beta}}g(\beta)(H)g(x)=h(\beta)(H)\sum_{g\in
G_{\beta}}g(x)=h(\beta)(H)\Sigma(x)$
for all $H\in\mathcal{H}^{\overline{G}}$, since $h(g(\beta))=h(\beta)$ for any
$g\in G$. It follows that $\Sigma(x)$ lies in the weight space
$(\mathfrak{g}^{\overline{G}})_{h(\beta)}$. Note that each element in
$\mathfrak{g}^{\overline{G}}$ can be written as a sum of some $\Sigma(x)$ with
$x\in\mathfrak{g}_{\beta}$, $\beta\in\Delta_{\widehat{Q}}$, we obtain that
$\mathfrak{g}^{\overline{G}}$ is $\mathfrak{h}(\Gamma)$-diagonalisable.
Secondly, it is easy to see that the non-zero weights of
$\mathfrak{g}^{\overline{G}}$ must be roots of $\Gamma$ since
$h(\Delta_{\widehat{Q}})=\Delta_{\Gamma}$. On the other hand, every root of
$\Gamma$ is also a weight of $\mathfrak{g}^{\overline{G}}$ under the adjoint
action by the monomorphism
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$.
Finally, for any $\beta\in\Delta_{\Gamma}$, the set
$\\{\beta+k\overline{\varepsilon}_{i}\mid
k\in\mathbb{Z}\\}\cap\Delta_{\Gamma}$ is finite. Thus the action of
$\overline{E}_{i}$ and $\overline{F}_{i}$ are local nilpotent on
$\mathfrak{g}^{\overline{G}}$. The proof is completed. ∎
Following from the proof of Proposition 4.6,
$(\mathfrak{g}^{\overline{G}})_{\alpha}$ is spanned by the elements
$\Sigma(x)=\sum_{g\in G_{\beta}}g(x)$, where $x\in\mathfrak{g}_{\beta}$
satisfies $g(x)=x$ for any $g\in H_{\beta}$, and
$\beta\in\Delta_{\widehat{Q}}$ satisfies $h(\beta)=\alpha$. Thus, by the
action of $G$ on $\\{E_{i\rho}\\}$ and $\\{F_{i\rho}\\}$, $\mbox{ad}H_{\beta}$
acts on $\mathfrak{g}_{\beta}$ is identity and so that
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{h(\beta)}=1$
for any simple root $\beta$. That is,
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{\alpha}=1$ for all
simple root $\alpha\in\Delta_{\Gamma}$. Moreover, we have the following claim.
###### Claim 4.7.
${\rm dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{\alpha}=1$ for any real
root $\alpha\in\Delta_{\Gamma}$.
###### Proof.
We consider the automorphism
$\overline{r}_{i\rho}:=\mbox{exp}(\mbox{ad}F_{i\rho})\mbox{exp}(-\mbox{ad}E_{i\rho})\mbox{exp}(\mbox{ad}F_{i\rho})$
of $\mathfrak{g}$. Then
$\overline{r}_{i\rho}(\mathfrak{g}_{\beta})=\mathfrak{g}_{r_{i\rho}(\beta)}$
and $\overline{r}_{i\rho}(H)=H-\varepsilon_{i\rho}(H)H_{i\rho}$ for any
$H\in\mathfrak{h}$ (see [13, Lemma 3.8]). Note that $\overline{r}_{i\rho}$ and
$\overline{r}_{i\rho^{\prime}}$ commute for any
$\rho,\rho^{\prime}\in\mbox{irr}G_{i}$, we let
$\overline{S}_{i}:=\prod_{\rho\in{\rm irr}G_{i}}\overline{r}_{i\rho}.$
Then, for any $H\in\mathcal{H}^{\overline{G}}$, we have
$\overline{S}_{i}(H)=H-\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}(H)H_{i\rho}=H-\epsilon_{i}(H)\sum_{\rho\in{\rm
irr}G_{i}}H_{i\rho}=H-\epsilon_{i}(H)H_{i},$
and $\overline{S}_{i}(H)\in\mathcal{H}^{\overline{G}}$. Note that
$\overline{S}_{i}$ and $G$ commute on $\mathfrak{g}^{\prime}$, it deduces that
$\overline{S}_{i}$ can define an automorphism of $\mathfrak{g}^{\overline{G}}$
such that
$\overline{S}_{i}\big{(}(\mathfrak{g}^{\overline{G}})_{\alpha}\big{)}=(\mathfrak{g}^{\overline{G}})_{\widehat{S}_{i}(\alpha)}.$
Therefore, $\overline{S}_{i}$ is an extension of the automorphism
$\mbox{exp}(\mbox{ad}\overline{F}_{i})\mbox{exp}(-\mbox{ad}\overline{E}_{i})\mbox{exp}(\mbox{ad}\overline{F}_{i})$
of $\mathfrak{g}(\Gamma)$.
Let $\alpha\in\Delta_{\Gamma}$ be a real root. By Lemma 3.11 and Proposition
3.12, there exist a real root $\beta\in\Delta_{\widehat{Q}}$ and $\omega\in
C_{G}(\mathcal{W}(\widehat{Q}))$ such that $h(\beta)=\alpha$, $\omega(\beta)$
is a simple root and $H_{\omega(\beta)}=H_{\beta}$. Let
$\omega=\widehat{S}_{i_{1}}\widehat{S}_{i_{2}}\cdots\widehat{S}_{i_{r}}$ and
$\overline{\omega}=\overline{S}_{i_{1}}\overline{S}_{i_{2}}\cdots\overline{S}_{i_{r}}$,
then $\overline{\omega}(\mathfrak{g}_{\beta})=\mathfrak{g}_{\omega(\beta)}$
and hence $\mathfrak{g}_{\beta}$ is fixed by $H_{\beta}$. Finally, note that
all these $\beta$ are in the same $G$-orbit, we have
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{\alpha}=1$. ∎
In particular, if $Q$ is a finite union of Dynkin quivers, then $\mathfrak{g}$
is a direct sum of simple Lie algebras and all roots of $\Gamma$ are real. By
the claim, we have
###### Corollary 4.8.
If $Q$ is a finite union of Dynkin quivers and $G\subseteq{\rm
Aut}(\mathbbm{k}Q)$ is finite abelian, then there is a Lie algebra isomorphism
$\mathfrak{g}(\Gamma)\cong\mathfrak{g}^{\overline{G}}$.
## 5\. Examples
In this section, we give two examples to elucidate our results.
###### Example 5.1.
Let $Q=(I,E)$ be the quiver
The action of $G=\langle g\rangle\cong\mathbb{Z}/6\mathbb{Z}$ on
$\mathbbm{k}Q$ given by
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $\alpha$ | $\beta$ | $\gamma$
---|---|---|---|---|---|---|---
$g$ | $e_{1}$ | $e_{3}$ | $e_{4}$ | $e_{2}$ | $-\beta$ | $-\gamma$ | $-\alpha$
where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to
vertex $i$, $i\in\\{1,2,3,4\\}$. Then the Cartan matrix of $Q$ is
$A=(a_{ij})={\small\left(\begin{array}[]{cccc}2&-1&-1&-1\\\ -1&2&0&0\\\
-1&0&2&0\\\ -1&0&0&2\\\ \end{array}\right)}.$
Let $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}$ be all
the simple roots of the symmetric Kac-Moody algebra $\mathfrak{g}(Q)$. We
endow the root lattice $\mathbb{Z}I$ with a symmetric bilinear form
$(-,-)_{Q}$ via $(\varepsilon_{i},\varepsilon_{j})_{Q}=a_{ij}$ and define
reflection
$r_{i}:\alpha\mapsto\alpha-(\alpha,\varepsilon_{i})_{Q}\varepsilon_{i}$ for
each vertex $i\in I$. Then, it is well-knowen that Weyl group
$\mathcal{W}(Q)\cong(\mathbb{Z}/2\mathbb{Z})^{3}\rtimes S_{4}$, and one can
check that
$\Delta_{Q}=\pm\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4},\varepsilon_{1}+\varepsilon_{2},\varepsilon_{1}+\varepsilon_{3},\varepsilon_{1}+\varepsilon_{4},\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3},\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{4},\varepsilon_{1}+\varepsilon_{3}+\varepsilon_{4},\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4},2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\\}$
is the root system of $\mathfrak{g}(Q)$.
We get the generalized McKay quiver $\widehat{Q}=(\widehat{I},\widehat{E})$ of
$(Q,G)$ as follows.
where $\rho_{i}$ is the irreducible representation of $G=\langle
g\rangle\cong\mathbb{Z}/6\mathbb{Z}$ defined by
$a\cdot g=\xi^{i}a,\qquad a\in\rho_{i},$
$\sigma_{j}$ is the irreducible representation of $\langle
g^{3}\rangle\cong\mathbb{Z}/2\mathbb{Z}$ defined by
$b\cdot g^{3}=\xi^{3j}b,\qquad b\in\sigma_{j},$
and $\xi$ is a primitive $6$-th root of unity. As we have discussed in Section
3.2, by the group isomorphism
$\varphi:G\rightarrow\widetilde{G},\qquad\varphi(g^{i})=\chi_{g^{i}},\qquad\chi_{g^{i}}(g^{j})=\xi^{ij},$
we define the action of $G$ on $\mathbbm{k}Q\ast G$ by setting
$g^{i}(\lambda g^{j})=\xi^{ij}\lambda g^{j}$
for any $g^{i}\in G$, $\lambda g^{j}\in\mathbbm{k}Q\ast G$. This induces an
action of $G=\langle g\rangle\cong\mathbb{Z}/6\mathbb{Z}$ on
$\mathbbm{k}\widehat{Q}$ given by
| $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e^{\prime}_{0}$ | $e^{\prime}_{1}$ | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$g$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{0}$ | $e^{\prime}_{1}$ | $e^{\prime}_{0}$ | $\xi_{0}\alpha_{1}$ | $\xi_{1}\alpha_{2}$ | $\xi_{2}\alpha_{3}$ | $\xi_{3}\alpha_{4}$ | $\xi_{4}\alpha_{5}$ | $\xi_{5}\alpha_{0}$
where idempotent elements $e_{i}$, $e^{\prime}_{i}$ are corresponding to the
vertex $(1,\rho_{i})$, $(2,\sigma_{i})$ respectively, and
$\xi_{i}\in\mathbbm{k}$ satisfying $\xi_{0}\xi_{1}\cdots\xi_{5}=1$. One can
check that the generalized McKay quiver of $(\widehat{Q},G)$ is just the
quiver $Q$.
By the definition given in Section 2.3, we obtain the symmetrisable
generalized Cartan matrix $C$ corresponding to $(Q,G)$, i.e.,
$C={\small\left(\begin{array}[]{cc}2&-1\\\ -3&2\\\ \end{array}\right)}.$
Then the valued graph $\Gamma$ corresponding to $C$ is
Let $\overline{\varepsilon}_{1},\overline{\varepsilon}_{2}$ be all the simple
roots of $\Gamma$. Then the Weyl group
$\mathcal{W}(\Gamma)\cong D_{6}=\langle a,b\mid
a^{2}=1,b^{3}=1,ab=b^{-1}a\rangle$
and root system
$\Delta_{\Gamma}=\\{\overline{\varepsilon}_{1},\overline{\varepsilon}_{2},\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2},2\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2},3\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2},3\overline{\varepsilon}_{1}+2\overline{\varepsilon}_{2}\\}$.
See Section 2.3 for detail.
We consider the map
$h:\quad\mathbb{Z}\widehat{I}\longrightarrow\mathbb{Z}\mathcal{I},\qquad
h(\alpha)_{i}=\sum_{\rho\in{\rm irr}G_{i}}\alpha_{i\rho}$
for any
$\alpha=\sum_{(i,\rho)\in\widehat{I}}\alpha_{i\rho}\varepsilon_{(i\rho)\in\widehat{I}}\in\mathbb{Z}\widehat{I}$.
The restriction of $h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is a
surjective, this means that for any positive root $\beta$ of $\Gamma$, there
exists an indecomposable $\widehat{Q}$-representation $X$ such that $h({\bf
dim}X)=\beta$. For example, we consider the positive root
$\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2}\in\Delta_{\Gamma}$.
Then, we have the following indecomposable $\widehat{Q}$-representation
$X_{(\rho_{3}\sigma_{0})}$:
and obviously, $h({\bf
dim}X_{(\rho_{3}\sigma_{0})})=\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2}$.
Furthermore, for any $0\leq l\leq 5$, $0\leq j\leq 1$ and $l\not\equiv
j\mod{2}$, we define the $\widehat{Q}$-representation
$X_{(\rho_{l}\sigma_{j})}=(X_{i\rho},X_{\alpha})$ by
$X_{i\rho}=\left\\{\begin{array}[]{ll}\mathbbm{k},&\mbox{ if
}(i,\rho)=(1,\rho_{l})\mbox{ or }(2,\sigma_{j});\\\ 0,&\mbox{ otherwise.
}\end{array}\right.\qquad X_{\alpha}=\left\\{\begin{array}[]{ll}1,&\mbox{ if
}\alpha=\alpha_{l};\\\ 0,&\mbox{ otherwise. }\end{array}\right.$
Then, it is easy to see that the set of all indecomposable
$\widehat{Q}$-representations with $h({\bf
dim}X)=\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2}$ is the set
$\left\\{X_{(\rho_{l}\sigma_{j})}\mid 0\leq l\leq 5,~{}0\leq j\leq 1\mbox{ and
}l\not\equiv j\mod{2}\right\\},$
and which is just the orbit of $X_{(\rho_{1}\sigma_{0})}$ under that action of
$G$. Similarly, for any positive real root
$\beta=h(\alpha)\in\Delta_{\Gamma}$, there are $|H_{\alpha}|$ (up to
isomorphism) indecomposable $\widehat{Q}$-representations $X$ such that
$h({\bf dim}X)=\beta.$
###### Example 5.2.
Let $Q=(I,E)$ be the quiver
and $G=\langle a\rangle\times\langle
b\rangle\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. The action
of $G$ on $\mathbbm{k}Q$ is given as follows
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{5^{\prime}}$
---|---|---|---|---|---|---|---|---|---|---
$a$ | $e_{5}$ | $e_{4}$ | $e_{3}$ | $e_{2}$ | $e_{1}$ | $e_{5^{\prime}}$ | $e_{4^{\prime}}$ | $e_{3^{\prime}}$ | $e_{2^{\prime}}$ | $e_{1^{\prime}}$
$b$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{5^{\prime}}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$
| $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha^{\prime}_{1}$ | $\alpha^{\prime}_{2}$ | $\alpha^{\prime}_{3}$ | $\alpha^{\prime}_{4}$
---|---|---|---|---|---|---|---|---
$a$ | $\alpha_{4}$ | $\alpha_{3}$ | $\alpha_{2}$ | $\alpha_{1}$ | $\alpha^{\prime}_{4}$ | $\alpha^{\prime}_{3}$ | $\alpha^{\prime}_{2}$ | $\alpha^{\prime}_{1}$
$b$ | $\alpha^{\prime}_{1}$ | $\alpha^{\prime}_{2}$ | $\alpha^{\prime}_{3}$ | $\alpha^{\prime}_{4}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$
where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to the
vertex $i$. Taken $\mathcal{I}=\\{1,2,3\\}$, then the generalized McKay quiver
of $(Q,G)$ is
where $\rho_{0}$, $\rho_{1}$ are the non-isomorphism irreducible
representations of $G_{3}=\langle a\rangle\cong\mathbb{Z}/2\mathbb{Z}$.
Reindexing the vertex set $\widehat{I}=\\{1,2,(3,\rho_{0}),(3,\rho_{1})\\}$ by
$\\{1,2,3,4\\},$ the Cartan matrix of $\widehat{Q}$ is
$A=(a_{ij})={\small\left(\begin{array}[]{cccc}2&-1&0&0\\\ -1&2&-1&-1\\\
0&-1&2&0\\\ 0&-1&0&2\\\ \end{array}\right)}.$
The Lie algebra $\mathfrak{g}:=\mathfrak{g}(\widehat{Q})$ is generated by
$\\{x_{i},y_{i},h_{i}\mid 1\leq i\leq 4\\}$ satisfying the relations
$\begin{array}[]{llll}\quad[h_{i},h_{j}]=0,&[x_{i},y_{j}]=\delta_{ij}h_{i};\\\
\quad[h_{i},x_{j}]=a_{ij}x_{j},&[h_{i},y_{j}]=-a_{ij}y_{j};\\\
\quad(\mbox{ad}x_{i})^{1-a_{ij}}(x_{j})=0,&(\mbox{ad}y_{i})^{1-a_{ij}}(y_{j})=0,\qquad
i\neq j.\end{array}$
In this case, the valued graph $\Gamma$ of $(Q,G)$ is
with the Cartan Matrix
$C={\small\left(\begin{array}[]{ccc}2&-1&0\\\ -1&2&-1\\\ 0&-2&2\\\
\end{array}\right)}.$
The Lie algebra $\mathfrak{g}(\Gamma)$ is generated by
$\\{X_{i},Y_{i},H_{i}\mid 1\leq i\leq 3\\}$ satisfying the relations
(5.4)
$\displaystyle\begin{array}[]{llll}\quad[H_{i},H_{j}]=0,&[X_{i},Y_{j}]=\delta_{ij}H_{i};\\\
\quad[H_{i},X_{j}]=c_{ij}X_{j},&[H_{i},Y_{j}]=-c_{ij}Y_{j};\\\
\quad(\mbox{ad}X_{i})^{1-c_{ij}}(X_{j})=0,&(\mbox{ad}Y_{i})^{1-c_{ij}}(Y_{j})=0,\qquad
i\neq j.\end{array}$
As discussed in Section 3.2, we see that the vertices $(3,\rho_{0})$ and
$(3,\rho_{1})$ of $\widehat{Q}$ are in the same $G$-orbit. Therefore, the Lie
algebra $\mathfrak{g}^{\overline{G}}$ is generated by
$\\{\overline{x}_{i},\overline{y}_{i},\overline{h}_{i}\mid 1\leq i\leq 3\\},$
where $\overline{x}_{i}=x_{i}$, $\overline{y}_{i}=y_{i}$,
$\overline{h}_{i}=h_{i}$ for $i=1,2$, and $\overline{x}_{3}=x_{3}+x_{4}$,
$\overline{y}_{3}=y_{3}+y_{4}$, $\overline{h}_{3}=h_{3}+h_{4}$, satisfying the
relations (5.4). Then, it is easy to see that the map
$\Phi:\quad\mathfrak{g}(\Gamma)\longrightarrow\mathfrak{g}^{\overline{G}}$
given by
$\Phi(X_{i})=\overline{x}_{i},\quad\Phi(Y_{i})=\overline{y}_{i},\quad\Phi(H_{i})=\overline{h}_{i}$
is an Lie algebra isomorphism.
At last, we consider quivers of $A$-type and $D$-type,
They have the same quiver isomorphism group $G=\mathbb{Z}/2\mathbb{Z}$. In
these cases, we have
$Q$ | $G$ | $\Gamma$ | $\widehat{Q}$ | $\widehat{\Gamma}$ | Conclusion
---|---|---|---|---|---
$A_{2n+1}$ | $\mathbb{Z}/2\mathbb{Z}$ | $C_{n+1}$ | $D_{n+2}$ | $B_{n+1}$ | $\mathfrak{g}(C_{n+1})\cong\mathfrak{g}(D_{n+2})^{\mathbb{Z}/2\mathbb{Z}}$
$D_{n}$ | $\mathbb{Z}/2\mathbb{Z}$ | $B_{n-1}$ | $A_{2n-1}$ | $C_{n-1}$ | $\mathfrak{g}(B_{n-1})\cong\mathfrak{g}(A_{2n-1})^{\mathbb{Z}/2\mathbb{Z}}$
where $C_{n}$ and $B_{n}$ is the $C$-type and $B$-type Dynkin diagram,
respectively.
## References
* [1] I. Assem, D. Simson, A. SKowroński, Elements of representation theory of associative algebras, Cambridge University Press, 2006.
* [2] M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293(1986) 511-531.
* [3] M. Auslander, I. Reiten, S.O. Smalo, Representation theory of Artin algebras. Cambridge Stud. Adv. Math.,Vol.36, Cambridge University Press, 1995.
* [4] E. Brieskorn, Rationale singularitäten komplexer Fliichen, Invent. Math. 4(1968) 336-358.
* [5] W. Crawley-Boevey, M.P. Holland, Noncommutative deformations of Kleinan singularities. Duke Math. J. 92(1998) 605-635.
* [6] L. Demonet, Skew group algebras of path algebras and preprojective algebras, J. Algebra 323(2010) 1052-1059.
* [7] V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 173, 1976.
* [8] J. Guo, On the McKay quivers and $m$-Cartan Matrices, Science in China (Series A: Mathematics), 52(2009) 513-518.
* [9] J. Guo, Martiínez-Villa, Algebra pairs associated to McKay quivers. Comm. in Algebra 30(2002) 1017-1032.
* [10] B. Hou, S. Yang, Skew group algebras of deformed preprojective algebras, J. Algebra (2011), 10.1016/j.jalgebra.2011.02.007. see also: arXiv: 1003.1797.
* [11] A. Hubery, Representations of quiver respecting a quiver automorphism and a of Kac, Ph. D. thesis, Leeds Univeraity 2002.
* [12] A. Hubery, Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. London Math. Soc. 69(2004) 79-96.
* [13] V.G. Kac, Infinite dimensional Lie algebras, 3rd edn, Cambridge University Press, Cambridge, 1990.
* [14] V.G. Kac, S.P. Wang, On automorphisms of Kac-Moody algebras and groups, Adv. in Math. 92(1992) 129-195.
* [15] G. Lusztig, Affine quivers and canonical bases, Publ. Math. Inst. Hautes Études Sci. 76(1992) 111 C163.
* [16] J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R. I., 1980, pp. 183-186.
* [17] M. Reid, McKay correspondance, Preprint, math.AG/9702016.
* [18] I. Reiten, C. Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92(1)(1985) 224-282.
|
arxiv-papers
| 2011-02-19T02:12:36 |
2024-09-04T02:49:17.125307
|
{
"license": "Public Domain",
"authors": "Bo Hou and Shilin Yang",
"submitter": "Bo Hou",
"url": "https://arxiv.org/abs/1102.3951"
}
|
1102.4038
|
# Multi-scale Modeling Approach to Acoustic Emission during Plastic
Deformation
Jagadish Kumar and G. Ananthakrishna Materials Research Centre, Indian
Institute of Science, Bangalore 560012, India
###### Abstract
We address the long standing problem of the origin of acoustic emission
commonly observed during plastic deformation. We propose a frame-work to deal
with the widely separated time scales of collective dislocation dynamics and
elastic degrees of freedom to explain the nature of acoustic emission observed
during the Portevin-Le Chatelier effect. The Ananthakrishna model is used as
it explains most generic features of the phenomenon. Our results show that
while acoustic emission bursts correlated with stress drops are well separated
for the type C serrations, these bursts merge to form nearly continuous
acoustic signals with overriding bursts for the propagating type A bands.
###### pacs:
83.50.-v, *43.40.Le, 62.20.fq, 05.45.-a
Acoustic emission (AE) is observed in an unusually large number of situations.
For example, it is observed during crack nucleation and propagation in
fracture of solids Sam92 , micro-fracturing process Petri94 , martensite
transformation Vives94 ; Rajeevprl , peeling of an adhesive tape Cicc04 ;
Rumiprl , collective dislocation motion etc Miguel ; Weiss . Clearly, sources
that lead to AE signals in such widely different situations are system
specific even as the general mechanism attributed to AE is the abrupt release
of the stored strain energy. The phenomenon is used as a non-destructive tool
in understanding the sources and mechanisms generating AE.
Acoustic emission during plastic deformation refers to high frequency
transient elastic waves generated by abrupt motion of dislocations. AE studies
in plastically deforming metals and alloys have been reported for over four
decades FL67 . The changes in the AE signals during deformation differs from
one type of experiment to another and also on the sample type. Some
correlations has been established between AE signals and the nature of stress-
strain ($\sigma-\epsilon$) curves FL67 ; CR87 ; ZR90 . Conventional yield
phenomenon is accompanied by a peak in AE pattern just beyond the elastic
regime that decays for larger strains. In contrast distinct AE patterns are
observed in the case of unstable plastic deformations such as the Lüders band
and the different types bands in the Portevin - Le Chatelier (PLC) effect CR87
; ZR90 ; Ch02 ; Ch07 . Such differences in AE patterns in different
experimental conditions (and samples) can be attributed to the way
dislocations respond to external forces. Theoretical approaches to AE are
based on Green function approach that use specific model sources such as an
expanding loop which generate AE Malen74 . Clearly, such approaches cannot be
useful if one is interested in following the changes in AE occurring during
the course of deformation since AE signals (as also stress) are averages over
dislocation activity in the entire sample. Despite the vast literature on the
subject, we are not aware of any model that predicts the nature of acoustic
emission during the entire course of deformation.
The purpose of the paper is to propose a theoretical frame-work to describe
both dislocation dynamics and elastic degrees of freedom simultaneously since
it is the abrupt motion of dislocations that transmits the kinetic energy to
the surrounding elastic medium triggering the AE signals. We address the
problem in the context of the PLC effect where the signature of the AE signals
are well correlated with the types of serrations and band types observed at
different strain rates ZR90 ; CR87 ; Ch02 ; Ch07 . Our results show that for
type C serrations, the AE bursts which are correlated with stress drops, are
well separated. As strain rate is increased, the AE bursts tend to merge to
form nearly continuous acoustic signals with overriding bursts for the
propagating type A bands.
It is well known that AE signals in the case of the Lüders and the PLC bands
arise from collective behavior of dislocationsCR87 ; ZR90 ; Ch02 ; Ch07 ; GA07
. In such cases, it is necessary to simultaneously describe the collective
behavior of dislocations and the elastic degrees of freedom. This so far has
not been possible due to several difficulties. First, a major source of
difficulty common to all plastic deformation experiments, is the absence of
theoretical frame work to simultaneously treat the widely separated inertial
time scale and that of dislocation dynamics. Second, there is lack of
dislocation based models to describe collective behavior of dislocationsGA07 .
Third, there is no clarity on how to describe transient acoustic waves.
Finally, even in models describing collective dislocation motion, stress
equilibration is assumed. This, however, no longer holds during the process of
AE generation GA07 ; Bhar03 ; Anan04 .
The PLC instability is characterized by three types of bands and the
associated serrations GA07 . On increasing strain rate or decreasing
temperature, randomly nucleated static type C bands are seen, identified with
large stress drops. Then the type B ’hopping’ bands are seen. Here, a new band
is formed ahead of the previous one in a spatially correlated way giving the
visual impression of hopping propagation. The serrations are more irregular
with smaller amplitude compared to the type C serrations. Finally, the
continuously propagating type A bands associated with small stress drops are
seen.
Our basic idea is to obtain the local plastic strain rate from model equations
that describe the entire spatio-temporal evolution of plastic deformation and
use it as a source term in the wave equation for the elastic strain. Here we
use the Ananthakrishna (AK) model for the PLC effect Anan82 ; Bhar03 ; Anan04
as it reproduces the band types Bhar03 ; Anan04 ; GA07 , and several other
generic features such as the existence of the instability within a window of
strain rates, the negative strain rate behavior etc Anan82 ; Rajesh00 . The
model also predicts chaotic stress drops which has been subsequently verified
Anan83 ; Noro97 . The basic idea of the model is that all the qualitative
features of the PLC effect emerge from nonlinear interaction of a few
dislocation populations, assumed to represent the collective degrees of
freedom of the system. The model consists of densities of mobile, immobile,
and decorated (Cottrell) type dislocations denoted by $\rho_{m}(x,\tau)$,
$\rho_{im}(x,\tau)$ and $\rho_{c}(x,\tau)$ respectively, in the scaled form.
The scaled evolution equations are Anan04 :
$\displaystyle\frac{\partial\rho_{m}}{\partial\tau}$ $\displaystyle=$
$\displaystyle-
b_{0}\rho_{m}^{2}-\rho_{m}\rho_{im}+\rho_{im}-a\rho_{m}+\phi_{eff}^{m}\rho_{m}$
(1) $\displaystyle+$
$\displaystyle\frac{D}{\rho_{im}}\frac{\partial^{2}(\phi_{eff}^{m}(x)\rho_{m})}{\partial
x^{2}},$ $\displaystyle\frac{\partial\rho_{im}}{\partial\tau}$
$\displaystyle=$ $\displaystyle
b_{0}(b_{0}\rho_{m}^{2}-\rho_{m}\rho_{im}-\rho_{im}+a\rho_{c}),$ (2)
$\displaystyle\frac{\partial\rho_{c}}{\partial\tau}$ $\displaystyle=$
$\displaystyle c(\rho_{m}-\rho_{c}),$ (3)
$\displaystyle\frac{d\phi(\tau)}{d\tau}$ $\displaystyle=$ $\displaystyle
d[\dot{\varepsilon}_{a}-\frac{1}{l}\int_{0}^{l}\rho_{m}(x,\tau)\phi_{eff}^{m}(x,\tau)dx],$
(4)
where $\tau$ is the scaled time variable. The term $b_{0}\rho_{m}^{2}$ in Eq.
(1), refers to the formation of dipoles and other dislocation locks,
$\rho_{m}\rho_{im}$ refers to the annihilation of a mobile dislocation with an
immobile one and the source term $\rho_{im}$ represents the athermal or
thermal reactivation of the immobile dislocation. $a\rho_{m}$ represents the
immobilization of mobile dislocations due to aggregation of solute atoms. Once
a mobile dislocation starts acquiring solute atoms we regard it as Cottrell-
type of dislocation $\rho_{c}$. As more and more solute atoms aggregate, they
eventually stop, and are considered as immobile dislocations $\rho_{im}$. This
is the source term $a\rho_{c}$ in Eq. (2). $\phi_{eff}^{m}\rho_{m}$ in Eq. (1)
represents the rate of multiplication of dislocations due to cross slip. This
depends on the velocity of mobile dislocations taken to be
$V_{m}(\phi)=\phi_{eff}^{m}$, where $\phi_{eff}=(\phi-h\rho_{im}^{1/2})$ is
the scaled effective stress, $m$ the velocity exponent, and $h$ a work
hardening parameter. Further, cross-slip allows dislocations to spread into
neighboring spatial locations and thus gives rise to diffusive coupling (last
term in Eq. (1)). These equations are coupled to Eq. (4) that represents the
constant strain rate deformation experiment. In Eq. (4),
${\dot{\varepsilon}}_{a}$ is the scaled applied strain rate,
${\dot{\varepsilon}}(p,x,\tau)=\rho_{m}(x,\tau)\phi_{eff}^{m}(x,\tau)$ is the
local plastic strain rate, $d$ the scaled effective modulus of the machine and
the sample, and $l$ the dimensionless length of the sample. Note that Eq.(4)
assumes stress equilibration. The scaled constants, $a,c$ and $b_{0}$ refer,
respectively, to the concentration of solute atoms slowing down the mobile
dislocation, the diffusion rate of solute atoms to mobile dislocations and the
thermal and athermal reactivation of immobile dislocations. The relevant
parameter is the applied strain rate $\dot{\varepsilon}_{a}$ with respect to
which different types of serrations and the associated bands are observed. The
instability range is found in the interval $30<\dot{\varepsilon}_{a}<1000$.
Equations (1 -4) are discretized on a grid of $N$ points and solved using a
adaptive step size differential equation solver (“MATLAB” ‘ode15s’). In
experiments, bands cannot propagate into the sample due to large strains at
the grips. This is mimicked by choosing the boundary conditions
$\rho_{im}(1,\tau)$ and $\rho_{im}(N,\tau)$ to be two orders higher than the
rest of the sample. In addition, we impose
$\rho_{m}(1,\tau)=\rho_{c}(N,\tau)=0$. The initial values of the dislocation
densities are chosen to be uniformly distributed with a Gaussian spread along
the sample. For the numerical work, we use $a=0.8,b=5\times
10^{-4},c=0.08,d=6\times 10^{-5},m=3.0,h=0,D=0.25,N=100$.
The above equations (Eqs. (1\- 4)) are adequate to obtain the plastic strain
rate only. However, noting that the abrupt collective dislocation motion
triggers the transient elastic waves, we need to describe both elastic degrees
of freedom and dislocation dynamics. This also implies that instantaneous
stress following such an event will display fluctuations that damp-off in
course of time. Indeed, the abrupt slip process induces dissipative forces
that tend to oppose the accelerated motion of the slip interface. This is a
mechanism that ensures eventual approach to mechanical equilibrium. Following
Ref. Land , we represent this dissipation in terms of the Rayleigh dissipation
function (RDF) given by ${\cal R}_{AE}={\Gamma\over
2}\int\Big{[}{\partial\dot{\epsilon}_{e}(y)\over\partial y}\Big{]}^{2}dy$. We
identify ${\cal R}_{AE}$ with acoustic energy dissipated by noting that this
has the form of the energy associated with abrupt dislocation motion during
plastic deformation, i.e., ${\cal R}_{AE}\propto\dot{\epsilon}^{2}(r)$ Rumiepl
. Thus ${\cal R}_{AE}$ is taken to be the energy of the transient elastic
waves. We have shown that the choice of representing the acoustic energy
dissipated in terms of Rayleigh dissipation function has been successful in
predicting the nature of AE signals in varied situations such as the
martensite transformationRajeevprl ; Kalaprl , fracture Rumiepl and peeling
of an adhesive tapeRumiprl ; Jag08a ; Jag08b . Writing down the kinetic energy
($\frac{\rho}{2}\int(\dot{\epsilon}^{2}(y,t)dy$, where $\rho$ is the density),
the potential energy ($\frac{\mu}{2}\int\epsilon^{2}(y,t)dy$, where $\mu$ is
the elastic constant), dispersion of the elastic waves
($\frac{D}{2}[\frac{\partial^{2}\epsilon_{e}}{\partial y^{2}}]^{2}$ with $D$ a
constant), and dissipation ${\cal R}_{AE}$, we get (using Lagrange’s equations
motion),
$\displaystyle\rho\frac{\partial^{2}\epsilon_{e}}{\partial t^{2}}$
$\displaystyle=$ $\displaystyle\mu\frac{\partial^{2}\epsilon_{e}}{\partial
y^{2}}-D\frac{\partial^{4}\epsilon_{e}}{\partial
y^{4}}+\Gamma\frac{\partial^{2}{\dot{\epsilon}_{e}}}{\partial
y^{2}}-\rho\frac{\partial^{2}\epsilon_{p}}{\partial t^{2}}.$ (5)
The second and third terms (on the right hand side) arise from the dispersion
and dissipation terms respectively. In addition, we have included the plastic
strain rate (last term) calculated from Eqs. (1, 2, 3), and Eq. (4). This acts
as a source term in the wave equation for the elastic degrees of freedom that
is expected to generate transient elastic waves. Note that Eq. (5) is general
and applicable to any plastic deformation situation as long as the plastic
strain rate is supplied. Transforming this equation into scaled variables used
in the AK model, we have
$\displaystyle\frac{\partial^{2}\varepsilon_{e}}{\partial\tau^{2}}=\frac{c^{2}}{(\theta
V_{0})^{2}}\frac{\partial^{2}}{\partial
y^{2}}\Big{[}\varepsilon_{e}+\frac{\Gamma}{c^{2}}\dot{\varepsilon_{e}}-\frac{D}{c^{2}}\frac{\partial^{2}\varepsilon_{e}}{\partial
y^{4}}\Big{]}-\frac{\partial\dot{\varepsilon}_{p}(y,\tau)}{\partial\tau}.$ (6)
(The relations between the scaled and unscaled are
$\dot{\epsilon}_{k}(t)=\frac{bV_{0}\gamma}{\beta}{\dot{\varepsilon}}_{k}(\tau)$
and $\tau=\theta V_{0}t$ where $b$ is the Burgers vector,
$\beta,\gamma,\theta$ and $V_{0}$ are constants used in the unscaled AK model
equations. See Ref. Rajesh00 for details.)
Finally, appropriate boundary conditions needs to imposed on Eq. (6) that
should be consistent with those on Eqs. (1-4). This however is not
straightforward. To do this, we first note that numerical solution requires
discretization of Eqs. (1-4) and Eq. (6). Further, as one end of the sample is
fixed and a traction is applied to the other end, the total imposed strain
rate is shared by the machine and the sample. This implies that the machine
elastic element should be included at both ends, i.e., the discrete form of
the wave equation should contain equations of motion for the end points of the
sample and machine. Then, the stiffness of the machine enters naturally in the
equations for the end points. Then, boundary conditions of Eq. (1-4) are
automatically satisfied by these equations. The relevant boundary conditions
for discretized form of Eqs. (6) are $\varepsilon_{1}(\tau)=0,\,{\rm
and}\,\varepsilon_{N}(\tau)=\dot{\varepsilon}_{a}\tau$ for $\tau>0$ where the
subscript 1 and N refer to the end sites. The initial conditions are:
$\varepsilon_{i}(0)=0+\xi,\,\,i=2,..,N-1$ with the random number $\xi$ is
drawn from interval $-\frac{1}{2}<\xi<\frac{1}{2}$.
However, the time scale of plastic strain rate (i.e., Eqs.(1-4)) is typically
$\sim{\dot{\varepsilon}}_{a}$ while that of Eq. (6) is much smaller. Indeed,
the step size in an adaptive step size algorithm used for the solution of Eqs.
(1 \- 4) are significantly larger that the time step required for integrating
Eqs. (6). Thus, we need to ensure that the time variable in Eq. (6) and Eqs.
(1-4) are mapped correctly. Denoting the $i^{th}$ integration time step in the
AK model by $\Delta\tau_{i}$, for the time interval between
$\tau_{i+1}<\tau<\tau_{i}$, we need to ensure that
$m\delta\tau^{\prime}=\Delta\tau_{i}$ where $\delta\tau^{\prime}$ is the fixed
step size used for Eq. (6). Further, we use interpolated values for the
plastic strain rate $\dot{\varepsilon}_{p}(k,\tau)$ (for any $k^{th}$ spatial
element) obtained by using linear interpolation formula
$\varepsilon_{p}(k,\tau)=\varepsilon_{p}(k,\tau_{i})+\frac{\varepsilon_{p}(k,\tau_{i})-\varepsilon_{p}(k,\tau_{i+1})}{\tau_{i}-\tau_{i+1}}\tau$,
where $\tau_{i}<\tau<\tau_{i+1}$ where $\tau_{i}$ is $i^{th}$ time step of
integration of Eqs. (1-4). Moreover, the plastic strain rate calculated from
Eqs. (1-4) has a much coarser length scale compared to the fine length scale
required for wave propagation. Noting that the spatial coupling in the AK
model appears only in Eq. (1), it is easy to show that the strain rate
$\dot{\varepsilon}_{p}(x,\tau)$ in the AK model must be scaled by a factor
$\lambda^{2}$ (assumed to be constant) when used in the wave equation, i.e.,
$\dot{\varepsilon}_{p}(y,\tau)=\lambda^{2}\dot{\varepsilon}_{p}(x,\tau)$ where
$x$ and $y$ refer respectively to spatial coordinates in the AK model and Eq.
(6). (The range of $\lambda$ is $10^{3}-10^{6}$.) The results presented are
for $N=100,\lambda^{2}=10^{3},k_{m}=5k_{s},\frac{c^{2}}{(\theta
V_{0})^{2}}=1500,\gamma/(\theta V_{0})^{2}=10$ and $D/(\theta V_{0})^{2}=1$.
Note that the velocity of acoustic waves is of right order for $\theta
V_{0}\sim 100$.
Figure 1: (a) Uncorrelated type C bands for $\dot{\varepsilon}_{a}=40$. (b)
(Color online) Plots of stress and acoustic emission energy signals.
Figure 2: (a) Partially propagating type B bands for
$\dot{\varepsilon}_{a}=130$. (b) (Color online) Plots of stress and acoustic
emission energy signals in asymptotic regime. The region between the arrows in
figures (a) and (b) are identified.
Figure 3: (a) Fully propagating type A bands at $\dot{\varepsilon}_{a}=240$.
(b) (Color online) Plots of stress and acoustic emission energy signals in
asymptotic regime. The region between the arrows in figures (a) and (b) are
identified.
Equations (1-4) and Eq. (6) are in principle coupled since
${\dot{\varepsilon}}_{p}(y,\tau)$ is a function of stress $\phi(\tau)$. A self
consistent solution of these equations is equivalent to solving the full
dynamical problem involving both plastic deformation and elastic degrees of
freedom with the attendant difficulties. This will not be attempted here.
Instead we provide an approximate method akin to adiabatic methods. The
procedure adopted is to first calculate $\dot{\varepsilon}_{p}(k,\tau)$ for
the entire duration of time by solving Eqs. (1-4). Then, Eqs. (6) is solved
using $\dot{\varepsilon}_{p}(k,\tau)$ as a source term (along with the scale
factor $\lambda$). This gives the elastic strain $\varepsilon_{e}(y,\tau)$.
Then, the integral of $\varepsilon_{e}(y,\tau)$ over the specimen dimension
gives the transient stress $\phi_{tr}(\tau)$ explicitly. While
$\phi_{tr}(\tau)$ will be equal to $\phi$ within the elastic limit, it will be
different from $\phi$ beyond this limit.
The AK model predicts the three band types found with increasing strain rate
Bhar03 ; Anan04 ; GA07 . At low ${\dot{\varepsilon}}_{a}$, say
${\dot{\varepsilon}}_{a}=40$, the uncorrelated static C bands are seen as
shown in Fig. 1(a). The serrations are large and nearly regular. The scaled
acoustic energy dissipated is obtained using $R_{AE}={\Gamma^{s}\over
2}\int\Big{[}{\partial\dot{\varepsilon}_{e}(y)\over\partial y}\Big{]}^{2}dy$,
where $\Gamma^{s}={\Gamma}/{(\theta V_{0})^{2}}$. Since, stress drops in this
case are due to isolated band nucleation, the AE pattern consists of well
separated bursts that are well correlated with the stress drops. Figure 1(b)
shows a typical stress-strain curve along with the AE bursts for
${\dot{\varepsilon}}_{a}=40$. The post burst AE is continuous that gradually
increases until a new burst is seen ZR90 .
At intermediate strain rates, say ${\dot{\varepsilon}}_{a}=130$ hopping type B
bands are seen as shown in Fig. 2(a). These propagate partially and stop mid-
way. Another hopping band reappears in the neighborhood. Often, nucleation
occurs at more than one location. The corresponding asymptotic stress-time
plot is shown in Fig. 2(b). The associated serrations are irregular but are
smaller in magnitude compared to the type C. While the correlation between
stress drops and AE peaks still holds when the propagation is short, the AE
bursts are not as well separated as in the case of type C serrations. A plot
of the AE signal is shown in Fig. 2(b). During hoping propagation, low level
AE activity is seen in the region between two AE bursts (shown by arrows) CR87
; Ch02 ; Ch07 . (A few small bursts are also seen.) As we increase
${\dot{\varepsilon}}_{a}$, the extent of propagation increases with
concomitant decrease in stress drop magnitudes. At high
$\dot{\varepsilon}_{a}$ we find fully propagating type A bands. Figure 3(a)
shows dislocation bands nucleating at one end of the sample and propagating
continuously to other end for ${\dot{\varepsilon}}_{a}=240$. The corresponding
AE pattern [Fig. 3 (b)] appears nearly continuous with a few over-riding
bursts. Large bursts in AE are correlated with the nucleation of the band (or
the band reaching the edge or due to occasional intersection of two bands).
There is a low level AE activity during propagation (the region between the
arrows). In experiments, bands once nucleated trigger a burst in AE but during
propagation very low activity is seen CR87 ; Ch02 ; Ch07 . Thus, the generic
features of AE signals during the PLC effect are well captured.
In summary, we have developed a theoretical frame-work for dealing with widely
separated inertial time scale and that of collective dislocation modes to
explain the nature of acoustic emission patterns observed in the PLC effect.
This has been done by computing the plastic strain rate from the AK model for
the PLC effect and using it as a source term in the wave equation. An
important input in the theory is that the energy of the transient acoustic
wave dissipated caused by the abrupt slip (resulting from collective unpinning
of dislocations) is represented in terms of the Rayleigh dissipation function
Rumiprl ; Rajeevprl . The results show that for type C bands, well separated
burst type AE signals that are correlated with stress drops are seen. As we
increase the strain rate successive bursts tend to merge. For high
$\dot{\varepsilon}_{a}$ where type A propagating bands are seen, the bursts
merge to form continuous type of AE signal. Over riding this are AE bursts
that correspond to band nucleation or a band reaching the edge. These features
are consistent with experimental results ZR90 ; Ch02 ; Ch07 . Other features
such as those from hardening can not be captured here as there is very little
hardening in the AK modelCh02 ; Ch07 . However, an extension of the AK model
that removes this limitation can be used Ritupan . The frame-work is clearly
applicable to other deformation conditions as long as a dislocation based
model can be developed that captures major features of the phenomena. Finally,
better approximate schemes have been designed that also give similar results
Jag10 .
G. A acknowledges the Department of Atomic Energy grant through Raja Ramanna
Fellowship Scheme and and INSA for Senior Scientist postion, and also BRNS
Grant No. 2007/36/62-BRNS/2564. We thank Prof. A. S. Vasudeva Murthy for
useful discussions.
## References
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* (5) M. Ciccotti, B. Giorgini, D. Villet, and M. Barquins, Int. J. Adhes. Adhes. 24, 143 (2004).
* (6) Rumi De and G. Ananthakrishna, Phys. Rev. Lett. 97, 165503 (2006).
* (7) M. C. Miguel et al., Nature 410, 667 (2001).
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* (10) C. H. Caceres and Rodriguez, Acta Metall. 35, 2851 (1987).
* (11) F. Zeides and J. Roman, Scipta Metall. 24, 1919 (1990).
* (12) F. Chmelik et al., Mater. Sci. Eng. A 324, 200 (2002).
* (13) F. Chmelik et al., Mater. Sci. Eng. A 462, 53 (2007).
* (14) K. Malen and L. Bolin, Phys. Stat. Sol. (b) 61, 637 (1974); B. Tirbonod, Int. J. Fracture 58, 21 (1992); B. Polyzos and A. Trochidis, Wave Motion 21, 343 (1995).
* (15) G. Ananthakrishna, Phys. Rep. 440, 113 (2007).
* (16) M. S. Bharathi, S. Rajesh and G. Ananthakrishna, Scripta Mater. 48, 1355 (2003).
* (17) G. Ananthakrishna and M. S. Bharathi, Phys. Rev. E 70, 026111 (2004).
* (18) G. Ananthakrishna and M.C. Valsakumar, J. Phys. D 15, L171 (1982).
* (19) S. Rajesh and G. Ananthakrishna, Phys. Rev. E 61, 3664 (2000).
* (20) G. Ananthakrishna and M. C. Valsakumar, Phys. Lett. A95, 69 (1983).
* (21) G. Ananthakrishna et al.,Phys. Rev. E60,5455(1999); M.S.Bharathi,et al., Phys. Rev.Lett.87,165508(2001).
* (22) L. D. Landau and E. M. Lifschitz, Theory of Elasticity (Pergamon, Oxford, 1986).
* (23) Rumi De and G. Ananthakrishna, Europhys. Lett. 66, 715 (2004).
* (24) S. Sreekala and G. Ananthakrishna, Phys. Rev. Lett. 90, 135501 (2003).
* (25) Jagadish Kumar, M. Ciccotti and G. Ananthakrishna, Phys. Rev. E 77, 045202 (2008).
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* (27) R. Sarmah and G. Ananthakrishna, to be published.
* (28) Jagadish Kumar, Ph. D thesis, Indian Institute of Science, Bangalore (2010), chapter 6.
|
arxiv-papers
| 2011-02-20T02:31:34 |
2024-09-04T02:49:17.136542
|
{
"license": "Public Domain",
"authors": "Jagadish Kumar and G. Ananthakrishna",
"submitter": "G. Ananthakrishna",
"url": "https://arxiv.org/abs/1102.4038"
}
|
1102.4291
|
# On the high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves
A. S. Janfada and S. Salami
Department of Mathematics, Urmia University, Urmia, Iran
a.sjanfada@urmia.ac.ir
salami.sajad@gmail.com
###### Abstract
In this article, we try to find high rank elliptic curves in the family
$E_{n,{\theta}}$ defined over ${\mathbb{Q}}$ by the equation
$y^{2}=x^{3}+2snx-(r^{2}-s^{2})n^{2}x$, where $0<{\theta}<\pi$,
$\cos({\theta})=s/r$ is rational with $0\leq|s|<r$ and $\gcd(r,s)=1$. These
elliptic curves are related to the ${\theta}$-congruent number problem as a
generalization of the classical congruent number problem. We consider two
special cases ${\theta}=\pi/3$ and ${\theta}=2\pi/3$. Then by searching in a
certain known family of ${\theta}$-congruent numbers and using Mestre-Nagao
sum as a sieving tool, we find some square free integers $n$ such that
$E_{n,{\theta}}({\mathbb{Q}})$ has Mordell-Weil rank up to $7$ in the first
case and $6$ in the second case.
## 1 Introduction
Constructing high rank elliptic curves is one of the major problems concerned
the elliptic curves. Dujella [6] collected a list of known high rank elliptic
curves with prescribed torsion groups. The largest known rank, found by Elkies
[9] in 2006, is $28$. Several authors studied this problem for elliptic curves
with certain properties. For instance, we cite [6, 16] for the curves with
given torsion groups, [10, 21] for the curves $x^{3}+y^{3}=k$ related to the
so-called taxicab problem, [7] for the curves $y^{2}=(ax+1)(bx+1)(cx+1)(dx+1)$
induced by Diophantine quadruples $\\{a,b,c,d\\}$, [1] for the curves
$y^{2}=x^{3}+dx$, [8, 20] for the classical congruent number elliptic curves
$y^{2}=x^{3}-n^{2}x$.
In this paper we treat with special cases of a family of elliptic curves which
are closely related to the ${\theta}$-congruent numbers as an extension of the
classical congruent numbers. Let $0<{\theta}<\pi$ and $\cos({\theta})=s/r$ be
a rational number with $0\leq|s|<r$ and $\gcd(r,s)=1$. A positive integer $n$
is called a ${\theta}$-congruent number if there exists a triangle with
rational sides and area equal to $n{\alpha_{\theta}}$, where
${\alpha_{\theta}}=\sqrt{r^{2}-s^{2}}$. Note that for ${\theta}=\pi/2$, a
${\theta}$-congruent number is the ordinary congruent number. It is easy to
see that if a positive integer $n$ is $\theta$-congruent, then so is $nt^{2}$,
for any positive integer $t$. Throughout this paper, we assume $n$ is a square
free positive integer and concentrate on finding ${\theta}$-congruent number
elliptic curves with high Mordell-Weil rank for two special cases
${\theta}=\pi/3$ and $2\pi/3$.
In Section 2, we recall some known results about ${\theta}$-congruent number
elliptic curves; in particular, a criterion for a square free positive integer
to be ${\theta}$-congruent number, a result on which our work hinges. In
Section 3, we describe briefly the Mestre-Nagao sum and Birch and Swinnerton-
Dyer conjecture on any elliptic curves defined on ${\mathbb{Q}}$. In section
4, we describe our strategy for searching the high rank ${\theta}$-congruent
elliptic curves in two cases ${\theta}=\pi/3$ and ${\theta}=2\pi/3$ and then
collect the main results of our works, which includes elliptic curves
$E_{n,{\theta}}$ with high Mordell-Weil (algebraic) rank $r_{{\theta}}^{g}(n)$
in these cases. By an analytic methods, Yoshida [24] proved that
$r_{\pi/3}^{g}(6)=1$, $r_{\pi/3}^{g}(39)=2$ and $r_{2\pi/3}^{g}(5)=1$,
$r_{2\pi/3}^{g}(14)=2$. These integers, indeed, are the smallest ones by
moderate Mordell-Weil rank. Our searching leads to finding square free
integers $n$ such that $3\leq r_{\pi/3}^{g}(n)\leq 7$ and $3\leq
r_{2\pi/3}^{g}(n)\leq 6$.
In our computations we use the Pari/Gp software [2], William Stein’s SAGE
software [27] and Cremona’s MWrank program [4], which use the method of
descent via 2-isogeny for computing the Mordell-Weil rank of the elliptic
curves.
## 2 ${\theta}$-congruent numbers elliptic curves
The problem of determining ${\theta}$-congruent numbers is related to the
problem of finding a non-2-torsion points on the family of elliptic curves
$E_{n,{\theta}}:y^{2}=x^{3}+2snx-(r^{2}-s^{2})n^{2}x,$
called ${\theta}$-congruent number elliptic curves, where $r$ and $s$ are as
in the previous section. This family introduced and studied by Fujiwara [11],
for the first time, and some authors in various point of views. For any $n$
and ${\theta}$ with $0<{\theta}<\pi$, let $E_{n,{\theta}}({\mathbb{Q}})$ be
the group of rational points on $E_{n,{\theta}}$. Fujiwara [12] studied the
torsion groups of the curves $E_{n,{\theta}}$. Hibinio and Kan [13], using a
criterion of Birch, considering modular parameterizations, and studying
Heegner points on some modular curves, constructed some families of prime
$\pi/3$ and $2\pi/3$-congruent numbers. The most important results on
$E_{n,{\theta}}$ was proved by Yoshida [24, 25, 26]. In [24], he constructed
new families of $\pi/3$ and $2\pi/3$-congruent numbers using 2-descent
methods, Heegner points, and Waldesporger’s results on modular forms of half-
integeral weight. He also conjectured that:
1) $n$ is $\pi/3$-congruent number if $n\equiv 6,10,11,13,17,18,21,22$ or $23\
({\rm mod}\ 24)$;
2) $n$ is $2\pi/3$-congruent number if $n\equiv 5,9,10,15,17,19,21,22$ or $23\
({\rm mod}\ 24)$.
Using ternary quadratic forms, Yoshida [24] proved a theorem analogous to the
Tunnell’s theorem [28] for the classical $\pi/2$-congruent number problem. He
also constructed new families of $\pi/3$ and $2\pi/3$-congruent numbers with
two and three prime factors.
The curve $E_{n,\pi/2}$ is the well known congruent number elliptic curve
defined by $y^{2}=x^{3}-n^{2}x$. Finding high rank curves in this family is
due to Rogers [20, 21] and co-work of the present authors with Dujella [8] in
which reference, there is a list of congruent number elliptic curves with
$r_{\pi/2}^{g}(n)\leq 7$. In particular, it is shown that the integers $n=5$,
$34$, $1254$, $29297$, $48272239$, are the smallest $n$ with
$r_{\pi/2}^{g}(n)=1$, $2$, $3$, $4$, $5$, respectively. The smallest known
integer $n$ with $r_{\pi/2}^{g}(n)=6$ is $n=6611719866$, however, its
minimality is not proved yet. The largest known value for $r_{\pi/2}^{g}(n)$
is $7$ with $n=797507543735$, which is found by Rogers [21]. There is no other
known congruent number $n$ for which the Mordell-Weil rank of $E_{n,\pi/2}$ is
equal to $7$.
It is known [15] that $n$ is a congruent number if and only if
$r_{{\theta}}^{g}(n)>0$ for the congruent number elliptic curve $E_{n,\pi/2}$.
A similar result holds for ${\theta}$-congruent numbers.
###### Theorem 1.
(Fujiwara [11]) Let $n$ be any square free positive integer and consider the
elliptic curve $E_{n,{\theta}}$ as above. Then we have:
(i) $n$ is a ${\theta}$-congruent number if and only if there exists a
non-$2$-torsion point in $E_{n,{\theta}}({\mathbb{Q}})$;
(ii) If $n\neq 1,2,3,6$, then $n$ is a ${\theta}$-congruent number if and only
if $r_{{\theta}}^{g}(n)>0$.
Kan [14] proved the following result which gives a family of
$\theta$-congruent numbers. This result is an efficient tool in our work.
###### Lemma 2.
A square free positive integer $n$ is a ${\theta}$-congruent number if and
only if $n$ is the square free part of
$pq(p+q)(2rq+p(r-s)),$ (1)
for some positive integers $p$, $q$ with ${\rm gcd}(p,q)=1$.
## 3 Mestre-Nagao sum and analytic rank
We recall the Mestre-Nagao sum [17, 18, 19] for elliptic curves. Let $E$ be an
elliptic curve over ${\mathbb{Q}}$ and $p$ be any prime. There is both
theoretical and experimental evidence to suggest that elliptic curves of high
ranks have the property that $N_{p}$, the number of elements in
$E({\mathbb{F}}_{p})$, is large for finitely many primes $p$.
Let $N$ be a positive integer and let ${\bf P}_{N}$ be the set of all primes
less than $N$. Mestre-Nagao sum is defined by
$S(N,E)=\sum_{p\in{\bf P}_{N}}(1-\frac{p-1}{N_{p}})\log p=\sum_{p\in{\bf
P}_{N}}\frac{-a_{p}+2}{N_{p}}\log p,$
which can be computed for any elliptic curve. It is experimentally known [18,
19] to expect that high rank curves have large values $S(N,E)$. We cite [3]
for a heuristic argument which links this concept to the famous Birch and
Swinnerton-Dyer conjecture which is simply stated as follows.
###### Conjecture 3.
Let $E$ be an elliptic curve over ${\mathbb{Q}}$. Let $L(E,s)$ be the Hass-
Weil L-function of $E$ and denote by $r^{g}$ the Mordell-Weil rank of
$E({\mathbb{Q}})$. Then the Taylor expansion of $L(E,s)$ about $s=1$ has the
form
$L(E,s)=c(s-1)^{r^{a}}+higher\ order\ terms,$
with $c\neq 0$ and $r^{a}=r^{g}$.
The integer $r^{a}$ is called the analytic rank of elliptic curve $E$, which
is the order of $L(E_{n,{\theta}},s)$ at $s=1$. For an elliptic curve
$E_{n,{\theta}}$, denote $r^{a}$ by $r_{\theta}^{a}(n)$. There are some
algorithms [5] to compute the analytic rank of elliptic curves. In SAGE
software [27], there are three functions to compute the analytic rank of
elliptic curves with small coefficients. We shall use the following function
of SAGE in our computations:
lcalc.analytic_rank(E)
## 4 Our searching strategy and the main results
Now we attempt to find high rank elliptic curves $E_{n,{\theta}}$ when
${\theta}=\pi/3$ and $2\pi/3$. We divide our attempting into two steps
depending on the range of the square free positive integers $n$.
Step (I) $n\leq 5\times 10^{6}$. First of all, using the s-option of MWrank
program, we compute $s_{{\theta}}(n)$, the Selmer rank of $E_{n,{\theta}}$ for
all $3039633$ square free positive integers in this range. It is easily
checked that $r_{\theta}^{g}(n)\leq s_{{\theta}}(n)$. For more details on
Selmer groups of elliptic curves and their ranks we cite [23]. Table 1
distributes these square free integers through the various values of
$s_{\theta}(n)$ in two cases ${\theta}=\pi/3$ and $2\pi/3$. Using MWrank and
considering the Birch and Swinnerton-Dyer conjecture, we find the smallest
$n$’s with $r_{\pi/3}^{g}(n)=3,4,5$ and $r_{2\pi/3}^{g}(n)=3,4$.
$s_{\theta}(n)$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $\geq 6$ | Total
---|---|---|---|---|---|---|---|---
${\theta}=\pi/3$ | 783043 | 1401045 | 734290 | 116158 | 5045 | 52 | 0 | 3039633
${\theta}=2\pi/3$ | 760511 | 1374165 | 751192 | 144641 | 9038 | 86 | 0 | 3039633
Table 1: Distribution of square free integers less than $5\times 10^{6}$
through the various values of $s_{\theta}(n)$ in two cases ${\theta}=\pi/3$
and $2\pi/3$
Step (II) $n>5\times 10^{6}$. In this step, we search for $n$’s with
$r_{\pi/3}^{g}(n)\geq 6$ and $r_{2\pi/3}^{g}(n)\geq 5$. We consider all
different square free ${\theta}$-congruent numbers $n$ of the form (1) in
Lemma 2, where positive integers $p$ and $q$ satisfy in the following
conditions:
$1<p,q\leq 10^{4},\quad{\rm gcd}(p,q)=1,\quad w(n)\geq 4,$
where $w(n)$ is the number of odd prime factors of $n$. Then we get a list of
different $n$’s with more than $7\times 10^{6}$ elements for each of the cases
${\theta}=\pi/3$ and ${\theta}=2\pi/3$. Applying to Mestre-Nagao sum and using
the s-option of MWrank, we reduce the length of this list. In fact, we choose
$n$’s for which
$S(10^{3},E_{n,{\theta}})>15,\quad S(10^{4},E_{n,{\theta}})>20,\quad
S(10^{5},E_{n,{\theta}})>40,$
where $s_{\pi/3}(n)>5$, and $s_{2\pi/3}(n)>4$. These computations are done by
the Pari/Gp software [2]. After computing the value of $r_{\theta}^{g}(n)$ by
MWrank for these candidates, we can find $n$’s with $r_{\theta}^{g}(n)=6,7$
for ${\theta}=\pi/3$ and $n$’s with $r_{\theta}^{g}(n)=5,6$ for
${\theta}=2\pi/3$.
In the following subsections, we collect all $n$’s with $3\leq
r_{\pi/3}^{g}(n)\leq 7$ and $3\leq r_{2\pi/3}^{g}(n)\leq 6$. In each case,
using MWrank, we find a minimal generating set for the Mordell-Weil groups. To
improve the generators, we used the LLL-algorithm to find those generators
with smaller heights.
### 4.1 The case ${\theta}=\pi/3$
Rank 3: The integers $407$ and $646$ are the two smallest integers among
$116158$ integers $n$ less than $5\times 10^{6}$ with $s_{\pi/3}(n)=3$. We
have $r_{\pi/3}^{g}(407)=r_{\pi/3}^{a}(407)=1$, however, for $n=646$ these
ranks are both $3$ and the generators of
$E_{646,\pi/3}:y^{2}=x^{3}+1292x^{2}-1251948x$
are:
P1 = [-722, 34656],
P2 = [6137, 521645],
P3 = [-1216, 40432].
Rank 4: There are $63$ integers $n$ less than $172081$ with $s_{\pi/3}(n)=4$.
For $29$ cases we have $0\leq r_{\pi/3}^{g}(n)\leq 4$ and the others satisfy
$2\leq r_{\pi/3}^{g}(n)\leq 4$. Using SAGE, one can find that
$r_{\pi/3}^{a}(n)=0$ for the former $29$ integers, and $r_{\pi/3}^{a}(n)=2$
for the latter group. So by assuming Birch and Swinnerton-Dyer Conjecture, the
smallest positive integer with $r_{\pi/3}^{g}(n)=4$ is $172081$ whose related
curve
$E_{172081,\pi/3}:y^{2}=x^{3}+344162x^{2}-88835611683x$
has the generators:
P1 = [-505141, -61627202],
P2 = [-58621, -78669382],
P3 = [-440076,-143244738],
P4 = [224175, 92987790].
Rank 5: An easy computation shows that $221746$ is the smallest among $52$
integers $n$ with $s_{\pi/3}(n)=5$. By MWrank, one can see that
$r_{\pi/3}^{g}(221746)=5$, and the related elliptic curve
$E_{221746,\pi/3}:y^{2}=x^{3}+443492x^{2}-147513865548x$
has the following generators:
P1 = [345450, 207822720],
P2 = [-15792, 49357896],
P3 = [994896, 1130036040],
P4 = [-13254, -45063600],
P5 = [-386575, -255989965].
Rank 6: By part (II) of our searching technique, we can get finitely many $n$
with $s_{\pi/3}(n)=6$ and $n>5\times 10^{6}$. Using MWrank, we can find nine
$n$’s with $r_{\pi/3}^{g}(n)=6$ the smallest of which is $n=11229594411$ and
the related curve is of the form
$E_{11229594411,\pi/3}:y^{2}=x^{3}+22459188822x^{2}-378311371906687310763x$
whose generators are:
P1 = [904103532759/25, -992069570757491352/125],
P2 = [1541731888897/16, 2090318638263775025/64],
P3 = [265444083202036/2025, 4636387440736982658134/91125],
P4 = [719501508201/64, 40873417425022581/512],
P5 = [13006760076899764/269361, 1693181585331404000267498/139798359],
P6 = [50286669020153449/278784, 11896090671289659453790795/147197952].
Note that there are also some $n$’s (even smaller than 11229594411) with
$s_{\pi/3}(n)=6$, however, MWrank cannot give the exact values of
$r_{\pi/3}^{g}(n)$. The other $8$ square free numbers are as:
167514827545, 198606002595, 2713148227665, 3302971161265,
3492293850595, 6634009064865, 4058213000419, 455633303263450.
Rank 7: We can find only one $n$ with $r_{\pi/3}^{g}(n)=7$. This is
$n=365803464586$ and the corresponding curve is
$E_{365803464586,\pi/3}:y^{2}=x^{3}+731606929172x^{2}-401436524109362868454188x$
with the generators:
P1 = [433764757524, 212456676940982628],
P2 = [1291274050073, -1689545579159165609],
P3 = [-59335333874904423/3644281, -570541659890431976790514695/6956932429],
P4 = [11954902524369/4, -45277466996084516865/8],
P5 = [2138828658027602/5329, 56890395483549429623312/389017],
P6 = [786769181014433554/80089, 721982407380536692088852160/22665187],
P7 = [-562236028164373765342/540237049, 3617165210435366625559445197360/12556729729907].
Also we can find three integers $n=2185135410173$, $27441232583014$ and
$1892439367910454$ with $s_{\pi/3}(n)=7$ while, using MWrank gives only the
bound $1\leq r_{\pi/3}^{g}(n)\leq 7$ for all of them.
### 4.2 The case ${\theta}=2\pi/3$
Rank 3: There is no any positive square free integer less than $n=221$ for
which $r_{2\pi/3}^{g}(n)=r_{\pi/3}^{a}(n)=s_{2\pi/3}(n)=3$. So, we get the
curve
$E_{221,2\pi/3}:y^{2}=x^{3}-442x^{2}-146523x$
with the generators:
P1 = [-204, 1734],
P2 = [-169, 2704],
P3 = [4131, -249696].
Rank 4: The smallest $n$ with
$r_{2\pi/3}^{g}(n)=r_{\pi/3}^{a}(n)=s_{2\pi/3}(n)=4$ is $12710$. There are
only two integers, $n=4718$ and $6398$, less than $12710$ with
$s_{2\pi/3}(n)=4$, but for these integers we have
$r_{2\pi/3}^{g}(n)=r_{\pi/3}^{a}(n)=0$. Hence we have the curve
$E_{12710,2\pi/3}:y^{2}=x^{3}-25420x^{2}-484632300x$
with the generators:
P1 = [-310, 384400],
P2 = [-9920, -1153200],
P3 = [48050, 5381600],
P4 = [76880, 16337000].
Rank 5: By part (II), we get finitely many $n$’s with $r_{2\pi/3}^{g}(n)=5$
and $n>5\times 10^{6}$, the smallest of which is $n=16470069$. The
corresponding curve
$E_{16470069,2\pi/3}:y^{2}=x^{3}-32940138x^{2}-813789518594283x$
has the generators:
P1 = [-3115959/4, -198146948769/8],
P2 = [-16255958103/1024, -813789518594283/32768],
P3 = [118172745075/1849, -21701053829180880/79507],
P4 = [174895662711/3481, -10850526914590440/205379],
P5 = [18013358979/361, -275820552686448/6859].
Note that there are finitely many $n$’s less than $16470069$ with
$s_{2\pi/3}(n)=5$, but MWrank can not calculate the exact values of
$r_{2\pi/3}^{g}(n)$.
Rank 6: We found $29$ positive integers $n$ with $r_{2\pi/3}^{g}(n)=6$ such
that $n=4562490669$ is the smallest of them, which gives the curve
$E_{4562490669,2\pi/3}:y^{2}=x^{3}-912498132x^{2}-624489630677617068x$
with the generators:
P1 = [1372171206, 2930957696016],
P2 = [24303608784, 3714988879700280],
P3 = [1677715326, -33259028622624],
P4 = [3635049873, -183588193835865],
P5 = [27273656667348/18769, 39342846732689875284/2571353],
P6 = [36967427406/25, 2217080599939296/125].
The other $28$ square free numbers are as:
456249066, 764046470, 902472906, 5062245006, 9667090290, 11801899970,
19969987310, 20240772006, 23819599518, 24080567966, 30834423438, 39360775454,
58181539130, 64256704710, 98708770590, 106366008126, 148280772990,
181684390314, 292826163630, 309000045354, 333515184002, 685374515826,
713465075246, 685374515826, 713465075246, 860842004286, 1185986591790,
1248260820170, 1185986591790, 1248260820170.
Note that we can find two $n$’s with $s_{\theta}(n)=7$, but by MWrank one can
see that $1\leq r_{2\pi/3}^{g}(n)\leq 7$. These integers are $n=162552566$ and
$45010115083565$. Also, for $n=2118002187593054$, we have $s_{\theta}(n)=8$
but MWrank gives only the bound $1\leq r_{2\pi/3}^{g}(n)\leq 8$.
## 5 Acknowledgements
The authors would like to express their gratitude to Andrej Dujella for
reading the first version of this paper.
## References
* [1] J. Aguirre, F. Castaneda, and J. C. Peral, High rank elliptic curves of the form $y^{2}=x^{3}+Bx$, Revista. Math. compl., XIII (2000) no. 1, 1–15.
* [2] C. Batue, K. Belabas, D. Bernardi. H. Cohen, and M. Oliver, The computer aLgebraic system Pari/Gp, Universite Bordeaux I (1999).
http://pari.math.u-bordeaux.fr
* [3] G. Camplbell, Finding elliptic curves and families of elliptic curves over ${\mathbb{Q}}$ of large rank , PhD Thesis, Rutgers University (1999).
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http://www.maths.nott.ac.uk/personal/jec/MWrank
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* [8] A. Dujella, A. S. Janfada and S. Salami, A search for high rank congruent number Elliptic Curves, Journal of Integer Sequences, 12 (2009), Article 09.5.8 .
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http://www.math.harvard.edu/ elkies/compnt.html
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* [18] K. Nagao, An exampel of elliptic curve over ${\mathbb{Q}}$ with rank $\geq 20$, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993) 291–293.
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* [20] N. Rogers, Rank Computations for the congruent number elliptic curves, Exper. Math. 9 (2000) no. 4, 591–594.
* [21] N. Rogers, Elliptic curves $x^{3}+y^{3}=k$ with high rank, PhD Thesis in Mathematics, Harvard University (2004).
* [22] K. Rubin and A. Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soci. (New series), 39 (2002) no. 4, 455–474.
* [23] J. H. Silverman, The Aarithmetic of elliptic curves, Springer-Verlag, Graduate Texts in Mathematics 106, 2nd edition, (2009).
* [24] Shin-ichi Yoshida, Some variant of the congruent number problem, I, Kyushu J. Math. 55 (2001) 387–404.
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* [26] Shin-ichi Yoshida, Some variant of the congruent number problem. III, Electronic print in Chiba University.
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|
arxiv-papers
| 2011-02-21T17:54:33 |
2024-09-04T02:49:17.147513
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ali S. Janfada, Sajad Salami, andrej Dujella, Juan C. Peral",
"submitter": "Sajad Salami",
"url": "https://arxiv.org/abs/1102.4291"
}
|
1102.4293
|
0pt
#### Background:
The comparison of computer generated protein structural models is an important
element of protein structure prediction. It has many uses including model
quality evaluation, selection of the final models from a large set of
candidates or optimisation of parameters of energy functions used in template-
free modelling and refinement. Although many protein comparison methods are
available online on numerous web servers, they are not well suited for large
scale model comparison: (1) they operate with methods designed to compare
actual proteins, not the models of the same protein, (2) majority of them
offer only a single pairwise structural comparison and are unable to scale up
to a required order of thousands of comparisons. To bridge the gap between the
protein and model structure comparison we have developed the Protein Models
Comparator (pm-cmp). To be able to deliver the scalability on demand and
handle large comparison experiments the pm-cmp was implemented “in the cloud”.
#### Results:
Protein Models Comparator is a scalable web application for a fast distributed
comparison of protein models with RMSD, GDT_TS, TM-score and Q-score measures.
It runs on the Google App Engine cloud platform and is a showcase of how the
emerging PaaS (Platform as a Service) technology could be used to simplify the
development of scalable bioinformatics services. The functionality of pm-cmp
is accessible through API which allows a full automation of the experiment
submission and results retrieval. Protein Models Comparator is a free software
released under the Affero GNU Public Licence and is available with its source
code at: http://www.infobiotics.org/pm-cmp
#### Conclusions:
This article presents a new web application addressing the need for a large-
scale model-specific protein structure comparison and provides an insight into
the GAE (Google App Engine) platform and its usefulness in scientific
computing.
Protein structure comparison seems to be most successfully applied to the
functional classification of newly discovered proteins. As the evolutionary
continuity between the structure and the function of proteins is strong, it is
possible to infer the function of a new protein based on its structural
similarity to known protein structures. This is, however, not the only
application of structural comparison. There are several aspects of protein
structure prediction (PSP) where robust structural comparison is very
important.
The most common application is the evaluation of models. To measure the
quality of a model, the predicted structure is compared against the target
native structure. This type of evaluation is performed on a large scale during
the CASP experiment (Critical Assessment of protein Structure Prediction),
when all models submitted by different prediction groups are ranked by the
similarity to the target structure. Depending on the target category, which
could be either a template-based modelling (TBM) target or a free modelling
(FM) target, the comparison emphasis is put either on local similarity and
identification of well predicted regions or global distance between the model
and the native structure [1, 2, 3].
The CASP evaluation is done only for the final models submitted by each group.
These models have to be selected from a large set of computer generated
candidate structures of unknown quality. The most promising models are
commonly chosen with the use of clustering techniques. First, all models are
compared against each other and then, split into several groups of close
similarity (clusters). The most representative elements of each cluster (e.g.
cluster centroids) are selected as final models for submission [4, 5].
The generation of models in the free modelling category, as well as the
process of model refinement in both FM and TBM categories, requires a well
designed protein energy function. As it is believed that the native structure
is in a state of thermodynamic equilibrium and low free energy, the energy
function is used to guide the structural search towards more native-like
structures. Ideally, the energy function should have low values for models
within small structural distance to the native structure, and high values for
the most distinct and non-protein-like models. To ensure such properties, the
parameters of energy functions are carefully optimised on a training set of
models for which the real distances to the native structures are precomputed
[6, 7, 8, 9].
### Model comparison vs. protein alignment
All these three aspects of prediction: evaluation of models quality, selection
of the best models from a set of candidates and the optimisation of energy
functions, require a significant number of structural comparisons to be made.
However, this comparisons are not made between two proteins, but between two
protein models that are structural variants of the same protein and are
composed of the same set of atoms. Because of that, the alignment between the
atoms is known a priori and is fixed, in contrast to comparison between two
different proteins where the alignment of atoms usually has to be found before
scoring the structural similarity.
Even though searching for optimal alignment is not necessary in model
comparison, assessing their similarity is still not straightforward.
Additional complexity is caused in practice by the incompleteness of models.
For example, many CASP submitted models contain the atomic coordinates for
just a subset of the protein sequence. Often even the native structures have
several residues missing as the X-ray crystallography experiments not always
locate all of them. As the model comparison measures operate only on the
structures of equal length, a common set of residues have to be determined for
each pair of models before the comparison is performed (see Figure 1). It
should be noted that this is not an alignment in the traditional sense but
just a matching procedure that selects the residues present in both
structures.
Figure 1: Matching common residues between two structures. There are two
common cases when number of residues differs between the structures: (A) some
residues at the beginning/end of a protein sequence were not located in the
crystallography experiment and (B) structure was derived from templates that
did not cover the entire protein sequence. In both cases pm-cmp performs a
comparison using the maximum common subset of residues. Figure 2:
Application control flow. The interaction with a user is divided into 4 steps:
setup of the experiment options, upload of the structural models, start of the
computations and finally download of the results when ready.
### Comparison servers
Although many protein structure comparison web services are already available
online, they are not well suited for models comparison. Firstly, they do not
operate on a scale needed for such a task. Commonly these methods offer a
simple comparison between two structures (1:1) or in the best case, a
comparison between a single structure and a set of known structures extracted
from the Protein Data Bank (1:PDB). While what is really needed is the ability
to compare a large number of structures either against a known native
structure (1:N) or against each other (N:N). Secondly, the comparison itself
is done using just a single comparison method, which may not be reliable
enough for all the cases (types of proteins, sizes etc.).
An exception to this is the ProCKSI server [10] that uses several different
comparison methods and provides 1:N and N:N comparison modes. However, it
operates with methods designed to compare real proteins, not the models
generated in the process of PSP, and therefore it lacks the ability to use a
fixed alignment while scoring the structural similarity. Also the high
computational cost of these methods makes large-scale comparison experiments
difficult without a support of grid computing facilities (see our previous
work on this topic [11, 12]).
The only server able to perform a large-scale model-specific structural
comparison we are aware of, is the infrastructure implemented to support the
CASP experiment [13]. This service, however, is only available to a small
group of CASP assessors for the purpose of evaluation of the predictions
submitted for a current edition of CASP. It is a closed and proprietary system
that is not publicly available neither as an online server nor in a form of a
source code. Due to that, it cannot be freely used, replicated or adapted to
the specific needs of the users. We have created the Protein Models Comparator
(pm-cmp) to address these issues.
### Google App Engine
We implemented pm-cmp using the Google App Engine (GAE) [14], a recently
introduced web application platform designed for scalability. GAE operates as
a cloud computing environment providing Platform as a Service (PaaS), and
removes the need to consider physical resources as they are automatically
scaled up as and when required. Any individual or a small team with enough
programming skills can build a distributed and scalable application on GAE
without the need to spend any resources on the setup and maintenance of the
hardware infrastructure. This way, scientist freed from tedious configuration
and administration tasks can focus on what they do best, the science itself.
GAE offers two runtime environments based on Python or Java. Both environments
offer almost identical set of platform services, they only differ in maturity
as Java environment has been introduced 12 months after first preview of the
Python one. The environments are well documented and frequently updated with
new features. A limited amount of GAE resources is provided for free and is
enough to run a small application. This limits are consequently decreased with
each release of the platform SDK (Software Development Kit) as the stability
and performance issues are ironed out. There are no set-up costs and all
payments are based on the daily amount of resources (storage, bandwidth, CPU
time) used above the free levels.
In the next sections we describe the overall architecture and functionality of
our web application, exemplify several use cases, present the results of the
performance tests, discuss the main limitations of our work and point out a
few directions for the future.
## Implementation
The pm-cmp application enables users to set up a comparison experiment with a
chosen set of similarity measures, upload the protein structures and download
the results when all comparisons are completed. The interaction between pm-cmp
and the user is limited to four steps presented in Figure 2.
### Application architecture
The user interface (UI) and most of the application logic was implemented in
Python using the web2py framework [15]. Because web2py provides an abstraction
layer for data access, this code is portable and could run outside of the GAE
infrastructure with minimal changes. Thanks to the syntax brevity of the
Python language and the simplicity of web2py constructs the pm-cmp application
is also very easy to extend. For visualisation of the results the UI module
uses Flot [16], a JavaScript plotting library.
The comparison engine was implemented in Groovy using Gaelyk [17], a small
lightweight web framework designed for GAE. It runs in Java Virtual Machine
(JVM) environment and interfaces with the BioShell java library [18] that
implements a number of structure comparison methods. We decided to use Groovy
for the ease of development and Python-like programming experience, especially
that a dedicated GAE framework (Gaelyk) already existed. We did not use any of
the enterprise level Java frameworks such as Spring, Stripes, Tapestry or
Wicket as they are more complex (often require an sophisticated XML-based
configuration) and were not fully compatible with GAE, due to specific
restriction of its JVM. However, recently a number of workarounds have been
introduced to make some of this frameworks usable on GAE.
The communication between the UI module and the comparison engine is done with
the use of HTTP request. The request is sent when all the structures have been
uploaded and the experiment is ready to start (see Figure 3). The comparison
module organises all the computational work required for the experiment into
small tasks. Each task, represented as HTTP request, is put into a queue and
later automatically dispatched by GAE according to the defined scheduling
criteria.
Figure 3: Protein Models Comparator architecture. The application GUI was
implemented in the GAE Python environment. It guides the user through the
setup of an experiment and then sends HTTP request to the comparison engine to
start the computations. The comparison engine was implemented in the GAE Java
environment.
### Distribution of tasks
The task execution on GAE is scheduled with a token bucket algorithm that has
two parameters: a bucket size and a bucket refill rate. The number of tokens
in the bucket limits the number of tasks that could be executed at a given
time (see Figure 4). The tasks that were executed in parallel run on the
separate instances of the application in the cloud. New instances are
automatically created when needed and deleted after a period of inactivity
which enables the application to scale dynamically to the current demand.
Figure 4: Task queue management on Google App Engine. A) 8 tasks has been
added to a queue. The token bucket is full and has 3 tokens. B) Tokens are
used to run 3 tasks and the bucket is refilled with 2 new tokens.
Our application uses tasks primarily to distribute the computations, but also
for other background activities like deletion of uploaded structures or old
experiments data. The computations are distributed as separate structure vs.
structure comparison tasks. Each task reads the structures previously written
to the datastore by the UI module, performs the comparison and stores back the
results. This procedure is slightly optimised with a use of the GAE memcache
service and each time a structure is read for the second time it is served
from a fast local cache instead of being fetched from the slower distributed
datastore. Also to minimise the number of datastore reads all selected
measures are computed together in a single task.
The comparison of two structures starts with a search for the common
$C_{\alpha}$ atoms. Because the comparison methods require both structures to
be equal in length, a common atomic denominator is used in the comparison. If
required, the total length of the models is used as a reference for the
similarity scores, so that the score of a partial match is proportionally
lower than the score of a full length match. This approach makes the
comparison very robust, even for models of different size (as long as they
share a number of atoms).
## Results
The pm-cmp application provides a clean interface to define a comparison
experiment and upload the protein structures. In each experiment the user can
choose which measures and what comparison mode (1:N or N:N) should be used
(see Figure 5). Currently, four structure comparison measures are implemented:
RMSD, GDT_TS [19], TM-score [20] and Q-score [21]. These are the main measures
used in evaluation of CASP models.
Additionally, a user can choose the scale of reference for GDT_TS and TM-
score. It could be the number of matching residues or the total size of the
structures being compared. It changes the results only if the models are
incomplete. The first option is useful when a user is interested in the
similarity score regardless of the number of residues used in comparison. For
example, she submits incomplete models containing only coordinates of residues
predicted with high confidence and wants to know how good these fragments are
alone. On the other hand, a user might want to take into account all residues
in the structures being compared, not just the matching ones. For that, she
would use the second option where the similarity score is scaled by the length
of the target structure (in 1:N comparison mode) or by the length of the
shorter structure from a pair being compared (in N:N comparison mode). This
way a short fragment with a perfect match will have a lower score than a less
perfect full-length match.
After setting up the experiment, the next step is the upload of models. This
is done with the use of Flash to allow multiple file uploads. The user can
track the progress of the upload process of each file and the number of files
left in the upload queue. When the upload is finished a user can start the
computations, or if needed, upload more models.
The current status of recently submitted experiments is shown on a separate
page. Instead of checking the status there, a user can provide an e-mail
address on experiment setup to be notified when the experiment is finished.
The results of the experiment are presented in a form of interactive
histograms showing for each measure the distribution of scores across the
models (see Figure 6). Also a raw data file is provided for download and
possible further analysis (e.g. clustering). In case of errors the user is
notified by e-mail and a detailed list of problems is given. In most cases
errors are caused by inconsistencies in the set of models, e.g. lack of common
residues, use of different chains, mixing models of different proteins or non-
conformance to the PDB format. Despite the errors, the partial results are
still available and contain all successfully completed comparisons.
Figure 5: Experiment setup screen. To set up an experiment the user has to
choose a label for it, optionally provide an e-mail address (if she wants to
be notified about the experiment status), select one or more comparison
measures, and choose the comparison mode (1:N or N:N) and the reference scale.
Figure 6: Example of distribution plots. For a quick visual assessment of
models diversity the results of comparison are additionally presented as
histograms of the similarity/distance values.
URL | Method | Parameters | Return
---|---|---|---
/experiments/setup | POST | label \- string | 303 Redirect
measures \- subset of [RSMD, GDT_TS, TM-score, Q-score]
mode \- first against all or all against all
scale \- match length or total length
/experiments/structures/[id] | POST | file \- multipart/form-data encoded file | HTML link to the uploaded file
/experiments/start/[id] | GET | - | 200 OK
/experiments/status/[id] | GET | - | status in plain text
/experiments/download/[id] | GET | - | results file
Table 1: Description of the RESTful interface of pm-cmp.
There are three main advantages of pm-cmp over the existing online services
for protein structure comparison. First of all, it can work with multiple
structures and run experiments that may require thousands of pairwise
comparisons. Secondly, these comparisons are performed correctly, even if some
residues are missing in the structures, thanks to the residue matching
mechanism. Thirdly, it integrates several comparison measures in a single
service giving the users an option to choose the aspect of similarity they
want to test their models with.
### Application Programming Interface (API)
As Protein Models Comparator is build in the REST (REpresentational State
Transfer) architecture, it is easy to access programmatically. It uses
standard HTTP methods (e.g. GET, POST, DELETE) to provide services and
communicates back the HTTP response codes (e.g. 200 - OK, 404 - Not Found)
along with the content. By using the RESTful API summarised in Table 1, it is
possible to set up an experiment, upload the models, start the computations,
check the experiment status and download the results file automatically. We
provide pm-cmp-bot.py, an example of a Python script that uses this API to
automate the experiment submission and results retrieval. As we wanted to keep
the script simple and readable, the handling of connection problems is limited
to the most I/O intensive upload part and in general the script does not retry
on error, verify the response, etc. Despite of that, it is a fully functional
tool and it was used in several tests described in the next section.
### Performance tests
To examine the performance of the proposed architecture we ran a 48h test in
which a group of beta testers ran multiple experiments in parallel at
different times of a day. As a benchmark we used the models generated by
I-TASSER [22], one of the top prediction methods in the last three editions of
CASP.
From each set containing every 10th structure from the I-TASSER simulation
timeline we selected the top $n$ models, i.e. the closest to the native by
means of RMSD. The number of models was chosen in relation to the protein
length to obtain one small, two medium and one large size experiment as shown
in Table 2. The smallest experiment was four times smaller the the large one
and two times smaller than the medium one.
protein | 1b72A | 1kviA | 1egxA | 1fo5A
---|---|---|---|---
(models*length) | (350x49) | (500x68) | (300x115) | (800x85)
total size | 17150 | 34000 | 34500 | 68000
Table 2: Four sets of protein models used in the performance benchmark
(available for download on the pm-cmp website).
We observed a very consistent behaviour of the application, with a relative
absolute median deviation of the total experiment processing time smaller than
10%. The values reported in Table 3 show the statistics for 15 runs per each
of the four sets of models. The task queue rate was set to 4/s with a bucket
size of 10. Whenever execution of two experiments overlapped, we accounted for
this overlap by subtracting the waiting time from the execution time, so that
the time spent in a queue while the other experiment was still running was not
counted. Using GAE 1.2.7 we were able to run about 30 experiments per day
staying within the free CPU quota.
| | | processing time[s]
---|---|---|---
protein | models | length | median | mad∗ | min | max
1b72A | 350 | 49 | 178 | 17 | 108 | 272
1egxA | 300 | 115 | 195 | 17 | 125 | 274
1kviaA | 500 | 68 | 236 | 16 | 203 | 406
1fo5A | 800 | 85 | 369 | 33 | 307 | 459
*) mad (median absolute deviation) = $median_{i}(|x_{i}-median(X)|)$ Table 3: Results of the performance benchmark.
To test the scalability of pm-cmp we ran additional two large experiments with
approximately 2500 comparisons each (using GAE 1.3.8). We used the models
generated by I-TASSER again: 2500 models for [PDB:1b4bA] (every 5th structure
from the simulation timeline) and 70 models for [PDB:2rebA2] (top models from
every 10th structure sample set). The results of 11 runs per set are
summarised in Table 4. All runs were separated by a 15 minutes inactivity
time, to allow GAE to bring down all active instances. Thus each run activated
the application instances from scratch, instead of reusing instances activated
by the previous run. Because the experiments did not overlap and due to the
use of more mature version of the GAE platform, the relative absolute median
deviation was much lower than in the first performance benchmark and did not
exceed 3.5%.
| | | processing time[s]
---|---|---|---
experiment | models | length | median | mad | min | max
1b4bA (1:N 2501 cmp) | 2500 | 71 | 838.00 | 25.00 | 746 | 903
2reb_2 (N:N 2415 cmp) | 70 | 60 | 854.00 | 29.00 | 731 | 958
Table 4: Performance for large number of comparisons.
To relate the performance of our application to the performance of the
comparison engine executed locally we conducted another test. This time we
followed a typical CASP scenario and we evaluated 308 server submitted models
for the CASP9 target T0618 ([PDB:3nrhA]). The comparison against the target
structure was performed with the use of the pm-cmp-bot and two times were
measured: experiment execution time (as in previous test) and the total time
used by pm-cmp-bot (including upload/download times). The statistics of 11
runs are reported in Table 5. As the experiments were performed in 1:N mode
the file upload process took a substantial 30% of the total time. The local
execution of the comparison engine on a machine with Intel P8400 2.26GHz (2
core CPU) was almost 5 times slower than the execution in the cloud. We
consider this to be a significant speed up, especially having in mind the
conservative setting of the task queue rate (4/s while GAE allows a maximum of
100/s). Our preliminary experiments with GAE 1.4.3 showed that the speedup
possible with the queue rate of 100 tasks per second is at least an order of
magnitude larger.
| | processing time[s]
---|---|---
platform | time | median | mad | min | max
GAE | total | 135 | 4 | 127 | 146
GAE | execution | 89 | 2 | 86 | 97
local | execution | 413 | 8 | 394 | 422
Table 5: Performance compared to local execution.
## Discussion
The pm-cmp application is a convenient tool performing a comparison of a set
of protein models against a target structure (e.g. in model quality assessment
or optimisation of energy functions for PSP) or against each other (e.g. in
selection of the most frequently occurring structures). It is also an
interesting showcase of a scalable scientific computing on the Google App
Engine platform. To provide more inside on the usefulness of GAE in
bioinformatics applications in general, we discuss below the main limitations
of our approach, possible workarounds and future work.
### Response time limit
A critical issue in implementing an application working on GAE was to keep the
response time to each HTTP request below the 30s limit. This is why the
division of work into small tasks and extensive use of queues was required.
However, this might be no longer critical in the recent releases of GAE 1.4.x
which allow the background tasks to run 20 times longer. In our application,
where a single pairwise comparison with all methods never took longer than
10s, the task execution time was never an issue. The bottleneck was the task
distribution routine. As it was not possible to read more than 1000 entities
from a datastore within the 30s time limit, our application was not able to
scale up above the 1000 comparisons per experiment. However, GAE 1.3.1
introduced the mechanism of cursors to tackle this very problem. That is, when
a datastore query is performed its progress can be stored in a cursor and
resumed later. Our code distribution routine simply call itself (by adding
self to the task queue) just before the time limit is reached and continue the
processing in the next cycle. This way our application could scale up to
thousands of models. However, as it currently operates within the free CPU
quota limit, we do not allow very large experiments online yet. For practical
reasons we set the limit to 5000 comparisons. This allows us to divide the
daily CPU limit between several smaller experiments, instead of having it all
consumed by a single large experiment. In the future we would like to monitor
the CPU usage and adjust the size of the experiment with respect to the amount
of the resources left each day.
### Native code execution
Both environments available on GAE are build on interpreted languages. This is
not an issue in case of a standard web applications, however in scientific
computing the efficiency of the code execution is very important (especially
in the context of response time limits mentioned above). A common practice of
binding these languages with fast native modules written in C/C++ is
unfortunately not an option on GAE. No arbitrary native code can be run on the
GAE platform, only the pure Python/Java modules. Although Google extends the
number of native modules available on GAE it is rather unlikely that we will
see anytime soon modules for fast numeric computation such as NumPy. For that
reason we implemented the comparison engine on the Java Virtual Machine,
instead of using Python.
### Bridging Python and Java
Initially we wanted to run our application as a single module written in
Jython (implementation of Python in Java) that runs inside a Java Servlet and
then bridge it with web2py framework to combine Python’s ease of use with the
numerical speed of the JVM. However, we found that this is not possible
without mapping all GAE Python API calls made by web2py framework to its Java
API correspondents. As the amount of work needed to do that exceeded the time
we had to work on the project we attempted to join these two worlds
differently. We decided to implement it as two separate applications, each in
its own environment, but sharing the same datastore. This was not possible as
each GAE application can access only its own datastore. We had to resort to
the mechanism of versions. It was designed to test a new version of an
application while the old one is still running. Each version is independent
from the others in terms of the used environment and they all share the same
datastore. This might be considered to be a hack and not a very elegant
solution but it worked exactly as intended; we end up with two separate
modules accessing the same data.
### Handling large files
There is a hard 1MB limit on the size of a datastore entity. The dedicated
Blobstore service introduced in GAE 1.3.0 makes the upload of large files
possible but as it was considered experimental and did not provide at first an
API to read the blob content, we decided not to use it. As a consequence we
could not use a simple approach of uploading all experiment data in a single
compressed file. Instead, we decided to upload the files one by one directly
to the datastore, since a single protein structure file is usually much
smaller than 1MB. To make the upload process easy and capable of handling
hundreds of files, we used the Uploadify library [23] which combines
JavaScript and Flash to provide a multi-files upload and progress tracking.
Although since GAE 1.3.5 it is now possible to read the content of a blob, the
multiple file decompression still remains a complex issue because GAE lacks a
direct access to the file system. It would be interesting to investigate in
the future if a task cycling technique (used in our distribution routine)
could be used to tackle this problem.
### Vendor lock-in
Although the GAE code remains proprietary, the software development kit (SDK)
required to run the GAE stack locally is a free software licensed under the
Apache Licence 2.0. Information contained in the SDK code allowed the creation
of two alternative free software implementations of the GAE platform: AppScale
[24] and TyphoonAE [25]. The risk of vendor lock-in is therefore minimised as
the same application code could be run outside of the Google’s platform if
needed.
### Comparison to other cloud platforms
GAE provides an infrastructure that automates much of the difficulties related
to creating scalable web applications and is best-suited for small and medium-
sized applications. Applications that need high performance computing, access
to relational database or direct access to operating system primitives might
be better suited for more generic cloud computing frameworks.
There are two major competitors to the Google platform. Microsoft’s Azure
Services are based on the .NET framework and provide a less constrained
environment, but require to write more low level code and do not guarantee
scalability. Amazon Web Services are a collection of low-level tools that
provide Infrastructure as a Service (IaaS), that is storage and hardware.
Users can assign a web application to as many computing units (instances of
virtual machines) as needed. They also receive complete control over the
machines, at the cost of requiring maintenance and administration. Similarly
to Microsoft’s cloud, it does not provide automated scalability, so it is
clearly a trade-off between access at a lower and unconstrained level and the
scalability that has to be implemented by the user. Additionally, both these
platforms are fully paid services and do not offer free/start-up resources.
## Conclusions
Protein Models Comparator is filling the gap between commonly offered online
simple 1:1 protein comparison and the non-public proprietary CASP large-scale
evaluation infrastructure. It has been implemented using Google App Engine
platform that offers automatic scalability on the data storage and the task
execution level.
In addition to a friendly user web interface, our service is accessible
through REST-like API that allows full automation of the experiments (we
provide an example script for remote access). Protein Models Comparator is a
free software, which means anyone can study and learn from its source code as
well as extend it with his own modifications or even set up clones of the
application either on GAE or using one of the alternative platforms such as
AppScale or TyphoonAE.
Although GAE is a great platform for prototyping as it eliminates the need to
set up and maintain the hardware, provides the resources on demand and
automatic scalability, the task execution limit makes it suitable only for
highly parallel computations (i.e. the ones that could be split into small
independent chunks of work). Also a lack of direct disk access and inability
to execute the native code restricts the possible uses of GAE. However,
looking back at the history of changes it seems likely that in the future GAE
platform will become less and less restricted. For example, the long running
background tasks had been on the top of the GAE project roadmap [26] and
recently the task execution limit was raised in GAE 1.4.x making the platform
more suitable for scientific computations.
## Acknowledgements
We would like to thank the fellow researchers who kindly devoted their time to
testing the pm-cmp: E. Glaab, J. Blakes, J. Smaldon, J. Chaplin, M. Franco, J.
Bacardit, A.A. Shah, J. Twycross and C. García-Martínez.
This work was supported by the Engineering and Physical Sciences Research
Council [EP/D061571/1]; and the Biotechnology and Biological Sciences Research
Council [BB/F01855X/1].
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|
arxiv-papers
| 2011-02-21T17:57:04 |
2024-09-04T02:49:17.152313
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawe{\\l} Widera and Natalio Krasnogor",
"submitter": "Pawe{\\l} Widera",
"url": "https://arxiv.org/abs/1102.4293"
}
|
1102.4310
|
# Pentagonal Domain Exchange
Shigeki Akiyama Department of Mathematics, Faculty of Science, Niigata
University, Ikarashi-2 8050 Niigata, 950-2181 Japan
akiyama@math.sc.niigata-u.ac.jp and Edmund Harriss Department of
Mathematical Sciences, 1 University of Arkansas, Fayetteville, AR 72701, USA
edmund.harriss@mathematicians.org.uk
###### Abstract.
Self-inducing structure of pentagonal piecewise isometry is applied to show
detailed description of periodic and aperiodic orbits, and further dynamical
properties. A Pisot number appears as a scaling constant and plays a crucial
role in the proof. Further generalization is discussed in the last section.
The first author is supported by the Japanese Society for the Promotion of
Science (JSPS), grant in aid 21540010.
Adler-Kitchens-Tresser [1] and Goetz [10] initiated the study of piecewise
isometries. This class of maps shows the way to possible generalizations of
results on interval exchanges to higher dimensions [16, 30]. In this paper we
examine the detailed properties of the map shown in Figure 1 from an algebraic
point of view.
Figure 1. A piecewise rotation $T$ on two pieces. The triangle is rotated
$2\pi/5$ around $a$ and the trapezium is rotated $2\pi/5$ around $b$. Periodic
points with short periods are shown below, in two colours to illustrate that
they cluster into groups, each forming a pentagon.
The goal of this paper is to see how this map is applied to show number
theoretical results. First we reprove that almost all orbits in the sense of
Lebesgue measure are periodic, and in addition, there are explicit aperiodic
points. Second we show that aperiodic points forms a proper dense subset of an
attractor of some iterated function system and are recognized by a Büchi
automaton (c.f. Figure 14). The dynamics acting on this set of aperiodic
points are conjugate to the $2$-adic odometer (addition of one) whose explicit
construction is given (Theorem 3). As a result, we easily see that all
aperiodic orbits are dense and uniformly distributed in the attractor. We
finally give a characterization of points which have purely periodic
multiplicative coding by constructing its natural extension (Theorem 6). In
doing so we obtain an intriguing picture Figure 16 that emerges naturally from
taking algebraic conjugates, whose structure is worthy of further study. We
discuss possible generalizations for 7-fold and 9-fold piecewise rotations in
Section 3.
A dynamical system is self-inducing if the first return map to some subset has
the same dynamics as the full map. The most important example is the
irrational rotation, presented as exchange of two intervals. An elementary
example begins with $\Phi$ shown in Figure 2.
Figure 2. An interval exchange map $\Phi$, where
$\lambda=\frac{1+\sqrt{5}}{2}$
For this interval exchange, now consider the second interval $B$. As shown in
Figure 3 this interval is translated to the left once, and to the right. Thus
$\Phi^{2}(B_{1})$ is back in $B$, the interval $B_{2}$ requires one more step,
but $\Phi^{3}(B_{2})$ also lies within $B$. This first return dynamics on $B$
is therefore conjugate to the dynamics on $A\cup B$.
Figure 3. The interval exchange $\Phi$ is self-inducing. The intervals $B_{1}$
and $B_{2}$ are swapped by the first return map of $\Phi$ on the interval $B$.
Self-inducing subsystem of two interval exchange corresponds to purely
periodic orbits of continued fraction expansion and they are efficiently
captured by the continued fraction algorithm.
This gives a motivation to study the interval exchange transform (IET) of
three or more pieces, trying to find higher dimensional continued fraction
with good Diophantine approximation properties. The study of self-inducing
structure of IET’s was started by a pioneer work of Rauzy [26], now called
Rauzy induction, and got extended in a great deal by many authors including
Veech [31] and Zorich [33], see [32] for historical developments.
Self-inducing piecewise isometries emerged from dynamical systems as a natural
generalization of IET [21, 1, 5, 11, 12, 8, 22] and the first return dynamics
appears in outer billiards [28, 6]. Like IET they provide a simple setting to
study many of the deep and perplexing behaviors that can emerge from a
dynamical system.
The self-inducing structure links such dynamical systems to number theoretical
algorithms, such as, digital expansions and Diophantine approximation
algorithms, and allows us to study their periodic orbits by constructing their
natural extensions. This idea leads to complex and beautiful fractal behavior.
Our target is the piecewise isometry in Figure 1, but to illustrate the bridge
formed between the two fields let us begin with a simple conjecture from
number theory:
###### Conjecture 1.
For any $-2<\lambda<2$, each integer sequence defined by $0\leq
a_{n+1}+\lambda a_{n}+a_{n-1}<1$ is periodic.
Since $a_{n+2}\in\mathbb{Z}$ is uniquely determined by
$(a_{n},a_{n+1})\in\mathbb{Z}^{2}$, we treat this recurrence as a map
$(a_{n},a_{n+1})\mapsto(a_{n+1},a_{n+2})$ acting on $\mathbb{Z}^{2}$. It is
natural to set $\lambda=-2\cos(\theta)$ to view this map as a ‘discretized
rotation’:
$\begin{pmatrix}a_{n+1}\\\ a_{n+2}\end{pmatrix}\sim\begin{pmatrix}0&1\\\
-1&-\lambda\end{pmatrix}\begin{pmatrix}a_{n}\\\ a_{n+1}\end{pmatrix}$
with eigenvalues $\exp(\pm\sqrt{-1}\theta)$. As the matrix is conjugate to the
planar rotation matrix of angle $\theta$, putting
$P=\begin{pmatrix}1&0\cr\cos\theta&-\sin\theta\end{pmatrix}$, we have
$P\begin{pmatrix}a_{n+1}\cr
a_{n+2}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\cr\sin\theta&\cos\theta\end{pmatrix}P\begin{pmatrix}a_{n}\cr
a_{n+1}\end{pmatrix}+P\begin{pmatrix}0\cr\langle\lambda
a_{n+1}\rangle\end{pmatrix}$
where $\langle x\rangle$ is the fractional part of $x$. Therefore this gives a
rotation map of angle $\theta$ acting on a lattice $P\mathbb{Z}^{2}$ but the
image requires a bounded perturbation of modulus less than two to fit into
lattice points of $P\mathbb{Z}^{2}$. For conjecture 1 we expect that such
perturbations do not cumulate and the orbits stay bounded, equivalently, all
orbits become periodic.
A nice feature of the map $(a_{n},a_{n+1})\mapsto(a_{n+1},a_{n+2})$ is that it
is clearly bijective on $\mathbb{Z}^{2}$ by symmetry, while under the usual
round off scheme, the digital information should be more or less lost by the
irrational rotation. This motivates dynamical study of global stability of
this algorithm.
The conjecture is trivial when $\lambda=0,\pm 1$. Among non-trivial cases, the
second tractable case is when $\theta$ is rational and $\lambda$ is quadratic
over $\mathbb{Q}$. Akiyama, Brunotte, Pethő and Steiner [3] proved:
###### Theorem 1.
The conjecture is valid for $\lambda=\frac{\pm
1\pm\sqrt{5}}{2},\pm\sqrt{2},\pm\sqrt{3}.$
It seems hard to prove Conjecture 1 for other values. The case
$\lambda=\frac{1-\sqrt{5}}{2}$ was firstly shown by Lowenstein, Hatjispyros
and Vivaldi [21] with heavy computer assistance. A number theoretical proof
for $\frac{1+\sqrt{5}}{2}$ appeared in [2], whose proof is short but not so
easy to generalize. We try to give an accessible account using self-inducing
piecewise isometry in the case $\lambda=\omega=\frac{1+\sqrt{5}}{2}$, together
with its further dynamical behavior. The proof in Section 1 is basically in
[3]. However this version may elucidate the background idea and is directly
connected to the scaling constant of self-inducing structure of piecewise
isometry acting on a lozenge.
A Pisot number is an algebraic integer $>1$ whose conjugates have modulus less
than $1$. Throughout the paper, we will see the importance of the fact that
the scaling constant of self-inducing system is a Pisot number. Our all
discussions heavily depend on this fact. Indeed, Pisot scaling constants often
appear in self-inducing structures of several important dynamical systems, for
e.g., IET and substitutive dynamical systems. We discuss this point in Section
3. It is pretty surprising that we see this phenomenon in cubic piecewise
rotations as well. We hope this paper gives an easy way to access this
interesting area of mathematics.
We wish to show our gratitude to P.Hubert, W.Steiner and F.Vivaldi for helpful
comments and relevant literatures in the development of this manuscript.
## 1\. Proof of the periodicity for golden mean
Setting $\zeta=\exp(2\pi i/5)$, we have $\omega=-\zeta^{2}-\zeta^{-2}$ and
$1/\omega=\zeta+\zeta^{-1}$. The integer ring of $\mathbb{Q}(\zeta)$ coincides
with the ring $\mathbb{Z}[\zeta]$ generated by $\zeta$ in $\mathbb{Z}$,
$\mathbb{Z}[\zeta]$ is a free $\mathbb{Z}$-module generated by
$1,\zeta,\zeta^{2},\zeta^{3}$. Hereafter we use a different base as a
$\mathbb{Z}$-module:
###### Lemma 2.
$\mathbb{Z}[\zeta]$ is a free $\mathbb{Z}$-module of rank $4$ generated by
$1,\omega,\zeta,\omega\zeta$.
###### Proof.
From $\omega=-\zeta^{2}-\zeta^{-2}$, we have
$x_{1}+x_{2}\omega+(y_{1}+y_{2}\omega)\zeta=(x_{1}+y_{2})+(y_{1}+y_{2})\zeta+(y_{2}-x_{2})\zeta^{2}-x_{2}\zeta^{3}.$
On the other hand
$a_{0}+a_{1}\zeta+a_{2}\zeta^{2}+a_{3}\zeta^{3}=(a_{0}-a_{2}+a_{3})-a_{3}\omega+((a_{1}-a_{2}+a_{3})+(a_{2}-a_{3})\omega)\zeta.$
∎
Taking the complex conjugate, the same statement is valid with another basis
$1,\omega,\zeta^{-1},\omega\zeta^{-1}$. Thus each element in
$\mathbb{Z}[\zeta]$ has a unique expression:
$x-\zeta^{-1}y\qquad(x,y\in\mathbb{Z}[\omega]).$
Denote by $\langle x\rangle$ the fractional part of $x\in\mathbb{R}$. Then a
small computation gives
$\displaystyle 0\leq a_{n}+\omega a_{n+1}+a_{n+2}$ $\displaystyle<1$
$\displaystyle a_{n}+\omega a_{n+1}+a_{n+2}$ $\displaystyle=\langle\omega
a_{n+1}\rangle$ $\displaystyle\langle\omega
a_{n}\rangle-\frac{1}{\omega}\langle\omega a_{n+1}\rangle+\langle\omega
a_{n+2}\rangle$ $\displaystyle\equiv 0\pmod{\mathbb{Z}}$ $\displaystyle
x_{n}-(\zeta+\zeta^{-1})x_{n+1}+x_{n+2}$ $\displaystyle\equiv
0\pmod{\mathbb{Z}}$ $\displaystyle(x_{n+1}-\zeta^{-1}x_{n+2})$
$\displaystyle\equiv\zeta^{-1}(x_{n}-\zeta^{-1}x_{n+1})\pmod{\zeta^{-1}\mathbb{Z}}$
and $x_{n}=\langle\omega a_{n}\rangle$. Our problem is therefore embedded into
a piecewise isometry $T$ acting on a lozenge $[0,1)+(-\zeta^{-1})[0,1)$:
$T(x)=\begin{cases}x/\zeta&\text{Im}(x/\zeta)\geq 0\\\
(x-1)/\zeta&\text{Im}(x/\zeta)<0\end{cases}.$
The action of $T$ is geometrically described in Figure 1. The lozenge
$L=[0,1)+(-\zeta^{-1})[0,1)$ is rotated by the multiplication of $-\zeta^{-1}$
and then the trapezoid $\mathcal{Z}$ which falls outside $L$ is pulled back in
by adding $-\zeta^{-1}$. In total, the isosceles triangle $\Delta$ is rotated
clockwise by the angle $3\pi/5$ around the origin and the trapezoid
$\mathcal{Z}$ is rotated by the same angle but around the point
$\frac{1}{2}+i\frac{\sqrt{5(5+2\sqrt{5})}}{10}\simeq 0.5+0.6882i$ indicated by
a black spot, that is the intersection of two diagonals. Our aim is to show
that each point $x\in\mathbb{Z}[\zeta]\cap L$ gives a periodic $T$-orbit.
A. Goetz [10] gave a slightly different map. Ours is an ‘inclined’
modification of [16] and [3].
Clearly the map $T$ is bijective and preserves 2-dimensional Lebesgue measure
$\mu$. However the measure dynamical system $(L,\nu,\mathbb{B},T)$ (with the
$\sigma$-algebra $\mathbb{B}$ of Lebesgue measurable sets) is far from
ergodic. It turned out that orbits of $T$ is periodic for almost all points
but for an exceptional set of Lebesgue measure zero. Our goal is to prove that
the set $\mathbb{Z}[\zeta]$ has no intersection with this exceptional set.
This is not so obvious since $\mathbb{Z}[\zeta]$ is dense in $L$ because
$\mathbb{Z}[\omega]$ is dense in $\mathbb{R}$.
To illustrate the situation, it is instructive to describe an orbit of $1/3$.
See Figure 4.
Figure 4. The orbit of $1/3$
Later we will show that the orbit of $1/3$ is aperiodic and forms a dense
subset of the exceptional set of aperiodic points. Roughly speaking, our task
is to show that $\mathbb{Z}[\zeta]\cap L$ has no intersection with the fractal
set appeared Figure 4.
The key to the proof is a self-inducing structure with a scaling constant
$\omega^{2}$. We consider a region $L^{\prime}=\omega^{-2}L$ and consider the
first return map
$\hat{T}(x)=T^{m(x)}(x)$
for $x\in L^{\prime}$ where $m(x)$ is the minimum positive integer such that
$T^{m(x)}(x)\in L^{\prime}$. For any $x\in L^{\prime}$, the value $m(x)=1,3$
or $6$. We can show that
(1) $\omega^{2}\hat{T}(\omega^{-2}x)=T(x)$
for $x\in L$. The proof is geometric, shown in Figure 5. The return time
$m(x)=3$ in the open pentagon $\Delta^{\prime}=\omega^{-2}\Delta$ [this is
marked $\Delta$ in the figure] and $m(x)=6$ in the shaded pentagonal region
$D$ with three closed and two open edges. In the remaining isosceles triangle
in $\omega^{-2}L$ (whose two equal edges are closed and the other open), the
return time $m(x)$ is $1$.
Figure 5. Self Inducing structure
Note that the equation is valid for all $x\in L^{\prime}$. This makes the
later discussion very simple. Unfortunately this is not the case for other
quadratic values of $\gamma$ and we have to study the behavior of the boundary
independently, see [3].
Let $U$ be the 1-st hitting map to $L^{\prime}$ for $x\in L$, i.e.,
$U(x)=T^{m(x)}(x)$ for the minimum non-negative integer $m(x)$ such that
$T^{m(x)}(x)\in L^{\prime}$. Note that $U$ is a partial function, i.e., $U(x)$
is not defined when there is no positive integer $m$ such that $T^{m}(x)\in
L^{\prime}$. Since
$T(x)=\begin{cases}x/\zeta&x\in\Delta\\\ (x-1)/\zeta&x\in
T(\mathcal{Z})\setminus\Delta\end{cases},$
it is easy to make the map $U$ explicit:
$U(x)=\begin{cases}x&x\in L^{\prime}\\\ \left(x-1\right)/\zeta&x\in
T^{5}(D)\\\ \left(x-\zeta\right)/\zeta^{2}&x\in T^{4}(D)\\\
\left(x-\frac{\zeta}{\omega}\right)/\zeta^{3}&x\in T^{3}(D)\\\
\left(x+\frac{1}{\omega\zeta^{2}}\right)/\zeta^{4}&x\in T^{2}(D)\\\
x+\frac{1}{\omega\zeta}&x\in T(D)\\\ \text{Not defined}&x\in P_{0}\cup
P_{1}\cup P_{2}\end{cases}$
where $P_{0}$ is the largest open pentagon and $P_{1}$ and $P_{2}=P_{1}/\zeta$
are two second largest closed pentagons in Figure 6.
Figure 6. Period Pentagons
Set
$Q=\left\\{0,1,\zeta,\frac{\zeta}{\omega},-\frac{1}{\omega\zeta^{2}},-\frac{1}{\omega\zeta}\right\\}=\\{d_{0},d_{1},d_{2},d_{3},d_{4},d_{5}\\}\subset\mathbb{Z}[\zeta]$
to use later.
We introduce a crucial map $S$ which is the composition of the 1-st hitting
map $U$ and expansion by $\omega^{2}$, i.e. $S(x)=\omega^{2}U(x)$. Denote by
$\pi(x)$ the period of $T$-orbits of $x\in L$ and put $\pi(x)=\infty$ if $x$
is not periodic by $T$. (We easily see $\pi(x)=5$ in $P_{0}$ and $\pi(x)=10$
in $P_{1}\cup P_{2}$ unless $x$ is the centroid of the pentagon.) Then if
$\pi(x)$ and $\pi(S(x))$ are defined and finite, then we see that
$\pi(S(x))<\pi(x)$ which is a consequence of Equation (1). Therefore if
$\pi(x)$ is finite then we have a decreasing sequence
$\pi(x)>\pi(S(x))>\pi(S^{2}(x))>\dots$
of positive integers. This shows that there exists a positive integer $k$ such
that $S^{k}(x)$ is not defined. In this case we say that $S$-orbit of $x$ in
finite. We easily see that if $S$-orbit of $x\in L$ is finite, then clearly
$\pi(x)$ is finite by Equation (1). Thus we have a clear distinction: $x\in L$
is $T$-periodic if and only if its $S$-orbit is finite. Assume that $x\in
L\cap\mathbb{Z}[\zeta]$ gives an infinite $S$-orbit. When $U(x)$ is defined,
we have $U(x)=(x-d_{m(x)})/\zeta^{m(x)}$ with $m(x)=\\{0,1,2,3,4,5\\}$ and
$d_{i}\in Q$ for all $x\in L$. Thus we have
$S^{k}(x)=\omega^{2k}\frac{x}{\zeta^{\sum_{j=1}^{k}m_{j}}}-\sum_{i=1}^{k}\omega^{2(k-i+1)}\frac{d_{m_{i}}}{\zeta^{\sum_{j=i}^{k}m_{j}}}.$
By the assumption $S^{k}(x)$ is defined for $k=1,2,\dots$ and stays in $L$.
Consider the conjugate map $\phi$ which sends $\zeta\rightarrow\zeta^{2}$. As
$\phi(\omega)=-1/\omega$, we have
$\phi(S^{k}(x))=\frac{\phi(x)}{\omega^{2k}\zeta^{2\sum_{j=1}^{k}m_{j}}}-\sum_{i=1}^{k}\frac{d^{\prime}_{m_{i}}}{\omega^{2(k-i+1)}\zeta^{2\sum_{j=i}^{k}m_{j}}}$
with $d^{\prime}_{i}=\phi(d_{i})\in\phi(Q)$. Put $A=\max\\{|d^{\prime}_{i}|\
:\ d_{i}\in Q\\}$. Then we have
$|\phi(S^{k}(x))|\leq|\phi(x)|+\frac{A}{\omega^{2}-1}$
Thus we have $S^{k}(x),\phi(S^{k}(x))$ and their complex conjugates are
bounded by a constant which does not depend on $k$. This implies that the
sequence $(S^{k}(x))_{k}$ must be eventually periodic.
Summing up, for a point $x$ in $\mathbb{Z}[\zeta]$, its $S$-orbit is finite or
eventually periodic. When it is finite then its $T$-orbit is periodic and when
its $S$-orbit is eventually periodic then $T$-orbit is aperiodic. Thus we have
an algorithm for $x\in\mathbb{Z}[\beta]\cap L$ to tell whether its $T$-orbit
is periodic or not. Since
$|\phi(S^{k}(x))|\leq\frac{|\phi(x)|}{\omega^{2k}}+\frac{A}{1-\omega^{-2}},$
for any positive $\varepsilon$, the right hand side is bounded by
$\varepsilon+\frac{A}{\omega^{2}-1}$
for a sufficiently large $k$. This means that under the assumption that there
is an infinite $S$-orbit, the set
$\left\\{x\in\mathbb{Z}[\zeta]\cap L\ :\
|\phi(x)|\leq\varepsilon+\frac{A}{\omega^{2}-1}\right\\}$
contains $x$ with $\pi(x)=\infty$. Since this set is finite, it is equal to
$B=\left\\{x\in\mathbb{Z}[\zeta]\cap L\ :\
|\phi(x)|\leq\frac{A}{\omega^{2}-1}\right\\}$
for a sufficiently small $\varepsilon$. Since there are only finitely many
candidates in $B$, we obtain an algorithm to check whether an element
$x\in\mathbb{Z}[\zeta]\cap L$ with $\pi(x)=\infty$ exists. In fact, all
elements in $B$ gives a finite $S$-expansion, we are done.
The same algorithm applies to $\frac{1}{M}\mathbb{Z}[\zeta]$ with a fixed
positive integer $M$. In this way, we can also show that points in
$\frac{1}{2}\mathbb{Z}[\zeta]$ are periodic. We can find aperiodic orbits in
$\frac{1}{3}\mathbb{Z}[\zeta]$. For example, one can see that $1/3$ has an
aperiodic $T$-orbit because its $S$-orbit:
$\frac{1}{3},\frac{w^{2}}{3},-\frac{\zeta^{-1}}{3},-\frac{\omega^{2}\zeta^{-1}}{3}-\frac{2\zeta^{-1}}{3},-\frac{\omega^{-2}\zeta^{-1}}{3},-\frac{\zeta^{-1}}{3},\dots$
satisfies $S^{2}(1/3)=S^{6}(1/3)$.
It is crucial in the above proof that the scaling constant of the self-
inducing structure is a Pisot number. Scaling constants of piecewise
isometries often become Pisot numbers, moreover algebraic units. We discuss
these phenomena in Section 3.
## 2\. Coding of aperiodic $T$-orbits
Denote by $\mathbf{A}$ the set of all $T$-aperiodic points in $L$. By the
proof of the previous section, we have
$\mathbf{A}=\\{x\in L\ |\ S^{k}(x)\text{ is defined for all }k=1,2,\dots\\}.$
We also have $S(\mathbf{A})\subset\mathbf{A}$. This means that for
$x_{1}\in\mathbf{A}$, there is a $m_{i}\in\\{0,1,2,3,4,5\\}$ and
$x_{i}\in\mathbf{A}$ such that
$\omega^{2}\zeta^{-m_{i}}(x_{i}-d_{m_{i}})=x_{i+1}\in\mathbf{A}$ for
$i=1,2,\dots$. We therefore have an expansion
(2)
$x_{1}=d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\left(d_{m_{4}}+\frac{\zeta^{m_{4}}}{\omega^{2}}\dots\right.\right.\right.$
Conversely a sequence $\\{m_{i}\\}_{i=1,2,\dots}$ defines a single point of
$Y$. Therefore $\mathbf{A}$ must be a subset of the attractor $Y$ of the
iterated function system (IFS):
$Y=\bigcup_{i=0}^{5}\left(\frac{\zeta^{i}}{\omega^{2}}Y+d_{i}\right),$
an approximation of which is depicted in Figure 7(a).
(a) All digits (b) $d_{0},d_{2},d_{3},d_{5}$
Figure 7. Attractors containing $\mathbf{A}$
At this point we can assert that $2$-dimensional Lebesgue measure of aperiodic
points in $L$ must be zero, because $\omega^{4}\simeq 6.854\dots>6$.
We notice that the digits in $Q$ are not arbitrarily chosen because the image
of $S$ must be in $T(\mathcal{Z})$. Thus the digits $d_{1}$ and $d_{4}$
appears only at the beginning in the expression of Equation (2). Therefore it
is more suitable to study $\mathbf{A}\cap T(\mathcal{Z})$. The attractor
(3)
$Y^{\prime}=\left(\frac{1}{\omega^{2}}Y^{\prime}+d_{0}\right)\cup\left(\frac{\zeta^{2}}{\omega^{2}}Y^{\prime}+d_{2}\right)\cup\left(\frac{\zeta^{3}}{\omega^{2}}Y^{\prime}+d_{3}\right)\cup\left(\frac{\zeta^{5}}{\omega^{2}}Y^{\prime}+d_{5}\right)$
is depicted in Figure 7(b).
This iterated function system satisfies OSC by a pentagonal shape $K$ with
whose vertices are
$0,-\zeta^{-1},\zeta,-\zeta\omega^{-1}-\zeta^{-1},-\zeta^{2}\omega^{-1}$
as in Figure 8. We confirm that the pieces
$K_{m}=\frac{\zeta^{m}}{\omega^{2}}K+d_{m}$ do not overlap.
Figure 8. Open set condition
We consider the induced system of $(L,\mathbb{B},\nu,T)$ to $T(\mathcal{Z})$.
Denote by $\widetilde{T}$ the first return map on $T(\mathcal{Z})$. Then the
induced system $(T(\mathcal{Z}),\widetilde{T})$ is the domain exchange of two
isosceles triangle $A$ and $B$ depicted in Figure 9. The triangle $A$ has two
closed edges of equal length and one open edge, while $B$ has one closed edge
and two open edges of the same length. The open regular pentagon $P_{0}$ and
the triangle $B$ move together by $\widetilde{T}$ and can be merged into a
single shape.
Figure 9. Induced Rotation $\widetilde{T}$ on $T(\mathcal{Z})$
We see
(4) $\widetilde{T}(x)=\begin{cases}T^{2}(x)&x\in\Delta\\\ T(x)&x\in
T(\mathcal{Z})\setminus\Delta.\end{cases}$
Again we find self-inducing structure with the scaling constant $\omega^{2}$:
(5) $\omega^{2}\widetilde{T}(\omega^{-2}x)=\widetilde{T}(x)$
for all $x\in T(\mathcal{Z})$. This can be seen in Figure 10 with
$\alpha=\omega^{-2}A$, $\beta=\omega^{-2}B$ and $R=\omega^{-2}P_{0}$.
Figure 10. Self Inducing Structure of $(T(\mathcal{Z}),\widetilde{T})$
This induced dynamics $(T(\mathcal{Z}),\widetilde{T})$ is essential in
describing the set $\mathbf{A}$.
Readers may notice that we can find a self-inducing structure by smaller
scaling constant $\omega$ in Figure 7(b) by taking two connected pieces.
However this choice of inducing region is not suitable because the self-
inducing relation (with flipping) is measure theoretically valid, but has
different behavior on the boundary.
Let us introduce two codings. First is the coding of $T$-orbits of a point $x$
in $L$ in two symbols $\\{0,1\\}$:
$\mathbf{d}(x)=(\psi(T^{n}(x))_{n}\in\\{0,1\\}^{\mathbb{N}}$ where
$\psi(x)=\begin{cases}0&x\in\Delta\\\ 1&x\in\mathcal{Z}\end{cases}.$
For e.g., the $\mathbf{d}(1/3)=10110101011010101101101101\dots$ The second
coding is defined by
$\widetilde{\mathbf{d}}(x)=(\widetilde{\psi}(\widetilde{T}^{n}(x)))_{n}\in\\{a,b\\}^{\mathbb{N}}$
for $x\in T(\mathcal{Z})$ where
$\widetilde{\psi}(x)=\begin{cases}a&x\in\Delta\\\ b&x\in
T(\mathcal{Z})\setminus\Delta.\end{cases}$
For a point $x$ in $T(\mathcal{Z})$ we have two codings by $\\{a,b\\}$ and by
$\\{0,1\\}$. From Equation (4), the two codings are equivalent through the
substitution $a\rightarrow 01$, $b\rightarrow 1$. For a given coding of
$T$-orbit by $\\{0,1\\}$, there is a unique way to retrieve the coding of
$\widetilde{T}$-orbit by $\\{a,b\\}$, because the symbol $0$ must be followed
by $1$. For e.g., $T(1/3)=-2\zeta^{-1}/3\in T(\mathcal{Z})$ is coded in two
ways as:
$\widetilde{\mathbf{d}}(-2\zeta^{-1}/3)=a\>b\>a\>a\>a\>b\>a\>a\>a\>b\>a\>b\>a\>b\>a\>b\dots$
and
$\mathbf{d}(-2\zeta^{-1}/3)=01\>1\>01\>01\>01\>1\>01\>01\>01\>1\>01\>1\>01\>1\>01\>1\dots$
Hereafter we discuss the coding $\widetilde{\mathbf{d}}$. Observing the
trajectory of the region $\omega^{-2}(\Delta)$ and
$\omega^{-2}(T(\mathcal{Z})\setminus\Delta)$ by the first return map by the
iteration of $\widetilde{T}$ to the region $\omega^{-2}T(\mathcal{Z})$, it is
natural to introduce a substitution $\sigma_{0}$:
$a\rightarrow aaba,\quad b\rightarrow baba.$
on $\\{a,b\\}^{*}$ and we have
$\widetilde{\mathbf{d}}(\omega^{-2}x)=\sigma_{0}(\widetilde{\mathbf{d}}(x))$
for $x\in T(\mathcal{Z})$. More generally, following the analogy of the
previous section, the first hitting map to the region
$\omega^{-2}(T(\mathcal{Z}))$ provide us an expansion of a point $x\in
T(\mathcal{Z})$ exactly in the same form as (2) with restricted digits
$\\{d_{0},d_{2},d_{3},d_{5}\\}$. One can confirm that
(6)
$\widetilde{\mathbf{d}}\left(\frac{\zeta^{m}}{\omega^{2}}x+d_{m}\right)=\begin{cases}\sigma_{0}(\widetilde{\mathbf{d}}(x))&m=0\\\
a\oplus\sigma_{0}(\widetilde{\mathbf{d}}(\widetilde{T}(x)))&m=2\\\
ba\oplus\sigma_{0}(\widetilde{\mathbf{d}}(\widetilde{T}^{2}(x)))&m=3\\\
aba\oplus\sigma_{0}(\widetilde{\mathbf{d}}(\widetilde{T}^{3}(x)))&m=5\end{cases}$
where $\oplus$ is the concatenation of letters. Defining conjugate
substitutions by $\sigma_{1}=a\sigma_{0}a^{-1}$,
$\sigma_{2}=ba\sigma_{1}a^{-1}b^{-1}$ and
$\sigma_{3}=aba\sigma_{0}a^{-1}b^{-1}a^{-1}$, i.e.,
$\displaystyle\sigma_{0}(a)=aaba,$ $\displaystyle\quad\sigma_{0}(b)=baba$
$\displaystyle\sigma_{1}(a)=aaab,$ $\displaystyle\quad\sigma_{1}(b)=abab$
$\displaystyle\sigma_{2}(a)=baaa,$ $\displaystyle\quad\sigma_{2}(b)=baba$
$\displaystyle\sigma_{3}(a)=abaa,$ $\displaystyle\quad\sigma_{3}(b)=abab$
one may rewrite
$\widetilde{\mathbf{d}}\left(\frac{\zeta^{m}}{\omega^{2}}x+d_{m}\right)=\begin{cases}\sigma_{0}(\widetilde{\mathbf{d}}(x))&m=0\\\
\sigma_{1}(\widetilde{\mathbf{d}}(\widetilde{T}(x)))&m=2\\\
\sigma_{2}(\widetilde{\mathbf{d}}(\widetilde{T}^{2}(x)))&m=3\\\
\sigma_{3}(\widetilde{\mathbf{d}}(\widetilde{T}^{3}(x)))&m=5.\end{cases}$
We say that an infinite word $y$ in $\\{a,b\\}^{\mathbb{N}}$ is an $S$-adic
limit of $\sigma_{i}\ (i=0,1,2,3)$ if there exist
$y_{i}\in\\{a,b\\}^{\mathbb{N}}$ for $i=1,2,\dots$ such that
$y=\lim_{\ell\rightarrow\infty}\sigma_{m_{1}}\circ\sigma_{m_{2}}\circ\sigma_{m_{3}}\circ\cdots\circ\sigma_{m_{\ell}}(y_{\ell}).$
with $m_{i}\in\\{0,1,2,3\\}$. Since each element $x\in
T(\mathcal{Z})\cap\mathbf{A}$ has an infinite expansion (2) with digits
$\\{d_{0},d_{2},d_{3},d_{5}\\}$, we find $x_{i}\in\omega^{-2}T(\mathcal{Z})$
such that
$\widetilde{\mathbf{d}}(x)=\lim_{\ell\rightarrow\infty}\sigma_{m_{1}}\circ\sigma_{m_{2}}\circ\sigma_{m_{3}}\circ\cdots\circ\sigma_{m_{\ell}}(\widetilde{\mathbf{d}}(x_{\ell})).$
This shows that $\widetilde{\mathbf{d}}(x)$ is an $S$-adic limit of
$\sigma_{i}\ (i=0,1,2,3)$.
Note that from the definition (2) of $\sigma_{i}$, for a given $S$-adic limit
$y$ there is an algorithm to retrieve uniquely the sequence
$(\sigma_{m_{i}})_{i}$. Checking first four letters of $y$, we know the first
letter of $y_{1}$ and to determine $m_{1}$ we need first 6 letters. We can
iterate this process easily.
Summing up, we embedded the set $\mathbf{A}\cap T(\mathcal{Z})$ into the
attractor $Y^{\prime}$ of an IFS (3) and succeeded in characterizing the
coding of $\widetilde{T}$-orbits of points in this attractor as a set of
$S$-adic limits on $\\{\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}\\}$.
However recalling that points in closed pentagons $P_{1}$ and $P_{2}$ are
$T$-periodic and $Y^{\prime}$ is a non-empty compact set, we see from Figure
7(b) that $\mathbf{A}$ is a proper subset of $Y^{\prime}$.
We wish to characterize the set of aperiodic points in $Y^{\prime}$ and its
coding through $\widetilde{\mathbf{d}}$. Recalling the discussion in the
previous section, if $x\in T(\mathcal{Z})$ has periodic $T$-orbits if and only
if there exists a positive integer $k$ such that $S^{k}(x)\in P_{0}\cup
P_{1}\cup P_{2}$. The equivalent statement in the induced system
$(T(\mathcal{Z}),\widetilde{T})$ is that $x\in T(\mathcal{Z})$ is
$\widetilde{T}$-periodic if and only if there exists a positive integer $k$
such that $S^{k}(x)\in P_{0}\cup P_{1}$. Note that we have:
$\widetilde{T}(x)=\begin{cases}\zeta^{-1}(x-p)+p&x\in P_{0}\\\
\zeta^{-2}(x-q)+q&x\in P_{1}\\\ \end{cases}$
where $p=\frac{1}{2}+i\frac{\sqrt{5(5+2\sqrt{5})}}{10}$ (resp.
$q=i\sqrt{\frac{5+\sqrt{5}}{10}}$) is the center of $P_{0}$ (resp. $P_{1}$)
and consequently $\widetilde{T}^{5}(x)=x$ holds for $x\in P_{0}\cup P_{1}$. If
$x\in T(\mathcal{Z})$ and $x$ is $\widetilde{T}$-periodic, then there exist
$x_{i}\in T(\mathcal{Z})$ such that $x_{\ell}\in P_{0}\cup P_{1}$ and
$x=d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\dots\left(d_{m_{\ell}}+\frac{\zeta^{m_{\ell}}}{\omega^{2\ell}}x_{\ell}\right)\dots\right)\right),$
with $m_{i}\in\\{0,2,3,5\\}$. Thus the set of $\widetilde{T}$-periodic points
in $T(\mathcal{Z})$ consists of all the pentagons of the form
(7)
$d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\dots\left(d_{m_{\ell}}+\frac{\zeta^{m_{\ell}}}{\omega^{2\ell}}P_{j}\right)\dots\right)\right)$
with $j=0,1$ and $m_{j}\in\\{0,2,3,5\\}$. From the self-inducing structure
(5), it is easy to see that if two points $x,x^{\prime}$ are in the same
pentagon of above shape and none of them is the center, then they have exactly
the same periods. Moreover two $\widetilde{T}$-orbits keeps constant distance,
i.e.,
$\widetilde{T}^{n}(x)-\widetilde{T}^{n}(x^{\prime})=\zeta^{s}(x-x^{\prime})$
for some integer $s$. The period is completely determined in [3]. We have all
the periodic orbits in $T(\mathcal{Z})$ and therefore have a geometric
description of aperiodic points:
$\mathbf{A}\cap T(\mathcal{Z})=T(\mathcal{Z})\setminus\\{\text{All pentagons
of the form (\ref{Pent})}\\}.$
Subtraction of these pentagons from $T(\mathcal{Z})$ is described by an
algorithm. The initial set is $D_{0}=T(\mathcal{Z})\setminus P_{0}$ with two
open and three closed edges as in the left Figure 11. The interior ${\rm
Inn}(D_{0})$ gives another feasible open set to assure the open set condition
of the IFS of (3). Inductively we define the decreasing sequence of sets
$D_{i+1}=\bigcup_{m\in\\{0,2,3,5\\}}\left(\frac{\zeta^{m}}{\omega^{2}}D_{i}+d_{m}\right)$
for $i=0,2,\dots$. Then $D_{i}$ consists of $4^{i}$ pieces congruent to
$\omega^{-2i}D_{0}$ without overlapping. Note that since
$D_{1}=D_{0}\setminus(P_{1}\cup\omega^{-2}P_{0}\cup\frac{\omega^{-2}P_{0}-1}{\zeta}),$
$D_{1}$ is obtained by subtracting from $D_{0}$ one closed and two open
regular pentagons as in Figure 11.
Figure 11. Pentagon Removal Algorithm
To generate $D_{i+1}$, each $4^{i}$ pieces in $D_{i}$ are subdivided into $4$
sub-pieces by subtracting three small regular pentagons. Clearly all regular
pentagons of the shape (7) are subtracted by this iteration and we obtain
$\mathbf{A}\cap T(\mathcal{Z})=\bigcap_{i=0}^{\infty}D_{i}.$
This observation allows us to symbolically characterize aperiodic points in
$Y^{\prime}$. First, every point $x$ of $Y^{\prime}$ has an address
$d_{m_{1}}d_{m_{2}}\dots\in\\{d_{0},d_{2},d_{3},d_{5}\\}^{\mathbb{N}}$ by the
expansion (2). The address is unique but for countable exceptions. The
exceptional points forms the set of cut points of $Y^{\prime}$ having the
eventually periodic expansion:
$\displaystyle d_{0}d_{2}(d_{0})^{\infty}$ $\displaystyle\simeq$
$\displaystyle d_{3}d_{3}(d_{5})^{\infty}$ $\displaystyle
d_{3}(d_{0})^{\infty}$ $\displaystyle\simeq$ $\displaystyle
d_{2}(d_{5})^{\infty}$ $\displaystyle d_{2}d_{2}(d_{0})^{\infty}$
$\displaystyle\simeq$ $\displaystyle d_{5}d_{3}(d_{5})^{\infty}$
in the suffix of its address, which is understood by Figure 12 where
$K_{mn}=\frac{\zeta^{m}}{\omega^{2}}(\frac{\zeta^{n}}{\omega^{2}}K+d_{n})+d_{m}$.
Figure 12. Subdivision procedure
Note that if a point $x$ in $T(\mathcal{Z})$ is periodic, then there exists a
non-negative integer $k$ such that $S^{k}(x)\in P_{0}\cup P_{1}$. Moreover, if
$x\in Y^{\prime}\cap T(\mathcal{Z})$, then there exists a non-negative integer
$k$ such that $S^{k}(x)\in\partial(P_{1})$, because it can not be an inner
point of $P_{0}$ or $P_{1}$. In other words, such $x$ must be located in the
open edge of one of $4^{k}$ pieces of $D_{k}$. From Figure 11, one can
construct the following Figure 13 which recognize points of two open edges in
$\partial(D_{0})$. For construction, we introduce a new symbol set $\\{R,L\\}$
(right and left) to distinguish which open edge of $D_{k}$ is into focus.
To read the graph and obtain the previous sequences, ignore $\\{R,L\\}$ and
substitute $\\{0,2,3,5\\}$ with $\\{d_{0},d_{2},d_{3},d_{5}\\}$. A point $x\in
Y^{\prime}$ is periodic (or in the open edge of $D_{0}$) if and only if a
suffix of the address
$d_{m_{1}}d_{m_{2}}\dots\in\\{d_{0},d_{2},d_{3},d_{5}\\}^{\mathbb{N}}$ is in
Figure 13. Note that the points with double addresses are on the open edge of
some $D_{i}$ and consequently their suffixes are read in Figure 13. Figure 12
helps this construction. For e.g., the right open edge of $K_{5}$ consists of
the left open edge of $K_{53}$ and the right open edge of $K_{50}$, therefore
we draw outgoing edges from $5R$ to $3L$ and $0R$.
$\textstyle{2R}$$\textstyle{0R}$$\textstyle{0L,2L}$$\textstyle{3R,5R}$$\textstyle{5L}$$\textstyle{3L}$
Figure 13. $\widetilde{T}$-periodic expansions
As a result, the set of addresses of the points in $\mathbf{A}\cap
T(\mathcal{Z})$ are recognized by a Büchi automaton which is the complement of
the Büchi automaton of Figure 14. Here the double bordered states in Figure 14
are final states. Each infinite word produced by the edge labels
$\\{d_{0},d_{2},d_{3},d_{5}\\}$ on this directed graph is accepted, because it
visits infinitely many times the final states. We do not give here the exact
shape of its complement. It is known that complementation of a Büchi automaton
is much harder than the one of a finite automaton, because the subset
construction does not work (c.f. [29, 23]).
$\scriptstyle{3,5}$$\scriptstyle{0,2}$$\scriptstyle{0,2,3,5}$$\scriptstyle{3}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{3}$$\scriptstyle{5}$$\scriptstyle{2}$$\scriptstyle{3}$$\scriptstyle{0}$$\scriptstyle{5}$$\scriptstyle{2}$$\scriptstyle{5}$$\scriptstyle{2}$
Figure 14. Büchi automaton for periodic points in $Y^{\prime}$
Now consider the topology of $\\{a,b\\}^{\mathbb{N}}$ induced from the metric
defined by $2^{-\max_{x_{i}\neq y_{i}}i}$ for
$x=x_{1}x_{2}\dots,y=y_{1}y_{2}\dots\in\\{a,b\\}^{\mathbb{N}}$. Take a fixed
point $w=(w_{i})_{i=0,1,2,\dots}\in\\{a,b\\}^{\mathbb{N}}$ with
$\sigma_{0}(w)=w$. This is computed for e.g., by $\lim_{n}\sigma_{0}^{n}(a)$.
The shift map $V$ is a continuous map from $\\{a,b\\}^{\mathbb{N}}$ to itself
defined by $V((w_{i}))=(w_{i+1})$. Letting $X_{\sigma_{0}}$ be the closure of
the set $\\{V^{n}(w)\ |\ n=0,1,\dots\\}$, we can define the substitutive
dynamical system $(X_{\sigma_{0}},V)$ associated with $\sigma_{0}$. Since
$\sigma_{0}$ is primitive the set $X_{\sigma_{0}}$ does not depend on the
choice of the fixed point and $(X_{\sigma_{0}},V)$ is minimal and uniquely
ergodic (see [9]). Let $\tau$ be the invariant measure of
$(X_{\sigma_{0}},V)$. On the other hand, for the attractor $Y^{\prime}$ there
is the self-similar measure $\nu$, i.e., a unique probability measure (c.f.
Hutchinson [13]) satisfying
$\nu(X)=\frac{1}{4}\sum_{m\in\\{0,2,3,5\\}}\nu\left(\frac{\omega^{2}}{\zeta^{m}}(X-d_{m})\right)$
for $\nu$-measurable sets $\mathbb{B}_{Y^{\prime}}$ in $Y^{\prime}$.
###### Theorem 3.
The restriction of $\widetilde{T}$ to $Y^{\prime}$ is measure preserving and
$(Y^{\prime},\mathbb{B}_{Y^{\prime}},\nu,\widetilde{T})$ is isomorphic to the
$2$-adic odometer $(\mathbb{Z}_{2},x\mapsto x+1)$ as measure dynamical
systems:
(8) $\begin{CD}\mathbb{Z}_{2}@>{+1}>{}>\mathbb{Z}_{2}\\\
@V{\phi}V{}V@V{\phi}V{}V\\\ Y^{\prime}@>{\widetilde{T}}>{}>Y^{\prime}\end{CD}$
where $\phi:\mathbb{Z}_{2}\rightarrow Y^{\prime}$ is almost one to one and
measure preserving, which will be made explicit in the proof. Moreover the map
$\rho:x\mapsto\frac{x-(x\bmod{4})}{4}$
from $\mathbb{Z}_{2}$ to itself gives a commutative diagram:
(9) $\begin{CD}\mathbb{Z}_{2}@>{\rho}>{}>\mathbb{Z}_{2}\\\
@V{\phi}V{}V@V{\phi}V{}V\\\ Y^{\prime}@>{S}>{}>Y^{\prime}.\end{CD}$
The above theorem may be read that
$(Y^{\prime},\mathbb{B}_{Y^{\prime}},\nu,\widetilde{T})$ gives a one-sided
variant of numeration system in the sense of Kamae [15].
###### Proof.
First we confirm that $\widetilde{T}$ is measure preserving. Denote by
$[d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}]$ the cylinder set:
(10)
$d_{m_{1}}+\frac{\zeta^{m_{1}}}{\omega^{2}}\left(d_{m_{2}}+\frac{\zeta^{m_{2}}}{\omega^{2}}\left(d_{m_{3}}+\frac{\zeta^{m_{3}}}{\omega^{2}}\dots\left(d_{m_{\ell}}+\frac{\zeta^{m_{\ell}}}{\omega^{2\ell}}Y^{\prime}\right)\dots\right)\right)$
By the OSC, we have $\nu([d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}])=4^{-\ell}$.
From Figure 10, we see that $\widetilde{T}^{-1}([d_{3}])=[d_{5}]$,
$\widetilde{T}^{-1}([d_{2}])=[d_{3}]$, $\widetilde{T}^{-1}([d_{0}])=[d_{2}]$
but $\widetilde{T}^{-1}([d_{5}])$ intersects both $A$ and $B$. Hence if
$m_{1}=0,2,3$, then
$\nu(\widetilde{T}^{-1}([d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}]))=4^{-\ell}$.
By using the self-inducing structure in Figure 10, we also have
$\widetilde{T}^{-1}([d_{5}d_{3}])=[d_{0}d_{5}]$,
$\widetilde{T}^{-1}([d_{5}d_{2}])=[d_{0}d_{3}]$ and
$\widetilde{T}^{-1}([d_{5}d_{0}])=[d_{0}d_{2}]$. Thus if $m_{2}=0,2,3$, then
$\nu(\widetilde{T}^{-1}([d_{5},d_{m_{2}},\dots,d_{m_{\ell}}]))=4^{-\ell}$.
Repeating this, we can show that
$\nu(\widetilde{T}^{-1}([d_{m_{1}},d_{m_{2}},\dots,d_{m_{\ell}}]))=4^{-\ell}$
holds for all $m_{i}\in\\{0,2,3,5\\}$ but a single exception
$m_{1}=m_{2}=\dots=m_{\ell}=5$. Since $\ell$ is arbitrary chosen, a simple
approximation argument shows that $\widetilde{T}$ is measure preserving and
$(Y^{\prime},\mathbb{B}_{Y^{\prime}},\nu,\widetilde{T})$ forms a measure
dynamical system.
Let us define a map $\eta$ from $X_{\sigma_{0}}$ to $Y^{\prime}$. Take an
element $z=x_{1}x_{2}\dots\in X_{\sigma_{0}}$. Then each prefix
$x_{1}x_{2}\dots x_{\ell}$ with $\ell>3$ is a subword of the fix point $v$ of
$\sigma_{0}$ starting with $a$. Therefore there is a word
$y\in\\{\lambda,a,ba,aba\\}$ and $z_{1}\in X_{\sigma_{0}}$ such that
$x=y_{1}\sigma_{0}(z_{1})$. It is easy to see from (6) that this $y$ and
$z_{1}$ are unique.
Iterating this we have $z_{i}=y_{i+1}\sigma(z_{i+1})$ with
$y_{i}\in\\{\lambda,a,ba,aba\\}$, $z_{i+1}\in X_{\sigma_{0}}$ and $z_{0}=z$.
Thus we have for any $\ell$,
$\displaystyle z$ $\displaystyle=$ $\displaystyle
y_{1}\sigma_{0}(y_{2}\sigma_{0}(y_{3}\sigma_{0}\dots
y_{\ell}(\sigma_{0}(z_{\ell}))))$ $\displaystyle=$ $\displaystyle
y_{1}\sigma_{0}(y_{2})\sigma_{0}^{2}(y_{3})\dots\sigma_{0}^{\ell-1}(y_{\ell})\sigma_{0}^{\ell}(z_{\ell}).$
Define a map from $\\{\lambda,a,ba,aba\\}$ to $\mathbb{Z}$ by
$\kappa(\lambda)=0,\kappa(a)=1,\kappa(ba)=2,\kappa(aba)=3.$
Then $z_{i}=y_{i+1}\sigma(z_{i+1})$ is equivalent to
$z_{i}=\sigma_{\kappa(y_{i+1})}(z_{i+1})$ and $z$ is represented as an
$S$-adic limit:
$z=\lim_{\ell\rightarrow\infty}\sigma_{\kappa(y_{1})}\circ\sigma_{\kappa(y_{2})}\circ\dots\circ\sigma_{\kappa(y_{\ell})}(z_{\ell}).$
for $\ell=1,2,\dots$. This gives a multiplicative coding
$\mathbf{d^{\prime}}:X_{\sigma_{0}}\rightarrow\\{\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}\\}^{\mathbb{N}}$.
Let $\mathbf{A}^{\prime}$ be the points of $X_{\sigma_{0}}$ whose
multiplicative coding does not end up in an infinite word produced by reading
the vertex labels of Figure 15.
$\textstyle{\sigma_{1}}$$\textstyle{\sigma_{0}}$$\textstyle{\sigma_{0},\sigma_{1}}$$\textstyle{\sigma_{2},\sigma_{3}}$$\textstyle{\sigma_{3}}$$\textstyle{\sigma_{2}}$
Figure 15. Forbidden suffix of $\mathbf{A}^{\prime}$
Let us associate to $z$ a $2$-adic integer
$\iota(z)=-\sum_{i=0}\kappa(y_{i})2^{2i}\in\mathbb{Z}_{2}$. The map $\iota$ is
clearly bijective bi-continuous and the value $\iota(z)$ is also called the
multiplicative coding of $z$. We write down first several iterates of $V$ on
the fix point of $\sigma_{0}$, to illustrate the situation:
$\displaystyle\sigma_{0}\sigma_{0}\sigma_{0}\sigma_{0}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-0000\dots$
$\displaystyle\sigma_{3}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-3333\dots$
$\displaystyle\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-2333\dots$
$\displaystyle\sigma_{1}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-1333\dots$
$\displaystyle\sigma_{0}\sigma_{3}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-0333\dots$
$\displaystyle\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{3}\dots$
$\displaystyle\stackrel{{\scriptstyle\iota}}{{\rightarrow}}$
$\displaystyle-3233\dots$
One can see that the following commutative diagram (11) holds.
(11) $\begin{CD}X_{\sigma_{0}}@>{V}>{}>X_{\sigma_{0}}\\\
@V{\iota}V{}V@V{\iota}V{}V\\\ \mathbb{Z}_{2}@>{+1}>{}>\mathbb{Z}_{2}\end{CD}$
Therefore $(X_{\sigma_{0}},V)$ is topologically conjugate to the $2$-adic
odometer $(\mathbb{Z}_{2},x\mapsto x+1)$. Here the consecutive digits
$\\{0,1\\}$ in $\mathbb{Z}_{2}$ are glued together to give
$\\{0,1,2,3\\}=\\{0,1\\}+2\\{0,1\\}$. Indeed, $\sigma_{0}$ satisfies the
coincidence condition of height one in the sense of Dekking [25, 9] and above
conjugacy is a consequence of this. $(\mathbb{Z}_{2},x\mapsto x+1)$ is a
translation of a compact group $\mathbb{Z}_{2}$ which is minimal and uniquely
ergodic with the Haar measure of $\mathbb{Z}_{2}$. Moreover one can confirm
that $\iota$ preserves the measure and $(X_{\sigma_{0}},V)$ and
$(\mathbb{Z}_{2},x\mapsto x+1)$ are isomorphic through $\iota$ as measure
dynamical systems. In view of (6), we define
$\xi(i)=\begin{cases}0&i=\lambda\\\ 2&i=a\\\ 3&i=ba\\\ 5&i=aba\end{cases}$
and the map $\eta:X_{\sigma_{0}}\rightarrow Y^{\prime}$ by
(12)
$\eta(x)=d_{\xi(y_{1})}+\frac{\zeta^{\xi(y_{1})}}{\omega^{2}}\left(d_{\xi(y_{2})}+\frac{\zeta^{\xi(y_{2})}}{\omega^{2}}\left(d_{\xi(y_{3})}+\frac{\zeta^{\xi(y_{3})}}{\omega^{2}}\dots.\right.\right.$
Then $\eta$ is clearly surjective, continuous, and measurable because both
$\tau$ and $\nu$ are Borel probability measures. Since the set of points with
double addresses is on the open edge, the map $\eta$ is bijective from
$\mathbf{A^{\prime}}$ to $\mathbf{A}\cap T(\mathcal{Z})$. Since
$\widetilde{\mathbf{d}}(T(x))=V(\widetilde{\mathbf{d}}(x))$, we have a
commutative diagram:
(13) $\begin{CD}\mathbf{A^{\prime}}@>{V}>{}>\mathbf{A^{\prime}}\\\
@V{\eta}V{}V@V{\eta}V{}V\\\ \mathbf{A}\cap
T(\mathcal{Z})@>{\widetilde{T}}>{}>\mathbf{A}\cap T(\mathcal{Z}).\end{CD}$
From Figure 14, it is easy to see that the set $\mathcal{P}$ of
$\widetilde{T}$-periodic points in $Y^{\prime}$ is measure zero by $\nu$,
i.e., $\nu(\mathbf{A}\cap T(\mathcal{Z}))=\nu(Y^{\prime}\cap
T(\mathcal{Z}))=1$, because the number of words of length $n$ in Figure 14 is
$O(2^{n})$. Similarly as the Perron-Frobenius root of the substitution
$\sigma_{0}$ is $4$ and the number of words of lengths $n$ in Figure 15 are
$O(2^{n})$, we see that $\tau(\mathbf{A^{\prime}})=\tau(X_{\sigma_{0}})=1$.
From (13) the pull back measure $\nu\circ\eta^{-1}$ of $X_{\sigma_{0}}$ is
invariant by $V$, we have $\tau=\nu\circ\eta^{-1}$ by unique ergodicity.
Therefore by taking $\phi=\eta\circ\iota$, we have the commutative diagram (8)
with measure zero exceptions. Let $V^{\prime}$ be a map from $X_{\sigma_{0}}$
to itself which acts as the shift operator on the multiplicative coding
$\mathbf{d^{\prime}}$, i.e.,
$(\mathbf{d^{\prime}}(V^{\prime}(z))=\sigma_{n_{2}}\sigma_{n_{3}}\dots$ for
$\mathbf{d^{\prime}}(z)=\sigma_{n_{1}}\sigma_{n_{2}}\dots$. Then we see that
(14) $\begin{CD}\mathbf{A^{\prime}}@>{V^{\prime}}>{}>\mathbf{A^{\prime}}\\\
@V{\eta}V{}V@V{\eta}V{}V\\\ \mathbf{A}\cap
T(\mathcal{Z})@>{S}>{}>\mathbf{A}\cap T(\mathcal{Z}).\end{CD}$
and the commutative diagram (9) is valid but for measure zero exceptions. ∎
###### Corollary 4.
Each aperiodic point $x\in\mathbf{A}\cap T(\mathcal{Z})$, the
$\widetilde{T}$-orbit of $x$ is uniformly distributed in $Y^{\prime}$ with
respect to the self similar measure $\nu$.
###### Proof.
In the proof of Theorem 3 the map $\eta$ is bijective form
$\mathbf{A^{\prime}}$ to $\mathbf{A}\cap T(\mathcal{Z})$. Therefore if
$x\in\mathbf{A}\cap T(\mathcal{Z})$, then there exists a unique element in
$z\in X_{\sigma_{0}}$ with $\eta(z)=x$. Therefore there exist an element
$z_{0}\in\mathbb{Z}_{2}$ such that $\phi(z_{0})=x$. The Haar measure $\mu_{2}$
on $Z_{2}$ is given by the values on the semi-algebra:
$\mu_{2}([c_{0},c_{1},\dots,c_{\ell-1}])=4^{-\ell}$
for each cylinder set $[c_{0},c_{1},\dots,c_{\ell-1}]=\\{y\in\mathbb{Z}_{2}\
|\ y\equiv\sum_{i=0}^{\ell-1}c_{i}4^{i}\pmod{4^{\ell}}\\}$. Since
$(\mathbb{Z}_{2},x\mapsto x+1)$ is uniquely ergodic, the assertion follows
immediately from the commutative diagram (8). ∎
Not all points in $Y^{\prime}$ gives a dense orbit as we already mentioned
that $\mathbf{A}\cap Y^{\prime}$ is a proper dense subset of $Y^{\prime}$.
There are many periodic points in $Y^{\prime}$ as well. This gives a good
contrast to usual minimal topological dynamics given by a continuous map
acting on a compact metrizable space.
###### Corollary 5.
Each aperiodic point $x\in\mathbf{A}$, the $T$-orbit of $x$ is dense in the
set $X$.
###### Proof.
It is clear from the fact that $(T(\mathcal{Z}),\widetilde{T})$ is the induced
system of $(L,T)$. ∎
One can construct a dual expansion of the non-invertible dynamics
$(Y^{\prime},S)$ by the conjugate map $\phi:\zeta\rightarrow\zeta^{2}$ in
${\rm Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})$ and then make a natural extension:
an invertible dynamics which contains $(Y^{\prime},S)$. The idea comes from
symbolic dynamics. We wish to construct the reverse expansion of (12) to the
other direction. To this matter, we compute in the following way:
$\frac{\omega^{2}(\eta(x)-d_{\xi(y_{1})})}{\zeta^{\xi(y_{1})}}=d_{\xi(y_{2})}+\frac{\zeta^{\xi(y_{2})}}{\omega^{2}}\left(d_{\xi(y_{3})}+\frac{\zeta^{\xi(y_{3})}}{\omega^{2}}\left(d_{\xi(y_{4})}+\frac{\zeta^{\xi(y_{4})}}{\omega^{2}}\left(\dots\right.\right.\right.$
and
$\frac{\omega^{2}}{\zeta^{\xi(y_{2})}}\left(\frac{\omega^{2}(\eta(x)-d_{\xi(y_{1})})}{\zeta^{\xi(y_{1})}}-d_{\xi(y_{2})}\right)=d_{\xi(y_{3})}+\frac{\zeta^{\xi(y_{3})}}{\omega^{2}}\left(d_{\xi(y_{4})}+\frac{\zeta^{\xi(y_{4})}}{\omega^{2}}\left(\dots\right.\right.$
Therefore it is natural to introduce a left ‘expansion’:
$\frac{\omega^{2}}{\zeta^{i_{1}}}\left(\frac{\omega^{2}}{\zeta^{i_{2}}}\left(\frac{\omega^{2}}{\zeta^{i_{3}}}\left(\left(\dots\right)-d_{i_{3}}\right)-d_{i_{2}}\right)-d_{i_{1}}\right)$
with $i_{k}\in\\{0,2,3,5\\}$. As this expression does not converge, we take
the image of $\phi$ because $\phi(\omega)=-1/\omega$. Let us denote by
$u_{i_{k}}=\phi(d_{i_{k}})$. Then the expansion
$\frac{\zeta^{-2i_{1}}}{\omega^{2}}\left(\frac{\zeta^{-2i_{2}}}{\omega^{2}}\left(\frac{\zeta^{-2i_{3}}}{\omega^{2}}\left(\left(\dots\right)-u_{i_{3}}\right)\right)-u_{i_{2}}\right)-u_{i_{1}}$
converges and the closure of the set of such expansions gives a compact set
$\mathcal{Y}$ depicted in figure 16.
Figure 16. The dual attractor $\mathcal{Y}$
Of course the set is an attractor of the IFS:
$\mathcal{Y}=\frac{1}{\omega^{2}}(\mathcal{Y}-u_{0})\cup\frac{\zeta}{\omega^{2}}(\mathcal{Y}-u_{2})\cup\frac{\zeta^{-1}}{\omega^{2}}(\mathcal{Y}-u_{3})\cup\frac{1}{\omega^{2}}(\mathcal{Y}-u_{5}).$
Combining $\mathcal{Y}$ we can construct a natural extension of
$(Y^{\prime},S)$ as:
$Y^{\prime}\times\mathcal{Y}\ni\left(\eta,\theta\right)\stackrel{{\scriptstyle\hat{S}}}{{\mapsto}}\left(\frac{(\eta-
d_{i})\omega^{2}}{\zeta^{i}},\frac{\zeta^{-2i}(\theta-\phi(d_{i}))}{\omega^{2}}\right)\in
Y^{\prime}\times\mathcal{Y}$
On the other hand $(\mathbb{Z}_{2},\rho)$ have a natural extension:
$\mathbb{Z}_{2}\times[0,1)\ni(x,y)\stackrel{{\scriptstyle\hat{\rho}}}{{\mapsto}}\left(\frac{x-(x\bmod{4})}{4},\frac{y+(x\bmod{4})}{4}\right)\in\mathbb{Z}_{2}\times[0,1)$
and two systems are isomorphic both as topological and measure theoretical
dynamics:
(15)
$\begin{CD}\mathbb{Z}_{2}\times[0,1)@>{\hat{\rho}}>{}>\mathbb{Z}_{2}\times[0,1)\\\
@V{\phi\times\phi^{\prime}}V{}V@V{\phi\times\phi^{\prime}}V{}V\\\
Y^{\prime}\times\mathcal{Y}@>{\hat{S}}>{}>Y^{\prime}\times\mathcal{Y}.\end{CD}$
where $\phi^{\prime}$ is given as:
$\sum_{i=1}^{\infty}x_{i}4^{-i}\mapsto g_{x_{1}}(g_{x_{2}}(g_{x_{3}}(\dots)))$
where
$g_{0}(x)=(x-u_{0})/\omega^{2},g_{1}(x)=(x-u_{2})\zeta/\omega^{2},g_{2}(x)=(x-u_{3})\zeta^{-1}/\omega^{2}$
and $g_{3}(x)=(x-u_{5})/\omega^{2}$.
From this ‘algebraic’ natural extension construction, we can characterize
purely $S$-periodic points in $Y^{\prime}\cap\mathbb{Q}(\zeta)$.
###### Theorem 6.
A point $y$ in $Y^{\prime}\cap\mathbb{Q}(\zeta)$ has purely periodic
multiplicative coding with four digits
$\sigma_{0},\sigma_{2},\sigma_{3},\sigma_{5}$ if and only if $(y,\phi(y))\in
Y^{\prime}\times\mathcal{Y}$.
This is an analogy of the results [14] for $\beta$-expansion. The proof below
is on the same line.
###### Proof.
As $\omega$ is an algebraic unit and $d_{i}\in\mathbb{Z}[\zeta]$, the
denominator of $g_{i}(y)$ is the same as that of $y$ for $i=0,1,2,3$.
Therefore the module $y\in\mathcal{M}=\frac{1}{M}\mathbb{Z}[\zeta]$ is stable
by $g_{i}$ for some positive integer $M$. Note that points $y\in\mathcal{M}$
with $(y,\phi(y))\in Y^{\prime}\times\mathcal{Y}$ is finite, because
$y,\phi(y)$ and their complex conjugates are bounded in $\mathbb{C}$. One can
confirm that the map $\hat{S}$ becomes surjective from $\mathcal{M}$ to
itself. For a finite set, surjectivity implies bijectivity. Therefore a point
$y\in\mathcal{M}$ with $(y,\phi(y))\in Y^{\prime}\times\mathcal{Y}$ produces a
purely periodic orbit. On the other hand if $x$ has purely periodic
multiplicative coding, it is easy to see $(y,\phi(y))\in
Y^{\prime}\times\mathcal{Y}$. ∎
## 3\. Other self-similar systems
Pisot scaling constants appear in several important dynamics. For irrational
rotations (2IET), it is well known that scaling constants of self-inducing
systems must be quadratic Pisot units. A typical example Figure 2 was shown in
the introduction. They are computed by the continued fraction algorithm as
fundamental units of quadratic number fields. Poggiaspalla-Lowenstein-Vivald
[24] showed that the scaling constant must be an algebraic unit for self-
inducing uniquely ergodic IET. When the scaling constant of self-inducing IET
is a cubic Pisot unit, we have further nice properties [4, 19, 20].
A necessary condition that $1$-dimensional substitutive point sets give point
diffraction is that the scaling constant is a Pisot number [7]. Suspension
tiling dynamics of such substitution is conjectured to have pure discrete
spectrum if the characteristic polynomial of its substitution matrix is
irreducible. For higher dimensional tiling dynamics the Pisot (or Pisot
family) property is essential to have relatively dense point spectra, see for
e.g. [27, 17].
Pisot scaling properties seem to extend to the case of piecewise isometries.
To conclude we present some examples, though we do not make a systematic
study.
It is already observed in [16, 3] that Pisot scaling constants appear in our
problem if $\theta$ is the $n$-th root of unity for $n=4,6,8,10,12$ in the
same way as we did in $n=5$ but in a more involved manner. In each case they
are quadratic Pisot units. What about if $\lambda=-2\cos(\theta)$ is cubic? In
this case, the dynamics of Conjecture 1 are embedded into the piecewise affine
mapping acting on $(\mathbb{R}/\mathbb{Z})^{4}$ which is harder to visualize.
Instead let us consider formal analogies of piecewise isometries generated by
cubic $n$-th fold rotation in the plane. At the expense of losing connection
to Conjecture 1, we find many Pisot unit scaling constants! Being an algebraic
unit is natural and may be explained from invertibility of dynamics. However
we have no idea why the Pisot numbers turn up or even how to formulate these
phenomena as a suitable conjecture.
### 3.1. Seven-fold
We start with 7-fold case. Both pieces are rotated clockwise by $4\pi/7$ as in
Figure 17. The triangle is rotated around A and the trapezium around B. The
first return map to a region and a smaller region with the same first return
map (up to scaling) are described. Unlike the five fold case, returning to the
subregion does not cover the full region. A simple consequence is that there
are infinitely many possible orbit closures for non-periodic orbits in the
system. The scaling constant $\alpha\approx 5.04892$ is a Pisot number whose
minimal polynomial is $x^{3}-6x^{2}+5x-1$. Figure 18 shows how this remaining
space can be filled in. As this region is already a little small we will zoom
in and now consider just this induced sub-system in Figure 19. The smaller
substitutions are easier to see as there are two scalings giving the same
dynamics (A and B). The scaling constant $\beta\approx 16.3937$ for these
subregions is the Pisot number associated to $x^{3}-17x^{2}+10x-1$. The proof
that the remaining substitutions work is shown in Figure 20. The first return
map to the two lower triangles is shown. The same dynamics occur on a smaller
region. The orbit of the smaller region covers all the regions left out of
Figure 19 and so the substitution rule from that figure is now complete. The
scaling constant for this triangle is $\alpha$. This gives an example of
recursive tiling structure by Lowenstein-Kouptsov-Vivaldi [18]. Knowing that
every aperiodic orbits are in one of the above self-inducing structures, we
can show that
###### Theorem 7.
Almost all points of this $7$-fold lozenge have periodic orbits.
The argument is similar to that given around Figure 7(a). We easily find
decreasing series $X_{n}$ of union of polygons satisfying $\mu(\alpha
X_{n+1})<\alpha^{2}\mu(X_{n})$ (or $\mu(\beta X_{n+1})<\beta^{2}\mu(X_{n})$)
which cover all self-inducing structures.
The fundamental units of the maximal real subfield $\mathbb{Q}(\cos(4\pi/7))$
of the cyclotomic field $\mathbb{Q}(\zeta_{7})$ are given by $b$ and $b-1$
where $b=1/(2\cos(3\pi/7))\approx 2.24698$. Here $b$ is the Pisot number
satisfying $x^{3}-2x^{2}-x+1$. We see that $\alpha=b^{2}$ and
$\beta=b^{4}/(b-1)^{2}$ and thus $\alpha$ and $\beta$ generates a subgroup of
fundamental units of $\mathbb{Q}(\cos(4\pi/7))$. Note that both
$\sqrt{\alpha}=b$ and $\sqrt{\beta}=b^{2}/(b-1)$ are Pisot numbers but $b-1$
is not. Our piecewise isometry somehow selects Pisot units out of the unit
group!
Figure 17. A seven-fold piecewise isometry. Figure 18. The regions remaining
from the self-similarity shown in Figure 17 Figure 19. The substitution rule
of the induced subsystem shown in Figure 18. Figure 20. The final pieces of
the structure of the piecewise isometry found in Figure 17.
### 3.2. Nine-fold
The next example is 9-fold case in Figure 21. Both pieces are rotated anti-
clockwise by $4\pi/9$, the triangle around A and the trapezium around B. The
first return map ($\triangle$) to the triangle is also shown. In addition the
same dynamics are found on a smaller piece of the map. Like the 7-fold shown
in Figure 17 this does give a full description of the dynamics, but it is
$\triangle^{2}$ not $\triangle$. The scaling constant $\gamma\approx 8.29086$
is a Pisot unit defined by $x^{3}-9x^{2}+6x-1$. Unfortunately in this case we
were not able to find a complete description of the scaling structure.
The fundamental units of $\mathbb{Q}(\cos(4\pi/9))$ are $b$ and $b^{2}-2b-1$
where $b=1/(2\cos(4\pi/9))\approx 2.87939$ is a Pisot number given by
$x^{3}-3x^{2}+1$. We have $\gamma=b^{2}$ and are expecting to find another
Pisot unit $b^{2}/(b^{2}-2b-1)\approx 5.41147$ (or its square) as a scaling
constant in this dynamics, which would give an analogy to the seven-fold case.
Figure 21. A nine fold piecewise isometry.
## References
* [1] R.L. Adler, B.P. Kitchens, and C.P. Tresser, _Dynamics of non-ergodic piecewise affine maps of the torus_ , Ergodic Theory Dynam. Systems 21 (2001), 959–999.
* [2] S. Akiyama, H. Brunotte, A. Pethő, and W. Steiner, _Remarks on a conjecture on certain integer sequences_ , Periodica Math. Hungarica 52 (2006), 1–17.
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|
arxiv-papers
| 2011-02-21T18:59:06 |
2024-09-04T02:49:17.159073
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shigeki Akiyama and Edmund Harriss",
"submitter": "Edmund Harriss",
"url": "https://arxiv.org/abs/1102.4310"
}
|
1102.4425
|
Description of the characters and factor representations of the infinite
symmetric inverse semigroup***Partially supported by the RFBR grants
08-01-00379-a and 09-01-12175-ofi-m..
A. M. Vershik, P. P. Nikitin
St. Petersburg Department
of the Steklov Mathematical Institute
27, Fontanka, 191023 St.Petersburg, Russia
E-mail: vershik@pdmi.ras.ru, pnikitin0103@yahoo.co.uk
###### Abstract.
We give a complete list of indecomposable characters of the infinite symmetric
semigroup. In comparison with the analogous list for the infinite symmetric
group, one should introduce only one new parameter, which has a clear
combinatorial meaning. The paper relies on the representation theory of the
finite symmetric semigroups and the representation theory of the infinite
symmetric group.
## Introduction
In this paper, we describe the characters of the infinite symmetric semigroup.
The main result establishes a link between the representation theory of the
finite symmetric semigroups developed by Munn [13], [14], Solomon [17],
Halverson [12], Vagner [1], Preston [16], and Popova [11] on the one hand, and
the representation theory of locally finite groups (in particular, the
infinite symmetric group) and locally semisimple algebras developed in the
papers by Thoma [18], Vershik and Kerov [4]–[6], [20] on the other hand. The
below analysis of the Bratteli diagram for the infinite symmetric semigroup
reminds the analogous analysis in the more complicated case of describing the
characters of the Brauer–Weyl algebras [7]. The symmetric semigroup appeared
not only in the literature on the theory of semigroups and their
representations, but also in connection with the representation theory of the
infinite symmetric group [15] and the definition of the braid semigroup [19];
$q$-analogs of the symmetric semigroup were also considered [12]. Apparently,
the definition of the infinite symmetric semigroup given in this paper, as
well as problems related to representations of this semigroup, have not yet
been discussed in the literature.
Consider the set of all one-to-one partial transformations of the set
$\\{1,\dots,n\\}$, i.e., one-to-one maps from a subset of $\\{1,\dots,n\\}$ to
a subset (possibly, different from the first one) of $\\{1,\dots,n\\}$. We
define the product of such maps as their composition where it is defined. Thus
we obtain a semigroup with a zero (the map with the empty domain of
definition), which is usually called the symmetric inverse semigroup; denote
it by $R_{n}$ (there are also other notations, see [9], [17]).
Obviously, the symmetric group $S_{n}$ is a subgroup of the semigroup
$R_{n}\colon S_{n}\subset R_{n}$. Further, $R_{n}$ can be presented as the
semigroup of all $0$-$1$ matrices with at most one $1$ in each row and each
column equipped with matrix multiplication. This realization is similar to the
natural representation of the symmetric group. The matrices of this form are
in a one-to-one correspondence with all possible placements of nonattacking
rooks on the $n\times n$ chessboard, that is why Solomon called this monoid
(the semigroup with a zero) the rook monoid.
The following properties of inverse semigroups and, in particular, the
symmetric inverse semigroup are of great importance (see the Appendix).
(1) the complex semigroup algebra of every finite inverse semigroup is
semisimple ([10], [14]);
(2) every finite inverse semigroup can be isomorphically embedded into a
symmetric inverse semigroup ([1], [16]);
(3) the class of finite inverse semigroups generates exactly the class of
involutive semisimple bialgebras [2].
The following result, which describes the characters of a finite inverse
semigroup, was essentially discovered by several authors; its combinatorial
and dynamical characterization is given in [12].
The set of irreducible representations (and, consequently, the set of
irreducible characters) of the symmetric semigroup $R_{n}$ is indexed by the
set of all Young diagrams with at most $n$ cells. The branching of
representations in terms of diagrams looks as follows: when passing from an
irreducible representation of $R_{n}$ to representations of $R_{n+1}$, the
corresponding Young diagram either does not change, or obtains one new cell
(grows).
The infinite symmetric group $S_{\infty}$ is the countable group of all
finitary (i.e., nonidentity only on a finite subset) one-to-one
transformations of a countable set. In the same way one can define the
infinite symmetric inverse semigroup111Usually we omit the word “inverse” and
speak about the (infinite) symmetric semigroup. $R_{\infty}$ as the set of
partial one-to-one transformations of a countable set that are nonidentity
only on a finite subset.222Thus the infinite symmetric inverse semigroup does
not contain the zero map, since every element must be identity on the
complement of a finite set. The group $S_{\infty}$ is the inductive limit of
the chain $S_{n}$, $n=1,2,\dots$, with the natural embeddings of groups. In
the same way, the semigroups $R_{n}$, $n=1,2,\dots$, form a chain with respect
to the natural monomorphisms of semigroups333Under the monomorphism
$R_{n}\subset R_{n+1}$, the zero of $R_{n}$ is mapped not to a zero, but to a
certain projection; more exactly, to the generator $p_{n}\in R_{n+1}$, see
Theorem 1.7. $R_{0}\subset R_{1}\subset\dots\subset R_{n}\subset\dots$, and
its inductive limit is the infinite inverse symmetric semigroup. The
connection between the Bratteli diagram of the infinite symmetric group (the
Young graph) and that of the infinite symmetric inverse semigroup leads
naturally to introducing a new operation on graphs, which associates with
every Bratteli diagram its “slow” version. (Cf. the notion of the
“pascalization” of a graph introduced in [7].)
Our results rely on the well-developed representation theory of the infinite
symmetric group $S_{\infty}$ and, to some extent, generalize it. Recall that
the list of characters of the infinite symmetric group was found by Thoma
[18]. The new proof of Thoma’s theorem suggested by Vershik and Kerov [4] was
based on approximation of characters of the infinite symmetric group by
characters of finite symmetric groups and used the combinatorics of Young
diagrams, which, as is well known, parameterize the irreducible complex
representations of the finite symmetric groups. The parameters of
indecomposable characters in the exposition of [4] are interpreted as the
frequencies of the rows and columns of a sequence of growing Young diagrams.
The main result of this paper is that the list of parameters for the
characters of the infinite symmetric group is obtained from the list of Thoma
parameters by adding a new number from the interval $[0,1]$. The meaning of
this new parameter is as follows. The irreducible representations of the
finite symmetric semigroup $R_{n}$ are also parameterized by Young diagrams,
but with an arbitrary number of cells $k$ not exceeding $n$; thus, apart from
the limiting frequencies of rows and columns, a sequence of growing diagrams
has another parameter: the limit of the ratio $k/n$, which is the relative
velocity with which the corresponding path passes through the levels of the
branching graph; or, in other words, the deceleration of the rate of
approximation of a character of the infinite semigroup by characters of finite
semigroups.
The description of the characters allows us to construct a realization of the
corresponding representations. They live in the same space as the
corresponding representations of the infinite symmetric group. More exactly,
the space of the representation is constructed in exactly the same way as in
the model of factor representations of the infinite symmetric group suggested
in [5], but with the extended list of parameters, see Theorem 2.16.
In the first section, we give the necessary background on the representation
theory of the finite symmetric inverse semigroups. The second section is
devoted to the representation theory of the infinite symmetric semigroup
$R_{\infty}$ and contains our main results. In Appendix we collect general
information about finite inverse semigroups and some new facts about their
semigroup algebras regarded as Hopf algebras.
## 1\. The representation theory of the finite symmetric inverse semigroups
### 1.1. The semisimplicity of the semigroup algebra ${\mathbb{C}}[R_{n}]$.
The complete list of irreducible representations
We define the rank of a map $a\in R_{n}$ as the number of elements on which
this map is not defined. Each of the sets $A_{r}=\\{a\in R_{n}\mid$ the rank
of $a$ is at least $r\\}$ for $0\leq r\leq n$ is an ideal of the semigroup
$R_{n}$. The chain of ideals
$R_{n}=A_{0}\supset A_{1}\supset\dots\supset A_{n}$
is a principal series of the semigroup $R_{n}$, i.e., there is no ideal lying
strictly between $A_{r}$ and $A_{r+1}$, see Theorem 1.1.
Denote by ${\mathbb{C}}[S_{n}]$ the complex group algebra of the symmetric
group $S_{n}$. This algebra, as well as the group algebra of every finite
group, is semisimple, since in it there exists an invariant inner product.
The complex semigroup algebra of an inverse group is always semisimple too, as
follows from the general Theorem 3.3. An explicit decomposition of the algebra
${\mathbb{C}}[R_{n}]$ into matrix components was suggested by Munn [13].
###### Theorem 1.1 (Munn).
The algebra ${\mathbb{C}}[R_{n}]$ is semisimple and has the form
${\mathbb{C}}[R_{n}]\cong\bigoplus_{r=0}^{n}M_{\binom{n}{r}}({\mathbb{C}}[S_{r}]).$
Here $M_{l}(A)$ is the algebra of matrices of order $l$ over an algebra $A$. A
description of the representations of the algebra ${\mathbb{C}}[R_{n}]$ is
given by the following theorem.
###### Theorem 1.2 (Munn).
Let $S$ be a semigroup isomorphic to the semigroup $M_{n}(G)$ of $n\times n$
matrices with elements from a group $G$. Let $F$ be a field whose
characteristic is equal to zero or is a prime not dividing the order of $G$.
Let $\\{\gamma_{p}\\}_{p=1}^{k}$ be the complete list of nonequivalent
irreducible representations of the group $G$ over $F$. Denote by
$\gamma_{p}^{\prime}$ the map given by the formula
$\gamma_{p}^{\prime}(\\{x_{ij}\\})=\\{\gamma_{p}(x_{ij})\\}$
for every matrix $\\{x_{ij}\\}\in S=M_{n}(G)$. Then
$\\{\gamma_{p}^{\prime}\\}_{p=1}^{k}$ is the complete list of nonequivalent
irreducible representations of the semigroup $S$ over $F$.
Denote by $\mathscr{P}_{r}$ the set of all partitions of a positive integer
$r$. It follows from the previous theorem that the set of irreducible
representations of the semigroup $R_{n}$ can be naturally indexed by the set
$\bigcup_{r=0}^{n}\mathscr{P}_{r}$.
###### Remark 1.3.
As can be seen from the form of irreducible representations of the semigroup
$R_{n}$ described above, each such representation is an extension of a
uniquely defined induced representation of the group $S_{n}$. More exactly,
for the irreducible representation of $R_{n}$ corresponding to a partition
$\lambda\in\mathscr{P}_{r}$, consider the representation of the subgroup
$S_{r}\times S_{n-r}\subset S_{n}$ in which the action of $S_{r}$ corresponds
to the partition $\lambda$ and $S_{n-r}$ acts trivially. The corresponding
induced representation of $S_{n}$ can be extended to the original irreducible
representation of $R_{n}$. (This was also observed in [15].)
###### Remark 1.4.
On the semigroup algebra ${\mathbb{C}}[R_{n}]$ of the symmetric semigroup, as
well as on the group algebra ${\mathbb{C}}[S_{n}]$ of the symmetric group,
there is an involution, which, in particular, sends every irreducible
representation $\pi$ to the representation sgn$\pi$. It corresponds to the
natural involution on the Young graph and, consequently, of the slow Young
graph (for the definition, see Section 2.1) that sends a diagram to its
reflection in the diagonal. However, it is not an involution of the group
$S_{n}$ or the semigroup $R_{n}$.
### 1.2. A formula for the characters of the finite symmetric semigroup
Munn [13] also found a formula that expresses the characters of the symmetric
inverse semigroup in terms of characters of the symmetric groups. In order to
state the corresponding theorem, for every subset $K\subset\\{1,\dots,n\\}$,
$|K|=r$, fix an arbitrary partial bijection $\mu_{K}\colon
K\mapsto\\{1,\dots,r\\}$. By $\mu_{K}^{-}\colon\\{1,\dots,r\\}\mapsto K$ we
denote the map inverse to $\mu_{K}$ on $K$; thus $\mu_{K}^{-}\circ\mu_{K}$ is
the identity map on the set $K$.
###### Theorem 1.5 (Munn).
Let $\chi^{*}$ be the character of the irreducible representation of the
semigroup $R_{n}$ corresponding to a partition $\lambda\in\mathscr{P}_{r}$,
$1\leq r\leq n$. Let $\chi$ be the corresponding character of the symmetric
group $S_{r}$. Then for every element ${\sigma}\in R_{n}$,
$\chi^{*}({\sigma})=\sum\chi(\mu_{K}{\sigma}\mu_{K}^{-}),$
where the sum is taken over all subsets $K$ of the domain of definition of
${\sigma}$ such that $|K|=r$ and $K{\sigma}=K$.
### 1.3. Presentations of the semigroup $R_{n}$ by generators and relations
We are interested in families of generators of the semigroups
$\\{R_{n}\\}_{n=0}^{\infty}$ that increase under the embeddings $R_{n}\subset
R_{n+1}$. This condition is satisfied for the generators suggested by Popova
[11] and those suggested by Halverson [12]. In Halverson’s paper, the
generators and relations are described for a $q$-analog of the symmetric
inverse semigroup. Below we present the particular case of his result for
$q=1$.
Let ${\sigma}_{i}$, $1\leq i<n$, be the Coxeter generators of the group
$S_{n}$. By $p_{i}\in R_{n}$, $1\leq i\leq n$, we denote the following maps:
$p_{i}(j)$ is not defined if $j\leq i$, and $p_{i}(j)=j$ if $j>i$.
###### Theorem 1.6 (Popova).
The semigroup $R_{n}$ is generated by the elements ${\sigma}_{1}$, …,
${\sigma}_{n-1}$, $p_{1}$ with the following relations:
(1) the Coxeter relations for the group $S_{n}$;
(2)
${\sigma}_{2}p_{1}{\sigma}_{2}={\sigma}_{2}{\sigma}_{3}\cdots{\sigma}_{n-1}p_{1}{\sigma}_{2}{\sigma}_{3}\cdots{\sigma}_{n-1}=p_{1}=p_{1}^{2}$;
(3) $(p_{1}{\sigma}_{1})^{2}=p_{1}{\sigma}_{1}p_{1}=({\sigma}_{1}p_{1})^{2}$.
###### Theorem 1.7 (Halverson).
The semigroup $R_{n}$ is generated by the elements
${\sigma}_{1},\dots,\allowbreak{\sigma}_{n-1},p_{1},\dots,p_{n}$ with the
following relations:
(1) the Coxeter relations for the group $S_{n}$;
(2) ${\sigma}_{i}p_{j}=p_{j}{\sigma}_{i}=p_{j}$ for $1\leq i<j\leq n$;
(3) ${\sigma}_{i}p_{j}=p_{j}{\sigma}_{i}$ for $1\leq j<i\leq n-1$;
(4) $p_{i}^{2}=p_{i}$ for $1\leq i\leq n$;
(5) $p_{i+1}=p_{i}{\sigma}_{i}p_{i}$ for $1\leq i\leq n-1$.
An interesting presentation of the semigroup $R_{n}$ by generators and
relations was suggested by Solomon [17]: in addition to the Coxeter generators
of the group $S_{n}$, he considers also the “right shift” $\nu$ defined as
$\nu(i)=\begin{cases}i+1&\text{for $1\leq i<n$,}\\\ \textrm{is not
defined}&\text{for $i=n$.}\end{cases}$
###### Theorem 1.8 (Solomon).
The semigroup $R_{n}$ is generated by the elements ${\sigma}_{1}$, …,
${\sigma}_{n-1}$, $\nu$ with the following relations:
(1) the Coxeter relations for the group $S_{n}$;
(2) $\nu^{i+1}{\sigma}_{i}=\nu^{i+1}$;
(3) ${\sigma}_{i}\nu^{n-i+1}=\nu^{n-i+1}$;
(4) ${\sigma}_{i}\nu=\nu{\sigma}_{i+1}$;
(5) $\nu{\sigma}_{1}{\sigma}_{2}{\sigma}_{3}\cdots{\sigma}_{n-1}\nu=\nu$,
where $1\leq i\leq n-1$ in (1)–(3) and (5), and $1\leq i\leq n-2$ in (4).
## 2\. The representation theory of the infinite symmetric inverse semigroup
In this section we assume that the reader is familiar with the basic notions
and results of the theory of locally semisimple and AF algebras. Besides, we
use some facts from the representation theory of the finite symmetric groups
$S_{n}$ and the infinite symmetric group $S_{\infty}$. See, e.g., [20].
There is a natural embedding $R_{n}\subset R_{n+1}$ of semigroups under which
every map from $R_{n}$ goes to a map from $R_{n+1}$ that sends the element
$n+1$ to itself. Consider the inductive limit of the chain $R_{0}\subset
R_{1}\subset\dots\subset R_{n}\subset\dots$ of semigroups, which we will call
the infinite symmetric inverse semigroup $R_{\infty}$.
### 2.1. The branching graph of the algebra
$\boldsymbol{{\mathbb{C}}[R_{\infty}]}$
Let $\mathbb{Y}$ be the Young graph, and let $\mathbb{Y}_{n}$ be the level of
$\mathbb{Y}$ whose vertices are indexed by all partitions of the integer $n$
(Young diagrams with $n$ cells). By $|{\lambda}|$ we denote the number of
cells in a diagram ${\lambda}$ (the sum of the parts of the partition
${\lambda}$).
Denote by $\tilde{\mathbb{Y}}$ the branching graph of the semigroup algebra
${\mathbb{C}}[R_{\infty}]$. It was described by Halverson [12].
###### Theorem 2.1 (Halverson).
The branching graph $\tilde{\mathbb{Y}}$ can be described as follows:
(1) the vertices of the $n$th level are indexed by all Young diagrams with at
most $n$ cells: $\tilde{\mathbb{Y}}_{n}=\bigcup_{i=0}^{n}\mathbb{Y}_{i}$;
(2) vertices $\lambda\in\tilde{\mathbb{Y}}_{n}$ and
$\mu\in\tilde{\mathbb{Y}}_{n+1}$ are joined by an edge if either $\lambda=\mu$
or $\mu$ is obtained from $\lambda$ by adding one cell.
This leads us to the following definition of the slow graph $\tilde{\Gamma}$
constructed from a branching graph $\Gamma$:
(1) the set of vertices of the $n$th level of $\tilde{\Gamma}$ is the union of
the sets of vertices of all levels of the original graph $\Gamma$ with indices
at most $n$, i.e., $\tilde{\Gamma}_{n}=\bigcup_{i=0}^{n}\Gamma_{i}$;
(2) vertices $\lambda\in\tilde{\Gamma}_{n}$ and $\mu\in\tilde{\Gamma}_{n+1}$
are joined by an edge if either $\lambda=\mu$ or $\mu$ is joined by an edge
with $\lambda$ in the original graph.
Recall the definition of the Pascal graph $\mathbb{P}$:
(1) the set $\mathbb{P}_{n}$ of vertices of the $n$th level consists of all
pairs of integers $(n,k)$, $0\leq k\leq n$;
(2) vertices $(n,k)\in\mathbb{P}_{n}$ and $(n+1,l)\in\mathbb{P}_{n+1}$ are
joined by an edge if either $l=k$ or $l=k+1$.
Observe that if the original graph $\Gamma$ is the chain (the graph whose each
level consists of a single vertex), then the corresponding slow graph
$\tilde{\Gamma}$ coincides with $\mathbb{P}$. By analogy with the Pascal
graph, we index the vertices of the $n$th level $\tilde{\Gamma}_{n}$ of the
slow graph with the pairs $(n,{\lambda})$, where ${\lambda}\in\Gamma_{i}$,
$i\leq n$.
###### Remark 2.2.
Note that if $G=\mathbb{P}$ is the Pascal graph, then the corresponding slow
graph $\tilde{G}$ is the three-dimensional analog of the Pascal graph. For the
three-dimensional Pascal graph, the slow graph is the four-dimensional Pascal
graph, etc. For the definition of the multidimensional analogs of the Pascal
graph and a description of the traces of the corresponding algebras, see,
e.g., [6].
###### Remark 2.3.
The set of paths on the branching graph $\tilde{\mathbb{Y}}$ is in bijection
with the random walks on $\mathbb{Y}$ of the following form: at each moment,
we are allowed either to stay at the same vertex or to descend to the previous
level in an admissible way. In view of this description, graphs similar to
$\tilde{\mathbb{Y}}$ are called slow.
###### Remark 2.4.
In [7], the representation theory of the infinite Brauer algebra was studied.
As in the previous remark, one can construct a bijection between the paths on
the branching graph of the Brauer algebra and the random walks of a similar
form on the Young graph: starting from the empty diagram, at each step we can
move either to a vertex of the next level (joined by an edge with the current
vertex) or to a vertex of the previous level (joined by an edge with the
current vertex).
### 2.2. Facts from the theory of locally semisimple algebras
Given a branching graph $\Gamma$, denote by $T(\Gamma)$ the space of paths of
$\Gamma$. On $T(\Gamma)$ we have the “tail” equivalence relation (see [4]):
paths $x,y\in T(\Gamma)$ are equivalent, $x\sim y$, if they coincide from some
level on. The partition of $T(\Gamma)$ into the equivalence classes will be
denoted by $\xi=\xi_{\Gamma}$. Also, for every $k\in\mathbb{N}\cup{0}$ and
every path $s=(s_{0},s_{1},\dots,s_{k})$ of length $k$, denote by
$F_{s}\subset T(\Gamma)$ the cylinder $F_{s}=\\{t\in T\mid t_{i}=s_{i}\,\text{
for }\,0\leq i\leq k\\}$.
Given $x,y\in\Gamma$, by $\dim(x;y)$ denote the number of paths leading from
$x$ to $y$. By $\dim(y)=\dim(\varnothing;y)$ denote the total number of paths
leading to $y$. By $\mathscr{E}(\Gamma)$ denote the set of ergodic central
measures on $T(\Gamma)$. Given $\mu\in\mathscr{E}(\Gamma)$ and a vertex $y$,
by $\mu(y)$ denote the measure of the set of all paths passing through $y$,
i.e., the total measure of all cylinders $F_{s}$,
$s=(s_{0},s_{1},\dots,s_{|y|})$, $s_{|y|}=\nobreak y$.
We will use the following description of the characters of a locally
semisimple algebra and the central measures on its branching graph (ergodic
method).
###### Theorem 2.5 ([4]).
For every central ergodic measure $\mu$, the set of paths
$s=(s_{0},s_{1},\dots,s_{f},\dots)$ such that
$\mu(y)=\lim_{f\to\infty}\frac{\dim(y)\cdot\dim(y;s_{f})}{\dim s_{f}}$
for all vertices $y$ is of full measure.
###### Theorem 2.6 ([4]).
For every character $\phi$ of the algebra
$A=C^{*}(\bigcup_{f=0}^{\infty}A_{f})$, there exists a path
$\\{{\lambda}_{f}\\}_{f=0}^{\infty}$ in the Bratteli diagram such that
$\phi(a)=\lim_{f\to\infty}\frac{\chi_{{\lambda}_{f}}(a)}{\dim{\lambda}_{f}}$
for all $a\in A$. Here $\chi_{{\lambda}_{f}}$ is the character of the
representation ${\lambda}_{f}$ of the algebra $A_{f}$ and $\dim{\lambda}_{f}$
is its dimension.
### 2.3. Description of the central measures on slow graphs
The key property of an arbitrary slow graph $\tilde{\Gamma}$ is that we can
present the space of paths $T(\tilde{\Gamma})$ as the direct product of the
spaces of paths $T(\Gamma)$ and $T(\mathbb{P})$. The same is true for the sets
of paths between any two vertices. Moreover, the partition
$\xi_{\tilde{\Gamma}}$ and the central ergodic measures on $T(\tilde{\Gamma})$
can also be presented as corresponding products.
###### Lemma 2.7.
Let $\Gamma$ be the branching graph of a locally semisimple algebra and
$\tilde{\Gamma}$ be the corresponding slow graph. Then
1\. $T(\tilde{\Gamma})=T(\Gamma)\times T(\mathbb{P})$. Moreover, the number of
paths between any two vertices of the slow graph $\tilde{\Gamma}$ is the
product of the number of paths between the corresponding vertices of the
original graph $\Gamma$ and the number of vertices between the corresponding
vertices of the Pascal graph $\mathbb{P}$:
$\dim_{\tilde{\Gamma}}((n_{1},{\lambda}_{1});(n_{2},{\lambda}_{2}))=\dim_{\Gamma}({\lambda}_{1},{\lambda}_{2})\cdot\dim_{\mathbb{P}}((n_{1},|{\lambda}_{1}|);(n_{2},|{\lambda}_{2}|)).$
(1)
2\. Let $s_{\tilde{\Gamma}},t_{\tilde{\Gamma}}\in T(\tilde{\Gamma})$,
$s_{\Gamma},t_{\Gamma}\in T(\Gamma)$, $s_{\mathbb{P}},t_{\mathbb{P}}\in
T(\mathbb{P})$, and let $s_{\tilde{\Gamma}}$ correspond to the pair
$(s_{\Gamma},s_{\mathbb{P}})$ and $t_{\tilde{\Gamma}}$ correspond to the pair
$(t_{\Gamma},t_{\mathbb{P}})$. Then $s_{\tilde{\Gamma}}\sim
t_{\tilde{\Gamma}}$ (with respect to $\xi_{\tilde{\Gamma}}$) if and only if
$s_{\Gamma}\sim t_{\Gamma}$ (with respect to $\xi_{\Gamma}$) and
$s_{\mathbb{P}}\sim t_{\mathbb{P}}$ (with respect to $\xi_{\mathbb{P}}$).
###### Proof.
1\. To each path in the graph $\tilde{\Gamma}$ there corresponds a unique
strictly increasing sequence of vertices of the original graph $\Gamma$.
Moreover, to each path ${(i,{\lambda}_{i})}_{i=n_{1}}^{n_{2}}$ in the graph
$\tilde{\Gamma}$ we can associate the path
${(i,|{\lambda}_{i}|)}_{i=n_{1}}^{n_{2}}$ in the Pascal graph. It is easy to
see that the original path is uniquely determined by the constructed pair of
paths, whence $T(\tilde{\Gamma})=T(\Gamma)\times T(\mathbb{P})$.
Note that the constructed map determines a bijection between the paths from a
vertex $(n_{1},{\lambda}_{1})$ to a vertex $(n_{2},{\lambda}_{2})$ in the
graph $\tilde{\Gamma}$ and the pairs of paths between the corresponding
vertices in the original graph $\Gamma$ and in the Pascal graph $\mathbb{P}$,
which proves formula (1).
2\. The bijection in the proof of Claim 1 is constructed in such a way that
the tail of a path $t_{\tilde{\Gamma}}=(t_{\Gamma},t_{\mathbb{P}})$ depends
only on the tails of the paths $t_{\Gamma}$ and $t_{\mathbb{P}}$, and vice
versa. ∎∎
###### Theorem 2.8 (Description of the central measures).
There is a natural bijection
$\mathscr{E}(\tilde{\Gamma})\cong\mathscr{E}(\Gamma)\times\mathscr{E}(\mathbb{P})$.
Every central ergodic measure
$M_{\tilde{\Gamma}}\in\mathscr{E}(\tilde{\Gamma})$ is the product of central
ergodic measures $M_{\Gamma}\in\mathscr{E}(\Gamma)$ and
$M_{\mathbb{P}}\in\mathscr{E}(\mathbb{P})$; namely,
$M_{\tilde{\Gamma}}(F_{(n,{\lambda})})=M_{\Gamma}(F_{\lambda})\cdot
M_{\mathbb{P}}(F_{(n,|{\lambda}|)})$ for every cylinder $F_{(n,{\lambda})}$.
###### Proof.
In accordance with the decomposition $T(\tilde{\Gamma})=T(\Gamma)\times
T(\mathbb{P})$, given a central ergodic measure
$M_{\tilde{\Gamma}}\in\mathscr{E}(\tilde{\Gamma})$, consider the projections
$M_{\Gamma}\in\mathscr{E}(\Gamma)$ and
$M_{\mathbb{P}}\in\mathscr{E}(\mathbb{P})$ defined as follows:
$M_{\Gamma}(F_{\lambda})=\sum_{n\geq|{\lambda}|}M_{\tilde{\Gamma}}(F_{(n,{\lambda})}),\qquad
M_{\mathbb{P}}(F_{(n,k)})=\sum_{|{\lambda}|=k}M_{\tilde{\Gamma}}(F_{(n,{\lambda})}).$
The measures $M_{\Gamma}$ and $M_{\mathbb{P}}$ are central by the centrality
of $M_{\tilde{\Gamma}}$.
Further, according to formula (1) from Lemma 2.7 and Theorem 2.5,
$\displaystyle M_{\tilde{\Gamma}}(F_{(n,{\lambda})})$
$\displaystyle=\lim_{f\to\infty}\frac{\dim((n,{\lambda}_{n});(f,{\lambda}_{f}))}{\dim(f,{\lambda}_{f})}$
$\displaystyle=\lim_{f\to\infty}\frac{\dim_{\mathbb{P}}((n,|{\lambda}_{n}|);(f,|{\lambda}_{f}|))}{\dim_{\mathbb{P}}(f,|{\lambda}_{f}|)}\cdot\frac{\dim_{\Gamma}({\lambda}_{n};{\lambda}_{f})}{\dim_{\Gamma}({\lambda}_{f})}$
$\displaystyle=\lim_{f\to\infty}\frac{\dim_{\mathbb{P}}(n,|{\lambda}_{n}|);(f,|{\lambda}_{f}|))}{\dim_{\mathbb{P}}(f,|{\lambda}_{f}|)}\cdot\lim_{f\to\infty}\frac{\dim_{\Gamma}({\lambda}_{n};{\lambda}_{f})}{\dim_{\Gamma}({\lambda}_{f})}\,.$
(2)
The limits in the right-hand side of (2) exist and are equal to
$M_{\Gamma}(F_{\lambda})$ and $M_{\mathbb{P}}(F_{(n,k)})$, which proves the
required formula for $M_{\tilde{\Gamma}}$. The ergodicity of the measures
$M_{\Gamma}$ and $M_{\mathbb{P}}$ follows from the ergodicity of the measure
$M_{\tilde{\Gamma}}$.
Conversely, the product (in the above sense) of central ergodic measures
$M_{\Gamma}\in\mathscr{E}(\Gamma)$ and
$M_{\mathbb{P}}\in\mathscr{E}(\mathbb{P})$ is a central ergodic measure
$M_{\tilde{\Gamma}}\in\mathscr{E}(\tilde{\Gamma})$. Its centrality follows
from Lemma 2.7, and its ergodicity follows from equation (2).∎∎
Recall (see, e.g., [6]) that for the Pascal graph $\mathbb{P}$, the limits in
Theorem 2.5 exist if and only if for the path
$((0,k_{0}),(1,k_{1}),\dots,(f,k_{f}),\dots)$
the limit
$\lim_{f\to\infty}k_{f}/f=\delta,\qquad\delta\in[0;1],$ (3)
does exist, and to every $\delta\in[0;1]$ there corresponds a unique central
measure $M_{\mathbb{P}}=M_{\mathbb{P}}^{\delta}$.
###### Corollary 2.9.
Every measure $M_{\tilde{\Gamma}}\in\mathscr{E}(\tilde{\Gamma})$ is
parameterized by a pair $(\delta,M_{\Gamma})$, $\delta\in[0;1]$,
$M_{\Gamma}\in\mathscr{E}(\Gamma)$.
###### Corollary 2.10.
The measure $M_{\tilde{\Gamma}}=(\delta,M_{\Gamma})$ on $T(\tilde{\Gamma})$ is
concentrated on paths for which the corresponding paths in the graph $\Gamma$
lie in the support of the measure $M_{\Gamma}$ and, besides, the limit (3)
does exist.
In particular, consider an arbitrary central ergodic measure $M_{\mathbb{Y}}$
on the graph $\mathbb{Y}$ corresponding to parameters
$\alpha=\\{\alpha_{i}\\}$, $\beta=\\{\beta_{i}\\}$, $\gamma$. Then the measure
$M_{\tilde{\mathbb{Y}}}=(\delta,M_{\mathbb{Y}})$ on $T(\tilde{\mathbb{Y}})$ is
concentrated on paths of the form $\\{(f,{\lambda}_{f})\\}$ for which the
corresponding limits for the sequence $\\{{\lambda}_{f}\\}$ are equal to
$\\{\alpha_{i}\\}$ and $\\{\beta_{i}\\}$ and, besides,
$\lim_{f\to\infty}|{\lambda}_{f}|/f=\delta$.
### 2.4. A formula for the characters of the infinite symmetric semigroup
The bijection described above between the sets of central measures on the
spaces of paths of the graph $\Gamma$ and of the slow graph $\tilde{\Gamma}$
holds for an arbitrary graded graph $\Gamma$. This bijection can be translated
to the sets of characters of the algebras corresponding to these graphs (see
Corollary 2.11 below) via the correspondence between central measures and
characters; however, explicit formulas for characters substantially depend on
the graphs and algebras and have no universal meaning. Below we prove a
formula that expresses a character of the algebra ${\mathbb{C}}[R_{\infty}]$
in terms of the corresponding character of the algebra
${\mathbb{C}}[S_{\infty}]$. In this section, by a character we always mean an
indecomposable character.
###### Corollary 2.11.
The parametrization of the set of central measures described above determines
a bijection which sends every pair
$(\delta,\chi^{S_{\infty}}_{\alpha,\beta,\gamma})$, where $\delta\in[0,1]$ and
$\chi^{S_{\infty}}_{\alpha,\beta,\gamma}$ is a character of the algebra
${\mathbb{C}}[S_{\infty}]$, to the character
$\chi^{R_{\infty}}_{\alpha,\beta,\gamma,\delta}$ of the algebra
${\mathbb{C}}[R_{\infty}]$.
To simplify the notation, below we often omit the superscripts and the
parameter $\gamma$ (which can be expressed in terms of $\alpha$ and $\beta$),
setting
$\chi_{\alpha,\beta}\equiv\chi^{S_{\infty}}_{\alpha,\beta,\gamma},\qquad\chi_{\alpha,\beta,\delta}\equiv\chi^{R_{\infty}}_{\alpha,\beta,\gamma,\delta}.$
The conjugation of an element ${\sigma}\in R_{n}$ by an element of the
symmetric group does not change the value of a character, so it suffices to
consider reduced elements ${\sigma}^{\circ}\in R_{n}$, for which all fixed
points are at the end: for every ${\sigma}\in R_{n}$ there exist $g\in S_{n}$,
$n({\sigma})\in\mathbb{N}\cup 0$ such that ${\sigma}^{\circ}=g{\sigma}g^{-1}$
and ${\sigma}^{\circ}(i)\neq i$ for $i<n({\sigma})$ and
${\sigma}^{\circ}(i)=i$ for $i\geq n({\sigma})$. By the definition of the
embedding $R_{n}\subset R_{n+1}$, we may assume that ${\sigma}^{\circ}\in
R_{n({\sigma})}$. The order $n({\sigma})$ of the element ${\sigma}^{\circ}$ is
uniquely determined by the element ${\sigma}$.
Let us introduce a set $M_{k}({\sigma})\subset S_{n}$ whose elements are
indexed by all $k$-element subsets $K\subset\\{1,\dots,n\\}$ fixed under
${\sigma}$: to each such subset we associate the bijection $\tilde{\sigma}\in
S_{n}$ that coincides with ${\sigma}$ on $K$ and is identity at all other
points.
Note that for every element ${\sigma}$ of the semigroup $R_{n}$ we may
consider the maximal (possibly, empty) subset of $\\{1,\dots,n\\}$ that is
mapped by ${\sigma}$ to itself in a one-to-one manner. The restriction of
${\sigma}$ to this subset will be called the invertible part of ${\sigma}$.
The invertible part of every element ${\sigma}\in R_{n}$ can be regarded as an
element of some symmetric group $S_{r}$, $r\leq n$, and, consequently, it can
be written as a product of disjoint cycles. The set $M_{k}({\sigma})$ can also
be parameterized by the set of all subcollections of cycles of total length
$k$ from the cycle decomposition of the invertible part of ${\sigma}$.
In the next theorem, the value of an indecomposable character of the infinite
symmetric semigroup at an element ${\sigma}\in R_{n}$ is presented as a linear
combination of the values of the corresponding Thoma character at each of the
elements of the disjoint union $\bigsqcup_{k}M_{k}({\sigma})$ with
coefficients depending only on the parameter $\delta$.
###### Theorem 2.12 (A formula for the characters).
Let
$\chi^{R_{\infty}}_{\alpha,\beta,\gamma,\delta}\equiv\chi_{\alpha,\beta,\delta}$
be an indecomposable character of the algebra ${\mathbb{C}}[R_{\infty}]$,
$\chi^{S_{\infty}}_{\alpha,\gamma,\beta}\equiv\chi_{\alpha,\beta}$ be the
corresponding indecomposable character of the algebra
${\mathbb{C}}[S_{\infty}]$, and ${\sigma}\in R_{\infty}$ be a reduced element.
Then
$\chi_{\alpha,\beta,\delta}({\sigma})=\sum_{k=0}^{n{{\sigma}}}\bigg{(}\delta^{n({\sigma})-k}(1-\delta)^{k}\cdot\sum_{\tilde{\sigma}\in
M_{k}({\sigma})}\chi_{\alpha,\beta}(\tilde{\sigma})\bigg{)}.$
###### Proof.
By Theorem 2.6, there exists a path $\\{(f,{\lambda}_{f})\\}_{f=0}^{\infty}$
such that
$\chi_{\alpha,\beta,\delta}({\sigma})=\lim_{f\to\infty}\frac{\chi^{*}_{(f,{\lambda}_{f})}({\sigma})}{\dim(f,{\lambda}_{f})}\,.$
Recall that an element ${\sigma}\in R_{n}$ is regarded as an element of the
semigroup $R_{f}$ that is identity on the subset $\\{n+1,\dots,f\\}$. By
Theorem 1.5, in order to compute the character
$\chi^{*}_{(f,{\lambda}_{f})}({\sigma})$, it suffices to describe subsets of
size $|{\lambda}_{f}|$ in the set $\\{1,\dots,f\\}$ fixed under the action of
the element ${\sigma}\in R_{f}$. In order to completely describe such subsets,
it suffices to associate with every fixed subset of size $k$ in the set
$\\{1,\dots,n\\}$ all possible subsets of $|{\lambda}_{f}|-k$ fixed points in
the set $\\{n+1,\dots,f\\}$. Thus
$\chi^{*}_{(f,{\lambda}_{f})}({\sigma})=\sum_{k}\bigg{(}\binom{f-n}{|{\lambda}_{f}|-k}\cdot\sum_{\tilde{\sigma}\in
M_{k}({\sigma})}\chi_{{\lambda}_{f}}(\tilde{\sigma})\bigg{)}.$
By Claim 1 of Lemma 2.7,
$\displaystyle\chi_{\alpha,\beta,\delta}({\sigma})$
$\displaystyle=\lim_{f\to\infty}\frac{\sum_{k}\big{(}\binom{f-n}{|{\lambda}_{f}|-k}\cdot\sum_{\tilde{\sigma}}\chi_{{\lambda}_{f}}(\tilde{\sigma})\big{)}}{\dim(f,|{\lambda}_{f}|)\cdot\dim({\lambda}_{f})}$
$\displaystyle=\sum_{k}\bigg{(}\lim_{f\to\infty}\frac{\binom{f-n}{|{\lambda}_{f}|-k}}{\dim(f,|{\lambda}_{f}|)}\cdot\sum_{\tilde{\sigma}}\lim_{f\to\infty}\frac{\chi_{{\lambda}_{f}}(\tilde{\sigma})}{\dim({\lambda}_{f})}\bigg{)}.$
(4)
According to Corollary 2.10 and Theorem 2.6 applied to the infinite symmetric
group $S_{\infty}$, each of the summands in the right factor in the right-hand
side of (4) tends to the corresponding value of the character
$\chi_{\alpha,\beta}$. Besides, by Corollary 2.10,
$\lim|{\lambda}_{f}|/f=\delta$, whence
$\lim_{f\to\infty}\frac{\binom{f-n}{|{\lambda}_{f}|-k}}{\dim(f,|{\lambda}_{f}|)}=\delta^{n-k}(1-\delta)^{k},$
and this completes the proof.∎∎
###### Corollary 2.13.
For an arbitrary element ${\sigma}\in R_{n}\subset R_{\infty}$,
$\chi_{\alpha,\beta,\delta}({\sigma})=\sum_{k=0}^{n}\bigg{(}\delta^{n-k}(1-\delta)^{k}\cdot\sum_{\tilde{\sigma}\in
M_{k}({\sigma})}\chi_{\alpha,\beta}(\tilde{\sigma})\bigg{)}.$
###### Corollary 2.14.
The restriction of a character $\chi_{\alpha,\beta,\delta}$ of the algebra
${\mathbb{C}}(R_{\infty})$ to ${\mathbb{C}}(S_{\infty})$ is equal to
$\chi_{\alpha^{\prime},\beta^{\prime}}$, where $\alpha^{\prime}_{1}=\delta$,
$\alpha^{\prime}_{i}=(1-\delta)\alpha_{i-1}$ for $i>1$ and
$\beta^{\prime}=(1-\delta)\beta$.
###### Proof.
We will verify the assertion in the case $\beta=0$. Let
$\alpha^{\prime}_{1}=\delta$, $\alpha^{\prime}_{i}=(1-\delta)\alpha_{i-1}$ for
$i>1$, and ${\sigma}\in S_{n}$. Then
$\chi^{S_{\infty}}_{\alpha^{\prime},0}({\sigma})=\prod_{\gamma}\bigg{(}(1-\delta)^{k_{\gamma}}\cdot\sum_{i}\alpha_{i}^{k_{\gamma}}+\delta^{k_{\gamma}}\bigg{)},$
where the product is taken over all minimal cycles $\gamma$ in the cycle
decomposition of the element ${\sigma}$ and $k_{\gamma}$ are the lengths of
these cycles. Expanding the product, we obtain
$\chi^{S_{\infty}}_{\alpha^{\prime},0}({\sigma})=\sum_{k}\sum_{\tilde{\sigma}\in
M_{k}({\sigma})}\bigg{(}(1-\delta)^{k}\delta^{n-k}\cdot\prod_{\gamma}\bigg{(}\sum_{i}\alpha_{i}^{k_{\gamma}}\bigg{)}\bigg{)},$
where the internal product is taken over all minimal cycles $\gamma$ of the
subcollection $\tilde{\sigma}$. Writing the last equation in the form
$\chi^{S_{\infty}}_{\alpha^{\prime},0}({\sigma})=\sum_{k}\biggl{(}\delta^{n-k}(1-\delta)^{k}\cdot\sum_{\tilde{\sigma}\in
M_{k}({\sigma})}\chi^{S_{\infty}}_{\alpha,0}(\tilde{\sigma})\biggr{)}=\chi^{R_{\infty}}_{\alpha^{\prime},0,\delta}({\sigma}),$
we obtain the desired assertion.∎∎
###### Remark 2.15.
In the previous corollary, the parameters $\alpha$ and $\beta$ are not
symmetric, despite the fact that in the graph $\tilde{\mathbb{Y}}$ the
symmetry is present. The reason is as follows: under the embedding of the
group $S_{n}$ into the semigroup $R_{n}$, the restriction of an irreducible
representation of $R_{n}$ to $S_{n}$ is the representation induced from a
representation of the subgroup $S_{r}\times S_{n-r}\subset S_{n}$ that is
trivial on the second factor, see Remark 1.3. Hence the operation of
restricting a representation does not commute with the involution (see Remark
1.4), which breaks the symmetry between the parameters $\alpha$ and $\beta$.
### 2.5. Realization of representations
We turn our attention to the case where $\sum_{i}\alpha_{i}=1$, i.e.,
$\beta_{i}=0$ for all $i$. Consider a measure on $\mathbb{N}$ of the form
$\mu_{\alpha}(i)=\alpha_{i}$, the set of sequences
$\mathscr{X}=\prod\mathbb{N}$ equipped with the measure
$m_{\alpha}=\prod\mu_{\alpha}$, and the set $\tilde{\mathscr{X}}$ of pairs of
sequences coinciding from some point on. In the space
$L^{2}(\tilde{\mathscr{X}},m_{\alpha})$ we can realize the representation of
the symmetric group $S_{\infty}$ corresponding to the Thoma parameters
$(\alpha,0)$, see [5], [21].
###### Theorem 2.16.
The realization of the representation of the group $S_{\infty}$ corresponding
to the parameters $(\alpha^{\prime},0)$, where $\alpha^{\prime}$ is defined in
Corollary 2.14, in the space of functions
$L^{2}(\tilde{\mathscr{X}},m_{\alpha^{\prime}})$ can be extended to a
realization of the representation of the semigroup $R_{\infty}$ corresponding
to the parameters $(\alpha,0,\delta)$.
###### Proof.
Define the action of the projection $p_{1}$ from Theorem 1.6 as follows: it
maps every sequence $(a_{1},a_{2},a_{3},\dots)\in\mathscr{X}$ to the sequence
$(1,a_{2},a_{3},\dots)\in\mathscr{X}$. The relations from Theorem 1.6 are
obviously satisfied.
Thus it suffices to check that introducing an additional projection does not
lead beyond the space of the representation. But, as shown in [3], the space
of the factor representation of the symmetric group $S_{\infty}$ coincides
with the whole space $L^{2}(\tilde{\mathscr{X}},m_{\alpha^{\prime}})$, which
completes the proof. ∎∎
###### Corollary 2.17.
In terms of the realization described above, one can give a short formula for
the characters of $R_{\infty}$, similar to the formula for the characters of
the symmetric group (cf. [5]), which expresses the value of a character at an
element ${\sigma}$ as the measure of the set of fixed points of ${\sigma}$;
namely,
$\chi_{\alpha,0,\delta}({\sigma})=m_{\alpha^{\prime}}(\\{x:{\sigma}(x)=x\\}),$
where $\alpha^{\prime}$ is defined in Corollary 2.14. See also [3].
## 3\. Appendix. General information on finite inverse semigroups
In this section, we mainly follow the monograph [9] and the paper [2].
### 3.1. The definition of an inverse semigroup
###### Theorem 3.1.
The following two conditions on a semigroup $S$ are equivalent:
(1) for every $a\in S$ there exists $x\in S$ such that $axa=a$, and any two
idempotents of $S$ commute;
(2) every principal left ideal and every principal right ideal of $S$ is
generated by a unique idempotent;
(3) for every $a\in S$ there exists a unique $x\in S$ such that $axa=a$ and
$xax=x$.
A semigroup satisfying the conditions of Theorem 3.1 is called an inverse
semigroup. One says that the elements $a$ and $x$ from condition (1) of the
theorem are inverse to each other; sometimes, this is denoted as $x=a^{-1}$.
Note that $(ab)^{-1}=b^{-1}a^{-1}$ for any $a,b\in S$.
Let us prove that the symmetric inverse semigroup is an inverse semigroup.
Given a partial map ${\sigma}\in R_{n}$ that acts from a subset
$X\subset\\{1,\dots,n\\}$ to a subset $Y\subset\\{1,\dots,n\\}$, we construct
the map ${\sigma}^{-1}$ from $Y$ to $X$ inverse to ${\sigma}$ in the ordinary
sense, i.e., for $y\in Y$ and $x\in X$ we set ${\sigma}^{-1}(y)=x$ if
${\sigma}(x)=y$. The elements ${\sigma}$ and ${\sigma}^{-1}$ are obviously
inverse to each other. Besides, the idempotents of the symmetric inverse
semigroup are exactly those maps that send some subset
$X\subset\\{1,\dots,n\\}$ to itself and are not defined on
$\\{1,\dots,n\\}\backslash X$. Therefore, any two idempotents commute, and the
semigroup is inverse by Theorem 3.1.
### 3.2. An analog of Cayley’s theorem
Vagner [1] and Preston [16] proved for inverse semigroups an analog of
Cayley’s theorem for groups.
###### Theorem 3.2.
An arbitrary inverse semigroup $S$ is isomorphic to an inverse subsemigroup of
the symmetric inverse semigroup of all one-to-one partial transformations of
the set $S$.
The proof is much more difficult than in the group case, and we do not
reproduce it (see [9]). Note that the theorem holds both for finite and
infinite inverse semigroups.
### 3.3. The semisimplicity of the semigroup algebra
Given an arbitrary finite semigroup $S$ and a field $F$, one can consider the
semigroup algebra $F[S]$ of $S$ over $F$. The elements of $S$ form a basis in
$F[S]$, and the multiplication law for these basis elements coincides with the
multiplication law in $S$. Necessary and sufficient conditions for the
semisimplicity of the semigroup algebra $F[S]$ of a finite inverse semigroup
$S$ were obtained independently by Munn [14] and Oganesyan [10].
###### Theorem 3.3.
The semigroup algebra $F[S]$ of a finite inverse semigroup $S$ over a field
$K$ is semisimple if and only if the characteristic of $K$ is zero or a prime
that does not divide the order of any subgroup in $S$.
### 3.4. Involutive bialgebras and semigroup algebras of inverse semigroups
A bialgebra (see [8]) is a vector space over the field ${\mathbb{C}}$ equipped
with compatible structures of a unital associative algebra and a counital
coassociative coalgebra. Namely, the following equivalent conditions are
satisfied:
(1) the comultiplication and the counit are homomorphisms of the corresponding
algebras;
(2) the multiplication and the unit are homomorphisms of the corresponding
coalgebras.
Let us also introduce the notion of a weakened bialgebra for the case where
the multiplication and comultiplication are homomorphisms, but there is no
condition on the unit and counit.
The group algebra of a finite group with the convolution multiplication and
diagonal comultiplication is a cocommutative bialgebra (and even a Hopf
algebra). It is well known (see [8]) that the semigroup algebra of every
finite semigroup with identity (monoid) is also a cocommutative bialgebra with
the natural definition of the operations.
An involution of an algebra is a second-order antilinear antiautomorphism of
this algebra; a second-order antilinear antiautomorphism of a coalgebra is
called a coinvolution. A bialgebra equipped with an involution and a
coinvolution is called an involutive bialgebra, or a bialgebra with
involution, if the multiplication commutes with the coinvolution and the
comultiplication commutes with the involution.
In [2] it was shown that the class of finite inverse semigroups generates
exactly the class of involutive semisimple bialgebras.
###### Theorem 3.4.
The semigroup algebra of a finite inverse semigroup is a semisimple
cocommutative involutive algebra. Analogously, the dual semigroup algebra
${\mathbb{C}}[S]$ of a finite inverse semigroup $S$ with identity is a
commutative involutive bialgebra. Conversely, every finite-dimensional
semisimple cocommutative (in the dual case, commutative) involutive bialgebra
is isomorphic (as an involutive bialgebra) to the semigroup algebra
(respectively, dual semigroup algebra) of a finite inverse semigroup with
identity.
For inverse semigroups without identity, the semigroup bialgebra is a weakened
bialgebra (the counit is not a homomorphism).
Translated by N. V. Tsilevich.
## References
* [1] V. V. Vagner Generalized groups Doklady Akad. Nauk SSSR (N.S.), 84:24–43, 1952.
* [2] A. M. Vershik Krein’s duality, positive 2-algebras, and the dilation of comultiplications Funct. Anal. Appl., 41(2):99–114, 2007.
* [3] A. M. Vershik Nonfree actions of groups and the theory of characters, in preparation.
* [4] A. M. Vershik and S. V. Kerov Asymptotic theory of the characters of a symmetric group Funktsional. Anal. i Prilozhen., 15(4):15–27, 1981.
* [5] A. M. Vershik and S. V. Kerov Characters and factor representations of the infinite symmetric group Dokl. Akad. Nauk SSSR, 257(5):1037–1040, 1981.
* [6] A. M. Vershik and S. V. Kerov Locally semisimple algebras. Combinatorial theory and the $K_{0}$-functor Itogi Nauki i Tekhniki, Ser. Sovrem. Probl. Mat., VINITI, 26:3–56, 1985.
* [7] A. M. Vershik and P. P. Nikitin Traces on infinite-dimensional Brauer algebras Funct. Anal. Appl., 40(3):165–172, 2006.
* [8] C. Kassel Quantum groups. Springer-Verlag, New York, 1995.
* [9] A. H. Clifford and G. B. Preston The algebraic theory of semigroups. Amer. Math. Soc., Providence, R.I., 1961.
* [10] V. A. Oganesyan On the semisimplicity of a system algebra Akad. Nauk Armyan. SSR Dokl., 21:145–147, 1955.
* [11] L. I. Popova Defining relations for some subgroups of partial transformations of a finite set Uch. Zapiski Leningr. Gos. Ped. Inst. im. A. I. Gertsena, 218:191–212, 1961.
* [12] T. Halverson Representations of the q-rook monoid. J. Algebra, 273(1):227-251, 2004.
* [13] W. D. Munn The characters of the symmetric inverse semigroup. Proc. Camb. Phil. Soc., 53(1):13–18, 1957.
* [14] W. D. Munn On semigroup algebras. Proc. Cambridge Phil. Soc., 51:1–15, 1955.
* [15] G. Olshansky Unitary representations of the infinite symmetric group: a semigroup approach. Representations of Lie groups and Lie algebras, Académiai Kiadó, Budapest, 1985, pp. 181–197.
* [16] G. B. Preston Representations of inverse semigroups. J. London Math. Soc., 29:411–419, 1954.
* [17] L. Solomon Representations of the rook monoid, J. Algebra, 256(2):309–342, 2002.
* [18] E. Thoma Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe Math. Zeitschr., 85(1):40–61, 1964.
* [19] V. V. Vershinin On the inverse braid monoid. Topology Appl., 156(6):1153-1166.
* [20] A. M. Vershik, S. V. Kerov The Grothendieck group of the infinite symmetric group and symmetric Functions (with the elements of the theory $K_{0}$-functor of AF-algebras) Adv. Stud. Contemp. Math., Gordon and Breach, 7:39–118, 1990.
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|
arxiv-papers
| 2011-02-22T07:49:36 |
2024-09-04T02:49:17.167349
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anatoly Vershik, Pavel Nikitin",
"submitter": "Anatoly Vershik M",
"url": "https://arxiv.org/abs/1102.4425"
}
|
1102.4599
|
# Towards Unbiased BFS Sampling
Maciej Kurant EECS Dept
University of California, Irvine
maciej.kurant@gmail.com Athina Markopoulou EECS Dept
University of California, Irvine
athina@uci.edu Patrick Thiran School of Computer & Comm. Sciences
EPFL, Lausanne, Switzerland
patrick.thiran@epfl.ch
###### Abstract
Breadth First Search (BFS) is a widely used approach for sampling large
unknown Internet topologies. Its main advantage over random walks and other
exploration techniques is that a BFS sample is a plausible graph on its own,
and therefore we can study its topological characteristics. However, it has
been empirically observed that incomplete BFS is biased toward high-degree
nodes, which may strongly affect the measurements.
In this paper, we first analytically quantify the degree bias of BFS sampling.
In particular, we calculate the node degree distribution expected to be
observed by BFS as a function of the fraction $f$ of covered nodes, in a
random graph $RG(p_{k})$ with an arbitrary degree distribution $p_{k}$. We
also show that, for $RG(p_{k})$, all commonly used graph traversal techniques
(BFS, DFS, Forest Fire, Snowball Sampling, RDS) suffer from exactly the same
bias.
Next, based on our theoretical analysis, we propose a practical BFS-bias
correction procedure. It takes as input a collected BFS sample together with
its fraction $f$. Even though $RG(p_{k})$ does not capture many graph
properties common in real-life graphs (such as assortativity), our
$RG(p_{k})$-based correction technique performs well on a broad range of
Internet topologies and on two large BFS samples of Facebook and Orkut
networks.
Finally, we consider and evaluate a family of alternative correction
procedures, and demonstrate that, although they are unbiased for an arbitrary
topology, their large variance makes them far less effective than the
$RG(p_{k})$-based technique.
###### Index Terms:
BFS, Breadth First Search, graph sampling, estimation, bias correction,
Internet topologies, Online Social Networks.
## I Introduction
00footnotetext: This paper is a revised and extended version of [1].
A large body of work in the networking community focuses on Internet topology
measurements at various levels, including the IP or AS connectivity, the Web
(WWW), peer-to-peer (P2P) and online social networks (OSN). The size of these
networks and other restrictions make measuring the entire graph impossible.
For example, learning only the topology of Facebook social graph would require
downloading more than $250TB$ of HTML data [2, 3], which is most likely
impractical. Instead, researchers typically collect and study a small but
representative sample of the underlying graph.
In this paper, we are particularly interested in sampling networks that
naturally allow to explore the neighbors of a given node (which is the case in
WWW, P2P and OSN). A number of graph exploration techniques use this basic
operation for sampling. They can be roughly classified in two categories: (i)
random walks, and (ii) graph traversals.
In the first category, _random walks_ , nodes can be revisited. This category
includes the classic Random Walk (RW) [4] and its variations [5, 6], as well
as the Metropolis-Hastings Random Walk (MHRW). They are used for sampling of
nodes on the Web [7], P2P networks [8, 9, 10], OSNs [2, 11] and large graphs
in general [12]. Random walks are well studied [4] and result in samples that
have either no bias (MHRW) or a known bias (RW) that can be corrected for [13,
14, 15, 16]. In contrast to BFS, random walks collect a representative sample
of nodes rather than of topology, and are therefore not the focus of the
paper. However, we use them as baseline for comparison.
Figure 1: Overview of analytical results. We calculate the node degree
distribution $q_{k}$ expected to be observed by BFS in a random graph
$RG(p_{k})$ with a given degree distribution $p_{k}$, as a function of the
fraction of sampled nodes $f$. (In this plot, we show only its average
$\langle q_{k}\rangle$.) We show RW and MHRW as a reference. $\langle
k\rangle=\langle p_{k}\rangle$ is the real average node degree, and $\langle
k^{2}\rangle$ is the real average squared node degree. Observations: (1) For a
small sample size, BFS has the same bias as RW; with increasing $f$, the bias
decreases; a complete BFS ($f\\!\\!=\\!1$) is unbiased, as is MHRW (or uniform
sampling). (2) All common graph traversal techniques (that do not revisit the
same node) lead to the same bias. (3) The shape of the BFS curve depends on
the real node degree distribution $p_{k}$, but it is always monotonically
decreasing; we calculate it precisely in this paper. (4) We also calculate the
original distribution $p_{k}$ based on the sampled $q_{k}$ and $f$ (not shown
here).
In the second category, _graph traversals_ , each node is visited exactly once
(if we let the process run until completion and if the graph is connected).
These methods vary in the order in which they visit the nodes; examples
include BFS, Depth-First Search (DFS), Forest Fire (FF), Snowball Sampling
(SBS) and Respondent-Driven Sampling (RDS)111RDS is essentially SBS equipped
with some bias correction procedure (omitted in Fig. 1).. Graph traversals,
especially BFS, are very popular and widely used for sampling Internet
topologies, _e.g._ , in WWW [17] or OSNs [18, 19, 20]. [19] alone has about
380 citations as of December 2010, many of which use its Orkut BFS sample. The
main reason of this high popularity is that a BFS sample is a plausible graph
on its own. Consequently, we can study its topological characteristics (_e.g._
, shortest path lengths, clustering coefficients, community structure), which
is a big advantage of BFS over random walks. Of course, this approach is
correct only if the BFS sample is representative of the entire graph. At first
sight it seems true, _e.g._ , a BFS sample of a lattice is a (smaller)
lattice.
Unfortunately, this intuition often fails. It was observed empirically that
BFS introduces a bias towards high-degree nodes [17, 21, 22, 23]. We also
confirmed this fact in a recent measurement of Facebook [2, 3], where our BFS
crawler found the average node degree $324$, while the real value is only
$94$. This means that the average node degree is overestimated by BFS by about
250%! This has a striking effect not only on the node property statistics, but
also on the topological metrics.
Despite the popularity of BFS on the one hand, and its bias on the other hand,
we still know relatively little about the statistical properties of node
sequences returned by BFS. The formal analysis is challenging because BFS,
similarly to every sampling without replacement, introduces complex
dependencies between the sampled nodes difficult to deal with mathematically.
_Contributions._ Our work is a step towards understanding the statistical
characteristics of BFS samples and correcting for their biases, with the
following main contributions.
First, we focus on a random graph $RG(p_{k})$ with a given (and arbitrary)
degree distribution $p_{k}$. We calculate precisely the node degree
distribution $q_{k}$ expected to be observed by BFS as a function of the
fraction $f$ of sampled nodes. We illustrate this and related results in Fig.
1. To the best of our knowledge, this is the first analytical result
describing the bias of BFS sampling.
Second, based on our theoretical analysis, we propose a practical BFS-bias
correction procedure. It takes as input a collected BFS sample together with
the fraction $f$ of covered nodes, and estimates the mean of an arbitrary
function $x(v)$ defined on graph nodes. Even though $RG(p_{k})$ misses many
graph properties common in real-life graphs (such as assortativity), our
$RG(p_{k})$-based correction technique performs well on a broad range of
Internet topologies, and on two large BFS samples of Facebook and Orkut
networks. We make its ready-to-use python implementation publicly available at
[24].
Third, we complement the above findings by proposing a family of alternative
correction procedures that are unbiased for any arbitrary topology. Although
seemingly attractive, they are characterized by large variance, which makes
them far less effective than the $RG(p_{k})$-based correction technique.
Scope. Our theoretical results hold strictly for the random graph model
$RG(p_{k})$. (However, we show that they apply relatively well to a broad
range of real-life topologies.) We also restrict our attention to static
graphs with self-declared unweighted social links; dynamically varying graphs
[8, 25, 26, 27, 28, 10, 29, 30] and interaction graphs [31, 32, 33] are out of
the scope of this paper.
Finally, our $RG(p_{k})$-based bias-correction procedure is designed for local
graph properties, such as node statistics. Our analytical results can
potentially help the estimation of non-local graph properties (such as graph
diameter), which is our main direction for the future.
Outline. The outline of the paper is as follows. Section II discusses related
work. Section III presents BFS and other graph traversal algorithms under
study. We also briefly describe random walks that are used as baseline for
comparison throughout the paper. Section IV presents the random graph
$RG(p_{k})$ model used in this paper. Section V analyzes the degree bias of
BFS. Section VI shows how to correct for this bias. Section VII evaluates our
results in simulations and by sampling real world networks. Section VIII
introduces and evaluates alternative BFS-bias correction techniques. Section
IX gives some practical sampling recommendations, and Section X concludes the
paper.
## II Related Work
BFS used in practice. BFS is widely used today for exploring large networks,
such as OSNs. In [18], Ahn et al. used BFS to sample Orkut and MySpace. In
[19] and [27], Mislove et al. used BFS to crawl the social graph in four
popular OSNs: Flickr, LiveJournal, Orkut, and YouTube. [19] alone has about
380 citations as of December 2010, many of which use its highly biased Orkut
BFS sample. In [20], Wilson et al. measured the social graph and the user
interaction graph of Facebook using several BFSs, each BFS constrained in one
of the largest 22 regional Facebook networks. In our recent work [2, 3], we
have also crawled Facebook using various sampling techniques, including BFS,
RW and MHRW.
BFS bias. It has been empirically observed that incomplete BFS and its
variants introduce bias towards high-degree nodes [17][21, 22, 23]. We
confirmed this in Facebook [2, 3], which, in fact, inspired and motivated this
paper. Analogous bias has been observed in the field of social science, for
sampling techniques closely related to BFS, _i.e._ , Snowball Sampling and RDS
[34, 35, 15] (see Section III-B4).
Analyzing BFS. To the best of our knowledge, the sampling bias of BFS has not
been analyzed so far. [36] and [37] are the closest related papers to our
methodology. The original paper by Kim [36] analyzes the size of the largest
connected component in classic Erdös-Rényi random graph by essentially
applying the configuration model with node degrees chosen from a Poisson
distribution. To match the stubs (or “clones” in [36]) uniformly at random in
a tractable way, Kim proposes a “cut-off line” algorithm. He first assigns
each stub a random index from $[0,np]$, and next progressively scans this
interval. Achlioptas et al. used this powerful idea in [37] to study the bias
of traceroute sampling in random graphs with a given degree distribution. The
basic operation in [37] is traceroute (_i.e._ , “discover a path”) and is
performed from a single node to all other nodes in the graph. The union of the
observed paths forms a “BFS-tree”, which includes all nodes but misses some
edges (_e.g._ , those between nodes at the same depth in the tree). In
contrast, the basic operation in the traversal methods presented in our paper
is to discover all neighbors of a node, and it is applied to all nodes in
increasing distance from the origin. Another important difference is that [37]
studies a completed BFS-tree, whereas we study the sampling process when it
has visited only a fraction $f<1$ of nodes. Indeed, a completed BFS
($f\\!\\!=\\!1$) is trivial in our case: it has no bias, as all nodes are
covered.
In the field of social science, a significant effort was put to correct for
the bias of BFS’s close cousin - Snowball Sampling (SBS) [34]. SBS together
with a bias correction procedure is called Respondent-Driven Sampling (RDS)
[35]. The currently used correction technique [15, 16] assumes that nodes can
be revisited, which essentially approximates SBS by Random Walk (see Section
VI-A1). In this paper, we formally show that this approximation is valid if
the fraction $f$ of sampled nodes is relatively small. However, as [38] points
out, the current RDS methodology is systematically biased for larger $f$.
Consequently, [39] proposed an SBS bias correction method based on the random
graph $RG(p_{k})$. This is essentially the same basic starting idea as used in
our original paper published independently [1]. However, the two papers
fundamentally differ in the final solution: [39] proposes a simulation-aided
approach, whereas we solve the problem analytically.
Another recent and related paper is [40]. The authors propose and evaluate a
heuristic approach to correct the degree bias in the $i$th generation of SBS,
based on the values measured in the generation $i\\!-\\!1$. In practice, this
generation-based scheme may be challenging to implement, because the number of
nodes per generation may grow close to exponential with $i$. Consequently, we
are likely to face a situation where collecting the next generation is
prohibitively expensive, while the current generation has much fewer nodes
than our sampling capabilities allow for.
Probability Proportional to Size Without Replacement (PPSWOR). At a closer
look, our $RG(p_{k})$-based approach reduces BFS (and other graph traversals)
to a classic sampling design called Probability Proportional to Size Without
Replacement (PPSWOR) [41, 42, 43, 44, 45, 46, 47, 48]. Unfortunately, to the
best of our knowledge, none of the existing results is directly applicable to
our problem. This is because, speaking in the terms used later in this paper,
the available results either (i) require the knowledge of $q_{k}(f)$
(expected, not sampled) as an input, (ii) propose how to calculate $q_{k}(f)$
for the first two nodes only, or (iii) calculate $q_{k}(f)$ as an average of
many simulated traversals of the known graph (in contrast, we only have one
run on unknown graph) [48]. In fact, this work can be naturally extended to
address the problems with PPSWOR.
Previous version of this paper. This work is a revised and extended version of
our recent conference paper [1]. The main changes are: (i) a successful
application of our $RG(p_{k})$-based correction procedure to a wide range of
large-scale real-life Internet topologies (Table II, Fig. 5, Fig. 6(d),
Section VII-B), (ii) bias correction procedures for arbitrary node properties
(Section VI), (iii) a complementary BFS-bias correction technique (Section
VIII), and (iv) a publicly available ready-to-use python implementation of our
approach.
Finally, we would like to stress that our two other JSAC submissions [3, 49]
focus on sampling techniques based on random walks, which differ in
fundamental aspects (sampling with replacement vs without, sampling of nodes
vs of topology) from the BFS sampling addressed here.
## III Graph exploration techniques
Let $G=(V,E)$ be a connected graph with the set of vertices $V$, and a set of
undirected edges $E$. Initially, $G$ is unknown, except for one (or some
limited number of) seed node(s). When sampling through graph exploration, we
begin at the seed node, and we recursively visit (one, some or all) its
neighbors. We distinguish two main categories of exploration techniques:
random walks and graph traversals.
### III-A Random walks (baseline)
Random walks allow revisiting the same node many times. We consider222We
include random walks only as a useful baseline for comparison with graph
traversals (_e.g._ , BFS). The analysis of random walks does not count as a
contribution of this paper. the following classic examples:
#### III-A1 Random Walk (RW)
In this classic sampling technique [4], we start at some seed node. At every
iteration, the next-hop node $v$ is chosen uniformly at random among the
neighbors of the current node $u$. It is easy to see that RW introduces a
linear bias towards nodes of high degree [4].
#### III-A2 Metropolis Hastings Random Walk (MHRW)
In this technique, as in RW, the next-hop node $w$ is chosen uniformly at
random among the neighbors of the current node $u$. However, with a
probability that depends on the degrees of $w$ and $u$, MHRW performs a self-
loop instead of moving to $w$. More specifically, the probability
$P^{\scriptscriptstyle\textrm{MH}}_{u,w}$ of moving from $u$ to $w$ is as
follows [50]:
$P^{\scriptscriptstyle\textrm{MH}}_{u,w}=\left\\{\begin{array}[]{ll}\frac{1}{k_{u}}\cdot\min(1,\frac{k_{u}}{k_{w}})&\textrm{if
$w$ is a neighbor of $u$,}\\\ 1-\sum_{y\neq
u}P^{\scriptscriptstyle\textrm{MH}}_{u,y}&\textrm{if $w=u$,}\\\
0&\textrm{otherwise},\end{array}\right.$ (1)
where $k_{v}$ is the degree of node $v$. Essentially, MHRW reduces the
transitions to high-degree nodes and thus eliminates the degree bias of RW.
This property of MHRW was recently exploited in various network sampling
contexts [8, 11, 2, 10].
### III-B Graph traversals
In contrast, graph traversals never revisits the same node. At the end of the
process, and assuming that the graph is connected, all nodes are visited.
However, when using graph traversals for sampling, we terminate after having
collected a fraction $f<1$ (usually $f\ll 1$) of graph nodes.
#### III-B1 Breadth First Search (BFS)
BFS is a classic graph traversal algorithm that starts from the seed and
progressively explores all neighbors. At each new iteration the earliest
explored but not-yet-visited node is selected next. Consequently, BFS
discovers first the nodes closest to the seed.
#### III-B2 Depth First Search (DFS)
This technique is similar to BFS, except that at each iteration we select the
latest explored but not-yet-visited node. As a result, DFS explores first the
nodes that are faraway (in the number of hops) from the seed.
#### III-B3 Forest Fire (FF)
FF is a randomized version of BFS, where for every neighbor $v$ of the current
node, we flip a coin, with probability of success $p$, to decide if we explore
$v$. FF reduces to BFS for $p\\!\\!=\\!1$. It is possible that this process
dies out before it covers all nodes. In this case, in order to make FF
comparable with other techniques, we revive the process from a random node
already in the sample. Forest Fire is inspired by the graph growing model of
the same name proposed in [51] and is used as a graph sampling technique in
[12].
#### III-B4 Snowball Sampling (SBS) and Respondent-Driven Sampling (RDS)
According to a classic definition by Goodman [34], an $n$-name Snowball
Sampling is similar to BFS, but at every node $v$, not all $k_{v}$, but
exactly $n$ neighbors are chosen randomly out of all $k_{v}$ neighbors of $v$.
These $n$ neighbors are scheduled to visit, but only if they have not been
visited before.
Respondent-Driven Sampling (RDS) [35, 15, 16] adopts SBS to penetrate hidden
populations (such as that of drug addicts) in social surveys. In Section II,
we comment on current techniques to correct for SBS/RDS bias towards nodes of
higher degree.
## IV Graph model $RG(p_{k})$
$G=(V,E)$ | graph $G$ with nodes $V$ and edges $E$
---|---
$k_{v}$ | degree of node $v$
$p_{k}\ =\frac{1}{|V|}\sum_{v\in V}1_{k_{v}=k}$ | degree distribution in $G$
$\langle k\rangle\ =\ \langle p_{k}\rangle\ =\sum_{k}k\,p_{k}$ | average node degree in $G$
$q_{k}$ | expected sampled degree distribution
$\langle q_{k}\rangle\ =\sum_{k}k\,q_{k}$ | expected sampled average node degree
$\widehat{q}_{k}$ | sampled degree distribution
$\widehat{p}_{k}$ | estimated original degree distribution in $G$
$f$ | fraction of nodes covered by the sample
TABLE I: Notation Summary.
A basic, yet very important property of every graph is its node degree
distribution $p_{k}$, _i.e._ , the fraction of nodes with degree equal to $k$,
for all $k\geq 0$.333As we define $p_{k}$ as a ‘fraction’, not the
‘probability’, $p_{k}$ determines the degree sequence in the graph, and vice
versa. Depending on the network, the degree distribution can vary, ranging
from constant-degree (in regular graphs), a distribution concentrated around
the average value (_e.g._ , in Erdös-Rényi random graphs or in well-balanced
P2P networks), to heavily right-skewed distributions with $k$ covering several
decades (as this is the case in WWW, unstructured P2P, Internet at the IP and
Autonomous System level, OSNs). We handle all these cases by assuming that we
are given _any_ fixed node degree distribution $p_{k}$. Other than that, the
graph $G$ is drawn uniformly at random from the set of all graphs with degree
distribution $p_{k}$. We denote this model by $RG(p_{k})$.
Because $RG(p_{k})$ mimics an arbitrary node degree distribution $p_{k}$, it
can be considered a “first-order approximation” of real-life graphs. Of
course, there are many graph properties other than $p_{k}$ that are not
captured by $RG(p_{k})$. However, we show later that, with respect to the BFS
sampling bias, $RG(p_{k})$ approximates the real Internet topologies
surprisingly well.
We use a classic technique to generate $RG(p_{k})$, called the _configuration
model_ [52]: each node $v$ is given $k_{v}$ “stubs” or “edges-to-be”. Next,
all these $\sum_{v\in V}k_{v}=2|E|$ stubs are randomly matched in pairs, until
all stubs are exhausted (and $|E|$ edges are created). In Fig. 2 (ignore the
rectangular interval [0,1] for now), we present four nodes with their stubs
(left) and an example of their random matching (right).
## V Analyzing the Node Degree Bias
In this section, we study the node degree bias observed when the graph
exploration techniques of Section III are run on the random graph $RG(p_{k})$
of Section IV. In particular, we are interested in the node degree
distribution $q_{k}$ expected to be observed in the raw sample. Typically, the
observed distribution is different from the original one, $q_{k}\neq p_{k}$,
with higher average value $\langle q_{k}\rangle>\langle p_{k}\rangle$ (_i.e._
, average sampled and observed node degree, respectively). Below, we derive
$q_{k}$ as a function of $p_{k}$ and, in the case of BFS, of the fraction of
sampled nodes $f$.
### V-A Random walks (baseline)
We begin by summarizing the relevant results known for walks, in particular
for RW and MHRW. They will serve as a reference point for our main analysis of
graph traversals below.
#### V-A1 Random Walk (RW)
Random walks have been widely studied; see [4] for an excellent survey. In any
given connected and aperiodic graph, the probability of being at a particular
node $v$ converges at equilibrium to the stationary distribution
$\pi^{\scriptscriptstyle\textrm{RW}}_{v}\\!\\!=\\!\frac{k_{v}}{2|E|}$.
Therefore, the expected observed degree distribution
$q^{\scriptscriptstyle\textrm{RW}}_{k}$ is
$q^{\scriptscriptstyle\textrm{RW}}_{k}\ =\ \
\sum_{v}\pi^{\scriptscriptstyle\textrm{RW}}_{v}\cdot 1_{\\{k_{v}=k\\}}\
=\frac{k}{2|E|}\,p_{k}\,|V|\ =\ \frac{k\,p_{k}}{\langle k\rangle},$ (2)
where $\langle k\rangle$ is the average node degree in $G$. Eq.(2) is
essentially similar to calculation in [13, 14, 15, 16]. As this holds for any
fixed (and connected and aperiodic) graph, it is also true for all connected
graphs generated by the configuration model. Consequently, the expected
observed average node degree is
$\langle q_{k}^{\scriptscriptstyle\textrm{RW}}\rangle\ =\
\sum_{k}k\,q^{\scriptscriptstyle\textrm{RW}}_{k}\ =\
\frac{\sum_{k}k^{2}\,p_{k}}{\langle k\rangle}\ =\ \frac{\langle
k^{2}\rangle}{\langle k\rangle},$ (3)
where $\langle k^{2}\rangle$ is the average squared node degree in $G$. We
show this value $\frac{\langle k^{2}\rangle}{\langle k\rangle}$ in Fig. 1.
#### V-A2 Metropolis Hastings Random Walk (MHRW)
It is easy to show that the transition matrix
$P^{\scriptscriptstyle\textrm{MH}}_{u,w}$ shown in Eq.(1) leads to a uniform
stationary distribution
$\pi^{\scriptscriptstyle\textrm{MH}}_{v}\\!\\!=\\!\frac{1}{|V|}$ [50], and
consequently:
$\displaystyle q^{\scriptscriptstyle\textrm{MH}}_{k}$ $\displaystyle=$
$\displaystyle p_{k}$ (4) $\displaystyle\langle
q_{k}^{\scriptscriptstyle\textrm{MH}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k}k\,q^{\scriptscriptstyle\textrm{MH}}_{k}\ =\
\sum_{k}k\,p_{k}\ =\ \langle k\rangle.$ (5)
In Fig. 1, we show that MHRW estimates the true mean.
### V-B Graph traversals (Main Result)
In both RW and MHRW the nodes can be revisited. So the state of the system at
iteration $i\\!+\\!1$ depends only on iteration $i$, which makes it possible
to analyze with Markov Chain techniques. In contrast, graph traversals do not
allow for node revisits, which introduces crucial dependencies between all the
iterations and significantly complicates the analysis. To handle these
dependencies, we adopt an elegant technique recently introduced in [36] (to
study the size of the largest connected component) and extended in [37] (to
study the bias of traceroute sampling). However, our work differs in many
aspects from both [36] and [37], on which we comment in detail in the related
work Section II.
Figure 2: An illustration of the stub-level, on-the-fly graph exploration
without replacements. In this particular example, we show an execution of BFS
starting at node $v_{1}$. Left: Initially, each node $v$ has $k_{v}$ stubs,
where $k_{v}$ is a given target degree of $v$. Each of these stubs is assigned
a real-valued number drawn uniformly at random from the interval $[0,1]$ shown
below the graph. Next, we follow Algorithm 1 with a starting node $v_{1}$. The
numbers next to the stubs of every node $v$ indicate the order in which these
stubs are enqueued on $Q$. Center: The state of the system at time $t$. All
stubs in $[0,t]$ have already been matched (the indices of matched stubs are
set in plain line). All unmatched stubs are distributed uniformly at random on
$(t,1]$. This interval can contain also some (here two) already matched stubs.
Right: The final result is a realization of a random graph $G$ with a given
node degree sequence (_i.e._ , of the configuration model). $G$ may contain
self-loops and multiedges.
#### V-B1 Exploration without replacement at the stub level
We begin by defining Algorithm 1 (below) - a general graph traversal technique
that collects a sequence of nodes $S$, without replacements. To be compatible
with the configuration model (see Section IV), we are interested in the
process _at the stub level_ , where we consider one stub at a time, rather
than one node at a time. An integral part of the algorithm is a queue $Q$ that
keeps the discovered, but still not-yet-followed stubs. First, we enqueue on
$Q$ all the stubs of some initial node $v_{1}$, and by setting
$S\\!\leftarrow\\![v_{1}]$. Next, at every iteration, we dequeue one stub from
$Q$, call it $a$, and follow it to discover its partner-stub $b$, and $b$’s
owner $v(b)$. If node $v(b)$ is not yet discovered, _i.e._ , if $v(b)\notin
S$, then we append $v(b)$ to $S$ and we enqueue on $Q$ all other stubs of
$v(b)$.
Algorithm 1 Stub-Level Graph Traversal
1: $S\leftarrow[v_{1}]$ and $Q\leftarrow$ [all stubs of $v_{1}$]
2: while $Q$ is nonempty do
3: Dequeue $a$ from $Q$
4: Discover $a$’s partner $b$
5: if $v(b)\notin S$ then
6: Append $v(b)$ to $S$
7: Enqueue on $Q$ all stubs of $v(b)$ except $b$
8: else
9: Remove $b$ from $Q$
10: end if
11: end while
Depending on the scheduling discipline for the elements in $Q$ (line 3),
Algorithm 1 implements BFS (for a first-in first out scheduling), DFS (last-in
first-out) or Forest Fire (first-in first-out with randomized stub losses).
Line 9 guarantees that the algorithm never tracebacks the edges, _i.e._ , that
stub $a$ dequeued from $Q$ in line 3 never belongs to an edge that has already
been traversed in the opposite direction.
#### V-B2 Discovery on-the-fly
In line 4 of Algorithm 1, we follow stub $a$ to discover its partner $b$. In a
fixed graph $G$, this step is deterministic. In the configuration model
$RG(p_{k})$, a fixed graph $G$ is obtained by matching all the stubs uniformly
at random. Next, we can sample this fixed graph and average the result over
the space of all the random graphs $RG(p_{k})$ that have just been
constructed. Unfortunately, this space grows exponentially with the number of
nodes $|V|$, making the problem untractable. Therefore, we adopt an
alternative construction of $G$ \- by iteratively selecting $b$ on-the-fly
(_i.e._ , every time line 4 is executed), uniformly at random from all still
unmatched stubs. By the principle of deferred decisions [53], these two
approaches are equivalent.
With the help of the on-the-fly approach, we are able to write down the
equations we need. Indeed, let us denote by $X_{i}\in V$ the $i$th selected
node, and let $\mathbb{P}(X_{1}\\!\\!=\\!u)$ be the probability that node
$u\in V$ is chosen as a starting node. It is easy to show that with
$z\\!\\!=\\!2|E|$ we have
$\displaystyle\mathbb{P}(X_{2}\\!\\!=\\!v)$ $\displaystyle=$
$\displaystyle\sum_{u\neq
v}\frac{k_{v}}{z\\!-\\!k_{u}}\cdot\mathbb{P}(X_{1}\\!\\!=\\!u)$ (6)
$\displaystyle\mathbb{P}(X_{3}\\!\\!=\\!w)$ $\displaystyle=$
$\displaystyle\sum_{v\neq w}\sum_{u\neq
w,v}\frac{k_{w}}{z\\!-\\!k_{v}\\!-\\!k_{u}}\cdot\frac{k_{v}}{z\\!-\\!k_{u}}\cdot\mathbb{P}(X_{1}\\!\\!=\\!u),\quad$
(7)
and so on. Theoretically, these equations allow us to calculate the expected
node degree at any iteration, and thus the degree bias of BFS.
#### V-B3 Breaking the dependencies
There is still one problem with the equations above. Due to the increasing
number of nested sums, the results can be calculated in practice for a first
few iterations only. This is because we select stub $b$ uniformly and
independently at random from all the _unmatched_ stubs. So the stub selected
at iteration $i$ depends on the stubs selected at iterations $1\ldots
i\\!-\\!1$, which results in the nested sums. We remedy this problem by
implementing the on-the-fly approach as follows. First, we assign each stub a
real-valued index $t$ drawn uniformly at random from the interval $[0,1]$.
Then, every time we process line 4, we pick $b$ as the unmatched stub with the
smallest index. We can interpret this as a continuous-time process, where we
determine progressively the partners of stubs dequeued from $Q$, by scanning
the interval from time $t\\!\\!=\\!0$ to $t\\!\\!=\\!1$ in a search of
unmatched stubs. Because the indices chosen by the stubs are independent from
each other, the above trick breaks the dependence between the stubs, which is
crucial for making this approach tractable.
In Fig. 2, we present an example execution of Algorithm 1, where line 4 is
implemented as described above.
#### V-B4 Expected sampled degree distribution
$q^{\scriptscriptstyle\textrm{BFS}}_{k}$
Now we are ready to derive the expected observed degree distribution $q_{k}$.
Recall that all the stub indices are chosen independently and uniformly from
$[0,1]$. A vertex $v$ with degree $k$ is not sampled yet at time $t$ if the
indices of all its $k$ stubs are larger than $t$, which happens with
probability $(1\\!-\\!t)^{k}$. So the probability that $v$ is sampled before
time $t$ is $1\\!-\\!(1\\!-\\!t)^{k}$. Therefore, the expected fraction of
vertices of degree $k$ sampled before $t$ is
$f_{k}(t)=p_{k}(1\\!-\\!(1\\!-\\!t)^{k}).$ (8)
By normalizing Eq.(8), we obtain the expected observed (_i.e._ , sampled)
degree distribution at time $t$:
$q^{\scriptscriptstyle\textrm{BFS}}_{k}(t)\ =\
\frac{f_{k}(t)}{\sum_{l}f_{l}(t)}\ =\
\frac{p_{k}(1-(1\\!-\\!t)^{k})}{\sum_{l}p_{l}(1-(1\\!-\\!t)^{l})}.$ (9)
Unfortunately, it is difficult to interpret
$q^{\scriptscriptstyle\textrm{BFS}}_{k}(t)$ directly, because $t$ is
proportional neither to the number of matched edges nor to the number of
discovered nodes. Recall that our primary goal is to express
$q^{\scriptscriptstyle\textrm{BFS}}_{k}$ as a function of fraction $f$ of
covered nodes. We achieve this by calculating $f(t)$ \- the expected fraction
of nodes, of any degree, visited before time $t$
$f(t)=\sum_{k}f_{k}(t)=1-\sum_{k}p_{k}(1\\!-\\!t)^{k}\ .$ (10)
Because $p_{k}\geq 0$, and $p_{k}>0$ for at least one $k>0$, the term
$\sum_{k}p_{k}(1\\!-\\!t)^{k}$ is continuous and strictly decreasing from 1 to
0 with $t$ growing from 0 to 1. Thus, for $f\in[0,1]$ there exists a well
defined $t\\!\\!=\\!t(f)$ that satisfies Eq.(10), _i.e._ , the inverse of
$f(t)$. Although we cannot compute $t(f)$ analytically (except in some special
cases such as for $k\leq 4$), it is straightforward to find it numerically.
Now, we can rewrite Eq. (9) as
$q^{\scriptscriptstyle\textrm{BFS}}_{k}(f)\ =\
\frac{p_{k}(1-(1\\!-\\!t(f))^{k})}{\sum_{l}p_{l}(1-(1\\!-\\!t(f))^{l})},$ (11)
which is the expected observed degree distribution after covering fraction $f$
of nodes of graph $G$. Consequently, the expected observed average degree is
$\langle q_{k}^{\scriptscriptstyle\textrm{BFS}}\rangle(f)\ =\ \sum_{k}k\cdot
q^{\scriptscriptstyle\textrm{BFS}}_{k}(f).$ (12)
In other words, Eq.(11) and Eq.(12) describe the bias of BFS sampling under
$RG(p_{k})$, which was our first goal in this paper. Below, we further analyze
these equations to get more insights in the nature of BFS bias.
#### V-B5 Equivalence of traversal techniques under $RW(p_{k})$
An interesting observation is that, under the random graph model $RW(p_{k})$,
all common traversal techniques (BFS, DFS, FF, SBS, etc) are subject to
exactly the same bias. The explanation is that the sampled node sequence $S$
is fully determined by the choice of stub indices on $[0,1]$, independently of
the way we manage the elements in $Q$.
#### V-B6 Equivalence of traversals to weighted sampling without replacement
Consider a node $v$ with a degree $k_{v}$. The probability that $v$ is
discovered before time $t$, given that it has not been discovered before
$t_{0}\leq t$, is
$\mathbb{P}(\textrm{$v$ before time $t$ $|$ $v$ not before
$t_{0}$})=1-\left(\frac{1\\!-\\!t}{1\\!-\\!t_{0}}\right)^{k_{v}}$ (13)
We now take the derivative of the above equation with respect to $t$, which
results in the conditional probability density function
$k_{v}(\frac{1\\!-\\!t}{1\\!-\\!t_{0}})^{k_{v}\\!-\\!1}$. Setting
$t\\!\\!\rightarrow\\!t_{0}$ (but keeping $t\\!>\\!\\!t_{0}$), reduces it to
$k_{v}$, which is the probability density that $v$ is sampled at $t_{0}$,
given that it has not been sampled before. This means that at every point in
time, out of all nodes that have not yet been selected, the probability of
selecting $v$ is proportional to its degree $k_{v}$. Therefore, this scheme is
equivalent to node sampling weighted by degree, without replacements.
#### V-B7 Equivalence of traversals with $f\\!\\!\rightarrow\\!0$ to RW
Finally, for $f\\!\\!\rightarrow\\!0$ (and thus $t\\!\\!\rightarrow\\!0$), we
have $1\\!-\\!(1\\!-\\!t)^{k}\simeq kt$, and Eq. (9) simplifies to Eq. (2).
This means that in the beginning of the sampling process, every traversal
technique is equivalent to RW, as shown in Fig. 1 for
$f\\!\\!\rightarrow\\!0$.
#### V-B8 $\langle q_{k}^{\scriptscriptstyle\textrm{BFS}}\rangle$ is
decreasing in $f$
As in Section V-B2, let $X_{i}\in V$ be the $i$th selected node, and let
$z\\!\\!=\\!2|E|$. We have shown above that our procedure is equivalent to
weighted sampling without replacements, thus we can write
$\mathbb{P}(X_{1}\\!\\!=\\!u)=\frac{k_{u}}{z}$. Now, it follows from Eq. (6)
that $\mathbb{P}(X_{2}\\!\\!=\\!w)=\frac{k_{w}}{z}\cdot\alpha_{w}$, where
$\alpha_{w}=\sum_{u\neq w}\frac{k_{u}}{z-k_{u}}$. Because for any two nodes
$a$ and $b$, we have
$\alpha_{b}\\!-\\!\alpha_{a}=z(k_{a}\\!-\\!k_{b})/((z\\!-\\!k_{a})(z\\!-\\!k_{b})),$
$\alpha_{w}$ strictly decreases with growing $k_{w}$. As a result,
$\mathbb{P}(X_{2})$ is more concentrated around nodes with smaller degrees
than is $\mathbb{P}(X_{1})$, implying that
$\mathbb{E}[k_{X_{2}}]<\mathbb{E}[k_{X_{1}}]$. We can use an analogous
argument at every iteration $i\leq|V|$, which allows us to say that
$\mathbb{E}[k_{X_{i}}]<\mathbb{E}[k_{X_{i-1}}]$. In other words, $\langle
q_{k}^{\scriptscriptstyle\textrm{BFS}}\rangle(f)$ is a decreasing function of
$f$.
A practical consequence is that many short traversals are more biased than a
long one, with the same total number of samples.
#### V-B9 Comments on the graph connectivity
Note that the configuration model $RG(p_{k})$ might result in a graph $G$ that
is not connected. In this case, every exploration technique covers only the
component $C$ in which it was initiated; consequently, the process described
in Section V-B3 stops once $C$ is covered.
In practice, it is also possible to efficiently generate a simple and
connected random graph with a given degree sequence [54].
## VI Correcting for node degree bias
In the previous section we derived the expected observed degree distribution
$q_{k}$ as a function of the original degree distribution $p_{k}$. The
distribution $q_{k}$ is usually biased towards high-degree nodes, _i.e._ ,
$\langle q_{k}\rangle\\!>\\!\langle p_{k}\rangle$. Moreover, because many node
properties are correlated with the node degree [2], their estimates are also
potentially biased. For example, let $x(v)$ be an arbitrary function defined
on graph nodes $V$ (_e.g._ , node age) and let its mean value
$x_{\scriptstyle\textrm{av}}=\frac{1}{|V|}\sum_{v\in V}x(v)$ (14)
be the value we are trying to estimate. If $x(v)$ is somehow correlated with
node degree $k_{v}$, then the straightforward estimator
$\widehat{x}^{\,naive}_{\scriptstyle\textrm{av}}=1/|S|\cdot\sum_{v\in S}x(v)$
is subject to the same bias as is $\langle q_{k}\rangle$. In this section, we
derive unbiased estimators $\widehat{x}_{\scriptstyle\textrm{av}}$ of
$x_{\scriptstyle\textrm{av}}$. We also directly apply
$\widehat{x}_{\scriptstyle\textrm{av}}$ to obtain the estimators
$\widehat{p}_{k}$ and $\langle\widehat{p}_{k}\rangle$ of the original node
degree distribution and its mean, respectively.
Let $S\subset V$ be a sequence of vertices that we sampled. Based on $S$, we
can estimate $q_{k}$ as
$\displaystyle\widehat{q}_{k}$ $\displaystyle=$
$\displaystyle\frac{\textrm{number of nodes in $S$ with degree $k$}}{|S|}.$
(15)
### VI-A Random walks (baseline)
#### VI-A1 Random Walk (RW)
Under RW, the sampling probability of a node $v$ is proportional to its degree
$k_{v}$. Because the sampling is done with replacements, we can apply the
Hansen-Hurwitz estimator [55] to obtain the following unbiased estimator [13,
14, 15, 16]
$\widehat{x}^{\,{\scriptscriptstyle\textrm{RW}}}_{\scriptstyle\textrm{av}}\ =\
\frac{\sum_{v\in S}x(v)/k_{v}}{\sum_{v\in S}1/k_{v}}.$ (16)
For example, if $x(v)\\!\\!=\\!1_{\\{k_{v}=k\\}}$ then
$\widehat{x}^{\,{\scriptscriptstyle\textrm{RW}}}_{\scriptstyle\textrm{av}}$
estimates the proportion of nodes with degree equal to $k$, _i.e._ , exactly
$p_{k}$. In that case, Eq.(16) simplifies to
$\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{RW}}}\ =\
\frac{\widehat{q}_{k}}{k}\ \cdot\
\left(\sum_{l}\frac{\widehat{q}_{l}}{l}\right)^{-1}$ (17)
where we used the fact that $\sum_{v\in
S}1_{\\{k_{v}=k\\}}=|V|\cdot\widehat{q}_{k}$. From Eq.(17), we can estimate
the average node degree as
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{RW}}}\rangle\ =\
\sum_{k}k\,\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{RW}}}\ =\
1\cdot\left(\sum_{l}\frac{\widehat{q}_{l}}{l}\right)^{-1}=\frac{|S|}{\sum_{v\in
S}\frac{1}{k_{v}}}$ (18)
#### VI-A2 Metropolis Hastings Random Walk (MHRW)
Under MHRW, we trivially have
$\displaystyle\widehat{x}^{\,{\scriptscriptstyle\textrm{MH}}}_{\scriptstyle\textrm{av}}$
$\displaystyle=$ $\displaystyle\frac{1}{|S|}\sum_{v\in S}x(v),$ (19)
$\displaystyle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{MH}}}$
$\displaystyle=$ $\displaystyle\widehat{q}_{k},$ (20)
$\displaystyle\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{MH}}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{k}k\,\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{MH}}}\
=\ \sum_{k}k\,\widehat{q}_{k}.$ (21)
### VI-B Graph traversals
Under BFS and other traversals, the inclusion probability
$\pi^{\scriptscriptstyle\textrm{BFS}}_{v}$ (_i.e._ , the probability of node
$v$ being included in sample $S$) of node $v\in V$ is proportional to
$\pi^{\scriptscriptstyle\textrm{BFS}}_{v}\ \ \sim\ \
\frac{q^{\,{\scriptscriptstyle\textrm{BFS}}}_{k_{v}}}{p_{k_{v}}}\ \ \sim\ \
1-(1\\!-\\!t(f))^{k_{v}},$
where the second relation originates from Eq.(11). Consequently, an
application of the Horvitz-Thompson estimator [56], designed typically for
sampling without replacement, leads to
$\widehat{x}^{\,{\scriptscriptstyle\textrm{BFS}}}_{\scriptstyle\textrm{av}}\
=\ \left(\sum_{v\in
S}\frac{x(v)}{1\\!-\\!(1\\!-\\!t(f))^{k_{v}}}\right)\cdot\left(\sum_{v\in
S}\frac{1}{1\\!-\\!(1\\!-\\!t(f))^{k_{v}}}\right)^{-1}.$ (22)
Now, similarly to the analysis of RW (above), we obtain
$\displaystyle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}$
$\displaystyle=$ $\displaystyle\frac{\widehat{q}_{k}}{1-(1\\!-\\!t(f))^{k}}\
\cdot\ \left(\sum_{l}\frac{\widehat{q}_{l}}{1-(1\\!-\\!t(f))^{l}}\right)^{-1}$
(23)
$\displaystyle\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{k}k\,\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}.$
(24)
However, in order to evaluate these expressions, we need to evaluate $t(f)$,
that, in turn, requires $p_{k}$. We can solve this chicken-and-egg problem
iteratively, if we know the real fraction
$f^{\scriptscriptstyle\textrm{real}}$ of covered nodes, or equivalently the
graph size $|V|$. First, we evaluate Eq.(23) for some values of $t$ and feed
the resulting $\widehat{p}_{k}$’s into Eq. (10) to obtain the corresponding
$f$’s. By repeating this process, we can efficiently drive the values of $f$
arbitrarily close to $f^{\scriptscriptstyle\textrm{real}}$, and thus find the
desired $\widehat{p}_{k}$.
In summary, for BFS, we showed how to estimate the mean
$x_{\scriptstyle\textrm{av}}$ of an arbitrary function $x(v)$ defined on graph
nodes, with the estimator of the original degree distribution $p_{k}$ as a
special case. Note that our approach is feasible, as it requires only the
sample $S$ (with value $x(v)$ and degree $k_{v}$ for every node $v\in S$) and
the fraction $f$ of sampled nodes. In [24], we make a python implementation of
all the above estimators publicly available.
### VI-C Alternative approach
In Section VIII, we propose and evaluate a family of alternative correction
procedures that are _unbiased for any arbitrary topology_. Although seemingly
attractive, they are characterized by large variance, which makes them far
less effective than our $RG(p_{k})$-based correction technique.
Figure 3: Comparison of sampling techniques in theory and in simulation. Left:
Observed (sampled) average node degree $\langle q_{k}\rangle$ as a function of
the fraction $f$ of sampled nodes, for various sampling techniques. The
results are averaged over 1000 graphs with 10000 nodes each, generated by the
configuration model with a fixed heavy-tailed degree distribution $p_{k}$
(shown on the right). Right: Real, expected, and estimated (corrected) degree
distributions for selected techniques and values of $f$ (other techniques
behave analogously). We obtained analogous results for other degree
distributions and graph sizes $|V|$. The term $\langle k\rangle$ is the real
average node degree, and $\langle k^{2}\rangle$ is the real average squared
node degree. Figure 4: The effect of assortativity $r$ on the results. First,
we use the configuration model with the same degree distribution $p_{k}$ as in
Fig. 3 (and the same number of nodes $|V|=10000$) to generate a graph $G$.
Next, we apply the pairwise edge rewiring technique [57] to change the
assortativity $r$ of $G$ without changing node degrees. This technique
iteratively takes two random edges $\\{v_{1},w_{1}\\}$ and
$\\{v_{2},w_{2}\\}$, and rewires them as $\\{v_{1},w_{2}\\}$ and
$\\{v_{2},w_{1}\\}$ only if it brings us closer to the desired value of
assortativity $r$. As a result, we obtain graphs with a positive (left) and
negative (right) assortativity $r$. Note that for a better readability, we
present only the values of $f\in[0,0.1]$, _i.e._ , ten times smaller than in
Fig. 3.
## VII Simulation results
In this section, we evaluate our theoretical findings on random and real-life
graphs.
### VII-A Random graphs
Fig. 3 verifies all the formulae derived in this paper, for the random graph
$RG(p_{k})$ with a given degree distribution. The analytical expectations are
plotted in thick plain lines in the background and the averaged simulation
results are plotted in thinner lines lying on top of them. We observe almost a
perfect match between theory and simulation in estimating the sampled degree
distribution $q_{k}$ (Fig. 3, right) and its mean $\langle q_{k}\rangle$ (Fig.
3, left). Indeed, all traversal techniques follow the same curve (as predicted
in Section V-B5), which initially coincides with that of RW (see Section V-B7)
and is monotonically decreasing in $f$ (see Section V-B8). We also show that
degree-weighted node sampling without replacements exhibits exactly the same
bias (see Section V-B6). Finally, applying the estimators $\widehat{p}_{k}$
derived in Section VI perfectly corrects for the bias of $q_{k}$.
Of course, real-life networks are substantially different from $RG(p_{k})$.
For example, depending on the graph type, nodes may tend to connect to similar
or different nodes. Indeed, in most social networks high-degree nodes tend to
connect to other high-degree nodes [58]. Such networks are called
_assortative_. In contrast, biological and technological networks are
typically _disassortative_ , _i.e._ , they exhibit significantly more high-
degree-to-low-degree connections. This observation can be quantified by
calculating the _assortativity coefficient_ $r$ [58], which is the correlation
coefficient computed over all edges (_i.e._ , degree-degree pairs) in the
graph. Values $r\\!<\\!0$, $r\\!>\\!0$ and $r\\!=\\!0$ indicate
disassortative, assortative and purely random graphs, respectively.
For the same initial parameters as in Fig. 3 ($p_{k}$, $|V|$), we simulated
different levels of assortativity. Fig. 4 shows the results. Graph
assortativity $r$ strongly affects the first iterations of traversal
techniques. Indeed, for assortativity $r>0$ (Fig. 4, left), the degree bias is
even stronger than for $r=0$ (Fig. 3, left). This is because the high-degree
nodes are now interconnected more densely than in a purely random graph, and
are thus easier to discover by sampling techniques that are inherently biased
towards high-degree nodes. Interestingly, Forest Fire is by far the most
affected. A possible explanation is that under Forest Fire, low-degree nodes
are likely to be completely skipped by the first sampling wave. Not
surprisingly, a negative assortativity $r<0$ has the opposite effect: every
high-degree node tends to connect to low-degree nodes, which significantly
slows down the discovery of the former.
In contrast, random walks RW and MHRW are not affected by the changes in
assortativity. This is expected, because their stationary distributions hold
for _any_ fixed (connected and aperiodic) graph regardless of its topological
properties.
### VII-B Real-life fully known topologies
Recall, that our analysis is based on the random graph model $RG(p_{k})$ (see
Section IV), which is only an approximation of a typical real-life network
$G$. Indeed, $RG(p_{k})$ follows the node degree distribution of $G$, but is
likely to miss other important properties such as assortativity [58], whose
effect on the BFS process we have just demonstrated. For this reason, one may
expect that the technique based on $RG(p_{k})$ performs poorly on real-life
graphs. Surprisingly, this is not the case.
We evaluated our approach on a broad range of large, real-life, fully known
Internet topologies. As our main source of data we use SNAP Graph Library
[59]; Table II overviews these datasets. We present the results in Fig. 5.
Interestingly, in most cases the sampled average node degree
$\langle\widehat{q}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$ closely
matches the prediction $\langle
q_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$ of the random graph model
$RG(p_{k})$. More importantly, applying our BFS estimator
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$ of real
average node degree corrects for the bias of
$\langle\widehat{q}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$
surprisingly well. Some significant differences are visible only for
$f\\!\rightarrow\\!0$ and for some specific topologies (the last two in Fig.
5), which is exactly because the real-life graphs are not fully captured by
graph model $RG(p_{k})$.
Finally, we also study the RW estimator Eq.(18), as a simpler alternative to
the BFS one Eq.(24). Although they coincide for $f\\!\rightarrow\\!0$, the RW
estimator systematically and significantly underestimates the average node
degree $\langle k\rangle$ for larger values of $f$.
Dataset | # nodes | # edges | $\langle k\rangle\\!\\!=\\!\langle p_{k}\rangle$ | $\frac{\langle k^{2}\rangle}{\langle k\rangle}$ | Description
---|---|---|---|---|---
ca-CondMat | 21 363 | 91 341 | 8.6 | 22.5 | Collaboration network of Arxiv Condensed Matter [60]
email-EuAll | 224 832 | 340 794 | 3.0 | 567.9 | Email network of a large European Research Institution [60]
Facebook-New-Orleans | 63 392 | 816 885 | 25.8 | 88.1 | Facebook New Orleans network [33]
wiki-Talk | 2 388 953 | 4 656 681 | 3.9 | 2705.4 | Wikipedia talk (communication) network [61]
p2p-Gnutella31 | 62 561 | 147 877 | 4.7 | 11.6 | Gnutella peer to peer network from August 31 2002 [60]
soc-Epinions1 | 75 877 | 405 738 | 10.7 | 183.9 | Who-trusts-whom network of Epinions.com [62]
soc-Slashdot0811 | 77 360 | 546 486 | 14.1 | 129.9 | Slashdot social network from November 2008 [63]
as-caida20071105 | 26 475 | 53 380 | 4.0 | 280.2 | CAIDA AS Relationships Datasets, from November 2007
web-Google | 855 802 | 4 291 351 | 10.0 | 170.4 | Web graph from Google [63]
TABLE II: Real-life Internet topologies used in simulations. All graphs are
connected and undirected (which required preprocessing in some cases). Figure
5: BFS in real-life (fully known) Internet topologies described in Table II.
The blue circles represent the average node degree
$\langle\widehat{q}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$ sampled
by BFS, as the function of the fraction of covered nodes $f$. The thin lines
are the corrected values
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$ resulting
from the BFS estimator Eq.(24) (plain line) and the RW estimator Eq.(18)
(dashed). Results are averaged over 1000 randomly seeded BFS samples. The
thick lines are the analytical expectations assuming the random graph model
$RG(p_{k})$. Thick red line (top) is the expectation of $\langle
q_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$, calculated with Eq.(12)
given the knowledge of the true node degree distribution $p_{k}$. Thick gray
line (bottom) is the expectation of corrected
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$, Eq.(24),
_i.e._ , precisely $\langle k\rangle$. Figure 6: BFS in on-line (not fully
known) topologies. As in Fig. 5, except that the plots are based on BFS
samples taken in Facebook with 28 (random) seeds (a) and one seed (b), as well
as in Orkut with one seed (d). Additionally, we show in (c) the full node
degree distributions for Facebook. Because we do not have the true degree
distribution $p_{k}$ of Orkut, we cannot calculate its analytical curve
$\langle q_{k}^{\scriptscriptstyle\textrm{BFS}}\rangle$. Nevertheless, we show
in (d) our best guess of Orkut’s average node degree $\langle k\rangle$
learned by other means, as explained in Footnote 2.
### VII-C Sampling Facebook and Orkut
In this section, we apply and test the previous ideas in sampling real-life,
large-scale, and not fully known online social networks: Facebook and Orkut.
#### VII-C1 Facebook
We have implemented a set of crawlers to collect the samples of Facebook (FB)
following the BFS, RW, MHRW techniques. The data sets are summarized in Table
III. BFS28 consists of 28 small BFS-es initiated at 28 different nodes, which
allowed us to easily parallelize the process. Moreover, at the time of data
collection, we (naively) thought that this would reduce the BFS bias. After
gaining more insight (which, nota bene, motivated this paper), we collected a
single large BFS1. UNI represents the ground truth. The details of our
implementation are described in [2, 3].
_Results._ We present the Facebook sampling results in Fig. 6(a-c) and in
Table III. First, we observe that under BFS28, our estimators
$q_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}$ and
$\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}$ perform very well. For
example, we obtain
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle\\!\\!=\\!85.4$
compared with the true value $\langle k\rangle\\!\\!=\\!94.1$. In contrast,
BFS1 yields
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle\\!\\!=\\!72.7$
only. Most probably, this is because BFS1 consists of a single BFS run that
happens to begin in a relatively sparse part of Facebook. Indeed, note that
this run starts at
$\widehat{q}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\\!\\!=\\!50$ for
$f\\!\\!=\\!0$, and systematically grows with $f$ instead of falling.
Finally, note that both BFS28 and BFS1 are very short compared to the Facebook
size, with $f<1\%$ in both cases. For this reason, we observe almost no drop
in the sampled average node degre $\langle
q_{k}^{\scriptscriptstyle\textrm{BFS}}\rangle$ in Fig. 6(a,b). For the same
reason, both the BFS and RW estimators yield almost identical results.
All the above observations hold also for the _entire_ degree distribution,
which is shown in Fig. 6(c).
#### VII-C2 Orkut
Finally, we apply our methodology to a single BFS sample of Orkut collected in
2006 and described in [19]. It contains $|S|=3072K$ nodes, which accounts for
$f\\!\\!=\\!11.3\%$ of entire Orkut size.
We show the results in Fig. 6(d). Similarly to Facebook BFS1, the sampled
average node degree
$\langle\widehat{q}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$ does not
decrease monotonically in $f$. Again, the underlying reason might be the
arbitrary choice of the starting node (in sparsely connected India in this
case). Nevertheless, the estimator
$\langle\widehat{p}_{k}^{\,{\scriptscriptstyle\textrm{BFS}}}\rangle$
approximates the average node degree444Unfortunately, according to our
personal communication with Orkut administrators, there is no ground truth
value of the Orkut’s average node degree $\langle k\rangle$ for October 2006,
_i.e._ , the period when the BFS sample of [19] was collected. However, many
hints point to a number close to $\langle k\rangle\\!\\!=\\!30$, _e.g._ , [18]
reports $\langle k\rangle=30.2$ in June-September 2006, and [64] reports
$\langle k\rangle=19$ in late 2004 (which is in agreement with the
densification law [51, 60]). But, as these studies may potentially be subject
to various biases, we cannot take these numbers for granted. relatively well.
Facebook | UNI | RW | BFS28 | BFS1 | MHRW
---|---|---|---|---|---
$|S|$ | 982K | 2.26M | 28$\times$81K | 1.19M | 2.26M
$f$ | 0.44% | 1.03% | 28$\times$0.04% | 0.54% | 1.03%
$\langle\widehat{q}_{k}\rangle$ | 94.1 | 338.0 | 323.9 | 285.9 | 95.2
$\langle q_{k}\rangle$ | - | 329.8 | 329.1 | 328.7 | 94.1
$\langle\widehat{p}_{k}\rangle$ | - | 93.9 | 85.4 | 72.7 | 95.2
Orkut | | | | |
$|S|$ | - | - | - | 3.07M | -
$f$ | - | - | - | 11.3% | -
$\langle\widehat{p}_{k}\rangle$ | 30 2 | | | 33.1 |
TABLE III: Facebook and Orkut data sets and measurements.
## VIII Arbitrary-topology BFS estimators
The $RG(p_{k})$-based BFS-bias correction procedure is, by construction,
unbiased for random graphs $RG(p_{k})$. However, when applied to arbitrary
graphs, in particular to real-life Internet topologies, our $RG(p_{k})$-based
estimators are potentially subject to some bias (_i.e._ , may be not perfect).
Fortunately, we have seen in Section VII-B that this bias is usually very
limited. This is because $RG(p_{k})$ mimics an arbitrary node degree
distribution $p_{k}$, which is, by far, the most crucial parameter affecting
the BFS degree bias.
Interestingly, it is possible to derive estimators that are _unbiased in any
arbitrary topology_. Unfortunately, these _arbitrary-topology estimators_ are
characterized by a very large variance, which makes them, in practice, less
effective than the $RG(p_{k})$-based estimators.
In this section we show examples of arbitrary-topology estimators and compare
them with $RG(p_{k})$-based estimators in simulations.
### VIII-A Goal
Let $G=(V,E)$ be a connected undirected graph. A typical (incomplete) graph
traversal, such as BFS, is determined by the first node. So we can denote by
$S(v)\subset V$ the set of sampled nodes, given that we started at node $v\in
V$. Our goal is to use $S(v)$ to estimate the total
$x_{\scriptstyle\textrm{tot}}=\sum_{v\in V}x(v)\,,$
where $x$ is a finite measurable function defined on graph nodes.
### VIII-B General arbitrary-topology estimator
Let $U\in V$ be a random variable representing the first node in our sample,
following the probability distribution
$\Pr[U\\!\\!=\\!w]\ =\ p(w)\ >\ 0.$
Let $Q(w)\subseteq V$ be a set of nodes uniquely defined by $G$ and $w$.
Define
$\widehat{x}_{\scriptstyle\textrm{tot}}=\sum_{v\in Q(U)}\frac{x(v)}{\pi(v)},$
(25)
where
$\pi(v)=\sum_{w\in V:\ v\in Q(w)}p(w).$ (26)
###### Lemma 1
$\widehat{x}_{\scriptstyle\textrm{tot}}$ is an unbiased estimator of
$x_{\scriptstyle\textrm{tot}}$.
_Proof:_ In order to prove Lemma 1, we have to show that
$\mathbb{E}[\widehat{x}_{\scriptstyle\textrm{tot}}]=\sum_{v\in V}x(v).$
Indeed:
$\displaystyle\mathbb{E}[\widehat{x}_{\scriptstyle\textrm{tot}}]$
$\displaystyle=\sum_{w\in V}p(w)\sum_{v\in Q(w)}\frac{x(v)}{\pi(v)}\ =$
$\displaystyle=\sum_{v\in V}\ \sum_{w\in V:\ v\in
Q(w)}\frac{x(v)}{\pi(v)}p(w)\ =$ $\displaystyle=\sum_{v\in
V}\frac{x(v)}{\pi(v)}\sum_{w\in V:\ v\in Q(w)}p(w)\ =$
$\displaystyle=\sum_{v\in V}\frac{x(v)}{\pi(v)}\pi(v)\ =$
$\displaystyle=\sum_{v\in V}x(v).$
(Note that the sums were swapped and appropriately updated after the first
step.)
$\boxempty$
### VIII-C Practical requirements
We have just shown that $\widehat{x}_{\scriptstyle\textrm{tot}}$ in Eq.(25) is
an unbiased estimator of $x_{\scriptstyle\textrm{tot}}$. This is true for _any
choice_ of $Q(w)\subseteq V$, regardless of our sampling method. By defining
$Q(w)$, we define the estimator. However, there are two requirements that we
should take into account.
First, our estimator must be _feasible_ , _i.e._ , we must be able to
calculate $\widehat{x}_{\scriptstyle\textrm{tot}}(v)$ from our sample $S(U)$.
This means that all nodes whose values are needed to calculate
$\widehat{x}_{\scriptstyle\textrm{tot}}$ must be known (sampled). One obvious
necessary condition is that $Q(U)\subset S(U)$, because $Q(U)$ is the set of
nodes whose values $x(v)$ are used in the estimator
$\widehat{x}_{\scriptstyle\textrm{tot}}$ in Eq.(25). However, usually we have
to know many nodes from beyond $Q(U)$ in order to evaluate Eq.(26). We give
some examples below.
Second, the estimator $\widehat{x}_{\scriptstyle\textrm{tot}}$ should be
characterized by a _small variance_.
### VIII-D Arbitrary-topology estimators for BFS
Let $B_{i}(u)$ be a ball of size $k$ around vertex $u\in V$, _i.e._ , the set
of all vertices within $i$ hops from $u$. For simplicity, we define our
sampling technique as a $i$-stage BFS, _i.e._ , $S(u)=B_{i}(u)$. Depending on
our choice of $Q(u)$, we may obtain various feasible arbitrary-topology
estimators:
#### VIII-D1 Trivial
The simplest choice of $Q(v)$ is
$Q(v)=\\{v\\}.$
This estimator makes use of the first sampled node only, which naturally
results in a huge variance.
#### VIII-D2 Extreme
We can extend trivial for one specific node $v^{*}$ to obtain
$Q(v)=\left\\{\begin{array}[]{cl}B_{i}(v)&\textrm{if\quad$v=v^{*}$}\\\
\\{v\\}&\textrm{otherwise.}\end{array}\right.$
#### VIII-D3 Half-radius
A more balanced approach is
$Q(v)=B_{\lfloor i/2\rfloor}(v).$
In other words, out of the collected $i$-stage BFS sample $S(v)$, we use for
estimation only the nodes collected in the first $i/2$ stages of our BFS. It
is easy to verify that the half-radius estimator is feasible.
#### VIII-D4 Half-radius extended
Finally, we can extend the half-radius estimator to potentially cover some
more nodes, as follows.
$Q(u)=B_{\lfloor k/2\rfloor}(u)\ \cup\ \\{v\in V:\ B_{i}(v)\subseteq
B_{i}(u)\\}.$
### VIII-E Evaluation
We have tried the above approaches in simulations to estimate the average node
degree $\langle k\rangle=x_{\scriptstyle\textrm{tot}}/|V|$.555For simplicity,
we considered the total number of nodes $|V|$ as known. As our error metric,
we used Root Mean Square Error (RMSE), which is appropriate in our case, as it
captures both the estimator bias and its variance. RMSE is defined as:
$RMSE\ =\
\sqrt{\mathbb{E}\left[(\widehat{x}_{\scriptstyle\textrm{tot}}/|V|-\langle
k\rangle)^{2}\right]}.$
In our simulations, we calculated the mean $\mathbb{E}$ over 1000 BFS samples
initiated at nodes chosen uniformly at random, _i.e._ , with probability
$p(v)=1/|V|$. In Table IV, we show the results for the half-radius estimator
with $i\\!\\!=\\!2$. Other values of $i$ and other estimators do not improve
the results compared to the $RG(p_{k})$-based estimator.
Although unbiased, all the proposed arbitrary-topology estimators have very
large RMSE compared to the $RG(p_{k})$-based estimators. There are two main
reasons for that. First, in order to guarantee feasibility, we usually have
$|Q(v)|\ll|S(v)|$, which results in a “waste” of values $x(v)$ of most of the
sampled nodes. Second, the sizes $|Q(v)|$ may significantly differ for
different nodes $v$, which translates to differences in particular estimates
$\widehat{x}_{\scriptstyle\textrm{tot}}(v)$.
To summarize, the arbitrary-topology estimator is unbiased but has a huge
variance, which makes it much worse than the potentially slightly biased (for
real-life topologies) but much more concentrated $RG(p_{k})$-based estimator.
It is an instance of the well-known “accuracy vs precision” trade-off. Indeed,
in the statistics terminology, we could say that the arbitrary-topology
estimator is “accurate but very imprecise”, whereas the $RG(p_{k})$-based
estimator is “slightly inaccurate but precise”.
Dataset | $\langle p_{k}\rangle$ | correction method | $\langle\widehat{p}_{k}\rangle$ | RMSE
---|---|---|---|---
ca-CondMat | 8.6 | arbitrary-topology | 8.5 | 10.3
$RG(p_{k})$-based | 7.6 | 3.3
email-EuAll | 3.0 | arbitrary-topology | 3.1 | 17.3
$RG(p_{k})$-based | 1.7 | 1.5
Facebook-New-Orleans | 25.8 | arbitrary-topology | 25.6 | 33.5
$RG(p_{k})$-based | 21.5 | 11.8
wiki-Talk | 3.9 | arbitrary-topology | 3.8 | 27.9
$RG(p_{k})$-based | 2.4 | 1.9
p2p-Gnutella31 | 4.7 | arbitrary-topology | 4.8 | 4.6
$RG(p_{k})$-based | 3.7 | 1.6
soc-Epinions1 | 10.7 | arbitrary-topology | 10.3 | 29.3
$RG(p_{k})$-based | 9.7 | 6.6
soc-Slashdot0811 | 14.1 | arbitrary-topology | 14.5 | 40.5
$RG(p_{k})$-based | 17.3 | 6.8
as-caida20071105 | 4.0 | arbitrary-topology | 3.9 | 4.7
$RG(p_{k})$-based | 2.9 | 1.5
web-Google | 10.0 | arbitrary-topology | 10.6 | 55.2
$RG(p_{k})$-based | 6.1 | 5.1
TABLE IV: Comparison of the arbitrary-topology estimator derived in this
section with the $RG(p_{k})$-based estimator proposed in the paper. We used
the real-life Internet topologies described in Table II. Here, we use the
half-radius arbitrary-topology estimator with depth $i=2$. The results are
averaged over 1000 seed nodes chosen uniformly at random from the graph.
## IX Practical recommendations
In order to sample _node properties_ , we recommend using RW. RW is simple,
unbiased for arbitrary topologies (assuming that we use correction procedures
summarized in Section VI-A1), and practically unaffected by the starting
point. RW is also typically more efficient than MHRW [10, 2, 3].
In contrast, RW and MHRW are not useful when sampling _non-local graph
properties_ , such as the graph diameter or the average shortest path length.
In this case, BFS seems very attractive, because it produces a full view of a
particular region in the graph, which is usually a plausible graph for which
the non-local properties can be easily calculated. However, all such results
should be interpreted very carefully, as they may be also strongly affected by
the bias of BFS. For example, the graph diameter drops significantly with
growing average node degree of a network. Whenever possible, it is a good
practice to restrict BFS to some well defined community in the sampled graph.
If the community is small enough, we may be able to exhaust it (at least its
largest connected component), which automatically makes our BFS sample
representative of this community. For example, [20, 33] collected full samples
of several Facebook regional networks, and [65, 63] completely covered the WWW
graph restricted to one or few domains. When such communities are not
available (_e.g._ , regional networks are not accessible anymore in Facebook),
we are left with a regular unconstrained BFS sample. In that case, we
recommend applying the $RG(p_{k})$-based correction procedure presented in
this paper to quantify the node degree bias, which may help us evaluate the
bias introduced in the topological metrics.
## X Conclusion
To the best of our knowledge, this is the first work to quantify the node-
degree bias of BFS. In particular, we calculated the node degree distribution
$q_{k}$ expected to be observed by BFS as a function of the fraction $f$ of
covered nodes, in a random graph $RG(p_{k})$ with a given degree distribution
$p_{k}$. We found that for a small sample size, $f\\!\rightarrow\\!0$, BFS has
the same bias as the classic Random Walk, and with increasing $f$, the bias
monotonically decreases.
Based on our theoretical analysis, we proposed a practical $RG(p_{k})$-based
procedure to correct for this bias when calculating any node statistics. Our
technique performed very well on a broad range of Internet topologies. Its
ready-to-use implementation can be downloaded from [24].
In this paper, we used our $RG(p_{k})$-based correction procedure to estimate
local graph properties, such as node statistics. An interesting direction for
future is to exploit the node degree-biases calculated here to develop
estimators of non-local graph properties, such as graph diameter.
## Acknowledgments
We would like to thank Bruno Ribeiro for useful discussions and the initial
idea of the unbiased estimator in Section VIII; Alan Mislove for custom-
prepared Orkut BFS sample; and Minas Gjoka for collecting the Facebook BFS
sample.
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|
arxiv-papers
| 2011-02-22T20:19:40 |
2024-09-04T02:49:17.178076
|
{
"license": "Public Domain",
"authors": "Maciej Kurant, Athina Markopoulou, Patrick Thiran",
"submitter": "Maciej Kurant",
"url": "https://arxiv.org/abs/1102.4599"
}
|
1102.4777
|
CERN–2011–00131 January 2011Sir John Adams: his legacy to the world of
particle acceleratorsJohn Adams Memorial Lecture 2009E. J. N. WilsonJohn Adams
Institute, University of Oxford, UKGENEVA2011
ISBN | 978–92–9083–356-7
---|---
ISSN | 0007–8328
Copyright © CERN, 2011
Creative Commons Attribution 3.0
Knowledge transfer is an integral part of CERN’s mission.
CERN publishes this report Open Access under the Creative Commons Attribution
3.0 license (http://creativecommons.org/licenses/by/3.0/) in order to permit
its wide dissemination and use.
This monograph should be cited as:
E. J. N. Wilson, Sir John Adams: his legacy to the world of particle
accelerators,
John Adams Memorial Lecture, 2009, CERN-2011-001 (CERN, Geneva, 2011).
Abstract
John Adams acquired an unrivalled reputation for his leading part in designing
and constructing the Proton Synchrotron (PS) in CERN’s early days. In 1968,
and after several years heading a fusion laboratory in the UK, he came back to
Geneva to pilot the Super Proton Synchrotron (SPS) project to approval and
then to direct its construction. By the time of his early death in 1984 he had
built the two flagship proton accelerators at CERN and, during the second of
his terms as Director-General, he laid the groundwork for the
proton–antiproton collider which led to the discovery of the intermediate
vector boson. How did someone without any formal academic qualification
achieve this? What was the magic behind his leadership? The speaker, who
worked many years alongside him, will discuss these questions and speculate on
how Sir John Adams might have viewed today’s CERN.
###### Contents
1. 0.1 Introduction
2. 0.2 How John Adams viewed building accelerators
3. 0.3 His first success
4. 0.4 Telecommunications Research Establishment Malvern—his university
5. 0.5 Harwell
6. 0.6 CERN
1. 0.6.1 The PS Parameter Committee
7. 0.7 Plasma research—the move to Culham
8. 0.8 The 300 GeV machine and the ISR
1. 0.8.1 Redesigning the 300 GeV machine
2. 0.8.2 Difficulties with the 300 GeV Project
3. 0.8.3 Designing the new machine
4. 0.8.4 Design improvements
5. 0.8.5 Bringing the 300 back to CERN—‘Project B’
6. 0.8.6 Highlights of SPS construction
7. 0.8.7 Magnet problems
8. 0.8.8 Commissioning
9. 0.8.9 He becomes Director-General a second time
10. 0.8.10 R. R. Wilson
9. 0.9 Since he left us …
10. 0.10 Conclusion
## 0.1 Introduction
Twenty-five years ago, in the year of John Adams’s death, Edoardo Amaldi gave
the first talk in this series of John Adams Memorial Lectures [1]. Amaldi’s
subject was, like mine, the life of this great man. In 1959, in the very early
days of CERN, Amaldi had recruited John Adams to build the Proton Synchrotron
(PS) and had remained his friend and supporter throughout his career. His
account was from the viewpoint of a senior figure in European accelerator
science.
My own account is written from the very different viewpoint of a member of
John Adams’s team. My personal experience of the man dates from 1969 when he
returned to CERN for a second time as Project Leader Designate of the 300 GeV
Machine (or Super Proton Synchrotron (SPS) as it was to become). I was a
research fellow at CERN when he recruited me as his technical assistant. I was
given the job of adapting the lattice of the SPS and coordinating its design
to the point that, in 1971, CERN’s Member States were finally able to approve
the project and agree that it should be built at CERN. I then continued to
work in day-to-day contact with John Adams as his Head of Parameters during
the design of the SPS and throughout its commissioning in 1976. I was
therefore fortunate enough to see him mastermind a huge project and deal with
the many obstacles that must be overcome in such an endeavour. It is my hope
that these two accounts complement each other to give a full picture of the
ingredients of his greatness.
John Adams was at the heart of CERN’s proud boast that its accelerators are
finished on time and work reliably, and he should be remembered as an example
for all future machine builders and project directors. In the course of
writing this account, several questions occurred to me. How did someone like
John Adams without any formal academic qualification achieve this? What was
the style and method behind his leadership? How did he achieve political
success with the Member States of CERN in turning the almost hopeless quest
for approval of the SPS to CERN’s advantage? I will also attempt to compare
him with his US counterpart R. R. Wilson, and imagine what he would now have
to say about CERN’s last 25 years. I believe the answers to these questions
will go a long way to understanding his mastery of the field and I will
therefore use italics to emphasize them.
## 0.2 How John Adams viewed building accelerators
Let me return to the matter of John Adams’s style of building machines that
were reliable and which cost no more than promised. He attempted to summarize
how he achieved this in some of his final words to the CERN Council:
_…The question of how much flexibility to build into a machine is obviously a
matter of judgment, and sometimes the machine designers are better judges than
the physicists who are anxious to start their research as soon as possible.
But whatever compromise is reached about flexibility, one should certainly
avoid taking risks with the reliability of the machine because then all its
users suffer for as long as it in service and the worst thing of all is to
launch accelerator project, irrespective of whether or not one knows how to
overcome the technical problems. That is the surest way of ending up with an
expensive machine of doubtful reliability, later than was promised, and a
physicist community which is thoroughly dissatisfied. CERN, I am glad to say,
has avoided this trap and has consistently built machines which operate
reliably, are capable of extensive development, and have been constructed
within the times promised and within the estimated costs._
## 0.3 His first success
[b]fig1 is a picture of John Adams at a high point in his career. It was taken
on 25 November 1959, in the CERN Auditorium—fifty years ago (almost to the day
of this lecture) as he announced to CERN Staff that the PS had accelerated
beam to 24 GeV. The November 2009 issue of the CERN Courier [2] contains an
extract from a lively contemporary account by Hildred Blewett of the previous
night’s excitement in the Control Room.
Figure 1: John Adams announces that the PS had accelerated beam to 24 GeV
In his hand can be seen an (empty) vodka bottle, which he had received from
Yu. P. Nikitin with the message that it was to be drunk when CERN passed
Dubna’s world record energy of 10 GeV. The bottle contains a Polaroid
photograph of the 24 GeV pulse ready to be sent to the Soviet Union.
Figure 2: The label of the vodka bottle (from the John Lawson Archives [3])
The label which we see in fig2 is itself a piece of history—a testament to an
international meeting at Dubna some months earlier and one of the early cracks
in the ice of the Cold War. The names include Mullet, Pickavance, Crowley-
Milling, Snowdon, Lawson, and many others in Cyrillic script.
And this brings us to the first question: How did this young man of 33,
without university education, come to lead such a project? Certainly he was
not coached in physics, mathematics or management at a prestigious university.
He had left school in 1936 without wishing to go on to university. Rather, he
sought practical employment as a student apprentice at the Siemens
Laboratories at Woolwich. He took a Higher National Certificate (HNC) night
school diploma in electronics to become a member of the Institution of
Electrical Engineers, but at this point his formal education came to an end.
When asked this question many years later, John Adams said, _“If university
means that you learn from capable men—I had ample opportunity_.”
## 0.4 Telecommunications Research Establishment Malvern—his university
He first began to meet these capable men when he joined up for the war effort
in 1940 and was posted to Telecommunications Research Establishment (TRE)
Swanage and later to Malvern where Radio Direction Finding (RDF) or radar was
under development. The staff and advisors of TRE included John Cockcroft,
Robert Watson-Watt, Henry Tizard, Alan Blumlein, Bernard Lovell, P. I. Dee, W.
E. Burcham, and E. D. Fry. Many of the accelerator builders of the post-war
years were also there including Hine, Crowley-Milling, Shersby-Harvie,
Snowdon, Mullet, Walkinshaw, and J. D. Lawson.
John Adams was in a group responsible for transmitter–receiver cells and
diodes for 3 cm radar. His boss and mentor then was Herbert Skinner who had
worked at the Cavendish Laboratory under Rutherford. His contemporaries said
Adams had an instinctive feeling for what was needed—a comment that appears
again and again during his later career. Adams’s roommate also said he was so
good at doing sketches he could design a complete three-dimensional circuit
layout on paper. It was during this time that he met Mervyn Hine—later to be
his closest collaborator in the design of the PS, and Michael Crowley-Milling
who was then working for Metropolitan Vickers building linacs for medical
purposes. Michael was to become part of the Adams team that built the SPS and
has written a book about John Adams which I commend to you as a more complete
account than space allows me here [4].
## 0.5 Harwell
After the war, the Atomic Energy Research Establishment (AERE) Harwell
Laboratory was set up at the initiative of Sir John Cockcroft, Mark Oliphant,
and James Chadwick so that the contributions made by Britons to the nuclear
effort in the US might continue in Europe. As part of this it was decided to
build a 100-inch cyclotron (fig3). Herbert Skinner, John Adams’s boss from
TRE, was in charge of General Physics at Harwell and invited him to join the
project. He was to work under Gerry Pickavance who had been part of a team
that had already built a cyclotron at Liverpool University. At the time, Gerry
had a reputation for assuming an importance above his station. It is said his
Liverpudlian colleagues once nailed him to the floor by the sleeves of his
lab-coat to teach him a lesson in modesty. As a Liverpool man myself, I can
attest to this being quite within the bounds of possibility, though by the
time I met Gerry Pickavance as Leader of the Rutherford Laboratory, he had
obviously learned his lesson in restraint, and had become an excellent senior
manager who was later to become a staunch supporter of John through the period
leading up to the SPS.
Figure 3: The Harwell cyclotron
It was at Harwell that John cut his teeth on project management. The Harwell
cyclotron [5] was challenging—a synchrocyclotron with 110-inch poles, closely
modelled on Stan Livingston’s design for the Massachusetts Institute of
Technology. When Gerry spent months at a time visiting the US, John was left
in charge of everything except the RF systems. He found he was taking more and
more of the crucial design decisions himself as he thoroughly worked his way
through a multitude of sketches and calculations as diverse as heat transfer
and particle orbits. This was a considerable responsibility for a young man
and here perhaps is another clue to his success as he _seized this unusual
opportunity to develop his skills and experience._
Even great men need a role model. For John Adams it was Harwell’s Director,
Sir John Cockcroft, who had been awarded a Nobel Prize for his atom-splitting
at the Cavendish Laboratory in the 1930s (fig4). Sir John Cockcroft was much
revered by John who in later life displayed a portrait of him behind his desk.
Cockcroft is said to have been a modest man who managed his team with quiet
authority. His management style was to let people get on with what they were
good at, but to show an almost daily interest in their progress. He would
often appear at the beginning of a day’s work behind the shoulder of a humble
lathe operator to ask him “How is it going?” He gave his staff considerable
freedom to follow their own line, but would be quick to support them by
shouldering the responsibility, should they need to be rescued. How different
from the aggressively critical attitude taught to today’s managers who, all
too often, are ready to dismiss ‘the weakest link’ rather than correct and
reform. John’s style was very much that of Cockcroft— _a style which I commend
to those who might wish to emulate him._
Figure 4: Sir John Cockcroft
Harwell was part of John’s learning curve and it was there that he first
tasted failure when he tried to persuade Skinner to give him an extra £50,000
to enlarge the yoke of the magnet and reach a higher energy. (See fig5 from
his notebook.) He lost the battle only to see the finished cyclotron end up
with not quite enough energy to produce the new ‘mesons’ which it might have
discovered. This may have been in his mind as he later pressed for 400 GeV
rather than 300 GeV for the SPS. Perhaps here he learned another lesson— _not
to give up on something your gut feeling tells you is correct._
Figure 5: A page from John Adams’s meticulously kept Harwell notebook
Once the Harwell cyclotron was finished, he had another setback as he was
reassigned to work on a fast breeder reactor. Knowing very little nuclear
physics, he had to work night and day to catch up but found it frustratingly
difficult. He became seriously depressed and was sent away for six months by
his wife Renie to stay with an uncle who was a pig farmer. He returned in
better spirits and with the courage to discuss his future with Cockcroft.
Cockcroft, who firmly believed in matching the man to the job, was sympathetic
and as a temporary measure set him to work on a klystron together with Mervyn
Hine. Soon after, Cockcroft saw a real chance to rescue John by setting him
off on an international venture that brought him to CERN. Here he learned two
more lessons— _don’t force yourself to do things which do not match your
skills_ and _at crucial times seek help from your mentor._
## 0.6 CERN
In May 1952 The CERN Council met for the first time in Paris. CERN’s initial
idea for a Proton Synchrotron (PS) was a 10 GeV weak focusing machine—a
scaled-up version of the 3 GeV Cosmotron at Brookhaven in the US, which had
recently become the first proton synchrotron to operate. A Norwegian, Odd
Dahl, was the CERN PS Project leader together with Frank Goward who was later
to become his deputy in Geneva. Very soon after this, in August 1952, Dahl,
Goward, and Rold Wideröe visited the Cosmotron and learned of the new idea of
strong focusing from Courant, Livingston, and Snyder. They returned to
immediately change the CERN plan for a 25 GeV alternating-gradient machine.
At that time, the UK was suspicious of its continental neighbours. After all,
it had benefited from a vigorous partnership with the US on nuclear matters
during World War II and saw little advantage in joining CERN. It fell upon
Edoardo Amaldi and Cockcroft to persuade a reluctant Ben Lockspeiser, then the
UK minister in charge of the Department of Scientific and Industrial Research,
to join. They also had to persuade Lord Cherwell, Churchill’s scientific
advisor, to withdraw his objections to CERN. In this they eventually
succeeded. Amaldi described in his first John Adams Memorial Lecture how he
then wished to meet some young British physicists and engineers, whereupon
Cockcroft brought John Adams to meet him at lunch. Afterwards Amaldi had an
extended interview with the young man as they travelled by car to Harwell and
chose to recommend John (and Frank Goward) for places in the new team to build
the PS. This set the seal on the career that was to lead John to his first
triumph. Amaldi, in ref1, recalls that John was surprisingly ready to move to
Europe. He already realised the role of international science in keeping
nations from warfare and wanted to be part of it. This seemed one of John
Adams’s guiding principles destined to steer his life towards CERN and later
to world projects: ‘ _International common ventures prevent wars._ ’ Frankly,
as a child of wartime United Kingdom, I appreciate how such thoughts were far
in advance of their time.
Figure 6: Agenda of a meeting to decide PS parameters
The UK was therefore still not immediately a signatory to CERN and, not for
the last time, John found himself working on a major European project without
the support of his own government. But Frank Goward and John Adams were seen
as experts in circular machines and they met frequently in Harwell and other
laboratories to discuss the new idea with the nascent PS team.
Among such discussions there was a crucial meeting at Harwell at the end of
1952—just after Amaldi’s visit, and before Adams officially worked for CERN.
Those at the meeting included J. D. Lawson, Kjell Johnsen, Mervyn Hine and
John Adams. It was not minuted, but in fig6 (from ref3) we see the agenda for
a subsequent meeting which gives a clue as to the contributions of the various
participants.
It was John’s job to help resolve the many doubts there still were about this
decision to change to alternating-gradient focusing. John Lawson had warned of
the dangers of non-linear resonances and Kjell Johnsen had to be persuaded
that transition would not be a problem. John and Mervyn Hine studied the non-
linear resonances driven by magnet imperfections using ACE, one of the first
computers available in the UK. It seemed that because of the high field
gradient (n-value) of the first design, magnet construction tolerances would
need to be unrealistically tight to avoid these resonances. Hine writes: “I
remember at the end of the Harwell meeting John summarized and took over. He
stepped into the authority position and wrote a summary on the blackboard in
his wonderfully clear left-hand writing.” In retrospect this seems to be a
crucial turning point at which _he seized the opportunity to assume authority
over the new project’s design_.
At subsequent meetings John was able to report that a set of compromise
parameters had been found. The n-value was to be reduced by a factor 4 and the
magnet aperture would have to be three times larger—but still tiny compared
with the Cosmotron. This was typical of the kind of approach that John brought
to the design of accelerators. Each effect had to be analysed and calculated
and its effect on the chances of a successful outcome had to be balanced
against the need to be economical in construction. His notebooks contained
logical lists of arguments for and against each compromise. He was to extend
this careful elimination of all risk to many other parts of the project and he
recruited incomparable teams of engineers to ensure the highest quality of
design and construction. Sometimes the workshops resembled a Swiss watch
factory, but it paid off and set the CERN standard for completing on time and
within budget. The secret was in the many long hours of discussion with others
and the analytical tool provided by his notebook.
### 0.6.1 The PS Parameter Committee
Figure 7: PS Group Leaders— From left to right we see Ed. Regenstreif, Pierre
Germain, Kjell Johnsen, Arnold Schoch, Mervyn Hine, John Adams, Franco
Bonaudi, Fritz Grutter, Kees Zilverschoon, and Colin Ramm
Not long after, in October 1953, the PS team gathered in premises lent to them
by the University of Geneva whilst awaiting the construction of the first
buildings on the new CERN site in Meyrin. Goward was the Project leader and he
assembled a formidable team of experts to design and construct the machine. In
the photograph (fig7) we see almost all the PS Group Leaders, each responsible
for an aspect of the machine.
John Adams and Mervyn Hine were known at this time as the ‘terrible twins’
using their experience with earlier projects to enliven the proceedings of the
Parameter Committee which met once a week to put flesh on the bones of the
design. (Later I will say more about the Parameter Committee and its role in
Adams’s method of project management.)
To Giorgio Brianti, arriving in November 1953, it was clearly John who,
chairing the Parameter Committee, masterminded the design and construction.
Brianti recalls that then John was already ‘the Boss’. Goward had fallen ill
and died in early 1954 leaving John to take over full responsibility for the
project, eventually steering to the moment of his triumph in 1959.
## 0.7 Plasma research—the move to Culham
Near the end of his first period at CERN he began to take an interest in
plasma research, attending a number of meetings with a view to setting up
another international organization. He was interested in plasma accelerators
and some experimental work started at CERN.
In 1958 a lot of plasma work in the UK was declassified and shared with the
Russians during a momentary thaw in the Cold War. It was in the days of ZETA,
the Harwell machine which prematurely announced the dawn of energy from
thermonuclear fusion. These hopes proved false, but in spite of this, the UK
was keen to set up a new laboratory at Culham to develop the field. Even
before the PS was finished, they tried to persuade John to return as Director
of this new laboratory. He was anxious that his children attend schools in the
UK academic system to prepare them for university and he agreed to head the
new laboratory once the PS was finished. However, following the death of Jan
Bakker, then CERN’s Director-General, in a plane accident in April 1960, John
Adams was appointed Director-General of CERN. His return to the UK had to be
delayed until he had not only finished the work of getting the PS running
properly but had seen the physics programme take off.
Figure 8: ZETA
In fig8 we see ZETA, and in fig9 the Culham laboratory near Abingdon. He was
to spend the next five years in the UK, eventually being asked to advise Frank
Cousins, a minister in the government of Harold Wilson. I remember him later
being very critical of the quality of the administration over which Frank
Cousins and Anthony Wedgwood-Ben presided. John’s advice was often not taken
and influenced government thinking only many years later. Frustrating as this
experience was, it left John with a clear idea of how politicians worked and
how they might or might not be influenced—an experience that was to be
invaluable in persuading Member States to support moving the SPS to CERN.
Figure 9: The Culham Laboratory
## 0.8 The 300 GeV machine and the ISR
In 1960 when he was still Director-General and just before he left for Culham,
John recommended to Council, “that CERN should plan to build a machine to
replace the Proton Synchrotron. It should be ready in 1970 therefore plans
should be prepared for consideration in 1962 or 1963.” A study group was set
up under Kjell Johnsen to look into the feasibility of a collider
(Interesecting Storage Rings, the ISR) and a 150–300 GeV proton synchrotron.
Council approved the ISR, appointing Johnsen to head its construction (see
fig10). At the same time a detailed design study of a new proton synchrotron
was published in a substantial report ‘A Design Study for a 300 GeV Proton
Synchrotron’ [6], commonly referred to as the ‘Grey Book’.
This machine proved to be a scaled-up version of the PS and ISR, incorporating
the lessons learned from their construction and applied with all the
conservatism that experienced engineers tend to bring to new projects. It
would have taken eight years to construct and cost about 1800 MCHF.
Figure 10: The ISR
### 0.8.1 Redesigning the 300 GeV machine
It had not been the aim of Kjell Johnsen’s team, who had written the Grey
Book, to be economical in either time or money. In the USA a similar proposal
was made for the 200 BeV accelerator, authored by many who had contributed to
CERN’s Grey Book and incorporating many of the same conventional features.
After its publication, the construction of the American machine was approved
at the US Fermi National Laboratory (Fermilab) near Chicago and work started
under the leadership of R. R. Wilson. He tore up the 200 BeV design and
proposed a much leaner design which could be constructed in only four years
and which would operate at 400 GeV—twice the energy originally proposed. The
most striking innovation was to change the lattice from combined-function
magnets to one in which the functions of bending and focusing were performed
by separate and quite different magnets.
The change from combined-function to separated-function lattice was later to
be so fundamental to securing approval for the SPS that it is worth a short
explanation. We recall that focusing in synchrotrons is achieved by a field
gradient across the mid plane of the magnets. This, together with the
centrifugal force on the particles, forces errant particles which swing away
either upwards, downwards, or on either side of the ideal central orbit around
the machine to be pushed back towards the central orbit. In early synchrotrons
and cyclotrons this gradient was uniform around the machine. In the AGS, PS,
and ISR the sign of the gradient alternated from magnet to magnet to produce a
much stronger focusing effect called alternating-gradient focusing. The
magnets had tapered gaps between the poles so that they both focused and bent
the particles at the same time. The direction of the taper alternated from
magnet to magnet. Although the gradient alternated in these machines, one kind
of magnet combined the function of bending and focusing which had a certain
simplicity. These were the magnets John and Kjell knew and loved from the PS
and ISR.
However, in such magnets the central field, which determines the central orbit
and the radius of the machine, cannot be as high as it is at the edge of the
poles where saturation limits the field to 1.8 T. Allowing for the gradient,
the field on the centre line of a combined-function magnet can only be about
1.3 T. In the separated-function idea there are two kinds of magnet: ‘pure
dipole’ bending magnets with uniform field of about 1.8 T and special ‘pure
gradient’ quadrupole magnets to provide the focusing. Bob Wilson had shown
that, by using such a separated-function design, there could be a considerable
saving in total bending magnet length—more than enough room to place special
dedicated quadrupole magnets to provide the focusing.111The first proposal of
separate-function magnets (i.e., the separation of dipoles and quadrupoles)
for an accelerator lattice was made by T. Kitagaki in 1953 [7]. Among other
implications, this separation allowed for smaller magnets and for the
introduction of long straight sections without dipole fields.
This change from combined to separated function happened in1967 just when I
had come to CERN on a fellowship and was encouraged by Roy Billinge (then
about to leave to build the Booster at Fermilab) to join the Accelerator
Research Department (AR) and work on improving the Grey Book. Roy and I (we
were both just 30) were both ‘rebels with a cause’ convinced that, by adopting
some of the radical simplifications that Bob Wilson was adopting for the
Fermilab machine, CERN’s 300 GeV machine would become faster to build and
cheaper. Roy went off to Fermilab while I taught myself the rudimentary skills
of lattice design and set about designing a separated-function version of the
Grey Book.
When I applied this to the 300 GeV machine, the energy jumped from 300 GeV to
400 GeV but when I showed this proudly to Kjell Johnsen and Kees Zilverschoon
(caretakers of the 300 GeV project, but still busy finishing the ISR), I was
told not to rock the boat. To be fair, the main concern at that time was to
decide it was to be built. Council delegates spent much of the time viewing
and reviewing the 22 and the finally 5 possible sites scattered across Europe
in countries who all hoped to benefit from the local trade and prestige. It
took the arrival of John Adams to give the revised design the attention it
deserved.
In 1969 John Adams returned to CERN, appointed by the CERN Council to lead the
300 GeV/SPS Project. Finding that I was the only full-time person working on
the lattice for the machine, he invited me to several one-on-one discussions
about the design of the new machine and listened with enthusiasm to my
separated-function version of the Grey Book. As a very junior visiting fellow
to CERN at the time, I was both surprised and flattered by the attention of
one of the ‘great men of science’. I expected it would be too revolutionary
and might seem to him to prejudice the operational reliability of the machine,
but it clearly fitted in with what turned out later to be his grand scheme for
CERN. We worked hard on redesigning the machine, and our proposal is to be
found in ref7.
### 0.8.2 Difficulties with the 300 GeV Project
Just before John arrived in CERN for the second time, the UK had dealt a blow
below the belt to the ‘300’. The Labour government, who were strapped for
cash, were not very interested in pure science and saw no financial advantage
in the project even if their site were to be chosen. In June 1968, Sir Brian
Flowers announced to the CERN Council that they should not count the UK among
the participating countries. This was just after John had given up his
influential responsibilities in London to move to CERN and become Project
Director Designate. He found himself once again playing a crucial role in
starting a project without the support of his own country.
Apart from the withdrawal of the UK from the new project and the difficulties
in settling on a site for the ‘300 Machine’, the new Project Director
Designate had to pay more than lip service to Fermilab ideas. Bob Wilson was
by then boasting a five-year construction time for his machine, an almost
unbelievable cost profile, and 400 GeV to boot. Adams was under considerable
pressure from certain German physicists and Citron (of the PS days) to move
away from the “lavish practices” of the PS and the ISR and take heed of the
wind of change blowing strong from Fermilab in Illinois.
He had to cut the cost of the 300 GeV proposal without sacrificing
reliability, resolve the question of where it would be built, and defuse the
feeling that Member State opinions were not being taken into account. I have
no doubt that these aims were listed on the first page of his notebook soon
after he returned and it is clear to me in retrospect that he lost no time in
devising a strategy to find a way through the minefield. It can hardly have
taken him too long, because there seemed to be no preliminary exploration of
blind alleys on the way—and all of this from a man who _seemed not to have
made up his mind about anything until he had heard all sides of the argument_
—masterly leadership by any standards. After the event John Adams explained
the difficulties he faced thus:
> “Looking back, I think one can discern a number of reasons why our Member
> States hesitated to reach a decision on the 300 GeV Programme in the form it
> was presented at that time.
>
> In the first place the economic situation in 1969 for science in general and
> nuclear physics in particular was very different from the ebullient years
> around 1964 and 1965 when the 300 GeV Programme was first put forward. It
> was evident that several Member States of CERN and possibly all of them
> found the cost of the Programme too high compared with their other
> investments in science and with the growth rates in their total science
> investments, which had dropped from figures around 15% per annum in 1965 to
> a few per cent per annum in 1969.
>
> In the second place, the idea of constructing a second European laboratory
> for nuclear physics remote from the existing one, which had seemed
> attractive in 1965, looked inappropriate in 1969, particularly since it
> implied running down the existing CERN laboratory when the new one got under
> way.
>
> In the third place, so many delays had occurred in the 300 GeV Programme and
> the American machine was coming along so fast that an eight-year Programme
> to reach experimental exploitation seemed too long.
>
> Fourthly, it turned out that choosing one site amongst five technically
> possible sites presented non-trivial political problems for the Member
> States of CERN.”
This quotation is typical of the point-by-point analysis that he used to
summarize his view on any argument. When he sat down in the afternoon to
update his notebook with his resolution of the arguments presented to him by
others, he would light his pipe and often compose just such a summary. He
found this kind of logical analysis led him to the most reasonable and
sensible decisions and, when presented to others, was persuasive and almost
irresistible in its clarity of thought and logic.
This is frankly not the kind of rhetoric that a politician might use to sway a
crowd but it is calm, reflective, designed to raise the minimum of eyebrows,
and, above all, be persuasive in its relentless logic. Brian Southworth, then
Editor of the CERN Courier once said: “John has the astonishing gift of
delivering absolute truth in the manner of Farmer Giles leaning over his gate
to comment on the weather.”
The ideas seem just to have occurred to him after the project was approved, as
a clever way of summarizing, but he surely arrived at these conclusions almost
immediately he arrived at CERN as Project Director Designate since it so well
summarizes what everyone was experiencing at that time—problems which only he
knew how to resolve. I believe he decided then to adopt it as a to-do list
and, playing his cards close to his chest, tackled each item in turn.
Figure 11: “Farmer Giles” (as sketched by John Adams)
### 0.8.3 Designing the new machine
His first step towards securing the CERN Council’s approval for the SPS was to
set up a Machine Committee to involve as many of the senior accelerator
experts from Member States as he could. The choice of this committee was
masterly—that of a benign Machiavelli. He needed the help of a number of his
old PS group leaders to ensure that high standards of engineering were not
sacrificed for the most important components, and to ensure the maximum
probability of success. Magnet, radio frequency, survey, and extraction were
looked after in this way. There were other major systems, among them the power
supply and the control system where he found those in Member States who felt
they could make an innovative contribution. Crowley-Milling’s control system
based on mini computers from Norway was doubly salutary, as were John Fox’s
power supplies based not on rotating machinery but saturable reactors. Others
were recruited to the committee to reassure the sceptics in Germany and the UK
that cost and manpower estimation was done in the way they would like to see
it. They were encouraged to hang the redesign of the machine on this new
separated-function lattice that I had designed.
John had headed the Parameter Committee in the days of the PS and he clearly
expected the Machine Committee to operate in the same fashion to ensure the
consistency of the design and the success of the project. Each of the meetings
followed an agenda which was principally a series of reports by those
responsible for the major systems of the machine. Each system was worked out
in greater engineering detail following suggestions at the previous meeting.
At the heart of the business was keeping a list of all the relevant design
data from top energy, through number of bending and focusing magnets, their
length and peak field, the injection and extraction systems, together with the
frequency and voltage applied to the accelerating cavities, and even the
diameter of the tunnel and the load on the cooling system. Each time anything
was changed, its impact on cost, performance, reliability, and of course
implications for other systems would be discussed with all the hardware
specialists present. Any changes had to be incorporated in a master list of
parameters and in the lattice design. For the design and later construction of
the SPS, I was lucky enough to be in charge of both parameters and lattice. Of
course it was John, always at the head of the table, who presided. I sat at
the other end keeping the minutes. He encouraged me to ask questions which
would provoke discussion and reveal weaknesses in the design which needed to
be debated and resolved. This method of managing a project had the great
advantage that there was only one meeting at which technical matters of
accelerator design and engineering would be decided in the presence of all
component group leaders who might be affected.
I remember the lively, heated discussions between new members and some of the
members from his past PS team who had moved on meanwhile to build the ISR.
They were at pains to squash the ideas of Bob Wilson in the United States,
often expressed by this younger “upstart Wilson” (perhaps they at first
believed I was a relative) in their midst. Meetings were not without their
explosive exchanges—not surprising considering my own brash inexperience. They
also hoped to be asked to build these components and did not want to make it
hard for themselves.
But having a variety of opinions put forward around the table suited John’s
style of facilitating a meeting. He was able to judiciously move the project
from the Grey Book towards a leaner design without apparently taking sides.
John’s moderate and reasonable interventions were usually in the form of a
simple question. “Wouldn’t that mean that…”, or as he turned to someone not
already part of the combat, “How would that affect the magnet/power
supply/schedule?—How would these new magnets look?—What tolerances would be
necessary in their construction?—Would they reduce cost?—How would they
compare with the PS and ISR? and How might one inject and extract the beam?”
In this way he would orchestrate the discussion by asking for opinions until
he heard one which matched his new way of thinking. Then he would summarize
the ‘consensus’ he had sculpted for us and define what was to be studied next.
Of course there were, meanwhile, many pipes of tobacco to be prepared after
meticulous cleaning of the instrument to provide a pretext for reflexion.
When all of this had been debated, I would be expected to ask myself in the
minutes of each meeting to accommodate new aspects of the design in an ever
increasing series of lattices, each with new sets of parameters.
Later, when we came to construct the SPS, the Machine Committee became the
Design Committee and the debates about magnets, cavities, injection,
extraction, power supplies, and civil engineering were again heated. The more
controversial decisions were often concerned with the lattice (myself) and the
magnets (represented by Roy Billinge, recently returned from the US). Both of
us had a preference for the new ways of building synchrotrons pioneered at
Fermilab which were often at odds with the ideas of the more experienced
members of the team. At the time the discussions in the SPS Design Committee
were taking place, the news from Fermilab was not good and it became clear
that their magnets had not been made to the standards of electrical integrity
established at CERN. But John was not to be put off from taking what was best
from Fermilab and imposing CERN standards on its construction.
In managing the construction, John Adams followed closely the precepts of his
mentor Sir John Cockcroft. He gave his group leaders, the members of his
Monday Morning Design Committee, considerable latitude to manage their own
groups. His interest was always on achieving performance goals on time and
without over expenditure. Subsequently I have heard it said that his budget
was generous compared with later machines. All I can say is that he made
strenuous efforts to build the SPS for much less than the unit costs achieved
in the PS and ISR days. True there was, wisely, a contingency in the funding,
but this was not needed for the SPS and at the end of the construction was
reallocated to provide a new North Experimental Hall.
_To have such a weekly meeting with the heads of your hardware groups and have
them inform everyone on progress in all aspects of the machine seems so
fundamental to John Adams’s style that would-be project leaders should depart
from this practice at their peril._
As I prepared this talk, I struggled to describe the particular method that
John Adams used to run a meeting. He hardly said anything, but would steer the
opinion of the members in the direction he wanted simply by asking questions.
An Oxford philosopher friend tells me this is exactly the method used by
Socrates and Plato in the School of Athens (See fig12). _Maieutics_ (its name
in Greek means helping give birth—in this case to ideas) is a disciplined
questioning that can be used to pursue thought in many directions and for many
purposes, including: to explore complex ideas, to get to the truth of things,
to open up issues and problems, to uncover assumptions, to analyse concepts,
to distinguish what we know from what we don’t know, and to follow out logical
implications of thought. I suppose our budding project manager should read a
bit of Plato now and again—though I have no evidence that John Adams did—he
was probably hard-wired to act in this way.
Figure 12: Raphael’s fresco ‘The School of Athens’
### 0.8.4 Design improvements
But our narrative has run on and we must now return to the days when his plan
to secure the SPS for CERN was taking shape. He expected the Machine Committee
to think of improvements and to incorporate the new ideas of Fermilab.
The first system to be scrutinized was the lattice—the pattern of magnets
around the ring. Whether this is combined- or separated-function, it always
has to be consistent with the parameters of the hardware. If it is decided to
add more RF cavities to accelerate faster or to increase beam capture
efficiency, the lattice has to be adjusted to make room for it. The lattice
determines the dynamics of particles within the beam pipe, if it has many
cells the focusing will be stronger, the beam envelope smaller, and both
magnet dimensions and even that of the tunnel can be made smaller. Of course,
even if the logic of the mathematics tells you that the tunnel need only be
150 cm in diameter, you can be sure that someone in the committee will remind
you that no one would be able to walk there, let alone drive a lorry full of
rock through it. In fig13 we see one cell of the lattice (out of 100 or so
around the circumference).
Figure 13: One cell of a separated-function lattice (showing the missing
magnet option)
[b]fig13 also shows another new idea: missing magnets. If only half of the
bending magnets are built and installed in the first stage but more money
becomes available, you add the second half to double the energy (from 200 GeV
to 400 GeV). I’m not sure where this idea came from—it was perhaps prompted by
Bob Wilson’s ‘energy saver’ which was a ‘missing power’ machine—but it later
proved very useful in countering the Member States when they complained they
were in financial straights. In fact it was only when the final prices came in
for the first set of magnets that we knew we could move directly to exercise
an option to order the rest.
In the days before computer controls, synchrotrons were designed with magnet
gaps between the poles large enough to accommodate not only the beam but a
generous safety margin to accommodate all the orbit distortions due to the
tiny errors in magnet construction and alignment with 98% probability. We
invented a strategy based on how orbit correction had been applied to the PS
to liberate aperture by correcting orbits. By the time the SPS was discussed,
the PS had successfully corrected a large fraction of this orbit distortion,
liberating more aperture for the beam. Why not therefore rely on using the
same kind of correction to reduce the SPS aperture (see fig14)? I’m not sure
if it was my idea but it was one that I championed. Perhaps I did not realise
it at the time but this was in danger of pulling the design in the direction
of making it less likely to work first time—one of John’s major concerns.
However, it brought about considerable cost savings.
Figure 14: Correcting orbit distortion liberates aperture
Magnet design is a subject dear to the heart of all accelerator builders and
each (including John) had their idea of how best to do this. Earlier I
explained how Bob Wilson had replaced 1.3 T combined-function magnets with 1.8
T pure dipoles. But combined-function were the magnets John and Kjell knew and
loved from the PS and ISR and had spent many years perfecting. Moreover, some
were still sceptical of Bob Wilson (who the unkind said ran a ranch of cowboys
in the States).
We spent many meetings (then and later when the machine was approved)
discussing the virtues and vices of the new magnet designs. Many in the
Machine Committee remembered their experiences with similar and dissimilar
magnets that they themselves had built or seen built. It was perhaps John’s
biggest challenge to resolve this issue and in the end it was settled by
designing the best lattice for 300 GeV using combined- and then separated-
function principles and looking at the cost implications using a computer
program supplied by the laboratory that was one of our most vehement
critics—Karlsruhe. John rightly insisted that everything had to be included in
the program. If the field in a magnet was lowered, the ring became longer and
more RF would have to be added to accelerate in the time defined by the
parameter list. The tunnel would be longer but stored energy which had to be
shipped in and out from the electricity grid would be reduced—and there were
many more such considerations. The energy dissipated would also change,
causing more or less cooling capacity to be installed. When all this was
costed and optimized we clearly saw that a separated-function ring would cost
no more, but would be more compact. Little did we know at the time that this
matched John’s master plan to fit the machine back on the molasse plateau at
CERN, and had the added advantage, vis-à-vis his critics, that Bob Wilson’s
innovations had not been ignored but exploited.
When all this was over and the Design Report for a 400 GeV machine written
[9], it turned out that the Machine Committee had done its job well. The cost
savings were important because of the criticisms of many of the Member States
concerning the generous and expensive safety margins that the Praetorian Guard
of old PS designers had sustained in the ‘300 GeV Proposal’. Previous visitors
from German and UK laboratories being shown around the ISR had marvelled
openly at the vast space around the machine—the air conditioning—gold-plated
connections (so it was said) and the absence of any attempt to learn from
earlier experience. The new design at least seemed to have answered their
technical objections.
### 0.8.5 Bringing the 300 back to CERN—‘Project B’
Member States had still to choose somewhere to put it and Member States were
determined to build the next machine anywhere else, but not at CERN! This was
in part fed by the feeling that many physicists had not succeeded in getting
their experiments approved at CERN while other ‘residents’ had been preferred.
You perhaps saw in John Adams’s _a posteriori_ analysis of the situation, how,
in spite of these objections, it was his aim to bring the machine back to
CERN. Studies of the separated-function lattice showed that this might now be
possible (at least for 300 GeV). The deciding argument was to be that if the
machine were built at CERN it would not be necessary to set up a whole new
laboratory and build a new linac and an injector synchrotron. The 25 GeV PS
was ready and waiting. This idea came to be known as ‘Project B’.
For several months in early 1970, Project B had to be kept secret while he
politically manoeuvred the Member States to accept the idea. They must be
attracted by the cost saving. Every week he set off each day to a new capital,
appropriately dressed and coiffed to impress the local audience—with the aim
of gradually coaxing them into this new way of thinking.
At first only John and Pepi Dokheer (his secretary) knew about Project B.
However, to check his ideas he had to enlist my help to calculate the lattice.
Unfortunately for both me and Project B, I had just broken my leg skiing and
lay for six weeks with a weight strapped to my foot in the Cantonal Hospital
in Geneva. Computing was out of the question. I was surprised one afternoon to
have a distinguished visitor at my bedside when John arrived complete with
secretary and chauffeur. He politely enquired about when I might return to my
computer terminal and said that he would have something very important for me
to do when I did.
Sure enough, once I was able to do the calculations, Project B seemed
eminently possible on the CERN site but he still wondered if the molasse
(sandstone) was extensive enough to contain the whole tunnelling operation and
then one day he said, “I suppose I have to let Jean Gervaise into the secret
so that we may look at the borings in his filing cabinet.” (Jean Gervaise was
in charge of surveying the site.)
It was, of course, exciting to work on such a secret project, but fending off
helpful enquires was not easy. Giuseppe Coconni asked me one day (and I think
it was his own idea), “Has anyone thought of putting the 300 at CERN?” I had
to pretend that no-one had considered it, but one might have a look.
Finally, it was time to spill the beans to the Scientific Policy Committee and
then to Council. John cleverly asked Bernard Gregory (then Director-General)
to make the first presentation while he, John, was in the US, safe from the
storm he expected to break, and ready to return and dampen the flames. As
expected, there was quite a lot of resistance from the physics community who
had been hoping for 400 GeV and, with good luck, closer to their home. There
were also many who probably felt somewhat cheated to hear what had been going
on without their knowledge, and it is debatable whether the secrecy was not
counterproductive.
All this came to a head a few weeks later when the European Committee for
Future Accelerators were asked to approve the new ‘Project B’. They met on a
Saturday, spending the morning complaining that the energy was too low, and
everything was going rather badly by the time they adjourned for lunch. After
lunch John took me on one side. He had skipped the lunch, returning to his
office to ponder over the cardboard model on his filing cabinet, which showed
the contours of the rock beneath CERN. He said, “I think we can just find room
for an 1100 m radius ring—will this be big enough for 400 GeV?” I confirmed
that it would, and he offered it to the afternoon session (with the proviso,
to satisfy his principle of caution, that there were still some crucial
borings to be done which might yet bring a nasty surprise). It was enough to
turn the tide in his favour and save the day for Project B.
There were still many Member States to be convinced to join. This took until
the Council meeting of December 1970 and even that had to be adjourned and
reconvened on 19th February 1971 before the last couple of Member States could
be persuaded. That afternoon, after a particularly good Council lunch, John
lost no time in returning to his office to start the business of recruiting
the new staff. There were 600 farmers who owned the land on which the new ring
was to be built. His first appointment that afternoon was with André Klein, a
high official from the Prefecture of the region whom he persuaded to join the
team to deal with any dissent from the landowners.
### 0.8.6 Highlights of SPS construction
The offices of the new Laboratory II were in a barrack as far from the centre
of CERN as possible, and later were moved over into France near Prévessin. It
was clear he wanted to put his imprint on a new style. Again he had only one
weekly technical meeting, like the Parameter Meetings he had chaired for the
PS. He chaired this Design Committee every Monday morning until the machine
was finished. The team he assembled over the few weeks following SPS approval
was a healthy mixture of those who had helped him with the PS and who had gone
on to build the ISR, and new blood from other Member State laboratories.
Figure 15: The 300 GeV Design Committee
[b]fig15 is a photograph taken on the occasion of the first meeting of the 300
GeV Design Committee from my viewpoint opposite John. On his left is his
second-in-command Hans-Otto Wuester, a charismatic but explosive German from
DESY Hamburg who had, as he reminded anyone who was slow to respond to his
encouragement, “one shoe that is sharpened to be used where it hurts.” He was
the foil to John’s gentlemanly manner and was often sent over by John to
Laboratoy I to “sort them out”. Usually the threat was enough!
We also see, going round clockwise from the left, Hans Horisberger
(engineering), Clemens Zettler (radio frequency), Roy Billinge (magnets),
Norman Blackburne (personnel), Bas De Raad (extraction), Klaus Goebel (health
and safety), and Simon van der Meer (power supplies). Others who came later
included Boris Milman (finance and planning), Giorgio Brianti (experimental
areas), Michael Crowley-Milling (control system), and Robert Lévy-Mandel
(civil engineering). Wuester, Billinge, Milman, Lévy-Mandel and Crowley-
Milling came from outside. Others: Zettler, Blackburne, and Goebel, were
second-in-command to CERN group leaders who presumably chose to stay where
they were.
Figure 16: The SPS ring tunnel is completed
John was particularly interested in keeping an eye on civil engineering. On
Saturdays he would tour the site with Robert Lévy-Mandel, noting where work
might be falling behind schedule. By the time Monday came around again Robert
would usually be able to report that he had talked with the contractors and
found a solution. Placing large contracts for the magnets was another major
concern. If the second half of the magnets for SPS were to be ordered, the
contract for the first half would have to come in at a low price and options
to build the second half would have to be written into the agreement.
### 0.8.7 Magnet problems
There were from time to time, as in any project, unforeseen technical
setbacks. One such was the discovery that 100 of the 700 bending magnets
already installed in the tunnel had developed short circuits to ground. This
was deeply shocking to all concerned and it looked as if the SPS was no better
than the Fermilab main ring where magnets failed at the rate of one a day in
the early tests. Had we been wise to emulate the methods of Fermilab? we
wondered.
The whole team of group leaders was summoned to meet every day for a week to
investigate the cause and plan a remedy. It was in the spirit of putting their
heads together. There might well have been shouting or admonishment, but with
John Adams in the chair there was instead, as there should be, just logic and
science.
Rather soon Billinge and Bob Sheldon, who was a chemist, and whose first
instinct was to lick a finger and to taste the tag ends of the coil
conductors, established that they had been prepared for brazing by cleaning
with phosphoric acid by an overzealous welder. The acid, it was discovered,
could fill up the hollow glass fibres which loaded the insulation and provide
a conducting path for short circuits. Fortunately there was time, without
delaying the start-up, to take out the infected magnets, rebuild the coils,
and wrap them in Kapton to prevent any other shorts.
Delays to several large projects (not least, the magnet insulation of the
Fermilab Main Ring, the niobium welds on the vacuum chamber of the Large
Electron–Positron machine (LEP), the busbar connections of the Tevatron, and
recently the interconnects in the Large Hadron Collider (LHC)) have regularly
been caused by the unpredictable consequences of engineering solutions. It was
fortunate that no delay resulted for the SPS—perhaps thanks to John Adams’s
rigourous analysis of possible difficulties and their solutions—but more
realistically because of the thorough pre-start-up tests he insisted upon
after installation in the tunnel.
### 0.8.8 Commissioning
The SPS was finished five years after the team first assembled in Prévessin.
Such was the thoroughness of the preparation that John had expected from his
team that each stage in the commissioning programme worked like clockwork.
Once again John left those in charge to do their stuff, but I do remember one
moment before the beam was injected when he asked everyone “Are you sure you
have not forgotten something?”.
Figure 17: SPS control room—first beams accelerated
The contrast with the commissioning of Fermilab, which I had lived through a
couple of years earlier, was clear. Everyone in the SPS control room had done
their professional job and knew enough about accelerator physics to diagnose
any little misbehaviour of the beam. To be fair, it also helped to be able to
learn from Fermilab experience. There was one little hiccup when we tried to
accelerate for the first time and the beam just disappeared. Within the same
day we tracked down a fault in the numerical program of the power supplies for
the focusing system and went on to accelerate.
Figure 18: The CERN Council is asked to approve 400 GeV
The 200 GeV acceleration came easily and the first pulse was synchronized to
be announced to the Council at the end of their morning session. At some time
in the past, the Council had insisted they be asked permission before moving
from 200 GeV to 400 GeV. It had been something to do with ordering the missing
magnets. After reporting acceleration to 200 GeV, John wryly asked their
authorization to accelerate to 400 GeV and by the tea break in the afternoon
he announced the first 400 GeV pulse—on time and of course—on budget.
Figure 19: The first 400 GeV pulse
### 0.8.9 He becomes Director-General a second time
During the construction period he had been Director-General of Laboratory II,
which included the SPS and the Prévessin site in France, while Willi Jenschke
had looked after the main Laboratory I site at Meyrin including the ISR, PS,
and their experiments. The time came, just before the SPS was finished, to
merge the two laboratories together under a single Director-General. I
remember meeting him then in the corridor (during the Council meeting where
this was to be decided). He confided in me disconsolately that, “they were
taking a long time over it—for some reason I do not understand, they think
they need to have two DG’s.” It turned out that, while they recognized he was
the man to look after the accelerators, they wanted an eminent physicist
rather than an engineer to manage the research programme. And so it was that
they decided that John would be one Director-General who would concentrate on
the accelerators, while Léon van Hove, a second Director-General, looked after
the physics programme. It is greatly to John’s credit that he was able to
accept this arrangement and together they made it work. During his final term
as Director-General he visited China. It was 1977 and before the iron grip of
the Gang of Four began to slacken. China was keen to build a large proton ring
near the Great Wall as a statement of China’s progress towards western
prosperity. John met Deng Xiaoping. “Very smart,” said John, “perhaps I made a
mistake to tell him the big proton machine would be no use to them and what
they really needed was a synchrotron light source.” And of course that is what
they did.
### 0.8.10 R. R. Wilson
Throughout the construction of the SPS, Robert Rathbun Wilson was John’s US
counterpart whom he rarely mentioned. Bob Wilson had set about constructing
the Fermilab main ring with very similar design aims to the SPS. He was
fortunate to be able to start about five years before SPS approval and had
finished it (though there were still some things to tidy up) about five years
before the SPS. His style could not have been more different from that of John
Adams. I had the privilege of working closely with both these men for, in the
middle of SPS construction, I was dispatched by John Adams to help sort out
some of the difficulties that Bob Wilson was having in commissioning his 400
GeV Main Ring. The fact that it needed someone from another laboratory to help
in this way is perhaps a comment on the risks that Bob Wilson was prepared to
take to save time and money in construction. This was something that John
Adams, in his desire to be careful and not prejudice the reliable operation
for the machine, was at pains to avoid.
However, it must be said that Bob Wilson inspired younger members of his team
(and ours) with his bold initiatives. Many of the ideas which simplified the
design of the SPS and assured its success had been copied from innovations he
pioneered at Fermilab. We have seen that John Adams had embraced these ideas
with enthusiasm, provided they did not put the outcome at risk.
Once, while visiting Fermilab, I can remember being asked by a resident
historian to compare these two great men. My answer was:
John Adams had artistic talent but had never had the time to follow his talent
to its conclusion—Bob Wilson on the other hand had managed to achieve an
international reputation as a sculptor and architect.
John Adams persuaded through reason and was always a gentleman —Bob Wilson
challenged his team with his own inspiration and rode roughshod over their
objections.
John Adams was careful—Bob Wilson deliberately took risks (but was prepared to
fix them afterwards).
John Adams was ideal for Europe whose politicians are used to allowing
themselves to be persuaded by the reason and common sense of their own
scientific advisors. Bob Wilson’s passionate rhetoric often rivalled the
Fathers of his Nation and was finely tuned to the ear of a Washington
politician or media magnate.
Both would have been a disaster had they exchanged the old world for the new,
or vice versa. Perhaps John Adams would have found it even easier to establish
his technical dominance in the USA and without formal qualifications, but the
US has little time for the staff management methods of Cockcroft or the
cerebral exercises of Socrates. Bob Wilson would probably have been judged
rash by European politicians and scientists, but his artistic gifts would have
found more nourishment in the richer soil of Paris, Florence, or Rome than in
Illinois.
_Both felt their career should have gone on longer, and I agree!_
## 0.9 Since he left us …
In preparing this talk I was asked to answer the question “How would CERN be
different if John Adams was still alive today?” I will attempt to answer this,
but emphasize that this is merely a personal view. I have not been as closely
involved in CERN’s recent projects as I was when John Adams was alive and I
expect that those who were may disagree with my conclusions. I still think
that the points I raise are worthy of debate and should be taken on board by
leaders of future projects.
The first accelerator project to follow in the wake of his years as Director-
General was the Antiproton Accumulator (AA) using the SPS for colliding
antiprotons with protons. This involved also the construction of two large
detectors, UA1 and UA2. John Adams was still with us when these projects and
LEP were started, and accelerator engineers and physicists who had been
schooled in his way of doing things were largely responsible for their
execution. These projects still bore his footprint.
The development of a more intense antiproton source to follow AA was perhaps
something he might have restrained, given that the Tevatron with twice the
centre-of-mass energy of the SP-PbarS was about to put antiproton physics with
the SPS out of business. However, the cost of the new Antiproton Accumulator
turned out later to be a small price to pay for the improved supply of
antiprotons for the low-energy LEAR programme. About this time there was an
upgrade to UA1 which did not materialize. John Adams might have seen this
coming but would probably not have been able to restrain it even if he had
wanted, since it was outside the field of accelerators.
I am tempted to think that the teething troubles due to the use of magnetic
material (niobium) in the finishing of the LEP vacuum chamber might have been
prevented by a Design Committee with John to guide them—or maybe it was just
bad luck. Anyway, the delay it caused was minimal and LEP proved a great
success in spite of it.
LEP had expensive delays due to fountains of water springing from the walls
and floor during the tunnelling. With hindsight it should never have
encroached upon the Jura limestone. John would certainly have been aware of
the dangers of leaving the molasse and tunnelling into water-bearing rock but
it would have required all his skills to persuade the physics community to
sanction a smaller and less energetic LEP.
The next phase that John Adams might have had an influence upon was the race
for approval between the Large Hadron Collider (LHC) and the Superconducting
Supercollider (SSC). Once the US and Texas had decided they could not foot the
bill for the SSC, he would have been in his element trying to arrive at a
machine which the world might afford. Had this come to pass and had he gone on
to have a leading role in a Super Collider’s construction, he would have kept
a tight control on the tenders for major hardware components—a scrutiny which
was very much needed at the SSC. Both the SSC and LHC used superconducting
magnets, and it would have been interesting to see if John Adams could have
found a way to curb the fears of industrial firms whose tenders for
superconducting magnets mainly reflected their caution in bidding for an
unfamiliar technology.
Approval for the LHC took a long time, but then so did approval for the SPS.
After the demise of the SSC, Chris Llewellyn-Smith and Giorgio Brianti finally
took the Council by the horns, and got them to agree to the LHC. Their
approach used many of the techniques that Adams had deployed in 1971 to secure
the SPS for CERN.
As for the future linear collider, I like to think that John would have seen
the virtue of a common cause which spanned the various laboratories involved
earlier, and used collaboration to push CLIC more rapidly towards becoming a
project rather than a research and development exercise.
Of course, the big question at the moment of writing is—Would John and his
Design Committee have seen the troubles with the LHC interconnects which
caused so much sorrow in the last twelve months? This is in many ways
reminiscent of the SPS history of magnet insulation problems. The only
difference perhaps is that the SPS problem became apparent during routine
electrical tests rather than in the glare of the spotlight of the world press.
Nevertheless, there is a strong probability that John (always on the lookout
for engineering weaknesses) would not have let it creep under the radar of his
Design Committee.
## 0.10 Conclusion
It was in 1981 that Sir John Adams received his knighthood from the Queen, but
he modestly never asked to be called Sir John by his colleagues. Once his term
of office was over, he moved back to his old office on the Prévessin site and
began to make himself available as an advisor to a number of European and
other international bodies. He would really like to have built LEP but as he
said, “Schopper was keen to do it.” His brilliant career was at an end, and in
the last few years he missed the bustle of building accelerators and the long
queue of those waiting to see him, but I suppose that comes to all as they
approach retirement, and what a career he had had! And what a legacy he left
behind at CERN! There is so much in his career that those at CERN would do
well to remember every time they start a new accelerator project. Not all of
us can have his gifts but we may aspire to them.
## References
* [1] E. Amaldi, John Adams and His Times, John Adams Memorial Lecture, 1985, CERN 86-04 (CERN, Geneva, 1986).
* [2] CERN Courier, November 2009, extracted from CERN Courier, November 1969, pp. 331–336, H. Blewett.
* [3] John Lawson Archives at the John Adams Institute, Oxford.
* [4] M. C. Crowley-Milling, John Bertram Adams: Engineer Extraordinary (Gordon and Breach, Yverdon, 1993).
* [5] J. B. Adams and A. O. Edmunds, The performance of the Harwell 110 inch synchrocyclotron, AERE Report G/R 568 (1950).
* [6] CERN Study Group on New Accelerators, Report on the Design Study of a 300 GeV Proton Synchrotron, CERN-AR/Int. SG/64-15, CERN/563 (CERN, Geneva, 1964), 2 vols., also in French.
* [7] T. Kitagaki, A focusing method for large accelerators, Phys. Rev. 89 (1953) 1161–2.
* [8] J. B. Adams and E. J. N. Wilson, Design studies for a large proton synchrotron and its laboratory, Nucl. Instrum. Methods, 87 (1970) 157–179.
* [9] The 300 GeV Programme, Ed. E. J. N. Wilson, CERN/1050 (CERN, Geneva, 1972), also in French.
|
arxiv-papers
| 2011-02-23T16:13:30 |
2024-09-04T02:49:17.188126
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E.J.N. Wilson",
"submitter": "Scientific Information Service Cern",
"url": "https://arxiv.org/abs/1102.4777"
}
|
1102.5002
|
# ÆTHEREAL MULTIVERSE
Selected Problems of Lorentz Symmetry Violation, Quantum Cosmology, and
Quantum Gravity
ŁUKASZ ANDRZEJ GLINKA
International Institute for
Applicable Mathematics & Information Sciences
Hyderabad (India) & Udine (Italy)
(Draft of Book)
###### Contents
1. Preface
2. Prologue
3. I Lorentz Symmetry Violation
1. 1 Deformed Special Relativity
1. A The linear deformation
1. A1 The Dirac equation and the new algebra
2. A2 Another Identifications
2. B The Snyder–Sidharth Hamiltonian
1. B1 The Case of Fermions
2. B2 The Case of Bosons
3. C The Modified Compton Effect
1. C1 The Relativistic Approach
2. C2 The Relativistic Limit. The Lensing Hypothesis.
3. C3 Bounds on the Modified Compton Equation
4. D The Dispersional Generalization
2. 2 The Neutrinos: Masses & Chiral Condensate
1. A Outlook on Noncommutative Geometry
2. B Massive neutrinos
3. C The Compton–Planck Scale
4. D The Global Effective Chiral Condensate
5. E Conclusion
3. 3 The Neutrinos: Energy Renormalization & Integrability
1. A Introduction
2. B Energy renormalization
3. C The Integrability Problem for the Dirac equations
4. D The Integrability Problem for the massive Weyl equations
1. D1 The Dirac basis
2. D2 The Weyl basis
3. D3 The space-time evolution
4. D4 Probability density. Normalization
5. E The Ultra-Relativistic Massive Neutrinos
4. II Quantum General Relativity
1. 4 The Quantum Cosmology
1. A Introduction
2. B The Classical Universe
3. C Quantization of Hamiltonian Constraint
4. D The Multiverse Thermodynamics
5. E The Early Light Multiverse
6. F Summary
2. 5 The Inflationary Multiverse
1. A The Inflationary Cosmology
2. B The Power Law Inflaton
3. C The Higgs–Hubble Inflaton
4. D The Chaotic Slow–Roll Inflation
5. E The Phononic Hubble Inflaton
6. F The Inflaton Constant
3. 6 Review of Quantum General Relativity
1. A 3+1 Splitting of General Relativity
2. B Geometrodynamics: Classical and Quantum
3. C The Wheeler Superspace
4. D The Problems of Geometrodynamics
5. E Other Approaches
4. 7 Global One-Dimensionality Conjecture
1. A Introduction
2. B The $\Gamma$-Scalar-Flat Space-times
3. C The Ansatz for Wave Functionals
4. D Field Quantization in Static Fock Space
5. E Several Implications
1. E1 The Global 1D Wave Function
2. E2 The Unitary Three-Manifolds
3. E3 The Fourier Analysis
4. E4 Quantum Correlations
5. 8 The Invariant Global Dimension
1. A The Invariant Global Quantum Gravity
2. B The One-Dimensional Dirac Equation
3. C The Cauchy-Like Wave Functionals
4. D Problem I: Inverted Transformation
5. E Problem II: The Hilbert Space and Superposition
6. F Problem III: The Problem of Time
6. 9 Examples of Invariant 1D Wave Functions
1. A The Minkowski Space-time
2. B The Kasner Space-time
3. C The Schwarzschild Space-time
4. D The (Anti-) De Sitter Space-time
5. E The (Anti-) De Sitter–Schwarzschild Space-time
6. F The Kerr Space-Time
7. G The Kerr–Newman Space-time
8. H The Reissner–Nordström Space-time
9. I The Gödel Space-time
10. J The Einstein–Rosen Gravitational Waves
11. K The Taub–Newman–Unti–Tamburino Space-time
7. 10 The Functional Objective Geometry
1. A Effective Scalar Curvature
2. B The Newton–Coulomb Potential
3. C Boundary Conditions for The Wave Functionals
8. 11 _Ab Initio_ Thermodynamics of Space Quanta Æther
1. A Entropy I: The Analytic Approach
2. B Entropy II: The Algebraic Approach
9. Epilogue
10. References
### Preface
A thought is an idea in transit.
Pythagoras
_Æthereal Multiverse_ is my personal research attempt to demonstrate the
productive realization of the fusion of two fundamental concepts of Antiquity,
i.e. Æther and Multiverse which lay the foundations of Aristotelian and
Epicurean–Islamic systems, respectively. It is my deep conviction that this
combination enables the fruitful description of physical Reality, which allows
to understand constructively the anthropic everything, i.e. all what can be
observed and detected by a man and all devices produced by humankind. From my
standpoint the constructive proposal for the fusion involves the essential
theoretical symbols of the physical Reality: Quantum Mechanics, Quantum Field
Theory, General Relativity, and Thermodynamics.
On the one hand this monograph collects advanced developments of certain
elementary knowledge of theoretical and mathematical physics. On the other
hand, however, the presented deductions are performed step-by-step and often
include detailed calculations. In this manner this book is available for the
readers interested in development and applications of the fundamental
knowledge. Intentionally the content is divided onto two independent lines
which have arose in my research work of the years 2006-2010. Part I, _Lorentz
Symmetry Violation_ , contains 3 chapters and is devoted to discussion of
several applications of the noncommutative geometry of the Snyder quantized
space-time. Strictly speaking I shall focus in some detail on the basic
consequences of what I call the Snyder–Sidharth Hamiltonian constraint, i.e.
the modification of Special Relativity arising from the noncommutative
geometry. Particularly the Compton effect and the massive neutrinos model are
discussed. Part II, _Quantum General Relativity_ , contains 8 chapters and is
much more inhomogeneous. This part presents my approach to quantum cosmology
and quantum gravity, and my point of view on inflationary cosmology. I present
the version of quantum geometrodynamics strictly based on the Wheeler–DeWitt
equation, which leads to constructive and consistent deductions. Particularly
two approaches to entropy calculation, which lead to distinguished
formulations of thermodynamics of space quanta Æther, are discussed.
The greatest motivation to this book is the 2010’ book of an experienced
mathematical physicist Robert W. Carroll [1] who expressed his opinion about
certain part of my research results
(…) we sketch some work of L. Glinka et al on thermodynamics and quantum
gravity. This involves bosonic strings and quantum field theory (QFT) and is
speculative (but very interesting). (…) we deal with some fundamental articles
by L. Glinka for which some general theory is also motivated by theoretical
material involving second quantization and Bogoliubov transformations (…) we
suggest printing out the latest versions and working from them - we can only
give a sketch here and remark that some of the work has a visionary nature
which is valuable in itself.
This book collects all my non-coauthored research results rigorously revisited
and enriched by necessary updates. Numerous typos and technical mistakes are
improved. The necessity of the improvement follows from my deep conviction
that the only mathematical truth results in the constructive theoretical and
mathematical physics. This book develops also my philosophical standpoint on
theoretical physics involving the philosophical interpretation needed for the
new physics.
My grateful acknowledgements belong to numerous senior scientists and scholars
who helped me kindly in the various aspects related to this book. Professor
Robert W. Carroll granted me by a number of valuable discussions and comments,
and mailed a hard copy of his 2010’ book. Professor Sir Harold W. Kroto gave
matter-of-fact discussions, and included my views into his GEOSET programme.
Essential comments and discussions due to Professor Burra G. Sidharth
significantly helped in edition of the primary manuscript of this book.
Discussions with Dr. Andrej B. Arbuzov and Professor Alessandro De Angelis
benefitted during my research work were helpful. Comments from Professor
Wojciech H. Zurek were also valuable and constructive.
I dedicate this book to all Friends of mine. Comments and discussion are
welcome and invited to laglinka@gmail.com.
Łukasz Andrzej Glinka
### Prologue
It is a dogma of the Roman Church that the existence of God can be proved by
natural reason. Now this dogma would make it impossible for me to be a Roman
Catholic. If I thought of God as another being like myself, outside myself,
only infinitely more powerful, then I would regard it as my duty to defy him.
Ludwig Wittgenstein
#### The U-turn to Antiquity
In the second decade of the 21th century technological progress defines
evolution of humankind. Present civilizations are based on technocratic ideas
which are often overly formal. Such a state of things rejects freedom of
thinking and focuses on creation of new barriers, called standards and norms.
The obstacles follow from promotion of feudal submissiveness toward various
traditions. This fight with rationalism results in inhibition of mental growth
of individuals, and leads to evident intellectual poverty. The irrationalism
ruins efforts of rationalists, establishes and stabilizes illusions and
delusions, and effectively results in the permanent damage of individual
mentality. Such a regime has been produced the emergence of social multiple
disabilities, and above all a regular de-evolution of humankind.
On the other hand there is detectable the vast U-turn to the ideas of the most
fundamental epoch in the history of human reasoning, _Antiquity_. The
necessity of such a nontrivial turnabout is fully justified by the situation.
In fact, only the ancient ideas possess the natural ability to reconstruct the
most valuable heritage, because of Antiquity is not infected by certain
destructive effects of the modern civilization. A productive description of
the physical Reality requires fusion of two notions: Æther, the idea of the
Aristotelian system, and Multiverse which lays the foundations for the
Epicurean–Islamic system.
#### Æther
In the mid-1950s, when Albert Einstein abandoned the living nature, Cosmology
altered manifestly esoteric and religious countenance, and became a respected
scientific branch. The satisfactory explanation of observational and
experimental data consolidated both the physical and philosophical heritage of
General Relativity. Riemannian geometry apparently is able to describe a
number of astronomical phenomena like Mercury’s perihelion precession and
light rays deviation in Sun’s vicinity, and offers constructive generalization
of the Newton universal gravitation. This Einstein’s theory has been described
visible objects and predicted existence of new physical beings.
Albeit, Einstein’s legacy is wide spread [2] and deserves being called
phenomenology. Despite Einstein was established as the specialist in molecular
physics and thermodynamics, he gave trailblazing description of fast
particles, solids, opalescence, emission and absorption of electromagnetic
radiation, and corrected the Zero-Point Energy hypothesis of M. Planck. His
pioneering approach to the photoelectric effect used the wave-particle duality
which became the main stream of the 20th century theoretical physics. He was
the follower of unification of gravity and electromagnetism. Despite the
heritage incontrovertibly impacted on the mentality and the character of
theoretical physics, Einstein’s intellectual growth was diverse. It is an
appearance, because of Einstein always tried to describe Æther.
###### The Mythical Nature
The concept of Æther is strictly rooted within the mythology of Ancient
Greece. Ancient Greeks professed Protogenoi, which are the immortal
<<primordial deities>> born in the beginning of Universe. These primary gods
were Æther (Mists of Light, Upper Air), Ananke (Inevitability, Necessity,
Compulsion), Chaos (Void, Lower Air), Chronos (Time), Erebus (Mists of
Darkness), Eros (Generation), Gaea (Earth), Hemera (Day), Hydros (Water),
Nesoi (Islands), Nyx (Night), Oceanus (Ocean), Ourea (Mountains), Phanes
(Procreation), Phusis (Nature), Pontus (The Sea), Tartarus (Hell), Tethys
(Fresh Water), Thalassa (Sea Surface), Thesis (Creation), Uranus (Heaven),
which generated Giants, Titan, Olympian, Oceanic, and Chthonic gods. In
Homeric Greek language Æther $\alpha\iota\theta\eta\rho$ means ”pure, fresh
air” or ”clear sky”, which in Greek mythology is the pure essence where the
gods lived and which they breathed. Æther was a personified idea of the
cosmogony professed by ancient Greeks, and considered as one of the elementary
substances forming the Universe. According to the Orphic hymns Æther is the
soul of the world emanating all life.
In alchemy and natural philosophy Æther was originated by Aristotle as
<<quinta essentia>>, the cognate chemical fifth element of the heavens are
made and unifying the sublunary elements, i.e. Fire, Earth, Air, and Water.
Aristotle claimed that the four elements move rectilinearly, and because of
orbits of the heavenly bodies are circular and lie on the rotating spheres
surfaces, so the physical spheres must be a body. In other words, the
celestial motions required an existence of the superior element. Aristotle
gave several plausible arguments for existence of Æther. One of them, called
argument from incorruptibility, states that the sublunar elements are easy
transiting into each other, but because of the eternal heavens they must be
made of a different element. Therefore, existence of Æther follows from the
necessity for a body endowed with natural circular motion. By Aristotle the
fifth element is unborn, immortal, and invariable, and its name arises from
ancient <<aei thein>> what means eternal motion. Factually, he identified
Æther with the mind or soul, which is divine and does not corrupt with the
earthly elements. Aristotle wrote [3]
(…) That is why the upper part is moved in a circle, while the All is not
anywhere. For what is somewhere is itself something, and there must be
alongside it some other thing wherein it is and which contains it. But
alongside the All or the Whole there is nothing outside the All, and for this
reason all things are in the heaven; for the heaven, we may say, is the All.
Yet their place is not the same as the heaven. It is part of it, the innermost
part of it, which is in contact with the movable body; and for this reason the
earth is in water, and this in the air, and the air in the aether, and the
aether in heaven, but we cannot go on and say that the heaven is in anything
else.
###### From Boyle & Newton to Lorentz
In the 17th century R. Boyle [4] treated Æther as the material substance due
to what he called subtle particles. However, in the context of modern physics
I. Newton, in the corpuscular theory of light presented in the book _Opticks_
[5], proposed the pioneering idea of an æthereal medium carrying vibrations
traveling faster than light, which straightforward intervention results in
refraction and diffraction of light. Newton negated the idea proposed by Ch.
Huygens, claiming that light travels by Æther medium, and presented the
explanation of the phenomenon of gravitation via the pressure of an atomic
Æther impacting upon matter in abnormal state. Subtle æthereal molecules
entered into matter via the pores, and when approached a physical body became
less resilient and rarely distributed. A material body find itself under the
pressure due to Æther upon all sides. Therefore, between two bodies this
pressure is less and rare Æther distribution causes in compulsory gravitation.
Newton, however, did not justify the rarefaction mechanism emerging near
matter.
We should reference also so called Le Sage’s kinetic theory of gravitation,
proposed by N. Fatio de Duillier and G.L. Le Sage, which was Æther-based
constructive explanation of the Newton universal gravitation. Le Sage [6]
suggested purely mechanical nature of the universal gravitation which in an
effect due to streams of ultra-mundane tenuous Æther corpuscules moving at the
speed of light and acting on all matter from all directions. In ”Lucrèce
Newtonien” Le Sage expressed the following standpoint
I am well convinced that since the law governing the intensity of universal
gravitation is similar to that for light, the thought will have occurred to
many physicists that an ethereal substance moving in rectilinear paths may be
the cause of gravitation, and that they may have applied to it whatever of
skill in the mathematics they have possessed.
W. Thomson the 1st Baron Kelvin [7] introduced another impacting concept of
Æther. He treated Æther as an elastic solid medium transmitting the
electromagnetic waves, and proposed its mechanical model. Such a proposition
led him to explanation of the nature of radiation.
J.C. Maxwell [8] was a follower of so called Luminiferous Æther, i.e. the
cosmic medium physically transmitting light. He applied this notion to deduce
equations of electrodynamics, called the Maxwell equations. Maxwell showed
that light is an electromagnetic wave, and thought about the physical lines of
forces of electric and magnetic fields as the lines within the Æther. His idea
based on the Poisson equation is referred as the Maxwell Vacuum.
Another diverse concepts of Æther were widely propagated and developed by
numerous eminent scientists and scholars at the turn of the 19th and 20th
centuries (See e.g. the contributions in the Ref. [9]). Exceptionally detailed
analysis of the historical development of Æther till discovery of quantum
mechanics was elegantly performed by E.T. Whittaker [10]. The best example is
the Æther model investigated by J.J. Thomson, who considered the context of a
hypothetical radiation that could be more penetrating then Röntgen’s X-rays.
Another significant investigation applying certain Æther model was performed
by G.H. Darwin in computation of a geometric deviation from Newton’s law of
universal gravitation.
The most intriguing idea was the concept of Æther wind called also Æther drag
and Æther drift. This hypothetic phenomenon has the place when Luminiferous
Æther is dragged by motion of matter or entrained by matter. A.J. Fresnel
proposed Æther wind with partial entraining, which was empirically confirmed
by the 1851 experiment of H. Fizeau. Different version of the Æther drag
hypothesis, founded by G. Stokes in 1845, was experimentally confirmed by A.A.
Michelson and E.W. Morley [11] in 1881 and 1887. Fifteen years after
interpretation of the results of the Michelson–Morley experiment as the
confirmation of Einstein’s Special Relativity, Michelson published the book
[12] in which one can find several interesting looking reflections
The standard light waves are not alterable; they depend on the properties of
the atoms and upon the universal ether; and these are unalterable. It may be
suggested that the whole solar system is moving through space, and that the
properties of ether may differ in different portions of space. I would say
that such a change, if it occurs, would not produce any material effect in a
period of less than twenty millions of years, and by that time we shall
probably have less interest in the problem. (…) the vibrations of these
particles, or of their electric charges, produce the disturbance in the ether
which is propagated in the form of light waves; and that the period of any
light wave corresponds to the period of vibration of the electric charge which
produces it. (…) the ether itself is electricity; a much more probable one is
that electricity is an ether strain - that a displacement of the ether is
equivalent to an electric current. If this is true, we are returning to our
elastic-solid theory. I may quote a statement which Lord Kelvin made in reply
to a rather skeptical question as to the existence of a medium about which so
very little is supposed to be known. The reply was: "Yes, ether is the only
form of matter about which we know anything at all." In fact, the moment we
begin to inquire into the nature of the ultimate particles of ordinary matter,
we are at once enveloped in a sea of conjecture and hypotheses - all of great
difficulty and complexity. One of the most promising of these hypotheses is
the "ether vortex theory," which, if true, has the merit of introducing
nothing new into the hypotheses already made, but only of specifying the
particular form of motion required. The most natural form of such vortex
motions with which to deal is that illustrated by ordinary smoke rings, such
as are frequently blown from the stack of a locomotive. Such vortex rings may
easily be produced by filling with smoke a box which has a circular aperture
at one end and a rubber diaphragm at the other, and then tapping the rubber.
The friction against the side of the opening, as the puff of smoke passes out,
produces a rotary motion, and the result will be smoke rings or vortices. (…)
Investigation shows that these smoke rings possess, to a certain degree, the
properties which we are accustomed to associate with atoms, notwithstanding
the fact that the medium in which these smoke rings exists is far from ideal.
If the medium were ideal, it would be devoid of friction, and then the motion,
when once started, would continue indefinitely, and that part of the ether
which is differentiated by this motion would ever remain so. (…) Another
peculiarity of the ring is that it cannot be cut - it simply winds around the
knife. Of course, in a very short time the motion in a smoke ring ceases in
consequence of the viscosity of the air, but it would continue indefinitely in
such a frictionless medium as we suppose the ether to be. (…) Suppose that an
ether strain corresponds to an electric charge, an ether displacement to the
electric current, these ether vortices to the atoms - if we continue these
suppositions, we arrive at what may be one of the grandest generalizations of
modern science - of which we are tempted to say that it ought to be true even
if it is not - namely, that all the phenomena of the physical universe are
only different manifestations of the various modes of motions of one all-
pervading substance - the ether. (…) Then the nature of the atoms, and the
forces called into play in their chemical union; the interactions between
these atoms and the non-differentiated ether as manifested in the phenomena of
light and electricity ; the structures of the molecules and molecular systems
of which the atoms are the units; the explanation of cohesion, elasticity, and
gravitation all these will be marshaled into a single compact and consistent
body of scientific knowledge. (…) In all probability, it not only exists where
ordinary matter does not, but it also permeates all forms of matter. The
motion of a medium such as water is found not to add its full value to the
velocity of light moving through it, but only such a fraction of it as is
perhaps accounted for on the hypothesis that the ether itself does not partake
of this motion. (…) The phenomenon of the aberration of the fixed stars can be
accounted for on the hypothesis that the ether does not partake of the earth’s
motion in its revolution about the sun. All experiments for testing this
hypothesis have, however, given negative results, so that the theory may still
be said to be in an unsatisfactory condition.
H.A. Lorentz, one of the most eminent theoretical physicists of the turn of
the 19th and 20th centuries, also manifestly professed Æther [13]. In his
lectures he straightforwardly supports Æther. In the lectures delivered at
Caltech one finds
Nowadays we are concerned only with the electromagnetic theory of light, in
which there is no longer any discussion of a density or elasticity of the
ether. In the electromagnetic theory of light attention is fixed on the
electric and magnetic fields that can exist in the "ether". (…) the state of
the ether is the same at all points of a plane perpendicular to the direction
of propagation, and so the waves may be called plane waves.
Similarly, Leiden lectures of Lorentz contain the ambiguous opinion
(…)[W]hether there is an aether or not, electromagnetic fields certainly
exist, and so also does the energy of electrical oscillations. If we do not
like the name of aether, we must use another word as a peg to hang all these
things upon. It is not certain whether space can be so extended as to take
care not only of the geometrical properties but also of the electric ones. One
cannot deny to the bearer of these properties a certain substantiality, and if
so, then one may, in all modesty, call true time the time measured by clocks
which are fixed in this medium, and consider simultaneity as a primary
concept.
The Lorentz transformations were deduced on base of Æther, and Lorentz
defended his own standpoint called Lorentz Æther theory. Lorentz pointed out
inconsistency between results of the Michelson–Morley and the Fizeau
experiments. Basing on the stationary Æther arising from the theory of
electrons Lorentz removed Luminiferous Æther drag.
G.F. FitzGerald, who was under the influence of calculations performed by O.
Heaviside [14] including deformations of magnetic and electric fields
surrounding a moving charge and the effects of it entering a denser medium
like e.g. what is today called Cherenkov’s radiation, in Science article [15]
wrote Æther-based conclusion
[T]he length of material bodies changes, according as they are moving through
the ether or across it, by an amount depending on the square of the ratio of
their velocities to that of light.
Another example is the 1915’ paper of W.J. Spillman, in which reasoning is
based on the _Fundamental Assumptions_ involving Æther
The ether. \- The ether is assumed to exist in the interstices of matter and
in open space. It is assumed to be capable of distortion by finite force, and
to oppose such distorting force with an equal opposite force. In other words,
every point of the ether has a position which it normally occupies, and when
removed from that position tends forcibly to return to it, the force being
proportional to the distortion. It is further assumed that distortion at a
given point in the ether tends to become distributed in the surrounding ether
according to the law of inverse squares, and that such distribution occurs at
a finite rate (the velocity of light).
The electron. \- It is assumed that in the immediate vicinity of the electron
there is a region of maximum permanent ether distortion, the distortion at
other points in the surrounding ether varying inversely as the square of the
distance from the center of the electron, and that pressures are transmitted
at a higher velocity through distorted than through non-distorted ether.
The permanent distortion of the ether in the vicinity of the electron may be
conceived of as a pushing back of the ether radially from the center of the
electron, as if an impenetrable and inelastic body were injected into the
midst of an elastic body; or it may be conceived of as being circular, in two
hemispheres facing each other, and opposite in direction in the two
hemispheres.
In explaining the phenomena of inertia, electric currents, magnetism, chemical
affinity, and radiant energy, the above alternative assumptions concerning the
character of the distortion lead to essentially similar lines of reasoning,
but the treatment on the assumption of radial distortion is very much simpler.
For this reason it is used here. In the case of static electricity only the
assumption of circular distortion, opposite in direction in the two kinds of
electric elements, will explain the facts. The extension of the theory to
static electricity is left for future treatment. It must be remembered,
however, that the development of the theory on the basis of radial distortion
differs only in detail, not in principle, from that on the basis of circular
distortion.
The Atom. \- The atom is assumed to consist, at least in part, of a Saturnian
system of electrons in rapid orbital motion. It will be shown that such
orbital motion, with the assumptions here made, would give rise to a pressure
in the ether such as Newton showed would account for gravitation.
Physicists are agreed that the phenomena of inertia, the electric current,
magnetism, and possibly also chemical affinity are probably related to each
other in such manner that when we find the explanation of one of them this
explanation will also throw light on the others.
###### Æther Drag
The most intriguing phenomena related to the concept of Æther is Æther
drag/drift/wind. Both Lorentz Æther theory and Einstein’s Special Relativity,
the theories of stationary Æther, are interpreted are the theories
extraordinary strongly antagonistic to the Æther wind. The essence of the
history of 20th century physics and the milestone of empirical negating of
Æther is the Michelson–Morley experiment which purpose was to detect Earth’s
motion with respect to Luminiferous Æther via comparison of speed of light in
diverse directions with respect to Earth. On the one hand, the theory of
stationary Æther propagated by H.A. Lorentz this attempt manifestly and
straightforwardly negated existence of Æther drag because of contradiction
with the result of the Fizeau experiment. On the other hand, the
Michelson–Morley experiment is consistently explained and interpreted within
Einstein’s Special Relativity with no reference to Æther drag.
There were performed also another experiments [16] having the purpose to
verify empirically the Æther drift: 1903’ F.T. Trouton and H.R. Noble, 1908’
F.T. Trouton and A.O. Rankine, 1913’ G. Sagnac, 1925’ A.A. Michelson and H.G.
Gale, 1932’ R.J. Kennedy and E.M. Thorndike, and 1935’ G.W. Hammar. Their
results suffered the fate analogous to the Michelson–Morley experiment, i.e.
were interpreted as negation of Æther wind and Æther in general. Transparent
discussion of these results can be found in the book of W. Pauli [17].
The Trouton–Noble experiment was strict realization of the idea due to
FitzGerald. He concluded that motion of a charged flat condenser through the
Æther should result in its perpendicular orientation to the motion. The
experimenters detected the lack of the relative motion to the Æther, what was
interpreted as the negative result.
The Trouton–Rankine experiment was performed for detection of ”preferred
frame” which would be the syndrome of existence of the Luminiferous Æther. The
measurement assumed that the length contraction produce a measurable effect in
the rest frame of the object observed in other frame. Both Special Relativity
and Lorentz Æther theory predicted that length contraction is non measurable.
Trouton and Rankine, however, applying the Ohm law and the Maxwell equations
theoretically predicted that the effect, change of resistance, is measurable
in the laboratory frame. The experimenters used the Wheatstone bridge, and the
change of resistance was not detected.
Sagnac applied a rotating interferometer, and observed a dependence of
interference fringe position on the angular velocity. The result of this
experiment was theoretically predicted in 1911 by M. von Laue [18], who proved
manifestly its consistence with Special Relativity and numerous models of
stationary Æther, including the Lorentz Æther theory. Factually, results of
this experiment confirmed existence of the stationary Æther, even in the sense
of the Lorentz Æther theory, but Laue’s calculations were interpreted as the
reflection of Special Relativity correctness. The Sagnac effect is probably
the most positive result for the Æther drag, and was also treated as the
straightforward demonstration of existence of the Æther. Moreover, the Sagnac
effect was constructively explained also within General Relativity which was
claimed by Einstein to be the theory of non-physical Æther.
The Michelson–Gale experiment modified the Michelson–Morley experiment by
application of enlarged Sagnac’s ring interferometer. The purpose was to find
out the relation between the Earth rotation motion and the light propagation
in the vicinity of the Earth. Similarly as in the case of the Michelson–Morley
experiment, the Michelson–Gale version compared the light from a single source
after two directional travel. The difference was replacement of the two the
Michelson–Morley arms with two different rectangles. The obtained results were
compatible with both Special Relativity and models of stationary Æther.
However, Lorentz Æther theory contradicted the Michelson–Morley experiment,
and therefore the Michelson–Gale version was interpreted also as the
confirmation of correctness of Special Relativity.
The investigators of the Kennedy–Thorndike experiment also interpreted their
results in terms of relativistic effects. They modified the Michelson–Morley
experiment via application of the interferometer’s in which one arm is very
short in comparison with the second arm. the Michelson–Morley experiment
constructively verified the length contraction hypothesis, whereas in the
Kennedy–Thorndike version time dilation was straightforwardly examined. The
much shorter arm and maximal stabilization of the apparatus enabled the verify
existence of a specifical fringe shift, which theoretically should change of
the light frequency. According to theoretical predictions the shift resulted
from a change of speed of the Earth with respect to Æther, would result in
changes of time of light travel. Albeit, the shift was not detected, what was
interpreted as the confirmation of Special Relativity correctness.
The purpose of the Hammar experiment was emergence of the Æther drag asymmetry
by application of massive lead blocks on both side of only one the
Michelson–Morley interferometer’s arm. The idea of this investigation was
similar to the tests performed by O.J. Lodge. The blocks absence should result
in equal affect of both arms by Æther, while in the blocks presence the one
arm should be affected. Hammer reported independence of fringe displacements
on the blocks absence/presence, what was the argument against the Æther wind.
The conclusion is unambiguous. Stigmatization of Æther and Æther drag based on
application of the Ockham razor. The physical interpretation used _ad hoc_
absence of the Æther drag, and non-existence of Æther was concluded. The
purpose was intentional interpretation of experimental data supporting
elimination of Æther and related phenomena. In other words exclusion of one
thing was performed via imposing of absence of another one. However, the
matter is much more sophisticated because of in general the things must not be
correlated.
It is worth stressing that computed effects of the Æther drag, obtained from
diverse Æther models, were checked in numerous experiments. Since 1904 D.C.
Miller and collaborators [19] performed over 200,000 perspective empirical
investigations devoted to the verification. Basing on the results, Miller
propagated existence of the Æther wind, which effects were much smaller than
predictions due to stationary Æther. Miller’s work has been inspired a lot of
scholars and researchers [20]. In 1955 R.S. Shankland et al [21] devaluated
the Miller research, and claimed that the only statistical fluctuation due to
the local temperature conditions and systematic error generated the Miller
effect. However, in 1983 W. Broad et al suggested to review with attention the
results received by D.C. Miller, and straightforwardly negated the refusal due
to Shankland. Interestingly, R.A. Müller [22] constructively explained
anisotropies of the Cosmic Microwave Background Radiation via application of
the concept of the Æther drag.
###### Einstein’s Visions
We refer for detailed discussion of Einstein’s opinions which recently has
been performed by L. Kostro in his book [23]. It looks like that Einstein as
the creator of Special Relativity, against his will, was stigmatized as the
killer of Æther. This opinion was based on the intentional interpretation of
the very small fragment of his 1905 Special Relativity paper [24], which can
be found in introduction to this paper
The introduction of a ’luminiferous ether’ will prove to be superfluous
inasmuch as the view here to be developed will not require an <<absolutely
stationary space>> provided with special properties, nor assign a velocity-
vector to a point of the empty space in which electromagnetic processes take
place.
Nevertheless, he was also stigmatized as a resurrector of the Æther on the
base of _Sidelights of Relativity_ [25] in which one finds
We may say that according to the general theory of relativity space is endowed
with physical qualities; in this sense, therefore, there exists an aether.
According to the general theory of relativity space without aether is
unthinkable; for in such space there not only would be no propagation of
light, but also no possibility of existence for standards of space and time
(measuring-rods and clocks), nor therefore any space-time intervals in the
physical sense. But this aether may not be thought of as endowed with the
quality characteristic of ponderable media, as consisting of parts which may
be tracked through time. The idea of motion may not be applied to it.
Also in the article _Concerning the Aether_ Einstein supported Æther
When we speak here of aether, we are, of course, not referring to the
corporeal aether of mechanical wave-theory that underlines Newtonian
mechanics, whose individual points each have a velocity assigned to them. (…)
Instead of ’aether’, one could equally well speak of ’the physical quantities
of space’. (…) So we are effectively forced by the current state of things to
distinguish between matter and aether, even though we may hope that future
generations will transcend this dualistic conception and replace it with a
unified theory, as the field theoreticians of our day have tried in vain to
accomplish. (…) It is usually believed that aether is foreign to Newtonian
physics and that it was only the wave theory of light which introduced the
notion of an omnipresent medium influencing, and affected by, physical
phenomena. (…) Newtonian mechanics had its aether in the sense indicated,
albeit under the name absolute space. To get a clear understanding of this
and, at the same time, to explore more fully the concept of aether, we must
take a step back. (…) The kinematics, or phoronomy, of classical physics had
as little need of an aether as (physically interpreted) Euclidean geometry
has. (…) We will call this physical reality which enters the Newtonian law of
motion alongside the observable, ponderable real bodies, the aether of
mechanics. The occurrence of centrifugal effects with a (rotating) body, whose
material points do not change their distances from one another, shows that
this aether is not to be understood as a mere hallucination of the Newtonian
theory, but rather that it corresponds to something real that exists in
nature. (…) The mechanical aether - which Newton called absolute space - must
remain for us a physical reality. Of course, one must not be tempted by the
expression aether into thinking that, like the physicists of the 19th century,
we have in mind something analogous to ponderable matter. (…) When Newton
referred to the space of physics as absolute, he was thinking of yet another
property of what we call here aether. Every physical thing influences others
and is, it its turn, generally influenced by other things. This does not
however apply to the aether of Newtonian mechanics. For the inertia-giving
property of this aether is, according to classical mechanics, not susceptible
to any influence, neither from the configuration of matter nor anything else.
Hence the term absolute. (…) Viewed historically, the aether hypothesis has
emerged in its present form by a process of sublimation from the mechanical
aether hypothesis of optics. After long and fruitless efforts, physicists
became convinced that light was not to be understood as the motion of an
inertial, elastic medium, that the electromagnetic fields of Maxwells theory
could not be construed as mechanical. So under the pressure of this failure,
the electromagnetic fields had gradually come to be regarded as the final,
irreducible physical reality, as states of the aether, impervious to further
explanation. (…) While at least in Newtonian mechanics all inertial systems
were equivalent, it seemed that, in the Maxwell-Lorentz theory, the state of
motion of the preferred coordinate system (at rest with respect to the aether)
was completely determined. It was accepted implicitly that this preferred
coordinate system was also an inertial system, i.e. that the principle of
inertia [Newtons first law] applied relative to the electromagnetic aether.
(…) No longer was a special state of motion to be ascribed to the
electromagnetic aether. Now, like the aether of classical mechanics, it
resulted not in the favoring of a particular state of motion, only the
favoring of a particular state of acceleration. Because it was no longer
possible to speak, in any absolute sense, of simultaneous states at different
locations in the aether, the aether became, as it were, four dimensional,
since there was no objective way of ordering its states by time alone.
According to special relativity too, the aether was absolute, since its
influence on inertia and the propagation of light was thought of as being
itself independent of physical influence. (…) Thus geometry, like dynamics,
came to depend on the aether. (…) Thus the aether of general relativity
differs from those of classical mechanics and special relativity in that it is
not absolute but determined, in its locally variable characteristics, by
ponderable matter. (…) On the one hand, the metric tensor, which codetermines
the phenomena of gravitation and inertia and, on the other, the tensor of the
electromagnetic field appear still as different expressions of the state of
the aether, whose logical independence one is inclined to attribute rather to
the incompleteness of our theoretical ediface than to a complex structure of
reality. (…) But even if these possibilities do mature into an actual theory,
we will not be able to do without the aether in theoretical physics, that is,
a continuum endowed with physical properties; for general relativity, to whose
fundamental viewpoints physicists will always hold fast, rules out direct
action at a distance. But every theory of local action assumes continuous
fields, and thus also the existence of an aether.
Therefore, Einstein’s point of view was that Æther is the core fundament of
physics. In fact, he never neglected and negated Æther existence, and moreover
he developed this concept. In the famous book coauthored with L. Infeld [26],
he presented development of physics with respect to the concept of Æther.
Several fragments are cited below
Our picture of ether might very probably be something like the mechanical
picture of a gas that explains the propagation of sound waves. It would be
much more difficult to form a picture of ether carrying transverse waves. To
imagine a jelly as a medium made up of particles in such a way that transverse
waves are propagated by means of it is no easy task. (…) Yet we know from
mechanics that interstellar space does not resist the motion of material
bodies. The planets, for example, travel through the ether-jelly without
encountering any resistance such as a material medium would offer to their
motion. If ether does not disturb matter in its motion, there can be no
interaction between particles of ether and particles of matter. Light passes
through ether and also through glass and water, but its velocity is changed in
the latter substances. How can this fact be explained mechanically? Apparently
only by assuming some interaction between ether particles and matter
particles. We have just seen that in the case of freely moving bodies such
interactions must be assumed not to exist. In other words, there is
interaction between ether and matter in optical phenomena, but none in
mechanical phenomena! This is certainly a very paradoxical conclusion! (…) We
may still use the word ether, but only to express some physical property of
space. This word ether has changed its meaning many times in the development
of science. At the moment it no longer stands for a medium built up of
particles. Its story, by no means finished, is continued by the relativity
theory. (…) For the time being, we shall continue to believe that the ether is
a medium through which electromagnetic waves, and thus also light waves, are
propagated, even though we are fully aware of the many difficulties connected
with its mechanical structure.
###### Theoretical Objections
It is evident that Einstein’s attempts were focused on explanation of a whole
Universe via using of the concept of <<non physical>> Æther. In Special
Relativity he introduced the non-Euclidean Minkowski Space-time that agreed
with the Lorentz transformations. Recall that Lorentz performed Æther-based
deduction of these transformations. It looks like that Einstein manifestly
swept out Æther under space-time carpet, and unexpectedly reinvented Æther
when his position was consolidated. Possibly, the impacting personality and
authority of H.A. Lorentz caused such a situation. Saying <<non physical>> in
the context of Æther looks rather like diplomacy then physics. Unfortunately,
physics is often based on diplomatic truth, and diplomacy is limitlessly
applied within physics.
The negating, which is irrelevant to General Relativity, manifestly supports
Special Relativity. The empirical results, however, can be reinterpreted as
the support of Æther existence and the phenomenon of the Æther drag. Perhaps
the concept of electromagnetic Luminiferous Æther is incorrect, but in general
it does not exclude another form or forms of Æther. For instance Æther treated
as the primordial cause can exist only initially and must not exist in later
stages of Universe evolution. The best example of theoretical investigation of
Æther was done more than 40 years ago. In mid-1920’s E. Cartan [27] by
application of connections formulated General Relativity in terms of Newtonian
dynamics. In 1966 A.M. Trautman [28] showed by straightforward calculation
that the Einstein field equations are the special case of the Newtonian
gravitation equations coupled to a thing which Trautman called Luminiferous
Æther. Soon after these results C.W. Misner in _Gravitation_ coauthored with
K.S. Thorne and J.A. Wheeler [29] axiomatized the Trautman approach to show
that Newtonian dynamics consistently joints General Relativity with the
Cartan–Trautman Æther.
P.C.W. Davies [30] interviewed J.S. Bell, one of founders of quantum physics
and originator of Bell’s theorem/inequality. Bell straightforwardly expressed
the opinion that the concept of Æther can be very useful tool in resolving the
Einstein–Podolsky–Rosen paradox, regarding measurements of microscopic
systems, by involving a reference frame in which signals go faster than light.
In his view the Lorentz length contraction is correct but inconsistent with
Special Relativity, but can result in the theory of Æther which is consistent
with the results of Michelson–Morley experiment. Bell manifestly stated
wrongness of rejection of the concept of Æther from physics, and proposed
resurrection of the Æther because of a number of unsolvable issues is very
easy to solve by imaging of existence of Æther. In [31] Bell discussed Æther.
Another attempt, which is good candidate for the model of Æther, was made by
R.P. Feynman [32]. Feynman proposed that the partial-differential equations
are able to describe classical macroscopic motion of X-ons, i.e. certain very
small entities. The medium created by these entities can be treated as the
model of Æther. Similarly the action on distance approach to electrodynamics,
proposed by Wheeler and Feynman [33], gives great hopes for Æther.
In this manner the situation of Æther is non-established. Moreover,
intentional interpretation of experimental data, so widely applied by
antagonists of Æther, enables to consider Æther as a physical being. On the
other hand, as suggested J. Bell, Æther may be a helpful tool in constructive
and consistent explanation of numerous phenomena and effects. A number of
heightened attempts manifestly rejecting Æther from description of Nature is
based on the methodology which in itself is the selection and propagating of
preferred interpretation. In fact, all presented justifications of non-
existence of Æther are easy to straightforward invalidation by suing of the
theoretical as well as the empirical arguments. The question is whether the
physical truth should be technocratic or diplomatic. Of course, Nature is
neither technocratic nor diplomatic, and with no ideological constraints tells
what is the truth.
###### Dirac Æther
In 1951 P.A.M. Dirac, regarded by Einstein the founder of relativistic quantum
mechanics and quantum field theory, concluded existence of Æther reflecting
the nature of four-velocity in the context of theory of electrons following
from his new electrodynamics [34]
It was soon found that the existence of an æther could not be fitted in with
relativity, and since relativity was well established, the æther was
abandoned. (…) If one reexamines the question in the light of present-day
knowledge, one finds that the æther is no longer ruled out by relativity, and
good reasons can now be advanced for postulating an æther. (…) at the present
time it needs modification, because we have to apply quantum mechanics to the
æther. The velocity of the æther, like other physical variables, is subject to
uncertainty relations. For a particular physical state the velocity of the
æther at a certain point of space-time will not usually be a well-defined
quantity, but will be distributed over various possible values according to a
probability law obtained by taking the square of the modulus of a wave
function. We may set up a wave function which makes all values for the
velocity of the æther equally probable. Such a wave function may well
represent the perfect vacuum state in accordance with the principle of
relativity. (…) A thing which cannot be symmetrical in the classical model may
very well be symmetrical after quantization. This provides a means of
reconciling the disturbance of Lorentz symmetry in space-time produced by the
existence of an æther with the principle of relativity. (…) we may very well
have an æther, subject to quantum mechanics and conforming to relativity,
provided we are willing to consider the perfect vacuum as an idealized state,
not attainable in practice. (…) We have now the velocity at all points of
space-time, playing a fundamental part in electrodynamics. It is natural to
regard it as the velocity of some real physical thing. Thus with the new
theory of electrodynamics we are rather forced to have an æther.
One year later Nature magazine published interesting looking polemics between
L. Infeld and P.A.M. Dirac [35]. Infeld jointed the formulas of Dirac in a
certain intentional way and used of rather laconic then logical arguments to
point out that the new electrodynamics does not need the concept of Æther if
<<all>> its conclusions will be accepted. Infeld, as the typical
representative of the scholastics of Soviet block, did not precise what means
the word <<all>> in such a context. In other words, factually even Infeld did
not accept <<all>> conclusions of the Dirac electrodynamics and selected the
only these ones which were adequate for elimination of Dirac Æther. In this
manner Infeld used intentional interpretation to argue his beliefs, and did
not focus attention on the physical aspects of Dirac Æther. Moreover, such a
negative opinion was also straightforwardly opposite to the efforts of Albert
Einstein which nota bene were supported by Infeld several years earlier.
Deduction of the existence of Æther performed by Dirac used purely formal
aspects of the Hamiltonian approach to the Maxwell electrodynamics, i.e. the
constraints and the action. Recall that over 25 years earlier Dirac discovered
the linkage between classical and quantum mechanics via the Poisson brackets
correspondence. Maxwell electrodynamics was unsatisfactory formulated because
of the correspondence works for the only Hamiltonian version of a classical
theory. Dirac performed the Hamiltonian formulation of Maxwell
electrodynamics, and discerned Æther in this theory. Applying the typical
arguments of the Ockham razor based on personal beliefs, L. Infeld manifestly
discredited the purposes and efforts due to P.A.M. Dirac.
Such a situation strengthened the Dirac strategy which was a follower of the
revisionist approach with respect to even well-established and accepted
physical knowledge. The selfish and unnatural selection performed by Infeld
quickly obtained an adequate and constructive reply due to Dirac. The reply is
brief and can be cited entirely
Infeld has shown how the field equations of my new electrodynamics can be
written so as not to require an æther. This is not sufficient to make a
complete dynamical theory. It is necessary to set up an action principle and
to get a Hamiltonian formulation of the equations suitable for quantization
purposes, and for this the æther velocity is required.
The existence of an æther has not been proved, of course, because of my new
electrodynamics has not yet justified itself. It will probably have to be
modified by the introduction of spin variables before a satisfactory quantum
theory of electrons can be obtained from it, and only after this has been
accomplished will one to be able to give a definite answer to the æther
question.
The method of L. Infeld was the tip of the iceberg and reflected the true
countenance of the regional standpoint based on the fossilized traditional
beliefs. At this time Soviet school of physics dominated European science and
the Marxist–Leninist scholastics was one of the most fashionable streams. Such
people like P.A.M. Dirac and A. Einstein were the pioneers who wanted to
change this manifestly irrational status quo. In fact, the opinion due to L.
Infeld about Dirac Æther straightforwardly crossed the efforts of both the
Nobel laureates. Albeit, above all Infeld negated his own opinions published
several years earlier together with Albert Einstein. This controversy was too
serious for the science of the region and, in fact, resulted in some kind of
hackwork within the Polish physics. Evidently seen lack of Nobel Prizes in
Natural Sciences in Poland is the most gross syndrome of the fossilized
reasoning and approach to science, and labels the civilization stagnation.
Recall that Infeld was the only one of 11 signatories to 1955’s
Russell–Einstein Manifesto who never received a Nobel Prize. His creative
efforts with respect to Poland started when in 1950 he left Canada, where by
12 years worked at the University of Toronto. He came back to communist Poland
and decided to help in reconstruction of the Polish science which during World
War II lost few generations of scholars. Admittedly his dictatorial approach
to science resulted in tremendous contribution to the Warsaw school of
physics, but transformed this school into the sanctuary of the Soviet
scholastics.
###### Zero Point Energy & Planck Scale
In 1900 Max Planck [36] published the revolutionary formula for energy of a
single vibrating atom. Several years after, in 1913, Einstein together with
his another collaborator O. Stern [37] modified the Planck formula by
involving of the concept of cosmic heat bath. This concept was directly
related to the universal frequency field associated with the Zero Point
Energy. From the modern point of view one can say that Einstein and Stern
renormalized the Planck energy at at absolute zero temperature via using of
the residual oscillating energy. In fact, the cosmic bath heat is the model of
Æther which explains numerous experimental data, like e.g. the Casimir effect,
the Lamb shift.
Factually, the Maxwell model of Æther has never been experimentally refuted.
In the context of quantum geometrodynamics due to J.A. Wheeler [38] Maxwell
vacuum can be regarded as quantum foams, i.e. a subquantum sea of Zero Point
Energy fluctuations. According to Wheeler space-time warps, called wormholes,
follow from tremendous densities of local energy due to the high energetic
modes of the universal frequency field. Wormholes are the tunnels transmit
electricity between two separate spatial places or, in more general context,
between different universes creating the superspace – configurational space of
General Relativity. Wheeler proposed to think in terms of mini holes, i.e.
primitive charged particles, as the wormholes related to the local space. In
his view the electricity goes orthogonally via our universe from a fourth
dimension. The mechanism of electron-positron pairs production follows by
black and white mini holes.
Interestingly, H. Aspden [39] proposed the hadronic model based on near-
balanced continuum and quons, i.e. massless Æther particles giving a charge
and condensing electron-positron pairs. Such a line of thinking was prolonged
also by H. Puthoff [40] who applied quantum theory to redefinition of the Zero
Point Energy hypothesis.
Quantum mechanics is also referred as a theory of Æther based on quantum
foams, leading to fluctuations of small scales which generate quick creations
and annihilations of particle pairs. In modern cosmology, the fifth element
unifying other ones, called oftentimes Dark Energy or quintessence, has been
considered as Zero-Point Field or quantum vacuum and already identified with
Æther by B.G. Sidharth [41]. Recently, also Einstein Æther theory, i.e.
generally covariant generalization of General Relativity describing space-time
and endowed both a metric and a unit time-like vector field manifestly
violating Lorentz invariance, has became popular (See e.g. papers in the Ref.
[42]).
#### Multiverse
###### Epicurus and Eastern Cosmologies
Epicurus [43] was probably the first ancient philosopher who openly propagated
the concept of Multiverse. In his Letter to Herodotus one finds manifestly
expressed his standpoint
Moreover, there is an infinite number of worlds, some like this world, others
unlike it. For the atoms being infinite in number, as has just been proved,
are borne ever further in their course. For the atoms out of which a world
might arise, or by which a world might be formed, have not all been expended
on one world or a finite number of worlds, whether like or unlike this one.
Hence there will be nothing to hinder an infinity of worlds.
(…) After the foregoing we have next to consider that the worlds and every
finite aggregate which bears a strong resemblance to things we commonly see
have arisen out of the infinite. For all these, whether small or great, have
been separated off from special conglomerations of atoms; and all things are
again dissolved, some faster, some slower, some through the action of one set
of causes, others through the action of another.
And further, we must not suppose that the worlds have necessarily one and the
same shape. For nobody can prove that in one sort of world there might not be
contained, whereas in another sort of world there could not possibly be, the
seeds out of which animals and plants arise and all the rest of the things we
see.
Al-Qur’an, the holy book of Islam, also directly refers to multiple worlds.
Sūratu Al-Fātihah, called <<The Seven Verses of Repetition>>, translated into
English language by Hafiz Abdullah Yusuf Ali [44] sounds
${}^{1}\textit{\leavevmode\nobreak\ In\leavevmode\nobreak\
the\leavevmode\nobreak\ name\leavevmode\nobreak\ of\leavevmode\nobreak\
Allah,\leavevmode\nobreak\ Most\leavevmode\nobreak\
Gracious,\leavevmode\nobreak\ Most\leavevmode\nobreak\ Merciful.}$
${}^{2}\textit{\leavevmode\nobreak\ Praise\leavevmode\nobreak\
be\leavevmode\nobreak\ to\leavevmode\nobreak\ Allah,\leavevmode\nobreak\
the\leavevmode\nobreak\ Cherisher\leavevmode\nobreak\ and\leavevmode\nobreak\
Sustainer\leavevmode\nobreak\ of\leavevmode\nobreak\ the\leavevmode\nobreak\
worlds!}$ ${}^{3}\textit{\leavevmode\nobreak\ Most\leavevmode\nobreak\
Gracious,\leavevmode\nobreak\ Most\leavevmode\nobreak\ Merciful.}$
${}^{4}\textit{\leavevmode\nobreak\ Master\leavevmode\nobreak\
of\leavevmode\nobreak\ the\leavevmode\nobreak\ Day\leavevmode\nobreak\
of\leavevmode\nobreak\ Judgement.}$ ${}^{5}\textit{\leavevmode\nobreak\
Thee\leavevmode\nobreak\ we\leavevmode\nobreak\ do\leavevmode\nobreak\
worship,\leavevmode\nobreak\ and\leavevmode\nobreak\ Thine\leavevmode\nobreak\
aid\leavevmode\nobreak\ we\leavevmode\nobreak\ seek.}$
${}^{6}\textit{\leavevmode\nobreak\ Show\leavevmode\nobreak\
us\leavevmode\nobreak\ the\leavevmode\nobreak\ straight\leavevmode\nobreak\
way,}$ ${}^{7}\textit{\leavevmode\nobreak\ The\leavevmode\nobreak\
way\leavevmode\nobreak\ of\leavevmode\nobreak\ those\leavevmode\nobreak\
on\leavevmode\nobreak\ whom\leavevmode\nobreak\ Thou\leavevmode\nobreak\
hast\leavevmode\nobreak\ bestowed\leavevmode\nobreak\ Thy\leavevmode\nobreak\
Grace,}$ ${}^{\leavevmode\nobreak\ }\textit{\leavevmode\nobreak\
those\leavevmode\nobreak\ whose\leavevmode\nobreak\
(portion)\leavevmode\nobreak\ is\leavevmode\nobreak\ not\leavevmode\nobreak\
wrath,\leavevmode\nobreak\ nor\leavevmode\nobreak\ of\leavevmode\nobreak\
those\leavevmode\nobreak\ who\leavevmode\nobreak\ go\leavevmode\nobreak\
astray.}$
One of the thinkers and philosophers straightforwardly inspired by Al-Qur’an
was the muslim polymath Fakhr Al-Din Al-Razi. His point of view rejected the
Aristotelian-Avicennian single universe revolving around a single world. A.
Setia [45] referred fragments of the unpublished manuscript _al-Matalib
al-’Aliyah_ of Razi
It is established by evidence that there exists beyond a void without a
terminal limit, and it is established as well by evidence that God Most High
has power over all contingent beings. Therefore he the Most High has the power
to create a thousand thousand worlds beyond this world such that each one of
those worlds be bigger and more massive than this world as well as having the
like of what this world has of the throne, the chair, the heavens,and the
earth, and the sun and the moon. The arguments of the philosophers for
establishing that the world is one are weak, flimsy arguments founded upon
feeble premises.
Multiverse is also present in Puranas, the generic texts of Hinduism, Jainism
or Buddhism. For example in Bhagavata Purana 9.4.56 one finds the direct
reference to multiple universes
Lord Śiva said: My dear son, I, Lord Brahmā and the other devas, who rotate
within this universe under the misconception of our greatness, cannot exhibit
any power to compete with the Supreme Personality of Godhead, for innumerable
universes and their inhabitants come into existence and are annihilated by the
simple direction of the Lord.
###### Modal Realism
In the most general formulation the Multiverse hypothesis takes into account
the scenario in which there exists, numerable or innumerable, collection of
multiple possible universes. These worlds may include a whole Nature, the
concepts of space, time, matter, light, and even its psychological aspects
related to the concept of mind. Multiverse as the scientific concept was
introduced by American psychologist and philosopher W. James [46], who
included into human psychology the influence of divine and mystic experiences.
The structure of Multiverse, which in fact defines the nature of a possible
universe as well as the various relationships between distinguishable
universes, are not rigidly established and manifestly depend on a model of
Multiverse.
From the philosophical point of view the concept of Multiverse naturally
belongs to the logical system investigated by L. Wittgenstein in his famous
_Tractatus Logico-Philosophicus_ [47]. In this logic the logical truth is
defined as a statement true in all possible worlds or under all possible
interpretations, and a fact is only true in this world as it has historically
unfolded. This ontological system continues the program investigated by G.
Frege [48], but manifestly neglects the Frege axiom which semantic form is
_all true (and, similarly all false) sentences describe the same state of
affairs, that is, they have a common referent_. The pioneering formalization
of Wittgenstein’s _Tractatus_ was performed by Polish logician R. Suszko [49],
and resulted in so called Non-Fregean Logic in which there are no theorems
asserting how many semantic correlates of sentences there can be. Recently
this logic has been expanded by M. Omyła [50] onto the logic connecting
situations and objects. Wittgensteinian metaphysics, however, leads to
emergence of identical objects existence in diverse worlds.
Counterpart theory of D.K. Lewis [51] showed that such objects should be
regarded as similar rather than identical. Lewis elucidated the role of
probability and hypothetical statements. His version of modal realism led to
all possible worlds possessing equally realistic character like the actual
world. In _Parts of Classes_ , Lewis applied the pluralistic approach to the
foundations of mathematics. He considered such issues like set theory, the
Peano arithmetic, and the Gödel incompleteness theorems to mereology and
plural quantification. In Lewis’s approach such a word like ”actual” is merely
indexing procedure, labeling of position within a world. He proposed also the
definition of truth, strictly based on Multiverse nature of modal realism,
which states that things are necessarily true when they are true in all
possible worlds. Lewis was not the first philosopher studying possible worlds,
but contributed the essential idea about equally concreteness of all possible
worlds, and created the concept of the world in which an existence of the
object is no more real than an existence of this object in different possible
world. Similarly as in the case Æther the concept of Multiverse manifestly
violates the maxim due to English theologian and a member of the the mendicant
Order of Friars Minor (Franciscan) William Ockham, called the Ockham Razor.
The Ockham Razor says <<no>> to multiply entities, because of the Multiverse
hypothesis is beyond being a necessary explanation of the facts which theories
want to describe.
Different possible worlds are propagated also by modern American philosopher
and logician S.A. Kripke [52]. He has described modality via using of a
metaphysical route, and employed them to semantics, what resulted in so called
Kripke’s theory of truth. In this theory a natural language contains its own
truth predicate without rising contradiction. Involving the property of
partial definition of truth over the set of grammatically well-formed
sentences in the language, Kripke recursively showed that a language can
consistently contain its own truth predicate. In other words Kripke negated
the impossibility of such a situation deduced by A. Tarski [53]. In Kripke’s
view truth predicate adds new sentences to the language and truth is the union
of all the elements, i.e. is in turn defined for all of them. Infinite number
of steps establishes ”fixed point” in the language, which can be treated as
the fundamental natural language containing its own truth predicate.
Another pluralistic formulations of the problem of truth involve
correspondence, coherence and constructiveness. The approach due to C. Wright
[54] proposes that truth must not be a single discourse-invariant analog of
identity, and that there are the only certain principles of application the
truth predicate to a sentence, i.e. some platitudes about true sentences.
Wright emphasizes the crucial role of the context, and defines a truth
predicate as superassertible if and only if it is assertible in a certain
state of information. He did not proposed any mechanism for improving or
growing of such a state of information. Because arbitrary standards, norms,
and habits question the discourse, he gives the fundamental role for
assertiveness. The approach proposed by M.P. Lynch [55] claims that truth is
multiply functional property. In _Truth in Context_ he proposed a path where
metaphysical pluralism is consistent with robust realism about truth. His
studies on investigated so called _relativistic Kantianism_ , i.e. taking of
facts and propositions as relative without implications about relativity of an
ordinary truth. According to Lynch truths are relative, but individual
concepts of truth must not be. In _True to Life_ Lynch discussed basic truisms
about truth: objectivity, goodness, and arising by worthiness of requesting.
He considered mental origins of cynicism, and presented inadequacy of numerous
theories of truth. Lynch defends caring about truth, and argues that truth has
real value for a happy life.
H.N. Goodman [56] gave far from modal realism contributions to the Multiverse
hypothesis. He exalted artistry in human-world cognitive relationship, and
argued artworks as symbols referring and constructing diverse worlds. Because
any human activity is an artistry the Goodman approach is general. According
to this the interpretation is fundamentally unified to the world via the
symbols. The worlds demand interpretation of the symbols they contribute to
construct, and with no interpretation of the symbols the worlds do exist.
Perception, understanding, experience, and discovering use symbols. The
interest in symbols is cognitive, what Goodman advocates as cognitivism.
###### Many-Worlds Interpretation
The straightforward philosophical implication of the Multiverse hypothesis to
theoretical physics is the metatheory of quantum theory called _relative
state_ (RS) formulation of quantum mechanics. The foundations of this theory
were presented by H. Everett [57] in his doctoral dissertation supervised by
B.S. DeWitt. On pp. 8-9, within the introductory part of the Everett thesis,
one can find the statement
Since the universal validity of the state function description is asserted,
one can regard the state functions themselves as the fundamental entities, and
one can even consider the state function of the entire universe. In this sense
this theory can be called the theory of the ”universal wave function,” since
all of physics is presumed to follow from this function alone.
By direct application of the classical mechanical procedure for defining
probability, Everett derived the Born rule describing probabilities in quantum
mechanics and proved its universality. DeWitt [58] reincarnated relative state
formulation as Many-Worlds Interpretation (MWI), and together with his another
PhD student R.N. Graham alternatively derived the Born rule showing that for
infinite number of worlds, i.e. in the situation for which the statistical
laws of quantum theory are inadequate, their norm becomes infinite. Everett’s
thesis, M. Born paper and several another papers on MWI are collected in the
book [59].
In addition A.M. Gleason [60] and J.B. Hartle [61] independently obtained the
results of Everett’s thesis. J.B. Hartle and S.W. Hawking [62] used MWI
results to description of initial conditions for Big Bang cosmology by the
solution of Wheeler–DeWitt equation. Applying cold chaotic/eternal inflation
A. Linde [63] proposed the first Multiverse cosmology, where randomly emerging
events have independent initial conditions, and partially nucleate in space-
time foam as bubbles. J.S. Bell in [31] straightforwardly supported Many-
Worlds Interpretation. The direct cosmological results following from MWI
context so called is Anthropic Cosmological Principle, discussed by in the
book of J.D. Barrow & F.J. Tipler [64], and in the book of Tipler [65]. In the
book edited by Penrose & Isham one finds topical Tipler article in which he
expresses the following opinion about Many-Worlds Interpretation
I then asked, ’Who does not believe in the many-worlds interpretation?’. About
30 hands went up (including those of Roger Penrose and Bob Wald); clearly
Bryce [DeWitt], David [Deutsch] (and I) were in a minority at this meeting.
Finally I asked, ’Who is neutral on the many-worlds interpretation?’. The
remaining 20 hands went up. I shall do my best in this short paper to persuade
both the sceptics and those have not yet formed an opinion as to the validity
of the many-worlds interpretation (MWI) that this interpretation is
philosophically more beautiful than competing interpretations, and that it can
be used in quantum cosmology as a powerful tool not only to interpret the wave
function of the universe, but also to give us some information about the
equation which this wave function obeys. (…) Most sceptics, I’ve found, have a
mistaken idea of what the MWI really means, so it behoves me to review it
briefly. The MWI is a theory of measurement, so it is concerned with
describing how the universe looks to us qua human beings. At some stage during
any measurement, the information is digitalized, and this is true even for the
measurement of continuous variables, for example position or momentum. A
typical position measurement of an a-particle nucleus, say would be carried
out by letting the a-particle pass through an array of atoms such as those of
a photographic plate. The array cannot make position measurements of unlimited
accuracy; at best, the accuracy would be limited by the size of the atom. Even
if we were to improve the accuracy of the position measurement at this level
of the measuring process, the position measurement would in the end be
digitalized when it is transmitted to human beings, for the data corresponding
to an arbitrarily precise position measurement would in general exceed the
storage capacity of a human brain. Hence we can model any measurement by a
measurement of a discrete variable.
Many-Worlds Interpretation possesses numerous applications and references.
J.R. Gribbin [66] discussed Schrödinger’s cat paradox and Multiverse. M.
Lockwood [67], M. Gell-Mann & J.B. Hartle [68], D. Albert [69], R. Penrose
[70], D.J. Chalmers [71], and D.E. Deutsch [72] attempted to construct theory
of evolution directly based on MWI. M. Kaku [73] applied the idea of parallel
worlds, i.e. worlds within the Multiverse, to speculations within String
Theory. R. Plaga [74] proposed the empirical test of Many-Worlds
Interpretation. J.A. Barrett [75] discussed details of both the Everett and
”no collapse” interpretations of quantum mechanics. Deutsch [76] also
suggested to examine MWI by using of quantum computer. Applying MWI he derived
the information-theoretic Born rule, and determined Multiverse by information
flow. D. Page [77] sees the essential support of MWI in cosmological
observations. L. Polley [78] derived the Born rule by symmetry arguments
instead of Deutsch’s assumptions. Derivation of the Born rule by W.H. Zurek
[79] involved envariance, while he deduced probabilities from entanglement. In
another considerations [80] Zurek discussed also the problem of causality,
interaction with environment, and what he calls quantum darwinism describing
proliferation, in the environment, of multiple records of selected states of a
quantum system. The arguments due to Deutsch were improved on by D. Wallace
[81] and S. Saunders [82]. J.A. Wheeler [83] also expressed his views on
Everettian relative state. L. Smolin [84] and B. Greene [85] supported
Multiverse in the context of String Theory. M. Gardner [86] did critical
analysis of Many-Worlds Interpretation. C. Bruce [87] focused on Schrödinger’s
cat paradox, and L. Randall [88] studied the context of brane worlds. Various
experienced scientists and scholars expressed their opinions about Multiverse
in the book edited by B. Carr [89]. M. Tegmark [90] provided the
classification of multiple universes. P. Byrne [91] performed a detailed
analysis of the Everett heritage. Interesting looking studies which look like
topically have been presented recently by V. Allori et al [92] and by S.
Osnaghi et al [93]. A. Jenkins & G. Perez [94], and J. Feng & M. Trodden [95]
have discussed the observational context of MWI. Also recently I have
supported a certain particular context of the Multiverse hypothesis [96].
There are also another various intriguing contexts of Many-World
Interpretation and Multiverse. Interestingly recently, A. Kent [97] and N.P.
Landsman [98] have criticized the foundations of the Born rule applied in the
context of Many-Worlds Interpretation. The constructive application of MWI was
performed by D. Parfit [99], who discussed the concept of personal identity.
His conclusions and deductions are essentially intriguing and are questioning
the fundamental status quo of mental health. Namely, Parfit presented certain
sample situations in which a unified person splits into several copies, and
justified ambiguousness in fixation of the state of personality. He concluded
that dividing of ”I” does leads to inadequacy of the concept of personal
identity, which is the most celebrated and well-established concept of
psychology and psychiatry. However, productive introduction of medical norms
and social standards based on MWI is rather far perspective.
###### String Theory and Anthropic Principle
The Anthropic Principle leads to straightforward various implications of Many-
World Interpretation and Multiverse hypothesis within String Theory. The
contributions based on or related to Multiverse and MWI were presented by S.
Weinberg [100], G. ’t Hooft [101], S.W. Hawking [102], M.J. Rees [103], J.D.
Bekenstein [104], and L. Susskind [105]. Weinberg, in his famous and
intriguing book _Dreams of a Final Theory_ [106], unambiguously and manifestly
expressed the beliefs which are strictly related to the Many-Worlds
Interpretation
The final approach is to take the Schrodinger equation seriously (…) In this
way, a measurement causes the history of the universe for practical purposes
to diverge into different non-interacting tracks, one for each possible value
of the measured quantity. (…) I prefer this last approach.
There are also critical standpoints about the String Theory context of
Multiverse hypothesis. Smolin in his another book [107] is too critical
The search for quantum gravity is a true quest. The pioneers were explorers in
a new landscape of ideas and possible worlds. (…) The scenario of many
unobserved universes plays the same logical role as the scenario of an
intelligent designer. Each provides an untestable hypothesis that, if true,
makes something improbable seem quite probable. (…) The anthropic principle
that Susskind refers to is an old idea proposed and explored by cosmologists
since the 1970s, dealing with the fact that life can arise only in an
extremely narrow range of all possible physical parameters and yet, oddly
enough, here we are, as though the universe had been designed to accommodate
us (hence the term "anthropic"). The specific version that Susskind invokes is
a cosmological scenario that has been advocated by Andrei Linde for some time,
called eternal inflation. According to this scenario, the rapidly inflating
phase of the early universe gave rise not to one but to an infinite population
of universes. You can think of the primordial state of the universe as a phase
that is exponentially expanding and never stops. Bubbles appear in it, and in
these places the expansion slows dramatically. Our world is one of those
bubbles, but there are an infinite number of others. To this scenario,
Susskind adds the idea that when a bubble forms, one of the vast number of
string theories is chosen by some natural process to govern that universe. The
result is a vast population of universes, each of which is governed by a
string theory randomly chosen from the landscape of theories. Somewhere in the
so-called multiverse is every possible theory in the landscape. (…) I find it
unfortunate that Susskind and others have embraced the anthropic principle,
because it has been understood for some time that it is a very poor basis for
doing science. Since every possible theory governs some part of the
multiverse, we can make very few predictions. (…) It is not hard to see why.
To make a prediction in a theory that posits a vast population of universes
satisfying randomly chosen laws, we would first have to write down all the
things we know about our own universe. These things would apply to some number
of other universes as well, and we can refer to the subset of universes where
these facts are true as _possibly true universes_.
P. Woit in his book _Not Even Wrong_ [108] is critical too, but less radical
The anthropic principle comes in various versions, but they all involve the
fact that the laws of physics must be of a nature that allows the development
of intelligent beings such a ourselves. Many scientists believe that this is
nothing more than a tautology, which while true, can never be used to create a
falsifiable prediction, and thus can not be part of scientific reasoning.
Controversy has arisen as a significant group of superstring theorists have
begun to argue that superstring theory’s inability to make predictions is not
a problem with the theory, but a reflection of the true nature of the
universe. (…) Weinberg suggested that perhaps the explanation of the problem
of the small size of the cosmological constant was the anthropic principle.
The idea is that there are huge number of consistent possible universes, and
that our universe is part of some larger multiverse or megaverse. Quite
naturally, we find ourselves in a part of this multiverse in which galaxies
can be produced and thus intelligent life can evolve. If this is the case,
there is no hope of ever predicting the value of the cosmological constant,
since all one can do is note the tautology that it has a value consistent with
one’s existence.
Anyway, in theoretical physics the crucial question is whether arbitrary
mathematical creations can be applied to effective and constructive
description of the physical Reality. In this book I shall present
straightforwardly that certain selected ideas lead to such a constructive
approach. It must be emphasized that in general it is clear what is the best
argument for applicability of any mathematics to making of a constructive
physical scenarios. Namely, this is a problem of choice which always should be
verified empirically. A mathematics is physical if and only if it leads to a
physical truth, which is independent on various diplomatic operations. Always,
however, constructive failures are much more valuable then nonconstructive
successes. Regarding the opinion due to Wolfgang Pauli: _not even wrong_.
## Part I Lorentz Symmetry Violation
### Chapter 1 Deformed Special Relativity
Special Relativity can be formulated basing on the momentum space, in which
the Einstein energy-momentum relation holds
$E^{2}=m^{2}c^{4}+p^{2}c^{2},$ (1.1)
where $c$ is speed of light in vacuum, $p$ the momentum value of a
relativistic particle possessing mass $m$. Factually, the relation (1.1) can
be rewritten in the more conventional form
$E^{2}-p^{2}c^{2}-m^{2}c^{4}=0,$ (1.2)
which defines the Einstein Hamiltonian constraint of Special Relativity.
Solving of this constraint with respect to a particle energy $E$ leads to
fixation of the energy as the approximation. For Special Relativity the
Lorentz symmetry holds and (1.1), as a quadratic form on the Minkowski energy-
momentum space of a particle $p^{\mu}=[E,p^{i}c]$, is Lorentz invariant.
From the modal realism point of view, the Lorentz invariance of the constraint
(1.1) leads to the fundamental problem. Namely, the equation (1.1) is not the
only one possible such a quadratic form. Naturally, the constructive
generalization of Special Relativity is
$E^{2}=m^{2}c^{4}+p^{2}c^{2}+\Delta(E,p),$ (1.3)
where the deformation $\Delta(E,p)$ contains also a set of free parameters,
which I shall call deformation parameters. Such an extension, however,
determines the Multiverse in which the possible worlds are diverse Æther
theories, and such an Æther theory defines new physical beings and effects.
For Lorentz invariance a whole expression (1.3) must be a quadratic form on
the Minkowski space. This chapter presents the updated results of the author
paper [109]. Certain part of these results is removed and replaced by more
adequate ideas.
#### A The linear deformation
Let us consider first the deformation of Special Relativity due to a simple
linear term in a particle momentum $p$
$E^{2}=m^{2}c^{4}+p^{2}c^{2}+\mathcal{P}^{i}p_{i}c^{2},$ (1.4)
where $\mathcal{P}^{i}$, $i=1,2,3$ is a three-vector of deformation
parameters, which in fact is certain constant reference momentum 3-vector
distinguishing axes related to its direction. Let us introduce also the length
of the 3-momentum vector $\mathcal{P}^{i}$ as
$\mathcal{P}=\sqrt{\mathcal{P}^{i}\mathcal{P}_{i}}.$ (1.5)
It is not difficult to deduce that the deformation momentum three-vector
$\mathcal{P}^{i}$ can be physically understood as the Æther momentum vector.
Therefore, the linear deformation of Special Relativity (1.4) describes the
simplest type interaction between a particle and the Æther in the Minkowski
energy-momentum space . In such a situation the Æther is characterized by a
constant momentum vector field dealt by the direction of the three momentum
$\mathcal{P}^{i}$.
From the Minkowski space point of view the modified Special Relativity (1.4)
corresponds to nontrivially deformed invariant hyperboloid. The theory (1.4)
can be described by a quadratic form which by elementary algebraic
manipulation can be obtained from (1.4) as its canonical form
$E^{2}+\dfrac{1}{4}\mathcal{P}^{2}c^{2}=\left(\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c+\dfrac{\mathcal{P}c}{2}\right)^{2}+m^{2}c^{4},$
(1.6)
which can be worked out by multiple ways, i.e. in itself determines the
Multiverse of possible physical theories. By the identification method there
is a lot of routes of possible interpretations between all the parts of the
equation (1.6). Let us consider here the three basic interpretations.
##### A1 The Dirac equation and the new algebra
The first case is the following identification
$\left\\{\begin{array}[]{l}m^{2}c^{4}=\dfrac{\mathcal{P}^{2}c^{2}}{4}\\\
E^{2}=\left(\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c+\dfrac{\mathcal{P}c}{2}\right)^{2}\end{array}\right.,$
(1.7)
which leads to the mass values
$\pm mc=\dfrac{\mathcal{P}}{2},$ (1.8)
where we have included negative sign mass as physical. The second equation is
not difficult to solve. The solution determines energy as
$\gamma^{0}E=\gamma^{i}p_{i}c\pm mc^{2},$ (1.9)
where $\gamma$’s are defined by the following relations
$\displaystyle{\gamma^{0}}^{2}$ $\displaystyle=$ $\displaystyle 1,$ (1.10)
$\displaystyle\gamma^{i}$ $\displaystyle=$
$\displaystyle\dfrac{\mathcal{P}^{i}}{\mathcal{P}}.$ (1.11)
where the equality (1.11) for the present case (1.8) takes the form
$\gamma^{i}=\dfrac{\mathcal{P}^{i}}{2mc}.$ (1.12)
In other words from Eq. (1.11) one has $\mathcal{P}^{i}=\mathcal{P}\gamma^{i}$
what, after application of the usual raising the lower index
$\gamma_{i}=\delta_{ij}\gamma^{j}$, allows to write
$\mathcal{P}^{i}\mathcal{P}_{i}=\mathcal{P}^{2}\gamma^{i}\gamma_{i},$ (1.13)
i.e. for correctness with the definition (1.5) the identity
$\gamma^{i}\gamma_{i}\equiv 1$ must hold in arbitrary case. It is, albeit
blatantly incorrect if one treats $\gamma^{i}$ as the Dirac gamma matrices
obeying the Clifford algebra
$\left\\{\gamma^{i},\gamma^{j}\right\\}_{C}=2\delta^{ij},$ (1.14)
where $\delta_{ij}$ is $D\times D$ unit matrix, for
$\gamma^{i}\gamma_{i}=\delta^{i}_{i}=D$ and therefore in 3-dimensional space
$\gamma^{i}\gamma_{i}=3$. Factually such an algebraical treatment (1.14) was
investigated by W. Pauli [110], and is commonly used in both relativistic
quantum mechanics as well as quantum field theory [111]. Moreover, the
Clifford algebra is the fundamental computational rule in particle physics,
for instance for cross sections for reactions. Alternatively, the relations
(1.5) and (1.13) can be understood as the suggestion that the linear
deformation is true for the only $D=1$ dimensional space, i.e. that the Æther
exists only in one-dimensional space. It looks like, however, that such an
explanation is logically inconsistent and the Clifford algebra (1.14) must be
exchanged for the more adequate one.
Because, however, the definition (1.11) imparts the Æther momentum
noncommutative nature there is difference between expressions
$\mathcal{P}^{i}\mathcal{P}_{i}$ and $\mathcal{P}_{i}\mathcal{P}^{i}$ which in
the Abelian case are identical. By this reason let us introduce the definition
of the Æther momentum square including the noncommutative nature of
$\mathcal{P}^{i}$
$\mathcal{P}^{2}=\dfrac{1}{2}\left(\mathcal{P}^{i}\mathcal{P}_{i}+\mathcal{P}_{i}\mathcal{P}^{i}\right),$
(1.15)
which in the commutative situation leads to (1.5). Thus one obtains
$\mathcal{P}^{2}=\dfrac{1}{2}\left(\mathcal{P}^{2}\gamma^{i}\gamma_{i}+\mathcal{P}^{2}\gamma_{i}\gamma^{i}\right),$
(1.16)
what after using the identity $\gamma_{i}=\delta_{ij}\gamma^{j}$ gives
$\mathcal{P}^{2}=\dfrac{\mathcal{P}^{2}}{2}\delta_{ij}\left(\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}\right),$
(1.17)
and results in the basic relation
$1=\dfrac{1}{2}\delta_{ij}\left\\{\gamma^{i},\gamma^{j}\right\\},$ (1.18)
which can be expressed in more conventionally
$\left\\{\gamma^{i},\gamma^{j}\right\\}=\dfrac{2}{\delta_{ij}}.$ (1.19)
The Clifford algebra (1.14) can be reconstructed from (1.18) if and only if
one put by hands the identity $\delta^{ij}=\dfrac{1}{\delta_{ij}}$, i.e.
$\delta^{ij}\delta_{ij}=\delta^{i}_{i}=1$. However, such an algebraical
strategy is in general incorrect because of in the $D$ dimensional space case
$\dfrac{1}{\delta_{ij}}=\dfrac{\delta^{ij}}{\delta_{ij}\delta^{ij}}=\dfrac{\delta^{ij}}{\delta^{i}_{i}}=\dfrac{1}{D}\delta^{ij}$.
Strictly speaking it means that the spatial gamma matrices (1.11) do not
belong to the Clifford algebra for $D\neq 1$.
In itself, however, the obtained problem is solved by introduction of the new
algebra, which can be established straightforwardly and rather easy. Namely,
application of the relation $\dfrac{1}{\delta_{ij}}=\dfrac{1}{D}\delta^{ij}$
within the equation (1.19) results in the following anticommutator
$\left\\{\gamma^{i},\gamma^{j}\right\\}=\dfrac{2}{D}\delta^{ij},$ (1.20)
which for 3-dimensional case gives the rule
$\left\\{\gamma^{i},\gamma^{j}\right\\}=\dfrac{2}{3}\delta^{ij}.$ (1.21)
It is not difficult to see straightforwardly that the extension of spatial
gamma matrices to the space-time version
$\gamma^{i}\rightarrow\gamma^{\mu}=\left(-\gamma^{0},\gamma^{i}\right),$
(1.22)
leads to simple generalization of the basic formula (1.18)
$1=\dfrac{1}{2}\eta_{\mu\nu}\left\\{\gamma^{\mu},\gamma^{\nu}\right\\},$
(1.23)
where $\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$ is metric the Minkowski space-
time. Therefore, one obtains the new four-dimensional gamma matrix algebra
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=\dfrac{2}{4}\eta^{\mu\nu}=\dfrac{1}{2}\eta^{\mu\nu},$
(1.24)
which is distinguishable from the space-time Clifford algebra
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}_{C}=2\eta^{\mu\nu}.$ (1.25)
It is not difficult to proof that for $D+1$ dimensional space-time, where $D$
is the spatial dimension, the new algebra is
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=\dfrac{2}{D+1}\eta^{\mu\nu}.$
(1.26)
Let us consider now the constraint (1.9), and apply to this the canonical
relativistic quantization procedure
$(E,p_{i}c)\rightarrow
i\hslash\partial_{\mu}=i\hslash(-\partial_{0},c\partial_{i}).$ (1.27)
The resulting equation
$-i\hslash\gamma^{0}\partial_{0}\psi=(ic\hslash\gamma^{i}\partial_{i}\pm
mc^{2})\psi,$ (1.28)
in fact is the Dirac equation
$(i\hslash\gamma^{\mu}\partial_{\mu}\pm mc^{2})\psi=0,$ (1.29)
for which the Lorentz symmetry is fully validate. In other words we have
obtained the Dirac equation, where however, the gamma matrices do not belong
to the Clifford algebra but obey the new algebra (1.24). Interestingly,
factually we have generated the Dirac equation independently on the gamma
matrix algebra, what suggests that the Dirac equation must not be obtained as
”square-root taking” of the Klein–Gordon equation, like Dirac originally
deduced and applied [112]. Dirac’s computations did not involve relations
between gamma matrices manifestly, and therefore his deductions are true.
Interestingly, the general relation between the new algebra (1.20) and the
Clifford algebra (1.14) can be established straightforwardly
$\left\\{\gamma^{i},\gamma^{j}\right\\}=\dfrac{1}{D}\left\\{\gamma^{i},\gamma^{j}\right\\}_{C}.$
(1.30)
In other words, because of the Clifford algebra is here a unit $D\times D$
matrix, the limit $D\rightarrow\infty$ gives trivially
$\lim_{D\rightarrow\infty}\left\\{\gamma^{i},\gamma^{j}\right\\}=0,$ (1.31)
while the Clifford algebra is conserved in such a limit situation. Albeit, in
the light of the general relation (1.18) this case generates the blatantly
incorrect equality $1=0$. It suggests that $D\rightarrow\infty$ is a
nonphysical situation when the spatial metric is a constant $D\times D$ unit
matrix $\delta_{ij}$. Possibly, such a infinite limit has a sense if and only
if the space/space-time metric depends on the number of dimensions $D$.
However, we shall not discuss such examples in this book. It is easy to see
that similar situation is present for the space-time new algebra (1.26) and
the space-time Clifford algebra (1.25)
$\displaystyle\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}$ $\displaystyle=$
$\displaystyle\dfrac{1}{D+1}\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}_{C},$
(1.32)
$\displaystyle\lim_{D\rightarrow\infty}\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}$
$\displaystyle=$ $\displaystyle 0.$ (1.33)
Let us establish certain identity important for this chapter. Namely, let us
compute $\gamma^{i}\gamma_{i}$ in the light of the basic relation (1.18)
$\displaystyle\gamma^{i}\gamma_{i}=\gamma^{i}\delta_{ij}\gamma^{j}=\delta_{ij}\gamma^{i}\gamma^{j}=\dfrac{1}{2}\left(\delta_{ij}+\delta_{ji}\right)\gamma^{i}\gamma^{j}=\dfrac{1}{2}\left(\delta_{ij}\gamma^{i}\gamma^{j}+\delta_{ji}\gamma^{i}\gamma^{j}\right)=$
$\displaystyle=\dfrac{1}{2}\left(\delta_{ij}\gamma^{i}\gamma^{j}+\delta_{ij}\gamma^{j}\gamma^{i}\right)=\dfrac{1}{2}\delta_{ij}\left(\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}\right)=\dfrac{1}{2}\delta_{ij}\left\\{\gamma^{i},\gamma^{j}\right\\}=1,$
(1.34)
what differs from the $D$-dimensional Clifford algebra result
$\gamma^{i}\gamma_{i}=D$. The result (1.34) holds for any $D$, and first of
all also for the space-time version of the new algebra
$\gamma^{\mu}\gamma_{\mu}=1$, and is independent on the spatial dimension.
This is an important elucidation.
Because of the presented approach, based on _identification method_ , is
essentially new and evidently changes deductions and explanations related to
the Clifford algebra, we shall call the new algebra _the Æther algebra_ . The
presented way of reasoning shows that the Dirac equation is not related to the
Clifford algebra only, but factually can be deduced by a way very far from the
Dirac ”square-root” technique and produce other algebras of gamma matrices. We
showed here that in general gamma matrices can be deduced by techniques
different from the methods of relativistic physics, propagated by Dirac in his
contributions to quantum mechanics. Albeit, we regard Dirac’s results, and
particularly their diverse consequences for particle physics, as the
inspiration. The Æther algebra proposed above can be related to other,
possibly unknown, particles and forces. Possibly, the Dirac equation with non-
Dirac gamma matrices defines an effective theory.
##### A2 Another Identifications
The second possible identification is
$\left\\{\begin{array}[]{l}m^{2}c^{4}=E^{2}\\\
\dfrac{\mathcal{P}^{2}c^{2}}{4}=\left(\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c+\dfrac{\mathcal{P}c}{2}\right)^{2}\end{array}\right..$
(1.35)
In such a situation the first equality leads to the relation
$\gamma^{0}E=\pm mc^{2},$ (1.36)
where $\gamma^{0}$ is defined by the relation (1.10). The formula (1.36)looks
like the Einstein mass-energy relation . Similarly the second equality in
(1.35) leads to the following nontrivial and manifestly distinguishable
physical situations
$\displaystyle\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c$ $\displaystyle=$
$\displaystyle 0,$ (1.37)
$\displaystyle\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c$ $\displaystyle=$
$\displaystyle-\mathcal{P}c.$ (1.38)
Similarly as in the previous case one can introduce the Clifford algebra of
spatial gamma matrices given by the relations (1.10) and (1.11), and applying
the canonical relativistic quantization procedure (1.27) one obtains the
appropriate projections conditions. The first such a condition follows from
(1.36) and has a form
$\left(i\hslash\gamma^{0}\partial_{0}\mp mc^{2}\right)\psi=0,$ (1.39)
while the second one, following from (1.37) and (1.38), has the form
$\displaystyle ic\hslash\gamma^{i}\partial_{i}\psi$ $\displaystyle=$
$\displaystyle 0,$ (1.40)
$\displaystyle\left(ic\hslash\gamma^{i}\partial_{i}+\mathcal{P}c\right)\psi$
$\displaystyle=$ $\displaystyle 0.$ (1.41)
Interestingly, the equation (1.39) added to the condition (1.40) leads to the
usual Dirac equation (1.29), while addition of the condition (1.40) to the
equation (1.39) allows to establish the new quantum relativistic equation
jointing a Dirac particle and the Æther
$\left(i\hslash\gamma^{\mu}\partial_{\mu}+Mc^{2}\right)\psi=0,$ (1.42)
where $M$ is the effective mass of the particle-Æther system
$M=\mp m+\dfrac{1}{c}\mathcal{P},$ (1.43)
i.e. for $[\mathcal{P}]\sim[c]=3\cdot 10^{8}$ the correction due to the Æther
plays an essential physical role. Interestingly, for the positive sign near
particle mass $m$ in (1.43) the effective mass $M$ is always positive, while
for the negative sign the effective mass $M$ is positive for $mc<\mathcal{P}$,
negative for $mc>\mathcal{P}$, and vanishes when $mc=\mathcal{P}$.
On the one hand the effective mass (1.43) manifestly contains the correction
to a particle mass due to the Æther momentum value $\mathcal{P}$ but not due
to $\mathcal{P}_{i}$. Such a property involves a situation when the Æther
momentum vector $\mathcal{P}_{i}$ is nontrivial but its length $\mathcal{P}$
vanishes. In such a case an arbitrary component of the Æther momentum vector
is determined by the two remained components which are still arbitrary. It can
be seen that then a classical theory is a deformed Special Relativity (1.4)
while, because $M=\pm m$ by (1.43), quantum theory (1.42) is the Dirac
relativistic quantum mechanics. In other words the Dirac theory is related not
only to the Einstein theory, but possesses wider sense.
On the other hand, however, the quantum theory given by the projections
(1.39), and (1.40) and (1.41) carries different content than the usual Dirac
relativistic quantum mechanics. In the Dirac theory there is the only one
cumulative projection condition (1.29), while in the our theory the conditions
(1.39), and (1.40) and (1.41) in general are not cumulative. We mean that we
have obtained the condition for time evolution (1.39) and two alternative
conditions for spatial evolution (1.40) and (1.41), while in the Dirac theory
there is unified space-time evolution (1.29). By this reason the our situation
is physically distinguished from the theory based on four-dimensional Dirac
equation . However, the unification obtained by simple algebraic sum of the
time projection and the spatial projection led us to the usual Dirac theory
and the Dirac theory with the effective mass (1.43). This particular case is
within the general theory given by the projection conditions (1.39), and
(1.40) and (1.41).
Interestingly, also the classical physics context of the conditions (1.37) and
(1.38) is nontrivial. Namely, these relations establish two possible
constraints for a particle momentum components, what allows to express an
arbitrary one component of a particle momentum via the Æther momentum. In
other words the constraints (1.37) and (1.38) joint a classical particle with
the Æther. Factually, the first relation is
$\mathcal{P}^{1}p_{1}+\mathcal{P}^{2}p_{2}+\mathcal{P}^{3}p_{3}=0,$ (1.44)
and the latter one is
$\mathcal{P}^{1}(p_{1}+\mathcal{P}_{1})+\mathcal{P}^{2}(p_{2}+\mathcal{P}_{2})+\mathcal{P}^{3}(p_{3}+\mathcal{P}_{3})=0.$
(1.45)
There are in general three types of solutions for each of these constraints,
in which a one component of a particle momentum is dependent on two other
(arbitrary) components of particle momentum and all components of the
reference momentum 3-vector. The constraint (1.44) can be solved by
$\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(-\dfrac{\mathcal{P}^{2}}{\mathcal{P}^{1}}p_{2}-\dfrac{\mathcal{P}^{3}}{\mathcal{P}^{1}}p_{3},p_{2},p_{3}\right),$
(1.46) $\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(p_{1},-\dfrac{\mathcal{P}^{1}}{\mathcal{P}^{2}}p_{1}-\dfrac{\mathcal{P}^{3}}{\mathcal{P}^{2}}p_{3},p_{3}\right),$
(1.47) $\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(p_{1},p_{2},-\dfrac{\mathcal{P}^{1}}{\mathcal{P}^{3}}p_{1}-\dfrac{\mathcal{P}^{2}}{\mathcal{P}^{3}}p_{2}\right).$
(1.48)
Similarly, the constraint given by (1.45) possesses in general three possible
solutions
$\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(-\mathcal{P}_{1}-\dfrac{\mathcal{P}^{2}}{\mathcal{P}^{1}}(p_{2}+\mathcal{P}_{2})-\dfrac{\mathcal{P}^{3}}{\mathcal{P}^{1}}(p_{3}+\mathcal{P}_{3}),p_{2},p_{3}\right),$
(1.49) $\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(p_{1},-\mathcal{P}_{2}-\dfrac{\mathcal{P}^{1}}{\mathcal{P}^{2}}(p_{1}+\mathcal{P}_{1})-\dfrac{\mathcal{P}^{3}}{\mathcal{P}^{2}}(p_{3}+\mathcal{P}_{3}),p_{3}\right),$
(1.50) $\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(p_{1},p_{2},-\mathcal{P}_{3}-\dfrac{\mathcal{P}^{1}}{\mathcal{P}^{3}}(p_{1}+\mathcal{P}_{1})-\dfrac{\mathcal{P}^{2}}{\mathcal{P}^{3}}(p_{2}+\mathcal{P}_{2})\right).$
(1.51)
Particularly, the solution of the first constraint can be trivial, i.e.
$p_{i}=0$, and the second constraint can be solved simply by
$p_{i}=-\mathcal{P}_{i}$. Both these cases have a physical interpretation of
an inertial reference frame of a particle: either rest frame or motion of a
particle under the constant momentum opposite to the Æther momentum. For both
these particular solutions the Lorentz symmetry also holds.
The third interesting identification is
$\left\\{\begin{array}[]{l}-E^{2}=\dfrac{\mathcal{P}^{2}c^{2}}{4}\\\
-m^{2}c^{4}=\left(\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c+\dfrac{\mathcal{P}c}{2}\right)^{2}\end{array}\right..$
(1.52)
The first constraint can be resolved straightforwardly with the result
$\pm i\gamma^{0}E=\dfrac{\mathcal{P}c}{2}.$ (1.53)
The solution of the second equation also can be easy established
$\pm
imc^{2}=\dfrac{\mathcal{P}^{i}}{\mathcal{P}}p_{i}c+\dfrac{\mathcal{P}c}{2}.$
(1.54)
Employing (1.53) this solution can be written as
$\pm imc^{2}=\gamma^{i}p_{i}c\pm i\gamma^{0}E.$ (1.55)
After the canonical relativistic quantization the solution (1.55) leads to the
equation
$\left(ic\hslash\gamma^{i}\partial_{i}\mp\hslash\gamma^{0}\partial_{0}\mp
imc^{2}\right)\psi=0,$ (1.56)
having blatantly real and imaginary parts which are
$\displaystyle\mp\hslash\gamma^{0}\partial_{0}\psi$ $\displaystyle=$
$\displaystyle 0,$ (1.57)
$\displaystyle\left(c\hslash\gamma^{i}\partial_{i}\mp mc^{2}\right)\psi$
$\displaystyle=$ $\displaystyle 0,$ (1.58)
and must be treated as the system of equations.
With using of (1.53) the solution (1.54), however, can be rewritten in another
form and understood alternatively. Namely, because one has determined the
Æther momentum value via a particle energy and non-determined the Æther
momentum vector, the equation (1.54) can be presented in an equivalent form
$\pm imc^{2}=\dfrac{\mathcal{P}^{i}c}{\pm 2i\gamma^{0}E}p_{i}c\pm
i\gamma^{0}E.$ (1.59)
One sees straightforwardly, however, that the equation (1.59) after elementary
algebraic manipulations can be presented in a form of a quadratic equation
$(\pm\gamma^{0}E)^{2}\mp
mc^{2}(\pm\gamma^{0}E)-\dfrac{\mathcal{P}^{i}c}{2}p_{i}c=0.$ (1.60)
Now it is easy to conclude that essentially for arbitrary sign of $m$ this
equation can be rewritten as
$(\gamma^{0}E)^{2}\pm
mc^{2}(\gamma^{0}E)-\dfrac{\mathcal{P}^{i}c}{2}p_{i}c=0,$ (1.61)
while one can treat the positive mass case in the constraint (1.59) as the
physical situation. The canonical relativistic quantization applied to the
constraint (1.61) results in the following projection condition
$\left(-\hslash^{2}\partial_{0}^{2}\pm i\hslash
mc^{2}\gamma^{0}\partial_{0}-ic\hslash\dfrac{\mathcal{P}^{i}c}{2}\partial_{i}\right)\psi=0,$
(1.62)
which after multiplication by $-1/mc^{2}$ and taking into account _ad hoc_ the
following identification of the spatial gamma matrices
$\gamma^{i}=\dfrac{\mathcal{P}^{i}}{2mc}.$ (1.63)
can be easily led to more convenient form
$\left(\dfrac{\hslash^{2}}{mc^{2}}\partial_{0}^{2}\mp
i\hslash\gamma^{0}\partial_{0}+ic\hslash\gamma^{i}\partial_{i}\right)\psi=0.$
(1.64)
Interestingly, when one takes the plus sign in (1.64) as the physical case
than this condition can be written as
$\left(i\hslash\gamma^{\mu}\partial_{\mu}+\dfrac{\hslash^{2}}{mc^{2}}\partial_{0}^{2}\right)\psi=0,$
(1.65)
and then the Dirac equation is recovered
$(i\hslash\gamma^{\mu}\partial_{\mu}+Mc^{2})\psi=0,$ (1.66)
where $M$ is an effective mass term
$Mc^{2}\psi\equiv\dfrac{\hslash^{2}}{mc^{2}}\partial_{0}^{2}\psi.$ (1.67)
It is easy to see that the Lorentz symmetry is validate in such a mass
generation mechanism.
The case of the minus sign in the equation (1.64) can be also considered in
terms of the Dirac equation (1.66), but then the mass term is determined via
one of two equivalent conditions
$\displaystyle Mc^{2}\psi$ $\displaystyle\equiv$
$\displaystyle\left(-\dfrac{\hslash^{2}}{mc^{2}}\partial_{0}^{2}-2ic\hslash\gamma^{i}\partial_{i}\right)\psi,$
(1.68) $\displaystyle Mc^{2}\psi$ $\displaystyle\equiv$
$\displaystyle\left(\dfrac{\hslash^{2}}{mc^{2}}\partial_{0}^{2}-2i\hslash\gamma^{0}\partial_{0}\right)\psi.$
(1.69)
From the classical physics point of view, however, the mass term conditions
(1.68) and (1.69) are respectively
$\displaystyle Mc^{2}$ $\displaystyle=$
$\displaystyle-\dfrac{E^{2}}{mc^{2}}-2\dfrac{\mathcal{P}^{i}}{2mc}p_{i}c,$
(1.70) $\displaystyle Mc^{2}$ $\displaystyle=$
$\displaystyle\dfrac{E^{2}}{mc^{2}}-2\gamma^{0}E,$ (1.71)
where we used ${\gamma_{0}}^{2}=1$, and their equivalence leads to the
constraint
$\dfrac{E^{2}}{mc^{2}}-\gamma^{0}E+\dfrac{\mathcal{P}^{i}}{2mc}p_{i}c=0,$
(1.72)
which can be rewritten in more conventional form
$\gamma^{0}E=\dfrac{E^{2}}{mc^{2}}+\dfrac{\mathcal{P}^{i}}{2mc}p_{i}c.$ (1.73)
Applying the basic constraint (1.4) to the right hand side of (1.73) one
receives the relation
$\gamma^{0}E=mc^{2}+\dfrac{3\mathcal{P}^{i}cp_{i}c}{2mc^{2}}+\dfrac{p^{i}cp_{i}c}{mc^{2}},$
(1.74)
that defines the resolution of the constraint (1.4) in this case. Employing
once again the canonical relativistic quantization to (1.74) and identifying
the spatial gamma matrices as (1.63) one obtains the following equation
$i\hslash\gamma^{0}\partial_{0}\psi=\left(mc^{2}+3ic\hslash\gamma^{i}\partial_{i}-\dfrac{\hslash^{2}}{mc^{2}}\triangle\right)\psi,$
(1.75)
where
$\triangle=\gamma^{i}\gamma_{i}\partial^{i}\partial_{i}=\partial^{i}\partial_{i}$
is the Laplace operator, which can be presented as the Dirac equation (1.66)
with the effective mass given by two equivalent mass generation rules
$\displaystyle
Mc^{2}\psi=\left(-mc^{2}-4ic\hslash\gamma^{i}\partial_{i}+\dfrac{\hslash^{2}}{mc^{2}}\triangle\right)\psi,$
(1.76) $\displaystyle
Mc^{2}\psi=\dfrac{1}{3}\left(mc^{2}-4ic\hslash\gamma^{0}\partial_{0}-\dfrac{\hslash^{2}}{mc^{2}}\triangle\right)\psi.$
(1.77)
It must be emphasized that the Laplace operator
$\triangle=\gamma^{i}\gamma_{i}\partial^{i}\partial_{i}$ has the form
$\triangle=\partial^{i}\partial_{i}$ for the new algebra, while for the
Clifford algebra it is manifestly different
$\triangle=D\partial^{i}\partial_{i}$. It is the confirmation of the
correctness of the new algebra.
Another route to a new equation in the case under considering, can be obtained
e.g. by application of the first relation (1.52) within the constraint (1.73).
First one obtains
$\gamma^{0}E=-\dfrac{\mathcal{P}^{i}}{2mc}\dfrac{\mathcal{P}_{i}}{2mc}mc^{2}+\dfrac{\mathcal{P}^{i}}{2mc}p_{i}c,$
(1.78)
then by using of (1.63) and the identity
$\mathcal{P}_{i}=\delta_{ij}\mathcal{P}^{j}$ one has
$\gamma^{0}E=-\delta_{ij}\gamma^{i}\gamma^{j}mc^{2}+\gamma^{i}p_{i}c,$ (1.79)
and by the new algebra result (1.34) one receives finally
$\gamma^{0}E-\gamma^{i}p_{i}c+mc^{2}=0.$ (1.80)
In this manner, applying once again the canonical relativistic quantization
one obtains the equation
$\left(i\hslash\gamma^{0}\partial_{0}-ic\hslash\gamma^{i}\partial_{i}+mc^{2}\right)\psi=0,$
(1.81)
which differs from the Dirac equation for particle with mass $m$ by the minus
sign presence near spatial derivative. However, one sees straightforwardly
that the equation (1.81) can be interpreted as the Dirac equation (1.66) with
the effective mass $M$, if and only if the effective mass term is given by one
of the two equivalent relations
$\displaystyle Mc^{2}\psi$ $\displaystyle\equiv$
$\displaystyle\left(-2ic\hslash\gamma^{i}\partial_{i}+mc^{2}\right)\psi,$
(1.82) $\displaystyle Mc^{2}\psi$ $\displaystyle\equiv$
$\displaystyle\left(-2i\hslash\gamma^{0}\partial_{0}-mc^{2}\right)\psi.$
(1.83)
The obtained Dirac-like quantum theories can be always presented in the form
of the Schrödinger equation, i.e.
$i\hslash\partial_{0}\psi=H\psi,$ (1.84)
where $H=H(\partial_{0},\partial_{i})$ is the Hamilton operator describing a
quantum system. In comparison to the standard Dirac theory in the our case the
operator $H$, however, in general must not be always a hermitean operator. The
equation (1.84) should be solved with the usual spatial normalization
condition for the wave function $\psi$
$\int d^{D}x|\psi(x)|^{2}=1,$ (1.85)
where $1$ is the D-dimensional unit matrix. The question of solvability of the
systems deduced above, however, is not the main theme of this book. It is good
exercise for a reader.
As we proposed initially, the linear deformation of Special Relativity
constraint in (1.4) couples a particle and the Æther in the Minkowski energy-
momentum space . The reasoning done in the spirit of Dirac’s relativistic
quantum mechanics led us to the linkage (1.11) between the new algebra of the
spatial gamma matrices, and the Æther momentum vector, and its particular case
(1.63) was also discussed. Straightforward application of the Æther algebra
(1.20) results in the following noncommutative algebra of the Æther momentum
$\left\\{\dfrac{\mathcal{P}^{i}}{\mathcal{P}},\dfrac{\mathcal{P}^{j}}{\mathcal{P}}\right\\}=\dfrac{2}{D}\delta^{ij}.$
(1.86)
Let us generalize this algebra to the Minkowski space-time case
$\left\\{\dfrac{\mathcal{P}^{\mu}}{\mathfrak{P}},\dfrac{\mathcal{P}^{\nu}}{\mathfrak{P}}\right\\}=\dfrac{2}{D+1}\eta^{\mu\nu},$
(1.87)
where $\mathfrak{P}$ is the length of the Æther four-momentum
$\mathfrak{P}^{2}=\dfrac{1}{2}\left(\mathcal{P}^{\mu}\mathcal{P}_{\mu}+\mathcal{P}_{\mu}\mathcal{P}^{\mu}\right),$
(1.88)
which can be initially postulated as
$\mathcal{P}^{\mu}=\left(\mathcal{P}^{0},\mathcal{P}\gamma^{i}\right),$ (1.89)
where $\mathcal{P}^{0}$ is a time component of $\mathcal{P}^{\mu}$, and
$\mathcal{P}$ is length of the Æther three-momentum. Let us compute (1.88)
straightforwardly
$\mathfrak{P}^{2}=\dfrac{1}{2}\left(\mathcal{P}^{0}\mathcal{P}_{0}+\mathcal{P}_{0}\mathcal{P}^{0}+\mathcal{P}^{2}\delta_{ij}\left\\{\gamma^{i},\gamma^{j}\right\\}\right),$
(1.90)
and postulate $\mathcal{P}^{0}=-\alpha\gamma^{0}$, where $\alpha$ is an
unknown multiplier. Than, because of by definition one has
$\mathcal{P}_{0}=\eta_{00}\mathcal{P}^{0}=\alpha\gamma_{0}$, one receives
$\mathfrak{P}^{2}-\dfrac{1}{2}\mathcal{P}^{2}\delta_{ij}\left\\{\gamma^{i},\gamma^{j}\right\\}=-\alpha^{2}\left(1-\dfrac{1}{2}\delta_{ij}\left\\{\gamma^{i},\gamma^{j}\right\\}\right),$
(1.91)
what in the light of the spatial algebra (1.18) is satisfied if and only if
$\mathfrak{P}^{2}=\mathcal{P}^{2}\quad,\quad\mathfrak{P}^{2}=-\alpha^{2},$
(1.92)
and consequently time component of the Æther four-momentum is
$\mathcal{P}^{0}=\pm i\mathfrak{P}\gamma^{0}=\pm i\mathcal{P}\gamma^{0}.$
(1.93)
Therefore, $\mathcal{P}^{0}=0$ if and only if $\mathfrak{P}=\mathcal{P}=0$.
For the particular situation $\gamma^{i}=\dfrac{\mathcal{P}^{i}}{2mc}$ a
particle mass can be expressed via the Æther momentum vector
$m=\dfrac{1}{2}\dfrac{\mathcal{P}}{c},$ (1.94)
so in fact the modified Einstein Hamiltonian constraint (1.4) can be expressed
via the Æther and a particle momenta only
$E^{2}=\left(\dfrac{1}{4}\mathcal{P}^{i}\mathcal{P}_{i}+p^{i}p_{i}+\mathcal{P}^{i}p_{i}\right)c^{2}.$
(1.95)
The fixed value of square of the Æther momentum vector allows to establish
three equivalent forms of the the momentum three-vector
$\displaystyle\mathcal{P}^{i}$ $\displaystyle=$
$\displaystyle\left[\sqrt{(2mc)^{2}-\mathcal{P}^{2}\mathcal{P}_{2}-\mathcal{P}^{3}\mathcal{P}_{3}},\mathcal{P}^{2},\mathcal{P}^{3}\right]=$
(1.96) $\displaystyle=$
$\displaystyle\left[\mathcal{P}^{1},\sqrt{(2mc)^{2}-\mathcal{P}^{1}\mathcal{P}_{1}-\mathcal{P}^{3}\mathcal{P}_{3}},\mathcal{P}^{3}\right]=$
(1.97) $\displaystyle=$
$\displaystyle\left[\mathcal{P}^{1},\mathcal{P}^{2},\sqrt{(2mc)^{2}-\mathcal{P}^{1}\mathcal{P}_{1}-\mathcal{P}^{2}\mathcal{P}_{2}}\right],$
(1.98)
what allows also to derive easily the normalized Æther momentum vector
$\dfrac{\mathcal{P}^{i}}{\mathcal{P}}$. Moreover, one can reconsider the
situations defined by the constraints (1.37) and (1.38). In the first case the
energetic constraint (1.95) takes the following form
$E^{2}=\left(\dfrac{1}{4}\mathcal{P}^{i}\mathcal{P}_{i}+p^{i}p_{i}\right)c^{2},$
(1.99)
with the particle momentum defined by the solutions (1.46), (1.47) or (1.48),
while in the second situation one obtains
$E^{2}=\left(-\dfrac{3}{4}\mathcal{P}^{i}\mathcal{P}_{i}+p^{i}p_{i}\right)c^{2}.$
(1.100)
where the particle momentum is given by (1.49), (1.50) or (1.51). It is
visible that in the case (1.99) the energy is always nonzero. However, in the
case (1.100) the energy can be trivially vanishing if and only if the particle
momentum is constrained by
$p_{i}=\sqrt{\dfrac{3}{4}}\mathcal{P}_{i}.$ (1.101)
It is easy to see for (1.49), (1.50) or (1.51) that in the case (1.101) holds
$\mathcal{P}^{i}\mathcal{P}_{i}=0.$ (1.102)
Particularly, the constraint (1.102) is satisfied when the Æther momentum
vanishes identically $\mathcal{P}^{i}=0$, but by (1.101) such a situation
implies
$m=\dfrac{1}{\sqrt{3}}\dfrac{p}{c},$ (1.103)
where $p=\sqrt{p^{i}p_{i}}$ is the momentum value of a particle. Moreover, the
vanishing Æther momentum implies that the deformation vanishes, i.e. Special
Relativity should be reconstructed. Using of both $\mathcal{P}^{i}=0$ and
(1.103) in the light of the Hamiltonian constraint (1.4) one obtains
$E^{2}=4m^{2}c^{4},$ (1.104)
but by (1.101) the energy square (1.100) vanishes, so reconstruction of
Special Relativity gives finally $m^{2}=0$. Application of the constraint
(1.102) within the solutions (1.49)-(1.51) reconstructs the case
(1.46)-(1.48), so (1.102) is equivalent to $\mathcal{P}^{i}p_{i}=0$ in this
case.
Interestingly, one can also consider other type linear deformations of Special
Relativity. For example
$\displaystyle\Delta_{1}$ $\displaystyle=$
$\displaystyle\mathcal{P}_{i}p^{i}c^{2},$ (1.105) $\displaystyle\Delta_{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\mathcal{P}_{i}p^{i}c^{2}+\mathcal{P}^{i}p_{i}c^{2}\right),$
(1.106) $\displaystyle\Delta_{3}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\mathcal{P}_{i}p^{i}c^{2}+\mathcal{P}^{i}p_{i}c^{2}\right)\pm\mathcal{P}^{2}c^{2},$
(1.107)
etc., and perform analogous considerations. The deformations (1.105) and
(1.106) are equivalent to (1.4) one if and only if
$\mathcal{P}_{i}p^{i}=\mathcal{P}^{i}p_{i}$, i.e.
$\delta_{ik}\gamma^{k}p^{i}=\gamma^{j}p_{j}$ (1.108)
what, after multiplication of both sides by $\delta^{ik}\gamma_{k}=\gamma^{i}$
and taking into account the identity $\delta_{ik}\delta^{ik}=D$, takes the
form
$p^{i}=\dfrac{1}{D}\gamma^{i}\gamma^{j}p_{j}.$ (1.109)
In this section we have presented the approach based on the linear deformation
of the Einstein Hamiltonian constraint of Special Relativity which generates
new equations in frames of relativistic quantum mechanics. The linear
deformation can be generalized for other deformations, and similar strategy
can be used. All these new models can be treated as Dark Matter and/or Dark
Energy models.
#### B The Snyder–Sidharth Hamiltonian
Let us consider now more complex situation. Namely, we shall focus our
attention on the following deformation of the Einstein Hamiltonian constraint
(1.1) of Special Relativity
$E^{2}=m^{2}c^{4}+c^{2}p^{2}+\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}c^{2}p^{4},$
(1.110)
where $\ell$ is any minimal physical scale. This deformation was investigated
by H. Snyder [113] in the context of the infrared catastrophe of soft photons
in the Compton scattering, and in general to renormalize quantum field theory
by application of the noncommutative quantum space-time, as it is widely
studied by numerous authors and scholars [114]. In fact such a modification
follows from the nontrivial manipulation in phase space of any special
relativistic particle
$\displaystyle\dfrac{i}{\hslash}[p,x]$ $\displaystyle=$ $\displaystyle
1+\alpha\dfrac{\ell^{2}}{\hslash^{2}}p^{2}\quad,\quad\alpha\sim 1,$ (1.111)
$\displaystyle\left[x,y\right]$ $\displaystyle=$ $\displaystyle O(\ell^{2}),$
(1.112)
and therefore one has to deal with the structure of a non-differentiable
manifold, or lattice model of space-time. We shall discuss wider mathematical
details of the Snyder space-time (1.111) in next chapters of this part. It
must be emphasized that the deformation (1.111) reveals Lorentz invariance.
Factually B.G. Sidharth (Refs. [41]) first accepted the Snyder noncommutative
geometry as the serious argument for physics, has been studied the modified
Einstein Hamiltonian constraint (1.110) in the astroparticle physics context.
He proposed taking into account the Hamiltonian constraint (1.110) and
treating this deformation in generalized sense as a type of perturbational
series in the minimum scale $\ell$, that can be e.g. the Planck scale or the
Compton scale. By this reason we shall call the constraint (1.110) _the
Snyder–Sidharth Hamiltonian constraint_ or briefly the Snyder–Sidharth
constraint/Hamiltonian.
A number of distinguished scholars like S. Glashow, S. Coleman, and others
have considered diverse schemes which manifestly depart from the Einstein
Special Relativity. It must be stressed here that these all schemes are purely
_ad hoc_. Recently observations of ultra-high-energy cosmic rays and rays from
gamma bursts seem to suggest Lorentz symmetry violation [115], and the
Hamiltonian constraint (1.110) violates Lorentz symmetry. In this particular
context the author [109] has proposed to call the Hamiltonian constraint
(1.110) the Snyder–Sidharth Hamiltonian constraint what is also preferred in
this book. It is important to emphasize here, that in general situation the
Hamiltonian constraint is not Hamiltonian, i.e. energy. In general an energy
can be obtained by resolving a Hamiltonian constraint, as it was shown in the
previous section, and shall be continued in next chapters of this book.
Interestingly, the Snyder space-time has certain context in string theory,
where is often referred as stringy-like space-time.
Similarly as in the case of the linear deformation analyzed in the previous
section, the Snyder–Sidharth Hamiltonian constraint (1.110) as a quadratic
form can be seen easily lead to the canonical form
$E^{2}+\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}=\alpha\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\right)^{2}+m^{2}c^{4},$
(1.113)
and as previously there are in general three possible mathematical
interpretations of this relation. However, in the case of the fourth-order
deformation (1.110) one has no linear terms in particle momentum $p$, but
there are factually powers of $p^{2}$. In the light of the principles of
relativistic quantum theory (See e.g. the Refs. [116] and [117]) it means that
in such a situation the structure of Clifford algebra must be at least hidden
if no hidden manifestly. Let us consider these identifications to the case of
fermions and the case of bosons.
##### B1 The Case of Fermions
1. 1.
First, we can interpret the constraint equation (1.113) as system of two
equations
$\left\\{\begin{array}[]{l}m^{2}c^{4}=\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}\\\
E^{2}=\alpha\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\right)^{2}\end{array}\right.$
(1.114)
The first equation leads to solution that looks like formally as the bosonic
string tension
$m=\dfrac{\hslash}{2\sqrt{{\alpha}}c\ell}.$ (1.115)
Expressing $\ell$ via $m$ and constants, one can write the solution of the
second equality in the following way
$E=\dfrac{p^{2}}{2m}+mc^{2},$ (1.116)
where we have omitted the solution with minus sign as non physical. This is
the Hamiltonian of a free point particle in semi-classical mechanics, i.e. the
Newtonian kinetic energy corrected by the Einstein–Poincarè rest energy term.
Interestingly (1.115) and (1.116) are consistent if $m$ is the Planck mass,
i.e. $m=M_{P}=\sqrt{\dfrac{\hslash c}{G}}$, and $\ell$ is the Planck length,
i.e. $\ell_{P}=\sqrt{\dfrac{\hslash G}{c^{3}}}$. Factually the Planck mass
determines a unifying scale where the classical and the quantum meet and
collaborate. The Schwarzschild radius $r_{S}(m)=\dfrac{Gm}{c^{2}}$ evaluated
on the Planck mass equals to the Compton wavelength of a hypothetical particle
possessing such a value of mass [118]. In comparison with Special Relativity,
one has no here higher relativistic corrections to the semi-classical case.
After the primary canonical quantization one obtains exactly the Schrödinger
equation of free quarks in Quantum Chromodynamics (QCD) because quarks are
non-relativistic and massive [119]. For the case of vanishing scale and
nonzero $\alpha$ as well as for vanishing $\alpha$ and fixed nonzero scale
$\ell$, formally $m\equiv\infty$ and energy is also infinite, therefore it is
a nonphysical black-hole type singularity. For the large scale limit and non
vanishing $\alpha$, the mass spectrum is point-like $m=0$. For nonzero
momentum energy is also infinite, i.e. it is a nonphysical case. Anyway, it
shows manifestly that (1.116) is compatible with (1.111).
2. 2.
The second case changes the role of energy and mass
$\left\\{\begin{array}[]{l}-m^{2}c^{4}=\alpha\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\right)^{2}\\\
-E^{2}=\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}\end{array}\right.,$ (1.117)
and leads to discrete energy spectrum for fixed value of scale $\ell$. Because
(See Sidharth’s paper in [115]), however, the constraint (1.110) with positive
$\alpha$ is true for fermions, and with negative $\alpha$ for bosons, for the
case of fermions one has here
$iE=\dfrac{\hslash c}{2\sqrt{{\alpha}}\ell},$ (1.118)
as well as the mass one
$imc^{2}=\sqrt{\alpha}\dfrac{c\ell}{\hslash}p^{2}+\dfrac{\hslash
c}{2\sqrt{\alpha}\ell},$ (1.119)
i.e. rejecting the negative value from. However, one can eliminate scale via
using energy (1.118) with the result
$mc^{2}=-\dfrac{p^{2}c^{2}}{2E}+E,$ (1.120)
what can be rewritten in the form of the quadratic equation
$E^{2}=mc^{2}E+\dfrac{p^{2}c^{2}}{2},$ (1.121)
and by using of the deformed constraint (1.110) it yields
$m^{2}c^{4}+c^{2}p^{2}+\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}c^{2}p^{4}=mc^{2}E+\dfrac{p^{2}c^{2}}{2}.$
(1.122)
One can find now the energy (not square of energy!), that is $4th$-power in
momentum
$E=mc^{2}+\dfrac{p^{2}}{2m}+\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{p^{4}}{m}.$
(1.123)
So, again one can apply factorization of a quadratic form
$E+\dfrac{\hslash^{2}}{16\alpha\ell^{2}}\dfrac{1}{m}=\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}+\dfrac{\hslash^{2}}{4\alpha\ell^{2}}\right)^{2}+mc^{2},$
(1.124)
and consider the identification method for the relation mass-energy.
1. (a)
The first obvious interpretation yields
$\left\\{\begin{array}[]{l}mc^{2}=\dfrac{\hslash^{2}}{16\alpha\ell^{2}}\dfrac{1}{m}\vspace*{5pt}\\\
E=\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}+\dfrac{\hslash^{2}}{4\alpha\ell^{2}}\right)^{2}\end{array}\right.$
(1.125)
and again one can extract trivially the solution of the first equation
$m=\dfrac{\hslash}{4\sqrt{\alpha}c\ell},$ (1.126)
and solution of the latter equality can be written in the form of the Pauli
Hamiltonian constraint
$E=mc^{2}+\dfrac{p^{2}}{2m}+\dfrac{p^{4}}{16m^{3}c^{2}},$ (1.127)
or after elimination of mass via using (1.126) one receives
$E=\dfrac{\hslash
c}{4\sqrt{\alpha}\ell}+2c\sqrt{\alpha}\dfrac{\ell}{\hslash}p^{2}+4c\left(\sqrt{\alpha}\dfrac{\ell}{\hslash}\right)^{3}p^{4}.$
(1.128)
However, it can be easily seen that the relation (1.118) jointed with (1.126)
determines energy as
$iE=2mc^{2},$ (1.129)
or equivalently the mass square value
$m^{2}=-\dfrac{E^{2}}{2c^{4}}<0,$ (1.130)
what means that momentum values are non hermitian, so one has to deal with
tachyon. Moreover, by straightforward application of the formula (1.126)
together with the relation (1.119) one establishes that
$p^{2}=\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\left(-1+\dfrac{1}{2}i\right).$
(1.131)
Using the polar form of the complex number in brackets
$-1+\dfrac{1}{2}i=\dfrac{\sqrt{5}}{2}\exp\left(-i\arctan\dfrac{1}{2}+2ni\pi\right),$
(1.132)
where $n\in\mathbb{Z}$, one can take its square root
$\sqrt{-1+\dfrac{1}{2}i}=\dfrac{1}{\sqrt{2}}\left(\sqrt{\dfrac{\sqrt{5}}{2}-1}+i\sqrt{\dfrac{\sqrt{5}}{2}+1}\right)e^{ni\pi},$
(1.133)
and obtains the momentum spectrum in dependence on $\ell$
$p_{n}=\pm\dfrac{(-1)^{n}}{2}\left(\sqrt{\dfrac{\sqrt{5}}{2}-1}+i\sqrt{\dfrac{\sqrt{5}}{2}+1}\right)\dfrac{\hslash}{\sqrt{\alpha}\ell}.$
(1.134)
2. (b)
The latter subcase is
$\left\\{\begin{array}[]{l}mc^{2}=E\\\
\dfrac{\hslash^{2}}{16\alpha\ell^{2}}\dfrac{1}{m}=\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}+\dfrac{\hslash^{2}}{4\alpha\ell^{2}}\right)^{2}\end{array}\right.$
(1.135)
Again, solution of the first equation via using (1.118) is rather simple
$m=-i\dfrac{\hslash}{2\sqrt{{\alpha}}c\ell}\quad,\quad m^{2}<0,$ (1.136)
and once again the tachyon is obtained - there are particles with momentum
spectrum
$p=\left\\{0,\pm\dfrac{\hslash}{\sqrt{{2\alpha}}\ell}\right\\}.$ (1.137)
This is discrete momenta spectrum for fixed scale $\ell$. For running scale
this is non compact spectrum, but compactification to the point is done in the
large scale limit
$\lim_{\ell\rightarrow\infty}p=0,$ (1.138)
and it is the rest. For $\alpha=0$ and fixed scale $\ell$ there are two
singular values of the momentum $p$. For all $\ell\neq 0$ and $\alpha\neq 0$,
the case of nonzero $p$ is related to the existence of tachyon.
3. (c)
The third interpretation yields
$\left\\{\begin{array}[]{l}-E=\dfrac{\hslash^{2}}{16\alpha\ell^{2}}\dfrac{1}{m}\vspace*{5pt}\\\
-mc^{2}=\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}+\dfrac{\hslash^{2}}{4\alpha\ell^{2}}\right)^{2}\end{array}\right..$
(1.139)
Similarly as in the previous subcase, employing the relation (1.118) one
obtains from the first equation the mass
$m=-i\dfrac{\hslash}{8\sqrt{\alpha}c\ell},$ (1.140)
and again one has tachyon, i.e. $m^{2}<0$. Consequently the momentum spectrum
can be established as
$p=\left\\{\pm i\sqrt{{\dfrac{1}{8\alpha}}}\dfrac{\hslash}{\ell},\pm
i\sqrt{{\dfrac{3}{8\alpha}}}\dfrac{\hslash}{\ell}\right\\},$ (1.141)
and is non hermitian, as one has expected. By this reason this particular
case, that is the Pauli Hamiltonian constraint with mass related to minimum
scale, describes tachyon, the hypothetical particles moving with velocity
faster then light.
3. 3.
The third possible solution of the Snyder–Sidharth Hamiltonian constraint can
be constructed by the system of equations
$\left\\{\begin{array}[]{l}E^{2}=m^{2}c^{4}\vspace*{5pt}\\\
\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}=\alpha\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\right)^{2}\end{array}\right..$
(1.142)
First equation gives the standard Einstein mass-energy relation
$E=mc^{2},$ (1.143)
and the latter equality results in the discrete momentum spectrum
$p=\left\\{0,\pm\dfrac{1}{\sqrt{{\alpha}}}\dfrac{\hslash}{\ell}\right\\},$
(1.144)
for fixed scale $\ell$ value. For running scale this spectrum is non compact,
but in the large scale limit the spectrum is manifestly compactified to the
point
$\lim_{\ell\rightarrow\infty}p=0,$ (1.145)
and it is the rest. For $\alpha=0$ and fixed scale $\ell$ there are two
singular values of the momentum $p$. For all $\ell\neq 0$ and $\alpha\neq 0$,
the case of nonzero $p$ is related to the existence of a relativistic
particle.
The other situation is when particle mass vanishes identically, i.e. $m\equiv
0$. In such a case one sees that the constraint (1.113) takes the form
$E^{2}+\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}=\alpha\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\right)^{2},$
(1.146)
and by the non-trivial identification
$\displaystyle E^{2}+\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}$
$\displaystyle=$ $\displaystyle 0,$ (1.147) $\displaystyle
p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}$ $\displaystyle=$ $\displaystyle
0,$ (1.148)
can be solved as the tachyonic case
$\displaystyle E^{2}$ $\displaystyle=$
$\displaystyle-\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}},$ (1.149)
$\displaystyle p^{2}$ $\displaystyle=$
$\displaystyle-\dfrac{\hslash^{2}}{2\alpha\ell^{2}},$ (1.150)
which can be combined into a one constraint
$E^{2}=\dfrac{1}{2}p^{2}c^{2},$ (1.151)
which can be solved immediately
$E=\pm\dfrac{1}{\sqrt{2}}pc,$ (1.152)
The equation (1.152) differs from the usual Special Relativity condition for
massless particle, i.e. $E=pc$. The difference reflects the fact that the
Snyder–Sidharth deformation is an algebraic deformation in the Minkowski
energy-momentum space . It is easy to see from the relations (1.149) and
(1.150) that energy and momentum of the massless particles are purely
imaginary quantities, and therefore such a situation corresponds to massless
fermionic tachyon.
##### B2 The Case of Bosons
Fermions obey the Dirac equation which is a square root of the Klein–Gordon
equation ruling bosons. Therefore fermions are approximation and bosons are
fundamental. Previous section results work for fermions when $\alpha>0$ and
for $\alpha<0$ are true for boson, i.e. the case of bosons arises by the
exchange
$\alpha\longrightarrow-|\alpha|,$ (1.153)
within mass and energy formulas. The basic relation (1.113) then is
$E^{2}-\dfrac{\hslash^{2}c^{2}}{4|\alpha|\ell^{2}}=-|\alpha|\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}-\dfrac{\hslash^{2}}{2|\alpha|\ell^{2}}\right)^{2}+m^{2}c^{4}.$
(1.154)
1. 1.
In the first case one has the system
$\left\\{\begin{array}[]{l}m^{2}c^{4}=-\dfrac{\hslash^{2}c^{2}}{4|\alpha|\ell^{2}}\\\
E^{2}=-|\alpha|\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}-\dfrac{\hslash^{2}}{2|\alpha|\ell^{2}}\right)^{2}\end{array}\right.$
(1.155)
which defines the tachyon with mass and energy
$\displaystyle m$ $\displaystyle=$ $\displaystyle
i\dfrac{\hslash}{2\sqrt{{|\alpha|}}c\ell},$ (1.156) $\displaystyle E$
$\displaystyle=$ $\displaystyle\dfrac{p^{2}}{2m}+mc^{2}.$ (1.157)
2. 2.
The second case is the system
$\left\\{\begin{array}[]{l}m^{2}c^{4}=|\alpha|\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}-\dfrac{\hslash^{2}}{2|\alpha|\ell^{2}}\right)^{2}\\\
E^{2}=\dfrac{\hslash^{2}c^{2}}{4|\alpha|\ell^{2}}\end{array}\right.,$ (1.158)
which can be solved by
$\displaystyle mc^{2}$ $\displaystyle=$
$\displaystyle\dfrac{p^{2}c^{2}}{2E}-E,$ (1.159) $\displaystyle E$
$\displaystyle=$ $\displaystyle\dfrac{\hslash c}{2\sqrt{{|\alpha|}}\ell}.$
(1.160)
However, the relation (1.159) is in itself the quadratic equation
$E^{2}=-mc^{2}E+\dfrac{p^{2}c^{2}}{2},$ (1.161)
which, by using (1.110) and (1.153), is the constraint
$E=-mc^{2}-\dfrac{p^{2}}{2m}+|\alpha|\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{p^{4}}{m},$
(1.162)
similar to the Pauli Hamiltonian constraint, and can be easy factorized
$E-\dfrac{\hslash^{2}}{16|\alpha|\ell^{2}}\dfrac{1}{m}=-|\alpha|\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}-\dfrac{\hslash^{2}}{4|\alpha|\ell^{2}}\right)^{2}+mc^{2},$
(1.163)
and considered by three way.
1. (a)
In the first one has the system
$\left\\{\begin{array}[]{l}mc^{2}=-\dfrac{\hslash^{2}}{16|\alpha|\ell^{2}}\dfrac{1}{m}\vspace*{5pt}\\\
E=-|\alpha|\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}-\dfrac{\hslash^{2}}{4|\alpha|\ell^{2}}\right)^{2}\end{array}\right.$
(1.164)
which gives
$\displaystyle m$ $\displaystyle=$ $\displaystyle
i\dfrac{\hslash}{4\sqrt{|\alpha|}c\ell},$ (1.165) $\displaystyle E$
$\displaystyle=$ $\displaystyle
mc^{2}+\dfrac{p^{2}}{2m}+\dfrac{p^{4}}{16m^{3}c^{2}}.$ (1.166)
Via using the mass (1.165) the energy (1.166) reads
$E=i\dfrac{\hslash
c}{4\sqrt{|\alpha|}\ell}-i2c\sqrt{|\alpha|}\dfrac{\ell}{\hslash}p^{2}+i4c\left(\sqrt{|\alpha|}\dfrac{\ell}{\hslash}\right)^{3}p^{4}.$
(1.167)
Jointing of the energy (4.127) and the mass (1.165) leads to
$iE=2mc^{2},$ (1.168)
what means that
$m^{2}=-\dfrac{E^{2}}{2c^{4}}<0,$ (1.169)
i.e. this case describes tachyon. Using of the first equation in (1.158)
together with the mass (1.165) allows to establish
$p^{2}=\dfrac{\hslash^{2}}{2|\alpha|\ell^{2}}\left(1+\dfrac{1}{2}i\right),$
(1.170)
what after using the fact that
$\sqrt{-1+\dfrac{1}{2}i}=\dfrac{1}{\sqrt{2}}\left(\sqrt{\dfrac{\sqrt{5}}{2}+1}+i\sqrt{\dfrac{\sqrt{5}}{2}-1}\right)e^{ni\pi},$
(1.171)
where $n\in\mathbb{Z}$ leads to the momentum spectrum
$p_{n}=\pm\dfrac{(-1)^{n}}{2}\left(\sqrt{\dfrac{\sqrt{5}}{2}+1}+i\sqrt{\dfrac{\sqrt{5}}{2}-1}\right)\dfrac{\hslash}{\sqrt{|\alpha|}\ell}.$
(1.172)
Interestingly, writing (1.134) as $p_{n}(|\alpha|)$ and (1.172) as
$p_{n}(-|\alpha|)$ one can define the spectral mean
$\langle
p_{n}(|\alpha|)p_{n}(-|\alpha|)\rangle=\dfrac{1}{|\alpha|-|\alpha_{0}|}\int_{|\alpha_{0}|}^{|\alpha|}p_{n}(x)p_{n}(-x)dx,$
(1.173)
which is easy to derive
$\langle p_{n}(|\alpha|)p_{n}(-|\alpha|)\rangle=\pm
i\dfrac{\sqrt{5}}{4}\dfrac{\hslash^{2}}{\ell^{2}}\dfrac{\ln\left|\dfrac{\alpha}{\alpha_{0}}\right|}{|\alpha|-|\alpha_{0}|},$
(1.174)
and it is not difficult to see that
$\lim_{|\alpha|\rightarrow|\alpha_{0}|}\langle
p_{n}(|\alpha|)p_{n}(-|\alpha|)\rangle=\pm\sqrt{5}\dfrac{\hslash}{2\sqrt{|\alpha_{0}|}\ell}\dfrac{\hslash}{2\sqrt{-|\alpha_{0}|}\ell}.$
(1.175)
Now it is not difficult to establish also the spectral means
$\displaystyle\langle|p_{n}(|\alpha|)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{|\alpha|-|\alpha_{0}|}\int_{|\alpha_{0}|}^{|\alpha|}|p_{n}(x)|^{2}dx,$
(1.176) $\displaystyle\langle|p_{n}(-|\alpha|)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{|\alpha|-|\alpha_{0}|}\int_{|\alpha_{0}|}^{|\alpha|}|p_{n}(-x)|^{2}dx,$
(1.177)
which are equal to
$\displaystyle\langle|p_{n}(|\alpha|)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{5}}{4}\dfrac{\hslash^{2}}{\ell^{2}}\dfrac{\ln\left|\dfrac{\alpha}{\alpha_{0}}\right|}{|\alpha|-|\alpha_{0}|},$
(1.178) $\displaystyle\langle|p_{n}(-|\alpha|)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{5}}{4}\dfrac{\hslash^{2}}{\ell^{2}}\dfrac{\ln\left|\dfrac{\alpha}{\alpha_{0}}\right|}{|\alpha|-|\alpha_{0}|},$
(1.179)
and have the limiting values
$\displaystyle\lim_{|\alpha|\rightarrow|\alpha_{0}|}\langle|p_{n}(|\alpha|)|^{2}\rangle$
$\displaystyle=$
$\displaystyle\dfrac{\sqrt{5}}{4}\dfrac{\hslash^{2}}{|\alpha_{0}|\ell^{2}},$
(1.180)
$\displaystyle\lim_{|\alpha|\rightarrow|\alpha_{0}|}\langle|p_{n}(-|\alpha|)|^{2}\rangle$
$\displaystyle=$
$\displaystyle\dfrac{\sqrt{5}}{4}\dfrac{\hslash^{2}}{|\alpha_{0}|\ell^{2}}.$
(1.181)
The formulas (1.178)-(1.179) and (1.174) allows to construct the variance
$\sigma^{2}=\langle|p_{n}(|\alpha|)|^{2}\rangle\langle|p_{n}(-|\alpha|)|^{2}\rangle-|\langle
p_{n}(|\alpha|)p_{n}(-|\alpha|)\rangle|^{2},$ (1.182)
which vanishes in general
$\sigma^{2}=0,$ (1.183)
as well as in the limiting case
$\lim_{|\alpha|\rightarrow|\alpha_{0}|}\sigma^{2}=0.$ (1.184)
One can compute also the following means
$\displaystyle\langle p_{n}(|\alpha|)\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{|\alpha|-|\alpha_{0}|}\int_{|\alpha_{0}|}^{|\alpha|}p_{n}(x)dx,$
(1.185) $\displaystyle\langle p_{n}(-|\alpha|)\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{|\alpha|-|\alpha_{0}|}\int_{|\alpha_{0}|}^{|\alpha|}p_{n}(-x)dx,$
(1.186)
and obtain the quantity
$\displaystyle\langle p_{n}(|\alpha|)\rangle\langle
p_{n}(-|\alpha|)\rangle=\pm
i\sqrt{5}\dfrac{\hslash^{2}}{\ell^{2}}\dfrac{1}{\left(\sqrt{|\alpha|}+\sqrt{|\alpha_{0}|}\right)^{2}},$
(1.187)
which, together with (1.174), allows to establish the variance
$\sigma^{2}=\langle p_{n}(|\alpha|)\rangle\langle
p_{n}(-|\alpha|)\rangle-\langle p_{n}(|\alpha|)p_{n}(-|\alpha|)\rangle,$
(1.188)
with the final result
$\sigma^{2}=\pm
i\dfrac{\sqrt{5}}{4}\dfrac{\hslash^{2}}{\ell^{2}}\left[\dfrac{4}{\left(\sqrt{|\alpha|}+\sqrt{|\alpha_{0}|}\right)^{2}}-\dfrac{\ln\left|\dfrac{\alpha}{\alpha_{0}}\right|}{|\alpha|-|\alpha_{0}|}\right].$
(1.189)
It is easy to derive now the limiting case of (1.190)
$\lim_{|\alpha|\rightarrow|\alpha_{0}|}\sigma^{2}=i0.$ (1.190)
2. (b)
The latter subcase is
$\left\\{\begin{array}[]{l}mc^{2}=E\\\
\dfrac{\hslash^{2}}{16\alpha\ell^{2}}\dfrac{1}{m}=\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}+\dfrac{\hslash^{2}}{4\alpha\ell^{2}}\right)^{2}\end{array}\right.$
(1.191)
Again, solution of the first equation via using (1.118) is rather simple
$m=-i\dfrac{\hslash}{2\sqrt{{\alpha}}c\ell}\quad,\quad m^{2}<0,$ (1.192)
and once again the tachyon is obtained - there are particles with momentum
spectrum
$p=\left\\{0,\pm\dfrac{\hslash}{\sqrt{{2\alpha}}\ell}\right\\}.$ (1.193)
This is discrete momenta spectrum for fixed scale $\ell$. For running scale
this is non compact spectrum, but compactification to the point is done in the
large scale limit
$\lim_{\ell\rightarrow\infty}p=0,$ (1.194)
and it is the rest. For $\alpha=0$ and fixed scale $\ell$ there are two
singular values of the momentum $p$. For all $\ell\neq 0$ and $\alpha\neq 0$,
the case of nonzero $p$ is related to the existence of tachyon.
3. (c)
The third interpretation yields
$\left\\{\begin{array}[]{l}-E=\dfrac{\hslash^{2}}{16\alpha\ell^{2}}\dfrac{1}{m}\vspace*{5pt}\\\
-mc^{2}=\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}\dfrac{1}{m}\left(p^{2}+\dfrac{\hslash^{2}}{4\alpha\ell^{2}}\right)^{2}\end{array}\right..$
(1.195)
Similarly as in the previous subcase, employing the relation (1.118) one
obtains from the first equation the mass
$m=-i\dfrac{\hslash}{8\sqrt{\alpha}c\ell},$ (1.196)
and again one has tachyon, i.e. $m^{2}<0$. Consequently the momentum spectrum
can be established as
$p=\left\\{\pm i\sqrt{{\dfrac{1}{8\alpha}}}\dfrac{\hslash}{\ell},\pm
i\sqrt{{\dfrac{3}{8\alpha}}}\dfrac{\hslash}{\ell}\right\\},$ (1.197)
and is non hermitian, as one has expected. By this reason this particular
case, that is the Pauli Hamiltonian constraint with mass related to minimum
scale, describes tachyon, the hypothetical particles with velocity faster then
light.
3. 3.
The third possible solution of the Snyder–Sidharth Hamiltonian constraint can
be constructed by the system of equations
$\left\\{\begin{array}[]{l}E^{2}=m^{2}c^{4}\vspace*{5pt}\\\
\dfrac{\hslash^{2}c^{2}}{4\alpha\ell^{2}}=\alpha\left(\dfrac{c\ell}{\hslash}\right)^{2}\left(p^{2}+\dfrac{\hslash^{2}}{2\alpha\ell^{2}}\right)^{2}\end{array}\right..$
(1.198)
First equation gives the standard Einstein mass-energy relation
$E=mc^{2},$ (1.199)
and the latter equality results in the discrete momentum spectrum
$p=\left\\{0,\pm\dfrac{1}{\sqrt{{\alpha}}}\dfrac{\hslash}{\ell}\right\\},$
(1.200)
for fixed scale $\ell$ value. For running scale this spectrum is non compact,
but in the large scale limit the spectrum is manifestly compactified to the
point
$\lim_{\ell\rightarrow\infty}p=0,$ (1.201)
and it is the rest. For $\alpha=0$ and fixed scale $\ell$ there are two
singular values of the momentum $p$. For all $\ell\neq 0$ and $\alpha\neq 0$,
the case of nonzero $p$ is related to the existence of a relativistic
particle.
#### C The Modified Compton Effect
Particle astrophysics has a great interest in diverse situations between light
and particles, and factually a lot of its conclusions arise via analysis of
this type phenomena from both the theoretical and the experimental points of
view (See e.g. general books in Ref. [120]), which give a physical information
about Cosmos. One of such phenomena is the Compton scattering, discovered by
A.H. Compton in 1923 [121], which is a mid-energy interaction of light and
matter. The scattering is realized via the electromagnetic radiation, X-rays
and gamma ($\gamma$) rays, undergo in matter, i.e. electrons. A decrease in
photon energy/wavelength due to the inelastic scattering is the point called
the Compton effect. We shall not discuss the detailed classical analysis of
the Compton scattering, because of factually the analysis is based on a
framework involving law of conservation of energy, law of conservation of
momentum, and concepts of Einstein Special Relativity. Let us consider the
case of the Compton effect within the framework employing the Snyder–Sidharth
Hamiltonian constraint (2.32). This can be done, however, in three different
routs minimally. In the case of lack of deformation all the ways lead to the
same result, called the Compton equation. However, in presence of the
deformation due to the Snyder geometry one obtains three various results. It
is not clear which of the formulations is correct, and also in general it is
not known are there other alternatives. Let us consider the three approaches
step by step.
##### C1 The Relativistic Approach
The our approach is based on the standard Special Relativity formulation (See
e.g. Ref. [122]). Let us consider a photon and an electron having the energy-
momentum four-vectors
$\displaystyle p^{\mu}_{\gamma}$ $\displaystyle=$
$\displaystyle[E_{\gamma}\leavevmode\nobreak\ ,\leavevmode\nobreak\
p_{\gamma}^{i}c],$ (1.202) $\displaystyle p^{\mu}_{e}$ $\displaystyle=$
$\displaystyle[E_{0}\leavevmode\nobreak\ ,\leavevmode\nobreak\ 0],$ (1.203)
where $E_{0}=m_{e}c^{2}$ is the electron rest energy. An electron at rest is
scattered by an incoming photon, and an outgoing photon is observed under the
scattering angle $\theta$ relatively to the incident direction of an incoming
photon. The final energy-momentum four-vectors are
$\displaystyle p^{\mu}_{\gamma^{\prime}}$ $\displaystyle=$
$\displaystyle[E_{\gamma^{\prime}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
p_{\gamma^{\prime}}^{i}c],$ (1.204) $\displaystyle p^{\mu}_{e^{\prime}}$
$\displaystyle=$ $\displaystyle[E_{e}\leavevmode\nobreak\
,\leavevmode\nobreak\ p_{e}^{i}c].$ (1.205)
Let us preserve unchanged the Planck–Einstein relations of the wave-particle
duality. Thus for a photon one has
$\displaystyle E_{\gamma}$ $\displaystyle=$ $\displaystyle\hslash\omega,$
(1.206) $\displaystyle p_{\gamma}^{i}$ $\displaystyle=$ $\displaystyle\hslash
k^{i},$ (1.207)
where $\omega$ and $k_{\gamma}$ are angular frequency and wave vector, and the
value of wave vector of a photon is
$k=\sqrt{k_{i}k^{i}}=\dfrac{2\pi}{\lambda}=\dfrac{1}{\not{\lambda}},$ (1.208)
where $\not{\lambda}$ is reduced wavelength of a photon. Similarly for a
matter particle, including electrons, one has
$\displaystyle E_{e}$ $\displaystyle=$ $\displaystyle\hslash\omega_{e},$
(1.209) $\displaystyle p_{e}^{i}$ $\displaystyle=$ $\displaystyle\hslash
k_{e}^{i},$ (1.210)
where $\omega_{e}$ and $k_{e}$ are angular frequency and wave vector of an
electron, and the value of wave vector of an electron is
$k_{e}=\sqrt{{k_{e}}_{i}k_{e}^{i}}=\dfrac{2\pi}{\lambda_{e}}=\dfrac{1}{\not{\lambda}_{e}},$
(1.211)
where $\not{\lambda}_{e}$ is reduced wavelength of an electron.
In addition we assume that Special Relativity is deformed due to the Snyder
noncommutative geometry, i.e. that for any photon $\gamma$ and any electron
$e$ are satisfied the Snyder–Sidharth Hamiltonian constraints
$\displaystyle E^{2}_{\gamma}$ $\displaystyle=$ $\displaystyle
p_{\gamma}^{2}c^{2}+\dfrac{1}{\epsilon^{2}}\left(p_{\gamma}^{2}c^{2}\right)^{2},$
(1.212) $\displaystyle E^{2}_{e}$ $\displaystyle=$ $\displaystyle
E_{0}^{2}+p_{e}^{2}c^{2}+\dfrac{1}{\epsilon^{2}}\left(p_{e}^{2}c^{2}\right)^{2},$
(1.213)
where $p_{e}=\sqrt{{p_{e}}_{i}p_{e}^{i}}$ and
$p_{\gamma}=\sqrt{{p_{\gamma}}_{i}{p_{\gamma}}^{i}}$ are values of momenta of
an electron and a photon, respectively, and for shortened notation we have
introduced the following energy parameter
$\epsilon=\dfrac{\hslash c}{\sqrt{\alpha}\ell}=\dfrac{\hslash
c}{\ell_{P}}\dfrac{1}{\sqrt{\alpha}}\dfrac{\ell_{P}}{\ell}=E_{P}\dfrac{1}{\sqrt{\alpha}}\dfrac{\ell_{P}}{\ell},$
(1.214)
where $E_{P}=\sqrt{\dfrac{\hslash c^{5}}{G}}$ is the Planck energy. By
application of the wave-particle duality the Snyder–Sidharth Hamiltonian
constraints for a photon and an electron can be presented as
$\displaystyle\dfrac{\omega^{2}}{c^{2}}$ $\displaystyle=$ $\displaystyle
k^{2}+\dfrac{k^{4}}{\kappa^{2}},$ (1.215)
$\displaystyle\dfrac{\omega^{2}_{e}}{c^{2}}$ $\displaystyle=$ $\displaystyle
k_{C}^{2}+k_{e}^{2}+\dfrac{k_{e}^{4}}{\kappa^{2}},$ (1.216)
where $k_{C}$ is the wave vector
$k_{C}=\dfrac{2\pi}{\lambda_{C}}=\dfrac{1}{\not{\lambda}_{C}},$ (1.217)
related to the Compton wavelength $\lambda_{C}$ of an electron, and its
reduced form
$\not{\lambda}_{C}=\dfrac{\hslash c}{E_{0}}=\dfrac{\hslash}{m_{e}c},$ (1.218)
and $\kappa$ is the parameter
$\kappa=\dfrac{\epsilon}{\hslash c}=\dfrac{1}{\sqrt{\alpha}\ell},$ (1.219)
which can be interpreted as a wave vector value associated to the (non-
reduced!) wavelength $\lambda_{\kappa}=2\pi\sqrt{\alpha}\ell$, which for
$\alpha=\dfrac{1}{(2\pi)^{2}},$ (1.220)
becomes $\lambda_{\kappa}\equiv\ell$. The parameter (1.214) can be physically
interpreted as the Æther energy, so $\kappa$ is the wave vector value of the
Æther wave-particle.
In our interesting is the relation for the angle $\theta$ of scattered photon,
i.e. the angle between the wave vectors $k_{i}$ and $k^{\prime}_{i}$. Let us
apply first the law of conservation of energy-momentum, i.e.
$p^{\mu}_{\gamma}+p^{\mu}_{e}=p^{\mu}_{\gamma^{\prime}}+p^{\mu}_{e^{\prime}}.$
(1.221)
For the momentum this principle gives
$\displaystyle p_{\gamma}^{i}c=p_{\gamma^{\prime}}^{i}c+p_{e}^{i}c,$ (1.222)
or in terms of the wave vectors
$\displaystyle k^{i}=k^{\prime i}+k_{e}^{i},$ (1.223)
and by elementary algebraic operations leads to the result
$k_{e}^{2}=k^{2}+k^{\prime 2}-2kk^{\prime}\cos\theta.$ (1.224)
Next one can use the law of conservation of energy which, in the light of
(1.221), for the present case has the following form
$E_{\gamma}+E_{0}=E_{\gamma^{\prime}}+E_{e},$ (1.225)
and by this reason one obtains
$\dfrac{\omega_{e}}{c}=k_{C}+\dfrac{\omega}{c}-\dfrac{\omega^{\prime}}{c}.$
(1.226)
Taking square of both sides of the equation (1.226)
$\dfrac{\omega_{e}^{2}}{c^{2}}=k_{C}^{2}+\dfrac{\omega^{2}}{c^{2}}+\dfrac{\omega^{\prime
2}}{c^{2}}-2\dfrac{\omega}{c}\dfrac{\omega^{\prime}}{c}+2k_{C}\left(\dfrac{\omega}{c}-\dfrac{\omega^{\prime}}{c}\right),$
(1.227)
and applying the Snyder–Sidharth Hamiltonian constraint for photons and an
electron, one receives the relation
$\displaystyle k_{C}^{2}+k_{e}^{2}+\dfrac{k_{e}^{4}}{\kappa^{2}}$
$\displaystyle=$ $\displaystyle
k_{C}^{2}+k^{2}+\dfrac{k^{4}}{\kappa^{2}}+k^{\prime 2}+\dfrac{k^{\prime
4}}{\kappa^{2}}-2\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}\sqrt{k^{\prime
2}+\dfrac{k^{\prime 4}}{\kappa^{2}}}+$ (1.228) $\displaystyle
2k_{C}\left(\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}-\sqrt{k^{\prime
2}+\dfrac{k^{\prime 4}}{\kappa^{2}}}\right),$
or after small reduction
$\displaystyle k_{e}^{2}+\dfrac{k_{e}^{4}}{\kappa^{2}}$ $\displaystyle=$
$\displaystyle k^{2}+k^{\prime 2}+\dfrac{k^{4}}{\kappa^{2}}+\dfrac{k^{\prime
4}}{\kappa^{2}}-2\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}\sqrt{k^{\prime
2}+\dfrac{k^{\prime 4}}{\kappa^{2}}}+$ (1.229) $\displaystyle
2k_{C}\left(\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}-\sqrt{k^{\prime
2}+\dfrac{k^{\prime 4}}{\kappa^{2}}}\right),$
Applying the relation (1.224) within LHS of the equation (1.229), some
algebraic identities, and factorization one obtains
$\displaystyle-\cos\theta+\dfrac{\kappa^{2}}{2kk^{\prime}}\dfrac{\left(k^{2}+k^{\prime
2}\right)^{2}}{\kappa^{4}}-2\dfrac{k^{2}+k^{\prime
2}}{\kappa^{2}}\cos\theta+\dfrac{2kk^{\prime}}{\kappa^{2}}\cos^{2}\theta=$
$\displaystyle\dfrac{\kappa^{2}}{2kk^{\prime}}\dfrac{k^{4}+k^{\prime
4}}{\kappa^{4}}-\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}+k_{C}\left(\dfrac{1}{k^{\prime}}\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}-\dfrac{1}{k}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}\right).$ (1.230)
Now by ordering in $\cos\theta$ powers one receives the quadratic equation
$\displaystyle\beta_{\kappa}\cos^{2}\theta-\delta_{\kappa}\cos\theta+\eta_{\kappa}=0,$
(1.231)
where we have introduced the notation
$\displaystyle\beta_{\kappa}=\dfrac{2kk^{\prime}}{\kappa^{2}},$ (1.232)
$\displaystyle\delta_{\kappa}=1+2\dfrac{k^{2}+k^{\prime 2}}{\kappa^{2}},$
(1.233)
$\displaystyle\eta_{\kappa}=\dfrac{kk^{\prime}}{\kappa^{2}}+\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}-k_{C}\left(\dfrac{1}{k^{\prime}}\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}-\dfrac{1}{k}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}\right).$ (1.234)
First of all, one sees straightforwardly that in the Special Relativity limit
$\ell\rightarrow 0$, i.e. $\kappa\rightarrow\infty$, the coefficients (1.232),
(1.233), and (1.234) reads
$\displaystyle\beta_{\infty}=0,$ (1.235) $\displaystyle\delta_{\infty}=1,$
(1.236)
$\displaystyle\eta_{\infty}=1-k_{C}\left(\dfrac{1}{k^{\prime}}-\dfrac{1}{k}\right),$
(1.237)
so that the modified Compton equation (1.231) becomes linear in the cosinus
$\displaystyle-\cos\theta+1-k_{C}\left(\dfrac{1}{k^{\prime}}-\dfrac{1}{k}\right)=0,$
(1.238)
and its solution can be presented in the form
$\not{\lambda}_{C}(1-\cos\theta)=\dfrac{1}{k^{\prime}}-\dfrac{1}{k},$ (1.239)
called the Compton equation, describing the photon-electron scattering in
frames of Special Relativity as originally deduced by A.H. Compton. Because
the boundaries $-1\leq\cos\theta\leq 1$ hold, one obtains that in the Compton
effect the nontrivial restriction
$\lambda\leq\lambda^{\prime}\leq\lambda+2\lambda_{C},$ (1.240)
for the outgoing photon wavelength, holds.
##### C2 The Relativistic Limit. The Lensing Hypothesis.
However, in general the Snyder–Sidharth deformation results in the modified
Compton equation (1.231) which has solutions different from the Compton
equation (1.239). By straightforward easy computation one obtains formally two
mathematically and physically distinguish solutions of the equation (1.231)
$\cos\theta=\dfrac{1}{2\beta_{\kappa}}\left(\delta_{\kappa}\pm\sqrt{\delta^{2}_{\kappa}-4\beta_{\kappa}\eta_{\kappa}}\right),$
(1.241)
and in this way one can determine the exact formulas for the cosinus
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\cos\theta=\left[1\pm\sqrt{g^{2}\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)+\dfrac{2k_{C}}{\kappa}g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)+1}\right]\dfrac{\kappa^{2}}{4kk^{\prime}}+\dfrac{1}{2}\left(\dfrac{k}{k^{\prime}}+\dfrac{k^{\prime}}{k}\right),$
(1.242)
where we have introduced the dimensionless quantity
$g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)=\dfrac{2E_{\gamma}}{\kappa}-\dfrac{2E_{\gamma^{\prime}}}{\kappa},$
(1.243)
where we have used the Snyder–Sidharth Hamiltonian constraint for photons
(1.212). In the light of the law of conservation of energy (1.225) one sees
that the function (4.87) in fact can be completely expressed via energies of
rest and scattered electron
$g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)=\dfrac{2(E_{\gamma}-E_{\gamma^{\prime}})}{\kappa}=\dfrac{2(E_{e}-E_{0})}{\kappa},$
(1.244)
where $E_{e}$ is given by the Snyder–Sidharth constraint for electrons
(1.213). It means that the function (4.87) is always positively defined
$g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)\geq 0$. It can be
seen straightforwardly that if one exchanges momenta of photons
$k^{\prime}\leftrightarrow k$ then the function (4.87) changes the sign
$g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)\rightarrow-g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)$,
i.e. $g\left(\dfrac{k}{\kappa},\dfrac{k^{\prime}}{\kappa}\right)$ is an odd
function with respect to such an exchange. Therefore, the presence of the
square root term in the cosinus formula (1.242) breaks crossing symmetry
manifestly. This fact contradicts to the nature of cosinus which is an even
function. It results in the conclusion that presence of a minimal scale breaks
crossing symmetry. Moreover, in general the cosinus (1.242) does not belong to
the range of cosinus $[-1,1]$, but we shall discuss and solve this problematic
point later.
In the relativistic limit $\kappa\rightarrow\infty$ the Eqs. (1.242) behave
like
$\displaystyle\cos\theta=(1\pm
1)\dfrac{\kappa^{2}}{4kk^{\prime}}+\dfrac{1}{2}\left(\dfrac{k}{k^{\prime}}+\dfrac{k^{\prime}}{k}\right).$
(1.245)
For the minus sign case the second term in (1.245) vanishes as
$\kappa\rightarrow\infty$, while the plus sign case leads to non
renormalizable divergence. In other words the physical solution is the
solution with the minus sign, and when one applies the limit
$\kappa\rightarrow\infty$ it gives the finite result
$\not{\lambda}_{C}(1-\cos\theta)=\dfrac{1}{k_{C}}\left(1-\dfrac{1}{2}\left(\dfrac{k}{k^{\prime}}+\dfrac{k^{\prime}}{k}\right)\right),$
(1.246)
which blatantly differs from the Compton equation (1.239). It means that
despite using of the relativistic limit to the deformed scattering the
received result does not reconstruct the relativistic scattering.
Anyway, for conceptual correctness one can compare _ad hoc_ the equation
(1.246) with the Compton equation (1.239)
$\dfrac{1}{k^{\prime}}-\dfrac{1}{k}=\dfrac{1}{k_{C}}\left(1-\dfrac{1}{2}\left(\dfrac{k}{k^{\prime}}+\dfrac{k^{\prime}}{k}\right)\right),$
(1.247)
and easy obtain the condition for which the relativistic limit agrees with the
Compton equation
$k^{\prime 2}-2(k_{C}+k)k^{\prime}+k^{2}+2k_{C}k=0.$ (1.248)
This condition is satisfied for two cases $k^{\prime}=k_{C}+k\pm k_{C}$, i.e.
$k^{\prime}-k=2k_{C}\qquad\textrm{or}\qquad k^{\prime}-k=0,$ (1.249)
which can be expressed by incoming and outgoing photons wavelengths
$\dfrac{1}{\lambda^{\prime}}-\dfrac{1}{\lambda}=\dfrac{2}{\lambda_{C}}\qquad\textrm{or}\qquad\dfrac{1}{\lambda^{\prime}}-\dfrac{1}{\lambda}=0.$
(1.250)
The first solution looks like special case of the lensmaker’s equation,
$\dfrac{1}{r_{1}}-\dfrac{1}{r_{2}}+\left(1-\dfrac{1}{n}\right)\dfrac{\delta}{r_{1}r_{2}}=\dfrac{1}{(n-1)f},$
(1.251)
for the refractive index $n=3/2=c/v$, focal length of the lens
$f=\lambda_{C}$, and lens radii of curvature $r_{1}=\lambda^{\prime}$ and
$r_{2}=\lambda$, and the thickness of the lens $\delta=0$, i.e. the thin lens
creating by the medium in which speed of light is $v=2c/3$. The second one
corresponds with the telescopic lens case $f=\infty$ in this medium, but
because photon wavelength does not change in this case we shall consider this
solution as nonphysical.
In this way one can take into account the _ad hoc_ generalization
$\dfrac{\omega^{\prime}}{c}-\dfrac{\omega}{c}+\delta\dfrac{n-1}{n}\dfrac{\omega}{c}\dfrac{\omega^{\prime}}{c}=\dfrac{k_{C}}{n-1},$
(1.252)
which in the relativistic limit $\kappa\rightarrow\infty$ leads to
$k^{\prime}-k+\delta\dfrac{n-1}{n}kk^{\prime}=\dfrac{k_{C}}{n-1},$ (1.253)
i.e. with $k=1/\not{\lambda}=\dfrac{2\pi}{\lambda}$, $\delta=0$, and $n=3/2$
reconstructs the first solution (1.250). The problem is to establish the
linkage between the thickness $d$ and a minimal scale $\ell$, and in general
$n>1$ as a numerical coefficient. We shall call (1.252) _the lensing
hypothesis_.
##### C3 Bounds on the Modified Compton Equation
Let us construct straightforwardly the general solution of the modified
Compton equation (1.231). Because $\delta_{\kappa}\neq 0$ always holds, let us
extract $\cos\theta$ manifestly from the Eq. (1.231)
$\cos\theta=\dfrac{\beta_{\kappa}}{\delta_{\kappa}}\cos^{2}\theta+\dfrac{\eta_{\kappa}}{\delta_{\kappa}}.$
(1.254)
Then however, one must involve the fact that $\cos\theta$ is bounded function,
i.e. $-1\leq\cos\theta\leq 1$. Application of these boundaries to (1.254)
leads to the following restrictions for $\cos^{2}\theta$
$-\dfrac{\delta_{\kappa}+\eta_{\kappa}}{\beta_{\kappa}}\leq\cos^{2}\theta\leq\dfrac{\delta_{\kappa}-\eta_{\kappa}}{\beta_{\kappa}}.$
(1.255)
The equation (1.255), however, must be considered with taking into account the
fact $\cos^{2}\theta$ is naturally bounded to $0\leq\cos^{2}\theta\leq 1$.
Employment of these boundary values results in the system of constraints
$\displaystyle-\dfrac{\delta_{\kappa}+\eta_{\kappa}}{\beta_{\kappa}}$
$\displaystyle\equiv$ $\displaystyle 0,$ (1.256)
$\displaystyle\dfrac{\delta_{\kappa}-\eta_{\kappa}}{\beta_{\kappa}}$
$\displaystyle\equiv$ $\displaystyle 1,$ (1.257)
which factually reduces momentum space of outgoing and incoming photons. In
other words, we consider the equation (1.231) as the basic relation for the
scattering, but no its solution which is secondary result. Let us see the
consequences of such an approach.
For $\beta_{\kappa}\neq 0$, i.e. factually for nonzero values of incoming and
outgoing photon momenta, the reduction is given by the equivalent conditions
$\displaystyle\delta_{\kappa}+\eta_{\kappa}$ $\displaystyle=$ $\displaystyle
0,$ (1.258) $\displaystyle\delta_{\kappa}-\eta_{\kappa}$ $\displaystyle=$
$\displaystyle\beta_{\kappa},$ (1.259)
which are not difficult to solve. The results are
$\displaystyle\delta_{\kappa}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\beta_{\kappa},$ (1.260)
$\displaystyle\eta_{\kappa}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\beta_{\kappa}.$ (1.261)
It is easy to see that the first condition says that
$2\dfrac{k^{\prime 2}+k^{2}}{\kappa^{2}}-\dfrac{kk^{\prime}}{\kappa^{2}}+1=0,$
(1.262)
while the second constraint can be written as
$2\dfrac{kk^{\prime}}{\kappa^{2}}+\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}-k_{C}\left(\dfrac{1}{k^{\prime}}\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}-\dfrac{1}{k}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}\right)=0.$ (1.263)
Application of (1.262) and (1.263) allows to reduce the equation (1.231)
$\cos^{2}\theta-\dfrac{1}{2}\cos\theta-\dfrac{1}{2}=0.$ (1.264)
Because of (1.238) does not contain any wave vectors therefore its solutions
are constant and independent on photons wavelengths. In the usual Compton
effect the cosinus depends on the photon wavelengths. In the modified case the
cosinus (1.242) in general does not satisfy the cosinus variability range
$[-1,1]$, and application of these limits to the equation (1.231) led us to
the bounds (1.262)-(1.263), what resulted in the equation (1.264). In this way
the Compton effect modified due to the Snyder–Sidharth Hamiltonian constraint
is solved by a constant scattering angle which solves the equation (1.264),
and the values of the wave vectors $k$ and $k^{\prime}$ following from the
bounds (1.262)-(1.263). The constant angle values are easy to derive
$\displaystyle\cos\theta$ $\displaystyle=$ $\displaystyle
1\longrightarrow\theta=2n\pi,$ (1.265) $\displaystyle\cos\theta$
$\displaystyle=$
$\displaystyle-\dfrac{1}{2}\longrightarrow\theta=\pm\dfrac{\pi}{3}+2n\pi,$
(1.266)
where $n\in\textbf{Z}$. The first solution (1.265) means no scattering or
backward scattering. Solutions of the system (1.262)-(1.263) are not easy to
extract. With no additional constraint(s) between $k$ and $k^{\prime}$ this
system leads to a polynomial equation of more than 40 degree which must be
treated by complicated methods of the Galois group.
However, one can use the lensing hypothesis (1.252) as the additional
constraint which written out explicitly is
$\sqrt{k^{\prime 2}+\dfrac{k^{\prime
4}}{\kappa^{2}}}-\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}+\delta\dfrac{n-1}{n}\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}\sqrt{k^{\prime
2}+\dfrac{k^{\prime 4}}{\kappa^{2}}}=\dfrac{k_{C}}{n-1},$ (1.267)
what after elementary algebraic manipulations takes the form
$k_{C}\left(\dfrac{1}{k^{\prime}}\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}-\dfrac{1}{k}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}\right)=k_{C}\delta\dfrac{n-1}{n}\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}-\dfrac{\kappa^{2}}{kk^{\prime}}\dfrac{1}{n-1}\dfrac{k_{C}^{2}}{\kappa^{2}}.$
(1.268)
In other words, using of the additional constraint (1.268) within the
constraint (1.263) gives the result
$2\dfrac{kk^{\prime}}{\kappa^{2}}+\left(1+k_{C}\delta\dfrac{n-1}{n}\right)\sqrt{1+\dfrac{k^{2}}{\kappa^{2}}}\sqrt{1+\dfrac{k^{\prime
2}}{\kappa^{2}}}-\dfrac{\kappa^{2}}{kk^{\prime}}\dfrac{1}{n-1}\dfrac{k_{C}^{2}}{\kappa^{2}}=0,$
(1.269)
which can be presented as
$\sqrt{1+\dfrac{k^{2}+k^{\prime
2}}{\kappa^{2}}+\left(\dfrac{kk^{\prime}}{\kappa^{2}}\right)^{2}}=\dfrac{n}{(1+k_{C}\delta)n-k_{C}\delta}\left(\dfrac{1}{n-1}\dfrac{k_{C}^{2}}{\kappa^{2}}\dfrac{\kappa^{2}}{kk^{\prime}}-2\dfrac{kk^{\prime}}{\kappa^{2}}\right).$
(1.270)
Now one can apply the constraint (1.262) to the LHS of the equation (1.270).
It gives
$\sqrt{\dfrac{1}{2}+\dfrac{1}{2}\dfrac{kk^{\prime}}{\kappa^{2}}+\left(\dfrac{kk^{\prime}}{\kappa^{2}}\right)^{2}}=\dfrac{n}{(1+k_{C}\delta)n-k_{C}\delta}\left(\dfrac{1}{n-1}\dfrac{k_{C}^{2}}{\kappa^{2}}\dfrac{\kappa^{2}}{kk^{\prime}}-2\dfrac{kk^{\prime}}{\kappa^{2}}\right).$
(1.271)
By squaring both sides of the equation (1.271), doing very few elementary
algebraic manipulations, and grouping the result with respect to powers of
$x=\dfrac{kk^{\prime}}{\kappa^{2}}$ one obtains finally the condition
$Ax^{4}+Bx^{3}+Cx^{2}-D=0$ (1.272)
where we have introduced the coefficients
$\displaystyle A$ $\displaystyle=$ $\displaystyle
2\dfrac{(n-1)^{3}}{n}k_{C}\delta\left(2+\dfrac{n-1}{n}k_{C}\delta\right),$
(1.273) $\displaystyle B$ $\displaystyle=$
$\displaystyle(n-1)^{2}\left(1+\dfrac{n-1}{n}k_{C}\delta\right)^{2},$ (1.274)
$\displaystyle C$ $\displaystyle=$
$\displaystyle(n-1)^{2}\left(1+\dfrac{n-1}{n}k_{C}\delta\right)^{2}+4(n-1)\dfrac{k_{C}^{2}}{\kappa^{2}},$
(1.275) $\displaystyle D$ $\displaystyle=$ $\displaystyle
2\dfrac{k_{C}^{4}}{\kappa^{4}},$ (1.276)
which are always positive and nonzero. Naturally, the equation (1.272) in
general can be solved and possesses solutions
$k^{\prime}=x\dfrac{\kappa^{2}}{k},$ (1.277)
where $x$ is a coefficient. Because of $k^{\prime}\geqslant 0$ and $k>0$, the
real and positive values of $x$ are physical. In this way there are two
physical $x$
$x=\dfrac{1}{4A}\left(\sqrt{2a-b-c\pm\dfrac{d}{\sqrt{a+b+c}}}-\sqrt{a+b+c}-B\right),$
(1.278)
with the conditions
$\displaystyle a+b+c$ $\displaystyle>$ $\displaystyle 0,$ (1.279)
$\displaystyle 2a\pm\dfrac{d}{\sqrt{a+b+c}}$ $\displaystyle\geqslant$
$\displaystyle b+c,$ (1.280)
$\displaystyle\sqrt{2a-b-c\pm\dfrac{d}{\sqrt{a+b+c}}}$
$\displaystyle\geqslant$ $\displaystyle B+\sqrt{a+b+c},$ (1.281)
where we have introduced the shortened notation
$\displaystyle a$ $\displaystyle=$ $\displaystyle B^{2}-\frac{8}{3}CA,$
(1.282) $\displaystyle b$ $\displaystyle=$
$\displaystyle\dfrac{4}{3}2^{1/3}A\beta\left(\alpha+\sqrt{\alpha^{2}-4\beta^{3}}\right)^{-\frac{1}{3}},$
(1.283) $\displaystyle c$ $\displaystyle=$ $\displaystyle
2^{1/3}A\left(\alpha+\sqrt{\alpha^{2}-4\beta^{3}}\right)^{\frac{1}{3}},$
(1.284) $\displaystyle d$ $\displaystyle=$
$\displaystyle\dfrac{2B}{A}\left(-B^{2}+4CA\right),$ (1.285)
and $\alpha=2C^{3}-27B^{2}D+72ACD$, $\beta=C^{2}-12AD$.
Interestingly, one can see easy that the case $n=1$ applied to (1.272) leads
to $D=0$, what is true for the only relativistic limit
$\kappa\rightarrow\infty$. However, the result of the relativistic limit
applied to the equation (1.272) has also different countenance. Before taking
the limit one must reduce this equation maximally with respect to powers of
$1/\kappa\rightarrow 0$, i.e.
$A\dfrac{\left(kk^{\prime}\right)^{4}}{\kappa^{4}}+B\dfrac{\left(kk^{\prime}\right)^{3}}{\kappa^{2}}+C\left(kk^{\prime}\right)^{2}-2k_{C}^{4}=0,$
(1.286)
and by this reason in such a limit one obtains the condition
$(n-1)^{2}\left(1+\dfrac{n-1}{n}k_{C}\delta\right)^{2}\left(kk^{\prime}\right)^{2}-2k_{C}^{4}=0,$
(1.287)
which is easy to solve
$k^{\prime}=\dfrac{\dfrac{\sqrt{2}}{n-1}}{1+\dfrac{n-1}{n}k_{C}\delta}\dfrac{k_{C}^{2}}{k}\sim\dfrac{1}{k},$
(1.288)
where we neglected negative solution as non physical. It is seen that $n=1$ is
not appropriate for such a solution.
#### D The Dispersional Generalization
Let us construct finally the other approach to the modified Compton effect
argued by the generalization due to the specific form of the dispersion
relations. For this let us consider first the deduction due to the unmodified
case, i.e. Special Relativity. In such a situation the dispersion relations
for a photon and an electron have the form
$\displaystyle\dfrac{\omega^{2}}{c^{2}}$ $\displaystyle=$ $\displaystyle
k^{2},$ (1.289) $\displaystyle\dfrac{\omega^{2}_{e}}{c^{2}}$ $\displaystyle=$
$\displaystyle k_{C}^{2}+k_{e}^{2}.$ (1.290)
By this reason the laws of conservation of momentum (1.224) and energy (1.226)
expressed via the dispersion relations (1.289) and (1.290) have the following
form
$\displaystyle\dfrac{\omega^{2}_{e}}{c^{2}}-k_{C}^{2}$ $\displaystyle=$
$\displaystyle\dfrac{\omega^{2}}{c^{2}}+\dfrac{\omega^{\prime
2}}{c^{2}}-\dfrac{\omega}{c}\dfrac{\omega^{\prime}}{c}\cos\theta,$ (1.291)
$\displaystyle\dfrac{\omega_{e}}{c}$ $\displaystyle=$ $\displaystyle
k_{C}+\dfrac{\omega}{c}-\dfrac{\omega^{\prime}}{c}.$ (1.292)
and after elementary algebraic manipulations lead to the result
$\not{\lambda}_{C}(1-\cos\theta)=\dfrac{c}{\omega^{\prime}}-\dfrac{c}{\omega},$
(1.293)
which after application of the explicit form of the dispersion relation
(1.289) gives the Compton equation (1.239). In this manner we shall call this
equation _the generalized Compton equation_.
Similar reasoning can be performed to the modified dispersion relations (1.215
and (1.216). Then the laws of conservation of momentum and energy expressed
via the dispersion relations are preserved in the form (1.291)-(1.292)
obtained by Special Relativity. Therefore, also the Compton equation (1.293)
is preserved. However, the change is due to the specific dispersion relations
(1.215) and (1.216), so that the modified Compton effect is described by the
equation
$\not{\lambda}_{C}(1-\cos\theta)=\dfrac{1}{\sqrt{k^{\prime 2}+\dfrac{k^{\prime
4}}{\kappa^{2}}}}-\dfrac{1}{\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}}.$ (1.294)
Interestingly, in the relativistic limit $\kappa\rightarrow\infty$ the
modified Compton equation (1.294) coincides with the usual Compton equation
(1.293) with no additional presumptions. We shall call such an approach _the
dispersional generalization_. It is clear that this generalization should be
working also for elementary processes other than the Compton scattering. By
this reason let us express the proposition
###### Proposition (The Dispersional Generalization).
Let us presume that there is an initial theory of elementary precesses having
established results, which is characterized by certain dispersion relations
and the laws of conservation. Let us consider the theory due to a modification
of an initial theory. The consistent analysis of an arbitrary elementary
process within the modified theory is based on the three-step procedure:
1. 1.
Derivation of the dispersion relations due of the modified theory,
2. 2.
Application of the dispersion relations to the laws of conservation of the
modified theory,
3. 3.
Using of the explicit form of the dispersion relations within the results
obtained due to the laws of conservation in frames of the modified theory.
The analysis is consistent if and only if the results of the modified theory
coincide with the results of an initial theory for lack of the modification.
Because of $-1\leqslant\cos\theta\leqslant 1$, the relation (1.294) allows to
establish the bounds for wave vector of an outgoing photon
$k^{\prime}_{min}\leqslant k\leqslant k^{\prime}_{max}$, where
$\displaystyle k^{\prime}_{min}$ $\displaystyle=$
$\displaystyle\dfrac{\kappa}{\sqrt{2}}\sqrt{\sqrt{1+\dfrac{4}{\kappa^{2}}\left(k^{2}+\dfrac{k^{4}}{\kappa^{2}}\right)}-1},$
(1.295) $\displaystyle k^{\prime}_{max}$ $\displaystyle=$
$\displaystyle\dfrac{\kappa}{\sqrt{2}}\sqrt{\sqrt{1+\dfrac{4}{\kappa^{2}}\dfrac{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}{\left(1+2\not{\lambda}_{C}\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}\right)^{2}}}-1}.$
(1.296)
Let us apply the lensing hypothesis to the obtained general result (1.293).
The formula (1.252) can be rewritten in the form
$\dfrac{c}{\omega^{\prime}}-\dfrac{c}{\omega}=\delta\dfrac{n-1}{n}-\dfrac{c}{\omega}\dfrac{c}{\omega^{\prime}}\dfrac{k_{C}}{n-1},$
(1.297)
what means that the generalized Compton equation (1.293) becomes
$\not{\lambda}_{C}(1-\cos\theta)=\delta\dfrac{n-1}{n}-\dfrac{c}{\omega}\dfrac{c}{\omega^{\prime}}\dfrac{k_{C}}{n-1}.$
(1.298)
In this manner, by application of the identification method , one can
establish the following relations
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\dfrac{n}{n-1}\not{\lambda}_{C},$ (1.299)
$\displaystyle\cos\theta$ $\displaystyle=$
$\displaystyle\dfrac{c}{\omega}\dfrac{c}{\omega^{\prime}}\dfrac{k_{C}^{2}}{n-1},$
(1.300)
what allows to derive
$\cos\theta=\dfrac{c}{\omega}\dfrac{c}{\omega^{\prime}}\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}}.$
(1.301)
Because, however, $-1\leq\cos\theta\leq 1$ one has the bound
$\dfrac{\omega^{\prime}}{c}\geqslant\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}}\dfrac{c}{\omega},$
(1.302)
which for the Compton effect means that
$k^{\prime}\geqslant\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}}\dfrac{1}{k},$
(1.303)
while for the modified Compton effect gives
$k^{\prime}\geqslant\dfrac{\kappa}{\sqrt{2}}\left[\sqrt{1+\dfrac{4}{\kappa^{2}}\left(\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}}\right)^{2}\dfrac{1}{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}}-1\right].$
(1.304)
In the case of the usual Compton effect the formula (1.301) gives
$\cos\theta=\dfrac{1}{k}\dfrac{1}{k^{\prime}}\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}},$
(1.305)
while in the modified case one receives
$\cos\theta=\dfrac{1}{\sqrt{k^{\prime 2}+\dfrac{k^{\prime
4}}{\kappa^{2}}}}\dfrac{1}{\sqrt{k^{2}+\dfrac{k^{4}}{\kappa^{2}}}}\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}}.$
(1.306)
The lensing hypothesis (1.297) together with (1.299) leads to
$\dfrac{\omega^{\prime}}{c}=\dfrac{1+\dfrac{c}{\omega}\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}^{2}}}{\not{\lambda}_{C}+\dfrac{c}{\omega}},$
(1.307)
what together with (1.302) leads to the another bound
$\dfrac{\omega}{c}\geqslant\dfrac{1}{\not{\lambda}_{C}}\sqrt{\dfrac{\delta}{\not{\lambda}_{C}}-1}=\dfrac{\sqrt{n-1}}{\not{\lambda}_{C}},$
(1.308)
which for the usual Compton effect means that
$k\geqslant\dfrac{1}{\not{\lambda}_{C}}\sqrt{\dfrac{\delta}{\not{\lambda}_{C}}-1},$
(1.309)
while for the modified Compton effect leads to
$k\geqslant\dfrac{\kappa}{\sqrt{2}}\sqrt{\sqrt{1+\dfrac{4}{\kappa^{2}}\dfrac{1}{\not{\lambda}_{C}}\sqrt{\dfrac{\delta}{\not{\lambda}_{C}}-1}}-1}.$
(1.310)
The formula (1.307) can be rewritten in the form
$\dfrac{\omega^{\prime}}{c}=\dfrac{1}{\not{\lambda}_{C}}+\left(\dfrac{1}{\not{\lambda}_{C}}\right)^{2}\dfrac{\delta-2\not{\lambda}_{C}}{1+\not{\lambda}_{C}\dfrac{\omega}{c}},$
(1.311)
and studied approximatively with respect to the point
$\dfrac{\omega}{c}=\dfrac{1}{\not{\lambda}_{C}}$, which defines the following
value of wave vector of incoming photon.
$k=\dfrac{\kappa}{\sqrt{2}}\sqrt{\sqrt{1+4\dfrac{k_{C}^{2}}{\kappa^{2}}}-1}.$
(1.312)
It is easy to see that, because of the formula (1.307), in such a situation
$\dfrac{\omega^{\prime}}{c}=\dfrac{1+\dfrac{\delta-\not{\lambda}_{C}}{\not{\lambda}_{C}}}{2\not{\lambda}_{C}}=\dfrac{\delta}{2\not{\lambda}_{C}^{2}},$
(1.313)
and by the this reason the cosinus formula (1.301) takes the form
$\cos\theta=2\left(1-\dfrac{\not{\lambda}_{C}}{\delta}\right)=\dfrac{2}{n},$
(1.314)
i.e. the scattering is possible when the thickness of the lens is
$\dfrac{2}{3}\not{\lambda}_{C}\leqslant\delta\leqslant 2\not{\lambda}_{C},$
(1.315)
or equivalently the refraction index of the medium is
$n\geqslant 2.$ (1.316)
For $\dfrac{\omega}{c}<\dfrac{1}{\not{\lambda}_{C}}$ the correct expansion of
the formula (1.311 is given by
$\dfrac{\omega^{\prime}}{c}=\dfrac{1}{\not{\lambda}_{C}}+(\delta-2\not{\lambda}_{C})\left(\dfrac{1}{\not{\lambda}_{C}}\right)^{2}\sum_{n=0}^{\infty}(-1)^{n}\not{\lambda}_{C}^{n}\left(\dfrac{\omega}{c}\right)^{n},$
(1.317)
while for $\dfrac{\omega}{c}>\dfrac{1}{\not{\lambda}_{C}}$ the appropriate
expansion has the following form
$\dfrac{\omega^{\prime}}{c}=\dfrac{1}{\not{\lambda}_{C}}+(\delta-2\not{\lambda}_{C})\left(\dfrac{1}{\not{\lambda}_{C}}\right)^{2}\sum_{n=1}^{\infty}(-1)^{n}\left(\dfrac{1}{\not{\lambda}_{C}}\right)^{n}\left(\dfrac{c}{\omega}\right)^{n}.$
(1.318)
In the neighborhood of the point
$\dfrac{\omega}{c}=\dfrac{1}{\not{\lambda}_{C}}$ the approximations (1.317)
and (1.318)
$\displaystyle\dfrac{\omega^{\prime}}{c}$ $\displaystyle\approx$
$\displaystyle\dfrac{\delta-2\not{\lambda}_{C}}{\not{\lambda}_{C}^{2}}-\dfrac{\delta-2\not{\lambda}_{C}}{\not{\lambda}_{C}}\dfrac{\omega}{c},$
(1.319) $\displaystyle\dfrac{\omega^{\prime}}{c}$ $\displaystyle\approx$
$\displaystyle\dfrac{1}{\not{\lambda}_{C}}-\dfrac{\delta-2\not{\lambda}_{C}}{\not{\lambda}_{C}^{3}}\dfrac{c}{\omega},$
(1.320)
must coincide. By this reason in such a situation one obtains
$\delta=3\not{\lambda}_{C},$ (1.321)
i.e. the refraction index $n$ near
$\dfrac{\omega}{c}=\dfrac{1}{\not{\lambda}_{C}}$ has the value
$n=1.5.$ (1.322)
In the light of the relation (1.321) the cosinus (1.314) becomes
$\cos\theta=\dfrac{4}{3},$ (1.323)
i.e. is non-physical. It means that near the point
$\dfrac{\omega}{c}=\dfrac{1}{\not{\lambda}_{C}}$ the modified Compton effect,
considered in frames of the lensing hypothesis, has no place because of lack
of scattering.
If $\theta$, $\omega$, and $\omega^{\prime}$ are established, e.g. via
experimental data, then the thickness of the lens $d$ and the refractive index
$n$ are
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\not{\lambda}_{C}\left(1+\not{\lambda}_{C}^{2}\dfrac{\omega}{c}\dfrac{\omega^{\prime}}{c}\cos\theta\right),$
(1.324) $\displaystyle n$ $\displaystyle=$ $\displaystyle
1+\dfrac{1}{\not{\lambda}_{C}^{2}}\dfrac{c}{\omega}\dfrac{c}{\omega^{\prime}}\sec\theta.$
(1.325)
Because, however, $n=c/v$ where $v$ is velocity of light in the medium in
which the refraction has a place, one can see that
$v=c\dfrac{\not{\lambda}_{C}^{2}\dfrac{\omega}{c}\dfrac{\omega^{\prime}}{c}\cos\theta}{1+\not{\lambda}_{C}^{2}\dfrac{\omega}{c}\dfrac{\omega^{\prime}}{c}\cos\theta}.$
(1.326)
In this section we presented the approach to the modified Compton effect.
First, we constructed the relativistic analysis which produced the modified
Compton equation coinciding to the Compton equation for lack of the
modification due to the Snyder–Sidharth deformation. Then, we solved the
modified Compton equation and showed that the result of lack of modification
differs from deductions of Special Relativity. By this reason we called out
the lensing hypothesis. Then, we showed that the scattered angle values are
independent on particle energies, and derived the equation jointing wave
vectors of incoming and outgoing photon. Finally, we called out the
dispersional generalization which led us to the correct result in lack of
modification. Application of the lensing hypothesis and the identification
method together with the dispersional generalization produced a number of new
results due to the modified Compton effect.
### Chapter 2 The Neutrinos: Masses & Chiral Condensate
#### A Outlook on Noncommutative Geometry
In 1947 an American physicist H.S. Snyder, for elimination of the infrared
catastrophe in the Compton effect and effectively resolving the problem of
ultraviolet divergences in quantum field theory, proposed employing the model
of space-time based on the commutators [113]
$\displaystyle\dfrac{i}{\hslash}[x,p]=1+\alpha\left(\dfrac{\ell}{\hslash}\right)^{2}p^{2},$
(2.1) $\displaystyle\dfrac{i}{\hslash}[x,y]=O(\ell^{2})\quad,$ (2.2)
where $p$ is three momentum of a particle, $x$ and $y$ are two different
points of space, $\ell$ is a fundamental length scale, $\hslash$ is the Planck
constant, $\alpha\sim 1$ is a dimensionless constant, $[\cdot,\cdot]$ is an
appropriate Lie bracket. For the Lorentz and Poincaré invariance modified due
to $\ell$, Snyder considered a momentum space of constant curvature isometry
group, _i.e._ the Poincaré algebra deformation into the De Sitter space.
The Snyder space-time (2.1)–(2.2) define a noncommutative geometry and a
deformation (for basics of the theory and applications see e.g. the
bibliography in Ref. [123]). Let us this in some detail. First, for better
insight, let us sketch the rules of noncommutative geometry in a certain
general outlook. Let us consider an associative Lie algebra $A$ for which
$\tilde{A}=A[[\lambda]]$ is the module due to the ring of formal series
$\mathbb{K}[[\lambda]]$ in a parameter $\lambda$. Let us call $\tilde{A}$ a
deformation of $A$ i.e. $\mathbb{K}[[\lambda]]$-algebra such that
$\tilde{A}/\lambda\tilde{A}\approx A$. If $A$ is endowed with a locally convex
topology with continuous laws, _i.e._ is a topological algebra, then
$\tilde{A}$ is called topologically free. We presume that in the Lie algebra
$A$ the law of composition is determined via an ordinary product and the
related bracket is $[\cdot,\cdot]$. In such a situation $\tilde{A}$ is an
associative Lie algebra if and only if for arbitrary two elements of the
algebra $f,g\in A$ a new product $\star$ and the related bracket
$[\cdot,\cdot]_{\star}$ are defined as follows
$\displaystyle f\star g$ $\displaystyle=$ $\displaystyle
fg+\sum_{n=1}^{\infty}\lambda^{n}C_{n}(f,g),$ (2.3)
$\displaystyle\left[f,g\right]_{\star}$ $\displaystyle\equiv$ $\displaystyle
f\star g-g\star f=\left[f,g\right]+\sum_{n=1}^{\infty}\lambda^{n}B_{n}(f,g),$
(2.4)
where $C_{n}$ and $B_{n}$ are the Hochschild and the Chevalley 2-cochains, and
for arbitrary three elements of the algebra $f,g,h\in A$ are satisfied two
conditions: the Jacobi identity
$[[f,g]_{\star},h]_{\star}+[[h,f]_{\star},g]_{\star}+[[g,h]_{\star},f]_{\star}=0$
(2.5)
and the law of associativity
$(f\star g)\star h=f\star(g\star h).$ (2.6)
If $b$ and $\partial$ are the Hochschild and the Chevalley coboundary
operators, i.e. such that $b^{2}=0$ and $\partial^{2}=0$, then for each $n$
and $j,k\geqslant 1$ such that $j+k=n$ the following relations hold
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!bC_{n}(f,g,h)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum_{j,k}\left[C_{j}\left(C_{k}(f,g),h\right)-C_{j}\left(f,C_{k}(g,h)\right)\right],$
(2.7) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\partial B_{n}(f,g,h)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum_{j,k}\left[B_{j}\left(B_{k}(f,g),h\right)+B_{j}\left(B_{k}(h,f),g\right)+B_{j}\left(B_{k}(g,h),f\right)\right],$
(2.8)
Let $C^{\infty}(M)$ be an algebra of smooth functions on a differentiable
manifold $M$. The law of associativity yields the Hochschild cohomologies. An
antisymmetric contravariant 2-tensor $\theta$, which trivialize the Schouten-
Nijenhuis bracket $[\theta,\theta]_{SN}=0$ on $M$, determines the Poisson
brackets $\\{f,g\\}=i\theta df\wedge dg$ satisfying the Jacobi identity and
the Leibniz rule. Than $(M,\\{\cdot,\cdot\\})$ is called a Poisson manifold.
In 1997 a Russian mathematician M.L. Kontsevich [124] defined deformation
quantization of a general Poisson differentiable manifold. Let
$\mathbb{R}^{d}$ be endowed with a Poisson brackets
$\alpha(f,g)=\sum_{1\leqslant i,j\leqslant n}\alpha^{ij}\dfrac{\partial
f}{\partial x^{i}}\dfrac{\partial g}{\partial x^{j}},$ (2.9)
where $1\leqslant k\leqslant d$. For $\star$-product and $n\geqslant 0$,
exists a family $G_{n,2}$ of $(n(n+1))^{n}$ oriented graphs $\Gamma$. Let
$V_{\Gamma}$ be the set of vertices of $\Gamma$. This set has $n+2$ elements
collected in two subsets: the first type $\\{1,\ldots,n\\}$ and the second
type $\\{\bar{1},\bar{2}\\}$. Let $E_{\Gamma}$ denotes the set of oriented
edges of $\Gamma$, having $2n$ elements. The rule is that there is no edge
starting at a second type vertex. Let $Star(k)$ denotes the set of oriented
edges starting at a first type vertex $k$ with cardinality $\sharp k=2$,
$\sum_{1\leqslant k\leqslant n}\sharp k=2n$. Than
$\\{e^{1}_{k},\ldots,e^{\sharp k}_{k}\\}$ are the edges of $\Gamma$ starting
at vertex $k$. Vortices starting and ending in the edge $v$ are
$v=(s(v),e(v))$ where $s(v)\in\\{1,\ldots,n\\}$ and
$e(v)\in\\{1,\ldots,n;\bar{1},\bar{2}\\}$. $\Gamma$ has no loop and no
parallel multiple edges. For arbitrary two elements $f,g\in
C^{\infty}(\mathbb{R}^{d})$ a bidifferential operator $(f,g)\mapsto
B_{\Gamma}(f,g)$ is associated to $\Gamma$. The symbols
$\alpha^{e^{1}_{k}e^{2}_{k}}$ are associated to each first type vertex $k$
from where the edges $\\{e^{1}_{k},e^{2}_{k}\\}$ start; $f$ is the vertex $1$,
and $g$ is the vertex $\bar{2}$. An edge $e^{1}_{k}$ acts like differentiation
operator $\partial/\partial x^{e^{1}_{k}}$ on its ending vertex. Than
$B_{\Gamma}$ is a sum over all maps $I:E_{\Gamma}\rightarrow\\{1,\ldots,d\\}$
$\\!\\!\\!\\!\\!\\!B_{\Gamma}(f,g)=\sum_{I}\left(\prod_{k=1}^{n}\prod_{k^{\prime}=1}^{n}\partial_{I(k^{\prime},k)}\alpha^{I(e^{1}_{k})I(e^{2}_{k})}\right)\\!\\!\left(\prod_{k_{1}=1}^{n}\partial_{I(k_{1},\bar{1})}f\right)\\!\\!\left(\prod_{k_{2}=1}^{n}\partial_{I(k_{2},\bar{2})}g\right).\\!\\!\\!$
(2.10)
Let us denote by $\mathcal{H}_{n}$ the configuration space of $n$ distinct
points in upper half-plane $\mathcal{H}=\\{z\in\mathbb{C}|\Im(z)>0\\}$ with
the Lobachevsky hyperbolic metric, which is an open submanifold of
$\mathbb{C}^{n}$. Let for the vertex $k$ such that $1\leqslant k\leqslant n$,
$z_{k}\in\mathcal{H}$ denotes a variable associated to $\Gamma$. The vertices
$1$ and $\bar{2}$ are associated to $0\in\mathbb{R}$ and $1\in\mathbb{R}$,
respectively. If $\tilde{\phi}_{v}=\phi(s(v),e(v))$ is a function on
$\mathcal{H}_{n}$, associated to $v$, determined by the angle function
$\phi:\mathcal{H}_{2}\rightarrow\mathbb{R}/2\pi\mathbb{Z}$ having the
following form
$\phi(z_{1},z_{2})=\mathrm{Arg}\dfrac{z_{2}-z_{1}}{z_{2}-\bar{z}_{1}}=\dfrac{1}{2i}\mathrm{Log}\dfrac{\bar{z}_{2}-z_{1}}{z_{2}-\bar{z}_{1}}\dfrac{z_{2}-z_{1}}{\bar{z}_{2}-\bar{z}_{1}},$
(2.11)
then the integral of $2n$-form $w(\Gamma)\in\mathbb{R}$ is a weight associated
to $\Gamma\in G_{n,2}$
$\displaystyle
w(\Gamma)=\dfrac{1}{n!(2\pi)^{2n}}\int_{\mathcal{H}_{n}}\bigwedge_{1\leqslant
k\leqslant n}\left(d\tilde{\phi}_{e^{1}_{k}}\wedge
d\tilde{\phi}_{e^{2}_{k}}\right),$ (2.12)
which does not depend on the Poisson structure or the dimension $d$. On
$(\mathbb{R}^{d},\alpha)$ the Kontsevich $\star$-product maps
$C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})\rightarrow
C^{\infty}(\mathbb{R})[[\lambda]]$
$(f,g)\mapsto f\star g=\sum_{n\geqslant 0}\lambda^{n}C_{n}(f,g),$ (2.13)
where $C_{0}(f,g)=fg$, $C_{1}(f,g)=\\{f,g\\}_{\alpha}=\alpha df\wedge dg$, and
in general
$C_{n}(f,g)=\sum_{\Gamma\in G_{n,2}}w(\Gamma)B_{\Gamma}(f,g).$ (2.14)
Equivalence classes of (2.13) are bijective to the equivalence classes of the
Poisson brackets $\alpha_{\lambda}=\sum_{k\geqslant 0}\lambda^{k}\alpha_{k}$.
For linear Poisson structures, _i.e._ on coalgebra $A^{\star}$, the weight
(2.12) of all even wheel graphs vanishes, and the Kontsevich star product
(2.13) coincides with the $\star$-product determined by the Duflo isomorphism.
This case allows to quantize the class of quadratic Poisson brackets belonging
to the image of the Drinfeld map which associates a quadratic to a linear
bracket.
Let us consider the deformations of phase-space and space given by the
parameters $\lambda_{ph}$, $\lambda_{s}$ being
$\displaystyle\lambda_{ph}=\dfrac{i\alpha}{2},$ (2.15)
$\displaystyle\lambda_{s}=\dfrac{i\beta}{2},$ (2.16)
where $\alpha\sim 1$ and $\beta\sim 1$ are dimensionless constants, and
resulting in the appropriate star product (2.3), or equivalently to the
Kontsevich star-product (2.13), on the phase space $(x,p)$ and between two
distinct space points $x$ and $y$
$\displaystyle x\star p$ $\displaystyle=$ $\displaystyle
px+\sum_{n=1}^{\infty}\left(\dfrac{i\alpha}{2}\right)^{n}C_{n}(x,p),$ (2.17)
$\displaystyle x\star y$ $\displaystyle=$ $\displaystyle
xy+\sum_{n=1}^{\infty}\left(\dfrac{i\beta}{2}\right)^{n}C_{n}(x,y),$ (2.18)
where $C_{n}(x,p)$ and $C_{n}(x,y)$ are the Hochschild cochains in the
Kontsevich formula (2.13) related to the phase space and the space
deformations, respectively. The Lie brackets arising from the star products
(2.17) and (2.18) has the following form
$\displaystyle\left[x,p\right]_{\star}$ $\displaystyle=$
$\displaystyle\left[x,p\right]+\sum_{n=1}^{\infty}\left(\dfrac{i\alpha}{2}\right)^{n}B_{n}(x,p),$
(2.19) $\displaystyle\left[x,y\right]_{\star}$ $\displaystyle=$
$\displaystyle\left[x,y\right]+\sum_{n=1}^{\infty}\left(\dfrac{i\beta}{2}\right)^{n}B_{n}(x,y),$
(2.20)
where $B_{n}(x,p)$ and $B_{n}(x,y)$ are the Chevalley cochains related to
phase-space and space, respectively. Taking the first approximation in the
formulas (2.19) and (2.20), and application of the non-deformed commutators
$[x,p]=-i\hslash$ and $[x,y]=0$ leads to
$\displaystyle\left[x,p\right]_{\star}$ $\displaystyle=$
$\displaystyle-i\hslash+\dfrac{i\alpha}{2}B_{1}(x,p),$ (2.21)
$\displaystyle\left[x,y\right]_{\star}$ $\displaystyle=$
$\displaystyle\dfrac{i\beta}{2}B_{1}(x,y).$ (2.22)
or after using of the Dirac ”method of classical analogy” [125]
$\displaystyle\dfrac{1}{i\hslash}\left[p,x\right]_{\star}$ $\displaystyle=$
$\displaystyle 1-\dfrac{\alpha}{2\hslash}B_{1}(x,p),$ (2.23)
$\displaystyle\dfrac{1}{i\hslash}\left[x,y\right]_{\star}$ $\displaystyle=$
$\displaystyle\dfrac{\beta}{2\hslash}B_{1}(x,y).$ (2.24)
Because of, however, for two arbitrary elements $f,g\in C^{\infty}(M)$ one has
$B_{1}(f,g)=2\theta(df\wedge dg)$, therefore the formulas (2.23) and (2.24)
can be rewritten in the following form
$\displaystyle\dfrac{1}{i\hslash}\left[p,x\right]_{\star}$ $\displaystyle=$
$\displaystyle 1-\dfrac{\alpha}{\hslash}(dx\wedge dp),$ (2.25)
$\displaystyle\dfrac{1}{i\hslash}\left[x,y\right]_{\star}$ $\displaystyle=$
$\displaystyle\dfrac{\beta}{\hslash}dx\wedge dy.$ (2.26)
Let us consider the space lattice characterized by an infinitesimal growth of
a space coordinate identified with the fundamental scale $\ell$
$dx=\ell,$ (2.27)
and presume that the momentum of a particle is related to the coordinate of a
particle via the De Broglie wave-particle duality formula
$p=\dfrac{\hslash}{x}.$ (2.28)
Application of the model (2.27)-(2.28) allows to derive straightforwardly an
infinitesimal growth of the momentum
$dp=-\dfrac{\hslash}{x^{2}}dx=-\dfrac{p^{2}}{\hslash}\ell.$ (2.29)
Therefore, the infinitesimal growths (2.27) and (2.29) applied to the deformed
brackets (2.25) and (2.26) allows to establish finally
$\displaystyle\dfrac{i}{\hslash}\left[x,p\right]_{\star}$ $\displaystyle=$
$\displaystyle 1+\dfrac{\alpha}{\hslash^{2}}\ell^{2}p^{2},$ (2.30)
$\displaystyle\dfrac{i}{\hslash}\left[x,y\right]_{\star}$ $\displaystyle=$
$\displaystyle-\dfrac{\beta}{\hslash}\ell^{2}.$ (2.31)
The relations (2.30) and (2.31) prove that the Snyder space-time (2.1)-(2.2)
is a noncommutative geometry obtained via the first approximation of the
Kontsevich deformation quantization.
In the 1960s a Soviet physicist M.A. Markov [126] proposed to take into
account a fundamental length scale as the minimal scale identified with the
Planck length, i.e. $\ell=\ell_{P}=\sqrt{{\dfrac{\hslash c}{G}}}$, and
expressed the hypothesis that a mass $m$ of any elementary particle is bounded
by the maximal mass identified with the Planck mass, i.e. $m\leqslant
M_{P}=\dfrac{\hslash}{c\ell_{P}}=\sqrt{{\dfrac{G\hslash}{c^{3}}}}$. Applying
this crucial idea, since 1978 a Soviet-Russian theoretician V.G. Kadyshevsky
and his collaborators [127] have studied widely certain aspects of the Snyder
model of noncommutative geometry strictly related to particle physics.
Recently also V.N. Rodionov has developed independently the stream of
Kadyshevsky [128]. The problems discussed in this chapter seem to be more
related to a general current [129], where particularly the Snyder space-time
(2.1)-(2.2) has been found a number of applications.
Beginning 2000 an Indian scholar and philosopher B.G. Sidharth [130] showed
that in spite of the self-evident Lorentz invariance of the deformation
(2.1)-(2.2), in general the Snyder noncommutative geometry breaks the two
fundamental paradigms celebrated in relativistic physics: the Einstein energy-
momentum relation as well as the Lorentz symmetry. Sidharth (Cf. Ref. [131])
concluded that in such a situation the Hamiltonian constraint of Special
Relativity is deformed due to the additional term proportional to the fourth
power of three-momentum of a relativistic particle and the second power of a
minimal scale $\ell$, which Sidharth has been identified with a minimal scale,
i.e. the Planck length or the Compton wavelength of an electron
$E^{2}=m^{2}c^{4}+c^{2}p^{2}+\alpha\left(\dfrac{c}{\hslash}\right)^{2}\ell^{2}p^{4}.$
(2.32)
Neglecting negative mass states as nonphysical, Sidharth established a number
of intriguing new facts [132]. Particularly, by straightforward application of
Dirac ”square-root” technique to the Hamiltonian constraint of the modified
Special Relativity (2.32) he concluded the corresponding modified Dirac
equation
$\left(\gamma^{\mu}\hat{p}_{\mu}+mc^{2}+\sqrt{\alpha}\dfrac{c}{\hslash}\ell\gamma^{5}\hat{p}^{2}\right)\psi=0.$
(2.33)
which differs from the conventional Dirac relativistic quantum mechanics by a
correction due to the $\gamma^{5}$-term proportional to the second power of
the three momentum of a particle and to a minimal scale $\ell$.
However, it looks like that Sidharth has been neglected the fact that the
modified Special Relativity (2.32) leads to a one more additional possibility
which is physically nonequivalent to the modified Dirac equation (2.33)
considered by him as the physical quantum theory. Namely, he omitted the Dirac
Hamiltonian constraint with the negative $\gamma^{5}$-term
$\left(\gamma^{\mu}\hat{p}_{\mu}+mc^{2}-\sqrt{\alpha}\dfrac{c}{\hslash}\ell\gamma^{5}\hat{p}^{2}\right)\psi=0.$
(2.34)
Fortunately, however, such an issue seems to be easy to solve because of the
possible physical results of the quantum theory (2.34) can be
straightforwardly concluded from the results following from the modified Dirac
equation possessing the positive $\gamma^{5}$-term (2.33) by application of
the mirror reflection in a minimal scale $\ell\rightarrow-\ell$. We are not
going to neglect also the negative mass states in the modified Dirac theory as
nonphysical, because this situation strictly corresponds with the results
obtained from the equation (2.33) transformed via a mirror reflection in mass
of a relativistic particle $m\rightarrow-m$. Moreover, it must be emphasized
that in the standard relativistic quantum mechanics the negative sign
corresponds to the Antimatter, which recently has been considered as the
element of Reality (See e.g. Ref. [133]).
Therefore we propose to consider the result of the canonical relativistic
quantization
$p_{\mu}=(E,p_{i}c)\rightarrow\hat{p}_{\mu}=i\hslash(\partial_{0},c\partial_{i})$
(2.35)
applied to the deformed Special Relativity (2.32) linearized by the Dirac
”square-root” technique. In general such a procedure leads to the four
possible physically nonequivalent quantum theories which can be presented in a
form of one compact equation
$\left(\gamma^{\mu}\hat{p}_{\mu}\pm
mc^{2}\pm\sqrt{\alpha}\dfrac{c}{\hslash}\ell\gamma^{5}\hat{p}^{2}\right)\psi=0.$
(2.36)
We shall presume that on the analogy of the standard Dirac relativistic
quantum mechanics, a wave function $\psi$ of the modified Dirac equation
(2.36) is a four-component spinor
$\psi=\left[\phi_{0},\phi_{1},\phi_{2},\phi_{3}\right]^{\mathrm{T}}$, and that
the Dirac gamma matrices satisfy the four-dimensional Æther algebra
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=\dfrac{1}{2}\eta_{\mu\nu}$
introduced in the previous chapter. It must be emphasized that a presence of
the $\gamma^{5}$-term in the modified Dirac equation (2.36) results in
manifest violation of parity symmetry, and therefore also the Lorentz symmetry
is violated due to such a correction. For simplicity, however, we shall
consider one of the four cases (2.36) given by the Sidharth’s Dirac equation
(2.33). The results due to the three remained situations can be described by
straightforward application of the mentioned mirror transformations in the
mass of a particle and a minimal scale to the results due to the generic
theory (2.33).
Both this chapter and the next one are strictly based on the recent results of
the author [134] enriched by necessary minor updates. The research value of
these two chapters is justified by the fact that the approach based on
deformations of Special Relativity, including the Snyder–Sidharth deformation
(2.32), recently has became one of the most intensively developing and
fruitful research direction in astrophysics of gamma rays, especially in the
context of CP violation [135]. Interestingly, the modified Dirac equation
(2.33) was originally proposed [132] as the idea for ultra-high energy
physics. Sidharth, however, has not presented computations based on this idea,
which could result in experimentally verifiable physical predictions. In
communication with the author [136], Sidharth has presented a number of
intriguing and interesting looking speculations about the extra mass terms
deforming Special Relativity and the corresponding Dirac equations. Also we
discussed a lot of philosophical issues and suggestions about the foundational
role of noncommutative geometry for new physics based on the Lorentz symmetry
violation. The discussions did not established a physical truth, and therefore
derivation of the modified Dirac equation and the reasoning performed by
Sidharth in general possesses philosophical countenance. Albeit, the physical
role of deformation theory and noncommutative geometry is still a great riddle
to the same degree as it is an amazing hope, and factually nobody established
real meaning of such an abstractive mathematics for theoretical physics. In
the author opinion the most hopeful observational research region to
verification of the theories (2.36) is high-energy and ultra-high-energy
astrophysics. The astrophysical phenomena are probably the best test for the
Planck scale. Particularly, ultra-high-energy cosmic rays coming from gamma
bursts sources, neutrinos coming from supernovas, and other effects observed
in this energy region, are the most fruitful research material for tests of
the modified theories. This cognitive aspect of the thing is both the most
logical and rational justification for considering the equation (2.33),
arising due to the Snyder noncommutative geometry (2.1)-(2.2), and trying pull
out possibly novel valuable extensions of the well-grounded physical
knowledge. It must be emphasized that an arbitrary abstractive mathematics
creates potentially new physical theories, but extraction of physics is
usually a heroic work. By its simplicity the Snyder noncommutative geometry is
one of themes of this book, but another models are not forbidden. Possibly,
however, the only Snyder noncommutative geometry possesses clear physical
meaning. Such a hypothesis is also the good point for experimental
verification.
#### B Massive neutrinos
In fact the Sidharth $\gamma^{5}$-term, emerging from the Snyder
noncommutative geometry of phase space (2.1), is the shift of the conventional
Dirac relativistic quantum mechanics. Let us presume that new physics arises
from the physical picture in which the modified Dirac equation holds, but
Special Relativity stays _non-modified_. In other words, we shall preserve
Einstein’s Special Relativity unchanged, but change the Dirac relativistic
quantum mechanics by the Snyder noncommutative geometry. Such a modification
is an algebra deformation. It is easy to deduce that such a deformation can be
realized by preservation of the hyperbolic geometry of both the Minkowski
energy-momentum space as well as the space-time. While the physical foundation
of the Einstein theory is dynamics of a relativistic particle, the physical
foundations of an algebra deformation are based on a non-dynamical
justification. For example a deformation can be due to finite sizes of a
particle. In this manner, in our view while the Snyder–Sidharth deformation of
Special Relativity (2.32) can be interpreted as a dynamical result, the
corresponding modification of the Dirac equation (2.33) is due to the non-
dynamical $\gamma^{5}$-term despite this term is explicitly dependent on a
particle three-momentum. To make such a constructive strategy evident we
propose to apply the formalism of the Minkowski space, despite a presence of
the $\gamma^{5}$-term, straightforwardly to both the modified Special
Relativity (2.32) and the modified Dirac equation (2.33).
The standard identity of the Minkowski energy-momentum space
$p_{\mu}p^{\mu}=\left(\gamma^{\mu}p_{\mu}\right)^{2}=E^{2}-c^{2}p^{2},$ (2.37)
allows to extract square of three momentum which, together with the mass shell
condition $p_{\mu}p^{\mu}=mc^{2}$, applied to the modified Special Relativity
(2.32) results in the equation
$\alpha\left(\dfrac{c}{\hslash}\right)^{2}\ell^{2}p^{4}=0,$ (2.38)
which for nonzero momentum has the unique and unambiguous solution $\ell=0$.
In other words, the hyperbolic identity (2.37) on the mass shell leads to
direct reconstruction of Special Relativity.
Let us now apply the identity (2.37) to the modified Dirac equation (2.33).
Square of three-momentum can be extracted via the identity (2.37) and applied
within the equation (2.33)
$\left[\gamma^{\mu}\hat{p}_{\mu}+mc^{2}+\dfrac{\sqrt{\alpha}}{\hslash
c}\ell\gamma^{5}\left[E^{2}-\left(\gamma^{\mu}\hat{p}_{\mu}\right)^{2}\right]\right]\psi=0,$
(2.39)
results in the quadratic equation
$\left[-\dfrac{\sqrt{\alpha}}{\hslash
c}\ell\gamma^{5}\left(\gamma^{\mu}\hat{p}_{\mu}\right)^{2}+\gamma^{\mu}\hat{p}_{\mu}+mc^{2}+\dfrac{\sqrt{\alpha}}{\hslash
c}\ell\gamma^{5}E^{2}\right]\psi=0,$ (2.40)
which after multiplication of both sides by $\gamma^{5}$ and using of the
combination $\gamma^{5}\gamma^{\mu}p_{\mu}$ can be rewritten in the following
form
$\left[\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)^{2}-\epsilon\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)+E^{2}-\epsilon
mc^{2}\gamma^{5}\right]\psi=0,$ (2.41)
where $\epsilon$ is a maximal energy due to a minimal scale $\ell$
$\epsilon=\dfrac{\hslash c}{\sqrt{\alpha}\ell}.$ (2.42)
The equation (2.41) expresses projection of the operator
$\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)^{2}-\epsilon\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)+E^{2}-\epsilon
mc^{2}\gamma^{5},$ (2.43)
on the spinor $\psi$. By application of elementary algebraic manipulations,
however, the quadratic operator (2.43) can be factorized straightforwardly to
the form
$(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}-\epsilon_{+})(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}-\epsilon_{-}),$
(2.44)
where $\epsilon_{\pm}$ are the manifestly non-hermitian energies
$\epsilon_{\pm}=\dfrac{\epsilon}{2}\left(1\pm\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\sqrt{{1+\dfrac{4\epsilon
mc^{2}}{\epsilon^{2}-4E^{2}}\gamma^{5}}}\right).$ (2.45)
Principally the quantities (2.45) are due to the order reduction, and also
cause the Dirac-like linearization.
Let us treat a particle energy $E$, a particle mass $m$, and a maximal energy
$\epsilon$ (or equivalently a minimal scale $\ell$) in the formula (2.45) as
free parameters. It can be observed straightforwardly that the modified Dirac
equation (2.33) is equivalent to the two nonequivalent relativistic quantum
theories
$\displaystyle\left(\gamma^{\mu}\hat{p}_{\mu}-M_{+}c^{2}\right)\psi=0,$ (2.46)
$\displaystyle\left(\gamma^{\mu}\hat{p}_{\mu}-M_{-}c^{2}\right)\psi=0,$ (2.47)
where $M_{\pm}$ are the generated mass matrices
$M_{\pm}=\dfrac{\epsilon}{2c^{2}}\left(-1\mp\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}+\dfrac{4mc^{2}}{\epsilon}\gamma^{5}}}\right)\gamma^{5}.$
(2.48)
The result (2.48) is in itself nontrivial. Factually, by application of the
Minkowski energy-momentum space the Dirac equation modified due to the Snyder
noncommutative geometry has been reduced to two distinguishable standard Dirac
theories describing a kind of effective particles characterized by manifestly
non-hermitian mass matrices $M_{\pm}$. Both these theories are all the more so
intriguing because the total information about a minimal scale $\ell$, and
therefore about the Snyder noncommutative geometry, was placed in the mass
matrices $M_{\pm}$ only, while the relativistic space-time and energy-momentum
formalisms are exactly the same as in the standard Dirac quantum theory. Note
that such a procedure is also correct from the methodological point of view.
We have applied the tensor formalism of the hyperbolic 4-dimensional energy-
momentum geometry within the Dirac equation modified due to the
$\gamma^{5}$-term. This point has not been noticed or has been omitted in the
analysis made by Sidharth. In this manner we have constructed new type _mass
generation mechanism_ which deduction is impossible to perform within the
frames of Special Relativity only, _i.e._ for in the situation when a minimal
scale is vanishing $\ell=0$ or equivalently a maximal energy is infinite
$\epsilon=\infty$. In fact, application of the noncommutative geometry results
in a finite value of a maximal energy, what is a kind of renormalization of
Special Relativity. Therefore the mass generation mechanism presented above
results in the effect of such a nontrivial renormalization, possesses purely
kinetic nature and, above all, is due to the factorization applied to the
operator (2.43) projecting onto the spinor wave function $\psi$. It must be
emphasized that this kinetic effect in the result due to the abstractive
mathematics of noncommutative geometry and algebra deformation. The key
problem is generalization of this mechanism to more general situations
described in frames of the Kontsevich deformation quantization. In other words
the crucial unsolved issue is a reply to the question: _does a physical
contribution from noncommutative geometry in general contains in a mass
generation mechanism only?_. The necessity of reply to this question is argued
above all by the problem of experimental verification of the theoretical
results due to noncommutative geometry, and possesses fundamental meaning for
philosophical foundations of the new physics obtained via constructive
applications of the abstractive mathematics. The reply to this question is,
however, far from this book content.
Let us present now the mass matrices $M_{\pm}$ in more convenient form
employing a linear dependence of the $\gamma^{5}$ matrix. Fist, let us apply
the Taylor series expansion to the square root part of the mass matrices
(2.48). The expansion can be performed in the following way
$\displaystyle\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}+\dfrac{4mc^{2}}{\epsilon}\gamma^{5}}}$
$\displaystyle=$
$\displaystyle\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\gamma^{5}}}=$
(2.49) $\displaystyle=$
$\displaystyle\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\sum_{n=0}^{\infty}\binom{1/2}{n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\gamma^{5}\right)^{n},$
where the generalized Newton binomial symbol was used
$\binom{n}{k}=\dfrac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n+1-k)}.$
By application of the basic properties of $\gamma^{5}$ matrix, _i.e._
$\left(\gamma^{5}\right)^{2n}=-1$ and
$\left(\gamma^{5}\right)^{2n+1}=-\gamma^{5}$, one can decompose of the sum
present in the last term of the formula (2.49) onto the two components related
to the odd and even powers of $\gamma^{5}$ matrix
$\displaystyle\sum_{n=0}^{\infty}\binom{1/2}{n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\gamma^{5}\right)^{n}=$
$\displaystyle=-\sum_{n=0}^{\infty}\binom{1/2}{2n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{2n}-\sum_{n=0}^{\infty}\binom{1/2}{2n+1}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{2n+1}\gamma^{5}.$
(2.50)
Straightforward application of the standard summation procedure allows to
establish the sums presented in both the components of the decomposition
(2.50)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{n=0}^{\infty}\binom{1/2}{2n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{\\!\\!\\!2n}$
$\displaystyle=$
$\displaystyle\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}+\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}},\vspace*{10pt}$
(2.51)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{n=0}^{\infty}\binom{1/2}{2n+1}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{\\!\\!\\!2n+1}$
$\displaystyle=$
$\displaystyle\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}-\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}.$
(2.52)
In this manner one sees easily that both the mass matrices $M_{\pm}$ possess
the following decomposition onto two components: the hermitian
$\mathfrak{H}(M_{\pm})$ and the antihermitian $\mathfrak{A}(M_{\pm})$
$M_{\pm}=\mathfrak{H}(M_{\pm})+\mathfrak{A}(M_{\pm}),$ (2.53)
where both the parts can be presented in a compact form
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathfrak{H}(M_{\pm})\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\pm\dfrac{\epsilon}{2c^{2}}\left[\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\left(\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}-\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}\right)\right],$
(2.54)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathfrak{A}(M_{\pm})\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!-\dfrac{\epsilon}{2c^{2}}\left[1\pm\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\left(\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}+\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}\right)\right]\gamma^{5}.$
(2.55)
By application of elementary algebraic manipulations one obtains equivalent
decomposition of the mass matrices $M_{\pm}$ into the basis of the commutating
projectors
$\left\\{\Pi_{i}:\dfrac{1+\gamma^{5}}{2},\dfrac{1-\gamma^{5}}{2}\right\\}$,
$M_{\pm}=\sum_{i}\mu_{i}^{\pm}\Pi_{i}=\mu_{R}^{\pm}\dfrac{1+\gamma^{5}}{2}+\mu_{L}^{\pm}\dfrac{1-\gamma^{5}}{2},$
(2.56)
where $\mu_{R}^{\pm}$ and $\mu_{L}^{\pm}$ are the projected masses related to
the Dirac theories with signs $\pm$ in agreement with the mass matrix signs.
The values of the projected masses can be established easy
$\displaystyle\mu_{R}^{\pm}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{c^{2}}\left(\dfrac{\epsilon}{2}\pm\sqrt{{\epsilon^{2}-4\epsilon
mc^{2}-4E^{2}}}\right),$ (2.57) $\displaystyle\mu_{L}^{\pm}$ $\displaystyle=$
$\displaystyle\dfrac{1}{c^{2}}\left(\dfrac{\epsilon}{2}\pm\sqrt{{\epsilon^{2}+4\epsilon
mc^{2}-4E^{2}}}\right).$ (2.58)
For physical correctness we shall presume that the masses of right-handed
neutrinos (2.57) and the left-handed neutrinos (2.58) are real numbers. It is
nontrivial conditions, which allows to establish the range of a maximal energy
$\epsilon$ via the energy and the mass of an original quantum state obeying
the Dirac equation. In the case of massive states one obtains
$\displaystyle\epsilon$ $\displaystyle\in$
$\displaystyle\left(-\infty,-2mc^{2}\left(1+\sqrt{{1+\left(\dfrac{E}{mc^{2}}\right)^{2}}}\right)\right]\cup$
(2.59) $\displaystyle\cup$
$\displaystyle\left[-2mc^{2}\left(1-\sqrt{{1+\left(\dfrac{E}{mc^{2}}\right)^{2}}}\right),2mc^{2}\left(1+\sqrt{{1+\left(\dfrac{E}{mc^{2}}\right)^{2}}}\right)\right]\cup$
$\displaystyle\cup$
$\displaystyle\left[2mc^{2}\left(1+\sqrt{{1+\left(\dfrac{E}{mc^{2}}\right)^{2}}}\right),\infty\right),$
while in the case of massless quantum states possessing an energy $E$
$\displaystyle\epsilon\in\left(-\infty,-2|E|\right]\cup\left[2|E|,\infty\right).$
(2.60)
The properties of the projectors $\Pi_{i}^{\dagger}\Pi_{i}=\mathbf{1}_{4}$,
$\Pi_{1}\Pi_{2}=\dfrac{1}{2}\mathbf{1}_{4}$, $\Pi_{1}^{\dagger}=\Pi_{2}$ and
$\Pi_{1}+\Pi_{2}=\mathbf{1}_{4}$ allows to derive the relation
$M_{\pm}M_{\pm}^{\dagger}=\dfrac{(\mu_{R}^{\pm})^{2}+(\mu_{L}^{\pm})^{2}}{2}\mathbf{1}_{4}.$
(2.61)
By introducing the right- and left-handed chiral Weyl fields
$\displaystyle\psi_{R}$ $\displaystyle=$
$\displaystyle\dfrac{1+\gamma^{5}}{2}\psi,$ (2.62) $\displaystyle\psi_{L}$
$\displaystyle=$ $\displaystyle\dfrac{1-\gamma^{5}}{2}\psi,$ (2.63)
where a wave function $\psi$ is a solution of the appropriate Dirac equations
(2.46) and (2.47), both the theories can be rewritten as the system of two
equations
$\displaystyle\left(\gamma^{\mu}\hat{p}_{\mu}+\mu^{+}c^{2}\right)\left[\begin{array}[]{c}\psi_{R}^{+}\\\
\psi_{L}^{+}\end{array}\right]=0,$ (2.66)
$\displaystyle\left(\gamma^{\mu}\hat{p}_{\mu}+\mu^{-}c^{2}\right)\left[\begin{array}[]{c}\psi_{R}^{-}\\\
\psi_{L}^{-}\end{array}\right]=0,$ (2.69)
where now the mass matrices $\mu^{\pm}$, related to the chiral fields
$\psi_{R,L}^{\pm}$, are manifestly hermitian quantities
$\mu^{\pm}=\left[\begin{array}[]{cc}\mu_{R}^{\pm}&0\\\
0&\mu_{L}^{\pm}\end{array}\right]=\left[\begin{array}[]{cc}\mu_{R}^{\pm}&0\\\
0&\mu_{L}^{\pm}\end{array}\right]^{\dagger}.$ (2.70)
Note that the masses (2.57) and (2.58) are invariant with respect to a choice
of representation of the Dirac matrices $\gamma^{\mu}$. By this reason these
quantities have physical character. It is interesting that for the mirror
reflection in a minimal scale $\ell\rightarrow-\ell$, or equivalently for the
mirror reflection $\epsilon\rightarrow-\epsilon$, one has the exchange between
the masses $\mu_{R}^{\pm}\leftrightarrow\mu_{L}^{\pm}$ while the chiral Weyl
fields stay unchanged. Similarly, in the situation of the mirror reflection in
the mass of an original quantum state $m\rightarrow-m$ one has the exchange
between the projected masses $\mu_{R}^{\pm}\leftrightarrow-\mu_{L}^{\pm}$. The
case of originally massless quantum states $m=0$ is also intriguing from a
theoretical point of view. From the mass formulas (2.57) and (2.58) one sees
straightforwardly that in such a case the equality $\mu_{R}=-\mu_{L}$ holds.
In the limiting case of generic Einstein Special Relativity $\ell=0$, the
projected masses have interesting properties
$\displaystyle\mu_{R}^{\pm}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}-\infty&\leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ +\\\ \infty&\leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ -\end{array}\right.,$ (2.73)
$\displaystyle\mu_{L}^{\pm}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\infty&\leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ +\\\ -\infty&\leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ -\end{array}\right..$ (2.76)
It means that in the case of Special Relativity the massive Weyl theories are
undetermined, i.e. the particles described by the Weyl equations (2.66) and
(2.69) do not exist. In general, however, for correctness of the projection
splitting (2.56) both the projected masses (2.57) and (2.58) must be real
numbers. Strictly speaking when the projected masses are complex numbers the
decomposition (2.56) does not yield hermitian mass matrices (2.70). Therefore,
in such a case the presented construction has no sense, and by this reason
should be replaced by another theory based on different arguments.
In the approach based on the Standard Model neutrinos are massless quantum
particles. In this manner, it is evident that application of the Snyder
noncommutative geometry generated the non-triviality which is _the kinetic
mass generation mechanism of the neutrino mass_. It must be emphasized that in
all the cited contributions Sidharth mentioned about a possibility of neutrino
mass ”due to the mass term”, where by the mass term he understands the
dynamical $\gamma^{5}$-term in the modified Dirac equation (2.33). Albeit,
regarding the Standard Model which is the theory of elementary particles and
fundamental interactions, a mass term can not be dynamical, and such a
nomenclature is misleading in further analysis of the philosophical ideas. We
have been received the massive neutrinos due to the two-step mass generation
mechanism. The first one is factorization of the modified Dirac equation
(2.33). The second one is the decomposition of the mass matrices (2.48) into
the projectors basis and introducing the chiral Weyl fields (2.62) and (2.63).
Calling such an unique procedure as the result ”due to the mass term” is at
least inaccurate, creates a number of inequivalent interpretations, and in
itself is escaping from physics to philosophy. It must be emphasized that any
mass generation mechanism is manifestly absent in Sidharth’s contributions and
the reasoning presented there completely differs from our analysis, omits
certain important physical and mathematical details (Cf. e.g. Ref. [137]). The
procedure proposed above, i.e. by the unique treatment of the
$\gamma^{5}$-term in the modified Dirac equation (2.33) and preservation the
hyperbolic geometry of Minkowski energy-momentum space (2.37) of the Einstein
Special Relativity, resulted in generation of the Weyl equations (2.66)-(2.69)
describing two left- $\psi_{L}^{\pm}$ and two right- $\psi_{R}^{\pm}$ chiral
fields possessing nonzero masses. In other words we have established the
constructive theory of massive neutrinos, related to both originally massive
$m\neq 0$ as well as massless $m=0$ quantum states described by relativistic
quantum mechanics. By this reason the our approach changes the physical
essence of the concepts _neutrino_ and _neutrino mass_. Our neutrino is a
chiral field due to any originally massive and massless quantum particle,
which in itself is also a quantum particle. Our neutrino mass is generated due
to a mass of originally quantum particle. Furthermore, it is easy to see that
the Weyl theories (2.66) and (2.69) associate 4 massive neutrinos with a one
original quantum state. Because of there are 4 possible theories (2.36) the
maximal number of the neutrinos due to the Snyder noncommutative geometry is
16. However, this number can be reduced due to experimental verification of
the results of the theories.
#### C The Compton–Planck Scale
The Planck scale, defined by taking into account a minimal scale $\ell$
identical with the Planck length $\ell_{P}=\sqrt{{\dfrac{\hslash G}{c^{3}}}}$,
in modern cosmology and particle physics is the energetic region at which
quantum physics meets classical physics. At this scale the standard
methodology of particle physics, i.e. the Standard Model, is manifestly
inadequate tool for constructive description because of necessity of a theory
of quantum gravity. There is few approaches to the adequate formalism of the
Planck scale physics, including string theory, M-theory, loop quantum gravity,
and noncommutative geometry (See e.g. the Ref. [138]). A theory of quantum
gravity, however, is the main requirement for the consistent description of
the physics at the Planck scale. We shall present certain proposal for such a
theory in the second part of this book, while this part is related to
noncommutative geometry.
Let us look on few coincidences at the Planck scale, which are informative for
general understanding. In such a situation a maximal energy (4.130) coincides
with the Planck energy divided by $\sqrt{\alpha}$
$\epsilon(\ell_{P})=\dfrac{1}{\sqrt{\alpha}}\sqrt{{\dfrac{\hslash
c^{5}}{G}}}=\dfrac{1}{\sqrt{\alpha}}M_{P}c^{2}=\dfrac{E_{P}}{\sqrt{\alpha}}.$
(2.77)
Let us take into account the Compton wavelength of a particle possessing the
rest mass $m$, i.e. $\lambda_{C}(m)=2\pi\dfrac{\hslash}{mc}$. A maximal energy
(4.130) computed at the scale identical to the Compton wavelength
$\lambda_{C}(m)$ is proportional to rest energy of a particle
$\epsilon(\lambda_{C}(m))=\dfrac{1}{2\pi\sqrt{\alpha}}mc^{2},$ (2.78)
and, if $\alpha$ has the proposed value (1.220), becomes simply the rest
energy of a particle $\epsilon=mc^{2}$.
Interestingly, in the situation when the rest mass of a particle equals the
Planck mass $m\equiv M_{P}=\sqrt{\dfrac{\hslash c}{G}}$ or equivalently the
rest energy of a particle equals to the Planck energy $mc^{2}=E_{P}$, there is
a number of non-trivialities. Let us denote the Compton wavelength of such a
Planckian particle by $\ell_{C}=\lambda_{C}(M_{P})$. In fact, this wavelength
defines the mixed scale which we propose to call _the Compton–Planck (CP)
scale_. It can be seen straightforwardly that in the CP scale holds
$\epsilon(\ell_{C})=\dfrac{\epsilon(\ell_{P})}{2\pi}.$ (2.79)
Moreover, when one regards (1.220), i.e. $\alpha=1/(2\pi)^{2}$, than also
$\alpha=\left(\dfrac{\epsilon(\ell_{C})}{\epsilon(\ell_{P})}\right)^{2}=\left(\dfrac{\ell_{P}}{\ell_{C}}\right)^{2}.$
(2.80)
Moreover, in the CP scale the doubled Compton wavelength equals to a
circumference of a circle with a radius of the Schwarzschild radius
$r_{S}(m)=\dfrac{2Gm}{c^{2}}$ of the Planck mass (Cf. Ref. [118])
$2\ell_{C}=2\pi r_{S}\left(M_{P}\right).$ (2.81)
Straightforward and easy computation shows that the Compton wavelength of the
Planck mass is identified with a circumference of a circle with a radius of
the Planck length, i.e.
$\ell_{C}=2\pi\ell_{P},$ (2.82)
and by this reason the doubled Planck length equals to the Schwarzschild
radius of the Planck mass
$2\ell_{P}=r_{S}(M_{P}).$ (2.83)
In general the ratio of the Planck length and the Compton wavelength of a
particle with mass $m$ is
$\dfrac{\ell_{P}}{\lambda_{C}(m)}=\dfrac{1}{2\pi}\dfrac{m}{M_{P}}.$ (2.84)
Let us generalize the last relation in the equation (2.80) as follows
$\alpha\equiv\left(\dfrac{1}{2\pi}\dfrac{m}{M_{P}}\right)^{2}.$ (2.85)
Taking $\alpha$ as (1.220) together with the presumption $\alpha\sim 1$, which
guarantees correctness of the Kontsevich deformation quantization. Such a
reasoning establishes the mass of a particle for which the Snyder
noncommutative geometry is adequate. It is not difficult to see that the mass
of a particle must be of an order of the Planck mass, i.e.
$m\sim M_{P}.$ (2.86)
If, however, one wishes to neglect (1.220) but preserve (2.85) together with
the condition $\alpha\sim 1$ then
$m\sim 2\pi M_{P},$ (2.87)
i.e. the mass of a particle described by the noncommutative geometry is of the
order $m\sim(10^{22}-10^{23})\mathrm{MeV/c^{2}}$.
#### D The Global Effective Chiral Condensate
Let us consider the meaning of the massive Weyl equations (2.66)-(2.66) in the
spirit of the gauge field theories [139], which are the base of the Standard
Model. The problem is to to construct the Lagrangian $\mathcal{L}$, revealing
Lorentz invariance, of the gauge field theory characterized by the massive
Weyl equations treated as the Euler-Lagrange equations of motion for the
chiral fields $\psi_{L}^{\pm}$ and $\psi_{R}^{\pm}$. In general, such a
construction is not easy to perform, but because in fact the massive Weyl
equations are the Dirac theories, it can be seen by straightforward
computations that the following four Dirac-like Lagrangians
$\displaystyle\mathcal{L}^{\pm}_{R}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{R}^{\pm}\left(\gamma^{\mu}\hat{p}_{\mu}+\mu_{R}^{\pm}c^{2}\right)\psi_{R}^{\pm},$
(2.88) $\displaystyle\mathcal{L}^{\pm}_{L}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{L}^{\pm}\left(\gamma^{\mu}\hat{p}_{\mu}+\mu_{L}^{\pm}c^{2}\right)\psi_{L}^{\pm},$
(2.89)
where
$\bar{\psi}_{R,L}^{\pm}=\left(\psi_{R,L}^{\pm}\right)^{\dagger}\gamma^{0}$ are
the Dirac adjoint of $\psi_{R,L}^{\pm}$, lead to the massive Weyl equations by
the appropriate principle of the least action. In other words, the Euler-
Lagrange equations of motion
$\dfrac{\partial\mathcal{L}^{\pm}_{R,L}}{\partial\psi_{R,L}^{\pm}}-\partial_{\mu}\dfrac{\partial\mathcal{L}^{\pm}_{R,L}}{\partial\left(\partial_{\mu}\psi_{R,L}^{\pm}\right)}=0,$
(2.90)
coincide with the massive Weyl equations (2.66)-(2.66). In this manner, the
appropriate full gauge field theory of massive neutrinos can be directly
constructed by using of the Lagrangian which is an algebraical sum of the four
gauge field theories (2.88)-(2.89)
$\mathcal{L}=\mathcal{L}^{+}_{R}+\mathcal{L}^{-}_{R}+\mathcal{L}^{+}_{L}+\mathcal{L}^{-}_{L},$
(2.91)
i.e. the massive Weyl equations are obtained from the Euler-Lagrange equations
of motion (2.90) with the exchange
$\mathcal{L}^{\pm}_{R,L}\rightarrow\mathcal{L}$. It must be emphasized that
the choice of the Lagrangians in the form (2.88)-(2.89) is due to
straightforward analogy between the massive Weyl equation and the Dirac
equation. The essential difference between these theories is the only number
of spinor components what, however, has no influence on the form of Lagrangian
and the principle of the least action. The Lagrangians (2.88) and (2.89)
describe the two components of the Weyl spinor. In other words, this choice is
both the most intuitive and the simplest. However, it does not mean that there
is no another, possibly more complicated, choice dictated by another
justification.
One can see easy that the gauge field theories (2.88) and (2.89) exhibit
several well-known gauge symmetries. Namely, the (local) chiral symmetry
$SU(2)_{R}^{\pm}\otimes SU(2)_{L}^{\pm}$ expressed via one of the
transformations
$\displaystyle\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\exp\left\\{i\theta_{R}^{\pm}\right\\}\psi_{R}^{\pm}\vspace*{5pt}\\\
\psi_{L}^{\pm}\rightarrow\psi_{L}^{\pm}\end{array}\right.,\vspace*{5pt}$
(2.94)
$\displaystyle\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\psi_{R}^{\pm}\vspace*{5pt}\\\
\psi_{L}^{\pm}\rightarrow\exp\left\\{i\theta_{L}^{\pm}\right\\}\psi_{L}^{\pm}\end{array}\right.,$
(2.97)
the vector symmetry $U(1)_{V}^{\pm}$
$\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi_{R}^{\pm}\vspace*{5pt}\\\
\psi_{L}^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi_{L}^{\pm}\end{array}\right.,$
(2.98)
and the axial symmetry $U(1)_{A}^{\pm}$
$\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\exp\left\\{-i\theta^{\pm}\right\\}\psi_{R}^{\pm}\vspace*{5pt}\\\
\psi_{L}^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi_{L}^{\pm}\end{array}\right..$
(2.99)
In this manner the total symmetry group is
$SU(3)_{C}^{+}\oplus SU(3)_{C}^{-},$ (2.100)
where $SU(3)_{C}^{\pm}$ are the global (chiral) 3-flavor gauge symmetries
related to each of the gauge theories (2.88) and (2.89), i.e.
$\displaystyle SU(2)_{R}^{+}\otimes SU(2)_{L}^{+}\otimes U(1)_{V}^{+}\otimes
U(1)_{A}^{+}\equiv SU(3)^{+}\otimes SU(3)^{+}=SU(3)_{C}^{+},$ (2.101)
$\displaystyle SU(2)_{R}^{-}\otimes SU(2)_{L}^{-}\otimes U(1)_{V}^{-}\otimes
U(1)_{A}^{-}\equiv SU(3)^{-}\otimes SU(3)^{-}=SU(3)_{C}^{-},$ (2.102)
describing 2-flavor massive free quarks - _the neutrinos_ in our proposition.
Because of, the group (2.100) does not possess a name in literature, we shall
call the group _the composite symmetry_ $SU(3)_{C}^{TOT}$.
By application of the definitions for the chiral Weyl fields (2.62) and (2.63)
one obtains
$\bar{\psi}_{R,L}^{\pm}\gamma^{\mu}p_{\mu}\psi_{R,L}^{\pm}=\bar{\psi}^{\pm}\dfrac{1\pm\gamma^{5}}{2}\gamma^{\mu}p_{\mu}\dfrac{1\pm\gamma^{5}}{2}\psi^{\pm}=\bar{\psi}^{\pm}\left(\dfrac{1\pm\gamma^{5}}{2}\gamma^{\mu}\dfrac{1\pm\gamma^{5}}{2}\right)p_{\mu}\psi^{\pm},$
(2.103)
where $\bar{\psi}^{\pm}=\left(\psi^{\pm}\right)^{\dagger}\gamma^{0}$ is the
Dirac adjoint of the Dirac fields $\psi^{\pm}$ related to the chiral Weyl
fields by the transformations (2.62) and (2.63). Because of the identity
$\dfrac{1\pm\gamma^{5}}{2}\gamma^{\mu}\dfrac{1\pm\gamma^{5}}{2}=\dfrac{\gamma^{\mu}\pm\left\\{\gamma^{\mu},\gamma^{5}\right\\}+\gamma^{5}\gamma^{\mu}\gamma^{5}}{4}=\dfrac{1-(\gamma^{5})^{2}}{4}\gamma^{\mu}=\dfrac{1}{2}\gamma^{\mu},$
(2.104)
where we have applied the properties of $\gamma^{5}$ matrix
$\left\\{\gamma^{\mu},\gamma^{5}\right\\}=0$ and $(\gamma^{5})^{2}=-1$, one
obtains finally
$\bar{\psi}_{R,L}^{\pm}\gamma^{\mu}p_{\mu}\psi_{R,L}^{\pm}=\dfrac{1}{2}\bar{\psi}^{\pm}\gamma^{\mu}p_{\mu}\psi^{\pm}.$
(2.105)
Similarly, applying the identity
$\displaystyle\dfrac{1\pm\gamma^{5}}{2}\dfrac{1\pm\gamma^{5}}{2}$
$\displaystyle=$ $\displaystyle\dfrac{1}{4}\left(1\pm
2\gamma^{5}+(\gamma^{5})^{2}\right)=\pm\dfrac{1}{2}\gamma^{5},$ (2.106)
one can establish the quantity
$\mu_{R,L}^{\pm}c^{2}\bar{\psi}^{\pm}_{R,L}\psi_{R,L}^{\pm}=\mu_{R,L}^{\pm}c^{2}\bar{\psi}^{\pm}\dfrac{1\pm\gamma^{5}}{2}\dfrac{1\pm\gamma^{5}}{2}\psi^{\pm},$
(2.107)
with the result
$\mu_{R,L}^{\pm}c^{2}\bar{\psi}^{\pm}_{R,L}\psi_{R,L}^{\pm}=\dfrac{1}{2}\bar{\psi}^{\pm}\left(\pm\mu_{R,L}^{\pm}c^{2}\gamma^{5}\right)\psi^{\pm},$
(2.108)
where the plus sign of the mass is appropriate for the right-handed neutrinos,
while the minus sign is appropriate for the left-handed neutrinos. In this way
one obtains finally the partial Lagrangians
$\displaystyle\mathcal{L}_{R,L}^{\pm}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{R,L}^{\pm}\left(\gamma^{\mu}p_{\mu}+\mu_{R,L}^{\pm}c^{2}\right)\psi_{R,L}^{\pm}=$
(2.109) $\displaystyle=$
$\displaystyle\bar{\psi}_{R,L}^{\pm}\gamma^{\mu}p_{\mu}\psi_{R,L}^{\pm}+\mu_{R,L}^{\pm}c^{2}\bar{\psi}^{\pm}_{R,L}\psi_{R,L}^{\pm}=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\bar{\psi}^{\pm}\gamma^{\mu}p_{\mu}\psi^{\pm}+\dfrac{1}{2}\bar{\psi}^{\pm}\left(\pm\mu_{R,L}^{\pm}c^{2}\gamma^{5}\right)\psi^{\pm}=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\bar{\psi}^{\pm}\left(\gamma^{\mu}p_{\mu}\pm\mu_{R,L}^{\pm}c^{2}\gamma^{5}\right)\psi^{\pm}.$
which reveal the Lorentz invariance. In this manner it can be seen
straightforwardly that with using of the partial Lagrangians (2.109) the
global chiral Lagrangian (2.91) takes the following form
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\bar{\psi}^{+}\left(\gamma^{\mu}p_{\mu}+\mu_{R}^{+}c^{2}\gamma^{5}\right)\psi^{+}+\dfrac{1}{2}\bar{\psi}^{+}\left(\gamma^{\mu}p_{\mu}-\mu_{L}^{+}c^{2}\gamma^{5}\right)\psi^{+}+$
(2.110) $\displaystyle+$
$\displaystyle\dfrac{1}{2}\bar{\psi}^{-}\left(\gamma^{\mu}p_{\mu}+\mu_{R}^{-}c^{2}\gamma^{5}\right)\psi^{-}+\dfrac{1}{2}\bar{\psi}^{-}\left(\gamma^{\mu}p_{\mu}-\mu_{L}^{-}c^{2}\gamma^{5}\right)\psi^{-},$
or after summation
$\mathcal{L}=\bar{\psi}^{+}\left(\gamma^{\mu}\hat{p}_{\mu}+\mu_{eff}^{+}c^{2}\right)\psi^{+}+\bar{\psi}^{-}\left(\gamma^{\mu}\hat{p}_{\mu}+\mu_{eff}^{-}c^{2}\right)\psi^{-},$
(2.111)
where $\mu_{eff}^{\pm}$ are the effective mass matrices of the gauge fields
$\psi^{\pm}$,
$\mu_{eff}^{\pm}=\dfrac{\mu_{R}^{\pm}-\mu_{L}^{\pm}}{2}\gamma^{5}.$ (2.112)
After introduction of the global effective 8-component field
$\Psi=\left[\begin{array}[]{c}{\psi^{+}}\\\ {\psi^{-}}\end{array}\right],$
(2.113)
the theory (2.111) becomes
$\mathcal{L}=\bar{\Psi}\left(\gamma^{\mu}\hat{p}_{\mu}+M_{eff}c^{2}\right)\Psi,$
(2.114)
where $M_{eff}$ is the mass matrix given by
$M_{eff}=\left[\begin{array}[]{cc}{\mu^{+}_{eff}}&0\\\
0&{\mu^{-}_{eff}}\end{array}\right],$ (2.115)
$\gamma^{\mu}$ are $8\times 8$ matrices, and
$\hat{p}_{\mu}=i\hslash\left[\begin{array}[]{c}\partial_{\mu}\\\
\partial_{\mu}\end{array}\right],$ (2.116)
is 8-component momentum operator. Therefore (2.114) can be associated with
octonions. Hermiticity of both the effective mass matrices $\mu^{\pm}_{eff}$,
and therefore also of the global effective mass matrix $M_{eff}$, depends on a
choice of representation of the $\gamma^{5}$ matrix. For consistency the
preferred representation of $\gamma^{5}$ matrix must be hermitian. It means
that the effective global gauge field theory (2.114) is physical for the only
such a representation.
Obviously, the global effective gauge field theory (2.114) demonstrates an
invariance with respect to action of the composite gauge vector symmetry
$SU(2)_{V}^{TOT}$
$SU(2)_{V}^{TOT}=SU(2)_{V}^{+}\oplus SU(2)_{V}^{-},$ (2.117)
where $SU(2)_{V}^{\pm}$ are the $SU(2)\otimes SU(2)$ group transformations
applied separately to each of the gauge fields $\psi^{\pm}$
$\left\\{\begin{array}[]{c}\psi^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi^{\pm}\vspace*{5pt}\\\
\bar{\psi}^{\pm}\rightarrow\bar{\psi}^{\pm}\exp\left\\{-i\theta^{\pm}\right\\}\end{array}\right..$
(2.118)
Such a situation means that there is realized the mechanism of spontaneous
symmetry breakdown for the global effective gauge field theory (2.114). The
broken symmetry is the composite global chiral symmetry $SU(3)_{C}^{TOT}$, and
the result of the symmetry breakdown is its subgroup the composite isospin
symmetry group $SU(2)_{V}^{TOT}$
$SU(3)_{C}^{TOT}\longrightarrow SU(2)_{V}^{TOT}.$ (2.119)
Such a situation possesses an unambiguous physical interpretation. Namely, it
is the syndrome of an existence of the global effective chiral condensate of
the massive neutrinos, being a composition of two independent chiral
condensates, which is the global effective gauge field theory invariant under
action of the gauge symmetry (See, e.g. the book of S. Weinberg in Ref. [139])
$SU(2)_{V}^{TOT}=(SU(2)^{+}\otimes SU(2)^{+})\oplus(SU(2)^{-}\otimes
SU(2)^{-}).$ (2.120)
However, because of action of the composite global chiral gauge symmetry
$SU(3)_{C}^{TOT}$ (2.100), the gauge field theories (2.88) and (2.89) looks
like formally as the theories of non-interacting massive free quarks. Such a
situation is very similar to the formalism of Quantum Chromodynamics (QCD)
[140], but factually in the presented physical scenario one has to deal with a
composition of two independent copies of QCD. For each of these theories the
space of fields is distinguishable for the space of fields of the single QCD.
The difference is contained, namely, in the fact that there are only two
massive chiral fields - the left- and the right-handed Weyl fields, which are
the neutrinos in our proposition. The global effective chiral condensate of
massive neutrinos (2.114) is the result manifestly beyond the Standard Model,
but essentially it can be included into the fundamental theory of particle
physics as the new contribution due to noncommutative geometry. Usually the
spontaneous symmetry breakdown results in the related Goldstone bosons
generated via the mechanism. However, in the situation presented above this
mechanism results in the chiral condensate, i.e. does not generate new
particles. In the context of the massive neutrinos, the result of the
mechanism of the spontaneous symmetry breakdown is the global effective chiral
condensate of the massive neutrinos. By this reason such a situation is
manifestly distinguishable, and is beyond methods of the Standard Model.
#### E Conclusion
The deformed Special Relativity given by the Snyder–Sidharth Hamiltonian
constraint (2.32) obtained due to the Snyder geometry of noncommutative space-
time (2.1)-(2.1) manifestly and essentially differs from the usual Einstein
energy-momentum relation well-known from Special Relativity. In particular as
is self-evident from the form of the Snyder–Sidharth Hamiltonian
constraint(2.32), the Snyder noncommutative geometry produces the extra
contribution to the Einstein energy-momentum relation due to the additional
$\ell^{2}$-term. As we have shown, this contribution can be neglected as the
result of the algebra deformation. This is brought out very clearly in the
Dirac equations (2.46)-(2.47) which are manifestly non-hermitian, as well as
in the massive Weyl equations (2.66)-(2.69) which are blatantly hermitian and
are responsible for description of the neutrinos in our proposition.
A massless neutrino, characteristic for both the conventional Weyl theory as
well as the Standard Model, is now seen to argue as mass, and further, this
mass has a two left-handed components and a two right-handed components, as it
is straightforwardly noticeable from the formulas (2.53) and (2.56). Once this
is recognized, the mass matrix which otherwise appears non-hermitian, turns
out to be actually hermitian, as seen in the formula (2.70), but if and only
if when the masses (2.53) and (2.56) of the neutrinos are real numbers. There
is no any restrictions, however, for their sign, i.e. the masses can be
positive as well as negative. In other words, the underlying Snyder
noncommutative geometry (2.1)-(2.1) is reflected in the modified Dirac
equation (2.33) and naturally and nontrivially gives rise to the mass of the
neutrino.
As we have mentioned in partial discussions within this chapter, in analogy
with the Standard Model Sidharth [137] suggested that such a situation is a
possible result ”due to mass term”, however, with no any concrete calculations
and proposals for the mass generation mechanism. The mass generation
mechanism, proposed in this chapter for such a constructive and consistent
formulation of this Sidharth idea, has purely kinetic nature, and moreover it
is formally the result of the first approximation of more general
noncommutative geometry determined by the Kontsevich deformation quantization.
In this manner we have shown that the mass generation mechanism ”due to mass
term” can be elegantly formulated in frames of noncommutative geometry,
particularly in frames of the Snyder space-time.
We have shown also that the model of massive neutrinos can be understood and
consistently described from the point of view of gauge field theories, which
naturally includes Lorentz invariance. Such a formulation leads to interesting
construction involving two independent copies of Quantum Chromodynamics and
non-interacting massive free quarks, which is also employing effective
composite isospin group resulting in the global effective chiral condensate of
the massive neutrinos. The mechanism of spontaneous symmetry breakdown
presented above, which is the tool to receiving the composite isospin group,
does not require existence of related Goldstone bosons, but the role of
Goldstone bosons plays the chiral condensate of massive neutrinos. In itself
this is new type of mass generation mechanism.
It must be remembered that in the Standard Model the neutrino is massless, but
the Super-Kamiokande experiments in the late nineties showed that the neutrino
does indeed have a mass and this is the leading motivation to an exploration
of models beyond the Standard Model. The model presented above is the b est
example of such a situation. In this connection it is also relevant to mention
that currently the Standard Model requires the Higgs mechanism for the
generation of mass in general, though the Higgs particle has been undetected
for forty five years and it is hoped will be detected by researchers of Fermi
National Accelerator Laboratory or the Large Hadron Collider. We hope for next
development within the proposed here model of massive neutrinos.
### Chapter 3 The Neutrinos: Energy Renormalization & Integrability
#### A Introduction
In the previous chapter we have established that the modified Dirac equation
arising due to the Snyder noncommutative geometry, yields the conventional
Dirac theory with non-hermitian mass, or equivalently to the Weyl equation
with a diagonal and hermitian mass matrices which describes the massive
neutrinos. The obtained model of massive neutrinos involves 4 massive chiral
fields related to any originally massive or massless quantum state obeying the
usual Dirac equation. By application of the mechanism of spontaneous symmetry
breakdown with respect to the global chiral symmetry the model was converted
into the form of the isospin-symmetric global effective gauge field theory of
the 8-component field $\Psi$ which is associated with the composed chiral
condensate of massive neutrinos.
All these results violate the Lorentz symmetry manifestly, albeit their
possible physical application can be considered in a diverse way. On the one
hand the global effective gauge field theory is beyond the Standard Model, yet
can be considered as its contributory part due to the Snyder noncommutative
geometry. On the other hand, in the model of massive neutrinos the masses of
the two left-handed and two right-handed chiral Weyl fields arise due to mass
and energy of an original state and a minimal scale, _e.g._ the Planck scale.
Therefore, its quantum mechanical countenance becomes almost a mystical
riddle. In fact, possible existence of the massive neutrinos would be the
logically consistent justification of physical correctness of the Snyder
noncommutative geometry.
This chapter is mostly focused on the quantum mechanical aspect of the model
of massive neutrinos. We shall present manifestly that the model in itself
yields consistent physical explanation of the Snyder noncommutative geometry
and consequently leads to energy renormalization of an original quantum
relativistic particle. We shall perform computations arising directly from the
Schrödinger equation formulation of both the modified Dirac equation and the
massive Weyl equation.
The first issue for discussion is the manifestly non-hermitian modified Dirac
Hamiltonian. Its integrability is formulated by straightforward application of
the Zassenhaus formula for exponentiation of sum of two non commuting
operators. It is shown directly, however, that this approach does not lead to
well-defined solutions, because of for such a formulation the Zassenhaus
exponents are still sums of two non commuting operators. Therefore such a
integrability procedure possesses a cyclic problem which can not be removed,
and by this reason is not algorithm. In this case the only approximations can
be studied, but extraction of full solution is an extremely difficult problem.
For solving the problem we shall change the integrability strategy, i.e.
instead of the modified Dirac equation we shall employ the Schrödinger
equation form of the Weyl equation with pure hermitian mass matrix.
Integration of this equation is straightforward, elementary, and analogous to
integration of the Dirac equation. Computations shall be presented in both the
Dirac and the Weyl representations of the Dirac gamma matrices.
We perform calculations in the Clifford algebra because of the representations
of gamma matrices obeying the Æther algebra are not established and are very
good problem for future research. This is caused by the fact, that the Æther
algebra was proposed first in this book, and was not considered in earlier
literature.
#### B Energy renormalization
Let us focus our attention on the masses of left-handed and right-handed
chiral Weyl fields (2.57) and (2.58). By straightforward elementary algebraic
manipulations these two relations can be rewritten as the following system of
equations
$\left\\{\begin{array}[]{c}\left(\mu_{R}^{\pm}c^{2}+\dfrac{\epsilon}{2}\right)^{2}=\epsilon^{2}-4\epsilon
mc^{2}-4E^{2}\\\
\left(\mu_{L}^{\pm}c^{2}-\dfrac{\epsilon}{2}\right)^{2}=\epsilon^{2}+4\epsilon
mc^{2}-4E^{2}\end{array}\right.$ (3.1)
which allows to study dependence between the deformation parameter, i.e. a
maximal energy $\epsilon$, and energy $E$ and mass $m$ of a particle and the
masses $\mu_{R}^{\pm}$ and $\mu_{L}^{\pm}$ of neutrinos treated as physically
measurable quantities which can be established via experimental data.
Subtraction of the second equation from the first one in the system of
equations (3.1), allows to obtain
$\left(\mu_{L}^{\pm}c^{2}-\dfrac{\epsilon}{2}\right)^{2}-\left(\mu_{R}^{\pm}c^{2}+\dfrac{\epsilon}{2}\right)^{2}=8\epsilon
mc^{2},$ (3.2)
or applying the difference of two squares $a^{2}-b^{2}=(a-b)(a+b)$
$\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}-\epsilon\right]\left(\mu_{L}^{\pm}+\mu_{R}^{\pm}\right)c^{2}=8\epsilon
mc^{2},$ (3.3)
what allows to derive a maximal energy as
$\epsilon=\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{1+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}.$
(3.4)
Because of $\epsilon\geqslant 0$ one has the condition of masses of neutrinos
$\mu_{L}^{\pm}\geqslant\mu_{R}^{\pm},$ (3.5)
what after using of the explicit formulas (2.57) and (2.58) leads to two
alternative conditions for $\epsilon$
$\epsilon\geqslant\mp\left(\sqrt{\epsilon^{2}+4\epsilon
mc^{2}-4E^{2}}+\sqrt{\epsilon^{2}-4\epsilon mc^{2}-4E^{2}}\right).$ (3.6)
The first condition leads to
$\epsilon\in\left(-\infty,-\dfrac{8}{3}mc^{2}-\dfrac{4}{3}\sqrt{3E^{2}+4m^{2}c^{4}}\right]\cup\left[-2mc^{2}+2\sqrt{E^{2}+m^{2}c^{4}},\infty\right),$
(3.7)
while the second one gives
$\epsilon\in\left[-2mc^{2}+2\sqrt{E^{2}+m^{2}c^{4}},-\dfrac{8}{3}mc^{2}+\dfrac{4}{3}\sqrt{3E^{2}+4m^{2}c^{4}}\right],$
(3.8)
and by taking these results together one obtains finally
$-2mc^{2}+2\sqrt{E^{2}+m^{2}c^{4}}\leqslant\epsilon\leqslant-\dfrac{8}{3}mc^{2}+\dfrac{4}{3}\sqrt{3E^{2}+4m^{2}c^{4}}.$
(3.9)
Because, however, $\epsilon\geqslant 0$ one has the conditions for mass and
energy of a particle
$\displaystyle-2mc^{2}+2\sqrt{E^{2}+m^{2}c^{4}}$ $\displaystyle\geqslant$
$\displaystyle 0,$ (3.10)
$\displaystyle-\dfrac{8}{3}mc^{2}+\dfrac{4}{3}\sqrt{3E^{2}+4m^{2}c^{4}}$
$\displaystyle\geqslant$ $\displaystyle 0,$ (3.11)
$\displaystyle-\dfrac{8}{3}mc^{2}+\dfrac{4}{3}\sqrt{3E^{2}+4m^{2}c^{4}}$
$\displaystyle\geqslant$ $\displaystyle-2mc^{2}+2\sqrt{E^{2}+m^{2}c^{4}}.$
(3.12)
The first and the second conditions leads to the trivial relation
$E^{2}\geqslant 0$, while the third one states that
$\dfrac{9}{32}E^{4}+m^{2}c^{4}E^{2}+m^{4}c^{8}\geqslant 0,$ (3.13)
what is satisfied if and only if mass $m$ and energy $E$ of a particle are
real numbers. By application of the bounds (3.9) to the relation (3.4) one
obtains the condition for masses
$m\leqslant\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{8},$ (3.14)
or equivalently the inequality for the ratio
$\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}\leqslant 1.$ (3.15)
By the definition (3.4) one has
$\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}=\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{\epsilon}-1,$
(3.16)
what in the light of the inequality (3.15) leads to the bound
$\epsilon\geqslant\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{2}c^{2},$ (3.17)
which can be translated into the language of a minimal scale
$\ell\leqslant\dfrac{\hslash}{\sqrt{\alpha}c}\dfrac{2}{\mu_{L}^{\pm}-\mu_{R}^{\pm}}.$
(3.18)
By using of the Compton wavelength of a neutrino
$\lambda_{C}(\mu_{R,L}^{\pm})=2\pi\dfrac{\hslash}{\mu_{R,L}^{\pm}c},$ (3.19)
the bound (3.25) can be written in the form
$\ell\leqslant\dfrac{1}{2\pi\sqrt{\alpha}}\dfrac{2\lambda_{C}(\mu_{L}^{\pm})\lambda_{C}(\mu_{R}^{\pm})}{\lambda_{C}(\mu_{R}^{\pm})-\lambda_{C}(\mu_{L}^{\pm})},$
(3.20)
which, if one wishes to apply $\alpha=\dfrac{1}{2\pi}$ established by (1.220),
becomes
$\ell\leqslant\dfrac{2\lambda_{C}(\mu_{L}^{\pm})\lambda_{C}(\mu_{R}^{\pm})}{\lambda_{C}(\mu_{R}^{\pm})-\lambda_{C}(\mu_{L}^{\pm})}.$
(3.21)
Equivalently, the bound (3.25) with the condition (1.220) can be presented in
most conventional form
$\ell\leqslant\lambda_{C}\left(\dfrac{\mu_{R}^{\pm}-\mu_{L}^{\pm}}{2}\right).$
(3.22)
If one wishes to do not preserve (1.220) then the bound (3.20) for fixed scale
value establishes the following inequality for $\alpha$
$\alpha\leqslant\left(\dfrac{1}{2\pi}\dfrac{1}{\ell}\dfrac{2\lambda_{C}(\mu_{L}^{\pm})\lambda_{C}(\mu_{R}^{\pm})}{\lambda_{C}(\mu_{R}^{\pm})-\lambda_{C}(\mu_{L}^{\pm})}\right)^{2},$
(3.23)
which in the light of the generalization (2.85) leads to
$\dfrac{m}{M_{P}}\leqslant\dfrac{1}{\ell}\dfrac{2\lambda_{C}(\mu_{L}^{\pm})\lambda_{C}(\mu_{R}^{\pm})}{\lambda_{C}(\mu_{R}^{\pm})-\lambda_{C}(\mu_{L}^{\pm})},$
(3.24)
or with using of (3.25)
$m\leqslant\left(\dfrac{\hslash}{c}\right)^{2}\dfrac{2\pi}{\ell\ell_{P}}\dfrac{2}{\mu_{L}^{\pm}-\mu_{R}^{\pm}}.$
(3.25)
This result in the light of the bound (3.14), however, leads to
$\left(\dfrac{\hslash}{c}\right)^{2}\dfrac{2\pi}{\ell\ell_{P}}\dfrac{2}{\mu_{L}^{\pm}-\mu_{R}^{\pm}}=\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{8},$
(3.26)
what results in the squared-mass difference
$\Delta\mu^{2}_{LR}=\left(\mu_{L}^{\pm}\right)^{2}-\left(\mu_{R}^{\pm}\right)^{2}=\left(\dfrac{\hslash}{c}\right)^{2}\dfrac{32\pi}{\ell\ell_{P}}=32\pi\dfrac{\ell_{P}}{\ell}M_{P}^{2}.$
(3.27)
In this manner if $\Delta\mu^{2}_{LR}$ is fixed by experimental data, then by
the equation (3.27) establishes the minimal scale
$\ell=32\pi\dfrac{M_{P}^{2}}{\Delta\mu^{2}_{LR}}\ell_{P},$ (3.28)
or approximatively
$\ell\approx 2.4220\cdot
10^{23}m\dfrac{1\dfrac{eV^{2}}{c^{2}}}{\Delta\mu^{2}_{LR}}.$ (3.29)
Interestingly, at the Planck scale $\ell=\ell_{P}$, then (3.28) generates
$\Delta\mu^{2}_{LR}=32\pi M_{P}^{2}\approx 1.4985\cdot
10^{10}\dfrac{YeV^{2}}{c^{2}},$ (3.30)
where $1YeV=10^{24}eV$, while at the Compton scale $\ell=\lambda_{C}(m_{e})$
$\Delta\mu^{2}_{LR}\approx 10^{5}\dfrac{PeV^{2}}{c^{2}},$ (3.31)
where $1PeV=10^{15}eV$. Similarly, at the Compton–Planck
scale$\ell=\lambda_{C}(M_{P})$
$\Delta\mu^{2}_{LR}=16M_{P}^{2}\approx 2.3850\cdot
10^{9}\dfrac{YeV^{2}}{c^{2}}.$ (3.32)
In the light of the mass formulas (2.57) and (2.58) one can deduce the
squared-mass difference
$\Delta\mu^{2}_{LR}=16\pi\dfrac{\ell_{P}}{\ell}M_{P}^{2}\left[1\pm\dfrac{\sqrt{\epsilon^{2}-4E^{2}+8\pi\dfrac{\ell_{P}}{\ell}M_{P}^{2}c^{4}}-\sqrt{\epsilon^{2}-4E^{2}-8\pi\dfrac{\ell_{P}}{\ell}M_{P}^{2}c^{4}}}{8mc^{2}}\right],$
(3.33)
where we have applied the generalization (2.85). Comparison of the equations
(3.27) and (3.33) gives
$1=\pm\dfrac{\sqrt{\epsilon^{2}-4E^{2}+8\pi\dfrac{\ell_{P}}{\ell}M_{P}^{2}c^{4}}-\sqrt{\epsilon^{2}-4E^{2}-8\pi\dfrac{\ell_{P}}{\ell}M_{P}^{2}c^{4}}}{8mc^{2}},$
(3.34)
what is the equation for a minimal scale $\ell$ as a function of mass $m$ and
energy $E$ of a particle. The solution of this equation is easy to establish
$\alpha\ell=\dfrac{\sqrt{3}}{16\pi}\dfrac{\ell_{P}}{\sqrt{1+\left(\dfrac{E}{2mc^{2}}\right)^{2}}},$
(3.35)
what can be equivalently treated as the formula for the energy of a particle
$\dfrac{E^{2}}{(2mc^{2})^{2}}=\dfrac{3}{256\pi^{2}}\dfrac{\ell_{P}^{2}}{\alpha^{2}\ell^{2}}-1,$
(3.36)
and application of the generalization (2.85) leads to
$E^{2}=\dfrac{3}{16}\dfrac{\ell_{P}^{2}}{\ell^{2}}E_{P}^{2}-(2mc^{2})^{2}=\dfrac{3}{16}\epsilon^{2}-(2mc^{2})^{2},$
(3.37)
where $E_{P}=M_{P}c^{2}$ is the Planck energy. Because, however,
$E^{2}\geqslant 0$ one obtains the inequality
$m\ell\leqslant\dfrac{\sqrt{3}}{8}M_{P}\ell_{P}=\dfrac{\sqrt{3}}{8}\dfrac{\hslash}{c},$
(3.38)
which for fixed mass of a particle gives the upper bound for a minimal scale
$\ell\leqslant\dfrac{2\pi\sqrt{3}}{8}\lambda_{C}(m),$ (3.39)
or the lower bound for maximal energy
$\epsilon\geqslant\dfrac{8}{\sqrt{3}}E_{P},$ (3.40)
while for fixed minimal scale leads to the upper bound for mass of a particle.
Applying a minimal scale (3.35) within the definition (4.130) allows to
eliminate a minimal scale dependence
$\epsilon=\dfrac{16\pi}{\sqrt{3}}E_{P}\sqrt{\alpha}\sqrt{1+\left(\dfrac{E}{2mc^{2}}\right)^{2}},$
(3.41)
what after application of the generalization (2.85) takes the form
$\epsilon=\dfrac{4}{\sqrt{3}}\sqrt{E^{2}+(2mc^{2})^{2}},$ (3.42)
and in the light of (3.40) leads to the bound for energy of a particle
$E^{2}\geqslant 4\left(E_{P}^{2}-m^{2}c^{4}\right),$ (3.43)
which by $E^{2}\geqslant 0$ gives the bound for mass of a particle
$m\leqslant M_{P},$ (3.44)
proving the Markov hypothesis expressing supposition that the upper bound for
mass of a particle is given by the Planck mass. Recall that we have deduced
(2.86) that the only case $m\sim M_{P}$ coincides with the Kontsevich
deformation quantization. In the light of the inequality (3.44) such a
situation suggests that a particle described by the Snyder noncommutative
geometry is the Planckian particle, i.e. the particle equipped with the mass
identical to the Planck mass $m=M_{P}$. In other words possible existence of
the Planckian particle will be establishing the physical sense of both the
Snyder noncommutative geometry and the Kontsevich deformation quantization. It
suggests also that physics at the Planck scale is the physics of the Planckian
particle.
A maximal energy (3.4) does not vanish for all
$\mu_{L}^{\pm}\neq\mu_{R}^{\pm}\neq 0$, and is finite if and only if
$\mu_{L,R}^{\pm}<\infty$. The mass $m$ of an original quantum state as well as
the masses of neutrinos $\mu_{R}^{\pm}$ and $\mu_{L}^{\pm}$ are presumed to be
physical quantities, which can be established by experimental data. In the
case, when an original quantum state is massless, a maximal energy has the
maximal value which equals to
$\epsilon(m=0)=\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\equiv\epsilon_{0},$
(3.45)
that is finite and non vanishing for finite $\mu_{R}^{\pm}\neq 0$ and
$\mu_{L}^{\pm}\neq 0$. In this manner one can study approximation around such
defined state. For $|\mu_{R}^{\pm}+\mu_{L}^{\pm}|>8m$ the appropriate Taylor
series expansion is
$\epsilon=\epsilon_{0}\left[1-\dfrac{8m}{\mu_{R}^{\pm}+\mu_{L}^{\pm}}+O\left(\left(\dfrac{8m}{\mu_{R}^{\pm}+\mu_{L}^{\pm}}\right)^{2}\right)\right],$
(3.46)
while for $|\mu_{R}^{\pm}+\mu_{L}^{\pm}|<8m$ one has the following expansion
$\epsilon=\epsilon_{0}\left[\dfrac{\mu_{R}^{\pm}+\mu_{L}^{\pm}}{8m}+O\left(\left(\dfrac{\mu_{R}^{\pm}+\mu_{L}^{\pm}}{8m}\right)^{2}\right)\right].$
(3.47)
For the case $|\mu_{R}^{\pm}+\mu_{L}^{\pm}|=8m$ both these series coincide and
$\epsilon=\dfrac{\epsilon_{0}}{2}.$ (3.48)
The established inequality (3.15), however, allows to remove from
considerations the case $|\mu_{R}^{\pm}+\mu_{L}^{\pm}|>8m$ given by the Taylor
series expansion (3.46).
On the other hand, however, addition of the second equation to the first one
in the system of equations (3.1) gives the relation
$\left(\mu_{L}^{\pm}c^{2}-\dfrac{\epsilon}{2}\right)^{2}+\left(\mu_{R}^{\pm}c^{2}+\dfrac{\epsilon}{2}\right)^{2}=2\left(\epsilon^{2}-4E^{2}\right).$
(3.49)
The LHS of the equation (3.49) is always positive as a sum of two squares of
real numbers, and therefore the RHS of this equation is always positive also.
In this manner, one obtains the renormalization of energy of a particle vie a
maximal energy
$-\dfrac{\epsilon}{2}\leqslant E\leqslant\dfrac{\epsilon}{2}.$ (3.50)
Naturally, for the generic case of Special Relativity we have
$\epsilon\equiv\infty$ and by this reason values of energy $E$ of a particle
are not bounded. Therefore, by the relation (3.50) it is evident that the
Snyder noncommutative geometry results in renormalization of energy of a
particle.
The relation (3.49) can be treated as the constraint for the energy $E$ of a
particle, and immediately solved with respect to $E$. The solution is a
quadratic form which can be presented in the canonical form with respect to a
maximal energy $\epsilon$
$E^{2}=\dfrac{3}{16}\left\\{\left[\epsilon+\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{3}c^{2}\right]^{2}-\left[\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{3}c^{2}\right]^{2}\left[7+\dfrac{12\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}\right]\right\\}.$
(3.51)
By explicit application of a maximal energy (3.4) within the energetic
constraint (3.51) one can present the particle energy via the only masses of
the neutrinos related to this particle
$E^{2}=\dfrac{\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2}}{48}\left\\{\left(\dfrac{4+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}{1+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}\right)^{2}-\left[7+\dfrac{12\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}\right]\right\\},$
(3.52)
which for the case of originally massless state has the value
$E^{2}(m=0)=\dfrac{1}{16}\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2}\left[3-4\dfrac{\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}\right]\equiv
E^{2}_{0},$ (3.53)
and by this reason energy of a particle is
$E^{2}=E_{0}^{2}+\Delta E^{2},$ (3.54)
where $\Delta E^{2}$ is the correction generated due to nonzero mass of an
original quantum state
$\Delta
E^{2}=-\dfrac{5}{16}\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2}\dfrac{\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}{1+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}\dfrac{\dfrac{8}{5}+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}{1+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}},$
(3.55)
which can be expanded into the Taylor series around the massless state
$\Delta
E^{2}=\dfrac{1}{16}\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2}\sum_{n=1}^{\infty}(-1)^{n}(3n+5)\left(\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}\right)^{n}.$
(3.56)
Because of the relation (3.15) it is more convenient to see the Taylor series
expansion of $E^{2}$ around the point $\mu_{L}^{\pm}+\mu_{R}^{\pm}=8m$, which
we shall call the $8m$ point, when $|\mu_{L}^{\pm}+\mu_{R}^{\pm}|>8m$. In such
a situation the decomposition, which we shall call _the $8m$ expansion_, has
somewhat different form
$E^{2}=E_{8m}^{2}+\Delta E^{2}_{8m},$ (3.57)
where
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!E_{8m}^{2}$ $\displaystyle=$
$\displaystyle
E_{0}^{2}+\dfrac{13}{64}\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2},$
(3.58) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Delta E^{2}_{8m}$
$\displaystyle=$
$\displaystyle\dfrac{\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2}}{64}\sum_{n=1}^{\infty}\left(-\dfrac{1}{2}\right)^{n}(3n+7)\left(\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}-1\right)^{n}.$
(3.59)
In this way, in the $8m$ expansion the leading correction to square of energy
of a particle is
${\Delta
E^{2}_{8m}}^{(1)}=-\dfrac{5}{64}\left[\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}\right]^{2}\left(\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}-1\right).$
(3.60)
For defined $E_{8m}^{2}$ the equation (3.58) establishes the relation between
the masses of the neutrinos
$\mu_{L}^{\pm}=\left(\dfrac{33}{25}\pm\sqrt{\dfrac{464}{625}+\left(\dfrac{8}{5}\dfrac{E_{8m}}{\mu_{R}^{\pm}c^{2}}\right)^{2}}\right)\mu_{R}^{\pm},$
(3.61)
and because of the condition $\mu_{L}^{\pm}-\mu_{R}^{\pm}\geqslant 0$
$\mu_{L}^{\pm}-\mu_{R}^{\pm}=\left(\dfrac{8}{25}\pm\sqrt{\dfrac{464}{625}+\left(\dfrac{8}{5}\dfrac{E_{8m}}{\mu_{R}^{\pm}c^{2}}\right)^{2}}\right)\mu_{R}^{\pm}\geqslant
0.$ (3.62)
the case of minus sign leads to the inequality
$\left(\dfrac{E_{8m}}{\mu_{R}^{\pm}c^{2}}\right)^{2}+\dfrac{1}{4}\leqslant 0,$
(3.63)
which does not possess solutions for real values of $E_{8m}$ and
$\mu_{R}^{\pm}$. Therefore, the physical solution is
$\mu_{L}^{\pm}=\left(\dfrac{33}{25}+\dfrac{4}{25}\sqrt{29+4\left(\dfrac{E_{8m}}{\mu_{R}^{\pm}c^{2}}\right)^{2}}\right)\mu_{R}^{\pm},$
(3.64)
and has minimal value for $E_{8m}=0$ with the value
$\mu_{L}^{\pm}=\left(\dfrac{33}{25}+\dfrac{4}{25}\sqrt{29}\right)\mu_{R}^{\pm}\approx
2.1816\mu_{R}^{\pm}.$ (3.65)
In fact, for given value of energy of massless state $E_{0}$ the equation
(3.53) can be used for establishment of the relation between masses of the
neutrinos. In result one receives two possible solutions
$\mu_{L}^{\pm}=\left(\dfrac{5}{3}\pm\dfrac{4}{3}\sqrt{{1+\dfrac{1}{3}\left(\dfrac{E_{0}}{\mu_{L}^{\pm}c^{2}}\right)^{2}}}\right)\mu_{R}^{\pm},$
(3.66)
and the physical solution is established by the condition (3.5)
$\mu_{L}^{\pm}-\mu_{R}^{\pm}=\left(\dfrac{2}{3}\pm\dfrac{4}{3}\sqrt{{1+\dfrac{1}{3}\left(\dfrac{E_{0}}{\mu_{L}^{\pm}c^{2}}\right)^{2}}}\right)\mu_{R}^{\pm}\geqslant
0,$ (3.67)
which in the case of the minus sign states that
$\left(\dfrac{E_{0}}{\mu_{L}^{\pm}c^{2}}\right)^{2}+\dfrac{9}{4}\leqslant 0,$
(3.68)
what is not satisfied for real $E_{0}$ and $\mu_{L}^{\pm}$. This argument
allows to generate the physical solution
$\mu_{L}^{\pm}=\left(\dfrac{5}{3}+\dfrac{4}{3}\sqrt{{1+\dfrac{1}{3}\left(\dfrac{E_{0}}{\mu_{L}^{\pm}c^{2}}\right)^{2}}}\right)\mu_{R}^{\pm}.$
(3.69)
which is minimized by $E_{0}=0$ with the value
$\mu_{L}^{\pm}=3\mu_{R}^{\pm}.$ (3.70)
The relation (3.52) for squared energy $E^{2}$ of a particle, i.e. factually
the constraint, is useful for analysis of certain situations. Albeit, in
general if one knows approximative value of $E^{2}$ the value of energy $E$ of
a particle can not be established by taking a square root of $E^{2}$. Square
root taking is in itself an approximation, and if one wishes to study
approximations of energy $E$ then this energy should be determined separately
via the appropriate Taylor series expansion. The problem is that the obtained
series for $E^{2}$ and for $E$ shall be in general different, and taking a
square root of arbitrary fixed order of approximation of $E^{2}$ does not
coincide with the same order of approximation of $E$.
In this manner, one can perform the deductions for energy $E$ of a particle
which are analogous to the deductions for its square. Let us write out the
formula for energy explicitly
$E=\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{4}\sqrt{\dfrac{1}{3}\left(\dfrac{4+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}{1+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}\right)^{2}-\dfrac{7}{3}-\dfrac{4\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}}.$
(3.71)
It is easy to see that for the originally massless case the value of energy is
$E=E_{0}$ where
$E_{0}=\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{4}\sqrt{3-\dfrac{4\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}},$
(3.72)
and in this way
$\displaystyle E-E_{0}$ $\displaystyle=$
$\displaystyle\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{4}\Bigg{\\{}\sqrt{\dfrac{1}{3}\left(\dfrac{4+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}{1+\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}}\right)^{2}-\left[\dfrac{7}{3}+\dfrac{4\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}\right]}-$
(3.73)
$\displaystyle\sqrt{3-\dfrac{4\mu_{L}^{\pm}\mu_{R}^{\pm}}{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)^{2}}}\Bigg{\\}}.$
Now one can apply the $8m$ expansion to the relation (3.73). In this case,
i.e. $E$ not $E^{2}$, the expansion is more difficult to apply, because one
has to deal with square roots. The expansion has the form
$E=\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{4}\sum_{n=0}^{\infty}A_{n}\left(\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}-1\right)^{n}.$
(3.74)
The explicit formula for the coefficients $A_{n}$ is not easy to extract,
because for $n\geqslant 0$ they satisfy the recurrence equation
$\displaystyle(66+22n+8(3+n)a^{2})A_{n+3}+(51+23n+12(2+n)a^{2})A_{n+2}+$
$\displaystyle+(9+8n+6(1+n)a^{2})A_{n+1}+n(1+a^{2})A_{n}=0,$ (3.75)
with the initial conditions
$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{11}{4}+a^{2}},$ (3.76) $\displaystyle A_{1}$
$\displaystyle=$ $\displaystyle-\dfrac{5}{8\sqrt{\dfrac{11}{4}+a^{2}}},$
(3.77) $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\dfrac{59+26a^{2}}{64\left(\dfrac{11}{4}+a^{2}\right)^{3/2}},$
(3.78)
where we have introduced the parameter
$a=\dfrac{4E_{0}/c^{2}}{\mu_{L}^{\pm}-\mu_{R}^{\pm}}.$ (3.79)
Similar situation has a place for expansion around massless state. In this
case the expansion has the form
$E=\dfrac{\left(\mu_{L}^{\pm}-\mu_{R}^{\pm}\right)c^{2}}{4}\sum_{n=0}^{\infty}B_{n}\left(\dfrac{8m}{\mu_{L}^{\pm}+\mu_{R}^{\pm}}\right)^{n},$
(3.80)
where in general for $n\geqslant 0$ the expansion coefficients $B_{n}$ satisfy
the following recurrence equation
$\displaystyle(18+6n+(3+n)a^{2})B_{n+3}+(24+10n+3(2+n)a^{2})B_{n+2}+$
$\displaystyle+(6+5n+3(1+n)a^{2})B_{n+1}+n(1+a^{2})B_{n}=0,$ (3.81)
having the following initial conditions
$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle\sqrt{6+a^{2}},$ (3.82)
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-\dfrac{4}{\sqrt{6+a^{2}}},$ (3.83) $\displaystyle B_{2}$
$\displaystyle=$ $\displaystyle-\dfrac{50+11a^{2}}{2(6+a^{2})^{3/2}}.$ (3.84)
#### C The Integrability Problem for the Dirac equations
The received Dirac equations
$\left(i\hslash\gamma^{\mu}\partial_{\mu}-M_{\pm}\right)\psi=0,$ (3.85)
where the mass matrices $M_{\pm}$ is given by the formula (2.56), can be
rewritten in the form of the Schrödinger equation
$i\hslash\partial_{0}\psi^{\pm}=\hat{H}\psi^{\pm},$ (3.86)
where in the present case the Hamilton operator $\hat{H}$ has the form
$\hat{H}=-i\hslash
c\gamma^{0}\gamma^{i}\partial_{i}-\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\gamma^{0}+\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{2}c^{2}\gamma^{0}\gamma^{5},$
(3.87)
which can be splitted into the hermitian $\mathfrak{H}(\hat{H})$ and the
antihermitian $\mathfrak{A}(\hat{H})$ components
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\mathfrak{H}(\hat{H})+\mathfrak{A}(\hat{H}),$ (3.88)
$\displaystyle\mathfrak{H}(\hat{H})$ $\displaystyle=$ $\displaystyle-i\hslash
c\gamma^{0}\gamma^{i}\partial_{i}-\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\gamma^{0},$
(3.89) $\displaystyle\mathfrak{A}(\hat{H})$ $\displaystyle=$
$\displaystyle\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{2}c^{2}\gamma^{0}\gamma^{5},$
(3.90)
with (anti)hermiticity defined standardly
$\displaystyle\int d^{3}x\bar{\psi}^{\pm}\mathfrak{H}(\hat{H})\psi^{\pm}$
$\displaystyle=$ $\displaystyle\int
d^{3}x\overline{\mathfrak{H}(\hat{H})\psi^{\pm}}\psi^{\pm},$ (3.91)
$\displaystyle\int d^{3}x\bar{\psi}^{\pm}\mathfrak{A}(\hat{H})\psi^{\pm}$
$\displaystyle=$ $\displaystyle-\int
d^{3}x\overline{\mathfrak{A}(\hat{H})\psi^{\pm}}\psi^{\pm}.$ (3.92)
Let us consider the situation when the masses of the neutrinos has the same
value, which we shall call $\mu$
$\mu_{R}^{\pm}=\mu_{L}^{\pm}\equiv\mu.$ (3.93)
It is easy to see that then the antihermitian component (3.90) vanishes
identically. However, the hermitian component (3.89) is still nontrivial.
Consequently, the Hamilton operator (3.88) takes the form of the conventional
Dirac Hamiltonian
$\hat{H}_{D}=-\gamma^{0}\left(i\hslash c\gamma^{i}\partial_{i}+\mu
c^{2}\right).$ (3.94)
In such a situation, however, by the relation (3.52) energy of a particle
vanishes identically
$E=0.$ (3.95)
Taking into account the bounds (3.50) one obtains that also a maximal energy
trivializes identically
$\epsilon\equiv 0,$ (3.96)
and therefore a minimal scale is infinite $\ell=\infty$. In the light of the
formulas (2.57)-(2.58) for the masses of the neutrinos, one obtains
$\mu_{R}^{\pm}=\mu_{L}^{\pm}\equiv 0,$ (3.97)
i.e. by the definition (3.93)
$\mu\equiv 0.$ (3.98)
Therefore, the Dirac Hamiltonian (3.94) becomes massless
$\hat{H}_{D}=-i\hslash c\gamma^{0}\gamma^{i}\partial_{i}.$ (3.99)
and consequently the Dirac equations (3.85) becomes the Weyl equation
$i\hslash\gamma^{\mu}\partial_{\mu}\psi=0,$ (3.100)
describing massless particle - the Weyl neutrino. It means that equality
between the masses of the neutrinos (3.93) defines the massless particle
obeying the Weyl equation. However, in the light of the Super-Kamiokande
results neutrino is equipped with nonzero mass. In this manner, such an
embarrassing situation created by the Weyl equation (3.100), and laying in the
foundations of the Standard Model, is manifestly non physical. In other words,
in the light of the Snyder noncommutative geometry the theory of massive
neutrinos is consistent if and only if the difference between masses of the
left-handed and the right-handed neutrinos is nonzero and positive.
The full modified Hamiltonian (3.87) possesses non-hermitian nature evidently.
Therefore consequently the Schrödinger equation form time evolution (3.86) is
non unitary manifestly. Its formal integration, however, can be carried out by
the standard method of quantum mechanics
$\psi^{\pm}(x,t)=G(t,t_{0})\psi^{\pm}(x,t_{0}),$ (3.101)
involving the following time evolution operator
$G(t,t_{0})\equiv\exp\left\\{-\dfrac{i}{\hslash}\int_{t_{0}}^{t}d\tau\hat{H}(\tau)\right\\}.$
(3.102)
By this reason, the integrability problem for the wave equation (3.86) with
the Hamilton operator (3.87) can be formulated in terms of the appropriate
Zassenhaus exponents
$\displaystyle\exp\left\\{A+B\right\\}$ $\displaystyle=$
$\displaystyle\exp(A)\exp(B)\prod_{n=2}^{\infty}\exp{C_{n}},$ (3.103)
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}C,$ (3.104)
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}(2[C,B]+[C,A]),$ (3.105) $\displaystyle C_{4}$
$\displaystyle=$
$\displaystyle-\frac{1}{24}([[C,A],A]+3[[C,A],B]+3[[C,B],B]),$
$\displaystyle\ldots$
where $C=[A,B]$. In the light of the definition (3.102) one can establish the
following identification
$\displaystyle A$ $\displaystyle\equiv$ $\displaystyle
A(t)=-\frac{i}{\hslash}\int_{t_{0}}^{t}d\tau\mathfrak{H}(\hat{H})(\tau),$
(3.107) $\displaystyle B$ $\displaystyle\equiv$ $\displaystyle
B(t)=-\frac{i}{\hslash}\int_{t_{0}}^{t}d\tau\mathfrak{A}(\hat{H})(\tau),$
(3.108)
and therefore the commutator $C$ can be derived straightforwardly and rather
easy. The result is
$C=-\dfrac{1}{\hslash^{2}}\int_{t_{0}}^{t}d\tau^{\prime}\int_{t_{0}}^{t}d\tau^{\prime\prime}\mathfrak{C}\left(\tau^{\prime},\tau^{\prime\prime}\right),$
(3.109)
where we have introduced the quantity
$\mathfrak{C}\left(\tau^{\prime},\tau^{\prime\prime}\right)\equiv\left[\mathfrak{H}(\hat{H})(\tau^{\prime}),\mathfrak{A}(\hat{H})(\tau^{\prime\prime})\right],$
(3.110)
which can be computed with using of elementary algebra
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathfrak{C}$
$\displaystyle=$
$\displaystyle\left(i\hslash\dfrac{\mu_{R}^{\pm}-\mu_{L}^{\pm}}{2}c^{3}\partial_{i}\right)\gamma^{0}\gamma^{i}\gamma^{0}\gamma^{5}+\left(\dfrac{(\mu_{R}^{\pm})^{2}-(\mu_{L}^{\pm})^{2}}{4}c^{4}\right)\gamma^{0}\gamma^{0}\gamma^{5}-$
(3.111) $\displaystyle-$
$\displaystyle\left(i\hslash\dfrac{\mu_{R}^{\pm}-\mu_{L}^{\pm}}{2}c^{3}\partial_{i}\right)\gamma^{0}\gamma^{5}\gamma^{0}\gamma^{i}-\left(\dfrac{(\mu_{R}^{\pm})^{2}-(\mu_{L}^{\pm})^{2}}{4}c^{4}\right)\gamma^{0}\gamma^{5}\gamma^{0}=$
$\displaystyle=$ $\displaystyle
2\left(i\hslash\dfrac{\mu_{R}^{\pm}-\mu_{L}^{\pm}}{2}c^{3}\partial_{i}\right)\gamma^{0}\gamma^{i}\gamma^{0}\gamma^{5}+2\left(\dfrac{(\mu_{R}^{\pm})^{2}-(\mu_{L}^{\pm})^{2}}{4}c^{4}\right)\gamma^{0}\gamma^{0}\gamma^{5},$
where we have applied the relations
$\displaystyle\gamma^{0}\gamma^{5}\gamma^{0}\gamma^{i}$ $\displaystyle=$
$\displaystyle-\gamma^{0}\gamma^{i}\gamma^{0}\gamma^{5},$ (3.112)
$\displaystyle\gamma^{0}\gamma^{5}\gamma^{0}$ $\displaystyle=$
$\displaystyle-\gamma^{0}\gamma^{0}\gamma^{5},$ (3.113)
arising from the property of the $\gamma^{5}$ matrix
$\left\\{\gamma^{5},\gamma^{\mu}\right\\}=0$. Therefore, consequently one
obtains finally the result
$\mathfrak{C}(\tau^{\prime},\tau^{\prime\prime})=2\mathfrak{H}(\hat{H})(\tau^{\prime})\mathfrak{A}(\hat{H})(\tau^{\prime\prime}),$
(3.114)
that leads to the equivalent statement - for arbitrary two times
$\tau^{\prime}$ and $\tau^{\prime\prime}$ the Poisson brackets of the
hermitian $\mathfrak{H}(\hat{H})(\tau^{\prime})$ and the antihermitian
$\mathfrak{A}(\hat{H})(\tau^{\prime\prime})$ components of the full
Hamiltonian (3.88) is trivial
$\left\\{\mathfrak{H}(\hat{H})(\tau^{\prime}),\mathfrak{A}(\hat{H})(\tau^{\prime\prime})\right\\}=0.$
(3.115)
Naturally, by simple factorization one obtains also
$C=2AB\quad,\quad\\{A,B\\}=0,$ (3.116)
and consequently
$\displaystyle\left[C,A\right]$ $\displaystyle=$ $\displaystyle CA,$ (3.117)
$\displaystyle\left[C,B\right]$ $\displaystyle=$ $\displaystyle CB,$ (3.118)
$\displaystyle\left[\left[C,A\right],A\right]$ $\displaystyle=$ $\displaystyle
2\left[C,A\right]A,$ (3.119) $\displaystyle\left[\left[C,A\right],B\right]$
$\displaystyle=$ $\displaystyle 2\left[C,A\right]B,$ (3.120)
$\displaystyle\left[\left[C,B\right],A\right]$ $\displaystyle=$ $\displaystyle
2\left[C,B\right]A,$ (3.121)
and so on. In this manner the 4th order approximation of the formula (3.103)
in the present case has the form
$\displaystyle\exp\left\\{A+B\right\\}$ $\displaystyle\approx$
$\displaystyle\exp(A)\exp(B)\exp{C_{2}}\exp{C_{3}}\exp{C_{4}},$ (3.122)
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}C,$ (3.123)
$\displaystyle C_{3}$ $\displaystyle=$ $\displaystyle-\frac{1}{6}(CA+2CB),$
(3.124) $\displaystyle C_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\left(CA^{2}+3CB^{2}+\dfrac{3}{2}C^{2}\right).$
(3.125)
For the case of constant in time masses $\mu_{R}^{\pm}$ and $\mu_{L}^{\pm}$
one can determine the relations
$\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{i}{\hslash}(t-t_{0})\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\gamma^{0},$
(3.126) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{i(\mu_{L}^{\pm}-\mu_{R}^{\pm})c^{2}}{2\hslash}(t-t_{0})\gamma^{5}\gamma^{0},$
(3.127) $\displaystyle C$ $\displaystyle=$
$\displaystyle\dfrac{(\mu_{L}^{\pm}-\mu_{R}^{\pm})c^{2}}{\hslash^{2}}(t-t_{0})^{2}\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\gamma^{5},$
(3.128)
and consequently by elementary algebraic manipulations one establishes the
Zassenhaus exponents as
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\dfrac{(\mu_{L}^{\pm}-\mu_{R}^{\pm})c^{2}}{2\hslash^{2}}(t-t_{0})^{2}\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\gamma^{5},$
(3.129) $\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle-\dfrac{i}{6\hslash^{3}}(\mu_{L}^{\pm}-\mu_{R}^{\pm})c^{2}(t-t_{0})^{3}\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\times$
(3.130) $\displaystyle\times$ $\displaystyle\left[\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\gamma^{5}+(\mu_{L}^{\pm}-\mu_{R}^{\pm})c^{2}\right]\gamma^{0},$
$\displaystyle C_{4}$ $\displaystyle=$
$\displaystyle\dfrac{(\mu_{L}^{\pm}-\mu_{R}^{\pm})c^{2}}{12\hslash^{4}}(t-t_{0})^{4}\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\times$
(3.131) $\displaystyle\times$ $\displaystyle\Bigg{\\{}\left[\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)^{2}+3\left(\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{2}c^{2}\right)^{2}\right]\gamma^{5}+$
$\displaystyle+$ $\displaystyle
3\dfrac{\mu_{L}^{\pm}-\mu_{R}^{\pm}}{2}c^{2}\left(-i\hslash
c\gamma^{i}\partial_{i}+\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}c^{2}\right)\Bigg{\\}}.$
This approximation is sufficient to conclude the general properties of the
procedure and the conclusions following from these features. The explicit form
of the Zassenhaus exponents $C_{n}$ shows manifestly that the integrability
problem formulated in terms of the Zassenhaus formula is not well defined.
Namely, the problem is that when the Hamilton operator is a sum of non-
commuting antihermitian and hermitian components then also the Zassenhaus
exponents $C_{n}$ obtain analogous legacy, i.e. are sums of two non-commuting
operators. The fundamental stage, _i.e._ the exponentiation procedure, must be
applied once again, and therefore consequently in the next step of the
procedure one meets the same property, i.e. sums of two non-commuting
operators. In this manner the problem is cyclic and can not be solved in any
approximation, while computation of full integration formula (3.103) becomes
the tremendous computational problem and its convergence is unclear. Therefore
for the case the Schrödinger equation (3.86) with the Hamilton operator
(3.87), such a recurrence integrability procedure based on the Zassenhaus
exponents is not an algorithm what results in the conclusion that the quantum
system is non integrable. By this reason one must construct any different
integrability procedure having finite number of steps and being an algorithm.
For realization of such a construction let us formulate the integrability
problem in a certain different form.
#### D The Integrability Problem for the massive Weyl equations
For constructive solving the problem, let us consider another integrability
procedure. Instead of the Dirac equation leads us focus on the massive Weyl
equations (2.66)-(2.69), which define the model of massive neutrinos. These
two equations can be straightforwardly rewritten in the form of the effective
two-component time evolution described by the Schrödinger equation
$i\hslash\partial_{0}\left[\begin{array}[]{c}\psi^{\pm}_{R}(x,t)\\\
\psi^{\pm}_{L}(x,t)\end{array}\right]=\hat{H}\left(\partial_{i}\right)\left[\begin{array}[]{c}\psi^{\pm}_{R}(x,t)\\\
\psi^{\pm}_{L}(x,t)\end{array}\right],$ (3.132)
where the Hamilton operator $\hat{H}$
$\hat{H}=-\gamma^{0}\left(i\hslash
c\gamma^{i}\partial_{i}+\left[\begin{array}[]{cc}\mu_{R}^{\pm}c^{2}&0\\\
0&\mu_{L}^{\pm}c^{2}\end{array}\right]\right),$ (3.133)
is manifestly hermitian and therefore the Schrödinger time evolution (3.132)
is unitary. In this manner the integration procedure can be performed in the
usual way well known from quantum mechanics.
Integrability of (3.132) is well defined. The solutions are
$\left[\begin{array}[]{c}\psi^{\pm}_{R}(x,t)\\\
\psi^{\pm}_{L}(x,t)\end{array}\right]=U(t,t_{0})\left[\begin{array}[]{c}\psi^{\pm}_{R}(x,t_{0})\\\
\psi^{\pm}_{L}(x,t_{0})\end{array}\right],$ (3.134)
where $U(t,t_{0})$ is the unitary time-evolution operator, that for the
constant masses is explicitly given by
$U(t,t_{0})=\exp\left\\{-\dfrac{i}{\hslash}(t-t_{0})\hat{H}\right\\},$ (3.135)
and $\psi^{\pm}_{R,L}(x,t_{0})$ are the initial time $t_{0}$ eigenstates with
defined momenta
$i\hslash\sigma^{i}\partial_{i}\psi^{\pm}_{R,L}(x,t_{0})=p_{R,L}^{\pm\leavevmode\nobreak\
0}\psi^{\pm}_{R,L}(x,t_{0}),$ (3.136)
where the initial momenta ${p_{R}^{\pm}}^{0}$ and ${p_{L}^{\pm}}^{0}$ are
related to the right-handed $\psi^{\pm}_{R}(x,t_{0})$ and the left-handed
$\psi^{\pm}_{L}(x,t_{0})$ chiral fields, respectively. The eigenequation
(3.136), however, can be straightforwardly integrated and the result will be
determining the spatial part of the evolution. The result can be presented in
the symbolic form
$\psi^{\pm}_{R,L}(x,t_{0})=\exp\left\\{-\dfrac{i}{\hslash}p_{R,L}^{\pm\leavevmode\nobreak\
0}(x-x_{0})_{i}\sigma^{i}\right\\}\psi^{\pm}_{R,L}(x_{0},t_{0}),$ (3.137)
which after direct exponentiation leads to
$\displaystyle\psi^{\pm}_{R,L}(x,t_{0})=\left(\mathbf{1}_{2}\cos\eta-i\eta_{i}\sigma^{i}\dfrac{\sin\eta}{\eta}\right)\psi^{\pm}_{R,L}(x_{0},t_{0}),$
(3.138)
where $\eta=|\eta_{i}|$ and $\eta_{i}$ is the three dimensionless vector
$\eta_{i}=\dfrac{p_{R,L}^{\pm\leavevmode\nobreak\ 0}}{\hslash}(x-x_{0})_{i}.$
(3.139)
In the present situation the embarrassing problem which emerges in the
integration procedure for the Dirac equation, discussed in the previous
section, is absent. Now the Zassenhaus exponents are not troublesome because
of, by definition, the $\gamma^{5}$ matrix is included into the chiral Weyl
fields. Therefore the Hamilton operator (3.133) is manifestly hermitian, and
consequently the exponentiation (3.135) can be straightforwardly performed by
the standard method of quantum mechanics. At first glance, however, the mass
matrix presence in the Hamilton operator (3.133) causes that one can choose
between at least two nonequivalent representations of the Dirac $\gamma$
matrices. On the one hand, the straightforward analogy to the Weyl equation
suggests that the appropriate choice is the Weyl basis. Albeit, on the other
hand, the Hamilton operator (3.133) can be treated as the usual hermitian
Dirac Hamiltonian, and therefore consequently the Dirac basis would be the
right representation for the Dirac $\gamma$ matrices. Other choices can be
also applied, but they have no unambiguous justification because of the mass
matrix presence in the Weyl equation. For example the Majorana basis, which is
an adequate choice for the case of massless neutrino, is not an adequate
choice for the case of massive neutrino. In this manner, in fact, one should
not prefer the representation but rather consider both the chiral fields and
the time evolution operator (3.135) in both the Weyl and the Dirac
representations. Let us denote by superscript $r$ the chosen representation.
By this reason the adequate labeling is
$\displaystyle U(t,t_{0})$ $\displaystyle\rightarrow$ $\displaystyle
U^{r}(t,t_{0}),$ (3.140) $\displaystyle\psi^{\pm}_{R,L}(x,t_{0})$
$\displaystyle\rightarrow$ $\displaystyle(\psi^{\pm}_{R,L})^{r}(x,t_{0}),$
(3.141) $\displaystyle\psi^{\pm}_{R,L}(x_{0},t_{0})$
$\displaystyle\rightarrow$ $\displaystyle(\psi^{\pm}_{R,L})^{r}(x_{0},t_{0})$
(3.142)
where the superscript $r=D,W$ means that the quantities are taken in the Dirac
and the Weyl basis, respectively. Interestingly, the eigenequation (3.136) is
independent on the representation choice, and therefore the initial momenta
$p_{R,L}^{\pm\leavevmode\nobreak\ 0}$ of the chiral Weyl fields are
measurable. For full consistency, let us test both the representations.
##### D1 The Dirac basis
The Dirac basis of the gamma matrices is defined as
$\gamma^{0}=\left[\begin{array}[]{cc}I&0\\\
0&-I\end{array}\right]\quad,\quad\gamma^{i}=\left[\begin{array}[]{cc}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right]\quad,\quad\gamma^{5}=\left[\begin{array}[]{cc}0&I\\\
I&0\end{array}\right],$ (3.143)
where $I$ is the $2\times 2$ unit matrix, and
$\sigma^{i}=[\sigma_{x},\sigma_{y},\sigma_{z}]$ is a vector of the $2\times 2$
Pauli matrices
$\sigma_{x}=\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right]\quad,\quad\sigma_{y}=\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right]\quad,\quad\sigma_{z}=\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right].$ (3.144)
Application of the Dirac basis (3.143) allows to express the Hamilton operator
(3.133) as follows
$\hat{H}=\left[\begin{array}[]{cc}\mu_{R}^{\pm}&i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\\\
i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}&-\mu_{L}^{\pm}\end{array}\right]c^{2},$
(3.145)
and for the case of constant in time neutrinos masses yields a solution
(3.134) with the unitary time evolution operator $U$
$U^{D}=\exp\left\\{-i\dfrac{c^{2}}{\hslash}(t-t_{0})\left[\begin{array}[]{cc}\mu_{R}^{\pm}&i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\\\
i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}&-\mu_{L}^{\pm}\end{array}\right]\right\\}.$
(3.146)
Straightforward exponentiation in (3.146) leads to the result
$\displaystyle U^{D}$ $\displaystyle=$
$\displaystyle\Bigg{\\{}\left[\begin{array}[]{cc}I&0\\\
0&I\end{array}\right]\cos\left[\dfrac{t-t_{0}}{\hslash}c^{2}\sqrt{{\left(\dfrac{\mu_{R}^{\pm}+\mu_{L}^{\pm}}{2}\right)^{2}}+\left(i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\right)^{2}}\right]-$
(3.149) $\displaystyle-$ $\displaystyle
i\left[\begin{array}[]{cc}\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}&i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\\\
i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}&-\dfrac{\mu_{L}^{\pm}+\mu_{R}^{\pm}}{2}\end{array}\right]\times$
$\displaystyle\times$
$\displaystyle\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}c^{2}\sqrt{{\left(\dfrac{\mu_{R}^{\pm}+\mu_{L}^{\pm}}{2}\right)^{2}+\left(i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\right)^{2}}}\right]}{\sqrt{{\left(\dfrac{\mu_{R}^{\pm}+\mu_{L}^{\pm}}{2}\right)^{2}+\left(i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\right)^{2}}}}\Bigg{\\}}\times$
$\displaystyle\times$
$\displaystyle\exp\left\\{-i\dfrac{(\mu_{R}^{\pm}-\mu_{L}^{\pm})c^{2}}{2\hslash}(t-t_{0})\right\\},$
(3.153)
where we understand that all the functions are treated by the appropriate
Taylor series expansions.
##### D2 The Weyl basis
As we have mentioned, however, application of the Weyl representation of the
Dirac $\gamma$ matrices is also justified by theoretical reasons. Such a basis
is defined as follows
$\gamma^{0}=\left[\begin{array}[]{cc}0&I\\\
I&0\end{array}\right]\quad,\quad\gamma^{i}=\left[\begin{array}[]{cc}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right]\quad,\quad\gamma^{5}=\left[\begin{array}[]{cc}-I&0\\\
0&I\end{array}\right],$ (3.154)
and the Hamilton operator (3.133) in this representation has the form
$\hat{H}=\left[\begin{array}[]{cc}i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}&-\mu_{L}^{\pm}\\\
-\mu_{R}^{\pm}&-i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\end{array}\right]c^{2}.$
(3.155)
Consequently, for the case of constant in time neutrinos masses one
establishes the unitary time evolution operator
$U^{W}=\exp\left\\{-i\dfrac{c^{2}}{\hslash}(t-t_{0})\left[\begin{array}[]{cc}i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}&-\mu_{L}^{\pm}\\\
-\mu_{R}^{\pm}&-i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\end{array}\right]\right\\},$
(3.156)
which after straightforward exponentiation becomes
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!U^{W}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}I&0\\\
0&I\end{array}\right]\cos\left[\dfrac{t-t_{0}}{\hslash}c^{2}\sqrt{{\mu_{L}^{\pm}\mu_{R}^{\pm}+\left(i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\right)^{2}}}\right]-$
(3.159) $\displaystyle-$ $\displaystyle
i\left[\begin{array}[]{cc}i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}&-\mu_{L}^{\pm}\\\
-\mu_{R}^{\pm}&-i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\end{array}\right]\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}c^{2}\sqrt{{\mu_{L}^{\pm}\mu_{R}^{\pm}+\left(i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\right)^{2}}}\right]}{\sqrt{{\mu_{L}^{\pm}\mu_{R}^{\pm}+\left(i\dfrac{\hslash}{c}\sigma^{i}\partial_{i}\right)^{2}}}}.$
(3.162)
Evidently, the time evolution operator evaluated in the Weyl representation
(3.159) has meaningfully simpler form then the result of the evaluation
performed in the Dirac basis (3.149). In this manner results of the choices
distinguishable and therefore are not physically equivalent, _i.e._ will yield
different solutions of the same equation. Albeit, it is not the strangest
property. Namely, both the choices can be related to physics in different
energetic regions, and therefore it is useful to solve the massive Weyl
equation in both the representations.
##### D3 The space-time evolution
It must be emphasized that the results obtained in both the previous
subsections are strictly related to the massive Weyl equations presented in
the form of the Schrödinger equation (3.132). Presently, one can
straightforwardly apply these key results, _i.e._ the momentum eigenequations
(3.136), the spatial evolutions (3.137), and the evaluations of the unitary
time evolution operators (3.149) and (3.159), to exact determination of the
corresponding wave functions of the massive Weyl equation (3.132) in both the
Dirac and the Weyl bases of the Dirac gamma matrices.
###### Dirac-like solutions
Let us derive first the wave functions in the Dirac basis. Employing the
shortened notation
$E^{D}({p_{R}^{\pm}}^{0})\equiv
c^{2}\sqrt{{\left(\mu_{\pm}^{D}\right)^{2}+\left(\dfrac{{p_{R}^{\pm}}^{0}}{c}\right)^{2}}},$
(3.163)
where $\mu_{\pm}^{D}$ is the arithmetic mean of the masses of the neutrinos
${\mu_{\pm}^{D}}=\dfrac{\mu_{R}^{\pm}+\mu_{L}^{\pm}}{2},$ (3.164)
by elementary algebraic manipulations one receives the right-handed chiral
Weyl fields
$\displaystyle(\psi^{\pm}_{R})^{D}(x,t)=\Bigg{\\{}\Bigg{[}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{D}({p_{R}^{\pm}}^{0})\right]-$
(3.165) $\displaystyle-$ $\displaystyle
i\mu_{\pm}^{D}c^{2}\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}({p_{R}^{\pm}}^{0})\right]}{E^{D}({p_{R}^{\pm}}^{0})}\Bigg{]}\exp\left\\{-\dfrac{i}{\hslash}{p_{R}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{R})^{D}_{0}-$
$\displaystyle-$ $\displaystyle
i{p_{L}^{\pm}}^{0}c\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}({p_{L}^{\pm}}^{0})\right]}{E^{D}({p_{L}^{\pm}}^{0})}\exp\left\\{-\dfrac{i}{\hslash}{p_{L}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{L})^{D}_{0}\Bigg{\\}}\times$
$\displaystyle\times$
$\displaystyle\exp\left\\{-i\dfrac{(\mu_{R}^{\pm}-\mu_{L}^{\pm})c^{2}}{2\hslash}(t-t_{0})\right\\},$
where $(\psi^{\pm}_{R,L})^{D}_{0}=(\psi^{\pm}_{R,L})^{D}(x_{0},t_{0})$.
Similarly, the left-handed chiral Weyl fields also can be also established in
an exact way
$\displaystyle(\psi^{\pm}_{L})^{D}(x,t)=\Bigg{\\{}\Bigg{[}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{D}({p_{L}^{\pm}}^{0})\right]+$
(3.166) $\displaystyle+$ $\displaystyle
i\mu_{\pm}^{D}c^{2}\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}({p_{L}^{\pm}}^{0})\right]}{E^{D}({p_{L}^{\pm}}^{0})}\Bigg{]}\exp\left\\{-\dfrac{i}{\hslash}{p_{L}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{L})^{D}_{0}-$
$\displaystyle-$ $\displaystyle
i{p_{R}^{\pm}}^{0}c\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}({p_{R}^{\pm}}^{0})\right]}{E^{D}({p_{R}^{\pm}}^{0})}\exp\left\\{-\dfrac{i}{\hslash}{p_{R}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{R})^{D}_{0}\Bigg{\\}}\times$
$\displaystyle\times$
$\displaystyle\exp\left\\{-i\dfrac{(\mu_{R}^{\pm}-\mu_{L}^{\pm})c^{2}}{2\hslash}(t-t_{0})\right\\}.$
###### Weyl-like solutions
Similar line of reasoning can be carried out for derivation of the wave
functions in the Weyl basis. Employing the following shortened notation
$E^{W}({p_{R}^{\pm}}^{0})\equiv
c^{2}\sqrt{{\left(\mu_{\pm}^{W}\right)^{2}+\left(\dfrac{{p_{R}^{\pm}}^{0}}{c}\right)^{2}}},$
(3.167)
where $\mu_{\pm}^{W}$ is the geometric mean of the masses of the neutrinos
${\mu_{\pm}^{W}}=\sqrt{{\mu_{R}^{\pm}\mu_{L}^{\pm}}},$ (3.168)
and performing elementary calculation one can deduce the right-handed chiral
Weyl fields
$\displaystyle(\psi^{\pm}_{R})^{W}(x,t)=\Bigg{\\{}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{W}({p_{R}^{\pm}}^{0})\right]-$
(3.169) $\displaystyle-$ $\displaystyle
i{p^{\pm}_{R}}^{0}c\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}({p_{R}^{\pm}}^{0})\right]}{E^{W}({p_{R}^{\pm}}^{0})}\Bigg{\\}}\exp\left\\{-\dfrac{i}{\hslash}{p_{R}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{R})^{W}_{0}+$
$\displaystyle+$ $\displaystyle
i\mu_{L}^{\pm}c^{2}\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}({p_{L}^{\pm}}^{0})\right]}{E^{W}({p_{L}^{\pm}}^{0})}\exp\left\\{-\dfrac{i}{\hslash}{p_{L}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{L})^{W}_{0},$
where similarly as in the case of the Dirac-like solutions we have introduced
$(\psi^{\pm}_{R,L})^{W}_{0}=(\psi^{\pm}_{R,L})^{W}(x_{0},t_{0})$. For the
left-handed chiral Weyl fields one obtains the formula
$\displaystyle(\psi^{\pm}_{L})^{W}(x,t)=\Bigg{\\{}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{W}({p_{L}^{\pm}}^{0})\right]-$
(3.170) $\displaystyle+$ $\displaystyle
i{p^{\pm}_{L}}^{0}c\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}({p_{L}^{\pm}}^{0})\right]}{E^{W}({p_{L}^{\pm}}^{0})}\Bigg{\\}}\exp\left\\{-\dfrac{i}{\hslash}{p_{L}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{L})^{W}_{0}+$
$\displaystyle+$ $\displaystyle
i\mu_{R}^{\pm}c^{2}\dfrac{\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}({p_{R}^{\pm}}^{0})\right]}{E^{W}({p_{R}^{\pm}}^{0})}\exp\left\\{-\dfrac{i}{\hslash}{p_{R}^{\pm}}^{0}(x-x_{0})_{i}\sigma^{i}\right\\}(\psi^{\pm}_{R})^{W}_{0}.$
In this manner one sees that the difference between obtained wave functions is
crucial. Straightforward comparison of the Weyl-like solutions (3.169) and
(3.170) with the Dirac-like solutions (3.165) and (3.166) shows that in the
case of the Dirac basis there are different coefficients of cosinuses and
sinuses, and there is an additional time-exponent. Moreover, the functions
$M^{D}({p_{R}^{\pm}}^{0})$ and $M^{W}({p_{R}^{\pm}}^{0})$ having the basic
status for both the received solutions also have different form which
manifestly depends on the choice of the Dirac representation of the $\gamma$
matrices. As we have suggested earlier, the difference is not a problem,
because the solutions can be related to different regions of energy. Anyway,
however, the validation of both the representations, and also other ones,
should be verified by experimental data.
##### D4 Probability density. Normalization
If one knows explicit form of the chiral Weyl fields then, applying the Dirac
basis, one can derive the usual Dirac fields by the following procedure
$(\psi^{\pm})^{D}=\left[\begin{array}[]{cc}\dfrac{(\psi^{\pm}_{R})^{D}+(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}&\dfrac{(\psi^{\pm}_{R})^{D}-(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}\\\
\dfrac{(\psi^{\pm}_{R})^{D}-(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}&\dfrac{(\psi^{\pm}_{R})^{D}+(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}\end{array}\right],$
(3.171)
where we have used the shortened notation
$(\psi^{\pm})^{D}=(\psi^{\pm})^{D}(x,t)$, and
$(\psi^{\pm}_{R,L})^{D}=(\psi^{\pm}_{R,L})^{D}(x,t)$. Similarly, employing the
Weyl basis, the Dirac fields can be determined as follows
$(\psi^{\pm})^{W}=\left[\begin{array}[]{cc}(\psi^{\pm}_{L})^{W}\mathbf{1}_{2}&\mathbf{0}_{2}\\\
\mathbf{0}_{2}&(\psi^{\pm}_{R})^{W}\mathbf{1}_{2}\end{array}\right],$ (3.172)
where like in the case of the Dirac basis we have applied the shortened
notation $(\psi^{\pm})^{W}=(\psi^{\pm})^{W}(x,t)$, and
$(\psi^{\pm}_{R,L})^{W}=(\psi^{\pm}_{R,L})^{W}(x,t)$. It is evident now, that
in general these two cases are different from physical, mathematical, and
computational points of view. In this manner, if we consider the quantum
mechanical probability density and its normalization, we are forced to relate
the probability density revealing Lorentz invariance to the chosen
representation
$\Omega^{D,W}\equiv(\bar{\psi}^{\pm})^{D,W}(\psi^{\pm})^{D,W},$ (3.173) $\int
d^{3}x\Omega^{D,W}=\mathbf{1}_{4}.$ (3.174)
Applying the Dirac field in the Dirac basis (3.171) by elementary derivation
one can obtain
$\Omega^{D}=\left[\begin{array}[]{cc}\dfrac{(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}+(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}&\dfrac{(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}-(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}\\\
\dfrac{(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}-(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}&\dfrac{(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}+(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}}{2}\mathbf{1}_{2}\end{array}\right],$
(3.175)
and similarly the probability density (3.173) computed for the Dirac field in
the Weyl basis (3.172) has the form
$\Omega^{W}=\left[\begin{array}[]{cc}(\bar{\psi}^{\pm}_{R})^{W}(\psi^{\pm}_{R})^{W}\mathbf{1}_{2}&\mathbf{0}_{2}\\\
\mathbf{0}_{2}&(\bar{\psi}^{\pm}_{L})^{W}(\psi^{\pm}_{L})^{W}\mathbf{1}_{2}\end{array}\right].$
(3.176)
Employing the normalization condition (3.174) in the Dirac representation one
obtains the system of equations
$\dfrac{1}{2}\left(\int
d^{3}x(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}+\int
d^{3}x(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}\right)=1,$ (3.177)
$\dfrac{1}{2}\left(\int
d^{3}x(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}-\int
d^{3}x(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}\right)=0,$ (3.178)
which leads to
$\displaystyle\int d^{3}x(\bar{\psi}^{\pm}_{R})^{D}(\psi^{\pm}_{R})^{D}$
$\displaystyle=$ $\displaystyle 1,$ (3.179) $\displaystyle\int
d^{3}x(\bar{\psi}^{\pm}_{L})^{D}(\psi^{\pm}_{L})^{D}$ $\displaystyle=$
$\displaystyle 1.$ (3.180)
In the case of Weyl representation one receives
$\displaystyle\int d^{3}x(\bar{\psi}^{\pm}_{R})^{W}(\psi^{\pm}_{R})^{W}$
$\displaystyle=$ $\displaystyle 1,$ (3.181) $\displaystyle\int
d^{3}x(\bar{\psi}^{\pm}_{L})^{W}(\psi^{\pm}_{L})^{W}$ $\displaystyle=$
$\displaystyle 1.$ (3.182)
In this manner one sees straightforwardly that the normalization conditions
(3.179), (3.180) and (3.181), (3.182)) are the same
$\int
d^{3}x(\bar{\psi}^{\pm}_{R,L})^{D,W}(x,t)(\psi^{\pm}_{R,L})^{D,W}(x,t)=1,$
(3.183)
i.e. are invariant with respect to the choice of the gamma matrices
representations, what means that they are physical conditions. Using of the
fact that full space-time evolution is determined as
$\displaystyle(\psi^{\pm}_{R,L})^{D,W}(x,t)=U^{D,W}(t,t_{0})(\psi^{\pm}_{R,L})^{D,W}(x,t_{0}),$
(3.184)
$\displaystyle\left[U^{D,W}(t,t_{0})\right]^{\dagger}U^{D,W}(t,t_{0})=\mathbf{1}_{2},$
(3.185)
one finds easily that
$\int
d^{3}x(\bar{\psi}^{\pm}_{R,L})^{D,W}(x,t_{0})(\psi^{\pm}_{R,L})^{D,W}(x,t_{0})=1.$
(3.186)
By using of the spatial evolution (3.138) one obtains the relation
$\left|(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})\right|^{2}\int
d^{3}x\left(\mathbf{1}_{2}+\dfrac{(x-x_{0})_{i}}{|x-x_{0}|}\Im\sigma^{i}\sin\left|2\dfrac{p_{R,L}^{\pm\leavevmode\nobreak\
0}}{\hslash}(x-x_{0})_{i}\right|\right)=1,$ (3.187)
where $\Im{\sigma^{i}}=\dfrac{\sigma^{i}-\sigma^{i\dagger}}{2i}$ is a
imaginary part of the vector $\sigma^{i}$. The decomposition
$\sigma_{i}=[\sigma_{x},0,\sigma_{z}]+i[0,-i\sigma_{y},0]$ yields
$\Im\sigma^{i}=[0,-i\sigma_{y},0]$, and the equation (3.187) becomes
$\left|(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})\right|^{2}\int
d^{3}x\left(\mathbf{1}_{2}-i\dfrac{(x-x_{0})_{y}}{|x-x_{0}|}\sigma_{y}\sin\left|2\dfrac{p_{R,L}^{\pm\leavevmode\nobreak\
0}}{\hslash}(x-x_{0})_{i}\right|\right)=1.$ (3.188)
Introducing the change of variables $(x-x_{0})_{i}\rightarrow{x^{\prime}}_{i}$
in the following way
${x^{\prime}}_{i}\equiv 2\dfrac{p_{R,L}^{\pm\leavevmode\nobreak\
0}}{\hslash}(x-x_{0})_{i},$ (3.189)
and the effective volume $V^{\prime}$ due to the vector ${x^{\prime}}_{i}$
$V^{\prime}\mathbf{1}_{2}=\int
d^{3}x^{\prime}\left\\{\mathbf{1}_{2}-i\sigma_{y}{x^{\prime}}_{y}\dfrac{\sin|x^{\prime}|}{|x^{\prime}|}\right\\},$
(3.190)
the equation (3.188) can be rewritten in the form
$\left|(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})\right|^{2}\left(\dfrac{2p_{R,L}^{\pm\leavevmode\nobreak\
0}}{\hslash}\right)^{3}V^{\prime}\mathbf{1}_{2}=\mathbf{1}_{2},$ (3.191)
and therefore one obtains finally
$(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})=\left(\dfrac{\hslash}{2p_{R,L}^{\pm\leavevmode\nobreak\
0}}\right)^{3/2}\dfrac{1}{\sqrt{V^{\prime}}}\exp{i\theta_{\pm}},$ (3.192)
where $\theta_{\pm}$ are arbitrary constant phases. The volume (3.190) differs
from the standard one by the presence of the extra axial (y) volume $V_{y}$
$V_{y}=-i\sigma_{y}\int
d^{3}x^{\prime}{x^{\prime}}_{y}\dfrac{\sin|x^{\prime}|}{|x^{\prime}|},$
(3.193)
which is the axial effect and has nontrivial feature, namely
$V_{y}=\left\\{\begin{array}[]{cc}0&\mathrm{on}\leavevmode\nobreak\
\mathrm{f\/inite}\leavevmode\nobreak\ \mathrm{symmetrical}\leavevmode\nobreak\
\mathrm{spaces}\\\ \infty&\mathrm{on}\leavevmode\nobreak\
\mathrm{inf\/inite}\leavevmode\nobreak\
\mathrm{symmetrical}\leavevmode\nobreak\ \mathrm{spaces}\\\
<\infty&\mathrm{on}\leavevmode\nobreak\ \mathrm{sections}\leavevmode\nobreak\
\mathrm{of}\leavevmode\nobreak\ \mathrm{symmetrical}\leavevmode\nobreak\
\mathrm{spaces}\end{array}\right..$ (3.194)
Now one sees straightforwardly that the normalization is strictly speaking
dependent on the choice of an appropriate region of integrability. For
infinite symmetric spatial regions such a normalization procedure is not well
defined, because of the axial volume effect is infinite. However, one can
study certain reasonable situations which consider solutions of the quantum
theory on finite symmetric spatial regions. Moreover, the problem of
integrability is defined with respect to the choice of the initial momentum of
the chiral Weyl fields $p_{R,L}^{\pm\leavevmode\nobreak\ 0}$. In fact, there
are many possible nonequivalent physical situations connected with the choice
of a concrete initial momentum eigenvalue. In the next section we are going to
discuss the one of such situations related to a finite symmetric spatial
region, the concrete example of the model of massive neutrinos, which in
general was solved in the present section.
#### E The Ultra-Relativistic Massive Neutrinos
As the final piece of this chapter let us consider the concrete application of
the general model presented in the previous section, which is based on the
normalization in a finite symmetrical box and putting _ad hoc_ the eigenvalue
of the initial momenta of the chiral Weyl fields according to the ultra-
relativistic limit of Special Relativity
$p_{R,L}^{\pm\leavevmode\nobreak\ 0}=\mu_{R,L}^{\pm}c.$ (3.195)
For such a simplified situation the normalization discussed in the previous
section leads to the following initial data condition
$(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})=\sqrt{\left(\dfrac{\hslash}{2\mu_{R,L}^{\pm}c}\right)^{3}\dfrac{1}{V^{\prime}}}\exp{i\theta_{\pm}}=\sqrt{\pi^{3}\dfrac{\lambda^{3}_{C}(\mu_{R,L}^{\pm})}{V^{\prime}}}\exp{i\theta_{\pm}},$
(3.196)
where $\lambda_{C}(\mu_{R,L}^{\pm})$ is the Compton wavelength of the right-
or left-handed neutrino. Because of normalization in the symmetrical box gives
$V^{\prime}=V=\int d^{3}x,$ (3.197)
one obtains finally
$(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})=\sqrt{\pi^{3}\dfrac{\lambda^{3}_{C}(\mu_{R,L}^{\pm})}{V}}\exp{i\theta_{\pm}}.$
(3.198)
When the theory is normalized in the region of the volume
$V=\pi^{3}\lambda^{3}_{C}(\mu_{R,L}^{\pm}),$ (3.199)
then initial values of the chiral Weyl fields determine the phase
$(\psi^{\pm}_{R,L})^{D,W}(x_{0},t_{0})=\exp{i\theta_{\pm}}.$ (3.200)
Interestingly, when one considers the normalization symmetrical spaces for the
left- and right-handed neutrino as a spheres of radiuses $R_{R,L}^{\pm}$ then
the normalization radiuses are
$R_{R,L}^{\pm}=\dfrac{\sqrt{3}}{2}\pi^{2/3}\lambda_{C}(\mu_{R,L}^{\pm})\approx
1.858\lambda_{C}(\mu_{R,L}^{\pm}).$ (3.201)
In this manner, if one can measure the normalization radius $R_{R,L}^{\pm}$
then the masses of the neutrinos can be established as
$\mu_{R,L}^{\pm}=\dfrac{\sqrt{3}}{4\pi^{1/3}}\dfrac{\hslash}{c}\dfrac{1}{R_{R,L}^{\pm}}=\dfrac{\sqrt{3}}{4\pi^{1/3}}M_{P}\dfrac{\ell_{P}}{R_{R,L}^{\pm}},$
(3.202)
what can be approximated as
$\mu_{R,L}^{\pm}\approx 1.04\cdot 10^{-43}\dfrac{1kg\cdot
1m}{R_{R,L}^{\pm}}=0.583\dfrac{1\dfrac{eV}{c^{2}}\cdot 1nm}{R_{R,L}^{\pm}}.$
(3.203)
If one wishes to establish the squared-mass difference
$\Delta\mu_{LR}^{2}=\dfrac{3}{16\pi^{2/3}}\left(\dfrac{\hslash}{c}\right)^{2}\left[\dfrac{1}{\left(R_{L}^{\pm}\right)^{2}}-\dfrac{1}{\left(R_{R}^{\pm}\right)^{2}}\right],$
(3.204)
or with using of the Planck mass and the Planck length
$\Delta\mu_{LR}^{2}=\dfrac{3M_{P}^{2}}{16\pi^{2/3}}\left[\left(\dfrac{\ell_{P}}{R_{L}^{\pm}}\right)^{2}-\left(\dfrac{\ell_{P}}{R_{R}^{\pm}}\right)^{2}\right].$
(3.205)
Because of the squared-mass difference is positive $\Delta\mu^{2}_{LR}>0$ one
has
$R_{R}^{\pm}>R_{L}^{\pm}.$ (3.206)
Moreover, application of the relation (3.27) leads to the conclusion
$\dfrac{3}{512\pi^{5/3}}\left[\dfrac{1}{\left(R_{L}^{\pm}\right)^{2}/\ell_{P}}-\dfrac{1}{\left(R_{R}^{\pm}\right)^{2}/\ell_{P}}\right]=\dfrac{1}{\ell},$
(3.207)
which can be treated as the lensmaker’s equation
$(n-1)\left[\dfrac{1}{R_{1}}-\dfrac{1}{R_{2}}+\dfrac{n-1}{n}\dfrac{d}{R_{1}R_{2}}\right]=\dfrac{1}{f},$
(3.208)
for the convergent lens of thickness $d$ small compared with the radiuses of
curvature $R_{1}$ and $R_{2}$. The lens has the focal length identical to a
minimal scale
$f=\ell,$ (3.209)
the radiuses of curvature strictly related to _the normalization radiuses_
$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\dfrac{\left(R_{L}^{\pm}\right)^{2}}{\ell_{P}},$ (3.210)
$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\dfrac{\left(R_{R}^{\pm}\right)^{2}}{\ell_{P}},$ (3.211)
and refractive index
$n=1+\dfrac{3}{512\pi^{5/3}}\approx 1.00087.$ (3.212)
Let us call such a lens _the neutrino lens_. The equation (3.207) expresses a
minimal scale via the normalization radiuses, i.e. if and only if
$R_{R,L}^{\pm}$ are established by experimental data then $\ell$ is also
established.
Let us consider such a normalization, i.e. the particular case of the general
solutions which describes the situation of the neutrino lens. First let us
derive the appropriate wave functions in the Dirac representation. Introducing
the function
$E^{D}(x,y)\equiv c^{2}\sqrt{{\left(\dfrac{x+y}{2}\right)^{2}}+x^{2}},$
(3.213)
the right-handed chiral Weyl fields are
$\displaystyle(\psi^{\pm}_{R})^{D}(x,t)=\Bigg{\\{}\Bigg{[}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{D}(\mu_{R}^{\pm},\mu_{L}^{\pm})\right]-$
(3.214) $\displaystyle-$ $\displaystyle
i\dfrac{\mu_{\pm}^{D}c^{2}}{E^{D}(\mu_{R}^{\pm},\mu_{L}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}(\mu_{R}^{\pm},\mu_{L}^{\pm})\right]\Bigg{]}\exp\left\\{-\dfrac{ic}{\hslash}\mu_{R}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}-$
$\displaystyle-$ $\displaystyle
i\dfrac{\mu_{L}^{\pm}c^{2}}{E^{D}(\mu_{L}^{\pm},\mu_{R}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}(\mu_{L}^{\pm},\mu_{R}^{\pm})\right]\exp\left\\{-\dfrac{ic}{\hslash}\mu_{L}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}\Bigg{\\}}\times$
$\displaystyle\times$
$\displaystyle\exp\left\\{i\left[\theta_{\pm}-\dfrac{(\mu_{R}^{\pm}-\mu_{L}^{\pm})c^{2}}{2\hslash}(t-t_{0})]\right]\right\\}.$
while the left-handed chiral Weyl fields are
$\displaystyle(\psi^{\pm}_{L})^{D}(x,t)=\Bigg{\\{}\Bigg{[}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{D}(\mu_{L}^{\pm},\mu_{R}^{\pm})\right]+$
(3.215) $\displaystyle+$ $\displaystyle
i\dfrac{\mu_{\pm}^{D}c^{2}}{E^{D}(\mu_{L}^{\pm},\mu_{R}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}(\mu_{L}^{\pm},\mu_{R}^{\pm})\right]\Bigg{]}\exp\left\\{-\dfrac{ic}{\hslash}\mu_{L}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}-$
$\displaystyle-$ $\displaystyle
i\dfrac{\mu_{R}^{\pm}c^{2}}{E^{D}(\mu_{R}^{\pm},\mu_{L}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{D}(\mu_{R}^{\pm},\mu_{L}^{\pm})\right]\exp\left\\{-\dfrac{ic}{\hslash}\mu_{R}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}\Bigg{\\}}\times$
$\displaystyle\times$
$\displaystyle\exp\left\\{i\left[\theta_{\pm}-\dfrac{(\mu_{R}^{\pm}-\mu_{L}^{\pm})c^{2}}{2\hslash}(t-t_{0})]\right]\right\\}.$
Similarly, one can derive the appropriate wave functions in the Weyl
representation. Introducing the function
$E^{W}(x,y)\equiv c^{2}\sqrt{xy+x^{2}},$ (3.216)
for the right-handed chiral Weyl fields are
$\displaystyle(\psi^{\pm}_{R})^{W}(x,t)=\exp{i\theta_{\pm}}\Bigg{\\{}\Bigg{[}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{W}(\mu_{R}^{\pm},\mu_{L}^{\pm})\right]-\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (3.217) $\displaystyle-$
$\displaystyle\dfrac{i\mu_{R}^{\pm}c^{2}}{E^{W}(\mu_{R}^{\pm},\mu_{L}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}(\mu_{R}^{\pm},\mu_{L}^{\pm})\right]\Bigg{]}\exp\left\\{-\dfrac{ic}{\hslash}\mu_{R}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}+\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle+$
$\displaystyle\dfrac{i\mu_{L}^{\pm}c^{2}}{E^{W}(\mu_{L}^{\pm},\mu_{R}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}(\mu_{L}^{\pm},\mu_{R}^{\pm})\right]\exp\left\\{-\dfrac{ic}{\hslash}\mu_{L}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}\Bigg{\\}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
while the left-handed chiral Weyl fields are
$\displaystyle(\psi^{\pm}_{L})^{W}(x,t)=\exp{i\theta_{\pm}}\Bigg{\\{}\Bigg{[}\cos\left[\dfrac{t-t_{0}}{\hslash}E^{W}(\mu_{L}^{\pm},\mu_{R}^{\pm})\right]-\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (3.218) $\displaystyle-$
$\displaystyle\dfrac{i\mu_{L}^{\pm}c^{2}}{E^{W}(\mu_{L}^{\pm},\mu_{R}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}(\mu_{L}^{\pm},\mu_{R}^{\pm})\right]\Bigg{]}\exp\left\\{-\dfrac{ic}{\hslash}\mu_{L}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}+\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle+$
$\displaystyle\dfrac{i\mu_{R}^{\pm}c^{2}}{E^{W}(\mu_{R}^{\pm},\mu_{L}^{\pm})}\sin\left[\dfrac{t-t_{0}}{\hslash}E^{W}(\mu_{R}^{\pm},\mu_{L}^{\pm})\right]\exp\left\\{-\dfrac{ic}{\hslash}\mu_{R}^{\pm}(x-x_{0})_{i}\sigma^{i}\right\\}\Bigg{\\}}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
The situation considered above is only the example following from the model of
massive neutrinos based on the massive Weyl equations (2.66)-(2.69) obtained
via application of the Snyder noncommutative geometry (2.1)-(2.2). Because of
such a physical situation is related to the ultra-relativistic limit,
therefore this type of massive neutrinos we shall call _ultra-relativistic
massive neutrinos_. There are many other possibilities for determination of
the relation between the initial values of eigenmomentum
$p_{R,L}^{\pm\leavevmode\nobreak\ 0}$ and the masses $\mu_{R,L}^{\pm}$ of the
right- and left-handed chiral Weyl fields $\psi_{R,L}^{\pm}$. However, the
choice (8.14) tested above presents the crucial reasonability which is the
straightforward consequence of its special-relativistic character. Such a
situation expresses validation of special equivalence principle for the
initial space-time evolution of the massive neutrinos, _i.e._
$E_{R,L}^{\pm}=\mu^{\pm}_{R,L}c^{2}=p_{R,L}^{\pm\leavevmode\nobreak\ 0}c$.
This case, however, is also nontrivial from the high energy physics point of
view [141]. Namely, it is strictly related to the ultra-high energy region,
widely studied in the modern astrophysics (See _e.g._ the Refs. [135] and
[142] and suitable references therein). The presented particular space-time
evolution describes physics of the massive neutrinos in such an energetic
region, and therefore its results should be verified by experimental data due
to ultra-high energy astrophysics.
## Part II Quantum General Relativity
### Chapter 4 The Quantum Cosmology
#### A Introduction
Quantum cosmology is one of the most important research tendencies within
modern theoretical gravitational physics. Its necessity is defined by problems
of formulation of the physics of early and very early stages of our Universe.
Its essence, however, is the problem of construction of the model of the
Universe, in which quantum theory and statistical mechanics meet and work
together. Such a strategy is well known as the crucial aspect of string
theory, however, and by this reason an arbitrary model of quantum cosmology
which is the linkage between the quantum physics and the statistical physics
will be possessing characteristic features and properties of string theory.
Moreover, it is possible even that any such a model will be defining new kinds
of string theories. Albeit, because of very abstractive mathematical form of
string theory, the problem is how to build the model of quantum cosmology
which could be manifestly presenting the value for phenomenology. In other
words, the crucial problem in formulation of quantum cosmology is an
experimental verification of any its model.
One of the most essential steps in the history of quantum cosmology was
quantum geometrodynamics (QGD), called also quantum General Relativity or
quantum gravity, which had the beginning in the works due to J.A. Wheeler and
B.S. DeWitt. Standardly, in such a strategy the primary canonical quantization
procedure is applied to arbitrary canonical formulation of classical theory of
gravitation, General Relativity (GR). The most spread canonical formulation of
General Relativity is the Hamiltonian approach which started by early works of
P.A.M. Dirac and obtained a final and commonly accepted appearance in the
works due to R. Arnowitt, S. Deser, and C.W. Misner. It is called the
Arnowitt–Deser–Misner decomposition or $3+1$ splitting.
In this part of this book we shall discuss certain both particular
consequences as well as development of the strategy of quantum gravity based
on the Wheeler–DeWitt equation . The main purpose of this chapter is, however,
the construction of such a model of quantum Universe, which will be leading to
plausible theoretical predictions possessing both clearly defined physical
sense as well as the possibility of verification by comparison of its results
with experimental data due to observations of the physical Universe. Applying
the well-known epistemological justifications of quantum theory, we propose to
describe quantum cosmology as quantum field theory formulated in terms of the
Fock space of creation and annihilation operators. The Fock space formalism
has exceptionally well established phenomenological meaning for quantum
physics. In this manner, the basic problem is the constructive application of
the method of secondary quantization, which would be resulting in
phenomenological deductions for quantum cosmology. The natural consequence of
the formalism of the Fock repère is the possibility for straightforward
construction of statistical mechanics of quantum states of Universe. Moreover,
the Fock space formalism applied in the context of quantum cosmology leads to
natural definition of Multiverse as the collection of quantum universes.
By this reason we shall employ the quantum field theoretical reasoning for
quantum cosmology. As the classical model of the Universe we shall consider
the conformal-flat metric first derived by A.A. Friedmann, and applied in the
cosmological context by G. Lemaître, H.P. Robertson, and A.G. Walker, which
recently has been obtained the fundamental status for a number of models of
observed Universe. We shall show that straightforward application of the
Hamiltonian approach to General Relativity for such a concrete solution of the
Einstein field equations results in the Hamiltonian constraint corresponding
to the fundamental state of a bosonic string.
This chapter is not too extensive. First, in the section B we shall sketch
very shortly the classical model of Universe described by the conformal-flat
Friedmann–Lemaître–Robertson–Walker metric for the case of spatially finite
volume, which in this book is called the Einstein–Friedmann Universe,
possessing the cylindrical or toroidal topology corresponding to open and
closed strings, respectively. We present the Hamiltonian approach which
connects both the Dirac approach and the Arnowitt–Deser–Misner Hamiltonian
formulation of General Relativity, in which there is splitting of evolution of
four-dimensional space-time into the dynamics of three-dimensional space in
one-dimensional time. There is received the Hamiltonian constraint, and the
Hubble law is obtained due to its resolution. Moreover, the dynamics of the
model is identified with the dynamics of boson having negative squared-mass,
i.e. with the tachyon. In the section C we perform the primary canonical
quantization procedure with respect to the classical model. By the primary
quantization of the Hamiltonian constraint we receive the appropriate
Wheeler–DeWitt equation. Applying the separation of variables based on the
Hamilton equations of motion we make reduction of order of such an evolution,
and therefore the appropriate one-dimensional Dirac equation for the Universe
is deduced. Employing the method of secondary quantization to such a reduced
evolution results in the quantum field theory formulated in the Fock space.
Application of the appropriate Bogoliubov transformation and the Heisenberg
equations of motion leads to the diagonalization of the equations of motion to
its canonical form. In this manner, we receive the static Fock repère and
quantization of the cosmological model is formulated in terms of the monodromy
in the Fock space. In the section D we shall construct the statistical
mechanics, especially thermodynamics, of quantum states of the Universe.
Computations are performed according to the Bose–Einstein statistics in frames
of the method of density matrix. We apply the approximation in which the
quantum Universe is a system with one degenerated state, i.e. one-particle
approximation. The section E is devoted to discussion of the particular
thermodynamical situation of the system of many quantum Universe, i.e. the
early light Multiverse corresponding to a cosmological radiation, which is
characterized by minimal entropy. In the final section F we summarize briefly
the results of this chapter and present the perspective for further
development of the proposed approach to quantum cosmology.
#### B The Classical Universe
General Relativity based on the Einstein field equations [143]
$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R=\kappa{T}_{\mu\nu},$ (4.1)
where $\kappa=\dfrac{8\pi G}{c^{4}}$ is constant, $R_{\mu\nu}$ is the Ricci
curvature tensor, $R$ is the Ricci scalar curvature, and $T_{\mu\nu}$ is the
stress-energy tensor, is commonly accepted as the classical theory of
gravitation describing the evolution of a metric tensor $g_{\mu\nu}$ of a
four-dimensional Riemannian space-time manifold $M$ [144, 145]. The field
equations (4.1) can be generated as the Euler–Lagrange equations of motion
obtained via the Hilbert–Palatini action principle [146, 147] $\delta
S_{EH}=0$ with respect to the fundamental field $g_{\mu\nu}$ applied to the
Einstein–Hilbert action
$\textit{S}_{\textrm{EH}}=\dfrac{1}{c}\int_{M}d^{4}x\sqrt{-g}\left(-\dfrac{1}{2\kappa\ell_{P}^{2}}R+\mathcal{L}_{M}\right),$
(4.2)
where $\ell_{P}$ is the Planck length, and the constant multiplier
$\dfrac{1}{\ell_{P}^{2}}$ in the geometric action arises from dimensional
correctness of the action, which should be $[E]\cdot[T]$. In the action (4.2)
we denoted $g=\det g_{\mu\nu}$, and $\mathcal{L}_{M}$ denotes the Lagrangian
density of Matter fields. The variational principle allows to express the
stress-energy tensor of Matter fields via the Lagrangian density of Matter
fields
$\displaystyle
T_{\mu\nu}=\dfrac{2}{\sqrt{-g}}\dfrac{\delta\left(\sqrt{-g}\mathcal{L}_{M}\right)}{\delta
g^{\mu\nu}}.$ (4.3)
Let us consider the exact solution of the Einstein field equations (4.1) first
derived by Friedmann, and studied in extensive cosmological context by
Lemaître, Robertson, and Walker [148], for which the space-time interval has
the following form
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-(dx^{0})^{2}+a^{2}(x^{0})\delta_{ij}dx^{i}dx^{j},$
(4.4)
where $a(x^{0})$ is the cosmic scale factor parameter, and $x^{\mu}$
$(\mu=0,1,2,3)$ is a Cartesian system of space-time coordinates in which the
time coordinate $x^{0}$ is the object of the diffeomorphisms [149, 150]
$x^{0}\rightarrow{x^{\prime}}^{0}=x^{\prime}(x^{0}).$ (4.5)
When volume of space is finite
$V=\int dx^{1}dx^{2}dx^{3}<\infty,$ (4.6)
then the interval (4.4) describes the Einstein–Friedmann Universe possessing a
topology related to finite space. Such a cosmology is very far from
triviality, because of the topological structure of finite space relates the
model to considerations of string theory. Finite space is associated with
cylindrical, toroidal, but also with more general stringy topologies and
orbifolds [151]. In other words, the Einstein–Friedmann Universe having finite
space looks like to be a topological string. However, such complicated
theoretical situations shall not be developed in this book. Also cylindrical
topology is far from our present considerations, because of presence of the
boundaries in general can be resulting in very nontrivial physical
consequences. The finite space possessing toroidal topology has no boundary
because of the ends of the cylinder are identified each other. By this reason,
for simplicity of the cosmological model, in this book we shall study the case
of the Einstein–Friedmann Universe equipped with the toroidal topology. Such a
choice is in itself non trivial, because of the toroidal topology can be
naturally identified with closed strings. In this manner, we shall consider
the cosmological model possessing more general significance, which also can be
extended on other topologies mentioned above.
Let us consider the Friedmann–Lemaître–Robertson–Walker metric (4.4) in frames
of the Hamiltonian approach jointing the Dirac approach [152] and the
Arnowitt–Deser–Misner Hamiltonian formulation of General Relativity [153]. In
this chapter we shall not present the detailed analysis and explanations which
can be found in more specialized student textbooks (See e.g. the Ref. [154]),
and shall be discussed in further part of this book. The Arnowitt–Deser–Misner
decomposition of any metric satisfying the Einstein field equation, in a given
coordinate system, has the form
$g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N^{i}\\\
N^{j}&h_{ij}\end{array}\right],$ (4.7)
where $N$ is the lapse function, $N_{i}$ is the shift vector, $h_{ij}$ is the
metric of 3-dimensional space embedded in the 4-dimensional space-time, and
$N^{j}=h^{ij}N_{i}$. The classical Universe described by the
Friedmann–Lemaître–Robertson–Walker metric (4.4) can be parametrized by
$\displaystyle N^{2}$ $\displaystyle=$ $\displaystyle 1,$ (4.8) $\displaystyle
N_{i}$ $\displaystyle=$ $\displaystyle[0,0,0],$ (4.9) $\displaystyle h_{ij}$
$\displaystyle=$ $\displaystyle a^{2}(x^{0})\delta_{ij}.$ (4.10)
According to the strategy propagated by Dirac, the lapse function should be
preserved explicitly but the shift vector, because of its trivialization,
becomes absent. Moreover, at the end of calculations the lapse function should
be putted explicitly. In this manner, the Hamiltonian approach involving the
Dirac and the ADM formulations of General Relativity allows to present the
Friedmann–Lemaître–Robertson–Walker metric in the following form
$g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}&0\\\
0&a^{2}(x^{0})\delta_{ij}\end{array}\right],$ (4.11)
what is associated with the space-time interval
$ds^{2}=-N^{2}(dx^{0})^{2}+a^{2}(x^{0})dx^{i}dx^{j}.$ (4.12)
The diffeomorphism
$ct=c\tau+x^{0},$ (4.13)
where $\tau$ is a reference constant, allows to transform the cosmological
time $t$ to the conformal time $\eta$ on the level of the integral measure
$d\eta=N(x^{0})dx^{0}\equiv\dfrac{dt}{a(t)},$ (4.14)
what allows to rewrite the interval (4.12) in the following form
$ds^{2}=a^{2}(\eta)\left[-c^{2}d\eta^{2}+\delta_{ij}dx^{i}dx^{j}\right].$
(4.15)
It means that, in the coordinate system $(c\eta,x^{i})$, the Universe is
described by the scaled Minkowski space-time
$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\Omega(\eta)\eta_{\mu\nu},$ (4.16)
where the scaling function $\Omega(\eta)$ is
$\Omega(\eta)=a^{2}(\eta).$ (4.17)
Derivation of the Christoffel symbols for the metric (4.11),
$\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\sigma}\left(\partial_{\nu}g_{\mu\sigma}+\partial_{\mu}g_{\sigma\nu}-\partial_{\sigma}g_{\mu\nu}\right),$
(4.18)
leads to the nontrivial components
$\displaystyle\Gamma^{0}_{00}$ $\displaystyle=$
$\displaystyle\frac{\dot{N}}{N},$ (4.19) $\displaystyle\Gamma^{0}_{ii}$
$\displaystyle=$ $\displaystyle\frac{a\dot{a}}{N^{2}},$ (4.20)
$\displaystyle\Gamma^{i}_{i0}$ $\displaystyle=$
$\displaystyle\frac{\dot{a}}{a},$ (4.21)
where dot means $x^{0}$-differentiation, and consequently the Ricci curvature
tensor
$\displaystyle
R_{\mu\nu}=\partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}-\partial_{\nu}\Gamma^{\alpha}_{\mu\alpha}+\Gamma^{\alpha}_{\beta\alpha}\Gamma^{\beta}_{\mu\nu}-\Gamma^{\alpha}_{\beta\nu}\Gamma^{\beta}_{\mu\alpha},$
(4.22)
takes the following form
$R_{\mu\nu}\\!=\\!\left[\begin{array}[]{cccc}-3\left(\dfrac{\ddot{a}}{a}-\dfrac{\dot{a}}{a}\dfrac{\dot{N}}{N}\right)\\!\\!\\!&\\!\\!\\!\mathbf{0}^{\mathrm{T}}\\\
\mathbf{0}\\!\\!\\!&\\!\\!\\!\dfrac{a^{2}}{N^{2}}\left(\dfrac{\ddot{a}}{a}-2\left(\dfrac{\dot{a}}{a}\right)^{2}-\dfrac{\dot{a}}{a}\dfrac{\dot{N}}{N}\right)\delta_{ij}\end{array}\right]\\!\\!.$
(4.23)
By straightforward calculation of the contravariant metric components and
taking into account the Ricci curvature tensor (4.47), one receives the Ricci
scalar curvature
$R=g^{\mu\nu}R_{\mu\nu}=\dfrac{6}{N^{2}}\left[\dfrac{\ddot{a}}{a}-\left(\dfrac{\dot{a}}{a}\right)^{2}-\dfrac{\dot{a}}{a}\dfrac{\dot{N}}{N}\right],$
(4.24)
what allows to establish evaluation of the Einstein–Hilbert action (4.2) for
the classical Universe. Let us present the procedure in some detail. First,
let us consider the geometric part of the Einstein–Hilbert action, i.e.
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{M}d^{4}x\sqrt{-g}R$
$\displaystyle=$
$\displaystyle\int_{M}d^{3}x\int{dx^{0}}Na^{3}\dfrac{6}{N^{2}}\left[\dfrac{\ddot{a}}{a}-\left(\dfrac{\dot{a}}{a}\right)^{2}-\dfrac{\dot{a}}{a}\dfrac{\dot{N}}{N}\right]=$
(4.25) $\displaystyle=$ $\displaystyle
6V\int{dx^{0}}\dfrac{\ddot{a}a^{2}N-\dot{a}^{2}aN-\dot{a}a^{2}\dot{N}}{N^{2}},$
(4.26)
where $V=\int{d^{3}x}$ is volume of space. Applying the total derivative
$\dfrac{d}{dx^{0}}\left(\dfrac{\dot{a}a^{2}}{N}\right)=\dfrac{\ddot{a}a^{2}N+\dot{a}^{2}aN+\dot{a}a^{2}\dot{N}}{N^{2}},$
(4.27)
one obtains
$\displaystyle\int_{M}d^{4}x\sqrt{-g}R$ $\displaystyle=$ $\displaystyle
6V\int{dx^{0}}\left[\dfrac{d}{dx^{0}}\left(\dfrac{\dot{a}a^{2}}{N}\right)-\dfrac{3\dot{a}^{2}a}{N}\right]=$
(4.28) $\displaystyle=$
$\displaystyle-18V\int{dx^{0}}\dfrac{\dot{a}^{2}a}{N},$ (4.29)
where the boundary term was omitted as vanishing. Because of the relation
holds
$\dot{a}=\dfrac{1}{c}\dfrac{a^{\prime}}{a},$ (4.30)
where prime means $\eta$-differentiation, one has
$\dot{a}^{2}a=\dfrac{1}{c^{2}}\dfrac{a^{\prime 2}}{a},$ (4.31)
and because of $dx^{0}=cdt$ one obtains finally
$\int_{M}d^{4}x\sqrt{-g}R=-\dfrac{18V}{c}\int{dt}\dfrac{1}{N}\dfrac{a^{\prime
2}}{a}=-\dfrac{18V}{c}\int{d\eta}\dfrac{a^{\prime 2}}{N}.$ (4.32)
Let us consider now the action of Matter fields
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{M}d^{4}x\sqrt{-g}\mathcal{L}_{M}$
$\displaystyle=$
$\displaystyle\int{d^{3}x}\int{dx^{0}}Na^{3}\mathcal{L}_{M}=c\int{d^{3}x}\int{dt}Na^{3}\mathcal{L}_{M}=$
(4.33) $\displaystyle=$ $\displaystyle
c\int{d^{3}x}\int{d\eta}Na^{4}\mathcal{L}_{M}=c\int{d\eta}Na^{4}\int{d^{3}x}\mathcal{L}_{M}.$
(4.34)
Let us take into account the Legendre transformation
$\mathcal{L}_{M}=p_{F}\dot{F}-\mathcal{H}_{M},$ (4.35)
where $F$ is a Matter field (for collection of Matter fields one has a sum
over fields), $p_{F}=\dfrac{\partial\mathcal{L}}{\partial\dot{F}}$ is the
conjugate momentum to $F$, and $\mathcal{H}_{M}$ is Hamiltonian density of
Matter fields. Taking into account the definition of the conjugate momentum
one can write
$p_{F}\dot{F}=\dot{F}\dfrac{\partial\mathcal{L}_{M}}{\partial\dot{F}}=\dfrac{\partial}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)-\mathcal{L}_{M},$
(4.36)
and therefore one receives the relation
$\mathcal{L}_{M}=p_{F}\dot{F}-\mathcal{H}_{M}=\dfrac{\partial}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)-\mathcal{L}_{M}-\mathcal{H}_{M},$
(4.37)
which allows to establish the Lagrangian density of Matter fields in the form
$\mathcal{L}_{M}=\dfrac{1}{2}\dfrac{\partial}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)-\dfrac{1}{2}\mathcal{H}_{M}.$
(4.38)
In this manner one can determine the volume integral of the Lagrangian density
of Matter fields
$\int{d^{3}x}\mathcal{L}_{M}=\dfrac{1}{2}\int{d^{3}x}\dfrac{\partial}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)-\dfrac{1}{2}\int{d^{3}x}\mathcal{H}_{M}.$
(4.39)
Let us consider the first term of this formula. It is evidently seen that the
integrand is a total derivative, and applying the chain rule of
differentiation one can write the integrand as
$\dfrac{\partial}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)=\dfrac{\partial{V}}{\partial\dot{F}}\dfrac{\partial}{\partial{V}}\left(\dot{F}\mathcal{L}_{M}\right).$
(4.40)
However, in the light of the fact $dV=d^{3}x$ the derivative identically
vanishes $\dfrac{\partial{V}}{\partial\dot{F}}=0$, so that in such a situation
one has
$\int{d^{3}x}\dfrac{\partial}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)=\dfrac{\partial{V}}{\partial\dot{F}}\int{dV}\dfrac{\partial}{\partial{V}}\left(\dot{F}\mathcal{L}_{M}\right)=\dfrac{\partial{V}}{\partial\dot{F}}\left(\dot{F}\mathcal{L}_{M}\right)=0,$
(4.41)
and therefore the volume integral of the Lagrangian density of Matter fields
is much more simplified
$\int{d^{3}x}\mathcal{L}_{M}=-\dfrac{1}{2}\int{d^{3}x}\mathcal{H}_{M}.$ (4.42)
In this manner, finally the action of Matter fields can be entirely expressed
via the only Hamiltonian density of Matter fields
$\displaystyle\int_{M}d^{4}x\sqrt{-g}\mathcal{L}_{M}$ $\displaystyle=$
$\displaystyle-\dfrac{c}{2}\int{d\eta}Na^{4}\int{d^{3}x}\mathcal{H}_{M}.$
(4.43)
Collecting all these facts one can establish evaluation of the
Einstein–Hilbert action on the Friedmann–Lemaître–Robertson–Walker metric.
Interestingly, in such a situation the Einstein–Hilbert action has the
Hamilton form in the conformal time, i.e.
$S_{EH}=\int{d\eta}\mathrm{L}(a,a^{\prime},\eta),$ (4.44)
and the Lagrangian $\mathrm{L}(a,a^{\prime},\eta)$ of the cosmological model
has the form
$\mathrm{L}(a,a^{\prime},\eta)=\dfrac{3}{2}M_{P}\ell_{P}^{2}\dfrac{V}{V_{P}}\dfrac{a^{\prime
2}}{N}-\dfrac{1}{2}Na^{4}\mathrm{H}_{M}(\eta).$ (4.45)
where $V_{P}=\dfrac{4}{3}\pi\ell_{P}^{3}$ is the volume of the Planck sphere,
and $\mathrm{H}_{M}(\eta)$ is the volume integral of the Hamiltonian density
of Matter fields
$\mathrm{H}_{M}(\eta)=\int d^{3}x\mathcal{H}_{M}(x,\eta),$ (4.46)
which has a natural interpretation of energy of Matter fields.
The momentum conjugated to the cosmic scale factor parameter can be derived
straightforwardly from the reduced action (4.44)
$\mathrm{P}_{a}=\dfrac{1}{\ell_{P}}\dfrac{\partial\mathrm{L}}{\partial(a^{\prime})}=3M_{P}\ell_{P}\dfrac{V}{V_{P}}\dfrac{a^{\prime}}{N},$
(4.47)
where the factor $\dfrac{1}{\ell_{P}}$ was putted _ad hoc_ for dimensional
correctness of the momentum, which should be $[E]\cdot[T]\cdot[L]^{-1}$. It
can be seen straightforwardly that by application of the Legendre
transformation to the reduced action (4.44) leads to
$S_{EH}=\int
d\eta\left\\{\ell_{P}\mathrm{P}_{a}a^{\prime}-N\mathrm{H}(\eta)\right\\},$
(4.48)
where $\mathrm{H}(\eta)$ is the Hamilton function of the cosmological model
$\mathrm{H}(\eta)=\dfrac{V_{P}}{3V}\dfrac{\mathrm{P}_{a}^{2}}{2M_{P}}+\dfrac{1}{2}a^{4}\mathrm{H}_{M}(\eta).$
(4.49)
According to the Arnowitt–Deser–Misner Hamiltonian formulation of General
Relativity, the Euler–Lagrange equations of motion obtained via vanishing of
functional derivatives of the total action with respect to the parameters $N$,
and $N_{i}$ are the constraints. In the present situation one has the only one
constraint – the Hamiltonian constraint
$\dfrac{\delta S_{EH}}{\delta N}=-\mathrm{H}(\eta)=0.$ (4.50)
In other words the Hamiltonian constraint to the case of the
Friedmann–Lemaître–Robertson–Walker metric is
$\dfrac{V_{P}}{3V}\dfrac{\mathrm{P}_{a}^{2}}{2M_{P}}+\dfrac{1}{2}a^{4}\mathrm{H}_{M}(\eta)\approx
0.$ (4.51)
Substitution of the explicit form of the conjugated momentum (4.47) to the
Hamiltonian constraint (4.51) leads to the equation
$a^{\prime
2}+\dfrac{N^{2}}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}a^{4}\mathrm{H}_{M}(\eta)=0.$
(4.52)
One can put now the value $N^{2}=1$ which is right for the present case. Using
of the Hubble parameter
$\displaystyle H(a)\equiv\dfrac{\dot{a}}{a}=\dfrac{a^{\prime}}{a^{2}},$ (4.53)
where in this context dot means $t$-differentiation, the Hamiltonian
constraint can be presented as the equation for the energy of Matter fields
$\mathrm{H}_{M}(\eta)=-3M_{P}\ell_{P}^{2}\dfrac{V}{V_{P}}H^{2}(a),$ (4.54)
which suggests that energy of Matter fields is negative. Therefore one can put
$\mathrm{H}_{M}(\eta)=-\left|\mathrm{H}_{M}(\eta)\right|,$ (4.55)
so that the equation (4.54) can be rewritten in the form
$H(a)=\pm\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}\right|},$
(4.56)
where $\mathrm{H}_{M}=\mathrm{H}_{M}(\bullet)$, and $\bullet=t,\eta$. The
solution (4.56) is the Hubble law.
The result (4.54) is in itself nontrivial. Namely, it means that the energy of
Matter fields explicitly depends not on the conformal time $\eta$, but on the
cosmic scale factor parameter. In other words
$\mathrm{H}_{M}(\eta)=\mathrm{H}_{M}(a)$. Applying the result (4.54) within
the Lagrangian (4.45) and Hamiltonian (4.49) of the cosmological model one
sees that explicit dependence on the conformal time transits into explicit
dependence on the cosmic scale factor parameter $a$, i.e.
$\displaystyle\mathrm{L}(a,a^{\prime})$ $\displaystyle=$
$\displaystyle\dfrac{M_{P}\ell_{P}^{2}}{2}\dfrac{3V}{V_{P}}\dfrac{1}{N}\left[a^{\prime
2}+N^{2}a^{4}H^{2}(a)\right],$ (4.57)
$\displaystyle\mathrm{H}(\mathrm{P}_{a},a)$ $\displaystyle=$
$\displaystyle\dfrac{V_{P}}{3V}\left[\dfrac{\mathrm{P}_{a}^{2}}{2M_{P}}-\dfrac{M_{P}\ell_{P}^{2}}{2}\left(\dfrac{3V}{V_{P}}\right)^{2}a^{4}H^{2}(a)\right].$
(4.58)
These formulas, however, show _the tautology of quantum cosmology_ , because
of in the light of the Dirac approach $\mathrm{H}(\mathrm{P}_{a},a)\approx 0$
generates the Hubble parameter $H(a)=\dfrac{1}{N}\dfrac{a^{\prime}}{a^{2}}$,
which after taking into account the value of $N=\pm 1$ proper for the
Friedmann–Lemaître–Robertson–Walker metric and the transformation to the
cosmological time $t$, $a^{\prime}=\dot{a}a$, becomes the usual Hubble
parameter, i.e. $H(a)=\dfrac{\dot{a}}{a}$.
The Hamiltonian (4.58), however in itself in nontrivial because of after
taking into account the fact $H(a)=\dfrac{a^{\prime}}{a^{2}}$ up to the
constant multiplier $\dfrac{V_{P}}{3V}$ it takes the form of the Hamiltonian
of a one-dimensional Euclidean oscillator
$\mathrm{H}(\mathrm{P}_{a},a)=\dfrac{V_{P}}{3V}\left[T(\mathrm{P}_{a})-V(\mathrm{P}_{a},a)\right].$
(4.59)
where the kinetic energy $T(\mathrm{P}_{a})$ and the potential energy
$V(\mathrm{P}_{a},a)$ are
$\displaystyle T(\mathrm{P}_{a})$ $\displaystyle=$
$\displaystyle\dfrac{\mathrm{P}_{a}^{2}}{2M_{P}},$ (4.60) $\displaystyle
V(\mathrm{P}_{a},a)$ $\displaystyle=$
$\displaystyle\dfrac{P_{a}^{2}}{2M_{P}}a^{2}.$ (4.61)
This oscillator is the one-dimensional Euclidean harmonic oscillator if one
identifies the _cosmological coordinate_ with
$x_{C}=\ell_{P}a$ (4.62)
and the potential energy with $V=\dfrac{kx_{C}^{2}}{2}$, then the constant $k$
is
$k=\dfrac{\mathrm{P}_{a}^{2}}{M_{P}\ell_{P}^{2}}.$ (4.63)
In other words the one-dimensional Euclidean harmonic oscillator is defined by
the conjugate momentum
$\mathrm{P}_{a}=\pm\sqrt{kM_{P}}\ell_{P},$ (4.64)
or equivalently
$x_{C}^{\prime}=\pm\sqrt{\dfrac{k}{M_{P}}}\ell_{P}\dfrac{V_{P}}{3V}.$ (4.65)
This equation can be solved immediately
$x_{C}(\eta)=\pm\sqrt{\dfrac{k}{M_{P}}}\ell_{P}\dfrac{V_{P}}{3V}\left(\eta-\eta_{I}\right).$
(4.66)
Applying the relation (4.64) and the solution (4.66) to the Hamiltonian (4.59)
one obtains the Hamiltonian of the one-dimensional Euclidean harmonic
oscillator
$\mathrm{H}(\eta)=\dfrac{k\ell_{P}^{2}}{2}\dfrac{V_{P}}{3V}\left[1-\dfrac{k}{M_{P}}\left(\dfrac{V_{P}}{3V}\right)^{2}\left(\eta-\eta_{I}\right)^{2}\right].$
(4.67)
Interestingly, when one identifies
$\eta_{I}=\sqrt{\dfrac{M_{P}}{k}}\dfrac{3V}{V_{P}},$ (4.68)
then the Hamiltonian of the one-dimensional Euclidean harmonic oscillator
becomes
$\mathrm{H}(\eta)=-\dfrac{k^{2}\ell_{P}^{2}}{2M_{P}}\left(\dfrac{V_{P}}{3V}\right)^{3}\left[\eta^{2}-2\sqrt{\dfrac{M_{P}}{k}}\dfrac{3V}{V_{P}}\eta\right].$
(4.69)
Its values are positive for $0<\eta<2\eta_{I}$, negative for $\eta>2\eta_{I}$,
and zero for $\eta=2\eta_{I}$. Interestingly, when one puts _ad hoc_ $V=V_{P}$
and
$k=k_{P}=\dfrac{M_{P}}{t_{P}^{2}}=M_{P}\omega_{P}^{2}=\dfrac{E_{P}}{\ell_{P}^{2}}\approx
7.4880571\cdot 10^{78}\dfrac{\mathrm{kg}}{\mathrm{s}^{2}},$ (4.70)
then $\eta_{I}=3t_{P}$, and the Hamiltonian of one-dimensional Euclidean
harmonic oscillatorr is simplified to
$\mathrm{H}(\eta)=-\dfrac{E_{P}}{54}\left[\left(\dfrac{\eta}{t_{P}}\right)^{2}-6\dfrac{\eta}{t_{P}}\right],$
(4.71)
so that initially, i.e. for $\eta=3t_{P}$, one obtains
$\mathrm{H}(\eta_{I})=\dfrac{E_{P}}{6}.$ (4.72)
Let us call such a case _the Planckian one-dimensional Euclidean harmonic
oscillator_.
In this manner, in the Hamiltonian approach presented above the
Einstein–Friedmann Universe is described by the Hamiltonian constraint (4.51),
and the Hubble law is obtained due to straightforward integration of this
constraint. The Hubble law can be expressed via both the cosmological as well
as the conformal time
$\displaystyle\int_{a_{I}}^{a}\dfrac{da^{\prime}}{a^{\prime 2}H(a^{\prime})}$
$\displaystyle=$ $\displaystyle\eta_{I}-\eta,$ (4.73)
$\displaystyle\int_{a_{I}}^{a}\dfrac{da^{\prime}}{a^{\prime}H(a^{\prime})}$
$\displaystyle=$ $\displaystyle t_{I}-t,$ (4.74)
where the subscript $I$ denotes the initial data values of given quantity. Let
us consider the cosmological redshift $z$ defined by the formula [155],
$\dfrac{a}{a_{I}}\equiv\dfrac{1}{1+z},$ (4.75)
where in terms of cosmological time $a=a(t)$, $a_{I}=a(t_{I})$,
$z=z(t_{I},t)$, while in in terms of conformal time $a=a(\eta)$,
$a_{I}=a(\eta_{I})$, $z=z(\eta_{I},\eta)$. Then the Hubble law (4.73)-(4.74)
can be expressed as
$\displaystyle\int_{z_{I}}^{z}\dfrac{dz^{\prime}}{H(z^{\prime})}$
$\displaystyle=$ $\displaystyle a(\eta_{I})\left(\eta-\eta_{I}\right),$ (4.76)
$\displaystyle\int_{z_{I}}^{z}\dfrac{dz^{\prime}}{(1+z^{\prime})H(z^{\prime})}$
$\displaystyle=$ $\displaystyle t-t_{I},$ (4.77)
where $z_{I}=z(t_{I},t_{I})=z(\eta_{I},\eta_{I})$, and by the equation (4.56)
and the definition (4.53) the cosmological redshift can be expressed by two
distinguishable ways
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!z(\eta_{I},\eta)$
$\displaystyle=$ $\displaystyle\pm
a(\eta_{I})\int_{\eta_{I}}^{\eta}d\eta^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}(\eta^{\prime})\right|}$
(4.78) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!z(t_{I},t)$
$\displaystyle=$
$\displaystyle\exp\left\\{\pm\dfrac{a(\eta_{I})}{a(t_{I})}\int_{t_{I}}^{t}dt^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}(t^{\prime})\right|}\right\\}-1.$
(4.79)
The relations (4.73) and (4.74), as well as, (4.76) and (4.77) allow to
establish the relation between conformal and cosmological time
$\displaystyle\eta-\eta_{I}$ $\displaystyle=$ $\displaystyle
t-t_{I}+\int_{a_{I}}^{a}\dfrac{a^{\prime}-1}{a^{\prime
2}}\dfrac{da^{\prime}}{H(a^{\prime})},$ (4.80) $\displaystyle
a(\eta_{I})(\eta-\eta_{I})$ $\displaystyle=$ $\displaystyle
t-t_{I}+\int_{z_{I}}^{z}\dfrac{z^{\prime}dz^{\prime}}{(1+z^{\prime})H(z^{\prime})}.$
(4.81)
Interestingly, because of the universal definition of the cosmological
redshift (4.75) one can put _ad hoc_ that
$z(\eta_{I},\eta)=z(t_{I},t),$ (4.82)
i.e. that cosmological redshift is invariant with respect to the exchange
between cosmological time and conformal time. In such a situation, for
consistency one should apply the measure (4.14) within the relation (4.78). It
can seen by straightforward calculation that such an invariance is nontrivial,
because of results in the Riccati equation
$\dot{g}=g+h,$ (4.83)
where $\dot{g}=\dfrac{dg}{dt}$, and we have introduced the notation
$\displaystyle g$ $\displaystyle=$ $\displaystyle g(t)=\exp[f(t)]\ln[f(t)],$
(4.84) $\displaystyle h$ $\displaystyle=$ $\displaystyle
h(t)=\dfrac{a_{I}}{a(t)},$ (4.85)
where $a_{I}=a(t_{I})$ and the unknown function $f(t)$ has the form
$f(t)=\pm\dfrac{a(\eta_{I})}{a(t_{I})}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}(t^{\prime})\right|}=H(t),$
(4.86)
where we have applied equality $a(\eta_{I})=a(t_{I})$. The Riccati equation
(4.83) can be solved immediately with the result
$g(t)=e^{t-t_{I}}\left(1+\int_{t_{I}}^{t}dt^{\prime}e^{-t^{\prime}}h(t^{\prime})\right).$
(4.87)
Differentiation of $g=\exp H\ln H$ leads to
$\dot{g}=\dot{H}\exp H\ln H+\exp H\dfrac{\dot{H}}{H},$ (4.88)
and applying of $\exp H=\dfrac{g}{\ln H}$ within (4.88) one obtains
$\dfrac{\dot{g}}{g}=\dot{H}\left(1+\dfrac{1}{H\ln H}\right).$ (4.89)
Differentiation of both the sides of the equation (4.89) leads to
$\displaystyle\dfrac{\ddot{g}}{g}-\dfrac{\dot{g}^{2}}{g^{2}}=\ddot{H}\left(1+\dfrac{1}{H\ln
H}\right)-\dfrac{\dot{H}^{2}}{H^{2}\ln H}\left(1+\dfrac{1}{\ln H}\right),$
(4.90)
or after using of the equation (4.89)
$\displaystyle\dfrac{\ddot{g}}{g}-\dfrac{\dot{g}^{2}}{g^{2}}=\dfrac{\ddot{H}}{\dot{H}}\dfrac{\dot{g}}{g}-\dfrac{\dot{H}^{2}}{H^{2}\ln
H}\left(1+\dfrac{1}{\ln H}\right).$ (4.91)
By this reason, one can apply the following identification
$\displaystyle\dfrac{\ddot{g}}{g}$ $\displaystyle=$
$\displaystyle\dfrac{\ddot{H}}{\dot{H}}\dfrac{\dot{g}}{g},$ (4.92)
$\displaystyle\dfrac{\dot{g}^{2}}{g^{2}}$ $\displaystyle=$
$\displaystyle\dfrac{\dot{H}^{2}}{H^{2}\ln H}\left(1+\dfrac{1}{\ln H}\right).$
(4.93)
The equation (4.92) can be rewritten as
$\dfrac{\ddot{H}}{\dot{H}}=\dfrac{\ddot{g}}{\dot{g}}.$ (4.94)
Its integration can be performed straightforwardly
$\ln\dot{H}=\ln\dot{g}+\ln C_{0},$ (4.95)
where $C_{0}$ is an integration constant, or equivalently
$\dot{H}=C_{0}\dot{g}.$ (4.96)
Integration of the equation (4.96) gives
$H=C_{0}g+C_{1},$ (4.97)
where $C_{1}$ ia an integration constant. In this manner
$\dfrac{\dot{H}^{2}}{H^{2}}=\left(\dfrac{C_{0}\dot{g}}{C_{0}g+C_{1}}\right)^{2}=\dfrac{\dot{g}^{2}}{\left(g+C_{1}^{\prime}\right)^{2}},$
(4.98)
where $C_{1}^{\prime}=\dfrac{C_{1}}{C_{0}}$, and the equation (4.93) can be
rewritten as
$\dfrac{\left(g+C_{1}^{\prime}\right)^{2}}{g^{2}}=\dfrac{1}{\ln
H}\left(1+\dfrac{1}{\ln H}\right),$ (4.99)
or in equivalent form of the algebraic equation for unknown $\ln H$
$\left(g+C_{1}^{\prime}\right)^{2}(\ln H)^{2}-g^{2}\ln H-g^{2}=0,$ (4.100)
which has two solutions
$\ln
H=\dfrac{1}{2}\left(\dfrac{g}{g+C_{1}^{\prime}}\right)^{2}\left[1\pm\sqrt{1+4\left(\dfrac{g+C_{1}^{\prime}}{g}\right)^{2}}\right],$
(4.101)
and by $\ln H=\dfrac{g}{\exp H}>1$ the correct solution is
$\ln
H=\dfrac{1}{2}\left(\dfrac{g}{g+C_{1}^{\prime}}\right)^{2}\left[1+\sqrt{1+4\left(\dfrac{g+C_{1}^{\prime}}{g}\right)^{2}}\right].$
(4.102)
Jointing of the solutions (4.97) and (4.102) allows to establish the algebraic
identity for the function $g$
$\ln(C_{0}g+C_{1})=\dfrac{1}{2}\left(\dfrac{C_{0}g}{C_{0}g+C_{1}}\right)^{2}\left[1+\sqrt{1+4\left(\dfrac{C_{0}g+C_{1}}{C_{0}g}\right)^{2}}\right],$
(4.103)
Equivalently, however, by using $\ln H=\dfrac{g}{\exp{H}}$ within the equation
(4.100) one obtains the algebraic equation for unknown function $\exp H$
$(\exp H)^{2}+g\exp H-(g+C_{1}^{\prime})^{2}=0,$ (4.104)
having also two possible solutions
$\exp
H=\dfrac{g}{2}\left(1\pm\sqrt{1+4\left(\dfrac{g+C_{1}^{\prime}}{g}\right)^{2}}\right),$
(4.105)
and because of $\exp H>1$, $g>1$ the correct solution is
$\exp
H=\dfrac{g}{2}\left(1+\sqrt{1+4\left(\dfrac{g+C_{1}^{\prime}}{g}\right)^{2}}\right),$
(4.106)
and by using of the solution (4.97) one obtains another identity for $g$
$\exp\left(C_{0}g+C_{1}\right)=\dfrac{g}{2}\left(1+\sqrt{1+4\left(\dfrac{C_{0}g+C_{1}}{C_{0}g}\right)^{2}}\right).$
(4.107)
Anyway, however, employing the solution (4.97) within the definition $g=\exp
H\ln H$ allows to establish one more identity for $g$
$g=\exp\left(C_{0}g+C_{1}\right)\ln\left(C_{0}g+C_{1}\right).$ (4.108)
Another identity can be established via application of the equation (4.89) and
the relation (4.96) which leads to
$\ln H=\dfrac{C_{0}g}{1-C_{0}g}\dfrac{1}{C_{0}g+C_{1}},$ (4.109)
what applied to the equation (4.100) gives the algebraic equation for the
function $g$
$C_{0}^{2}g^{3}+(C_{0}C_{1}-3C_{0})g^{2}+(2-C_{0}^{4}-2C_{1})g+C_{0}^{3}C_{1}-\dfrac{C_{1}}{C_{0}}=0,$
(4.110)
having in general three solutions. Taking into account the Riccati equation
(4.83), i.e. nonzero value of $\dot{g}=g+h$, one can differentiate both sides
of the equation (4.110) and obtain
$[3C_{0}^{2}g^{2}+2(C_{0}C_{1}-3C_{0})g+2-C_{0}^{4}-2C_{1}]\dot{g}=0,$ (4.111)
and conclude that the following constraint is fulfilled
$3C_{0}^{2}g^{2}+2(C_{0}C_{1}-3C_{0})g+2-C_{0}^{4}-2C_{1}=0.$ (4.112)
It allows to establish the value of function $g$
$g=g(C_{0},C_{1})=-\dfrac{C_{1}-3}{3C_{0}}\pm\dfrac{1}{3C_{0}}\sqrt{C_{1}^{2}+3C_{0}^{4}+3}=constans.$
(4.113)
Taking the positive solution as the physical
$g(C_{0},C_{1})=-\dfrac{C_{1}-3}{3C_{0}}+\dfrac{1}{3C_{0}}\sqrt{C_{1}^{2}+3C_{0}^{4}+3}.$
(4.114)
one can derive the value of the Hubble parameter
$H=1+\dfrac{2}{3}C_{1}+\dfrac{1}{3}\sqrt{C_{1}^{2}+3C_{0}^{4}+3}.$ (4.115)
Therefore, the problem is to establish the constants $C_{0}$ and $C_{1}$. It
can be seen straightforwardly that
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle H_{I}-C_{0}g_{I},$
(4.116) $\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\dfrac{\dot{H}_{I}}{\dot{g}_{I}},$ (4.117)
where $g_{I}=g(t_{I})$ and $\dot{g}_{I}=\dot{g}(t_{I})$. Because of the
function $g$ and its derivative are determined explicitly by the relation
(4.87) and the Riccati equation (4.83), one has
$\displaystyle g_{I}$ $\displaystyle=$ $\displaystyle 1,$ (4.118)
$\displaystyle\dot{g}_{I}$ $\displaystyle=$ $\displaystyle g_{I}+h(t_{I})=2,$
(4.119)
and by this reason one obtains finally
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
H_{I}-\dfrac{\dot{H}_{I}}{2},$ (4.120) $\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\dfrac{\dot{H}_{I}}{2}.$ (4.121)
In other words, in such a situation the Hubble parameter has a form
$H=1+\dfrac{2}{3}\left(H_{I}-\dfrac{\dot{H}_{I}}{2}\right)+\dfrac{1}{3}\sqrt{H_{I}^{2}-H_{I}\dot{H}_{I}+\dfrac{3}{16}\dot{H}_{I}^{4}-\dfrac{1}{4}\dot{H}_{I}^{2}+3}.$
(4.122)
Particularly, in the case when $C_{1}=0$ one obtains
$H_{I}=H_{0}\exp\left\\{2(t_{I}-t_{0})\right\\},$ (4.123)
and the actual value of the Hubble parameter becomes
$H=1+\dfrac{1}{\sqrt{3}}\sqrt{1+H_{0}^{4}\exp\left\\{8(t_{I}-t_{0})\right\\}}.$
(4.124)
It can be seen by straightforward computation that if one takes into account
the usual definition of the density of energy of Matter fields
$\mathcal{H}_{M}(x,\cdot)=\dfrac{d\epsilon_{M}(\cdot)}{dV(x)},$ (4.125)
where $\cdot=t,\eta$, $dV(x)=d^{3}x$ is an infinitesimal volume, and
$d\epsilon_{M}(\cdot)$ is an infinitesimal energy of Matter fields contained
in such a volume, then
$\epsilon_{M}=|\mathrm{H}_{M}|,$ (4.126)
and consequently the relations (4.78) and (4.79) allow to establish dependence
of energy of Matter fields from the redshift
$\displaystyle\epsilon_{M}(t)$ $\displaystyle=$ $\displaystyle
3M_{P}\ell_{P}^{2}\dfrac{V}{V_{P}}\left|\dfrac{a(t_{I})}{a(\eta_{I})}\right|^{2}\left|\dfrac{1}{1+z(t_{I},t)}\dfrac{dz(t_{I},t)}{dt}\right|^{2},$
(4.127) $\displaystyle\epsilon_{M}(\eta)$ $\displaystyle=$ $\displaystyle
3M_{P}\ell_{P}^{2}\dfrac{V}{V_{P}}\left|\dfrac{1}{a(\eta_{I})}\right|^{2}\left|\dfrac{dz(\eta_{I},\eta)}{d\eta}\right|^{2}.$
(4.128)
By application of the formulas (4.127) and (4.128), and definitions (4.14),
(4.56) and (4.75) it can be seen straightforwardly that $\mathcal{H}_{M}$,
$\epsilon_{M}$, and $H$ are diffeoinvariants of the coordinate time
$\mathcal{H}_{M}=\mathrm{inv.},\qquad\epsilon_{M}=\mathrm{inv.},\qquad
H(a)=\mathrm{inv.},$ (4.129)
and moreover the relation holds
$\epsilon_{M}(t)=3M_{P}\ell_{P}^{2}\dfrac{V}{V_{P}}\left|\dfrac{a(t_{I})}{a(\eta_{I})}\right|^{2}\left|\dfrac{\dot{a}(t)}{a(t)}\right|^{2}=3M_{P}\ell_{P}^{2}\dfrac{V}{V_{P}}\left|\dfrac{a(t_{I})}{a(\eta_{I})}\right|^{2}|H(a)|^{2}.$
(4.130)
Using of the relations (4.56) and (4.130) one obtains
$\left|\dfrac{a(t_{I})}{a(\eta_{I})}\right|^{2}=1.$ (4.131)
Integration of the equation (4.130) leads to the relation
$\dfrac{a(t)}{a(t_{I})}=\exp\left\\{\pm\dfrac{a(\eta_{I})}{a(t_{I})}\int_{t_{I}}^{t}dt^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(t^{\prime})}\right\\}.$
(4.132)
If one identifies $a(t_{I})=a(\eta_{I})=a_{I}$, and $H(a)$ as real and
positive then the energy of Matter fields is simply
$\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}=H^{2}(a),$ (4.133)
and the cosmic scale factor parameter takes the form
$a(t)=a_{I}\exp\left\\{\pm\int_{t_{I}}^{t}dt^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(t^{\prime})}\right\\}.$
(4.134)
In general the energy of Matter fields (4.133) defines the evolution of the
classical Universe. However, because of $\epsilon_{M}$ is invariant with
respect to exchange between the cosmological time and the conformal time, it
can be considered rather as function of the redshift $z$ or the cosmic scale
factor parameter $a$. Moreover, when there is several kinds of Matter fields
then $\epsilon_{M}$ can be considered as the total energy of Matter fields,
and by this reason is an algebraical sum of all contributions
$\epsilon_{M}=\sum_{i}\epsilon_{i},$ (4.135)
where the subscript $i$ is associated with a kind of Matter fields. In this
manner, the evolution (4.133) can be considered as the dependence of the
redshift on the cosmological as well as the conformal time
$\displaystyle\sqrt{\sum_{i}\epsilon_{i}(z)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{1+z}\dfrac{dz}{dt},$ (4.136)
$\displaystyle\sqrt{\sum_{i}\epsilon_{i}(z)}$ $\displaystyle=$
$\displaystyle\dfrac{dz}{d\eta},$ (4.137)
as well as as the dependence of the cosmic scale factor on the cosmological as
well as the conformal time
$\displaystyle\sqrt{\sum_{i}\epsilon_{i}(a)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{a}\dfrac{da}{dt},$ (4.138)
$\displaystyle\sqrt{\sum_{i}\epsilon_{i}(a)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{a^{2}}\dfrac{da}{d\eta}.$ (4.139)
Interestingly, the Hamiltonian constraint (4.51), describing the classical
toroidal Einstein–Friedmann Universe, can be parameterized in the way
characteristic for a bosonic string, i.e. by using of the Einstein energy-
momentum relation
$\mathrm{P}_{a}^{2}c^{2}+m^{2}(a)c^{4}=E^{2}(a),$ (4.140)
where the energy identically vanishes $E^{2}(a)=0$, and the squared mass is
manifestly negative
$m^{2}(a)=-\left(3M_{P}t_{P}\dfrac{V}{V_{P}}\right)^{2}a^{4}H^{2}(a)<0,$
(4.141)
Hence, it is easy to deduce that the Universe can be understood as the
tachyon, i.e. the groundstate of the classical bosonic string [156]. The mass
of the tachyon can be expressed via two ways
$\sqrt{|m^{2}(a)|}=3\dfrac{V}{V_{P}}M_{P}t_{P}a^{\prime}=\dfrac{3}{2}\dfrac{V}{V_{P}}M_{P}t_{P}\dot{\Omega},$
(4.142)
where $\Omega=a^{2}(t)$ is the scaling function (4.17) and
$S_{P}=4\pi\ell_{P}^{2}$ is the area of the Planck sphere. The Hubble law
(4.73)-(4.74) can be interpreted as the relation between volume of space and
the mass of the tachyon
$\displaystyle
M_{P}\int_{a_{I}}^{a}\dfrac{da^{\prime}}{\sqrt{|m^{2}(a^{\prime})|}}$
$\displaystyle=$ $\displaystyle\dfrac{V_{P}}{3V}\dfrac{\eta-\eta_{I}}{t_{P}},$
(4.143) $\displaystyle
M_{P}\int_{a_{I}}^{a}\dfrac{a^{\prime}da^{\prime}}{\sqrt{|m^{2}(a^{\prime})|}}$
$\displaystyle=$ $\displaystyle\dfrac{V_{P}}{3V}\dfrac{t-t_{I}}{t_{P}}.$
(4.144)
In this manner, we have received the classical point of view on the
Einstein–Friedmann Universe. Let us construct the quantum theory of the
Universe.
#### C Quantization of Hamiltonian Constraint
Application of the primary canonical quantization in the form
$\left[\hat{\mathrm{P}}_{a},\hat{a}\right]=-i\dfrac{\hslash}{\ell_{P}},$
(4.145)
where the parameter of quantization $\dfrac{\hslash}{\ell_{P}}=M_{P}c$ is the
Planck momentum having the value
$\dfrac{\hslash}{\ell_{P}}=M_{P}c\approx
6.524806271\dfrac{\mathrm{kg}\cdot\mathrm{m}}{\mathrm{s}}.$ (4.146)
In comparison to the standard quantum mechanics, in which the parameter of
quantization is the reduced Planck constant $\hslash$, this value of the
parameter of quantization is large, i.e. about $10^{34}$ times more. Such a
procedure allows to generate the operator of the momentum conjugated to the
cosmic scale factor parameter
$\hat{\mathrm{P}}_{a}=-i\dfrac{\hslash}{\ell_{P}}\dfrac{d}{da}.$ (4.147)
Applying such a primary canonical quantization within the Hamiltonian
constraint (4.140) of the Einstein–Friedmann Universe one receives the one-
dimensional Klein–Gordon equation
$\left(\dfrac{d^{2}}{da^{2}}+\omega^{2}(a)\right)\Psi(a)=0,$ (4.148)
where $\omega$ is the _cosmological dimensionless frequency_ given by the
formula
$\omega(a)=\dfrac{3V}{V_{P}}\dfrac{a^{2}H(a)}{\omega_{P}},$ (4.149)
where $V_{P}=\dfrac{4}{3}\pi\ell_{P}^{3}$ is volume of the Planck sphere, and
$\omega_{P}=\dfrac{1}{t_{P}}$ is the Planck frequency. In the context of
cosmology the Klein–Gordon equation (4.148) is the analog of the
Wheeler–DeWitt equation [157, 158] of more general quantum geometrodynamics.
Usually this equation identified with non relativistic quantum mechanics, i.e.
as the Schrödinger equation. Because of the cosmic scale factor parameter $a$
is defined as a function of cosmological or conformal time, in general the
wave function $\Psi(a)$ is a functional, and the equation (4.148) is a one-
dimensional functional differential equation. In general by the spirit of the
Wheeler superspace, the wave function is a wave functional on the space of
three-dimensional metrics characterized by the parameter $a$. We shall discuss
details of this approach in next chapters of this part. Albeit, it must be
emphasized that the approach based on the Schrödinger equation, which has been
worked out more than 50 years ago, since the mid-1980’s does not lead to new
constructive results.
A different possible interpretation of the quantum theory (4.148) is the
Klein–Gordon equation, i.e. the relativistic wave equation describing bosons.
It is not difficult to see by straightforward computations that the equation
(4.148) can be generated as the classical field theoretic Euler–Lagrange
equations of motion
$\left[\dfrac{\partial}{\partial\Psi}-\dfrac{\partial}{\partial
a}\dfrac{\partial}{\partial\left(\dfrac{d\Psi}{da}\right)}\right]L\left(\Psi,\dfrac{d\Psi}{da}\right)=0,$
(4.150)
via the variational principle
$\delta S[\Psi]=0,$ (4.151)
with respect to the action functional $S[\Psi]$ having the following form
$S[\Psi]=\int daL\left(\Psi,\dfrac{d\Psi}{da}\right),$ (4.152)
where the Lagrangian of the classical field, i.e. one-dimensional wave
function $\Psi=\Psi(a)$, is
$L\left(\Psi,\dfrac{d\Psi}{da}\right)=\dfrac{1}{2}\left(\dfrac{d\Psi}{da}\right)^{2}-\dfrac{1}{2}\omega^{2}(a)\Psi^{2}.$
(4.153)
The momentum $\Pi_{\Psi}$ canonically conjugated to the wave function $\Psi$
can be established immediately from the action (4.152) as
$\displaystyle\Pi_{\Psi}=\dfrac{\partial
L\left(\Psi,\dfrac{d\Psi}{da}\right)}{\partial\left(\dfrac{d\Psi}{da}\right)}=\dfrac{d\Psi}{da},$
(4.154)
and its straightforward application allows to rewrite the equation (4.148) in
the form
$\displaystyle\dfrac{d\Pi_{\Psi}}{da}+\omega^{2}(a)\Psi(a)=0.$ (4.155)
The equations of motion (4.154) and (4.155) can be treated as the system of
equations and employed for reduction of order of the Wheeler–DeWitt equation
(4.148). Such a reduction can be done by introducing to the theory the two-
component scalar field
$\Phi=\left[\begin{array}[]{c}\Pi_{\Psi}\\\ \Psi\end{array}\right],$ (4.156)
which allows to express the quantum cosmology given by the Klein–Gordon
equation(4.148), which describes the Einstein–Friedmann Universe as the
Multiverse of quantum universes, as the one-dimensional Dirac equation
$\left(-i\sigma_{2}\dfrac{d}{da}-M\right)\Phi=0,$ (4.157)
where $M$ is the mass matrix, and $\sigma_{2}$ is one of the Pauli matrices,
$\displaystyle M$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}-1&0\\\
0&-\omega^{2}\end{array}\right],$ (4.160) $\displaystyle\sigma_{2}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right].$ (4.163)
On the other hand the one-dimensional Dirac equation (4.157) describes two-
component one-dimensional scalar field, i.e. axion, and as one-dimensional
quantum mechanics is automatically supersymmetric. In this way the quantum
cosmology based on spatially finite and conformal-flat
Friedmann–Lemaître–Robertson–Walker metric , i.e. the Einstein–Friedmann
Universe, is strictly related to one-dimensional superstrings which we shall
call the Fermi–Bose superstrings. This name arises from the fact that in the
dimension $1$ there is no difference between Fermi–Dirac and Bose–Einstein
statistics because of the spin is zero. In this manner the quantum field
theory which shall be constructed relying on the Fock repère of creators and
annihilators will be describing the Multiverse of Fermi–Bose superstrings,
which we shall call the superstring Multiverse, which in the quantum cosmology
presented above is the system of multiple quantum Einstein–Friedmann
Universes.
Because of the quantum cosmology (4.148) has manifestly bosonic character, let
us perform the secondary canonical quantization of the one-dimensional Dirac
equation (4.157) appropriate to bosons [159]
$\left[\hat{\Pi}_{\Psi}[a],\hat{\Psi}[a^{\prime}]\right]=-i\delta\left(a-a^{\prime}\right),$
(4.164)
and other canonical commutation relations are trivial. In the explicit form
the bosonic field is
$\displaystyle\left[\begin{array}[]{c}\hat{\Psi}\\\
\hat{\Pi}_{\Psi}\end{array}\right]=\left[\begin{array}[]{cc}\dfrac{1}{\sqrt{2\omega(a)}}&\dfrac{1}{\sqrt{2\omega(a)}}\\\
-i\sqrt{\dfrac{\omega(a)}{2}}&i\sqrt{\dfrac{\omega(a)}{2}}\end{array}\right]\left[\begin{array}[]{c}{G}[a]\\\
{G}^{\dagger}[a]\end{array}\right],$ (4.171)
and is consistent with the algebraic approach to canonical commutation
relations due to von Neumann, and Araki and Woods [160]. In result the
Universe is described by the dynamical basis
${B}_{a}=\left\\{\left[\begin{array}[]{c}{G}[a]\\\
{G}^{\dagger}[a]\\!\\!\end{array}\right]:\left[{G}[a],{G}^{\dagger}[a^{\prime}]\right]=\delta\left(a-a^{\prime}\right),\left[{G}[a],{G}[a^{\prime}]\right]=0\right\\},$
(4.172)
satisfying non-Heisenberg equations of motion
$\dfrac{d{B}_{a}}{d{a}}=\left[\begin{array}[]{cc}-i\omega(a)&\dfrac{1}{2\omega(a)}\dfrac{d\omega(a)}{da}\\\
\dfrac{1}{2\omega(a)}\dfrac{d\omega(a)}{da}&i\omega(a)\end{array}\right]{B}_{a},$
(4.173)
and via perturbation theory holds
$\left|\dfrac{1}{2\omega(a)}\dfrac{d\omega(a)}{da}\right|\ll\dfrac{1}{a}.$
(4.174)
The equations (4.173) can be diagonalized via taking into account the new
basis ${B}_{a}^{\prime}$
${B}_{a}^{\prime}=\left\\{\left[\begin{array}[]{c}{G}^{\prime}[a]\\\
{G}^{\prime\dagger}[a]\end{array}\right]:\left[{G}^{\prime}[a],{G}^{\prime\dagger}[a^{\prime}]\right]=\delta\left(a-a^{\prime}\right),\left[{G}^{\prime}[a],{G}^{\prime}[a^{\prime}]\right]=0\right\\},$
(4.175)
obtained via taking together the Bogoliubov transformation and the Heisenberg
equations of motion
$\displaystyle{B}_{a}^{\prime}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}u&v\\\
v^{\ast}&u^{\ast}\end{array}\right]{B}_{a},$ (4.178)
$\displaystyle\dfrac{d{B}_{a}^{\prime}}{da}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}-i\omega^{\prime}&0\\\
0&i\omega^{\prime}\end{array}\right]{B}_{a}^{\prime},$ (4.181)
where the hyperbolic constraint holds
$|u|^{2}-|v|^{2}=1,$ (4.182)
and $\omega^{\prime}$ is an unknown frequency. It can be seen by
straightforward computation that such a procedure generates unambiguously
$\omega^{\prime}=0$, and therefore the new basis is the static Fock repère
${B}_{I}=\left\\{\left[\begin{array}[]{c}\mathrm{w}_{I}\\\
\mathrm{w}^{\dagger}_{I}\end{array}\right]:\left[\mathrm{w}_{I},\mathrm{w}^{\dagger}_{I}\right]=1,\left[\mathrm{w}_{I},\mathrm{w}_{I}\right]=0\right\\},$
(4.183)
and by this reason the static vacuum state is obtained
$\left\langle\textrm{0}\right|=\left\\{\left\langle\textrm{0}\right|:\mathrm{w}_{I}\left\langle\textrm{0}\right|=0\quad,\quad
0=\left|\textrm{0}\right\rangle\mathrm{w}_{I}^{\dagger}\right\\}.$ (4.184)
The system of operator equations (4.173) transits to the system of equations
for the Bogoliubov coefficients
$\dfrac{d}{da}\left[\begin{array}[]{c}v(a)\\\
u(a)\end{array}\right]=\left[\begin{array}[]{cc}-i\omega(a)&\dfrac{1}{2\omega(a)}\dfrac{\partial\omega(a)}{\partial
a}\\\ \dfrac{1}{2\omega(a)}\dfrac{\partial\omega(a)}{\partial
a}&i\omega(a)\end{array}\right]\left[\begin{array}[]{c}v(a)\\\
u(a)\end{array}\right],$ (4.185)
which is easy to solve in the superfluid parametrization [161]
$\displaystyle v(a)$ $\displaystyle=$
$\displaystyle\exp(i\theta(a))\sinh\phi(a),$ (4.186) $\displaystyle u(a)$
$\displaystyle=$ $\displaystyle\exp(i\theta(a))\cosh\phi(a),$ (4.187)
where $\theta$ and $\phi$ are the angles, which in the present situation
express via the mass of the tachyon as follows
$\displaystyle\theta(a)$ $\displaystyle=$ $\displaystyle\pm
i\int_{a_{I}}^{a}\omega(a^{\prime})da^{\prime},$ (4.188)
$\displaystyle\phi(a)$ $\displaystyle=$
$\displaystyle\ln{\sqrt{\dfrac{\omega_{I}}{\omega(a)}}},$ (4.189)
where $\omega_{I}=\omega(a_{I})$ is the initial data of $\omega(a)$.
In this manner, _the quantum cosmology is completely determined via the
monodromy matrix $C$ between the dynamical and the static Fock repère
$B_{a}=CB_{I}$ which is explicitly given by_
$C=\left[\begin{array}[]{cc}\left(\sqrt{\dfrac{\omega(a)}{\omega_{I}}}+\sqrt{\dfrac{\omega_{I}}{\omega(a)}}\right)\dfrac{e^{\lambda}}{2}&\left(\sqrt{\dfrac{\omega(a)}{\omega_{I}}}-\sqrt{\dfrac{\omega_{I}}{\omega(a)}}\right)\dfrac{e^{-\lambda}}{2}\\\
\left(\sqrt{\dfrac{\omega(a)}{\omega_{I}}}-\sqrt{\dfrac{\omega_{I}}{\omega(a)}}\right)\dfrac{e^{\lambda}}{2}&\left(\sqrt{\dfrac{\omega(a)}{\omega_{I}}}+\sqrt{\dfrac{\omega_{I}}{\omega(a)}}\right)\dfrac{e^{-\lambda}}{2}\end{array}\right],$
(4.190)
where $\lambda$ is integrated frequency
$\lambda=\lambda(a)=i\theta(a)=\mp\int_{a_{I}}^{a}\omega(a^{\prime})da^{\prime}.$
(4.191)
Now the quantum field $\hat{\Psi}$ can be computed straightforwardly
$\displaystyle\hat{\Psi}[a]=\dfrac{1}{\sqrt{2\omega_{I}}}\left(e^{\lambda(a)}\mathrm{w}_{I}+e^{-\lambda(a)}\mathrm{w}^{\dagger}_{I}\right),$
(4.192)
and if one introduces the state
$|n\rangle\equiv\left(\hat{\Psi}[a]\right)^{n}\left\langle\textrm{0}\right|=\dfrac{e^{-n\lambda(a)}}{\left(2\omega_{I}\right)^{n/2}}\mathrm{w}^{\dagger\leavevmode\nobreak\
n}_{I}\left|\textrm{0}\right\rangle,$ (4.193)
then the interesting relations can be derived
$\displaystyle\langle{m}|n\rangle$ $\displaystyle=$
$\displaystyle\dfrac{e^{(m-n)\lambda(a)}}{\left(2\omega_{I}\right)^{(m+n)/2}}\left\langle\textrm{0}\right|\mathrm{w}_{I}^{m}\mathrm{w}_{I}^{\dagger\leavevmode\nobreak\
n}\left|\textrm{0}\right\rangle,$ (4.194) $\displaystyle\langle{n}|n\rangle$
$\displaystyle=$
$\displaystyle\sum_{p=0}^{n}\dfrac{e^{i\pi(p+1)}}{\left(2\omega_{I}\right)^{n}}C^{n}_{n-p}\left\langle\textrm{0}\right|\left(\mathrm{w}_{I}^{\dagger}\mathrm{w}_{I}\right)^{p}\left|\textrm{0}\right\rangle=\dfrac{e^{i\left(2n+1\right)\pi}}{\left(2\omega_{I}\right)^{n}},$
(4.195)
where $C^{n}_{k}$ is the Newton binomial symbol, which in the present case is
$C^{n}_{n-p}=\dfrac{n!}{p!(n-p)!},$ (4.196)
and we have applied the normalization of the stable vacuum state
$\left\langle{\textrm{0}}|{\textrm{0}}\right\rangle=1.$ (4.197)
Obviously the state $|n\rangle$ is a $n$-particle state describing the
Multiverse. Its physical sense arises via the straightforward analogy with the
quantum theory of many body systems (See e.g. the Ref. [162]). Namely, in the
present case of the quantum cosmology one has to deal with the system of many
quantum universes. Such a specific quantum many body system, i.e. the
secondary-quantized Einstein–Friedmann Universe given by the quantum field
(4.192) and the many particle states (4.193), is manifestly a realization of
the Multiverse hypothesis. Let us consider now the thermodynamics of the
Multiverse.
#### D The Multiverse Thermodynamics
Because of we have derived the static basis (4.183), formally also exists the
thermal equilibrium for quantum states of the Einstein–Friedmann Universe
(4.4). In this section we shall formulate thermodynamics of many quantum
Universe. We shall apply the simplest approximation, i.e. we shall consider
the system with one degenerated state which is defined by the density operator
$\varrho_{{G}}$ having the form
$\displaystyle\varrho_{{G}}$ $\displaystyle=$ $\displaystyle{G}^{\dagger}{G}=$
(4.198) $\displaystyle=$ $\displaystyle{B}_{a}^{\dagger}\
\left[\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right]{B}_{a}=$ (4.201)
$\displaystyle=$
$\displaystyle{B}_{I}^{\dagger}\left[\begin{array}[]{cc}|u|^{2}&-uv\\\
-u^{\ast}v^{\ast}&|v|^{2}\end{array}\right]{B}_{I}\equiv{B}_{I}^{\dagger}\rho_{\mathrm{eq}}{B}_{I},$
(4.204)
where $\rho_{\mathrm{eq}}$ is the density operator in the thermal equilibrium.
By straightforward computation of the Boltzmann–von Neumann entropy
$\mathrm{S}$
$\displaystyle\mathrm{S}$ $\displaystyle=$
$\displaystyle\dfrac{\mathrm{tr}\left(\rho_{\mathrm{eq}}\ln\rho_{\mathrm{eq}}\right)}{\mathrm{tr}\rho_{\mathrm{eq}}}=$
(4.205) $\displaystyle=$
$\displaystyle\ln\left(2|u|^{2}-1\right)\equiv-\ln\Omega_{\mathrm{eq}},$
(4.206)
it is easy to deduce the distribution function $\Omega_{\mathrm{eq}}$:
$\Omega_{\mathrm{eq}}=\dfrac{1}{2|u|^{2}-1}=\dfrac{1}{2|v|^{2}+1}.$ (4.207)
On the other side, one can use the occupation number $n$
$\displaystyle n$ $\displaystyle\equiv$ $\displaystyle\langle
0|{G}^{\dagger}[a]{G}[a]|0\rangle=$ (4.208) $\displaystyle=$
$\displaystyle\dfrac{1}{4}\left(\dfrac{\omega(a)}{\omega_{I}}+\dfrac{\omega_{I}}{\omega(a)}\right)-\dfrac{1}{2}=|v|^{2},$
(4.209)
where $m_{I}=m(a_{I})$, to derivation of the entropy
$\mathrm{S}=\ln\langle\mathrm{n}\rangle,$ (4.210)
where $\langle\mathrm{n}\rangle$ is averaged occupation number
$\langle\mathrm{n}\rangle=2n+1=\dfrac{1}{2}\left(\dfrac{\omega(a)}{\omega_{I}}+\dfrac{\omega_{I}}{\omega(a)}\right).$
(4.211)
By combination of the formulas (4.206), (4.209), and (4.210), with the natural
conditions $n\geq 0$ and $\omega(a)\geq\omega_{I}$, one receives the mass
spectrum of the system of many Einstein–Friedmann Universes
$\dfrac{\omega(a)}{\omega_{I}}=\langle\mathrm{n}\rangle+\sqrt{\langle\mathrm{n}\rangle^{2}-1}.$
(4.212)
Similarly, by application of the averaging method one can derive the internal
energy $\mathrm{U}$
$\displaystyle\mathrm{U}$ $\displaystyle\equiv$
$\displaystyle\dfrac{\mathrm{tr}(\rho_{\mathrm{eq}}\mathrm{H}_{\mathrm{eq}})}{\mathrm{tr}\mathrm{\rho_{\mathrm{eq}}}}=$
(4.213) $\displaystyle=$
$\displaystyle\left(\langle\mathrm{n}\rangle+\dfrac{1}{2}\right)\left(\langle\mathrm{n}\rangle+\sqrt{\langle\mathrm{n}\rangle^{2}-1}\right)\omega_{I},$
(4.214)
and chemical potential $\mu$
$\displaystyle\mu$ $\displaystyle\equiv$
$\displaystyle\dfrac{\partial\mathrm{U}}{\partial\mathrm{n}}=$ (4.215)
$\displaystyle=$
$\displaystyle\left(\dfrac{\langle\mathrm{n}\rangle+\dfrac{1}{2}}{\sqrt{\langle\mathrm{n}\rangle^{2}-1}}+1\right)\left(\langle\mathrm{n}\rangle+\sqrt{\langle\mathrm{n}\rangle^{2}-1}\right)\omega_{I},$
(4.216)
where we have applied the Hamiltonian of the classical theory expressed in the
static Fock repère
$H=\left(G^{\dagger}[a]G[a]+G[a]G^{\dagger}[a]\right)\dfrac{\omega(a)}{2}=B_{I}^{\dagger}\mathrm{H}_{\mathrm{eq}}B_{I},$
(4.217)
where $H_{\mathrm{eq}}$ is the Hamiltonian in equilibrium
$\displaystyle
H_{\mathrm{eq}}=\left[\begin{array}[]{cc}|u|^{2}+|v|^{2}&-2uv\\\
-2u^{\ast}v^{\ast}&|u|^{2}+|v|^{2}\end{array}\right]\dfrac{\omega(a)}{2}.$
(4.220)
Let us establish temperature of the system by using of the _method of quantum
statistics_. For this one must take into account the concrete form of the
quantum statistics of the system. The bosonic character of the quantum
cosmology (4.148) suggests application of the Bose–Einstein statistics which
naturally describes bosonic systems. In this manner
$\Omega_{\mathrm{eq}}\equiv\dfrac{1}{\exp\left\\{\dfrac{\mathrm{U}-\mu
n}{\mathrm{T}}\right\\}-1}.$ (4.221)
Because of the quantum statistics derived in this section has the form
$\Omega_{\mathrm{eq}}=\dfrac{1}{|u|^{2}+|v|^{2}}=\dfrac{1}{2|u|^{2}-1},$
(4.222)
what after taking into account the fact $2|u|^{2}=\langle{n}\rangle+1$ results
in the relation
$\dfrac{\mathrm{U}-\mu n}{\mathrm{T}}=\ln\left(\langle{n}\rangle+1\right).$
(4.223)
Consequently such a procedure allows to determine the temperature of the
system of quantum states of the Universe as the fixed parameter.
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathrm{T}$
$\displaystyle=$
$\displaystyle\omega_{I}\dfrac{\langle\mathrm{n}\rangle+\sqrt{\langle\mathrm{n}\rangle^{2}-1}}{\ln(\langle\mathrm{n}\rangle+1)}\times$
(4.224) $\displaystyle\times$
$\displaystyle\left[\langle\mathrm{n}\rangle+\dfrac{1}{2}-\dfrac{1}{2}\left(\langle\mathrm{n}\rangle+\sqrt{\langle\mathrm{n}\rangle^{2}-1}+\dfrac{1}{2}\right)\sqrt{\dfrac{\langle\mathrm{n}\rangle-1}{\langle\mathrm{n}\rangle+1}}\right].$
(4.225)
#### E The Early Light Multiverse
Interestingly, there is a certain particular thermodynamical situation of the
system of many quantum Universes. Namely, these are the quantum Universes
possessing minimal entropy
$\mathrm{S}_{\mathrm{min}}=0.$ (4.226)
It can be deduced straightforwardly that such a Multiverse is characterized by
the following conditions
$n=0\quad,\quad\langle\mathrm{n}\rangle=1\quad,\quad\omega(a)=\omega_{I},$
(4.227)
and therefore its thermodynamical parameters have the values
$\displaystyle\mathrm{U}_{\mathrm{min}}$ $\displaystyle=$
$\displaystyle\dfrac{3}{2}\omega_{I},$ (4.228)
$\displaystyle\mu_{\mathrm{min}}$ $\displaystyle=$ $\displaystyle\infty,$
(4.229) $\displaystyle\mathrm{T}_{\mathrm{min}}$ $\displaystyle=$
$\displaystyle\dfrac{\mathrm{U}_{\mathrm{max}}}{\ln 2}.$ (4.230)
The relations (4.228), (4.229), and (4.230) shows that such a specific
collection of quantum Universes is determined by the only initial data, i.e.
the parameter $\omega_{I}$. The relation (4.230) nontrivially connects
temperature and internal energy of these quantum states, and therefore in
general spirit of statistical mechanics it is the equation of state of the
collection of quantum Universes. Infinite value of the chemical potential is
the only characteristic syndrome of openness of the system, and shows that in
the point $n=0$ a phase transition happens. Such a point can be understood as
the point in which quantum Universes start their existence. There is a
question about the physical meaning of the quantum states of Universe
characterized by the minimal entropy. It is not difficult to see that for such
Universes holds the relation
$\omega_{I}=\pm\dfrac{3V}{V_{P}}\dfrac{a^{2}_{I}H(a_{I})}{\omega_{P}},$
(4.231)
which defines the initial value of the Hubble parameter
$H(a_{I})=\dfrac{Q}{a^{2}_{I}},$ (4.232)
where $Q$ is a constant which depends on two free parameters: the initial data
of the mass and the volume of space
$Q=\pm\dfrac{V_{P}}{3V}\omega_{P}\omega_{I}=constans.$ (4.233)
Interestingly, when $\omega_{I}$ is finite and nonzero and volume of space is
infinite then identically $Q\equiv 0$, and therefore the initial value of the
Hubble parameter also vanishes. It proves that for consistency the volume of
space must be finite and nonzero. In such a situation the value of the Hubble
parameter (4.232) is associated for a radiation, and this is the physical
sense of this collection of quantum Universes. In this manner, the Multiverse
of such a quantum Universes expresses a cosmological nature of light, and by
this reason we propose to call this _the light Multiverse_. It can be seen by
straightforward derivation from the Hubble parameter (4.232) and its
definition
$H(a_{I})=\dfrac{1}{a_{I}}\dfrac{da_{I}}{dt_{I}}=\dfrac{1}{a_{I}^{2}}\dfrac{da_{I}}{d\eta_{I}},$
(4.234)
that the light Multiverse is described by the following initial values of the
cosmic scale factor parameter
$\displaystyle a_{I}(t_{I},t_{0})$ $\displaystyle=$
$\displaystyle\sqrt{a_{0}^{2}+2Q(t_{I}-t_{0})},$ (4.235) $\displaystyle
a_{I}(\eta_{I},\eta_{0})$ $\displaystyle=$ $\displaystyle
a_{0}+Q(\eta_{I}-\eta_{0}),$ (4.236)
where $t_{0}$, $\eta_{0}$, $a_{0}=a_{I}(t_{0},t_{0})=a_{I}(\eta_{0},\eta_{0})$
are the integration constants, and
$\displaystyle a_{I}(t_{I},t_{0})$ $\displaystyle\geqslant$ $\displaystyle
a_{0},$ (4.237) $\displaystyle a_{I}(\eta_{I},\eta_{0})$
$\displaystyle\geqslant$ $\displaystyle a_{0}.$ (4.238)
Because, however, both the scale factor parameters $a_{I}(t_{I},t_{0})$ and
$a_{I}(\eta_{I},\eta_{0})$ are the only constants, one can suggest _ad hoc_
that they are equal
$a_{I}(t_{I},t_{0})=a_{I}(\eta_{I},\eta_{0})\equiv a_{I}.$ (4.239)
Such a conjecture allows to establish the linkage between the cosmological and
the conformal times for the early evolution of the Multiverse
$\eta_{I}-\eta_{0}=\dfrac{a_{0}}{Q}\left(\sqrt{1+\dfrac{2Q}{a_{0}^{2}}(t_{I}-t_{0})}-1\right).$
(4.240)
There can be also interesting the Taylor expansion of this relation
$\eta_{I}-\eta_{0}=\sqrt{\pi}\sum_{n=0}^{\infty}\dfrac{(2Q/a_{0})^{n}}{\Gamma(n+2)\Gamma\left(\dfrac{1}{2}-n\right)}(t_{I}-t_{0})^{n+1}.$
(4.241)
In the sense of perturbation theory the proposed quantum cosmology is
consistent if and only if the condition (4.174) is satisfied. This inequality
can be expressed in terms of the Hubble parameter
$H(a)\ll H(a_{I}),$ (4.242)
and straightforwardly integrated. Application of the cosmological time allows
to rewrite the inequality (4.242) as
$\dfrac{1}{a}\dfrac{da}{dt}\ll\dfrac{1}{a_{I}}\dfrac{da_{I}}{dt_{I}},$ (4.243)
and straightforward integration in the region $a_{0}\leqslant a_{I}\leqslant
a$ gives
$\ln\left|\dfrac{a}{a_{I}}\right|\ll\ln\left|\dfrac{a_{I}}{a_{0}}\right|,$
(4.244)
what leads to the bound for cosmic scale factor parameter
$a\ll\dfrac{a_{I}^{2}}{a_{0}}.$ (4.245)
In the light of the inequalities (4.237) and (4.238) one has
$\dfrac{a_{I}^{2}}{a_{0}}=\dfrac{a_{I}}{a_{0}}a_{I}\geqslant a_{I},$ (4.246)
results in the bound for cosmic scale factor parameter
$a\geqslant a_{I},$ (4.247)
which can be expressed equivalently as the bound for redshift
$z(t_{I},t)\leqslant 0,$ (4.248)
and defines the early Universe. Similarly, the inequality (4.242) can be
rewritten in terms of conformal time
$\dfrac{1}{a^{2}}\dfrac{da}{d\eta}\ll\dfrac{1}{a^{2}_{I}}\dfrac{da_{I}}{d\eta_{I}}$
(4.249)
and straightforwardly integrated with the result
$-\dfrac{1}{a}+\dfrac{1}{a_{I}}\ll-\dfrac{1}{a_{I}}+\dfrac{1}{a_{0}},$ (4.250)
which finally gives the bound for cosmic scale factor parameter
$a\ll\dfrac{a_{0}a_{I}}{2a_{0}-a_{I}}.$ (4.251)
Interestingly, the results (4.245) and (4.251) coincide in the only one case
$a_{I}=a_{0}$.
By this reason, with using of the formulas (4.235) and (4.236), the early
light Multiverse in itself expresses the applicability conditions for the
model of quantum cosmology
$\displaystyle a$ $\displaystyle\ll$ $\displaystyle
a_{0}+\dfrac{2Q}{a_{0}}(t_{I}-t_{0}),$ (4.252) $\displaystyle a$
$\displaystyle\ll$ $\displaystyle
a_{0}+\dfrac{Q(\eta_{I}-\eta_{0})}{1-\dfrac{Q}{a_{0}}(\eta_{I}-\eta_{0})}.$
(4.253)
Because, however, one can suggest also that the cosmic scale factor parameter
expressed via cosmological and conformal time is the same, particularly in the
case of early Universe, one obtains the equation
$\dfrac{2Q}{a_{0}}(t_{I}-t_{0})=\dfrac{Q(\eta_{I}-\eta_{0})}{1-\dfrac{Q}{a_{0}}(\eta_{I}-\eta_{0})},$
(4.254)
having the solution
$\eta_{I}-\eta_{0}=\dfrac{2(t_{I}-t_{0})}{a_{0}+2\dfrac{Q}{a_{0}}(t_{I}-t_{0})},$
(4.255)
which compared with the relation (4.240)
$\dfrac{2(t_{I}-t_{0})}{a_{0}+2\dfrac{Q}{a_{0}}(t_{I}-t_{0})}=\dfrac{a_{0}}{Q}\left(\sqrt{1+\dfrac{2Q}{a_{0}^{2}}(t_{I}-t_{0})}-1\right),$
(4.256)
leads to the following equation
$2\dfrac{Q}{a_{0}}(t_{I}-t_{0})\left[\left(2\dfrac{Q}{a_{0}}(t_{I}-t_{0})\right)^{2}-2\dfrac{Q}{a_{0}}(t_{I}-t_{0})-1\right]=0.$
(4.257)
Interestingly, despite the equation (4.257) is satisfied for $t_{I}=t_{0}$,
i.e. if $a_{I}=a_{0}$, it also possesses two other solutions
$2\dfrac{Q}{a_{0}}(t_{I}-t_{0})=\varphi_{\pm},$ (4.258)
where $\varphi_{\pm}$ are the irrational constants
$\varphi_{\pm}=\dfrac{1\pm\sqrt{5}}{2},$ (4.259)
which in the case $\varphi_{+}=\varphi\approx 1.6180339887$ is the Fibonacci
golden ratio, and in the case $\varphi_{-}=1-\varphi=\varphi-\sqrt{5}$.
Employing the difference $t_{I}-t_{0}$ established via (4.258) within the
equation (4.255) one receives
$\eta_{I}-\eta_{0}=\dfrac{a_{0}}{Q}\dfrac{\varphi_{\pm}}{a_{0}+\varphi_{\pm}}.$
(4.260)
First let us consider the situation based on the cosmological time. When one
knows the value of the difference
$\tau=t_{I}-t_{0},$ (4.261)
the relation (4.258) can be used for determination of the constant $Q$
$Q=a_{0}\dfrac{\varphi_{\pm}}{2\tau}.$ (4.262)
Because of $Q=H_{I}a_{I}^{2}$ one has
$\dfrac{a_{I}^{2}}{a_{0}}=\dfrac{\varphi_{\pm}}{2H_{I}\tau},$ (4.263)
i.e. when one knows the initial value of the Hubble parameter $H_{I}$ and the
difference (4.261) then by the condition (4.245) one obtains
$a\ll\dfrac{\varphi_{\pm}}{2H_{I}\tau}.$ (4.264)
It is easy to see from (4.263) that the initial data of cosmic scale factor
parameter is given by the beginning value of this parameter
$a_{I}=\sqrt{\dfrac{\varphi_{\pm}}{2H_{I}\tau}a_{0}},$ (4.265)
what means that for consistency must be $a_{0}\neq 0$. The relations (4.263)
and (4.265) allow to derive the ratio
$\dfrac{a_{I}}{a_{0}}=\dfrac{\varphi_{\pm}}{2H_{I}\tau}\dfrac{1}{a_{I}}=\sqrt{\dfrac{\varphi_{\pm}}{2H_{I}\tau}\dfrac{1}{a_{0}}}$
(4.266)
and consequently the initial data of redshift $z_{I}=z(t_{0},t_{I})$
$\dfrac{a_{I}}{a_{0}}=\dfrac{1}{1+z_{I}},$ (4.267)
where
$z(t_{0},t)=\exp\left\\{\pm\dfrac{a(\eta_{0})}{a(t_{0})}\int_{t_{0}}^{t}dt^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}(t^{\prime})\right|}\right\\}-1,$
(4.268)
can be obtained straightforwardly
$z_{I}=\sqrt{\dfrac{2a_{0}}{\varphi_{\pm}}H_{I}\tau}-1,$ (4.269)
what after taking account that $a(\eta_{0})=a(t_{0})=a_{0}$ in the relation
(4.268) leads to the result
$\pm\int_{t_{0}}^{t_{I}}dt^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}(t^{\prime})\right|}=\dfrac{1}{2}\ln\left|\dfrac{2a_{0}}{\varphi_{\pm}}H_{I}(t_{I}-t_{0})\right|.$
(4.270)
After differentiating of both sides of the equation (4.270) with respect to
$t_{I}$ one obtains
$H_{I}=\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\left|\mathrm{H}_{M}(t_{I})\right|}=\dfrac{1}{2(t_{I}-t_{0})},$
(4.271)
what after straightforward integration
$\ln\dfrac{a_{I}}{a_{0}}=\dfrac{1}{2}\int_{0}^{t_{I}}\dfrac{dt^{\prime}}{t^{\prime}-t_{0}},$
(4.272)
on the one hand allows to establish the initial data of the cosmic scale
factor parameter
$a_{I}=a_{0}\sqrt{\dfrac{\tau}{t_{0}}},$ (4.273)
and on the other hand leads to the energy of Matter fields
$\epsilon_{M}(t_{I})=\dfrac{3M_{P}\ell_{P}^{2}}{4\tau^{2}}\dfrac{V}{V_{P}}.$
(4.274)
Taking into account the Planck sphere $V=V_{P}$ and $\tau=t_{P}$ one receives
$\epsilon_{M}(t_{I})=\dfrac{3}{4}E_{P}.$ (4.275)
Similarly for the Planck cube $V=\ell_{P}^{3}$ and $\tau=t_{P}$ one obtains
$\epsilon_{M}(t_{I})=\dfrac{9}{16\pi}E_{P}.$ (4.276)
Applying the formula (4.265) one receives noth the beginning value and the
initial data of the cosmic scale factor parameter
$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\dfrac{\varphi_{\pm}}{2H_{I}\tau}\dfrac{t_{0}}{\tau},$ (4.277)
$\displaystyle a_{I}$ $\displaystyle=$
$\displaystyle\dfrac{\varphi_{\pm}}{2H_{I}\tau}\sqrt{\dfrac{t_{0}}{\tau}}.$
(4.278)
Moreover, because of $Q=\pm\dfrac{V_{P}}{3V}\omega_{I}$ one can establish the
initial data
$\omega_{I}=\pm\dfrac{3}{2}a_{0}\varphi_{\pm}\dfrac{1}{\omega_{P}\tau}\dfrac{V}{V_{P}},$
(4.279)
which for the case of plus sign becomes
$\omega_{I}^{+}=\dfrac{3}{2}a_{0}\varphi\dfrac{1}{\omega_{P}\tau}\dfrac{V}{V_{P}},$
(4.280)
while for the case of the minus sign is
$\omega_{I}^{-}=\dfrac{3}{2}a_{0}(\varphi-1)\dfrac{1}{\omega_{P}\tau}\dfrac{V}{V_{P}}=\dfrac{\varphi-1}{\varphi}\omega_{I}^{+}.$
(4.281)
Interestingly, when one considers the Universe having volume of the Planck
cube $V=\ell_{P}^{3}$ and the time difference (4.261) equal to the Planck time
$\tau=t_{P}$
$t_{P}=\sqrt{\dfrac{\hslash G}{c^{5}}}\approx 5.39124\cdot 10^{-44}s,$ (4.282)
then one obtains approximatively
$\displaystyle\omega_{I}^{+}$ $\displaystyle=$
$\displaystyle\dfrac{9\varphi}{8\pi}a_{0},$ (4.283)
$\displaystyle\omega_{I}^{-}$ $\displaystyle=$
$\displaystyle\dfrac{9}{8\pi}(\varphi-1)a_{0}.$ (4.284)
Similarly, for the Planck sphere $V=\dfrac{4}{3}\pi\ell_{P}^{3}$ one receives
$\displaystyle\omega_{I}^{+}$ $\displaystyle=$
$\displaystyle\dfrac{3}{2}\varphi a_{0},$ (4.285)
$\displaystyle\omega_{I}^{-}$ $\displaystyle=$
$\displaystyle\dfrac{3}{2}(\varphi-1)a_{0}.$ (4.286)
Anyway, however, the relations (4.235) and (4.236) can be used to elimination
of the parameter $Q$ by the following combination
$\dfrac{a_{I}^{2}(t_{I},t_{0})-a_{0}^{2}}{2(t_{I}-t_{0})}=\dfrac{a_{I}(\eta_{I},\eta_{0})-a_{0}}{\eta_{I}-\eta_{0}},$
(4.287)
which after taking into account that
$a_{I}(t_{I},t_{0})=a_{I}(\eta_{I},\eta_{0})=a_{I}$ becomes
$\dfrac{a_{I}+a_{0}}{2}=\dfrac{t_{I}-t_{0}}{\eta_{I}-\eta_{0}},$ (4.288)
and leads to another relation between cosmological and conformal times
$\eta_{I}-\eta_{0}=\dfrac{2}{a_{I}+a_{0}}(t_{I}-t_{0}),$ (4.289)
which in the case $a_{0}=0$ becomes
$\eta_{I}-\eta_{0}=\dfrac{2}{a_{I}}(t_{I}-t_{0}).$ (4.290)
Applying the result (4.289) to the series (4.241) one obtains the rule
$\dfrac{1}{\sqrt{\pi}}=\dfrac{a_{I}+a_{0}}{2}\sum_{n=0}^{\infty}\dfrac{(2Q/a_{0})^{n}}{\Gamma(n+2)\Gamma\left(\dfrac{1}{2}-n\right)}(t_{I}-t_{0})^{n}.$
(4.291)
Let us consider the situation $\eta=\eta_{0}$ and $t=t_{0}$. Then the redshift
(4.78) and (4.79) must be redefined as follows
$\displaystyle z(\eta_{0},\eta)=\pm
a(\eta_{0})\int_{\eta_{0}}^{\eta}d\eta^{\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}|\mathrm{H}_{M}(\eta^{\prime})|},$
(4.292)
Let us denote by $z_{0}$ the value of the cosmological redshift at the
beginning of evolution of the classical Universe. Then by using of the
definition (4.292) one can establish
$z_{0}=z(t_{0},t_{0})\equiv 0,$ (4.293)
it is independent on a value of the cosmic scale factor parameter $a_{0}$ at
the beginning of evolution of the Universe. The relations (4.80) and (4.81)
computed for $\eta=\eta_{0}$ and $t=t_{0}$ are
$\displaystyle\eta_{I}-\eta_{0}$ $\displaystyle=$ $\displaystyle
t_{I}-t_{0}+\int_{a_{0}}^{a_{I}}\dfrac{a^{\prime}-1}{a^{\prime
2}}\dfrac{da^{\prime}}{H(a^{\prime})},$ (4.294) $\displaystyle
a_{0}(\eta_{I}-\eta_{0})$ $\displaystyle=$ $\displaystyle
t_{I}-t_{0}+\int_{0}^{z_{I}}\dfrac{z^{\prime}dz^{\prime}}{(1+z^{\prime})H(z^{\prime})},$
(4.295)
and by application of the result (4.289) become
$\displaystyle\left(\dfrac{2}{a_{I}+a_{0}}-1\right)(t_{I}-t_{0})$
$\displaystyle=$
$\displaystyle\int_{a_{0}}^{a_{I}}\dfrac{a^{\prime}-1}{a^{\prime
2}}\dfrac{da^{\prime}}{H(a^{\prime})},$ (4.296)
$\displaystyle\dfrac{a_{I}-a_{0}}{a_{I}+a_{0}}(t_{I}-t_{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{z_{I}}\dfrac{z^{\prime}dz^{\prime}}{(1+z^{\prime})H(z^{\prime})}.$
(4.297)
Interestingly, the LHS of the formula (4.296) vanishes identically when the
initial data $a_{I}$ and the beginning value $a_{0}$ of the cosmic scale
factor parameter are constrained by the equation
$a_{I}+a_{0}=2.$ (4.298)
If one expresses the integral on the RHS of (4.296) via time variables then
this equation says that
$0=\int_{t_{0}}^{t_{I}}dt^{\prime}-\int_{\eta_{0}}^{\eta_{I}}d\eta^{\prime},$
(4.299)
what leads to the equality between the differences
$\eta_{0}-\eta_{I}=t_{0}-t_{I},$ (4.300)
what is consistent with the relation (4.289). Such an equality means that if
the condition (4.298) holds then in such a region of evolution of the Universe
the conformal time and the cosmological time flow in such a way that their
difference is constant
$\eta-t=constans,$ (4.301)
and for convenience can be taken equal to zero. The equation (4.301) expresses
the law of conservation for the difference $\eta-t$. In such a situation the
second relation (4.297) becomes
$(1-a_{0})(t_{I}-t_{0})=\int_{0}^{z_{I}}\dfrac{z^{\prime}dz^{\prime}}{(1+z^{\prime})H(z^{\prime})},$
(4.302)
where the initial data of redshift is
$z_{I}=2\dfrac{a_{0}-1}{2-a_{0}}=2\dfrac{1-a_{I}}{a_{I}}.$ (4.303)
Interestingly, when $a_{0}=1$ then $a_{I}=1$ the initial data of redshift is
trivial $z_{I}=0$, and the LHS of the formula (4.302) identically vanishes.
After expression of the integral on the RHS of (4.302) via time variables one
obtains
$0=\int_{t_{0}}^{t_{I}}z(t_{0},t^{\prime})dt^{\prime}=\int_{t_{0}}^{t_{I}}\left(\dfrac{a(t_{0})}{a(t^{\prime})}-1\right)dt^{\prime}=a_{0}\int_{\eta_{0}}^{\eta_{I}}d\eta^{\prime}-\int_{t_{0}}^{t_{I}}dt^{\prime},$
(4.304)
what can be computed straightforwardly
$a_{0}(\eta_{I}-\eta_{0})=t_{I}-t_{0},$ (4.305)
and for $a_{0}=1$ leads once again to the law of conservation (4.300).
Let us see what happens in the particular situation for which $a_{0}=0$. First
let us establish the initial data of redshift $z_{I}$
$z_{I}=-1.$ (4.306)
Because of the constraint (4.298) one has $a_{I}=2$ and therefore the
relations (4.296) and (4.297) take the form
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{2}\dfrac{a^{\prime}-1}{a^{\prime
2}}\dfrac{da^{\prime}}{H(a^{\prime})},$ (4.307) $\displaystyle t_{0}-t_{I}$
$\displaystyle=$
$\displaystyle\int_{-1}^{0}\dfrac{z^{\prime}dz^{\prime}}{(1+z^{\prime})H(z^{\prime})}.$
(4.308)
Interestingly, in such a case the equation (4.254) is simplified
$\dfrac{Q}{2}(\eta_{I}-\eta_{0})^{2}-(t_{I}-t_{0})=0,$ (4.309)
and in the light of the relation (4.300) allows to establish the value of the
parameter $Q$ crucial for the initial data Hubble law (4.232)
$Q=2\dfrac{t_{I}-t_{0}}{(\eta_{I}-\eta_{0})^{2}}=\pm\dfrac{2}{\tau},$ (4.310)
where we have denoted $\tau=\eta_{I}-\eta_{0}=t_{I}-t_{0}$. In the light of
the definition (4.233) one can determine the initial data $m_{I}$
$\omega_{I}=\dfrac{6V}{V_{P}}\dfrac{1}{\omega_{P}\tau}.$ (4.311)
For finite volume of space $V$ and finite the time $\tau$ the initial data
$\omega_{I}$ are established consistently. The time $\tau$ can be interpreted
as the time between the beginning of the Universe and creation of initial
data. If its value is taken _ad hoc_ as identical to the Planck time
$\tau=t_{P}$ then
$\omega_{I}=\dfrac{6V}{V_{P}},$ (4.312)
and therefore if value of $\omega_{I}$ is small, $\omega_{I}\sim 1$ say, the
volume of space is
$V\sim\dfrac{V_{P}}{6}.$ (4.313)
Such a volume can be treated as definition of the early Multiverse.
In the light of the result (4.308) one obtains another definition of the time
$\tau$
$\tau=-\int_{-1}^{0}\dfrac{z^{\prime}dz^{\prime}}{(1+z^{\prime})H(z^{\prime})},$
(4.314)
which by using of the fact
$H(z^{\prime})=-\dfrac{1}{1+z^{\prime}}\dfrac{dz^{\prime}}{dt^{\prime}},$
(4.315)
and $z^{\prime}=z(t_{0},t^{\prime})$ becomes
$\tau=\int_{t_{0}}^{t_{I}}z(t_{0},t^{\prime})dt^{\prime},$ (4.316)
or after expressing via energy of Matter fields
$\tau=\int_{t_{0}}^{t_{I}}\left[\exp\left[\pm\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(t^{\prime\prime})}\right]-1\right]dt^{\prime}.$
(4.317)
finally results in the relation
$\tau=\dfrac{1}{2}\int_{t_{0}}^{t_{I}}\exp\left[\pm\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(t^{\prime\prime})}\right]dt^{\prime}.$
(4.318)
The natural generalization of the equation (4.318) is
$t_{2}-t_{1}=\dfrac{1}{2}\int_{t_{1}}^{t_{2}}\exp\left[\pm\int_{t_{1}}^{t^{\prime}}dt^{\prime\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(t^{\prime\prime})}\right]dt^{\prime},$
(4.319)
which in the case $t_{1}=0$, and $t_{2}=t$ can be used for determination of
the cosmological time
$t-t_{0}=\dfrac{1}{2}\int_{t_{0}}^{t}\exp\left[\pm\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(t^{\prime\prime})}\right]dt^{\prime},$
(4.320)
In the light of the relations (4.56) and (4.125) one has
$H(z^{\prime})=\sqrt{\dfrac{1}{M_{P}\ell_{P}^{2}}\dfrac{V_{P}}{3V}\epsilon_{M}(z^{\prime})},$
(4.321)
where $\epsilon_{M}(z^{\prime})$ is energy of Matter fields. For given energy
of Matter fields dependence on redshift $z^{\prime}$ the equation (4.321) can
be solved as a differential equation for the redshift as unknown function of
cosmological or conformal time. Application of the explicit form of
$\epsilon_{M}(z^{\prime})$ to the formula (4.314) shall be resulting in the
exact evaluation of the time $\tau$. Equivalently, the redshift derived from
the evolution (4.321) can be applied to the formula (4.316) for determination
of the time $\tau$.
Interestingly, one can apply the equation (4.321) to the generalized relation
(4.320)
$\displaystyle t-t_{0}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\int_{t_{0}}^{t}\exp\left[\pm\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\dfrac{1}{a(t^{\prime\prime})}\dfrac{da(t^{\prime\prime})}{dt^{\prime\prime}}\right]dt^{\prime}=$
(4.322) $\displaystyle=$
$\displaystyle\dfrac{1}{2}\int_{t_{0}}^{t}\exp\left[\pm\int_{a_{0}}^{a(t^{\prime})}\dfrac{da}{a}\right]dt^{\prime}=\dfrac{1}{2}\int_{t_{0}}^{t}\exp\left[\pm\ln\dfrac{a(t^{\prime})}{a_{0}}\right]dt^{\prime}=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\int_{t_{0}}^{t}\left(\dfrac{a(t^{\prime})}{a_{0}}\right)^{\pm
1}dt^{\prime},$
and it is evident that the case of the plus sign, i.e. an expanding Universe,
is distinguishable from the case of the minus sign, i.e. a collapsing
Universe. The case of plus sign gives
$\displaystyle(t-t_{0})^{+}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2a_{0}}\int_{t_{0}}^{t}a(t^{\prime})dt^{\prime}=\dfrac{1}{2a_{0}}\int_{t_{0}}^{t}a^{2}(t^{\prime})\dfrac{dt^{\prime}}{a(t^{\prime})}=$
(4.323) $\displaystyle=$
$\displaystyle\dfrac{1}{2a_{0}}\int_{\eta_{0}}^{\eta}a^{2}(\eta^{\prime})d\eta^{\prime},$
what after application of the definition of the Hubble parameter expressed in
terms of conformal time
$a^{2}(\eta)d\eta=\dfrac{da}{H(a)},$ (4.324)
allows to establish the result
$(t-t_{0})^{+}=\dfrac{1}{2a_{0}}\int_{a_{0}}^{a}\dfrac{da^{\prime}}{H(a^{\prime})}.$
(4.325)
The case of the minus sign leads to completely different result
$(t-t_{0})^{-}=\dfrac{a_{0}}{2}\int_{t_{0}}^{t}\dfrac{dt^{\prime}}{a(t^{\prime})}=\dfrac{a_{0}}{2}\int_{a_{0}}^{a}\dfrac{da^{\prime}}{a^{\prime
2}H(a^{\prime})},$ (4.326)
or after straightforward computation
$(t-t_{0})^{-}=\dfrac{a_{0}}{2}\int_{\eta_{0}}^{\eta}d\eta^{\prime}=\dfrac{a_{0}}{2}(\eta-\eta_{0}),$
(4.327)
which allows to determine the nontrivial relation between the cosmological and
the conformal time
$t=\dfrac{1}{2}a_{0}\eta.$ (4.328)
Because of the plus sign is related to expansion of the Multiverse and the
minus sign describes the opposite situation, the relations for the
cosmological time (4.323) and (4.326) mean that the flow of cosmological time
in an expanding Multiverse is distinguish then the flow in a collapsing
Multiverse. Strictly speaking the equation is satisfied
$a_{0}^{2}d(t-t_{0})^{+}=a^{2}d(t-t_{0})^{-},$ (4.329)
and if $t_{0}^{-}=t_{0}^{+}$ then one has the condition
$a_{0}^{2}dt^{+}-a^{2}dt^{-}=0.$ (4.330)
In other words the cosmological time flows in the same way if and only if for
arbitrary value of the conformal time or the cosmological time the relation
$a^{2}\equiv a_{0}^{2}$ holds identically. In such a situation the relation
(4.328) is automatically satisfied, while the value of the Hubble parameter is
trivial $H=\dfrac{1}{a^{2}}\dfrac{da}{d\eta}=0$. In this manner such an
approach suggests that the Multiverse has non dynamical nature or is dynamical
but considered in a fixed moment of time.
On the other side, however, it can be seen straightforwardly that application
of the trivial value $H=0$ of the Hubble parameter within the definitions
(4.325) and (4.326) of the cosmological time leads to manifestly divergent
integrands. Such a singular behavior suggests that the non dynamical
Multiverse has non physical nature or, in other words, that the fixation of
time has purely non physical nature. However, from the point of view of modern
physics fixation of time is the standard tool in the theoretical explanations
of quantum field theory.
There is another possible interpretation of the relations (4.329) and (4.330).
Namely, one can say that during the expansion of the Multiverse the beginning
value of the cosmic scale factor parameter is _ad hoc_ established to a
certain value $a_{0}$, while during a collapse of the Multiverse the beginning
value is undetermined. In such a situation the identification
$a^{2}=a_{0}^{2}$ means exactly that Multiverse is non expanding and is not
collapsing, i.e. is non dynamical .
#### F Summary
The Multiverse, which we understand as the collection of multiple quantum
universes, obtained via the quantization in the static Fock space of creators
and annihilators applied to the classical Einstein–Friedmann Universe, has
showed us the general way for the constructive scenario of the physics of the
observed Universe. By straightforward computations, involving mainly
elementary mathematical analysis, we have received the thermodynamics of the
system of many quantum universes, which in itself is the result possessing
both the most natural and strong value for phenomenology and empirical
verification of the theoretical results of the model. This elegant feature
allows to conclude that the proposed programme of construction of the model of
the quantum universe has been realized via using of the simple solution of the
Einstein field equations.
We have presented few important findings of such a formulation. These are
particularly:
1. 1.
Formulation of quantum cosmology via the models of theoretical physics having
well-established value for phenomenology.
2. 2.
Rational and simple scheme of natural emergence of observed Universe as the
system of multiple quantum universes - the Multiverse in our understanding.
3. 3.
Ideological unification of quantum cosmology with the most fundamental natural
sciences, like organic chemistry and evolutionary biology, in wider sense
expressing neglecting of interference of supernatural forces in creation and
development of Universe as misleading and groundless. Establishing of the
Multiverse hypothesis as the fundamental landscape for understanding of the
observed Universe.
4. 4.
Description of (very) early Universe as the static Multiverse of the
superfluid Fermi–Bose superstrings, which in general can be open or closed.
5. 5.
A conceptual way to understanding a physical role of quantum gravity, string
theory, and supersymmetry for birth and early evolution of the observed
Universe.
6. 6.
Essential cosmological role of superfluidity and light for the Multiverse
hypothesis.
To see the sense of the proposed strategy, let us sketch briefly the crucial
elements of the constructed model of quantum cosmology
1. 1.
Description of a certain selected solution of the Einstein field equations in
frames of the Hamiltonian approach jointing the Dirac approach and the
Arnowitt–Deser–Misner Hamiltonian formulation of General Relativity.
2. 2.
Application the received Hamiltonian constraint the methods of the primary and
the secondary quantization are applied, and the one-dimensional Dirac equation
is obtained.
3. 3.
Elementary formulation of the thermodynamics of quantum states of the selected
metric by application of the methods of statistical mechanics based on the
static Fock repère.
There is an obvious conjecture following from such a programme of quantum
cosmology. Namely, the proposed strategy of construction of quantum cosmology
can be generalized for another solutions of the Einstein field equations, and
in result the appropriate models of quantum gravity can be straightforwardly
obtained. In other words, the Multiverse hypothesis based on the one-
dimensional Dirac equation can be straightforwardly generalized onto all
solutions of the Einstein field equations which can be parametrized by
application of the Arnowitt–Deser–Misner decomposition. Such a strategy would
be resulting in the elegant transition between quantum cosmology and quantum
gravity. We shall present a certain idea for constructive realization of such
a general strategy in the next chapters of this book.
### Chapter 5 The Inflationary Multiverse
The quantum cosmology based on the one-dimensional Klein–Gordon equation can
be applied straightforwardly. Let us consider an application of the one-
dimensional quantum cosmology which leads to new results. It must be
emphasized that this chapter is rather far from the main stream of this part.
However, its content in itself is an essential linkage between the Multiverse
cosmology presented in this chapter and one of the most intriguing ideas of
the modern theoretical cosmology, which is inflation. In this section we shall
the particular situation within the general idea of inflation. This situation
is the inflation due to the Higgs inflaton.
#### A The Inflationary Cosmology
One of the main subjects of inflationary cosmology is the theory of
inflationary cosmological perturbations of quantum-mechanical origin (For
numerous details see e.g. the Ref. [163]). The idea to go beyond the isotropic
and homogeneous space-time given by the Friedmann–Lemaître–Robertson–Walker
metric, for which the interval expressed via the conformal time $\eta$ has the
form
$ds^{2}=a^{2}(\eta)\left(-c^{2}d\eta^{2}+\delta_{ij}dx^{i}dx^{j}\right).$
(5.1)
The question of is how small quantum perturbations around this solution of the
Einstein field equations behave during inflation, the phase of accelerated
expansion that took place in the early universe. It can be seen that the
corresponding physics is similar to the Schwinger effect[164]. In General
Relativity, inflation can be obtained by domination of a fluid which pressure
is negative. Since, at very high energies, quantum field theory is the natural
candidate to describe matter, it is natural and simple to postulate that a
scalar field, called _the inflaton_ was responsible for the evolution of the
universe in this regime. The action of Matter fields which is considered in
inflationary cosmology has the form
$\mathcal{S}=\dfrac{1}{c}\int_{M}d^{4}x\sqrt{-g}\left(\dfrac{E_{P}\ell_{P}}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+\dfrac{1}{\ell_{P}^{3}}V(\varphi)\right),$
(5.2)
where $\varphi$ is the inflaton field. It must be emphasized that the action
(5.2) of scalar field was complemented by the Planck units for dimensional
correctness of the action, such that the potential $V$ has a dimension of
energy. In the most general situation the interval of the perturbed
Friedmann–Lemaître–Robertson–Walker metric is [165]
$ds^{2}=a^{2}(\eta)\left(-c^{2}(1-2\phi)d\eta^{2}+2B_{,i}dx^{i}d\eta+\left[(1-2\psi)\delta_{ij}+2E_{,ij}+h_{ij}\right]dx^{i}dx^{j}\right),$
(5.3)
where the functions $\phi$, $B$, $\psi$ and $E$ represent the scalar sector
whereas the tensor $h_{ij}$, satisfying $h_{i}^{i}=h_{ij}^{;j}=0$ describes
the gravitational waves. There are no vector perturbations because a single
scalar field cannot seed rotational perturbations. At the linear level, the
two types of perturbations decouple and, therefore, can be treated separately.
In the case of scalar perturbations of the geometry evoked above, by freedom
to choose the coordinate system the four functions are in fact redundant, and
the scalar fluctuations of the geometry can be characterized by the gauge-
invariant Bardeen potential [166]
$\Phi_{B}=\phi+\dfrac{1}{a}[a(B-E^{\prime})]^{\prime},$ (5.4)
where prime means $\eta$-differentiation. The gauge-invariant perturbation
which characterizes the fluctuations in the inflaton scalar field is
$\delta\varphi^{g}(\eta,x)=\delta\varphi+\varphi^{\prime}(B-E^{\prime}).$
(5.5)
Because of the perturbed Einstein field equations couple the Bardeen potential
and the gauge-invariant perturbation one has one degree of freedom. Therefore
the scalar sector formalism is reduced to study of the Mukhanov–Sasaki
variable
$v(\eta,x)=a\sqrt{\dfrac{S_{P}}{\kappa
c}}\left[\delta\varphi^{g}+\varphi^{\prime}\dfrac{\Phi_{B}}{H}\right],$ (5.6)
where $S_{P}=4\pi\ell_{P}^{2}$ is the area of the Planck sphere, and
$H=\dfrac{a^{\prime}}{a}$ is the Hubble parameter. Usually inflationary
perturbations are formulated in terms of the variable
$\mu_{S}(\eta,x)=-\sqrt{\kappa\hslash}v(\eta,x)=-2a\sqrt{\gamma}\zeta(\eta,x),$
(5.7)
where $\zeta(\eta,x)$ is the conserved quantity
$\zeta=\dfrac{\mathcal{H}^{-1}\Phi_{B}^{\prime}+\Phi_{B}}{\dfrac{(\varphi^{\prime})^{2}}{2a^{2}}-\mathrm{V}(\varphi)}+\Phi_{B},$
(5.8)
and $\gamma$ is the background function
$\gamma=1-\dfrac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}.$ (5.9)
Here $\mathcal{H}=\dfrac{a^{\prime}}{a}$ is the conformal Hubble parameter,
which is connected with the Hubble parameter $H=\dfrac{\dot{a}}{a}$ by the
relation $\mathcal{H}=aH$. The automatically gauge-invariant tensor sector is
described by the quantity ${}_{T}(\eta,x)$ defined according by
$h_{ij}=\dfrac{\mu_{T}}{a}Q^{TT}_{ij},$ (5.10)
where $Q^{TT}_{ij}$ are the transverse and traceless eigentensors of the
Laplace operator on the space-like sections. Usually the perturbations are
studied mode by mode via using the Fourier transforms
$\widetilde{\mu}_{S}(\eta,k)$ and $\widetilde{\mu}_{T}(\eta,k)$
$\displaystyle\widetilde{\mu}_{S}(\eta,k)$ $\displaystyle=$ $\displaystyle\int
d^{3}x\mu_{S}(\eta,x)e^{-ikx},$ (5.11)
$\displaystyle\widetilde{\mu}_{T}(\eta,k)$ $\displaystyle=$ $\displaystyle\int
d^{3}x\mu_{T}(\eta,x)e^{-ikx},$ (5.12)
obeying the following Euler–Lagrange equations of motion
$\displaystyle\dfrac{d^{2}\widetilde{\mu}_{S}(\eta,k)}{d\eta^{2}}+\omega_{S}(k,\eta)\widetilde{\mu}_{S}(\eta,k)$
$\displaystyle=$ $\displaystyle 0,$ (5.13)
$\displaystyle\dfrac{d^{2}\widetilde{\mu}_{T}(\eta,k)}{d\eta^{2}}+\omega_{T}(k,\eta)\widetilde{\mu}_{T}(\eta,k)$
$\displaystyle=$ $\displaystyle 0,$ (5.14)
following from the second variation of the Einstein–Hilbert action. Here
$\omega_{S}(k,\eta)$ and $\omega_{T}(k,\eta)$ are the frequencies
$\displaystyle\omega^{2}_{S}(k,\eta)$ $\displaystyle=$ $\displaystyle
k^{2}c^{2}-\dfrac{(a\sqrt{\gamma})^{\prime\prime}}{a\sqrt{\gamma}},$ (5.15)
$\displaystyle\omega^{2}_{T}(k,\eta)$ $\displaystyle=$ $\displaystyle
k^{2}c^{2}-\dfrac{a^{\prime\prime}}{a}.$ (5.16)
The cosmological perturbations obey exactly the same type of equation as a
scalar field $\Phi(t,x)$ interacting with a classical electric field in the
Schwinger effect, i.e. the equation of a parametric oscillator
$\ddot{\widetilde{\Phi}}(t,k)+\omega^{2}(k,t)\widetilde{\Phi}(t,k)=0,$ (5.17)
where the frequency has the form
$\omega^{2}(k,t)=k^{2}c^{2}+\dfrac{m^{2}c^{4}}{\hslash^{2}}-2\dfrac{c^{2}}{\hslash}eEk_{z}t+\dfrac{1}{\hslash^{2}}e^{2}E^{2}t^{2},$
(5.18)
where $e$ is the elementary charge, and $E$ is the electric field. The only
difference is the classical source which, in the case of cosmological
perturbations, is the background gravitational field. Also the time dependence
of the frequencies is qualitatively different. The primary canonical
quantization of the theory proceeds as in the usual Schwinger effect. The
consequence of the interaction between the quantum cosmological perturbations
and the classical background is creation of particles, which in the context of
inflation are gravitons. Classically, this corresponds to the amplification
growing mode of the fluctuations.
Let us consider the slow-roll parameters [167]
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle
3\dfrac{\dfrac{M_{P}\ell_{P}}{2}\dot{\varphi}^{2}}{\dfrac{M_{P}\ell_{P}}{2}\dot{\varphi}^{2}+\dfrac{1}{\ell_{P}^{3}}\mathrm{V}(\varphi)}=-\dfrac{\dot{H}}{H^{2}},$
(5.19) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle-\dfrac{\ddot{\varphi}}{H\dot{\varphi}}=\epsilon-\dfrac{1}{2H}\dfrac{\dot{\epsilon}}{\epsilon},$
(5.20) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\dfrac{\dot{\epsilon}-\dot{\delta}}{H},$ (5.21)
obeying the equations of motion
$\displaystyle\dfrac{\dot{\epsilon}}{H}$ $\displaystyle=$ $\displaystyle
2\epsilon(\epsilon-\delta),$ (5.22) $\displaystyle\dfrac{\dot{\delta}}{H}$
$\displaystyle=$ $\displaystyle 2\epsilon(\epsilon-\delta)-\xi.$ (5.23)
It is convenient to express the slow-roll parameters via the horizon flow
functions $\epsilon_{1}$, $\epsilon_{2}$, and $\epsilon_{3}$
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle\epsilon_{1},$ (5.24)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\epsilon_{1}-\dfrac{1}{2}\epsilon_{2},$ (5.25)
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\epsilon_{2}\epsilon_{3}.$ (5.26)
It is easy to see that $\dfrac{\epsilon_{1}}{3}$ measures the ratio of of the
kinetic energy to the total energy, whereas $\epsilon_{2}$ represents a model
where the kinetic energy itself increases, when $\epsilon_{2}>0$, or decreases
when $\epsilon_{2}<0$ with respect to the total energy. Provided the slow-roll
conditions are valid $\epsilon_{1,2}\ll 1$ one can express the slow-roll
parameters via inflaton potential
$\displaystyle\epsilon_{1}$ $\displaystyle\simeq$
$\displaystyle\dfrac{1}{4S_{P}}\left(\dfrac{\mathrm{V}^{\prime}}{\mathrm{V}}\right)^{2},$
(5.27) $\displaystyle\epsilon_{2}$ $\displaystyle\simeq$
$\displaystyle\dfrac{1}{S_{P}}\left[\left(\dfrac{\mathrm{V}^{\prime}}{\mathrm{V}}\right)^{2}-\dfrac{\mathrm{V}^{\prime\prime}}{\mathrm{V}}\right],$
(5.28)
where prime denotes $\varphi$-differentiation, and $S_{P}=4\pi\ell_{P}^{2}$ is
the area of the Planck sphere. Derivation of the third horizon flow function
$\epsilon_{3}$ is rather tedious. Straightforward application of the
definitions (5.24), (5.25), and (5.20) to the definition (5.21), and taking
into account the definition (5.26) leads to the result
$\epsilon_{3}=2\epsilon_{1}\dfrac{\epsilon_{2}^{\prime}}{\epsilon_{1}^{\prime}}.$
(5.29)
while by using the definitions (5.35) and (5.36) one receives
$\dfrac{\epsilon_{2}^{\prime}}{\epsilon_{1}^{\prime}}=\left(1-\dfrac{1}{2}\dfrac{\dfrac{\mathrm{V}^{\prime\prime\prime}}{\mathrm{V}^{\prime}}-\dfrac{\mathrm{V}^{\prime\prime}}{\mathrm{V}}}{\dfrac{\mathrm{V}^{\prime\prime}}{\mathrm{V}}-\left(\dfrac{\mathrm{V}^{\prime}}{\mathrm{V}}\right)^{2}}\right)^{-1}.$
(5.30)
Therefore the final result can be presented in the compact form
$\epsilon_{3}\simeq\dfrac{1}{2S_{P}}\left(\dfrac{\mathrm{V}^{\prime}}{\mathrm{V}}\right)^{2}\left(1-\dfrac{1}{2}\dfrac{\dfrac{\mathrm{V}^{\prime\prime\prime}}{\mathrm{V}^{\prime}}-\dfrac{\mathrm{V}^{\prime\prime}}{\mathrm{V}}}{\dfrac{\mathrm{V}^{\prime\prime}}{\mathrm{V}}-\left(\dfrac{\mathrm{V}^{\prime}}{\mathrm{V}}\right)^{2}}\right)^{-1}.$
(5.31)
The frequencies of the gauge-invariant cosmological perturbations can be
written as
$\displaystyle\omega_{S}^{2}(k,\eta)$ $\displaystyle=$
$\displaystyle\omega_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+3\delta\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right],$
(5.32) $\displaystyle\omega_{T}^{2}(k,\eta)$ $\displaystyle=$
$\displaystyle\omega_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+3\epsilon\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right].$
(5.33)
#### B The Power Law Inflaton
Let us consider first the inflaton potential in the form of the power law
$\mathrm{V}(\varphi)=-\dfrac{\ell_{P}^{p}}{E_{P}}\dfrac{m^{2}c^{4}}{2}\varphi^{p},$
(5.34)
where $m$ is the mass of the power law inflaton $\varphi$.
In such a situation the horizon flow functions are easy to derive
$\displaystyle\epsilon_{1}$ $\displaystyle\simeq$
$\displaystyle\dfrac{1}{4S_{P}}\dfrac{p^{2}}{\varphi^{2}},$ (5.35)
$\displaystyle\epsilon_{2}$ $\displaystyle\simeq$
$\displaystyle\dfrac{1}{S_{P}}\dfrac{p}{\varphi^{2}},$ (5.36)
$\displaystyle\epsilon_{3}$ $\displaystyle\simeq$
$\displaystyle-\dfrac{1}{2S_{P}}\dfrac{p^{3}/(p+1)}{\varphi^{2}},$ (5.37)
so that the slow roll parameters have the form
$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\dfrac{1}{4S_{P}}\dfrac{p^{2}}{\varphi^{2}},$ (5.38)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\dfrac{1}{4S_{P}}\dfrac{p(p-2)}{\varphi^{2}},$ (5.39)
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2S_{P}^{2}}\dfrac{p^{4}/(p+1)}{\varphi^{4}}.$ (5.40)
In this manner one can establish straightforwardly the frequencies of the
scalar and the tensor cosmological perturbations
$\displaystyle\omega_{S}^{2}(k,\eta)$ $\displaystyle=$
$\displaystyle\omega_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+\dfrac{3}{4S_{P}}\dfrac{p(p-2)}{\varphi^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right],$
(5.41) $\displaystyle\omega_{T}^{2}(k,\eta)$ $\displaystyle=$
$\displaystyle\omega_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+\dfrac{3}{4S_{P}}\dfrac{p^{2}}{\varphi^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right].$
(5.42)
The problem is to establish the total frequency of the inflationary
cosmological perturbations of the power law inflaton. Let us postulate such an
effective frequency by the Pythagorean theorem
$\omega_{\textrm{eff}}^{2}=\omega_{S}^{2}(k,\eta)+\omega_{T}^{2}(k,\eta),$
(5.43)
which we shall call the _Pythagorean frequency_ , which has the following
explicit form
$\omega_{\textrm{eff}}=\sqrt{2\omega_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+\dfrac{3}{4S_{P}}\dfrac{p(p-1)}{\varphi^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right]}.$
(5.44)
One can consider the effective energy of the cosmological perturbations of the
power law inflaton. Let us propose _ad hoc_ that such an energy of the
inflationary cosmological perturbations is simply given by the Planck wave-
particle duality relation
$E_{\textrm{eff}}=\hslash\omega_{\textrm{eff}},$ (5.45)
so that applying the Pythagorean frequency (5.43) one obtains
$E_{\textrm{eff}}=\sqrt{2E_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+\dfrac{3}{4S_{P}}\dfrac{p(p-1)}{\varphi^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right]}.$
(5.46)
This chapter will be focused on discussion of several consequences for
possible physical meaning of the Higgs inflaton following from the energy
formula (5.46).
#### C The Higgs–Hubble Inflaton
Let us discuss first the nontrivial linkage of the power law inflaton with the
Multiverse model presented in the previous chapter of this part. For this let
us introduce the auxiliary field $\varphi=\varphi(\eta)$, which expressed in
terms of the Planck units is
$\varphi(\eta)=\varphi_{0}a(\eta),$ (5.47)
where $a(\eta)$ is the cosmic scale factor parameter, $\eta$ is the conformal
time describing the classical general relativistic evolution of the
Friedmann–Lemaître–Robertson–Walker metric, and
$\varphi_{0}=\dfrac{\alpha}{\ell_{P}}$ is the initial datum of the auxiliary
field $\varphi(\eta)$ (with $a(\eta_{0})=1$, and dimensionless $\alpha$).
The Einstein–Hilbert action of General Relativity evaluated on the
Friedmann–Lemaître–Robertson–Walker metric
$g_{\mu\nu}=\mathrm{diag}[-1,a^{2}(t)\delta_{ij}]$, which we established in
the previous section as (4.44), can be presented in the following form
$S[\varphi]=\int{d}\eta\left[M_{P}\ell_{P}V\dfrac{3}{2}\dfrac{\alpha^{2}\ell_{P}^{3}}{V_{P}}\varphi^{\prime
2}+\dfrac{\varphi^{4}}{\varphi_{0}^{4}}\epsilon_{M}(\eta)\right],$ (5.48)
where $V=\int d^{3}x<\infty$ is the spatial volume, $\eta$ is the conformal
time, $d\eta=\dfrac{dt}{a(t)}$, prime denotes $\eta$-differentiation,
$\epsilon_{M}(\eta)=\int d^{3}x\mathcal{H}_{M}(x,\eta)$ is the energy of
Matter fields. In this manner the most convenient choice of the constant
parameter $\alpha$ is
$\alpha=\dfrac{2}{3}\sqrt{\pi},$ (5.49)
so that the auxiliary field is
$\varphi(\eta)=\varphi_{0}a(\eta)\quad,\quad\varphi_{0}=\dfrac{2\sqrt{\pi}}{3\ell_{P}}\approx
7.3109596\cdot 10^{34}\dfrac{1}{\mathrm{m}},$ (5.50)
and the Einstein–Hilbert action becomes
$S[\varphi]=\int{d}\eta\left[\dfrac{M_{P}\ell_{P}}{2}V\varphi^{\prime
2}+\dfrac{\varphi^{4}}{\varphi_{0}^{4}}\epsilon_{M}(\eta)\right].$ (5.51)
Application of the conjugate momentum
$P_{\varphi}=\dfrac{1}{\ell_{P}^{2}}\dfrac{\delta
S[\varphi]}{\delta\varphi^{\prime}}=\dfrac{M_{P}V}{\ell_{P}}\varphi^{\prime},$
(5.52)
allows to present the action (5.51) in the Hamilton form by application of the
Legendre transformation
$S[\varphi]=\int
d\eta\left\\{\ell_{P}^{2}P_{\varphi}\varphi^{\prime}-H(\eta)\right\\},$ (5.53)
where $H(\eta)$ is the Hamiltonian
$H(\eta)=\dfrac{\ell_{P}^{3}}{V}\dfrac{P_{\varphi}^{2}}{2M_{P}}-\dfrac{\varphi^{4}}{\varphi^{4}_{0}}\epsilon_{M}(\eta)\approx
0,$ (5.54)
which vanishes automatically due to the Dirac method of canonical primary
quantization. The Hamiltonian constraint can be straightforwardly resolved in
the form
$P_{\varphi}=\pm\sqrt{\dfrac{V}{\ell_{P}^{3}}}\dfrac{\varphi^{2}}{\varphi^{2}_{0}}\sqrt{2M_{P}\epsilon_{M}(\eta)},$
(5.55)
which generates the solution in the form of the Hubble law
$\frac{\varphi(\eta)}{\varphi_{0}}=\frac{1}{1+z(\eta_{0},\eta)},$ (5.56)
where $z$ is the cosmological redshift
$z(\eta_{0},\eta)=\pm\frac{1}{\varphi_{0}}\int_{\eta_{0}}^{\eta}\sqrt{\dfrac{2\epsilon_{M}(\eta^{\prime})}{M_{P}\ell_{P}V}}d\eta^{\prime}.$
(5.57)
Application of the Dirac method of canonical primary quantization
$[\hat{P}_{\varphi},\varphi]=-i\dfrac{\hslash}{\ell_{P}^{2}},$ (5.58)
leads to the momentum operator
$\hat{P}_{\varphi}=-i\dfrac{\hslash}{\ell_{P}^{2}}\dfrac{d}{d\varphi},$ (5.59)
which applied to the Hamiltonian constraint, leads to the Wheeler–DeWitt
equation – the Klein–Gordon equation governing the Multiverse
$\left(\frac{d^{2}}{d\varphi^{2}}+\Omega_{\varphi}^{2}\right)\Psi(\varphi)=0,$
(5.60)
where $\Omega_{\varphi}$ is the frequency
$\Omega_{\varphi}=\ell_{P}\sqrt{\dfrac{2V}{\ell_{P}^{3}}}\dfrac{\varphi^{2}}{\varphi^{2}_{0}}\sqrt{\dfrac{\epsilon_{M}(\eta)}{E_{P}}}.$
(5.61)
The frequency (5.61) can be presented in the equivalent form
$\Omega_{\varphi}=\ell_{P}\sqrt{\dfrac{2V}{\ell_{P}^{3}}}\left(\dfrac{3E_{P}\mathrm{V}(\varphi)}{\sqrt{\pi}m^{2}c^{4}}\right)^{2/p}\sqrt{\dfrac{\epsilon_{M}(\eta)}{E_{P}}},$
(5.62)
which for the only $p=2$, i.e. for the case of the Higgs inflaton, becomes
proportional to the inflaton potential
$\mathrm{V}(\varphi)=\mathrm{V}_{H}(\varphi)=-\ell_{P}^{2}\dfrac{m^{2}c^{4}}{2}\varphi^{2}$.
In such a situation the meaning of the frequency (5.62) becomes much more
unambiguous. In other words such a Multiverse emerges due to _the Higgs–Hubble
inflaton_ , and by this reason we shall call it _the Higgs–Hubble Multiverse_.
In general the auxiliary field (5.47) satisfies the Hubble law (5.56), and
therefore we shall call it _the Hubble auxiliary field_ , what after
identification with the power law inflaton becomes _the Hubble inflaton_.
#### D The Chaotic Slow–Roll Inflation
Let us discuss in certain detail the Higgs–Hubble inflaton (5.47) in the
context of chaotic inflation (See e.g. the Ref. [168]). We shall work here in
frames of the standard scalar perturbation theory of the inflationary
cosmology in which a scalar field $\varphi$ in a curved space given by the
perturbed metric of the Friedmann–Lemaître–Robertson–Walker space-time.
The action of a scalar field $\sigma$ in an arbitrary curved space is
$S=\dfrac{1}{c}\int{d^{4}x}\sqrt{-g}\left(\dfrac{E_{P}\ell_{P}}{2}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma+\dfrac{1}{\ell_{P}^{3}}\mathrm{V}(\sigma)\right),$
(5.63)
and therefore $\varphi$ satisfies the Klein–Gordon equation
$\dfrac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\sigma\right)-\dfrac{2}{\ell_{P}^{4}E_{P}}\dfrac{d\mathrm{V}(\sigma)}{d\sigma}=0.$
(5.64)
If one performs the following perturbation
$\sigma(x,\eta)=\varphi(\eta)+\delta\varphi(x,\eta),$ (5.65)
then the Klein–Gordon equation for the unperturbated homogeneous scalar field
$\varphi=\varphi(\eta)$ takes the following form
$\varphi^{\prime\prime}+2\mathcal{H}\varphi^{\prime}+\dfrac{2a^{2}}{M_{P}\ell_{P}^{4}}\dfrac{d\mathrm{V}(\varphi)}{d\varphi}=0,$
(5.66)
where $a=a(\eta)$ is the cosmic scale factor parameter,
$\mathcal{H}=\dfrac{a^{\prime}(\eta)}{a(\eta)}$ is the conformal Hubble
parameter, and $\varphi_{0}=\dfrac{2\sqrt{\pi}}{3\ell_{P}}$ is the initial
datum of the Higgs–Hubble inflaton. The perturbation scalar field
$\delta\varphi(x,\eta)$ also possesses nontrivial dynamics which, however, we
shall not discuss here.
Our proposal is to interpret the Higgs–Hubble inflaton (5.47) as the
homogeneous unperturbated scalar field $\varphi$. In other words
$\varphi=\frac{\varphi_{0}}{1+z}=\varphi_{0}a.$ (5.67)
In such a situation one has
$\displaystyle\varphi^{\prime}$ $\displaystyle=$
$\displaystyle-\dfrac{z^{\prime}}{\varphi_{0}}\varphi^{2},$ (5.68)
$\displaystyle\varphi^{\prime\prime}$ $\displaystyle=$ $\displaystyle
2\left(\dfrac{z^{\prime}}{\varphi_{0}}\right)^{2}\varphi^{3},$ (5.69)
$\displaystyle a$ $\displaystyle=$
$\displaystyle\dfrac{\varphi}{\varphi_{0}},$ (5.70) $\displaystyle\mathcal{H}$
$\displaystyle=$ $\displaystyle\dfrac{\varphi^{\prime}}{\varphi},$ (5.71)
$\displaystyle\mathrm{V}(\varphi)$ $\displaystyle=$
$\displaystyle-\ell_{P}^{2}\dfrac{m^{2}c^{4}}{2E_{P}}\varphi^{2}.$ (5.72)
Therefore the Klein–Gordon equation (5.66) becomes
$\varphi^{\prime\prime}+2\dfrac{\varphi^{\prime
2}}{\varphi}-\dfrac{2}{\varphi_{0}^{2}}\left(\dfrac{mc^{2}}{\hslash}\right)^{2}\varphi^{3}=0,$
(5.73)
which after taking into account explicit form of the derivatives
$\varphi^{\prime}$ and $\varphi^{\prime\prime}$, and $\varphi\neq 0$, becomes
the differential equation for the cosmological redshift
$z^{\prime 2}-\dfrac{1}{2}\left(\dfrac{mc^{2}}{\hslash}\right)^{2}=0.$ (5.74)
The equation (5.74) can be solved straightforwardly
$z(\eta,\eta_{0})=\pm\dfrac{1}{\sqrt{2}}\dfrac{mc^{2}}{\hslash}(\eta-\eta_{0}).$
(5.75)
In this manner in general case the Higgs–Hubble inflaton has the form
$\varphi=\dfrac{\varphi_{0}}{1\pm\dfrac{1}{\sqrt{2}}\dfrac{mc^{2}}{\hslash}(\eta-\eta_{0})}.$
(5.76)
Because of in the light of the definition (5.57) one has
$z^{\prime}=\pm\dfrac{1}{\varphi_{0}}\sqrt{\dfrac{2\epsilon_{M}(\eta)}{2M_{P}\ell_{P}V}},$
(5.77)
what compared to the result of the equation (5.74)
$z^{\prime}=\pm\dfrac{1}{\sqrt{2}}\dfrac{mc^{2}}{\hslash},$ (5.78)
allows to establish the energy of Matter fields
$\epsilon_{M}(\eta)=\dfrac{4}{27}\pi^{2}\dfrac{V}{V_{P}}\dfrac{m^{2}c^{2}}{M_{P}},$
(5.79)
where $V_{P}=\dfrac{4}{3}\pi\ell_{P}^{3}$ is the volume of the Planck sphere.
Therefore the frequency (5.61) can be rewritten in the form
$\Omega_{\phi}=\sqrt{\dfrac{9}{8\pi}}V\dfrac{m}{M_{P}}\varphi^{2}.$ (5.80)
If one takes into account _ad hoc_ the relation
$\Omega_{\phi}=\ell_{P}\dfrac{\ell_{P}^{2}}{E_{P}^{2}}\dfrac{m^{2}c^{4}}{2}\varphi^{2},$
(5.81)
then one obtains the mass of the Higgs–Hubble inflaton
$m=\sqrt{2\pi}\dfrac{V}{V_{P}}M_{P}.$ (5.82)
Because of $\epsilon_{M}(\eta)$ is the energy of Matter fields one can write
_ad hoc_
$\epsilon_{M}(\eta)=\hslash\omega_{M}(\eta),$ (5.83)
where $\omega_{M}$ is the frequency field of the Matter fields, which in the
light of the formula (5.79) is
$\omega_{M}=\dfrac{4}{27}\pi^{2}\dfrac{V}{V_{P}}\left(\dfrac{mc^{2}}{\hslash}\right)^{2}\dfrac{1}{\omega_{P}},$
(5.84)
where $\omega_{P}=\dfrac{M_{P}c^{2}}{\hslash}$ is the Planck frequency. One
can suggest _ad hoc_ that $\omega_{M}=\omega_{P}$, and then the mass of the
Higgs–Hubble inflaton is
$m=\dfrac{3}{2\pi}\sqrt{\dfrac{3V_{P}}{V}}M_{P}.$ (5.85)
In such a situation comparison of the formulas (5.82) and (5.85) leads to the
following conclusion
$V=\dfrac{3}{2\pi}V_{P}=2\ell_{P}^{3}\approx 8.4483\cdot 10^{-105}m^{3}.$
(5.86)
In such a situation the Einstein–Hilbert action of the Einstein–Friedmann
Multiverse takes the form of the action of the one-dimensional
$\varphi^{4}$-theory evolving in the conformal time
$S[\varphi]=\dfrac{3}{2\pi}\int{d\eta}\left(\dfrac{M_{P}\ell_{P}V_{P}}{2}\varphi^{\prime
2}+\dfrac{3}{4}\ell_{P}^{4}\dfrac{m^{2}c^{2}}{M_{P}}\varphi^{4}\right).$
(5.87)
If one takes into account the usual action of a $\varphi^{4}$-theory expressed
in terms of the conformal time and complemented by the Planck units
$S[\varphi]=C\int{d\eta}\left(\dfrac{M_{P}\ell_{P}V_{P}}{2}\varphi^{\prime
2}-\ell_{P}^{4}\dfrac{g}{4!}\varphi^{4}\right),$ (5.88)
where $g$ is the coupling parameter and $C$ is a constant which has not
influence to the Euler–Lagrange equations of motion
$M_{P}V_{P}\varphi^{\prime\prime}+\ell_{P}^{3}\dfrac{g}{3!}\varphi^{3}=0,$
(5.89)
then one obtains the coupling parameter in the form
$g=-3\cdot 3!\dfrac{m^{2}c^{2}}{M_{P}}.$ (5.90)
In such a situation one can establish the beta function
$\beta(g)=\dfrac{dg}{d\ln m}$ (5.91)
which for the Higgs–Hubble inflaton is
$\beta(g)=m\dfrac{dg}{dm}=2g.$ (5.92)
Interestingly, this value of the beta function coincides with the asymptotics
$g\rightarrow\infty$ of $\beta(g)$ calculated from the duality relation for
the two-dimensional Ising model [169], where the dimension $D$ following from
the asymptotic relation $\beta(g)=Dg$ is $D=2$.
However, the situation is rather strange, because one one has to deal with the
dimension $1$. There is, however, different definition of the beta function
(See e.g. the Ref. [170])
$\beta(g)=\dfrac{dg}{d\ln m^{2}},$ (5.93)
which for the Higgs–Hubble inflaton is
$\beta(g)=m^{2}\dfrac{dg}{dm^{2}}=g.$ (5.94)
This value of $\beta(g)$ coincides with the asymptotics $g\rightarrow\infty$
of $\beta(g)$ in quantum electrodynamics. If one takes into account the
asymptotics $\beta(g)=Dg$, then $D=1$ coincides with the situation of the
Higgs–Hubble inflaton. In this manner the Einstein–Friedmann Universe as well
as the Higgs–Hubble inflaton obtained nontrivial physical meaning.
In our situation, however, the potential can be straightforwardly deduced from
the Einstein–Hilbert action (5.87) as
$V(\varphi,g,\ell_{P})=\ell_{P}^{4}\dfrac{g(\ell_{P})}{4!}\varphi^{4},$ (5.95)
where the coupling parameter is
$g(\ell_{P})=3\cdot 3!\dfrac{m^{2}c^{3}}{\hslash}\ell_{P}.$ (5.96)
Taking into account the scaling
$\varphi\rightarrow\varphi_{\lambda}=\dfrac{\varphi}{\lambda},$ (5.97)
for $\lambda=\ell_{P}$ one receives the property
$V(\varphi,g(\lambda),\lambda)=\hat{V}\left(\varphi_{\lambda},g(\lambda)\right),$
(5.98)
where the scaled potential is
$\hat{V}\left(\varphi_{\lambda},g(\lambda)\right)=\dfrac{g(\lambda)}{4!}\varphi^{4}.$
(5.99)
In general situation one can compute the Callan–Symanzik beta function by
application of the Callan–Symanzik function $\psi(g)$
$\dfrac{d\ln g}{d\ln\lambda}=\psi(g)=\dfrac{\beta(g)}{g}.$ (5.100)
It is easy to see that in our case $\ln g=\ln\lambda+\ln C$, where $C$ is
certain constant, and by this reason
$\psi(g)=1.$ (5.101)
In this manner one can establish the Callan–Symanzik beta function of the
Higgs–Hubble inflaton as
$\beta(g)=g.$ (5.102)
It is manifestly seen that this beta function coincides with the beta function
(5.94) obtained from the asymptotics $g\rightarrow\infty$ of quantum
electrodynamics, i.e. corresponds with the dimension $1$. The problem is to
construct the appropriate renormalization group equation. Its construction can
be performed by deformation of the usual renormalization group equation
$\left(\lambda\dfrac{\partial}{\partial\lambda}-\beta(g)\dfrac{\partial}{\partial{g}}+\mu\right)V(\varphi,g,\lambda)=0,$
(5.103)
where $\mu$ is some deformation parameter which can be established by
straightforward computation. Calculating the derivatives
$\displaystyle\dfrac{\partial}{\partial\lambda}V(\varphi,g,\lambda)$
$\displaystyle=$ $\displaystyle\dfrac{4}{\lambda}V(\varphi,g,\lambda),$
(5.104) $\displaystyle\dfrac{\partial}{\partial{g}}V(\varphi,g,\lambda)$
$\displaystyle=$ $\displaystyle\dfrac{1}{g}V(\varphi,g,\lambda),$ (5.105)
and taking into account the Callan–Symanzik beta function of the Higgs–Hubble
inflaton (5.102) established above, one obtains the following value of the
deformation parameter
$\mu=3.$ (5.106)
Therefore, the renormalization group equation of the Higgs–Hubble inflaton has
the form
$\left(\lambda\dfrac{\partial}{\partial\lambda}-\beta(g)\dfrac{\partial}{\partial{g}}+3\right)V(\varphi,g,\lambda)=0.$
(5.107)
#### E The Phononic Hubble Inflaton
Let us see in some detail what happens in the Hubble Multiverse and
Higgs–Hubble Multiverse, i.e. the Multiverse generated by the Hubble inflaton
and the Higgs–Hubble inflaton, respectively. Applying the explicit form of the
inflaton (5.47) to the inflaton energy (5.46) one can straightforwardly
express the inflaton energy in terms of the cosmic scale factor parameter. The
result is as follows
$E_{H}\equiv{E}_{\textrm{eff}}=\sqrt{2E_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+\dfrac{27}{64\pi^{2}}\dfrac{p(p-1)}{a^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right]}.$
(5.108)
First let us analyse this formula from the point of view of Special
Relativity, i.e. the Einstein energy-momentum relation
$E=\sqrt{p^{2}c^{2}+m^{2}c^{4}},$ (5.109)
and the wave-particle duality, i.e. the Planck–Einstein relations
$\displaystyle E$ $\displaystyle=$ $\displaystyle\hslash\omega,$ (5.110)
$\displaystyle p$ $\displaystyle=$ $\displaystyle\hslash k.$ (5.111)
It can be seen by direct computation from such a point of view the equation
(5.108) describes the particle-universe possessing the following values of
momentum and mass
$\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{2}\hslash k,$ (5.112)
$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle-2M_{P}^{2}\left(2+\dfrac{27}{64\pi^{2}}\dfrac{p(p-1)}{a^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}$
(5.113)
Because of the squared mass is manifestly negative, one has to deal with
_tachyon_ equipped with the mass
$m_{t}=im.$ (5.114)
Equivalently, the equation (5.113) can be understood as the expression for the
cosmic scale factor parameter via the mass of a particle, i.e.
$a^{2}(\eta)=\dfrac{27}{128\pi^{2}}\dfrac{p(p-1)}{\left(\dfrac{m_{t}c^{2}}{2\hslash}\eta\right)^{2}-1}.$
(5.115)
In such a context the Hubble inflaton becomes bosonic equipped with negative
squared mass. Applying the simple identification $E_{H}=\hslash\omega_{H}$
$\omega_{H}=\sqrt{2\omega_{P}^{2}\left[\ell_{P}^{2}k^{2}-\left(2+\dfrac{27}{64\pi^{2}}\dfrac{p(p-1)}{a^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}\right]},$
(5.116)
one can derive straightforwardly the group velocity
$v_{g}=\dfrac{d\omega_{H}}{dk}$ of the particle-universe
$v_{g}=\frac{c}{\sqrt{1+\left(\dfrac{mc}{p}\right)^{2}}}=\frac{c}{\sqrt{1-\dfrac{1}{(k\ell_{P}a)^{2}}\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}}},$
(5.117)
and similarly the phase velocity $v_{ph}=\dfrac{\omega_{H}}{k}$ can be
obtained
$v_{ph}=c\sqrt{1+\left(\frac{mc}{p}\right)^{2}}=c\sqrt{1-\dfrac{1}{(k\ell_{P}a)^{2}}\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}}.$
(5.118)
On the other hand, however, the inflaton energy (5.108) can be analyzed by the
point of view of phonons, i.e. quanta of sound in solids. In this context
application of the dispersion relation for phonons
$\omega_{k}=\sqrt{2\omega^{2}(k)(1-\cos(k\ell_{P}a))},$ (5.119)
leads to the identification
$\omega=\omega_{P}k\ell_{P},$ (5.120)
and the cosmic scale factor parameter $a$ becomes the lattice spacing. Then
the Hubble inflaton becomes phononic, and straightforward comparison of the
relations (5.108) and (5.119) leads to the following non-algebraic equation
$(k\ell_{P}a)^{2}\cos(k\ell_{P}a)=\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}},$
(5.121)
where for known value of $\eta$ the unknown is the lattice spacing $a$.
Solutions of the equation (5.121) can be found by the only numerical way. Let
us apply the principles of quantum solid state physics (For basics, advances,
and applications see e.g. the Ref. [171]).
According to the quantization rule for phonons, which we shall call _the
phononic quantization_ , the wave vector and the lattice spacing are
nontrivially jointed by the relation
$k\ell_{P}a=\frac{n}{N}\pi,$ (5.122)
where $N$ is a number of identical atoms, $n=0,\pm 1,\ldots,\pm N$, i.e. the
product $k\ell_{P}a$ takes integer values in the range
$\left[-\dfrac{\pi}{N},\dfrac{\pi}{N}\right]$. These integers, however, can
not be chosen arbitrary, because of they must solve the non algebraic equation
(5.121). Possibly there is no any integer solution of this equation. In such a
situation one must reinterpret the equation (5.121) as the equation for the
conformal time $\eta$ while the product $k\ell_{P}a$ is determined via the
phononic quantization (5.122).
Similarly as in the case of the particle-universe, one can derive
straightforwardly the group velocity $v_{g}=\dfrac{d\omega_{k}}{dk}$ and the
phase velocity $v_{ph}=\dfrac{\omega_{k}}{k}$ of the phonon-universe
$\displaystyle v_{g}$ $\displaystyle=$ $\displaystyle
v_{ph}\left(2+k\ell_{P}a\sqrt{1+\cos(k\ell_{P}a)}\right),$ (5.123)
$\displaystyle v_{ph}$ $\displaystyle=$ $\displaystyle
c\sqrt{1-\cos(k\ell_{P}a)},$ (5.124)
where
$\cos(k\ell_{P}a)=\dfrac{1}{k^{2}\ell_{P}^{2}}\left(2+\dfrac{27}{64\pi^{2}}\dfrac{p(p-1)}{a^{2}}\right)\dfrac{t_{P}^{2}}{\eta^{2}}.$
(5.125)
Applying the constraint (5.121) within the formulas (5.123) and (5.124) one
receives
$\displaystyle v_{ph}$ $\displaystyle=$ $\displaystyle
c\sqrt{1-\dfrac{1}{(k\ell_{P}a)^{2}}\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}},$
(5.126) $\displaystyle v_{g}$ $\displaystyle=$ $\displaystyle
v_{ph}\left(2+\sqrt{(k\ell_{P}a)^{2}+\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}}\right).$
(5.127)
Using the phononic quantization (5.122) result in the quantization of the
velocities
$\displaystyle v_{ph}$ $\displaystyle=$ $\displaystyle
c\sqrt{1-\left(\dfrac{N}{\pi{n}}\right)^{2}\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}},$
(5.128) $\displaystyle v_{g}$ $\displaystyle=$ $\displaystyle
v_{ph}\left(2+\sqrt{\left(\pi\dfrac{{n}}{N}\right)^{2}+\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}}\right).$
(5.129)
The phase velocity, however, is real if and only if
$k\ell_{P}a\geqslant\sqrt{2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)}\dfrac{t_{P}}{\eta},$
(5.130)
and therefore the quantization is not arbitrary, but restricted by the
inequality
$n\geqslant\dfrac{N}{\pi}\sqrt{2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)}\dfrac{t_{P}}{\eta}.$
(5.131)
It can be seen straightforwardly that also the following restriction holds
$n\geqslant\dfrac{N}{\sqrt{2}\pi}\dfrac{v_{g}-2v_{ph}}{v_{ph}}.$ (5.132)
In the limit situation
$k\ell_{P}a=\sqrt{2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)}\dfrac{t_{P}}{\eta}$ both
the group velocity and the phase velocity vanish identically, and by the
constraint (5.121) such a situation corresponds to the equation
$\cos\left[\sqrt{2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)}\dfrac{t_{P}}{\eta}\right]=1,$
(5.133)
or in other words with the following quantization of the cosmic scale factor
parameter
$a_{n}=\sqrt{\dfrac{1}{2}\left(2\pi
n\omega_{P}\eta\right)^{2}-\dfrac{27}{128\pi^{2}}p(p-1)},$ (5.134)
where $n\in\mathbf{Z}$. Interestingly, $a_{n}\equiv 0$, i.e. the Multiverse
evolution starts, if and only if the conformal time is quantized as follows
$\eta_{n}=\dfrac{3t_{P}}{8\pi^{2}}\dfrac{\sqrt{3p(p-1)}}{2n}.$ (5.135)
This situation means that such a Multiverse is cyclic, i.e. its evolution
begins few times.
It can be seen straightforwardly, however, that the phase velocities of the
particle and the phonon are identical, while the group velocities are
blatantly different. For full _the bosonic-phononic duality_ of the Hubble
inflaton, the most natural way is to put _ad hoc_ the equality between the
group velocities of the bosonic universe and the phononic universe. We shall
call _bonons_ the Hubble inflatons satisfying the bononic duality. This type
of duality is obviously nontrivial, because of in fact establishes the duality
between sound (phonons) and matter (bosons). In this manner bonons are the
Hubble inflatons following from _the matter-sound duality_.
It is easy to see that such a duality condition can be presented as the non-
algebraic equation
$\displaystyle(k\ell_{P}a)^{2}\cos^{3}(k\ell_{P}a)-(k\ell_{P}a)^{2}\cos^{2}(k\ell_{P}a)-(k\ell_{P}a)^{2}\cos(k\ell_{P}a)-$
$\displaystyle 2\cos(k\ell_{P}a)+(k\ell_{P}a)^{2}+1=0,$ (5.136)
which with using of the constraint (5.121) can be presented as the algebraic
equation of degree $3$
$x^{3}+(1-f)x^{2}-f(f+2)x+f^{3}=0,$ (5.137)
where we have introduced the notation
$\displaystyle x$ $\displaystyle=$ $\displaystyle(k\ell_{P}a)^{2}\geqslant 0,$
(5.138) $\displaystyle f$ $\displaystyle=$
$\displaystyle\left(2a^{2}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}.$
(5.139)
If one wishes to use the phononic quantization (5.122) then the equation
(5.137) can be used to determination of the quantization of the cosmic scale
factor parameter. This quantization can be obtained as the solution of the
equation
$f^{3}_{n}-x_{n}f^{2}_{n}-x_{n}(x_{n}+2)f_{n}+x_{n}^{2}+x_{n}^{3}=0,$ (5.140)
where
$\displaystyle x_{n}$ $\displaystyle=$
$\displaystyle\left(\pi\dfrac{n}{N}\right)^{2}\geqslant 0,$ (5.141)
$\displaystyle f_{n}$ $\displaystyle=$
$\displaystyle\left(2a^{2}_{n}+\dfrac{27}{64\pi^{2}}p(p-1)\right)\dfrac{t_{P}^{2}}{\eta^{2}}.$
(5.142)
The equation (5.140) does not possess real roots. It means the bononic duality
has non physical nature..
Let us consider, however, seriously the phononic Hubble inflaton.
Interestingly, after application of the phonon quantization, the inflaton
energy (5.108) in general is nontrivially quantized
$E_{H}^{(n)}=\sqrt{2\left(\pi\dfrac{n}{N}\right)^{2}\left[1-\cos\left(\pi\dfrac{n}{N}\right)\right]}\dfrac{E_{P}}{a},$
(5.143)
where $E_{P}$ is the Planck energy. Interestingly, in the most general
situation this inflaton energy is even function with respect to $n$, and
behaves as $E_{H}\sim\dfrac{1}{a}$. Therefore the total energy of the Hubble
Multiverse
$E_{H}^{\mathrm{TOT}}(a)=\int_{a_{I}}^{a}da^{\prime}E_{H}^{(n)}(a^{\prime})=\sqrt{2\left(\pi\dfrac{n}{N}\right)^{2}\left[1-\cos\left(\pi\dfrac{n}{N}\right)\right]}E_{P}\ln\dfrac{a}{a_{I}},$
(5.144)
has divergent behavior in the limit $a\rightarrow\infty$
$\lim_{a\rightarrow\infty}E_{H}^{\mathrm{TOT}}(a)=\infty.$ (5.145)
Another interesting quantity is the inflaton energy summarized with respect to
the number excitations $n$. The general formula can be deduced as follows
$\left\langle E_{H}\right\rangle_{N}=2\sum_{n=0}^{N}E_{H}^{(n)},$ (5.146)
where the multiplier $2$ follows from inclusion of the states with negative
$n$. In this manner one can establish the mean inflaton energy in the
Multiverse. The result van be presented in the form
$\left\langle E_{H}\right\rangle_{N}(a)=\Lambda_{N}\dfrac{E_{P}}{a},$ (5.147)
where $\Lambda_{N}$ is given by the formula
$\Lambda_{N}=2\sum_{n=0}^{N}\sqrt{2\left(\pi\dfrac{n}{N}\right)^{2}\left[1-\cos\left(\pi\dfrac{n}{N}\right)\right]}.$
(5.148)
For consistency one can also consider the mean inflaton energy averaged over
values of the cosmic scale factor parameter $a$
$\overline{\left\langle
E_{H}\right\rangle_{N}}=\dfrac{1}{a-a_{I}}\int_{a_{I}}^{a}da^{\prime}\left\langle
E_{H}\right\rangle_{N}(a^{\prime})=E_{P}\Lambda_{N}\dfrac{\ln\dfrac{a}{a_{I}}}{a-a_{I}},$
(5.149)
which gives the physical interpretation of the constant $\Lambda_{N}$
$\Lambda_{N}=\lim_{a\rightarrow a_{I}}\dfrac{\overline{\left\langle
E_{H}\right\rangle_{N}}}{E_{P}}.$ (5.150)
Interestingly, when the Multiverse becomes infinite, i.e.
$a\rightarrow\infty$, then the mean inflaton energy of the Hubble inflaton
averaged over $a$ (5.149) tends to zero.
#### F The Inflaton Constant
The question is, however, the convergence of quantity $\Lambda_{N}$ for the
huge $N$ limit, $\Lambda_{\infty}$, i.e. when the Multiverse is full of the
inflatons, for which
$\left\langle E_{H}\right\rangle_{\infty}=\Lambda_{\infty}\dfrac{E_{P}}{a},$
(5.151)
where formally
$\Lambda_{\infty}=\lim_{N\rightarrow\infty}\Lambda_{N}.$ (5.152)
In the other words, the problem is the value of $\Lambda_{\infty}$ and whether
$\Lambda_{\infty}$ is an universal constant. One can identify the identical
atoms with spatial dimensions $N\equiv D$, and treat the Multiverse model
presented above as the multidimensional Universe. Another interpretation is
that $N$ is a number of ”atoms of space”, and then the limit
$N\rightarrow\infty$ defines the classical space, i.e. the space which is a
solid-medium of the atoms - _the phononic Hubble inflatons_. The most stable
mean inflaton energy is obtained for infinite number of the identical atoms
$N=\infty$, i.e. when the solid-medium is the Æther model . Let us call
$\Lambda_{N}$ _the inflaton N-atomic constant_ , and $\Lambda_{\infty}$ _the
inflaton constant_.
The inflaton constant is useful. One can consider the frequency $\omega_{I}$
$\omega_{I}=\Lambda_{\infty}\omega_{P},$ (5.153)
where $\omega_{P}=E_{P}/\hslash$ is the Planck frequency, which can be
interpreted as the Zero-Point Frequency field. This leads to the
characteristic time of the inflation
$t_{I}=\dfrac{2\pi}{\omega_{I}}=\dfrac{2\pi}{\Lambda_{\infty}}t_{P},$ (5.154)
where $t_{P}$ is the Planck time. If one defines the cosmological potential
energy $V_{C}(x_{C})=\left\langle E_{H}\right\rangle_{\infty}$, where
$x_{C}=\ell_{P}a$ is the cosmological coordinate then one can determine the
force
$F_{H}=-\dfrac{dV_{C}(x_{C})}{dx_{C}}=-\dfrac{\Lambda_{\infty}\hslash{c}}{x_{C}^{2}}=-\dfrac{\Lambda_{\infty}}{4\pi}\dfrac{8}{\kappa}\left(\dfrac{p_{C}c}{E_{P}}\right)^{2}=-\dfrac{G\Lambda_{\infty}{M}_{P}^{2}}{x_{C}^{2}},$
(5.155)
where $p_{C}=h/x_{C}$ is De Broglie cosmological momentum and
$\kappa=8\pi\ell_{P}/E_{P}=8\pi G/c^{4}$ is the Einstein constant. This force
defines the Newton law of universal gravitation for
$\Lambda_{\infty}M_{P}^{2}=m_{1}m_{2}$, where $m_{1,2}$ are masses of two
interacting bodies.
Let us try to determine the value of the inflaton constant
$\Lambda_{N}=2\sum_{n=0}^{N}\sqrt{2\left(\pi\dfrac{n}{N}\right)^{2}\left[1-\cos\left(\pi\dfrac{n}{N}\right)\right]}.$
(5.156)
One can change this sum by introduction of the index
$k=\dfrac{n}{N}=\left\\{0,1\right\\}$. Then
$\Lambda_{N}=2\sum_{k=0}^{1}\sqrt{2\left(\pi k\right)^{2}\left[1-\cos\left(\pi
k\right)\right]},$ (5.157)
what is easy to establish straightforwardly
$\Lambda_{N}=4\pi,$ (5.158)
and is independent on $N$. In this manner the N-atomic inflaton constant is
the same as the inflaton constant. This result allows to write out the energy
$\left\langle E_{H}\right\rangle_{N}=\left\langle
E_{H}\right\rangle_{\infty}=4\pi\dfrac{E_{P}}{a}.$ (5.159)
By this reason one can evaluate the Zero-Point Frequency field (5.153), the
characteristic time of the inflation (5.154) and the Newton law
$\displaystyle\omega_{I}$ $\displaystyle=$ $\displaystyle
4\pi\omega_{P}\approx 2.3308857\cdot 10^{44}\mathrm{Hz},$ (5.160)
$\displaystyle t_{I}$ $\displaystyle=$ $\displaystyle t_{P}/2\approx
2.9656213\cdot 10^{-44}\mathrm{s},$ (5.161) $\displaystyle F_{H}$
$\displaystyle=$
$\displaystyle-\dfrac{32\pi^{2}}{\kappa}\left(\dfrac{p_{C}c}{E_{P}}\right)^{2}=-\dfrac{4\pi
G{M}_{P}^{2}}{x_{C}^{2}}.$ (5.162)
### Chapter 6 Review of Quantum General Relativity
In this chapter we shall present certain standard strategy having a basic
status for quantum General Relativity. Namely, these are the $3+1$
Arnowitt–Deser–Misner Hamiltonian formulation of General Relativity and the
Dirac method of canonical primary quantization which leads to the
Wheeler–DeWitt equation and the concept of the Wheeler superspace.
#### A 3+1 Splitting of General Relativity
Let us consider a four-dimensional pseudo-Riemannian manifold $(M,g)$ (For
differential geometric details see _e.g._ Refs. [144, 145, 238, 239]) equipped
with the 4-volume form $g=\det{g_{\mu\nu}}$ related to a metric tensor
$g_{\mu\nu}$ of signature $(1,3)$, the Christoffel symbols
$\Gamma^{\rho}_{\mu\nu}$, the Riemann–Christoffel curvature tensor
$R^{\lambda}_{\mu\alpha\nu}$, the Ricci curvature tensor $R_{\mu\nu}$, and the
Ricci scalar curvature ${{}^{(4)}}\\!R$
$\displaystyle\Gamma^{\rho}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}g^{\rho\sigma}\left(g_{\mu\sigma,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma}\right),$
(6.1) $\displaystyle R^{\lambda}_{\mu\alpha\nu}$ $\displaystyle=$
$\displaystyle\Gamma^{\lambda}_{\mu\nu,\alpha}-\Gamma^{\lambda}_{\mu\alpha,\nu}+\Gamma^{\lambda}_{\sigma\alpha}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\sigma\nu}\Gamma^{\sigma}_{\mu\alpha},$
(6.2) $\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R^{\lambda}_{\mu\lambda\nu}=\Gamma^{\lambda}_{\mu\nu,\lambda}-\Gamma^{\lambda}_{\mu\lambda,\nu}+\Gamma^{\lambda}_{\sigma\lambda}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\sigma\nu}\Gamma^{\sigma}_{\mu\lambda},$
(6.3) $\displaystyle{{}^{(4)}}\\!R$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}R_{\mu\nu},$ (6.4)
where a holonomic basis [29] was chosen. In General Relativity (For much more
detailed books in its basics and applications see e.g. the Refs. [29, 154,
155]) the manifold $M$ is identified with space-time, and presence of Matter
fields reflected by nonzero stress-energy tensor111Some authors call
$T_{\mu\nu}$ the energy-momentum tensor. $T_{\mu\nu}$ is then studied. In such
a situation the Einstein tensor
$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{{}^{(4)}}\\!R,$ (6.5)
allows to construct the Einstein field equations
$G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa\ell_{P}^{2}T_{\mu\nu},$ (6.6)
where $\kappa=\dfrac{8\pi G}{c^{4}}\approx 2.076\cdot
10^{-43}\leavevmode\nobreak\ \mathrm{N}^{-1}$ is the Einstein constant, and
$\Lambda$ is the cosmological constant. The constant
$\kappa\ell_{P}^{2}=\dfrac{6V_{P}}{E_{P}}=\dfrac{6}{\varrho_{P}c^{2}}$ up to
the constant multiplier is reciprocal of the Planck energy $E_{P}$ density
$\varrho_{P}=\dfrac{E_{P}}{V_{P}}$ in the volume
$V_{P}=\dfrac{4}{3}\pi\ell_{P}^{3}$ of the Planck sphere. This constant has
the value
$\kappa\ell_{P}^{2}\approx 5.424746\cdot
10^{-129}\dfrac{\mathrm{m}^{3}}{\mathrm{J}},$ (6.7)
so that its reciprocal has the value
$\dfrac{1}{\kappa\ell_{P}^{2}}=\dfrac{\varrho_{P}c^{2}}{6}\approx
1.843404\cdot 10^{128}\leavevmode\nobreak\
\dfrac{\mathrm{J}}{\mathrm{m}^{3}}.$ (6.8)
To construct the Hamilton formulation of General Relativity it is necessary to
foliate a space-time manifold $M$ with a family of space-like hypersurfaces,
called also _slices_. It is possible when $M$ is globally hyperbolic, i.e.
pseudo-Riemannian, manifold what is the usual situation in General Relativity.
Let $t(x^{\mu})$ be a scalar field, an arbitrary single-valued function of
coordinates $x^{\mu}$, such that the foliation $t=constans$ corresponds with a
family of nonintersecting space-like hypersurfaces $\Sigma(t)$. Let us denote
by $y^{i}$ the coordinates on all hypersurfaces $\Sigma(t)$. Let us choose a
concrete hypersurface $\Sigma$ defined by a parametric equations
$x^{\mu}=x^{\mu}(y^{i})$, where $i=1,\ldots,3$ indexes coordinates intrinsic
to $\Sigma$. Equivalently, hypersurface $\Sigma$ can be selected by any
restriction in the form $f(x^{\mu})=0$. Then $\partial_{\mu}f(x^{\mu})$ is a
normal to $\Sigma$ which if is not null allows to define the unit normal
vector field to $\Sigma$ as $n^{\mu}n_{\mu}=-1$. Then the normal vector field
is given by the formula
$n_{\mu}=-\dfrac{\partial_{\mu}f}{\sqrt{|\partial_{\mu}f\partial^{\mu}f|}}\quad,\quad
n^{\mu}\partial_{\mu}f>0.$ (6.9)
In other words $\Sigma(t)$ are such that the unit normal to the hypersurfaces
can be chosen to be future-directed time-like vector field
$n_{\mu}\sim\partial_{\mu}t$ satisfying the condition $n^{\mu}n_{\mu}=-1$.
Let $\gamma$ be a congruence of curves intersecting the space-like
hypersurfaces $\Sigma(t)$, which in general are not geodesics nor orthogonal
to $\Sigma(t)$. Let $t$ be a parameter on the congruence $\gamma$, and let us
denote by $t^{\mu}$ a tangent vector to $\gamma$. Then there is satisfied the
relation
$t^{\mu}\partial_{\mu}t=1.$ (6.10)
An arbitrary fixed curve $\gamma_{F}$ is a mapping between points on all
hypersurfaces $\Sigma(t)$
$\gamma_{F}:P\in\Sigma(t)\mapsto P^{\prime}\in\Sigma(t^{\prime})\mapsto
P^{\prime\prime}\in\Sigma(t^{\prime\prime})\mapsto\ldots
P^{(n)}\in\Sigma(t^{(n)}),$ (6.11)
where the index $n$ is an integer, and fixing the coordinates on arbitrary two
hypersurfaces leads to constant coordinates $y^{i}$ for arbitrary value of
$n$. In this manner the coordinate system $(t,y^{i})$ in $M$ is established.
Assuming a transformation between this coordinate system and the another
system $x^{\mu}$: $x^{\mu}=x^{\mu}(t,y^{i})$ one can determine the tangent
vector to the congruence $\gamma$
$t^{\mu}=(\partial_{t}x^{\mu})_{y^{i}}=\delta^{\mu}_{t},\leavevmode\nobreak\
\mathrm{in}\leavevmode\nobreak\ (t,y^{i})$ (6.12)
as well as the tangent vectors on hypersurfaces $\Sigma(t)$
$e^{\mu}_{i}=(\partial_{y^{i}}x^{\mu})_{t}=\delta^{\mu}_{i},\leavevmode\nobreak\
\mathrm{in}\leavevmode\nobreak\ (t,y^{i}).$ (6.13)
In any coordinates the relation is satisfied
$\mathcal{L}_{t}e^{\mu}_{i}=0.$ (6.14)
Let us use the unit normal vector field to the hypersurfaces in the form
$\displaystyle n_{\mu}$ $\displaystyle=$ $\displaystyle-N\partial_{\mu}t,$
(6.15) $\displaystyle n_{\mu}e^{\mu}_{i}$ $\displaystyle=$ $\displaystyle 0,$
(6.16)
where $N$ is called the lapse scalar, which is a function normalizing the
vector field $n_{\mu}$. In general $t^{\mu}\nparallel n^{\mu}$, and therefore
the tangent vector $t^{\mu}$ can be decomposed in the basis
$(n^{\mu},e^{\mu}_{i})$
$t^{\mu}=Nn^{\mu}+N^{i}e^{\mu}_{i},$ (6.17)
where $N^{i}$ is a three-vector valued function called the shift vector. The
coordinate transformation $x^{\mu}=x^{\mu}(t,y^{i})$ allows to write in
$(t,y^{i})$
$\displaystyle dx^{\mu}$ $\displaystyle=$ $\displaystyle
t^{\mu}dt+e^{\mu}_{i}dy^{i}=(Nn^{\mu}+N^{i}e^{\mu}_{i})dt+e^{\mu}_{i}dy^{i}$
(6.18) $\displaystyle=$
$\displaystyle(Ndt)n^{\mu}+(dy^{i}+N^{i}dt)e^{\mu}_{i},$
and hence one can establish the evaluation of the space-time interval
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}dx^{\mu}dx^{\nu}=dx^{\mu}dx_{\mu}$ (6.19) $\displaystyle=$
$\displaystyle[(Ndt)n^{\mu}+(dy^{i}+N^{i}dt)e^{\mu}_{i}][(Ndt)n_{\mu}+(dy^{j}+N^{j}dt)e_{\mu
j}]$ $\displaystyle=$
$\displaystyle(n^{\mu}n_{\mu})N^{2}dt^{2}+(dy^{i}+N^{i}dt)(dy^{j}+N^{j}dt)e^{\mu}_{i}e_{\mu
j}$ $\displaystyle=$
$\displaystyle-N^{2}dt^{2}+h_{ij}(dy^{i}+N^{i}dt)(dy^{j}+N^{j}dt)$
$\displaystyle=$
$\displaystyle-\left(N^{2}-N_{i}N^{i}\right)dt^{2}+N_{i}dx^{i}dt+N_{j}dx^{j}dt+h_{ij}dx^{i}dx^{j},$
where $h_{ij}$ is an induced metric on $\Sigma(t)$
$h_{ij}=g_{\mu\nu}e^{\mu}_{i}e^{\nu}_{j},$ (6.20)
which actually expresses the Pythagoras theorem between two points lying on
two distinguishable constant time hypersurfaces, and was investigated by R.
Arnowitt, S. Deser and C.W. Misner [153]. By this reason, a space-time metric
tensor $g_{\mu\nu}$ of a Lorentzian manifold satisfying the Einstein field
equations (6.6) obtains the following decomposition onto space and time
$g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right],$ (6.21)
where $N^{j}=h^{ij}N_{i}$ is the contravariant shift vector, and the spatial
metric satisfies the orthogonality condition
$h_{ik}h^{kj}=\delta_{i}^{j}.$ (6.22)
Completeness relations for the metric are
$g_{\mu\nu}=-n_{\mu}n_{\nu}+h_{ij}e^{i}_{\mu}e^{j}_{\nu}.$ (6.23)
It can be verified straightforwardly that the transformation between the four-
volume form and the three-volume form is
$\sqrt{-g}=N\sqrt{h},$ (6.24)
while the inverted metric has the form
$g^{\mu\nu}=\left[\begin{array}[]{cc}-\dfrac{1}{N^{2}}&\dfrac{N^{j}}{N^{2}}\\\
{}\dfrac{N^{i}}{N^{2}}&h^{ij}-\dfrac{N^{i}N^{j}}{N^{2}}\end{array}\right].$
(6.25)
Completeness relations for the inverse metric are
$g^{\mu\nu}=-n^{\mu}n^{\nu}+h^{ij}e^{\mu}_{i}e^{\nu}_{j}.$ (6.26)
The second fundamental form of a slice is called the extrinsic curvature
tensor or induced curvature and has the form
$K_{ij}=n_{\mu;\nu}e^{\mu}_{i}e^{\nu}_{j}=-\nabla_{(i}n_{j)}-n_{(i}a_{j)},$
(6.27)
where $a_{j}$ is called the acceleration of the unit normal vector field
$a_{j}=n^{i}n_{j|i},$ (6.28)
and its trace, called the intrinsic curvature, has a form
$K=K^{i}_{i}=h^{ij}K_{ij}=n^{\mu}_{;\mu}.$ (6.29)
The hypersurface is called convex when the congruence is diverging, i.e.
$K>0$, and concave when the congruence is converging, i.e. $K<0$. The tangent
vector satisfies the Gauss–Weingarten equation
$e^{\alpha}_{i;\beta}e^{\beta}_{j}=\Gamma^{k}_{ij}e^{\alpha}_{k}+K_{ij}n^{\mu},$
(6.30)
and the Gauss–Codazzi equations [240, 241, 242] can be derived by
straightforward computation
$\displaystyle
R_{\mu\nu\kappa\lambda}e^{\mu}_{i}e^{\nu}_{j}e^{\kappa}_{k}e^{\lambda}_{l}$
$\displaystyle=$ $\displaystyle R_{ijkl}-K_{il}K_{jk}+K_{ik}K_{jl},$ (6.31)
$\displaystyle
R_{\mu\nu\kappa\lambda}n^{\mu}e^{\nu}_{i}e^{\kappa}_{j}e^{\lambda}_{k}$
$\displaystyle=$ $\displaystyle K_{ij|k}-K_{ik|j},$ (6.32)
which via using of the decomposition of the Ricci curvature tensor and Ricci
scalar curvature
$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle-
R_{\kappa\mu\lambda\nu}n^{\kappa}n^{\lambda}+h^{ij}R_{\kappa\mu\lambda\nu}e^{\kappa}_{i}e^{\lambda}_{j},$
(6.33) $\displaystyle R$ $\displaystyle=$
$\displaystyle-2h^{kl}R_{\kappa\mu\lambda\nu}n^{\kappa}n^{\lambda}e^{\mu}_{k}e^{\nu}_{l}+h^{kl}h^{ij}R_{\kappa\mu\lambda\nu}e^{\kappa}_{i}e^{\lambda}_{j}e^{\mu}_{k}e^{\nu}_{l},$
(6.34)
can be presented in terms of the Einstein tensor (6.5)
$\displaystyle 2G_{\mu\nu}n^{\mu}n^{\nu}$ $\displaystyle=$
$\displaystyle{{}^{(3)}}R+K^{ij}K_{ij}+K^{2},$ (6.35) $\displaystyle
G_{\mu\nu}e^{\mu}_{j}n^{\nu}$ $\displaystyle=$ $\displaystyle
K^{i}_{j|i}-K_{,j}.$ (6.36)
Another identity, called the Ricci equation
$\mathcal{L}_{n}K_{ij}=n^{\mu}n^{\nu}e^{\kappa}_{i}e^{\lambda}_{j}R_{\mu\nu\kappa\lambda}-\dfrac{1}{N}N_{|ij}-K_{ik}K^{k}_{j},$
(6.37)
can be also derived by straightforward computation, which we omit here. By
using of the completeness relations (6.26) and the fact
$R_{\kappa\mu\lambda\nu}n^{\kappa}n^{\mu}n^{\lambda}n^{\nu}=0,$ (6.38)
one can see that the first term in (6.34) reduces to
$-2R_{\mu\nu}n^{\mu}n^{\nu}$. By using of the relations
$\displaystyle R_{\mu\nu}n^{\mu}n^{\nu}$ $\displaystyle=$ $\displaystyle
2\left(n^{\mu}_{;[\nu}n^{\nu}\right)_{;\mu]}+2n^{\mu}_{;[\mu}n^{\nu}_{;\nu]},$
(6.39) $\displaystyle n^{\mu}_{;\nu}n^{\nu}_{;\mu}$ $\displaystyle=$
$\displaystyle K^{ij}K_{ij},$ (6.40)
to the reduced first term in (6.34), and the Gauss–Codazzi equations (6.31) to
the second term in (6.34)
$\displaystyle
h^{kl}h^{ij}R_{\kappa\mu\lambda\nu}e^{\kappa}_{i}e^{\lambda}_{j}e^{\mu}_{k}e^{\nu}_{l}$
$\displaystyle=$ $\displaystyle
h^{kl}h^{ij}\left(R_{ijkl}-K_{il}K_{jk}+K_{ik}K_{jl}\right)=$ (6.41)
$\displaystyle=$ $\displaystyle{{}^{(3)}}R+K^{2}-K^{ij}K_{ij},$
one can obtains the three-dimensional evaluation of the four-dimensional Ricci
scalar curvature
${{}^{(4)}}R={{}^{(3)}}R+K^{2}-K^{ij}K_{ij}-2\left(n^{\mu}_{;\nu}n^{\nu}-n^{\mu}n^{\nu}_{;\nu}\right)_{;\mu}.$
(6.42)
We have denoted by stroke on the left of an index the intrinsic covariant
differentiation with respect to a coordinate labeled by this index. Two
indices before the stroke means taking two times the intrinsic covariant
derivative with respect to each of the indices. For instance for a vector
$V_{i}$ and a tensor $T_{ij}$ the intrinsic covariant differentiation is
defined as
$\displaystyle V_{i|j}$ $\displaystyle=$
$\displaystyle\nabla_{j}V_{i}=\partial_{j}V_{i}-\Gamma_{ji}^{k}V_{k},$ (6.43)
$\displaystyle T_{ij|k}$ $\displaystyle=$
$\displaystyle\nabla_{k}T_{ij}=\partial_{k}T_{ij}-T_{lj}\Gamma^{l}_{ik}-T_{il}\Gamma^{l}_{jk},$
(6.44)
where $\Gamma_{ij}^{k}$ are the spatial Christoffel symbols
$\Gamma^{k}_{ij}=\dfrac{1}{2}h^{kl}\left(h_{il,j}+h_{lj,i}-h_{ij,l}\right).$
(6.45)
The induced metric $h_{ij}$ and the extrinsic curvature $K_{ij}$ are the
dynamical variables which describe the geometry of a submanifold $\partial M$
by the $3+1$ decomposed Einstein field equations. The pair $(h_{ij},K_{ij})$
describes the local geometry of a single space-like (constant time)
hypersurface $\partial M$, and then the evolution of the global four-
dimensional geometry can be formulated in terms of the one-parameter family of
the dynamical variables $(h_{ij}(t),K_{ij}(t))$ describing evolution of the
local three-dimensional geometry of the space-like hypersurfaces $\partial
M_{t}$. For consistency one must also specify the relation between the time
evolution operator $\partial_{t}$ and the vector field $n$ normal to the
$\partial M_{t}$
$\partial_{t}=Nn+N^{i}\partial_{i},$ (6.46)
where $N$ and $N_{i}$ are called the lapse function and the shift vector,
respectively. Albeit, in general stationarity of Matter fields, i.e.
$T_{\mu\nu}\equiv 0$, results in existence of a global time-like Killing
vector field $\mathcal{K}_{\mu}$ for a metric tensor $g_{\mu\nu}$.
One can choose a coordinate system in such a way that the Killing vector field
equals to $\dfrac{\partial}{\partial t}$ and the foliation $t=constans$ is
space-like. In such a situation a metric tensor depends at most on a spatial
coordinates $x^{i}$, and therefore the time $t$ can be treated as a global
coordinate [243]. Let us introduce such a coordinate system chosen by this
gauge condition in such a way that an induced three-dimensional boundary space
is a constant time $t$ hypersurface. Then the space-time boundary $\partial M$
becomes an embedded space and satisfies the Nash embedding theorem (For
detailed discussion of the theorem, its consequences and advanced development
see _e.g._ the Refs. [244, 245, 246, 247]).
#### B Geometrodynamics: Classical and Quantum
Let the enveloping space-time manifold $M$ be compact and possesses a space-
like boundary $(\partial M,h)$ equipped with the 3-volume form
$h=\det{h_{ij}}$ related to the induced metric $h_{ij}$, and the second
fundamental form $K_{ij}$. Let has the topology of space-time will be
$\Sigma\times\mathbb{R}$ where $\Sigma$ is an unrestricted topology of the
three-dimensional space. Then the Einstein field equations (6.6) can be
generated as the Euler-Lagrange equations of motion via using of the
Hilbert–Palatini action principle [146, 147] with respect to the fundamental
field which for General Relativity is a metric tensor $g_{\mu\nu}$
$\dfrac{\delta S[g]}{\delta g_{\mu\nu}}=0,$ (6.47)
which must be complemented by the boundary condition
$\delta g_{\mu\nu}\left|{}_{\partial M}\right.=0,$ (6.48)
and applied to the Einstein–Hilbert action complemented by the
York–Gibbons–Hawking boundary action [223, 248], i.e. the action of a four-
geometry with fixed an induced three-geometry of a boundary
$S[g]=\dfrac{1}{2\kappa{c}\ell_{P}^{2}}\int_{M}d^{4}x\sqrt{-g}\left(-{{}^{(4)}}R+2\Lambda\right)+S_{\phi}[g]-\dfrac{1}{\kappa{c}\ell_{P}^{2}}\int_{\partial{M}}d^{3}x\sqrt{h}K,$
(6.49)
where $S_{\phi}[g]$ is the action of Matter fields
$S_{\phi}[g]=\dfrac{1}{c}\int_{M}d^{4}x\sqrt{-g}L_{\phi}.$ (6.50)
Einstein field equations (6.6) can be obtained via straightforward computation
of the variation $\delta S=\delta S_{EH}+\delta S_{YGH}+\delta S_{\phi}=0$ on
$\partial M$, where
$\displaystyle\delta S_{G}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\kappa{c}\ell_{P}^{2}}\int_{M}d^{4}x\sqrt{-g}\left(G_{\mu\nu}+\Lambda
g_{\mu\nu}\right)\delta g^{\mu\nu},$ (6.51) $\displaystyle\delta S_{YGH}$
$\displaystyle=$ $\displaystyle\dfrac{1}{2\kappa{c}\ell_{P}^{2}}\int_{\partial
M}d^{3}y\sqrt{|h|}h^{\mu\nu}n^{\rho}\delta g_{\mu\nu,\rho},$ (6.52)
$\displaystyle\delta S_{\phi}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2c}\int_{M}d^{4}x\sqrt{-g}T_{\mu\nu}\delta
g^{\mu\nu},$ (6.53)
where by $S_{G}=S_{EH}+S_{YGH}$ we have denoted the geometric part of the
total Lagrangian (6.49). Moreover, the variational principle allows to
establish the relation between the stress-energy tensor and the Lagrangian of
Matter fields
$T_{\mu\nu}=-\dfrac{2}{\sqrt{-g}}\dfrac{\delta}{\delta
g^{\mu\nu}}\left(\sqrt{-g}L_{\phi}\right).$ (6.54)
When the cosmological constant vanishes identically $\Lambda=0$, then a global
time-like Killing vector field $\mathcal{K}_{\mu}$ on a space-time manifold
$M$ exists. Recall (For more advanced and abstractive approach we suggest e.g.
the books in the Ref. [249]) that such a field follows from vanishing of the
Lie derivative with respect to this field of the metric tensor $g_{\mu\nu}$
$\mathcal{L}_{K}g_{\mu\nu}=\lim_{\epsilon\rightarrow
0}\dfrac{g_{\mu\nu}(\tilde{x})-\tilde{g}_{\mu\nu}(\tilde{x})}{\epsilon}=0,$
(6.55)
where $\tilde{g}_{\mu\nu}(\tilde{x})$ is the metric tensor $g_{\mu\nu}(x)$
transformed under the infinitesimal transformation - an isometric mapping
$\tilde{x}^{\mu}=x^{\mu}+\epsilon\mathcal{K}^{\mu},$ (6.56)
which is equivalent to the Killing equation
$\nabla_{(\mu}\mathcal{K}_{\nu)}(x)=0.$ (6.57)
In other words, the Killing vector fields are the infinitesimal generators of
isometries.
For positive value of the cosmological constant $\Lambda>0$ the Killing vector
field $\mathcal{K}_{\mu}$ does not exist, and space-like boundary $\partial M$
only foliates an exterior to the horizons on geodesic lines. Therefore in such
a situation the ADM decomposition (6.21) is a gauge of the field of metric. In
all these situations, however, the total action (6.49) evaluated for the $3+1$
decomposed metric tensor (6.21) takes the form of the Hamilton action
functional
$\displaystyle S[g]=\int dtL,$ (6.58)
where $L$ is the total Lagrangian expressed via the $3+1$ splitting. The most
important contribution is the geometric part of the Einstein–Hilbert
Lagrangian which is
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sqrt{g}\left(-{{}^{(4)}}R+2\Lambda\right)$
$\displaystyle=$ $\displaystyle
N\sqrt{h}\left(-K_{ij}K^{ij}+K^{2}-{{}^{(3)}}R+2\Lambda\right)$ (6.59)
$\displaystyle+$ $\displaystyle
2\partial_{0}\left(\sqrt{h}K\right)-2\partial_{i}\left(\sqrt{h}(KN^{i}-h^{ij}N_{|j})\right),$
and because the last two terms are total derivatives they can be dropped when
performing a canonical formulation. The Lagrangian related to the
York–Gibbons–Hawking boundary action in itself is total derivative, and
therefore this term does not play a role here. Analysis of both the Lagrangian
of Matter fields and the cosmological constant term can be done easily, and in
result one obtains the following Lagrangian of the total theory (6.58)
$L=\dfrac{1}{2\kappa\ell_{P}^{2}}\int_{\partial
M}d^{3}xN\sqrt{h}\left(K^{2}-K_{ij}K^{ij}-{{}^{(3)}}R+2\Lambda+2\kappa\ell_{P}^{2}\rho\right).$
(6.60)
The Einstein field equations can be decomposed in the $3+1$ splitting. In
result one obtains the evolutionary equations for the induced metric $h_{ij}$
and the intrinsic curvature $K_{ij}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}{h}_{ij}$
$\displaystyle=$ $\displaystyle N_{i|j}+N_{j|i}-2NK_{ij},$ (6.61)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}{K}_{ij}$
$\displaystyle=$ $\displaystyle-
N_{|ij}+N(R_{ij}+KK_{ij}-2K_{ik}K^{k}_{j})+N^{k}K_{ij|k}+K_{ik}N^{k}_{|j}+K_{jk}N^{k}_{|i}-$
(6.62) $\displaystyle-$
$\displaystyle\kappa\ell_{P}^{2}N\left[S_{ij}-\dfrac{1}{2}h_{ij}(S-\varrho)\right],$
where the dot means differentiation with respect to the time coordinate. It
can be seen by straightforward computation that the determinant of spatial
metric $h=\det h_{ij}$ and the extrinsic curvature $K=K^{i}_{i}$ satisfy the
equations
$\displaystyle\partial_{t}\ln\sqrt{h}$ $\displaystyle=$ $\displaystyle-
NK+N^{i}_{|i},$ (6.63) $\displaystyle\partial_{t}K$ $\displaystyle=$
$\displaystyle-h^{ij}N_{|ij}+N\left(K^{ij}K_{ij}+\dfrac{\kappa\ell_{P}^{2}}{2}(S+\varrho)\right)+N^{i}K_{|i}.$
(6.64)
Here $\varrho$, called the energy density, is double projection of the stress-
energy tensor onto the normal vector field
$\varrho=T(n,n)={T}_{\mu\nu}n^{\mu}n^{\nu},$ (6.65)
and $n^{\mu}$ is the normal vector field following from the $3+1$ splitting
$\displaystyle n^{\mu}$ $\displaystyle=$
$\displaystyle\left[\dfrac{1}{N},-\dfrac{N^{i}}{N}\right],$ (6.66)
$\displaystyle n_{\mu}$ $\displaystyle=$
$\displaystyle\left[-N,0_{i}\right]^{T},$ (6.67)
where $0_{i}=[0,0,0]^{T}$ is the null three-vector. The tensor $S_{ij}$,
called the spatial stress, is double projection of the stress-energy tensor
onto the spatial metric, and $S$ is its trace which we will call the spatial
stress density
$\displaystyle S_{ij}$ $\displaystyle=$ $\displaystyle
T(h,h)=T_{\mu\nu}h^{\mu}_{i}h^{\nu}_{j},$ (6.68) $\displaystyle S$
$\displaystyle=$ $\displaystyle h^{ij}S_{ij},$ (6.69)
where $h^{\mu}_{\nu}=\delta^{\mu}_{\nu}+n^{\mu}n_{\nu}$. Straightforward
calculation gives
$S-\varrho=T,$ (6.70)
where $T=g^{\mu\nu}T_{\mu\nu}$ is the trace of the stress-energy tensor. In
the light of the Einstein field equations (6.6) one obtains
$T=\dfrac{G+4\Lambda}{\kappa\ell_{P}^{2}},$ (6.71)
where $G$ is the trace of the Einstein tensor which can be computed
straightforwardly
$G=g^{\mu\nu}G_{\mu\nu}=g^{\mu\nu}\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{{}^{(4)}}R\right)={{}^{(4)}}R-2{{}^{(4)}}R=-{{}^{(4)}}R.$
(6.72)
In this manner one obtains the constraint between the spatial stress, the
energy density, the cosmological constant and the Ricci scalar curvature of an
enveloping space-time manifold
$S-\varrho=\dfrac{4\Lambda}{\kappa\ell_{P}^{2}}-\dfrac{{{}^{(4)}}R}{\kappa\ell_{P}^{2}}.$
(6.73)
It can be seen by straightforward computation that the total Lagrangian (6.60)
leads to the Euler–Lagrange equations of motion. The first equation is
${2c\kappa}h^{-1}\left(h_{ik}h_{jl}-\dfrac{1}{2}h_{ij}h_{kl}\right)\dfrac{\delta{S}}{\delta{h_{ij}}}\dfrac{\delta{S}}{\delta{h_{kl}}}-\dfrac{\ell_{P}^{2}}{2c\kappa}\left({{}^{(3)}}R-2\Lambda-2\kappa\ell_{P}^{2}\varrho\right)=0,$
(6.74)
while the second one has the form
$\dfrac{c}{\ell_{P}^{2}}\pi^{ij}_{|j}+J^{i}=0,$ (6.75)
where $\pi^{ij}$ is the momentum conjugated to the induced metric
$\pi^{ij}=\dfrac{1}{\ell_{P}}\dfrac{\delta{S[g]}}{\delta{h_{ij}}}=\dfrac{1}{\ell_{P}}\dfrac{\delta{L}}{\delta\left(\partial_{t}{h}_{ij}\right)}=-\dfrac{\ell_{P}}{2c\kappa}\sqrt{h}\left(K^{ij}-h^{ij}K\right),$
(6.76)
and $J^{i}$, called the momentum density, is the stress-energy tensor
projected onto the normal vector field and the spatial metric
$\displaystyle J^{i}=T(n,h)=T_{\mu\nu}n^{\mu}h^{\nu i}.$ (6.77)
The resulting dynamical equation (6.74) is the Hamilton–Jacobi equation (For
details of classical mechanics see _e.g._ the Ref. [250]) to the case of
General Relativity. Originally, the equation (6.74) divided by $\sqrt{h}$ with
$\Lambda=0$ and $\varrho=0$ was derived by A. Peres [251], and by this reason
we shall call it the Peres equation. Interestingly, Wheeler [157] called the
Hamilton–Jacobi equation of General Relativity (6.74) the
Einstein–Hamilton–Jacobi equation. The Peres equation defines the classical
geometrodynamics.
The total Lagrangian (6.60) can be analyzed by the Hamiltonian approach. Let
us determine the canonical momenta
$\displaystyle\pi_{\phi}$ $\displaystyle=$
$\displaystyle\dfrac{\beta}{\ell_{P}}\dfrac{\delta
L}{\delta\left(\partial_{t}{\phi}\right)},$ (6.78) $\displaystyle\pi$
$\displaystyle=$ $\displaystyle\dfrac{1}{\ell_{P}}\dfrac{\delta
L}{\delta\left(\partial_{t}{N}\right)}=0,$ (6.79) $\displaystyle\pi^{i}$
$\displaystyle=$ $\displaystyle\dfrac{1}{\ell_{P}}\dfrac{\delta
L}{\delta\left(\partial_{t}{N_{i}}\right)}=0,$ (6.80)
where $\beta$ is a constant of dimension of a Matter field $[\phi]$
constructed from the Planck units, conjugated to Matter fields, lapse
function, shift vector, respectively. Then the Legendre transformation [250]
allows to rewrite the total Lagrangian in the form
$L=\int_{\partial{M}}d^{3}x\left[\dfrac{1}{2\kappa\ell_{P}}\left(\pi_{\phi}\partial_{t}{\phi}+\pi\partial_{t}{N}+\pi^{i}\partial_{t}{N_{i}}+\pi^{ij}\partial_{t}{h}_{ij}\right)-NH-
N_{i}H^{i}\right],$ (6.81)
where the quantities $H$ and $H^{i}$ are defined as
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H$
$\displaystyle=$
$\displaystyle\dfrac{\sqrt{h}}{2\kappa\ell_{P}^{2}}\left(K^{2}-K_{ij}K^{ij}-{{}^{(3)}R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho\right),$
(6.82)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H^{i}$
$\displaystyle=$
$\displaystyle-2\dfrac{c}{\ell_{P}^{2}}\pi^{ij}_{\leavevmode\nobreak\
|j}-2{J^{i}}=-2\dfrac{c}{\ell_{P}^{2}}\partial_{j}\pi^{ij}-\dfrac{c}{\ell_{P}^{2}}h^{il}\left(2h_{jl,k}-h_{jk,l}\right)\pi^{jk}-2{J^{i}},$
(6.83)
where ${{}^{(3)}R}=h^{ij}R_{ij}$ is the Ricci scalar curvature of a three-
dimensional embedded space. Application of the time-preservation [152] to the
primary constraints
$\displaystyle\pi$ $\displaystyle\approx$ $\displaystyle 0,$ (6.84)
$\displaystyle\pi^{i}$ $\displaystyle\approx$ $\displaystyle 0,$ (6.85)
leads to the secondary constraints
$\displaystyle H$ $\displaystyle\approx$ $\displaystyle 0,$ (6.86)
$\displaystyle H^{i}$ $\displaystyle\approx$ $\displaystyle 0,$ (6.87)
called the Hamiltonian (scalar) constraint which yields the dynamics, and the
diffeomorphism (vector) constraint which merely reflects the spatial
diffeoinvariance. B.S. DeWitt [158] showed that the quantities $H^{i}$ are
generators of the spatial diffeomorphisms $\widetilde{x}^{i}=x^{i}+\xi^{i}$
$\displaystyle
i\dfrac{\ell_{P}}{\hslash}\left[h_{ij},\int_{\partial{M}}H_{a}\xi^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle
c\ell_{P}\left(-h_{ij,k}\xi^{k}-h_{kj}\xi^{k}_{\leavevmode\nobreak\
,i}-h_{ik}\xi^{k}_{\leavevmode\nobreak\ ,j}\right),$ (6.88) $\displaystyle
i\dfrac{\ell_{P}}{\hslash}\left[\pi^{ij},\int_{\partial{M}}H_{a}\xi^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle
c\ell_{P}\left[-\left(\pi^{ij}\xi^{k}\right)_{,k}+\pi^{kj}\xi^{i}_{\leavevmode\nobreak\
,k}+\pi^{ik}\xi^{j}_{\leavevmode\nobreak\ ,k}\right],$ (6.89)
where we have denoted the intrinsic covariant components $H_{i}=h_{ij}H^{j}$.
Application of the structure constants of the diffeomorphism group, which can
be presented in the most convenient compact form
$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-(x\rightarrow
y).$ (6.90)
Applying the relations (6.88) and (6.89) one can derive the first-class
constraints algebra
$\displaystyle i\dfrac{\ell_{P}}{\hslash}\left[H_{i}(x),H_{j}(y)\right]$
$\displaystyle=$
$\displaystyle\dfrac{c}{\ell_{P}^{5}}\int_{\partial{M}}H_{a}c^{a}_{ij}d^{3}z,$
(6.91) $\displaystyle i\dfrac{\ell_{P}}{\hslash}\left[H(x),H_{i}(y)\right]$
$\displaystyle=$
$\displaystyle\dfrac{c}{\ell_{P}^{5}}H\delta^{(3)}_{,i}(x,y),$ (6.92)
while involving of the elementary relation
$\delta\left(\sqrt{h}{{}^{(3)}R}\right)=\sqrt{h}h^{ij}h^{kl}\left(\delta
h_{ik,jl}-\delta
h_{ij,kl}\right)-\sqrt{h}\left[R^{ij}-\dfrac{{{}^{(3)}R}}{2}h^{ij}\right]\delta
h_{ij},$ (6.93)
allows to establish the third bracket
$i\dfrac{\ell_{P}}{\hslash}\left[\int_{\partial{M}}H\xi_{1}d^{3}x,\int_{\partial{M}}H\xi_{2}d^{3}x\right]=c\ell_{P}\int_{\partial{M}}H^{a}\left(\xi_{1,a}\xi_{2}-\xi_{1}\xi_{2,a}\right)d^{3}x.$
(6.94)
The constraints algebra (6.91)-(6.94) was derived first by B.S. DeWitt, and by
this reason we shall call it _the DeWitt algebra_.
The method of canonical primary quantization appropriate for constrained
systems was investigated by Dirac [152] and developed for needs of quantum
geometrodynamics by L.D. Faddeev [252] (For general analysis and discussion
see also the Refs. [189, 242]). In the light of the general method applied to
the present situation the canonical commutation relations are
$\displaystyle i\dfrac{\ell_{P}}{\hslash}\left[\pi^{ij}(x),h_{kl}(y)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y),$
(6.95) $\displaystyle
i\dfrac{\ell_{P}}{\hslash}\left[\pi^{i}(x),N_{j}(y)\right]$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\delta^{(3)}(x,y),$ (6.96) $\displaystyle
i\dfrac{\ell_{P}}{\hslash}\left[\pi(x),N(y)\right]$ $\displaystyle=$
$\displaystyle\delta^{(3)}(x,y).$ (6.97)
The solutions or rather representations of the momenta operators satisfying
the commutators (6.95)-(6.97) is the question of choice. In quantum
geometrodynamics the Wheeler metric representation is usually taken into
account. In such a representation the momenta operators are analogous to the
momentum operator in quantum mechanics
$\displaystyle\pi$ $\displaystyle=$
$\displaystyle-i\dfrac{\hslash}{\ell_{P}}\dfrac{\delta}{\delta N},$ (6.98)
$\displaystyle\pi^{i}$ $\displaystyle=$
$\displaystyle-i\dfrac{\hslash}{\ell_{P}}\dfrac{\delta}{\delta N_{i}},$ (6.99)
$\displaystyle\pi^{ij}$ $\displaystyle=$
$\displaystyle-i\dfrac{\hslash}{\ell_{P}}\dfrac{\delta}{\delta h_{ij}},$
(6.100)
and applied to the Hamiltonian constraint (6.82) yields the Wheeler–DeWitt
equation [158, 157]
$\left\\{2c\kappa\dfrac{\hslash^{2}}{\ell_{P}^{2}}G_{ijkl}\dfrac{\delta^{2}}{\delta
h_{ij}\delta
h_{kl}}+\dfrac{\ell_{P}^{2}}{2c\kappa}\sqrt{h}\left({{}^{(3)}R}-2\Lambda-2\kappa\varrho\right)\right\\}\Psi[h_{ij},\phi]=0,$
(6.101)
where $G_{ijkl}$ is the DeWitt supermetric on the configurational space of
General Relativity called the Wheeler superspace $S(\partial M)$ (For more
detailed analysis and discussion see _e.g._ Refs. [157, 158, 253, 254, 255,
256])
$G_{ijkl}=\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right).$
(6.102)
Other first-class constraints satisfy the canonical commutation relations
$\displaystyle\left[{\pi}(x),{\pi}^{i}(y)\right]$ $\displaystyle=$
$\displaystyle 0,$ (6.103) $\displaystyle\left[{\pi}(x),{H}^{i}(y)\right]$
$\displaystyle=$ $\displaystyle 0,$ (6.104)
$\displaystyle\left[{\pi}^{i}(x),{H}^{j}(y)\right]$ $\displaystyle=$
$\displaystyle 0,$ (6.105) $\displaystyle\left[{\pi}^{i}(x),{H}(y)\right]$
$\displaystyle=$ $\displaystyle 0,$ (6.106)
and after the canonical primary quantization are the supplementary conditions
on a wave functional $\Psi[h_{ij},\phi]$. The primary constraints lead to the
equations
$\displaystyle-i\dfrac{\hslash}{\ell_{P}}\dfrac{\delta\Psi[h_{ij},\phi]}{\delta
N}$ $\displaystyle=$ $\displaystyle 0,$ (6.107)
$\displaystyle-i\dfrac{\hslash}{\ell_{P}}\dfrac{\delta\Psi[h_{ij},\phi]}{\delta
N_{i}}$ $\displaystyle=$ $\displaystyle 0.$ (6.108)
The diffeomorphism constraint also leads to such a condition
$i\dfrac{E_{P}}{\ell_{P}^{2}}\left(\dfrac{\delta\Psi[h_{ij},\phi]}{\delta{h_{ij}}}\right)_{|j}={J^{i}}\Psi[h_{ij},\phi],$
(6.109)
which can be rewritten in explicit form
$\left[i\dfrac{E_{P}}{\ell_{P}^{2}}\dfrac{\partial}{\partial{x^{j}}}\dfrac{\delta}{\delta{h_{ij}}}+i\dfrac{E_{P}}{\ell_{P}^{2}}{h}^{il}\left(h_{jl,k}-\dfrac{1}{2}h_{jk,l}\right)\dfrac{\delta}{\delta{h_{jk}}}-{J^{i}}\right]\Psi[h_{ij},\phi]=0.$
(6.110)
The diffeomorphism constraint can be simply reduced
$\displaystyle-2\dfrac{c}{\ell_{P}^{2}}\partial_{j}\pi^{ij}-\dfrac{c}{\ell_{P}^{2}}h^{il}\left(2h_{jl,k}-h_{jk,l}\right)\pi^{jk}-2{J^{i}}$
$\displaystyle=$
$\displaystyle-2\dfrac{c}{\ell_{P}^{2}}\partial_{j}\pi^{ij}-\dfrac{c}{\ell_{P}^{2}}h^{il}\left(2h_{jl,k}-h_{jk,l}\right)h^{j}_{i}h^{k}_{j}\pi^{ij}-2{J^{i}}$
$\displaystyle=$
$\displaystyle-2\dfrac{c}{\ell_{P}^{2}}\left[\partial_{j}+h^{il}h^{j}_{i}h^{k}_{j}\left(h_{jl,k}-\dfrac{1}{2}h_{jk,l}\right)\right]\pi^{ij}-2{J^{i}}$
$\displaystyle=$
$\displaystyle-2\dfrac{c}{\ell_{P}^{2}}\left[\partial_{j}+h^{lk}\left(h_{jl,k}-\dfrac{1}{2}h_{jk,l}\right)\right]\pi^{ij}-2{J^{i}}.$
(6.111)
Application of the relations
$\displaystyle h^{lk}h_{jl,k}$ $\displaystyle=$
$\displaystyle\left(h^{lk}h_{jl}\right)_{,k}-h^{lk}_{,k}h_{jl}=\partial_{k}\delta^{k}_{j}-h^{lk}_{,k}h_{jl}=-h^{lk}_{,k}h_{jl}$
(6.112) $\displaystyle h^{lk}h_{jk,l}$ $\displaystyle=$
$\displaystyle\left(h^{lk}h_{jk}\right)_{,l}-h^{lk}_{,l}h_{jk}=\partial_{l}\delta^{l}_{j}-h^{lk}_{,l}h_{jk}=-h^{lk}_{,l}h_{jk},$
(6.113)
and manipulations in the indices
$\displaystyle-h^{lk}_{,l}h_{jk}=-h^{kl}_{,l}h_{jk}=-h^{lk}_{,k}h_{jl},$
(6.114)
allows to rewrite the diffeomorphism constraint in the form
$H^{i}=-2\dfrac{c}{\ell_{P}^{2}}\left(\partial_{j}-\dfrac{1}{2}h_{jl}h^{lk}_{,k}\right)\pi^{ij}-2\dfrac{\ell_{P}^{2}}{c\kappa}{J^{i}}\approx
0,$ (6.115)
which via using of
$h_{jl}h^{lk}_{,k}=\partial_{k}\delta^{k}_{j}-h_{jl,k}h^{kl}=-h_{jl,k}h^{kl}$
becomes
$H^{i}=-2\dfrac{c}{\ell_{P}^{2}}\left(\partial_{j}+\dfrac{1}{2}h_{jl,k}h^{kl}\right)\pi^{ij}-2{J^{i}}\approx
0.$ (6.116)
By this reason the canonical primary quantization of the diffeomorphism
constraint leads to the following equation
$\left[i\dfrac{E_{P}}{\ell_{P}^{3}}\left(\partial_{j}+\dfrac{1}{2}h_{jl,k}h^{kl}\right)\dfrac{\delta}{\delta{h_{ij}}}-{J^{i}}\right]\Psi[h_{ij},\phi]=0.$
(6.117)
The DeWitt supermetric (6.102) and the analogous metric following from the
Peres equation (6.74) by multiplication of both the sides by $2\sqrt{h}$ are
not the same despite the same procedure is applied. Factually, the relation
between both the metrics follows from the change
$h_{ik}h_{jl}\longleftrightarrow\dfrac{h_{ik}h_{jl}+h_{il}h_{jk}}{2}=h_{i(k}h_{jl)},$
(6.118)
what means that DeWitt applied symmetrization $h_{i(k}h_{jl)}$ instead of
$h_{ik}h_{jl}$, and by this reason included to the quantum geometrodynamics
one more stratum than Peres in derivation of the classical geometrodynamics.
The equation (6.118) is in itself non trivial, because it generates
$h_{ik}h_{jl}=h_{il}h_{jk},$ (6.119)
what after multiplication of both sides by $h^{ik}h^{jl}$ leads to
$D^{2}=h^{k}_{l}h^{l}_{k}=h^{k}_{k}=D,$ (6.120)
what is true if and only if the dimensionality of embedded space is $D=1$ or
$D=0$. Naturally, it is not true in general. However, in the case when one
uses the DeWitt method then one obtains the Peres equation with the DeWitt
supermetric, i.e.
${2c\kappa}G_{ijkl}\dfrac{\delta{S}}{\delta{h_{ij}}}\dfrac{\delta{S}}{\delta{h_{kl}}}+\dfrac{\ell_{P}^{2}}{2c\kappa}\sqrt{h}\left({{}^{(3)}}R-2\Lambda-2\kappa\varrho\right)=0,$
(6.121)
and canonical primary quantization of such a classical geometrodynamics leads
to the Wheeler–DeWitt equation. The classical geometrodynamics is argued by
the fact that the only such a procedure establishes straightforward
equivalence between quantization of the Hamiltonian constraint and
quantization of the Euler–Lagrange equations of motion for geometrodynamics of
an embedded space of arbitrary dimensionality $D$, including the situation
$D=3$ which we are studying in this book. Factually, Wheeler [157] did not
argued using of the symmetrization $h_{i(k}h_{jl)}$ and in this way he rather
studied the quantum Peres equation than the Wheeler–DeWitt equation. DeWitt
[158] established the supermetric (6.102) and derived the Wheeler–DeWitt
equation. The difference is crucial because in modern quantum geometrodynamics
the Wheeler superspace is defined by the DeWitt supermetric which possesses
non trivial properties. The diffeomorphism constraint (6.87) with (6.83) and
the equation of motion (6.75) differs only by the constant factor $-2$, and by
this reason lead to the same quantum and classical conditions independently on
the DeWitt supermetric. This is because of this condition merely reflects
diffeoinvariance, while the Wheeler–DeWitt equation or the Peres equation
define quantum and classical dynamics, respectively. In other words, in
quantum geometrodynamics studying diffeomorphism constraint is worthless from
the dynamical point of view.
DeWitt [158] argued that application of the Wheeler metric representation for
closed finite worlds results in the feature: the wave functional
$\Psi[h_{ij},\phi]$ depends only on components of three-metric, i.e. is
$\Psi[h_{ij}]$. In such a situation the equations (6.110) express the
necessary and sufficient conditions for diffeoinvariance of the wave
functional $\Psi[h_{ij}]$. For finite worlds it means that the wave functional
depends only on the geometry of an embedded space. He proposed to construct
the related structure of $\Psi[h_{ij}]$ via the hypersurface integrals which
can be constructed out of products of the Riemann-Christoffel tensor and its
covariant derivatives, with the topology of three-space itself being
separately specified. DeWitt discussed differences between infinite and finite
worlds. He proposed that in the finite case one can replace the wave
functional $\Psi[h_{ij}]$ by $\Psi[{{}^{(3)}}\mathfrak{G}]$, where
${{}^{(3)}}\mathfrak{G}$ is the three-geometry. For description of the quantum
gravity he introduced $\mathfrak{M}$ \- the set of all possible three-
geometries which a finite world may possess, called today _midisuperspace_ ,
and asked for topological issues related to $\mathfrak{M}$.
However, such topological arguments are rather misleading in the light of the
fact that the Wheeler–DeWitt equation has never been solved in general. The
only known solution is the Hartle–Hawking wave function [224], called the wave
function of the Universe, which is expressed via the Feynman path integral
technique
$\displaystyle\Psi[h_{ij}]$ $\displaystyle=$ $\displaystyle N\int_{C}\delta
g(x)\exp(iS_{E}[g]),$ (6.122) $\displaystyle\Psi[h_{ij},\phi]$
$\displaystyle=$ $\displaystyle N\int_{C}\delta g\delta\phi\exp(iS[g,\phi]),$
(6.123)
where $N$ is normalization factor, and $S_{E}[g]$ and $S[g,\psi]$ are the
geometric part of the total action and the total action, respectively. The
functional integral is over all four-geometries with a space-like boundary on
which the induced metric is $h_{ij}$. Albeit, the path integrals (6.122) and
(6.123) are the only a kind tautology following from the fact that the
Wheeler-DeWitt equation can be treated as the non relativistic quantum
mechanics, i.e. the Schrödinger equation. Such a strategy, however, has never
been lead to a general evaluation of the Hartle–Hawking wave function. In
other words, the Feynman path integrals of quantum geometrodynamics
(6.122)-(6.123) can be established straightforwardly and easy in very few
particular cases while its computation has never been performed for a general
case. Such a situation is the legacy of the fact that in general the
integration strategy based on functional integration carries a difficult
computational level, and above all in general functional integration is not
well-defined mathematical procedure. Such a wave functionals lead to the
Wheeler–DeWitt equation, and by this reason are its solutions. However, they
define the only class of solutions of the Wheeler–DeWitt equation.
#### C The Wheeler Superspace
The mathematical structure of geometrodynamics is determined by the Wheeler
superspace, i.e. the configurational space of General Relativity, which is a
space of all equivalence class of metric fields of General Relativity related
by action of the diffeomorphism group of a compact, connected, orientable,
Hausdorff, $C^{\infty}$ three-dimensional space-like manifold without boundary
$\partial M$. Superspace is the factor space
$S(\partial M)=\dfrac{Riem(\partial M)}{Dif\\!f(\partial M)},$ (6.124)
where $Riem(\partial M)$ is a space of all $C^{\infty}$ Riemannian metrics on
the boundary $\partial M$, while $Dif\\!f(\partial M)$ is the group of all
$C^{\infty}$ diffeomorphisms of $\partial M$ that preserve orientation.
Superspace as a space of orbits of the diffeomorphism group is in itself a
connected, second-countable, metrizeable space. From a topological point of
view $Riem(\partial M)$ is an open positive convex cone in the infinite
dimensional vector space of all smooth $C^{\infty}$ symmetric second-rank
tensor fields over $\partial M$ having the point-set topology, i.e.
$\forall\lambda\in\mathbb{R}_{+}\cap\forall h_{ij}\in Riem(\partial M):\lambda
h_{ij}\in Riem(\partial M).$ (6.125)
This vector space is a locally convex topological vector space possessing a
translation-invariant metric $\bar{d}$ which induces its topology and defines
completeness of the space, i.e. is a Fréchet space. The metric can be chosen
in such a way that $Dif\\!f(\partial M)$ preserves distances, and
$Riem(\partial M)$ inherits the metric and by this reason is metrizable
topological space that is also paracompact and second countable. Factoring out
$Dif\\!f(\partial M)$ transits the topological information concerning
$\partial M$ to the quotient space $S(\partial M)$. There are two problems.
The first is the case of closed $\partial M$ equipped with metrics with non-
trivial isometry group, for which $S(\partial M)$ is not manifold. As showed
Fischer [254] in such a situation $Dif\\!f(\partial M)$ does not act freely so
$S(\partial M)$ is stratified manifold with nested sets of strata ordered
according to the dimension of the isometry groups. However, then exists a way
to resolving the singularities [257] which involves the frame bundle
$F(\partial M)$ over $\partial M$ (For details of the theory of bundles see
e.g. the Ref. [258]) such that the quotient space
$\dfrac{Riem(\partial M)\times F(\partial M)}{Dif\\!f(\partial M)},$ (6.126)
is the refinement of superspace. The action of the diffeomorphism group
$Dif\\!f(\partial M)$ is now free because non-trivial isometries fixing a
frame are removed. If $\phi$ is such an isometry, one can apply the
exponential map and the relation valid for any isometry
$\phi\circ\exp=\exp\circ\phi_{\star}$ to show that the subset of points in
$\partial M$ fixed by $\phi$ is open. Since this set is also closed and
$\partial M$ is connected, $\phi$ must be the identity. The definition (6.124)
can be more refined if one restricts the group of diffeomorphisms to the
proper subgroup of those diffeomorphisms that fix a preferred point, called
$\infty\in\partial M$ and the tangent space at this point
$Dif\\!f_{F}(\partial M)=\left\\{\phi\in Dif\\!f(\partial
M)|\phi(\infty)=\infty,\phi_{\star}(\infty)=id|_{T_{\infty}\partial
M}\right\\}.$ (6.127)
Then the quotient $Riem(\partial M)\times F(\partial M)/Dif\\!f(\partial M)$
is isomorphic to
$S_{F}(\partial M)=\dfrac{Riem(\partial M)}{Dif\\!f_{F}(\partial M)},$ (6.128)
but a preferred point $\infty$ must be chosen arbitrary. $S_{F}(\partial M)$
is called _the extended superspace_. The problematic situation is also
asymptotic flatness. If one treats $\partial M$ as one-point compactification
of a manifold with one end then diffeomorphisms have to respect the asymptotic
geometry and by this reason extended superspace is right. The extended
superspace would have been unnecessary in the closed case if one restricted
attention to those manifolds $\partial M$ which do not allow for metrics with
continuous symmetries, i.e. which degree of symmetry is zero. Recall that the
degree of symmetry of a manifold $\partial M$ is defined as
$\deg(\partial M)=\sup_{h_{ij}\in Riem(\partial M)}\dim J(\partial M,h_{ij}),$
(6.129)
where $J(\partial M,h_{ij})$ is the isometry group of $(\partial M,h_{ij})$
$J(\partial M,h_{ij})=\left\\{\phi\in Dif\\!f(\partial
M)|\phi^{\star}h_{ij}=h_{ij}\right\\},$ (6.130)
and when the dimension of a manifold is $D=\dim\partial M$ then
$\dim J(\partial M,h_{ij})\leqslant\dfrac{D(D+1)}{2}.$ (6.131)
$J(\partial M,h_{ij})$ is compact if $\partial M$ is compact. If $\partial M$
allows for an effective action of a compact group $G$ then it clearly allows
for a metric $h_{ij}$ on which $G$ acts as isometries just average any
Riemannian metric over $G$. For compact $\partial M$ the degree of symmetry is
zero if and only if $\partial M$ cannot support an action of the circle group
$SO(2)$. A list of three-manifolds with $deg>0$ was done by A.E. Fischer
[259], and with $\deg=0$ by A.E. Fischer and V.E. Moncrief [257]. Because of
the projection
$Riem(\partial M)\rightarrow S_{F}(\partial M)$ (6.132)
is continuous, and $Dif\\!f_{F}(\partial M)$ acts continuously on
$Riem(\partial M)$, the topology of extended superspace is quotient and open.
For arbitrary two geometries $x,y\in S_{F}(\partial M)$ a metric on
$S_{F}(\partial M)$
$d(x,y)=\sup_{\phi_{x},\phi_{y}\in Dif\\!f_{F}(\partial
M)}\bar{d}(\phi_{x}^{\star}x,\phi_{y}^{\star}y),$ (6.133)
where $\bar{d}$ is mentioned above a translation-invariant metric on
$Riem(\partial M)$, turns $S_{F}(\partial M)$ into a connected, metrizable and
second countable topological space. Hence $Riem(\partial M)$ and
$S_{F}(\partial M)$ are perfectly decent connected topological spaces
satisfying the axioms of strongest separability and countability. The basic
geometric idea is to regard $Riem(\partial M)$ as principal fibre bundle with
structure group $Dif\\!f_{F}(\partial M)$ and the extended superspace
$S_{F}(\partial M)$
$\begin{CD}Dif\\!f_{F}(\partial M)@>{i}>{}>Riem(\partial
M)@>{p}>{}>S_{F}(\partial M)\end{CD}$
where the $i,p$ are the inclusion and projection maps, respectively. This is
made possible by the so-called slice theorems and the fact that the group acts
freely and properly. This bundle structure has two far-reaching consequences
regarding the geometry and topology of $S_{F}(\partial M)$.
The topologically trivial space $Riem(\partial M)$ can be visualized as the
box fibred by the action of the diffeomorphism group $Dif\\!f_{F}(\partial
M)\in Dif\\!f(\partial M)$ generated by the diffeomorphism constraint, where
orbits of the diffeomorphism group $Dif\\!f(\partial M)$ are represented by
straight-lines in the box. The quotient space $S(\partial M)$ obtains non-
trivial topology from $Diff(\partial M)$ with an orbit represented by one
point only. The subgroup $Dif\\!f_{F}(\partial M)$ acts as isometries on the
DeWitt supermetric on $Riem(\partial M)$. $S(\partial M)$ is the set of
geometries of an embedding $\partial M$, which are equivalence classes of
isometric Riemannian metrics. By the Metrization Theorem for Superspace,
$S(\partial M)$is a connected, second-countable, metrizeable space. In other
words a countable basis of open sets exists for its topology, and there also
exists a metric on $S(\partial M)$ inducting such a topology. The partially-
ordered set of conjugacy classes of compact subgroups of $Dif\\!f(\partial M)$
indexes a partition of $S(\partial M)$, i.e. is a set of nonempty subspaces
$\\{\Sigma_{\alpha}\\}$ such that
$\displaystyle S(\partial M)$ $\displaystyle=$
$\displaystyle\bigcup_{\alpha}\Sigma_{\alpha},$ (6.134)
$\displaystyle\Sigma_{\alpha}\cap\Sigma_{\beta}\neq\emptyset$
$\displaystyle\Rightarrow$ $\displaystyle\alpha=\beta.$ (6.135)
A partition is a manifold partition if each $\Sigma_{\alpha}$ is a manifold.
All geometries with the same kind of symmetry have homeomorphic
neighbourhoods, and therefore create a manifold in the Wheeler superspace
$S(\partial M)$. The neighbourhoods of all symmetric geometries are not
homeomorphic to the neighbourhoods of all nonsymmetric geometries, and
therefore $S(\partial M)$ is not a manifold. According to the Decomposition
Theorem of Superspace, $S(\partial M)$ can be decomposed by its subspaces
$S_{G}(\partial M)$ on a countable, partially-ordered, $C^{\infty}$-Fréchet
manifold partition. By Stratification Theorem for Superspace, $S_{G}(\partial
M)$ is an inverted stratification indexed by the type of symmetry, i.e.
geometries with a given symmetry are completely contained within the boundary
of less symmetric geometries. There is a theorem due to D. Giulini which
states that in a neighbourhood of the round three-sphere in $S(\partial M)$
the DeWitt supermetric is an infinite-dimensional Lorentzian metric, i.e. is
of signature $(-1,\infty)$. However, at each point of space $\partial M$ the
DeWitt supermetric defines a Lorentzian metric on the $1+5$ dimensional space
of symmetric second-rank tensors at that point, which can be identified with
the homogeneous quotient space
$\dfrac{GL(3,\mathbb{R})}{SO(3)}\cong\dfrac{SL(3,\mathbb{R})}{SO(3)\times\mathbb{R}_{+}}.$
(6.136)
As showed Giulini and Kiefer [260] the Lorentzian signature of the DeWitt
supermetric has nothing to do with the Lorentzian signature of the space-time
metric, i.e. it persists in Euclidean gravity. But it is related to the
attractivity of gravity. Any two points of the Wheeler superspace which differ
by an action of the diffeomorphism constraint are gauge equivalent and hence
physically indistinguishable. However, the question of whether and when the
diffeomorphism constraint actually generates all diffeomorphisms of $\partial
M$ is unsolved. In fact, General Relativity is a dynamical system on the
cotangent bundle, i.e. phase space, built over $S(\partial M)$. The topology
of superspace is inherited from the topology of $\partial M$. The Hamiltonian
evolution is varying embedding of space $\partial M$ into space-time $M$.
Hence the images of an embedded space have the same topological type, what
reflects the fact that the classical geometrodynamics transitions of topology
are impossible. This is not implied by the Einstein field equations, but is a
consequence of the restriction to space-times admitting a global space-like
foliation. There is a number of solutions to the Einstein field equations
which do not satisfy such a requirement, i.e. such space-times cannot be
constructed by integrating the Gauss–Codazzi equations with some reasonable
initial data. From the Hartle–Hawking wave function point of view topology
changing classical solutions should not be removed as possible contributors in
the Feynman path integral. In the evolutionary formulation of the Einstein
field equations, there is no space-time to start with. Only solutions of the
dynamical equations construct the space-time. Then one can interpret the time
dependence of the induced metric as being brought about by wafting three-space
through space-time via an embedding. Initially there is only a space-like
submanifold of unrestricted topology.
The deformations of the space-like hypersurfaces, i.e. infinitesimal changes
of embeddings $\mathcal{E}:\partial M\mapsto M$, possess nontrivial
kinematics. The generators of normal and tangential deformations are
$\displaystyle\mathcal{N}_{N}$ $\displaystyle=$ $\displaystyle\int_{\partial
M}d^{3}xN(x)n^{\mu}[y(x)]\dfrac{\delta}{\delta y^{\mu}(x)},$ (6.137)
$\displaystyle\mathcal{T}_{N^{i}}$ $\displaystyle=$
$\displaystyle\int_{\partial
M}d^{3}xN^{i}(x)y^{\mu}_{,i}(x)\dfrac{\delta}{\delta y^{\mu}(x)},$ (6.138)
where $y^{\mu}_{,i}=\partial_{i}y^{\mu}$, and $y^{\mu}$ and $x^{i}$ are local
coordinates on $M$ and $\partial M$, respectively. The generators (6.137) and
(6.138) can be understood as tangent vectors to the space of embeddings of
$\partial M$ into M. An embedding is locally given by four functions
$y^{\mu}(x)$, such that the $3\times 4$ matrix $y^{\mu}_{,i}$ has its maximum
rank 3. We have denoted by $n^{\mu}$ the components of the normal to the image
$\mathcal{E}(\partial M)\in M$, which are functionals of $y^{\mu}(x)$, i.e.
$n^{\mu}=n^{\mu}[y(x)]$. The generators (6.137) and (6.138) satisfy the
following commutation relations
$\displaystyle\left[\mathcal{T}_{N^{i}},\mathcal{T}_{{N^{\prime}}^{i}}\right]$
$\displaystyle=$
$\displaystyle-\mathcal{T}_{\left[N^{i},{N^{\prime}}^{i}\right]},$ (6.139)
$\displaystyle\left[\mathcal{T}_{N^{i}},\mathcal{N}_{N}\right]$
$\displaystyle=$ $\displaystyle-\mathcal{N}_{N^{i}(N)},$ (6.140)
$\displaystyle\left[\mathcal{N}_{N},\mathcal{N}_{N^{\prime}}\right]$
$\displaystyle=$
$\displaystyle-\mathcal{T}_{N\nabla_{h}N^{\prime}-N^{\prime}\nabla_{h}N},$
(6.141)
where $\nabla_{h}N=\left(h^{ab}\partial_{b}N\right)\partial_{a}$, and the Lie
brackets (6.140) and (6.141) were obtained via taking variations of the basic
identities $g_{\mu\nu}n^{\mu}n^{\nu}=-1$ and
$g_{\mu\nu}y^{\mu}_{,i}n^{\nu}=0$. The space-time vector field
$V=Nn^{\mu}\partial_{\mu}+N^{i}\partial_{i},$ (6.142)
induces the foliation-dependent decomposition of the tangent vector
$\mathrm{T}(V)$ at $Y\in\textrm{Emb}(\partial M,M)$
$\mathrm{T}(V)=\int_{\partial M}d^{3}xV^{\mu}(y(x))\dfrac{\delta}{\delta
y^{\mu}(x)},$ (6.143)
obeying the Lie algebra
$\left[\mathrm{T}(V),\mathrm{T}\left(V^{\prime}\right)\right]=\mathrm{T}\left(\left[V,V^{\prime}\right]\right),$
(6.144)
which means that $V\mapsto\mathrm{T}(V)$ is a Lie homomorphism from the
tangent-vector fields on $M$ to the tangent-vector fields on
$\textrm{Emb}(\partial M,M)$. In this sense, the Lie algebra of the four-
dimensional diffeomorphism group is implemented on phase space of arbitrary
generally covariant theory which phase space includes the embedding variables,
i.e. is so-called parametrized theory. Decomposing the vector (6.143) into
normal and tangential components with respect to the leaves of the embedding
at which the tangent-vector field to $Emb(\partial M,M)$ is evaluated, yields
an embedding-dependent parametrization of $\mathrm{T}(V)$ via $(N,N^{i})$
$\mathrm{T}(N,N^{i})=\int_{\partial
M}d^{3}x\left[Nn^{\mu}[y(x)]+N^{i}(x)y^{\mu}_{,i}(x)\right]\dfrac{\delta}{\delta
y^{\mu}(x)}.$ (6.145)
Computing the functional derivatives of $n$ with respect to $y$ one can
establish the commutator of deformation generators
$\left[\mathrm{T}\left(N,N^{i}\right),\mathrm{T}\left(N^{\prime},{N^{\prime}}^{i}\right)\right]=-\mathrm{T}\left(N^{\prime\prime},{N^{\prime\prime}}^{i}\right),$
(6.146)
where
$\displaystyle N^{\prime\prime}$ $\displaystyle=$ $\displaystyle
N^{i}(N)-{N^{\prime}}^{i}(N^{\prime}),$ (6.147)
$\displaystyle{N^{\prime\prime}}^{i}$ $\displaystyle=$
$\displaystyle[N^{i},{N^{\prime}}^{i}]+N\nabla_{h}N^{\prime}-N^{\prime}\nabla_{h}N.$
(6.148)
The situation can be easy visualized. Composition of two an infinitesimal
hypersurface deformations with parameters $(N,N^{i})$ and
$(N^{\prime},{N^{\prime}}^{i})$ that maps $\partial M\mapsto\partial M_{1}$
and $\partial M_{1}\mapsto\partial M_{12}$ respectively, differs by the
hypersurface deformation with parameters
$(N^{\prime\prime},{N^{\prime\prime}}^{i})$ given by the composition that maps
with the same parameters but in the opposite order.
${\partial{M_{1}}}$${\partial{M}}$${\partial{M_{2}}}$${\partial{M_{12}}}$${\partial{M_{21}}}$$(N^{\prime},{N^{\prime}}^{i})$$(N^{\prime},{N^{\prime}}^{i})$$(N,N^{i})$$(N^{\prime\prime},{N^{\prime\prime}}^{i})$$(N,N^{i})$
(6.149)
Hamiltonian General Relativity is a particular Lie-anti representation of the
algebra (6.139)-(6.140) as a Hamilton system on the phase space of physical
fields. When the latter merely depends on the induced metric $h_{ij}$ on
$\partial M$, then the unconstrained phase space is the cotangent bundle
$T^{\star}Riem(\partial M)$ over $Riem(\partial M)$, parameterized by the pair
$(h_{ij},\pi_{ij})$. In this simplest situation one can search for
$(N,N^{i})\mapsto\left(\mathcal{H}(N,N^{i}):T^{\star}Riem(\partial
M)\rightarrow\mathbb{R}\right)$ (6.150)
where $\mathcal{H}$ is a distribution,the test functions are $N$ and $N^{i}$,
with values in real-valued functions on $T^{\star}Riem(\partial M)$
$\mathcal{H}(N,N^{i})[h_{ij},\pi_{ij}]=\int_{\partial
M}d^{3}x\left(N(x)H(x)+N_{i}(x)H^{i}(x)\right).$ (6.151)
The fundamental condition is that the Poisson brackets between two values of
$\mathcal{H}(N,N^{i})$ is
$\left\\{\mathcal{H}\left(N,N^{i}\right),\mathcal{H}\left(N^{\prime},{N^{\prime}}^{i}\right)\right\\}=\mathcal{H}\left(N^{\prime\prime},{N^{\prime\prime}}^{i}\right)$
(6.152)
The essential question is recovering the action of the Lie algebra of four-
dimensional diffeomorphism on the extended phase space including embedding
variables.
The Hamiltonian ani-Lie representation of the algebra (6.139)-(6.140) can be
constructed via the identification
$\displaystyle\mathcal{N}_{N}\mapsto\mathcal{H}(N)$ $\displaystyle=$
$\displaystyle\int_{\partial M}d^{3}xN(x)H(x),$ (6.153)
$\displaystyle\mathcal{T}_{N^{i}}\mapsto\mathcal{D}(N^{i})$ $\displaystyle=$
$\displaystyle\int_{\partial M}d^{3}xN_{i}(x)H^{i}(x),$ (6.154)
and the resulting algebra is the Lie algebra of the diffeomorphism group
$Dif\\!f(\partial M)$
$\displaystyle\left\\{\mathcal{D}\left(N^{i}\right),\mathcal{D}\left({N^{\prime}}^{i}\right)\right\\}$
$\displaystyle=$
$\displaystyle\mathcal{D}\left(\left[N^{i},{N^{\prime}}^{i}\right]\right),$
(6.155)
$\displaystyle\left\\{\mathcal{D}\left(N^{i}\right),\mathcal{H}\left(N\right)\right\\}$
$\displaystyle=$ $\displaystyle\mathcal{H}\left(N^{i}\left(N\right)\right),$
(6.156)
$\displaystyle\left\\{\mathcal{H}\left(N\right),\mathcal{H}\left(N^{\prime}\right)\right\\}$
$\displaystyle=$
$\displaystyle\mathcal{D}\left(N\nabla_{h}N^{\prime}-N^{\prime}\nabla_{h}N\right),$
(6.157)
which means that geometrodynamics does not define the extraordinary situation.
Any four dimensional $Dif\\!f(\partial M)$-invariant theory will gives rise to
this same algebra. The Lie algebra of $Dif\\!f(\partial M)$, namely, is
universally satisfied for a theory considered in an arbitrary $space+time$
decomposition, which is formed by hypersurface foliations and is space-time
covariant. The universality of the diffeomorphism Lie algebra suggests
searching for its another, possibly more general, representations on a given
phase space. The theorem due to K. Kuchař, C. Teitelboim, and S.A. Hojman
[261], which states that for the the unique two-parameter family, given by
$\kappa$ and $\Lambda$, of realizations for the algebra (6.153)-(6.154)
equipped with the constraints $H$ and $H^{i}$, in which the conjugated
momentum can be expressed via the extrinsic curvature and the kinetic term is
multiplied by an overall $n^{\mu}n_{\mu}$, is the most general Hamiltonian
representation of the universal diffeomorphism Lie algebra (6.155)-(6.157) on
$T^{\ast}Riem(\partial M)$, i.e. the space tangent to all $C^{\infty}$
Riemannian metrics on the boundary $\partial M$, up to the residual canonical
transformations having the following form
$\pi^{ij}\mapsto\pi^{ij}+\dfrac{\delta F[h_{ij}]}{\delta h_{ij}},$ (6.158)
where $F$ is some function invariant with respect to action of the
diffeomorphism group $Dif\\!f(\partial M)$ determined on the space
$Riem(\partial M)$.
Superspace possesses analogous ambiguity to the Aharonov–Bohm effect [262]
following from the lack of simple connectedness of topology. This depends on
the topology of $\partial M$, i.e. contractibility of $Riem(\partial M)$
results in the relation for n-th homotopy group
$\pi_{n}\left(\dfrac{Riem(\partial M)}{Dif\\!f_{F}(\partial
M)}\right)\cong\pi_{n-1}\left(Dif\\!f_{F}(\partial M)\right),$ (6.159)
where $n\geqslant 1$. For $n=1$ one obtains the fundamental group of the
extended superspace
$\pi_{1}\left(S_{F}(\partial M)\right)\cong\pi_{0}\left(Dif\\!f_{F}(\partial
M)\right)=\dfrac{Dif\\!f_{F}(\partial M)}{Dif\\!f_{F}^{0}(\partial
M)}=MCG_{F}(\partial M),$ (6.160)
where $Dif\\!f_{F}^{0}(\partial M)$ is the identity component of
$Dif\\!f_{F}(\partial M)$, and $MCG_{F}(\partial M)$ denotes abbreviation of
the name mapping-class group for frame fixing diffeomorphisms [256].
Uniqueness of representations of the Poisson brackets (6.152) has much more
serious ambiguity. It is namely labeled by an additional $\mathbb{C}$-valued
parameter, called the Barbero–Immirzi parameter, obtained due to connection
variables (See the Ref. [263] and the books [118, 216]). In such a situation
the Poisson brackets (6.152) can not represented via a semi-direct product of
it with the Lie algebra of $SU(2)$ gauge transformations, and by taking the
quotient with respect to this algebra the Poisson brackets (6.152) are
represented non locally. The Poisson brackets (6.152) do not seem to apply in
case of connection variables, because of the connection variable does not
admit an interpretation as a space-time gauge field restricted to space-like
hypersurfaces and the dynamics generated by the constraints does not admit the
interpretation of being induced by appropriately moving a hypersurface through
a space-time with fixed geometric structures on it. The homotopy groups of the
extended superspace, given by the formula (6.159), encode much of its global
topology. They were investigated by D. Giulini [256] and D. Witt [264], and
its quantum gravity context was investigated in the contributions due to J.
Friedman and R. Sorkin [265], C.J. Isham [266], and R. Sorkin [267].
Topological invariant of $S_{F}(\partial M)$ are also topological invariants
of $\partial M$, while homotopy invariants of $\partial M$ in general are not
homotopy invariants of $S_{F}(\partial M)$. Such a situation means that one
can distinguish homotopy equivalent but non homeomorphic three-manifolds by
looking at homotopy invariants of their associated superspaces. When $\partial
M$ is connected closed orientable three-manifold then the homology and
cohomology groups are determined by the fundamental group. Namely, the first
four (zeroth to third, the only non-trivial ones) homology and cohomology
groups are
$\displaystyle H_{\star}$ $\displaystyle=$
$\displaystyle(\mathbb{Z},A\pi_{1},FA\pi_{1},\mathbb{Z}),$ (6.161)
$\displaystyle H^{\star}$ $\displaystyle=$
$\displaystyle(\mathbb{Z},FA\pi_{1},A\pi_{1},\mathbb{Z}),$ (6.162)
where $A$ and $F$ are the operation of abelianisation of a group and the
operation of taking the free part of a finitely generated abelian group,
respectively. Particularly interesting is the fundamental group of the
extended superspace. The analogy with quantum mechanics already suggests that
its classes of inequivalent irreducible unitary representations correspond to
a superselection structure which here might serve as fingerprint of the
topology of $\partial M$ in the quantum theory. The sectors might, for
instance, correspond to various statistics that preserve or violate a naively
expected spin-statistics correlation (See e.g. papers in the Ref. [268]).
General three-manifolds can be understood by surgery (For theory see e.g.
books in the Ref. [269], for excellent constructive application see e.g. the
paper due to G. Perelman [270]). Particularly cutting along certain embedded
two manifolds, such that the remaining pieces are simpler, is often applied
technique. As the crucial and essential example, let us consider those
simplifications that are achieved by cutting along embedded two-spheres. An
essential two-sphere is one which does not bound a three-ball and a splitting
two-sphere is one which complement has two connected components. Let us
consider a closed three-manifold $\partial M$, which is cutting along an
essential splitting two-sphere, capping off the two-sphere boundary of each
remaining component by a three-disk, and this process is repeating for each of
the remaining closed three-manifolds. This process stops after a finite number
of steps where the resulting components are uniquely determined up to
diffeomorphisms, orientation preserving if oriented manifolds are considered,
and permutation [271]. The process stops at that stage at which none of the
remaining components, $\Pi_{1},\ldots,\Pi_{n}$, allows for essential splitting
two-spheres, i.e. at which each $\Pi_{i}$ is a prime manifold, i.e. such a
manifold for which each embedded two-sphere either bounds a three-disc or does
not split; it is called irreducible if each embedded two-sphere bounds a
three-disc. In the latter case the second homotopy group, $\pi_{2}$, must be
trivial, since, if it were not, the so-called sphere theorem. The theorem
states that if for connected three-manifold $\partial M$ $\pi_{2}M=0$ then
there is either an embedded $S^{2}$ in $M$ representing a nontrivial element
in $\pi_{2}M$, or an embedded two-sided $\mathbb{R}\mathrm{P}^{2}$ in
$\partial M$ such that the composition of the cover
$S^{2}\rightarrow\mathbb{R}\mathrm{P}^{2}$ with the inclusion
$\mathbb{R}\mathrm{P}^{2}\hookrightarrow\partial M$ represents a nontrivial
element of $\pi_{2}M$. This theorem ensures the existence of a non-trivial
element of $\pi_{2}$ which could be represented by an embedded two-sphere.
Conversely, it follows from the validity of the Poincaré conjecture that a
trivial $\pi_{2}$ implies irreducibility. Hence irreducibility is equivalent
to a trivial $\pi_{2}$. There is precisely one non-irreducible prime three-
manifold, and that is the handle $S^{1}\times S^{2}$. Hence a three-manifold
is prime if and only if it is either a handle or if its $\pi_{2}$ is trivial.
For a general three-manifold $\partial M$ given as connected sum of primes
$\Pi_{1},\ldots,\Pi_{n}$, there is a general method to establish
$MCG_{F}(\partial M)$ in terms of $MCG_{F}(\Pi_{i})$. The strategy is to look
at the effect of elements in $MCG_{F}(\partial M)$ on the fundamental group of
$\partial M$. As $\partial M$ is the connected sum of primes, and as connected
sums in $D$ dimensions are taken along $D-1$ spheres which are simply-
connected for $D\geqslant 3$, the fundamental group of a connected sum is the
free product of the fundamental groups of the primes for $D\geqslant 3$. The
group $MCG_{F}(\partial M)$ now naturally acts as automorphisms of
$\pi(\partial M)$ by simply taking the image of a based loop that represents
an element in $\pi(\partial M)$ by a based (same basepoint) diffeomorphism
that represents the class in $MCG_{F}(\partial M)$. Hence there is a natural
map of $MCG_{F}(\partial M)$ into group of automorphisms of the fundamental
group of the D-dimensional manifold $\partial M$
$d_{F}:MCG_{F}(\partial M)\rightarrow\textrm{Aut}\left(\pi_{1}(\partial
M)\right).$ (6.163)
The known finite presentations of automorphism groups, i.e. its
characterization in terms of a finite number of generators and finite number
of relations, of free products in terms of presentations of the automorphisms
of the individual factors and additional generators, basically exchanging
isomorphic factors and conjugating whole factors by individual elements of
others, can now be used to establish finite presentations of $MCG_{F}(\partial
M)$, provided finite presentations for all prime factors are known. This
presentation of the automorphism group of free products is due to D.I.
Fouxe–Rabinovitch [272], and its modern forms with corrections were performed
by D. McCullough and A. Miller [273] and N.D. Gilbert [274]. This situation
would be more complicated if $Dif\\!f(\partial M)$ rather than
$Dif\\!f_{F}(\partial M)$, at least the diffeomorphisms fixing a preferred
point, had been considered. Only for $Dif\\!f_{F}(\partial M)$, or the
slightly larger group of diffeomorphisms fixing the point, is it generally
true that the mapping-class group of a prime factor injects into the mapping-
class group of the connected sum in which it appears. For more on this,
compare the discussion by D. Giulini [275]. Clearly, one also needs to know
which elements are in the kernel of the map (6.163).
The cotangent bundle over superspace is not the fully reduced phase space for
matter-free General Relativity. It only takes account of the vector
constraints and leaves the scalar constraint unreduced. However, under certain
conditions, the scalar constraints can be solved by the ”conformal method”
which leaves only the conformal equivalence class of three-dimensional
geometries as physical configurations. In those cases the fully reduced phase
space is the cotangent bundle over conformal superspace, which is given by
replacing $Dif\\!f(\partial M)$ by the semi-direct product
$Dif\\!f^{C}(\partial M)=C(\partial M)\rtimes Dif\\!f(\partial M),$ (6.164)
where $C(\partial M)$ is the abelian group of conformal re-scalings that acts
on $Riem(\partial M)$ via pointwise multiplication $(f,h_{ij})\mapsto
fh_{ij}$, where $f:\partial M\rightarrow\mathbb{R}_{+}$. The right action of
$(f,\phi)\in Dif\\!f_{C}(\partial M)$ on $h_{ij}\in Riem(\partial M)$ is then
given by $R_{(f,\phi)}(h_{ij})=f\phi^{\star}h_{ij}$, so that, using
$R_{(f,\phi)}R_{(f^{\prime},\phi^{\prime})}=R_{(f,\phi)(f^{\prime},\phi^{\prime})}$,
the semi-direct product structure is
$(f,\phi)(f^{\prime},\phi^{\prime})=(f^{\prime}(f\circ\phi),\phi\circ\phi^{\prime})$.
Because of $(ff^{\prime})\circ\phi=(f\circ\phi)(f^{\prime}\circ\phi)$
$Dif\\!f(\partial M)$ indeed acts as automorphisms of $C(\partial M)$.
Conformal superspace and extended conformal superspace would then be defined
as
$\displaystyle CS(\partial M)$ $\displaystyle=$
$\displaystyle\dfrac{Riem(\partial M)}{Dif\\!f^{C}(\partial M)},$ (6.165)
$\displaystyle CS_{F}(\partial M)$ $\displaystyle=$
$\displaystyle\dfrac{Riem(\partial M)}{Dif\\!f^{C}_{F}(\partial M)},$ (6.166)
where
$Dif\\!f^{C}_{F}(\partial M)=C(\partial M)\rtimes Dif\\!f_{F}(\partial M).$
(6.167)
Since $C(\partial M)$ is contractible, the topologies of $Dif\\!f^{C}(\partial
M)$ and $Dif\\!f^{C}_{F}(\partial M)$ are those of $Dif\\!f(\partial M)$ and
$Dif\\!f_{F}(\partial M)$ which also transcend to the quotient spaces whenever
the groups act freely. In the first case this is essentially achieved by
restricting to manifolds of vanishing degree of symmetry, whereas in the
second case this follows almost as before, with the sole exception being
$(S^{3},h_{ij})$ with $h_{ij}$ conformal to the round metric. Let
$CJ(\partial M,h_{ij}=\left\\{\phi\in Dif\\!f(\partial
M)|\phi^{\star}h_{ij}=fh_{ij},f:\partial M\rightarrow\mathbb{R}_{+}\right\\}$
(6.168)
be the group of conformal isometries. For compact $\partial M$ it is known to
be compact except if and only if $\partial M=S^{3}$ and $h_{ij}$ conformal to
the round metric [276]. Hence, for $\partial M\neq S^{3}$, we can average h
over the compact group $CJ(\partial M,h_{ij})$ and obtain a new Riemannian
metric $h^{\prime}_{ij}$ in the conformal equivalence class of $h_{ij}$ for
which $CJ(\partial M,h_{ij})$ acts as proper isometries. Therefore it cannot
contain non-trivial elements fixing a frame. Hence in contrast, the geometry
for conformal superspace differs insofar from that discussed above as the
conformal modes that formed the negative directions of the DeWitt supermetric.
The horizontal subspaces, orthogonal to the orbits of
$Dif\\!f^{C}_{F}(\partial M)$, are now given by the transverse and traceless
symmetric two-tensors. In that sense the geometry of conformal superspace, if
defined as before by some ultralocal bilinear form on $Riem(\partial M)$, is
manifestly positive due to the absence of trace terms and hence less
pathological than the superspace metric discussed above. It might seem that
its physical significance is less clear, as there is now no constraint left
that may be said to induce this particular geometry. Whether it is a realistic
hope to understand superspace and conformal superspace, its cotangent bundle
being the space of solutions to the Einstein field equations, well enough to
actually gain a sufficiently complete understanding of its automorphism group
is hard to say. An interesting strategy lies in the attempt to understand the
solution space directly in a group-, or Lie algebra-, theoretic fashion in
terms of a quotient $G^{\infty}/H^{\infty}$, where $G^{\infty}$ is an infinite
dimensional group (Lie algebra) that (locally) acts transitively on the space
of solutions and $H^{\infty}$ is a suitable subgroup (algebra), usually the
fixed-point set of an involutive automorphism of $G$. The basis for the hope
that this might work in general is the fact that it works for the subset of
stationary and axially symmetric solutions of the Einstein field equations for
which $G^{\infty}$ is the Geroch group [277].
One can distinguish homotopy equivalent but non homeomorphic three-dimensional
$\partial M$ by looking at homotopy invariants of their associated
$S_{F}(\partial M)$. Good example are certain types of lens spaces $L(p,q)$
(For detailed discussion see e.g. the Ref. [278]), which are prime manifolds.
Lens spaces were introduced by H. Tietze [279] as the simplest possible
examples of 3-manifolds obtained by identifying faces of a polyhedron. They
were both first known three-manifolds not determined only by homology and
fundamental group, as well as the most simple closed manifolds for which the
homotopy type does not determine the homeomorphism type. J.W. Alexander [280]
proved that $L(5;1)$ and $L(5;2)$ are not homeomorphic despite their
fundamental groups are isomorphic and their homology are identical, in spite
that their homotopy type is not the same. Homotopy type of lens spaces is the
same, and therefore another lens spaces have isomorphic fundamental groups and
homology. However, in general homeomorphism type is not the same for lens
spaces, and by this reason lens spaces can are the birth of geometric topology
of manifolds as distinct from algebraic topology. In dimension 3 are defined
as the quotient space of a 3-sphere $S^{3}$
$S^{3}=\left\\{(z_{1},z_{2})\in\mathbb{C}\times\mathbb{C}||z_{1}|^{2}+|z_{2}|^{2}=1\right\\},$
(6.169)
presented as the union of two solid tori
$\displaystyle A_{+}$ $\displaystyle=$
$\displaystyle\left\\{|z|=1-|w|^{2},|w|\leqslant\dfrac{\sqrt{2}}{2}\right\\},$
(6.170) $\displaystyle A_{-}$ $\displaystyle=$
$\displaystyle\left\\{|z|\leqslant\dfrac{\sqrt{2}}{2},|w|=1-|z|^{2}\right\\},$
(6.171)
whose common boundary torus is given as the zero level of the function
$f(z,w)=|z|^{2}-|w|^{2}$. These solid tori are invariant under isometric free
action of the cyclic group $\mathbb{Z}_{p}$ of order $p$
$\mathbb{Z}_{p}=\left\\{1,\epsilon,\epsilon^{2},\ldots,\epsilon^{p-1}\right\\},$
(6.172)
where $\epsilon$ is a primitive $p$-th root of unity
$\epsilon=\exp\dfrac{2\pi i}{p}.$ (6.173)
In other words the 3-dimensional lens space is the orbit space
$L(p,q)=\dfrac{S^{3}}{\mathbb{Z}_{p}},$ (6.174)
where $(p,q)$ is the pair of relatively prime (coprime) integers with $p>1$.
Moreover, after taking the quotient of the sphere, these solid tori are again
transformed into solid tori into which the lens space $L(p,q)$ splits. The
function $f(z,w)$ defined initially on the sphere generates a smooth function
on $L(p,q)$. The levels of this function define some foliation on the lens
space. The three-manifold $L(p,q)$ can be visualized easy. Namely, the equator
of a 3-ball, i.e. solid ball in $\mathbb{R}^{3}$, is divided into $p$ equal
segments, so that the upper and lower hemispheres become $p$-sided polygons.
These hemispherical faces are then identified by a rotation through
$2\pi\dfrac{q}{p}$ about the vertical symmetry axis, where $0\leqslant q<p$
and $(p,q)=1$. If a corner is introduced along the equator of the 3-ball it
assumes the lens-shaped appearance that gave these manifolds their name, the
term lens space being introduced by Seifert and Threlfall [281]. Tietze noted
that $L(p,q)$ may also be described as the manifold with a genus $1$ Heegaard
diagram consisting of a curve on the boundary of a solid torus which winds
around p times latitudinally and $q$ times meridionally.
The action of $\mathbb{Z}_{p}$ on $S^{3}$ is
$\displaystyle z_{1}^{\prime}$ $\displaystyle=$ $\displaystyle\epsilon z_{1},$
(6.175) $\displaystyle z_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\epsilon^{q}z_{2}.$ (6.176)
In this way each set of $p$ equidistant points on the equator is identified to
a single point. The fundamental group of $L(p,q)$ is independent on q
$\pi_{1}\left(L(p,q)\right)=\mathbb{Z}_{p},$ (6.177)
and the higher homotopy groups are those of its universal covering space is
$L(1,0)=S^{3}$ with $p$ sheets, i.e. $p$-fold 3-sphere. Therefore, also the
standard invariants (6.161) and (6.162) taken for $L(p,q)$ are sensitive only
to $p$. Two lens spaces $L(p,q)$ and $L(p,q^{\prime})$ are
1. 1.
homotopy equivalent if and only if $qq^{\prime}=\pm n^{2}\mod p$ where
$n\in\mathbb{Z}$ [282]
2. 2.
homeomorphic if and only if $q^{\prime}=\pm q^{\pm 1}\mod p$ [283]
3. 3.
orientation-preserving homeomorphic if and only if $q^{\prime}=q^{\pm 1}\mod
p$ [283].
The torsion linking form is the invariant allowing to perform the homotopy
classification of three-dimensional lens spaces, and the Reidemeister torsion
[283] allows to make the homeomorphism classification. This invariant was
formalized and generalized to higher dimensions by Reidemeister’s student
Franz [284]. The latter was performed by K. Reidemeister as a classification
up to piecewise linear homeomorphism (For PL topology see e.g. the Ref.
[285]), and E.J. Brody [286] showed that the Reidemeister construction is also
a homeomorphism classification. Lens spaces are determined by simple homotopy
type, and there are no normal invariants, like e.g. characteristic classes, or
surgery obstruction. J.H. Przytycki and A. Yasuhara [287] formulated the knot-
theoretic classification: for a closed curve $C$ in the universal cover of the
lens space which lifts to a knot having a trivial Alexander polynomial,
computation of the torsion linking form on the pair $(C,C)$ gives the
homeomorphism classification. P. Salvatore and R. Longoni [288] gave another
showed that homotopy equivalent but not homeomorphic lens spaces may have
configuration spaces with different homotopy types, which can be detected by
different Massey products.
The lens spaces were also natural subjects for investigations of a more
algebraic topological nature. In this vein, Rueff [289] showed that there
exists a degree 1 map $L(p,q)\rightarrow L(p,q^{\prime})$ if and only if
$qq^{\prime}\equiv r^{2}(\mod p)$, for some $r$. Lens space in the dimension 3
is the Seifert fiber space [290], i.e. a $S^{1}$-bundle (circle bundle) over a
2-dimensional orbifold. In the dimension three the equivalence of the
combinatorial and topological classifications, called _Hauptvermutung_ and
proved by E.E. Moise [291], is validate (For detailed discussion see e.g. the
Ref. [292]). For given $p$, we aim to attach a homotopy invariant to $L(p,q)$
that depends on $q$. For this, one needs the Bockstein homomorphism. The short
exact sequence of coefficient groups
${0}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}_{p}}$${0}$$p$ (6.178)
induces the long exact sequence
${\ldots}$${H^{n}(X;\mathbb{Z})}$${H^{n}(X;\mathbb{Z})}$${H^{n}(X;\mathbb{Z}_{p})}$${H^{n+1}(X;\mathbb{Z})}$${\ldots}$$p$$\rho$$\beta_{0}$
(6.179)
Let
$\beta=\rho\circ\beta_{0}:H^{n}(X;\mathbb{Z}_{p})\rightarrow
H^{n+1}(X;\mathbb{Z}_{p}).$ (6.180)
This, and $\beta^{0}$ itself, is called the Bockstein homomorphism. It is
natural in a space $X$, and it is called a cohomology operation. The Hurewicz
theorem, the Universal Coefficient Theorem, and the Poincaré duality allow to
compute the cohomologies ($L=L(p,q)$)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{1}(L;\mathbb{Z})\approx\mathbb{Z}_{p}\quad,\quad
H^{1}(L;\mathbb{Z})\approx 0\quad,\quad H_{1}(L;\mathbb{Z}_{p})\approx
H^{1}(L;\mathbb{Z}_{p})\approx\mathbb{Z}_{p},$ (6.181)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{2}(L;\mathbb{Z})\approx
0\quad,\quad H^{2}(L;\mathbb{Z})\approx\mathbb{Z}_{p}\quad,\quad
H_{2}(L;\mathbb{Z}_{p})\approx H^{2}(L;\mathbb{Z}_{p})\approx\mathbb{Z}_{p},$
(6.182)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H_{3}(L;\mathbb{Z})\approx\mathbb{Z}\quad,\quad
H^{3}(L;\mathbb{Z})\approx\mathbb{Z}\quad,\quad H_{3}(L;\mathbb{Z}_{p})\approx
H^{3}(L;\mathbb{Z}_{p})\approx\mathbb{Z}_{p}.$ (6.183)
and the exact sequence
${H^{1}(L;\mathbb{Z})}$${H^{1}(L;\mathbb{Z}_{p})}$${H^{2}(L;\mathbb{Z})}$${H^{2}(L;\mathbb{Z})}$${H^{2}(L;\mathbb{Z}_{p})}$$\beta_{0}$$0$$\rho$
(6.184)
shows that $\beta_{0}$ and $\rho$ are isomorphisms, and therefore
$\beta:H^{1}(L;\mathbb{Z}_{p})\rightarrow H^{2}(L;\mathbb{Z}),$ (6.185)
is an isomorphism.
#### D The Problems of Geometrodynamics
The key problem of quantum geometrodynamics are observables. The important
notion is the space of solutions of all constraints $\mathcal{F}_{0}$ which is
a subspace of a functional space $\mathcal{F}$ of quantum states represented
in the Schrödinger picture by wave functionals of the three-metric
$\Psi[h_{ij}]$. Physical states must belong to $\mathcal{F}_{0}$ in order to
be invariant under the symmetries encoded in constraints. The Dirac
observables must commute with all the first-class constraints generating gauge
transformations, $[O,H]\Psi=0$, so the action of an observable on a physical
state does not project the state out of the space of physical states
$\mathcal{F}_{0}$. An inner product must be defined on $\mathcal{F}_{0}$ in
order to obtain an Hilbert space of physical normalized state vectors.
Kuchař [293] considered the problem of observables, and reached the conclusion
that the observables are defined by non vanishing Poisson brackets with all of
the constraints, but claimed that rightness of such a treatment is justified
for the diffeomorphism constraint, while is manifestly wrong for the
Hamiltonian constraint because of $H$ generates the dynamics between the
hypersurfaces. Both a hypersurface as well as its points in itself are not
directly observable. Albeit, the values of the canonical pair
$(q_{ij},\pi^{ij})$ are evidently distinguishable on an initial hypersurface
and on an evolved hypersurface, and moreover the effects of the evolution are
not possible to observe when there is a lack of difference between these
values. For consistency Kuchař introduced two types of variable, i.e.
observables and perennials. In his approach observables are the
diffeoinvariant dynamical variables which do not commute with the Hamiltonian
constraint, meanwhile perennials are the observables commuting with the
Hamiltonian constraint and can not be observed because of the Hamiltonian
constraint should not be seen as a generator of the gauge transformations. By
this reason the observables do not act on the space of physical solutions
$\mathcal{F}_{0}$.
In spite of a number of details, in general the strategy proposed by Kuchař
can be summarized concisely. First of all one must find the four kinematic
variables $X^{A}:\partial M\mapsto M$ where $A=0,\ldots,3$ which represent a
space-like embedding of a space-like hypersurface $\partial M$ into the space-
time manifold $M$. These scalar fields represent the space-time positions and
observables evolving along $M$, which are the true gravitational degrees of
freedom, are the dynamical variables separated out from these fields on the
level of phase space. The second point is to interpret the constraints as
conditions and identify the momenta $P_{A}$ conjugated to $X_{A}$, which
determine the evolution of the degrees of freedom between hypersurfaces, with
the energy-momenta of the degrees of freedom. Such a procedure involves
solving the constraints on the classical level, necessity of the internal
time, and quantization formulated in terms of the Tomonaga–Schwinger equation
[294, 295]
$i\dfrac{\delta\Psi[\phi^{r}(x)]}{\delta
X^{A}(x)}=h_{A}\left(x;X^{B},\phi^{r},p^{s}\right)\Psi[\phi^{r}(x)],$ (6.186)
where $r,s=1,2$ and the variables $X^{A}$ are treated as classical, like time
in quantum mechanics. There arise problems which include multiple-choice, no
global time, problem in definition of the Hamiltonian $h_{A}$ and many others.
Brown and Kuchař [296] introduced matter variables, which label space-time
points and are coupled to space-time geometry, instead of functionals of the
gravitational variables. They proposed to take into account a dust field
filling all space and playing a role of time, what includes an internal time
variable against which systems can evolve, and which can play a role of the
fixed background for the construction of quantum gravity. In the Brown–Kuchař
formalism the Schrödinger equation can be written out and the emerging
Hamiltonian does not depend on the dust variables.
Another version of the solution of the problem of time, called unimodular
gravity was proposed by W.A. Unruh [297], who modified General Relativity such
that the cosmological constant is a dynamical variable for which the conjugate
is taken to be the cosmological time. The result is that the Hamiltonian
constraint is augmented by a cosmological constant term giving the modified
Hamiltonian constraint $\Lambda+\sqrt{h}H=0$. The presence of this extra term
and the cosmological time $\tau$ unfreezes the dynamics and leas to the
$\tau$-dependent Schrödinger equation.
Also DeWitt [158] tried to solve the problem of time in frames of quantum
geometrodynamics. It is the problem of extracting a notion of time from
timeless dynamics described by the Wheeler–DeWitt equation. A consequence of
the timeless nature of this equation is the problematic implementation of an
inner product for state vectors. In analogy to the inner product obtained from
the Klein–Gordon equation, DeWitt proposed to definition the inner product of
two solutions of the Wheeler–DeWitt equation
$(\Psi_{a},\Psi_{b})=Z\int\Psi^{\ast}_{a}[{{}^{(3)}}\mathfrak{G}]\prod_{x}\left(d\Sigma^{ij}G_{ijkl}\dfrac{\overrightarrow{\delta}}{i\delta{h_{kl}}}-\dfrac{\overleftarrow{\delta}}{i\delta{h_{kl}}}G_{ijkl}d\Sigma^{ij}\right)\Psi_{b}[{{}^{(3)}}\mathfrak{G}],$
(6.187)
where the product is taken over all the points of a three-dimensional embedded
space $\partial M$, the integration is over a $5\times\infty^{3}$-dimensional
surface in $S(\partial M)$, $d\Sigma^{ij}(x)$ is the surface element of the
topological product of a set of 5-dimensional hypersurfaces $\Sigma(x)$ one
chosen at each point of $\partial M$, and $Z$ is normalization constant. The
Klein–Gordon inner product (6.187) is invariant under the deformation of the
$5\times\infty^{3}$ surface, but is not positively defined and vanishes for
real solutions of the Wheeler–DeWitt equation. Moreover, by such a treatment,
all problems related to the Klein–Gordon equation, like e.g. no separation
into positive and negative frequencies and the negative probability, are
available in general for the Wheeler-DeWitt equation.
There are also another problems within quantum geometrodynamics following from
the Wheeler–DeWitt equation. First of all, this is the initial data problem.
Namely, by quantum geometrodynamics the classical space-time is the history of
space geometry governed by the deterministic evolution. There arises
uncertainty relation between intrinsic and extrinsic geometry due to the Lie
bracket of an induced metric and its conjugated momentum. By this reason
interpretation of the properties of quantum space-time is unclear. The second
important problem is that for consistency the standardly applied quantization
of the constraints needs a choice of a regularization method. However, factor
orderings lead to non-unique result and quantum anomalies, i.e. the most
terrible ambiguities. The problem of indefiniteness of measure in the Wheeler
superspace follows from the definition of the inner product. The canonical
variables are present in the non polynomial way in the Wheeler–DeWitt
equation, what in itself gives rise to problematic analysis. As we have
mentioned earlier, the Wheeler–DeWitt equation has not been solved in general,
the only simplest Feynman’s path integral solutions are discussed and the
general integrability problem seems to be omitted. Another question is the
interpretation of both the wave functionals solving the Wheeler–DeWitt
equation as well as their normalization and superpositions. Factually, no
Dirac observable of the quantum geometrodynamics is known. Factually, both
classical and quantum geometrodynamics are time-independent evolutions, but
both the problem of time and therefore also the quantum evolution are still
unsolved.
The model of Quantum Cosmology presented in the previous chapter possesses the
hidden structure of the Wheeler superspace. In such a situation the
configurational space is the stratum of superspace called _minisuperspace_. We
shall continue studying of the $3+1$ decomposed metric fields, that are all
isotropic solutions of the Einstein field equations, and in itself create
another stratum of superspace called _midisuperspace_. The midisuperspace
models are not the most popular in the modern theoretical gravitational
physics, and in general quantum geometrodynamics by its functional nature has
a status of rather not a very well-defined mathematical theory than a theory
of quantum gravity possessing physical significance. By this reason we shall
present the new constructive analysis of such theories based on well
established methods of quantum field theory, which leads to plausible sounding
phenomenology. The plausibility is not a coincidence, but is the consequence
of application of the models of quantum field theory having established
meaning for physics. It must be emphasized that such a strategy is fully
justified for one-dimensional quantum gravities. Albeit, its both
applicability to and usefulness for another possible situations are the good
question. One can suppose _ad hoc_ that the one-dimensionality of quantum
gravity is its universal physical feature, and other situations are non
physical. However, such a reasoning in itself is the attempt to preserve _ad
hoc_ applicability of quantum field theory for theory of quantum gravity,
while recently the legitimateness of quantum field theoretic methods applied
to rather non usual situations in itself is a moot point. On the other hand,
everything what is widely applied and developed in theoretical physics is
primarily rooted in methodology of quantum field theory. The best example is
string theory which is a quantum field theory. By this reason, the necessity
of doing the construction of the adequate quantum field theoretic formalism of
quantum gravity, i.e. quantum theory of gravitational field or quantum field
theory of gravity, is logically argued. The logical arguments, however, must
not be satisfied by Nature, and by this reason the results received via the
adequate formalism must be empirically verified. Otherwise, the physical
meaning of the theory will be unclear.
Factually the quantum geometrodynamics based on the Wheeler–DeWitt equation is
the first constructive attempt to formulation of quantum General Relativity.
Actually, however, QGD has became the most influential motivation for
development of both other theories of quantum gravity based on QGD as well as
building of completely different formulations. In the further part of this
part we are going to present the model of quantum gravity strictly based on
the Wheeler-DeWitt equation (6.101) presented above. Basics and applications
of the ADM Hamiltonian approach to General Relativity, the classical and the
quantum geometrodynamics, and the Wheeler–DeWitt equation have been studied
intensively in the scientific and research literature since more than 50 years
(See _e.g._ the Refs. [298]-[568]). In the lack of other constructive
competitors the theory still is the theory of quantum gravitational fields,
and factually the only one having real chances for predictions of constructive
phenomenology.
#### E Other Approaches
Another point of view on quantum gravity follows from application of the
Ashtekar Hamiltonian formulation of General Relativity [569], which applies
the Einstein–Cartan theory with a complex connection. Rovelli and Smolin [570]
used Ashtekar’s new variables to investigation of the loop representation of
quantum General Relativity. This direction was developed by Ashtekar, Rovelli,
Smolin, Jacobson, and Lewandowski [571] and in the quantum cosmological
context by Bojowald [572]. The resulting theories are called loop quantum
gravity/cosmology and take into account the fundamental role of
diffeomorphisms, including the diffeomorphism constraint which does not play a
crucial role for dynamics in the quantum geometrodynamics formulated in terms
of the Wheeler–DeWitt equation. In loop quantum gravity important role plays
the Ashtekar–Lewandowski group. Recently, this research direction has been
received the well-established research status and is still under intensive
development (See, _e.g._ papers in the Ref. [573]).
The Arnowitt–Deser–Misner and Ashtekar Hamiltonian formulations of General
Relativity present different strategies. This heritage reflects in evident
differences between quantum geometrodynamics and loop quantum
gravity/cosmology. Quantum gravity formulated by the Wheeler–DeWitt equation
is treated as established theory, while loop quantum gravity similarly to
string theory is presently intensively developed. The attempts of quantum
geometrodynamics were practically obscured by the alternative approach, while
in itself Wheeler–DeWitt equation still needs development and is a source of
hidden constructive phenomenology.
There is a number of alternative evolution schemes, called numerical
relativity (For modern analysis see _e.g._ the Ref. [574]), which is not taken
into account in construction of quantum gravity. The privileged strong
position of the ADM and the Ashtekar formulations follows from their
straightforward roots in the Hamiltonian analysis, because of the primary
canonical quantization procedure follows from the Hamiltonian analysis.
Usually alternative evolution schemes are strictly based on these two
canonical formulations, or are its particular cases. The crucial issue which
connects all these schemes is a formulation of the Cauchy problem for the
Einstein field equations.
The pioneering approach to the Cauchy problem for General Relativity in the
case of analytic initial data was proposed by Darmois [575] in 1927 and
Lichnerowicz [576] in 1939. In 1944 Lichnerowicz [577] proposed the first
$3+1$ formalism based on the conformal decomposition of a spatial metric. In
1952 Fourès-Bruhat [578] formulated the Cauchy problem for $C^{5}$ initial
data via using of the local existence and uniqueness in harmonic coordinates,
what in 1956 resulted in the $3+1$ formalism in moving frame. In 1962
Arnowitt, Deser, and Misner [153] proposed the $3+1$ formalism based on the
Hamiltonian analysis of General Relativity. Soon after, in 1972, York [579]
considered gravitational dynamical degrees of freedom carried by the conformal
spatial metric, and in 1974 Ó Murchadha and York [580] introduced the
conformal transverse-traceless (CTT) method for solving the constraint
equations. In 1977 Smarr [581] considered 2D axisymmetric head-on collision of
two black holes and produced the first numerical solution beyond spherical
symmetry of the Cauchy problem for asymptotically flat spacetimes. In 1978
Smarr and York [582] proposed radiation gauge for numerical relativity what
resulted in the elliptic-hyperbolic system with asymptotic TT behavior. In
1983 Bardeen and Piran [583] considered 2D computations of partially
constrained schemes. Nakamura [584] in 1983, and Stark and Piran [585] in 1985
applied 2D axisymmetric gravitational collapse to a black hole. In 1986
Ashtekar [569] proposed new variables. In 1987 Nakamura, Oohara, and Kojima
[586] tested evolution of pure gravitational wave spacetimes in spherical
coordinates. In 1989 Bona and Masso [587], in 1995 Choquet-Bruhat and York
[588], in 2001 Kidder, Scheel and Teukolsky [589] considered the first-order
symmetric hyperbolic formulations of the Einstein field equations within the
$3+1$ formalism. Shibata and Nakamura [590] in 1995, and Baumgarte and Shapiro
[591] in 1999 investigated so called BSSN formulation, i.e. conformal
decomposition of the $3+1$ equations and promotion of some connection function
as an independent variable. In 1999 York [592] introduced the conformal thin-
sandwich (CTS) method for solving the constraint equations. In 2000 Shibata
[593] performed 3D full computation of binary neutron star merger, what was
the first full GR 3D solution of the Cauchy problem in the astrophysical
context. In 2000 Hayward [594] proposed a new scheme involving a dual-null
decomposition of space-time and removing second-order terms from the Einstein
field equations, which would vanish in the case of spherically symmetric
space-time. In 2004 Bonazzola, Gourgoulhon, Grandclément and Novak [595]
proposed the constrained scheme based on maximal slicing and Dirac’s gauge.
### Chapter 7 Global One-Dimensionality Conjecture
#### A Introduction
This chapter is devoted to the our proposition for theory of quantum gravity.
The role of quantum gravity is a fundamental problem of modern theoretical
physics. For instance, for lack of the consistent theory of quantum gravity we
are not able to understand physics of our Universe at the Planck scale.
Factually, despite a number of significant efforts (For various approaches see
_e.g._ Refs. [118], [138] and [172]-[220]), we are still very far of
understanding the role of quantized gravitational fields for physical
phenomena at high and ultra-high energies. In this chapter we propose a very
simple model of quantum gravity which can be useful for clarifying its some
important aspects. However, the simplicity of the theory of quantum gravity
presented here is far from triviality and is non obvious argument. In fact,
the model is proposed _ad hoc_ , but is strictly based on the Wheeler–DeWitt
equation, and in itself is a certain particular realization of this rather
general theory. Albeit, the our model is significantly simpler and by this
reason is able to generate new facts and apply the straightforward analogy
with the established phenomenological models of theoretical physics.
The field-theoretic formalism, so celebrated in modern physics, yields a
plausible phenomenology for a number of experimental data coming from a rich
spectrum of observations. In this chapter such a point of view is applied as
the base for construction of a simple theory of quantum gravity. We shall
perform the construction via the standard strategy resulting in the
Wheeler–DeWitt equation with, however, modified treatment of Matter fields and
the wave functional solving the Wheeler–DeWitt equation. The $3+1$ splitting
of a general relativistic metric tensor and the canonical primary quantization
of the appropriate Hamiltonian and diffeomorphism constraints are employed in
the way well-grounded in numerous approaches to quantization of gravitation.
The modification of the standard quantum geometrodynamics is based on the
_global one-dimensionality conjecture_ , which in itself in not beyond the
quantum geometrodynamics and arises from the straightforward and strict
analogy with the generic cosmological model [221] presented in the Chapter 4.
The crucial idea of the model is the ansatz which can be summarized by the
four brief phrases
1. 1.
Investigation of the global one-dimensionality conjecture, _i.e._ taking into
account a certain specific one-dimensional nature of Matter fields and the
wave functional,
2. 2.
Reduction of quantum geometrodynamics, resulting in the one-dimensional theory
characteristic for bosonic fields,
3. 3.
Application of the Hamilton equations of motion, yielding the corresponding
one-dimensional Dirac equation,
4. 4.
Expression of the supposition that the quantum gravity is a one-dimensional
field theory, and performing its secondary quantization.
The Hamilton equations of motion allow to establish the appropriate one-
dimensional Dirac equation and the corresponding Clifford algebra. The
secondary quantization, based on the Fock space and the diagonalization
procedure consisting of the Bogoliubov transformation and the Heisenberg
equations of motion, yields correctly defined quantum field theory formulated
in terms of the static Fock repère associated with initial data. We derive the
1D wave functional and discuss the corresponding 3-dimensional manifolds.
Quantum correlations of the field are associated with physical scales.
Mathematically, we employ the one-dimensional functional integrals, and
therefore despite the model of quantum gravity corresponds to the trend
initiated by S.W. Hawking and his collaborators [222]-[237] derivation of its
solutions is significantly simplified.
#### B The $\Gamma$-Scalar-Flat Space-times
Let us consider first the relation (6.73), i.e.
$S-\varrho=\dfrac{4\Lambda}{\kappa\ell_{P}^{2}}-\dfrac{{{}^{(4)}}R}{\kappa\ell_{P}^{2}}.$
(7.1)
In the light of the rule (6.42) one has
${{}^{(4)}}R={{}^{(3)}}R+K^{2}-K^{ij}K_{ij},$ (7.2)
where we have omitted the total derivative, because of its vanishing in
Hamiltonian analysis. Applying the Hamiltonian constraint
${{}^{(3)}}R+K^{2}-K^{ij}K_{ij}-2\Lambda-2\kappa\ell_{P}^{2}\varrho\approx 0,$
(7.3)
one obtains another relation between the energy density and the spatial stress
density
$\varrho=\dfrac{2\Lambda}{\kappa\ell_{P}^{2}}-S,$ (7.4)
which can be presented as the equation for the cosmological constant
$\Lambda=\dfrac{\kappa\ell_{P}^{2}}{2}(S+\varrho),$ (7.5)
and gives the insight into the nature of the cosmological constant. In other
words, the cosmological constant is an arithmetic mean of the spatial stress
density and the energy density multiplied by the Einstein constant $\kappa$.
One can, however, also apply the difference (6.70) $S-\varrho=T$ together with
the equation (7.5) and establish
$\displaystyle\varrho$ $\displaystyle=$
$\displaystyle\dfrac{\Lambda}{\kappa\ell_{P}^{2}}-\dfrac{T}{2},$ (7.6)
$\displaystyle S$ $\displaystyle=$
$\displaystyle\dfrac{\Lambda}{\kappa\ell_{P}^{2}}+\dfrac{T}{2}.$ (7.7)
By taking into account the fact $\varrho=T_{\mu\nu}n^{\mu}n^{\nu}$ the
equation (7.6) can be rewritten in the form
$T_{\mu\nu}\left(g^{\mu\nu}+2n^{\mu}n^{\nu}\right)=\dfrac{2\Lambda}{\kappa\ell_{P}^{2}},$
(7.8)
and solved immediately with respect to the stress-energy tensor
$T_{\mu\nu}=\dfrac{2\Lambda}{\kappa\ell_{P}^{2}}\dfrac{1}{g^{\mu\nu}+2n^{\mu}n^{\nu}}.$
(7.9)
Let us consider the RHS of this expression. The tensor coefficient multiplied
by $\dfrac{2\Lambda}{\kappa}$ can be rewritten in the form
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\dfrac{1}{g^{\mu\nu}+2n^{\mu}n^{\nu}}$
$\displaystyle=$
$\displaystyle\dfrac{(g_{\mu\nu}+2n_{\mu}n_{\nu})}{(g^{\mu\nu}+2n^{\mu}n^{\nu})(g_{\mu\nu}+2n_{\mu}n_{\nu})}=$
(7.10) $\displaystyle=$
$\displaystyle\dfrac{(g_{\mu\nu}+2n_{\mu}n_{\nu})}{g^{\mu\nu}g_{\mu\nu}+4(n^{\mu}n_{\mu})^{2}+4n^{\mu}n_{\mu}}=\dfrac{1}{4}(g_{\mu\nu}+2n_{\mu}n_{\nu}),$
where we have used the identities
$g^{\mu\nu}n_{\mu}n_{\nu}=g_{\mu\nu}n^{\mu}n^{\nu}=n^{\mu}n_{\mu}$,
$n_{\mu}n^{\mu}=-1$, and $g^{\mu\nu}g_{\mu\nu}=4$. By this reason one obtains
the stress-energy tensor
$T_{\mu\nu}=\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}g_{\mu\nu}+\dfrac{\Lambda}{\kappa\ell_{P}^{2}}n_{\mu}n_{\nu},$
(7.11)
which covariant form also can be derived easy
$T^{\mu\nu}=\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}g^{\mu\nu}+\dfrac{\Lambda}{\kappa\ell_{P}^{2}}n^{\mu}n^{\nu}.$
(7.12)
and can be unambiguously recognized as the stress-energy tensor of the perfect
fluid (See e.g. the Ref. [596])
$T^{\mu\nu}=pg^{\mu\nu}+\left(\dfrac{p}{c^{2}}+\mu\right)u^{\mu}u^{\nu},$
(7.13)
for which the four-velocity $u^{\mu}$ equals to the unit normal vector field
multiplied by the speed of light $c$, i.e.
$\displaystyle u^{\mu}$ $\displaystyle=$ $\displaystyle cn^{\mu},$ (7.14)
$\displaystyle u^{\mu}u_{\mu}$ $\displaystyle=$ $\displaystyle-c^{2},$ (7.15)
and the isotropic pressure $p$ and the mass density $\mu$ are as follows
$\displaystyle p$ $\displaystyle=$
$\displaystyle\dfrac{\Lambda}{2\kappa\ell_{P}^{2}},$ (7.16) $\displaystyle\mu$
$\displaystyle=$ $\displaystyle\dfrac{\Lambda}{2c^{2}\kappa\ell_{P}^{2}}.$
(7.17)
Now the trace of the stress-energy tensor can be established by
straightforward easy computation
$\displaystyle T$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}T_{\mu\nu}=\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}g^{\mu\nu}g_{\mu\nu}+\dfrac{\Lambda}{\kappa\ell_{P}^{2}}g^{\mu\nu}n_{\mu}n_{\nu}=\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}4+\dfrac{\Lambda}{\kappa\ell_{P}^{2}}n^{\mu}n_{\mu}=$
(7.18) $\displaystyle=$
$\displaystyle\dfrac{2\Lambda}{\kappa\ell_{P}^{2}}-\dfrac{\Lambda}{\kappa\ell_{P}^{2}}=\dfrac{\Lambda}{\kappa\ell_{P}^{2}},$
and consequently the energy density (7.6) and the spatial stress density (7.7)
have the values
$\displaystyle\varrho$ $\displaystyle=$
$\displaystyle\dfrac{\Lambda}{2\kappa\ell_{P}^{2}},$ (7.19) $\displaystyle S$
$\displaystyle=$ $\displaystyle\dfrac{3\Lambda}{2\kappa\ell_{P}^{2}}.$ (7.20)
The momentum density related to such a situation can be established
straightforwardly
$\displaystyle J^{i}$ $\displaystyle=$ $\displaystyle T_{\mu\nu}n^{\mu}h^{\nu
i}=\left(\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}g_{\mu\nu}+\dfrac{\Lambda}{\kappa\ell_{P}^{2}}n_{\mu}n_{\nu}\right)n^{\mu}h^{\nu
i}=\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}n_{\nu}h^{\nu
i}-\dfrac{\Lambda}{\kappa\ell_{P}^{2}}n_{\nu}h^{\nu i}=$ (7.21)
$\displaystyle=$
$\displaystyle-\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}n_{\nu}h^{\nu
i}=-\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}n_{\nu}\left(g^{\nu
i}+n^{\nu}n^{i}\right)=-\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}\left(n^{i}-n^{i}\right)=0.$
By this reason the classical geometrodynamics becomes
${2c\kappa}G_{ijkl}\dfrac{\delta{S[g]}}{\delta{h_{ij}}}\dfrac{\delta{S[g]}}{\delta{h_{kl}}}+\dfrac{\ell_{P}^{2}}{2c\kappa}\sqrt{h}\left({{}^{(3)}}R-3\Lambda\right)=0,$
(7.22)
while the Wheeler–DeWitt equation is
$\left\\{2c\kappa\dfrac{\hslash^{2}}{\ell_{P}^{2}}G_{ijkl}\dfrac{\delta^{2}}{\delta
h_{ij}\delta
h_{kl}}+\dfrac{\ell_{P}^{2}}{2c\kappa}\sqrt{h}\left({{}^{(3)}R}-3\Lambda\right)\right\\}\Psi[h_{ij},\phi]=0,$
(7.23)
and the quantized diffeomorphism constraint is
$i\dfrac{E_{P}}{\ell_{P}^{2}}\left(\partial_{j}+\dfrac{1}{2}h_{jl,k}h^{kl}\right)\dfrac{\delta{\Psi}[h_{ij},\phi]}{\delta{h_{ij}}}=0.$
(7.24)
In other words, in such a situation both the classical and quantum
geometrodynamics become purely geometrical.
Good question is what is the Lagrangian of Matter fields describing such a
situation. Rewriting the stress-energy tensor (7.11) in the form
$T_{\mu\nu}=\dfrac{1}{2}\left(3p+\mu c^{2}\right)g_{\mu\nu},$ (7.25)
and using of the definition (6.54) of $T_{\mu\nu}$ following from the
Hilbert–Palatini action principle one obtains the equation
$-\dfrac{2}{\sqrt{-g}}\dfrac{\delta}{\delta
g^{\mu\nu}}\left(\sqrt{-g}L_{\phi}\right)=\dfrac{1}{2}\left(3p+\mu
c^{2}\right)g_{\mu\nu},$ (7.26)
which can be presented in equivalent form
$\delta\left(\sqrt{-g}L_{\phi}\right)=-\dfrac{1}{4}\left(3p+\mu
c^{2}\right)\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}.$ (7.27)
Using of the Jacobi formula for differentiating a determinant
$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}=-gg_{\mu\nu}\delta g^{\mu\nu},$ (7.28)
allows to write the equation (7.27) as
$L_{\phi}\delta\sqrt{-g}+\sqrt{-g}\delta{L_{\phi}}=\dfrac{1}{2}\left(3p+\mu
c^{2}\right)\delta\sqrt{-g}.$ (7.29)
Because, however, the parameters $p$ and $\mu$ are constant one has uniquely
$\displaystyle L_{\phi}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(3p+\mu c^{2}\right),$ (7.30)
$\displaystyle\delta{L_{\phi}}$ $\displaystyle=$ $\displaystyle 0.$ (7.31)
Applying the relations (7.16) and (7.17) one receives finally
$L_{\phi}=\dfrac{\Lambda}{\kappa\ell_{P}^{2}}=-\varrho.$ (7.32)
The spatial stress density $S=h^{ij}S_{ij}$ together with the formula (7.20)
can be used for derivation of the spatial stress tensor
$S_{ij}=\dfrac{S}{3}h_{ij}=\dfrac{\Lambda}{2\kappa\ell_{P}^{2}}h_{ij},$ (7.33)
and together with the difference
$S-\varrho=\dfrac{\Lambda}{\kappa\ell_{P}^{2}},$ (7.34)
allow to establish the tensor
$-\kappa\ell_{P}^{2}\left[S_{ij}-\dfrac{1}{2}h_{ij}(S-\varrho)\right]=0.$
(7.35)
Hence in such a situation the evolutionary equations for the extrinsic
curvature tensor and the intrinsic curvature are given by
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}{K}_{ij}$
$\displaystyle=$ $\displaystyle-
N_{|ij}+N\left(R_{ij}+KK_{ij}-2K_{ik}K^{k}_{j}\right)+N^{k}K_{ij|k}+K_{ik}N^{k}_{|j}+K_{jk}N^{k}_{|i},$
(7.36)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}K$
$\displaystyle=$
$\displaystyle-h^{ij}N_{|ij}+N\left(K^{ij}K_{ij}+\Lambda\right)+N^{i}K_{|i},$
(7.37)
while the evolutionary equations for induced metrics and its determinant
remain unchanged, i.e.
$\displaystyle\partial_{t}{h}_{ij}$ $\displaystyle=$ $\displaystyle
N_{i|j}+N_{j|i}-2NK_{ij},$ (7.38) $\displaystyle\partial_{t}\ln\sqrt{h}$
$\displaystyle=$ $\displaystyle-NK+N^{i}_{|i}.$ (7.39)
However, the spatial stress tensor (7.33) is not unique. Because of
$h^{ij}n_{i}n_{j}=n^{i}n_{i}=1$, there are another inequivalent choices
$\displaystyle S_{ij}$ $\displaystyle=$ $\displaystyle Sn_{i}n_{j},$ (7.40)
$\displaystyle S_{ij}$ $\displaystyle=$
$\displaystyle\dfrac{S}{4}\left(h_{ij}+n_{i}n_{j}\right),$ (7.41)
$\displaystyle S_{ij}$ $\displaystyle=$
$\displaystyle\dfrac{S}{2}\left(h_{ij}-n_{i}n_{j}\right),$ (7.42)
and also much more general form
$S_{ij}=\dfrac{S}{3\alpha+\beta}\left(\alpha h_{ij}+\beta
n_{i}n_{j}\right)=S_{ij}(\alpha,\beta),$ (7.43)
where $\alpha$ and $\beta$ are any numbers. All these forms of the spatial
stress tensor have the same value of trace, but in the basis $(h_{ij},n_{i})$
the two-parameter family (7.43) is the most general solution. In other words
all relations between the spatial stress density $S$ and the energy density
$\varrho$ are validate when the spatial stress tensor has a form (7.43). It is
easy to see that the following Poisson algebra is satisfied
$\left\\{S_{ij}(\alpha,\beta),S_{kl}(\alpha^{\prime},\beta^{\prime})\right\\}=S\dfrac{\alpha\alpha^{\prime}\chi_{ijkl}+\beta\beta^{\prime}\lambda_{ijkl}+\alpha\beta^{\prime}\zeta_{ijkl}+\alpha^{\prime}\beta\zeta_{klij}}{9\alpha\alpha^{\prime}+\beta\beta^{\prime}+3\alpha\beta^{\prime}+3\alpha^{\prime}\beta},$
(7.44)
where we have introduced the tensors
$\displaystyle\chi_{ijkl}$ $\displaystyle=$
$\displaystyle\left\\{h_{ij},h_{kl}\right\\},$ (7.45)
$\displaystyle\lambda_{ijkl}$ $\displaystyle=$
$\displaystyle\left\\{n_{i}n_{j},n_{k}n_{l}\right\\},$ (7.46)
$\displaystyle\zeta_{ijkl}$ $\displaystyle=$
$\displaystyle\left\\{h_{ij},n_{k}n_{l}\right\\}.$ (7.47)
By the context we shall call (7.44 _the stress algebra_. Let us consider the
RHS of the equation (7.44). When the bracketed quantities are classical (C),
i.e. are not operators corresponding to the classical quantities, the Poisson
brackets are easy to establish. Let us denote such a classical Poisson
brackets as
$\sigma_{ijkl}(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=\left\\{S_{ij}(\alpha,\beta),S_{kl}(\alpha^{\prime},\beta^{\prime})\right\\}_{C},$
(7.48)
or in explicit form
$\sigma_{ijkl}(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=2S\dfrac{\alpha\alpha^{\prime}h_{ij}h_{kl}+\beta\beta^{\prime}n_{i}n_{j}n_{k}n_{l}+\alpha\beta^{\prime}h_{ij}n_{k}n_{l}+\alpha^{\prime}\beta
n_{i}n_{j}h_{kl}}{9\alpha\alpha^{\prime}+\beta\beta^{\prime}+3\alpha\beta^{\prime}+3\alpha^{\prime}\beta}.$
(7.49)
It can be seen by straightforward computation that
$\displaystyle\sigma_{jl}(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=h^{ik}\sigma_{ijkl}=2S\dfrac{\alpha\alpha^{\prime}h_{jl}+\left(\beta\beta^{\prime}+\alpha\beta^{\prime}+\alpha^{\prime}\beta\right)n_{j}n_{l}}{9\alpha\alpha^{\prime}+\beta\beta^{\prime}+3\alpha\beta^{\prime}+3\alpha^{\prime}\beta},$
(7.50)
$\displaystyle\sigma(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=h^{jl}\sigma_{jl}=2S\dfrac{3\alpha\alpha^{\prime}+\beta\beta^{\prime}+\alpha\beta^{\prime}+\alpha^{\prime}\beta}{9\alpha\alpha^{\prime}+\beta\beta^{\prime}+3\alpha\beta^{\prime}+3\alpha^{\prime}\beta}.$
(7.51)
By this reason for generality let us consider the spatial stress tensor
(7.43). Then one has
$-\kappa\ell_{P}^{2}\left[S_{ij}-\dfrac{1}{2}h_{ij}(S-\varrho)\right]=\dfrac{\Lambda}{2}\dfrac{\beta}{3\alpha+\beta}\left(h_{ij}-3n_{i}n_{j}\right).$
(7.52)
In this manner the evolutionary equations (7.37), (7.38), and (7.39) remain
unchanged, but the evolution of extrinsic curvature tensor is
$\displaystyle\partial_{t}{K}_{ij}$ $\displaystyle=$ $\displaystyle-
N_{|ij}+N\left[R_{ij}+KK_{ij}-2K_{ik}K^{k}_{j}+\dfrac{\Lambda}{2}\dfrac{\beta}{3\alpha+\beta}\left(h_{ij}-3n_{i}n_{j}\right)\right]+$
(7.53) $\displaystyle+$ $\displaystyle
N^{k}K_{ij|k}+K_{ik}N^{k}_{|j}+K_{jk}N^{k}_{|i}$
However, in the light of the Einstein field equations one can express the
trace of stress-energy tensor via the Ricci scalar curvature and the
cosmological constant, i.e.
$T=-\dfrac{{{}^{(4)}}R}{\kappa\ell_{P}^{2}}+\dfrac{4\Lambda}{\kappa\ell_{P}^{2}}$
(7.54)
Applying this fact to the equation (7.18) one obtains
${{}^{(4)}}R\equiv 3\Lambda,$ (7.55)
what means that the space-time manifold is four-dimensional pseudo-Riemannian
manifold of constant Ricci scalar curvature given by the cosmological constant
up to constant multiplier equal to the dimensionality of an embedded space
$D=3$. In the light of the relation (7.11) the RHS of the Einstein field
equations (6.6) is
$\kappa\ell_{P}^{2}T_{\mu\nu}=\dfrac{1}{2}\Lambda g_{\mu\nu}+\Lambda
n_{\mu}n_{\nu},$ (7.56)
while the LHS of the Einstein field equations is
$R_{\mu\nu}-\dfrac{1}{2}{{}^{(4)}}Rg_{\mu\nu}+\Lambda
g_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}\Lambda g_{\mu\nu},$ (7.57)
and by this reason the Einstein field equations for such a situation are
$R_{\mu\nu}=\Lambda\left(g_{\mu\nu}+n_{\mu}n_{\nu}\right)=\Lambda h_{\mu\nu},$
(7.58)
where we have applied the completeness relations (6.23), in which
$h_{\mu\nu}=h_{ij}e^{i}_{\mu}e^{j}_{\nu}.$ (7.59)
Let us consider the contravariant form of $h_{\mu\nu}$
$h^{\mu\nu}=g^{\mu\kappa}g^{\nu\lambda}h_{\kappa\lambda}=h_{ij}g^{\mu\kappa}e^{i}_{\kappa}g^{\nu\lambda}e^{j}_{\lambda}=h^{kl}(h_{ik}g^{\mu\kappa}e^{i}_{\kappa})(h_{jl}g^{\nu\lambda}e^{j}_{\lambda})=h^{kl}e^{\mu}_{k}e^{\nu}_{l},$
(7.60)
where we have applied the notation
$e^{\mu}_{k}=h_{ik}g^{\mu\kappa}e^{i}_{\mu}.$ (7.61)
Applying the inverted way one obtains
$h_{\mu\nu}=g_{\mu\kappa}g_{\nu\lambda}h^{\kappa\lambda}=g_{\mu\kappa}g_{\nu\lambda}h^{kl}e^{\kappa}_{k}e^{\lambda}_{l}=h^{kl}(g_{\mu\kappa}e^{\kappa}_{k})(g_{\nu\lambda}e^{\lambda}_{l})=h^{kl}e_{\mu
k}e_{\nu l},$ (7.62)
where we have applied the notation
$e_{\nu k}=g_{\mu\nu}e^{\mu}_{k},$ (7.63)
following from transition between the completeness relations for metric (6.23)
and the completeness relations for inverse metric (6.26). In this manner the
equation (7.58) bacomes
$R_{\mu\nu}=\left(\Lambda
g_{\mu\kappa}g_{\nu\lambda}\right)h^{ij}e^{\kappa}_{i}e^{\lambda}_{j},$ (7.64)
and application of the Ricci curvature tensor evaluated on the three-boundary
(6.33)
$R_{\mu\nu}=-R_{\kappa\mu\lambda\nu}n^{\kappa}n^{\lambda}+R_{\kappa\mu\lambda\nu}h^{ij}e^{\kappa}_{i}e^{\lambda}_{j},$
(7.65)
to the equation (7.64) leads to the system of equations
$\displaystyle\left\\{\begin{array}[]{cc}R_{\kappa\mu\lambda\nu}n^{\kappa}n^{\lambda}=0\vspace*{10pt}\\\
\left(R_{\kappa\mu\lambda\nu}-\Lambda
g_{\kappa\mu}g_{\lambda\nu}\right)h^{\kappa\lambda}=0\end{array}\right..$
(7.68)
The first equation in (7.68) expresses the property that the double projection
of the Riemann–Christoffel curvature tensor onto the unit normal vector field
vanishes, while the second one expresses the fact that the projection onto the
metric $h^{\kappa\lambda}$ of the tensor
$A_{\kappa\mu\lambda\nu}:=R_{\kappa\mu\lambda\nu}-\Lambda
g_{\kappa\mu}g_{\lambda\nu},$ (7.69)
vanishes. In other words, in the space-time is characterized by the
Riemann–Christoffel curvature tensor
$R_{\kappa\mu\lambda\nu}=\Lambda
g_{\kappa\mu}g_{\lambda\nu}+A_{\kappa\mu\lambda\nu},$ (7.70)
where $A_{\kappa\mu\lambda\nu}$ is the tensor satisfying the equations
$\displaystyle\Lambda
n_{\mu}n_{\nu}+A_{\kappa\mu\lambda\nu}n^{\kappa}n^{\lambda}$ $\displaystyle=$
$\displaystyle 0,$ (7.71) $\displaystyle
A_{\kappa\mu\lambda\nu}h^{\kappa\lambda}$ $\displaystyle=$ $\displaystyle 0,$
(7.72)
where the first equation follows from the first equation of the system (7.68).
The equation (7.71) projected onto $n_{\kappa}n_{\lambda}$ and leads to
$A_{\kappa\mu\lambda\nu}=-\Lambda
n_{\mu}n_{\nu}n_{\kappa}n_{\lambda}+\Gamma_{\kappa\mu\lambda\nu},$ (7.73)
where the tensor $\Gamma_{\kappa\mu\lambda\nu}$ satisfying the equations
$\displaystyle\Gamma_{\kappa\mu\lambda\nu}n^{\kappa}n^{\lambda}$
$\displaystyle=$ $\displaystyle 0,$ (7.74)
$\displaystyle\Gamma_{\kappa\mu\lambda\nu}h^{\kappa\lambda}$ $\displaystyle=$
$\displaystyle 0,$ (7.75)
and the second equation was deduced from application of the tensor (7.73) to
the equation (7.72). In this manner, by application of the completeness
relations for the metric $g_{\mu\nu}$, the Riemann–Christoffel curvature
tensor describing the considered space-time has a form
$R_{\kappa\mu\lambda\nu}=\Lambda\left(g_{\kappa\mu}g_{\lambda\nu}-n_{\mu}n_{\nu}n_{\kappa}n_{\lambda}\right)+\Gamma_{\kappa\mu\lambda\nu},$
(7.76)
with the tensor $\Gamma_{\kappa\mu\lambda\nu}$ being a solution of the
equations (7.74) and (7.75). The problem is to solve the system (7.74)–(7.75)
in general, but we shall not perform this procedure in this book. The
Riemann–Christoffel curvature tensor (7.76) in general describes all four-
dimensional space-times for which the Ricci scalar curvature is
${{}^{(4)}}R=3\Lambda$ and the Ricci curvature tensor is
$R_{\mu\nu}=\Lambda(g_{\mu\nu}+n_{\mu}n_{\nu})$. Interestingly, one can
compute these curvatures immediately with using of (7.76)
$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\kappa\lambda}R_{\kappa\mu\lambda\nu}=\Lambda\left(g_{\mu\nu}+n_{\mu}n_{\nu}\right)+\Gamma_{\mu\nu},$
(7.77) $\displaystyle{{}^{(4)}}R$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}R_{\mu\nu}=3\Lambda+{{}^{(4)}}\Gamma,$ (7.78)
what gives equations for the contractions of the tensor
$\Gamma_{\kappa\mu\lambda\nu}$
$\displaystyle\Gamma_{\mu\nu}$ $\displaystyle:=$ $\displaystyle
g^{\kappa\lambda}\Gamma_{\kappa\mu\lambda\nu}=0,$ (7.79)
$\displaystyle{{}^{(4)}}\Gamma$ $\displaystyle:=$ $\displaystyle
g^{\kappa\lambda}\Gamma_{\mu\nu}=0.$ (7.80)
Moreover, double projection of the Ricci curvature tensor (7.77) onto the unit
normal vector field leads to
$R_{\mu\nu}n^{\mu}n^{\nu}=\Gamma_{\mu\nu}n^{\mu}n^{\nu},$ (7.81)
i.e. in this projective sense the $\Gamma_{\mu\nu}$ curvature tensor carries
the same information as the Ricci curvature tensor.
Interestingly, one can computed the Weyl curvature tensor
$\displaystyle W_{\mu\kappa\nu\lambda}$ $\displaystyle=$ $\displaystyle
R_{\mu\kappa\nu\lambda}+\dfrac{1}{2}\left(g_{\mu\lambda}R_{\nu\kappa}+g_{\kappa\nu}R_{\lambda\mu}-g_{\mu\nu}R_{\lambda\kappa}-g_{\kappa\lambda}R_{\nu\mu}\right)+$
(7.82) $\displaystyle+$
$\displaystyle\dfrac{1}{3}{{}^{(4)}}R\left(g_{\mu\nu}g_{\lambda\kappa}-g_{\mu\lambda}g_{\nu\kappa}\right),$
which for the considered situation takes the form
$\\!\\!\\!\\!\\!W_{\mu\kappa\nu\lambda}=\Gamma_{\kappa\mu\lambda\nu}+\Lambda\left(g_{\kappa\mu}g_{\lambda\nu}+2g_{\mu[\lambda}g_{\nu]\kappa}+4g_{[\mu(\lambda}n_{\nu)}n_{\kappa]}-n_{\mu}n_{\nu}n_{\kappa}n_{\lambda}\right).$
(7.83)
It means that the curvature tensor $\Gamma_{\kappa\mu\lambda\nu}$ is not the
Weyl tensor. The contractions of the Weyl curvature tensor are
$\displaystyle W_{\kappa\lambda}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}W_{\mu\kappa\nu\lambda}=\Gamma_{\kappa\lambda}-\Lambda\left(g_{\kappa\lambda}+n_{\kappa}n_{\lambda}\right)=2\Gamma_{\kappa\lambda}-R_{\kappa\lambda},$
(7.84) $\displaystyle{{}^{(4)}}W$ $\displaystyle=$ $\displaystyle
g^{\kappa\lambda}W_{\kappa\lambda}=2{{}^{(4)}}\Gamma-{{}^{(4)}}R=\Gamma-3\Lambda,$
(7.85)
what means that in the particular case considered in this section
$\displaystyle\Gamma_{\kappa\lambda}$ $\displaystyle=$
$\displaystyle\dfrac{R_{\kappa\lambda}+W_{\kappa\lambda}}{2},$ (7.86)
$\displaystyle{{}^{(4)}}\Gamma$ $\displaystyle=$
$\displaystyle\dfrac{{{}^{(4)}}R+{{}^{(4)}}W}{2}.$ (7.87)
Let us call $\Gamma_{\mu\nu}$ _the $\Gamma$ curvature tensor_, and
${{}^{(4)}}\Gamma$ _the $\Gamma$ scalar curvature_. Then the space-times
considered above is $\Gamma_{\mu\nu}$-flat manifold of zero $\Gamma$ scalar
curvature, which we shall call _the $\Gamma$-scalar-flat manifolds_. The
equations (7.77) and (7.78) can be used for construction of the LHS of the
Einstein field equations
$G_{\mu\nu}+\Lambda g_{\mu\nu}=\dfrac{1}{2}\Lambda g_{\mu\nu}+\Lambda
n_{\mu}n_{\nu}+\Gamma_{\mu\nu}+\dfrac{1}{2}{{}^{(4)}}\Gamma g_{\mu\nu},$
(7.88)
and because of the stress-energy tensor is given by (7.11) the RHS of the
Einstein field equations is
$\kappa\ell_{P}^{2}T_{\mu\nu}=\dfrac{1}{2}\Lambda g_{\mu\nu}+\Lambda
n_{\mu}n_{\nu},$ (7.89)
and by this reason the Einstein field equations expressed via the $\Gamma$
curvatures takes the form of the vacuum field equations
$\Gamma_{\mu\nu}+\dfrac{1}{2}{{}^{(4)}}\Gamma g_{\mu\nu}=0.$ (7.90)
In other words the $\Gamma$ curvatures are the curvatures which for blatantly
non stationary space-time given by the Ricci scalar curvature
${{}^{(4)}}R=3\Lambda$, the Ricci curvature tensor
$R_{\mu\nu}=\Lambda(g_{\mu\nu}+n_{\mu}n_{\nu})$, and the stress-energy tensor
(7.89) makes the non stationary solution of the Einstein field equations the
space-time obeying vacuum field equations (7.90). The constructive hypothesis
is
###### Hypothesis (The $\Gamma$ Curvatures Hypothesis).
In general the $\Gamma$ curvatures transforming non stationary four-
dimensional Einstein field equations to the vacuum field equations (7.90) can
be constructed the only via using of the Riemann–Christoffel curvature tensor,
its contractions with space-time metric, and combinations of all these
quantities.
The Ricci curvature tensor (7.58) can be presented in the form
$R_{\mu\nu}=\Lambda g_{\mu\nu}+\Lambda n_{\mu}n_{\nu},$ (7.91)
and by this reason the second term on RHS of the equation (7.58) can be
interpreted as the correction to the four-dimensional Einstein manifold, i.e.
the four-dimensional Riemannian manifold for which Ricci curvature tensor is
proportional to metric $R_{\mu\nu}=\lambda g_{\mu\nu}$ and therefore the
scalar curvature is constant ${{}^{(4)}}R=4\lambda$ (For advanced discussion
of general Einstein manifolds e.g. the well-known Besse’s book [597]), defined
by the sign identical to the cosmological constant $\lambda=\Lambda$. In other
words the situation presented in this section corresponds to deformation of
the four-dimensional Einstein manifolds of sign $\lambda=\Lambda$
$R_{\mu\nu}=\Lambda g_{\mu\nu}+\Delta_{\mu\nu},$ (7.92)
where $\Delta_{\mu\nu}=\Lambda n_{\mu}n_{\nu}$ is _the deformation curvature
tensor_ , for which
${{}^{(4)}}R=4\Lambda+\Delta,$ (7.93)
where $\Delta=g^{\mu\nu}\Delta_{\mu\nu}=-\Lambda$ is _the deformation scalar
curvature_. Comparison of the result (7.93) with the equation (7.78) leads to
expression of the $\Gamma$ scalar curvature via the deformation scalar
curvature
$\Gamma=\Lambda+\Delta.$ (7.94)
For vanishing cosmological constant $\Lambda\equiv 0$ one has to deal with the
four-dimensional Ricci-flat space-time manifold of zero Ricci scalar
curvature, which we shall call _the Ricci-scalar-flat manifold_. This is
however, the result of the fact that we have computed the value of the
cosmological constant by using of (6.42), i.e. the Ricci scalar curvature
evaluated on the boundary $\partial M$. It means that in such a particular
case the enveloping space-time is the Ricci-scalar-flat manifold from the
point of view of an embedded space. It does not mean, however, that then
space-time is flat in general, because of its the Riemann–Christoffel
curvature tensor must not be vanishing identically when both the Ricci
curvature tensor and the Ricci scalar curvature are trivialized. This is in
itself non trivial result because in such a situation both the cosmological
constant and as well as the stress-energy tensor are in general non vanishing
and arbitrary, what suggests that from the space point of view space-time
looks like vacuum space-time. Such a situation, however, should be rather
understood rather as a local property, i.e. related to quantum gravity given
by the Wheeler–DeWitt equation, than the classical space-time. Moreover, it
must be emphasized that topology of such a space-time is still unrestricted,
because of there is a lot of possible topologies of a four-dimensional Ricci-
scalar-flat manifold. The Ricci-flat manifolds are the particular case of the
Einstein manifolds, for which the sign is trivial $\lambda=0$. Such a class of
the Einstein manifolds include e.g. the Calabi–Yau manifolds [598] and the
hyper-Kähler manifolds [599] which are in intensive interest of mathematical
and theoretical physicists (For some particular applications see e.g. papers
in the Ref. [600]), especially in context of string theory. In a four-
dimensional case every Calabi–Yau manifolds is hyper-Kähler manifold. There
are also much more simpler solutions of the vacuum Einstein field equations.
For example the flat Minkowski space-time is the most simple vacuum solution,
and nontrivial situations include the Schwarzschild space-time and the Kerr
space-time describing the geometry of space-time around a non-rotating
spherical mass and a rotating massive body, respectively.
#### C The Ansatz for Wave Functionals
According to the evaluation (7.2) the Ricci scalar curvature ${{}^{(4)}}R$ of
the enveloping space-time manifold expresses via the Ricci scalar curvature
${{}^{(3)}}R$ of the embedded space, and its extrinsic $K_{ij}$ and intrinsic
$K$ curvatures. All these embedding characterizations in general are
functionals of an induced metric $h_{ij}$. Moreover, the trace of the spatial
stress $S\equiv T(h,h)$ as double projection of the stress-energy tensor on an
induced metric is also a functional of $h_{ij}$. The cosmological constant can
be treated as a constant functional of $h_{ij}$. In this manner, the energy
density $\varrho$ is at the most a functional of $h_{ij}$, and by this reason
in an arbitrary situation one has the functional dependence
$\varrho[h_{ij}]=-\dfrac{4\Lambda}{\kappa\ell_{P}^{2}}+S[h_{ij}]+\dfrac{1}{\kappa\ell_{P}^{2}}\left({{}^{(3)}}R+K^{2}-K^{ij}K_{ij}\right)=\dfrac{2\Lambda}{\kappa\ell_{P}^{2}}-S[h_{ij}],$
(7.95)
which allows to establish
$\displaystyle S[h_{ij}]$ $\displaystyle=$
$\displaystyle\dfrac{3\Lambda}{\kappa\ell_{P}^{2}}-\dfrac{1}{2\kappa\ell_{P}^{2}}\left({{}^{(3)}}R+K^{2}-K^{ij}K_{ij}\right),$
(7.96) $\displaystyle\varrho[h_{ij}]$ $\displaystyle=$
$\displaystyle-\dfrac{\Lambda}{\kappa\ell_{P}^{2}}+\dfrac{1}{2\kappa\ell_{P}^{2}}\left({{}^{(3)}}R+K^{2}-K^{ij}K_{ij}\right).$
(7.97)
Factually, both the relations (7.96) and (7.97) are the results of application
of the Hamiltonian constraint, i.e. strictly speaking they have a sense only
for geometrodynamics. The functional nature of their LHS is a straightforward
conclusion of the functional character of their RHS. Such a situation implies
non trivial physical content. Namely, because of both the spatial stress
density $S$ and the energy density $\varrho$ are projections of the stress-
energy tensor of Matter fields, they depend on Matter fields and their
derivatives. In this manner by the functional nature of (7.96) and (7.97) one
can conclude that such a situation is equivalent to the statement that Matter
fields are functionals of $h_{ij}$,
$\phi=\phi[h_{ij}],$ (7.98)
say. In this manner the DeWitt wave functional
$\Psi[h_{ij},\phi]=\Psi[h_{ij},\phi[h_{ij}]]\equiv\Psi[h_{ij}],$ (7.99)
is fully justified, and the Wheeler–DeWitt equation becomes
$\left\\{2c\kappa\dfrac{\hslash^{2}}{\ell_{P}^{2}}G_{ijkl}\dfrac{\delta^{2}}{\delta
h_{ij}\delta
h_{kl}}+\dfrac{\ell_{P}^{2}}{2c\kappa}\sqrt{h}\left({{}^{(3)}R}[h_{ij}]-2\Lambda-2\kappa\ell_{P}^{2}\varrho[h_{ij}]\right)\right\\}\Psi[h_{ij}]=0.$
(7.100)
Anyway, however, the crucial general problem is solving the Wheeler–DeWitt
equation in general. As we have mentioned earlier the Wheeler–DeWitt equation
has never been solved in general, and even taking into account the DeWitt wave
functional does not simplify this general problem because of $\Psi[h_{ij}]$ is
still a functional but not function. It means that it is not clear how to
treat $\Psi[h_{ij}]$ mathematically. We shall present here the strategy for
solution of the Wheeler–DeWitt equation which is based on the DeWitt wave
functional but reduces the functional $\Psi[h_{ij}]$ to a function. In itself
such a reduction defines a new model of quantum gravity within the quantum
geometrodynamics formulated in terms of the Wheeler–DeWitt equation.
To start the deductions, we should rethink the quantum geometrodynamics
(7.100), particularly the structure of the DeWitt wave functional
$\Psi[h_{ij}]$. The fundamental interpretation of the Wheeler–DeWitt equation,
as the result of the primary canonical quantization, is the Schrödinger
equation or the Klein–Gordon equation. In both these situations, however, a
wave function is always a scalar field. The operator acting on the wave
functional in the quantum geometrodynamics (7.100) is always scalar and is a
functional on the configurational space, i.e. here the Wheeler superspace
$2c\kappa\dfrac{\hslash^{2}}{\ell_{P}^{2}}G_{ijkl}\dfrac{\delta^{2}}{\delta
h_{ij}\delta
h_{kl}}+\dfrac{\ell_{P}^{2}}{2c\kappa}\sqrt{h}\left({{}^{(3)}R}[h_{ij}]-2\Lambda-2\kappa\ell_{P}^{2}\varrho[h_{ij}]\right)=\mathcal{\hat{O}}[h_{ij}],$
(7.101)
what is similar to the case of the Schrödinger or the Klein–Gordon equation,
in which the operator acting on the wave function is a functional on the
configurational space, i.e. the product space $\mathbb{R}^{4}$. Moreover, the
differential operator of the Wheeler–DeWitt equation is the $Dif\\!f(\partial
M)$-invariant. It suggests clearly that the DeWitt wave functional
$\Psi[h_{ij}]$ must be a function invariant with respect to action of the
diffeomorphism group, i.e. must be a function of another $Dif\\!f(\partial
M)$-invariant quantities. Furthermore, for full consistency these
diffeoinvariant quantities must be constructed via using of the induced metric
$h_{ij}$, $f=f(h_{ij})=inv$, say. Then, however, by the Kuchař formalism the
wave functional $\Psi(f)$ inevitably will be becoming an observable or a
perennial, and above all if one expresses the differential operator (7.101)
via these invariant quantities then one can treat these invariants as solution
of the problem of time in quantum geometrodynamics by identification of the
time $t$ with the invariant of an induced metric, i.e. $t\equiv f$. If a wave
functional is an usual function then also one can perform straightforwardly
and in extraordinary simply way the formalism of secondary quantization and
product the theory of quantum gravity which is the quantum field theory of
gravity. Let us apply such a strategy for quantum geometrodynamics.
###### Step 1: Global One-Dimensionality Conjecture
By the DeWitt construction based on the Wheeler metric representation
$\Psi[h_{ij}]$ is a functional of the $3\times 3$ symmetric matrix of an
induced metric. It suggests that the wave functional is a single functional
$\Psi[h_{ij}]=\Psi\left[\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\\\
h_{12}&h_{22}&h_{23}\\\ h_{13}&h_{23}&h_{33}\end{array}\right]\right].$
(7.102)
However, such a reasoning is not unique. The wave functional must not be a
single functional but rather is a $3\times 3$ symmetric matrix which elements
are dependent on a single element of an induced metric
$\Psi[h_{ij}]=\left[\begin{array}[]{ccc}\Psi[h_{11}]&\Psi[h_{12}]&\Psi[h_{13}]\\\
\Psi[h_{12}]&\Psi[h_{22}]&\Psi[h_{23}]\\\
\Psi[h_{13}]&\Psi[h_{23}]&\Psi[h_{33}]\end{array}\right].$ (7.103)
The still unsolved problem of quantum gravity is the reduction procedure
$\Psi\left[\left[\begin{array}[]{ccc}h_{11}&h_{12}&h_{13}\\\
h_{12}&h_{22}&h_{23}\\\
h_{13}&h_{23}&h_{33}\end{array}\right]\right]\rightarrow\left[\begin{array}[]{ccc}\Psi[h_{11}]&\Psi[h_{12}]&\Psi[h_{13}]\\\
\Psi[h_{12}]&\Psi[h_{22}]&\Psi[h_{23}]\\\
\Psi[h_{13}]&\Psi[h_{23}]&\Psi[h_{33}]\end{array}\right].$ (7.104)
This is evidently perfectionist situation, because in general the wave
functional can be considered as a $3\times 3$ symmetric matrix which elements
are functional of several elements of an induced metric.
Albeit, the way of straightforward analogy with quantum mechanics suggests
that the wave functional $\Psi[h_{ij}]$ is a classical scalar field like usual
wave function in quantum mechanics based on the Schrödinger equation, i.e. in
such a light $\Psi[h_{ij}]$ is a single functional. Let us accept such a state
of things. For realization of this idea the wave functional should be
dependent on a scalar function of an induced metric $h_{ij}$, which must be an
invariant of the induced matrix as well as invariant with respect to action of
the diffeomorphism group. The Cayley–Hamilton theorem for any $3\times 3$
square matrix $\mathbf{h}$ states that the matrix obeys its characteristic
equation
$\mathbf{h}^{3}-I_{\mathbf{h}}\mathbf{h}^{2}+II_{\mathbf{h}}\mathbf{h}-III_{\mathbf{h}}\mathbf{I}_{3\times
3}=0,$ (7.105)
where the coefficients of the polynomial
$\displaystyle I_{\mathbf{h}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\mathbf{h},$ (7.106) $\displaystyle II_{\mathbf{h}}$
$\displaystyle=$
$\displaystyle\dfrac{\left(\mathrm{Tr}\mathbf{h}\right)^{2}-\mathrm{Tr}\mathbf{h}^{2}}{2},$
(7.107) $\displaystyle III_{\mathbf{h}}$ $\displaystyle=$
$\displaystyle\det\mathbf{h},$ (7.108)
are the invariants of the matrix $\mathbf{h}$. A scalar valued matrix function
$\Psi(h_{ij})$ that depends merely on the three invariants of a symmetric
$3\times 3$ matrix
$\Psi\left(h_{ij}\right)=\Psi\left(I_{\mathbf{h}},II_{\mathbf{h}},III_{\mathbf{h}}\right),$
(7.109)
is independent on rotations of the coordinate system, is called _objective
function_. The invariants $I_{\mathbf{h}}$ and $II_{\mathbf{h}}$, however, are
irrelevant because of do not carry full information about $I_{\mathbf{h}}$.
The third invariant $III_{\mathbf{h}}$ as a function of all elements of a
matrix carries full information about the matrix. In $3+1$ decomposition
determinant is diffeoinvariant function of a $3\times 3$ induced metric
$h_{ij}$. It suggests that the invariant dimension is $\det h_{ij}$. Then wave
functional reduces to
$\Psi\left(h_{ij}\right)=\Psi\left(III_{\mathbf{h}}\right)=\Psi(h).$ (7.110)
and the quantum geometrodynamics becomes a one-dimensional quantum mechanics.
We shall call $\det h_{ij}$ _the global dimension_ , because it is a function
of local dimensions (coordinates) and some free parameters, and (7.110) _the
global one-dimensionality conjecture_. We shall call _generalized dimensions_
another, possibly more convenient, invariants constructed as $f(h)$. We shall
call _objective quantum gravity_ a theory of quantum gravity related to wave
functionals (7.109), and _global one-dimensional quantum gravity_ the theory
of quantum gravity related to the wave functionals (7.110).
Such a global one-dimensional wave function can be constructed in the
following way. Suppose that Matter fields in general are functionals dependent
on the one global variable
$\phi=\phi[h],$ (7.111)
which is the determinant $h=\det h_{ij}$ of an induced metric on $\partial M$.
Recall that in the dimension 3 one has
$h=\dfrac{1}{3}\epsilon^{ijk}\epsilon^{lmn}h_{il}h_{jm}h_{kn},$ (7.112)
where $\epsilon^{abc}$ is the three-dimensional Levi-Civita symbol
$\epsilon^{abc}=\dfrac{(a-b)(b-c)(c-a)}{2}.$ (7.113)
As the crucial point of the model let us assume that quantum gravity is
globally one-dimensional. In result the DeWitt wave functional becomes one-
dimensional wave function
$\Psi[h_{ij}]\rightarrow\Psi(h),$ (7.114)
and the Wheeler–DeWitt equation is
$\left\\{-2c\kappa\dfrac{\hslash^{2}}{\ell_{P}^{2}}G_{ijkl}\dfrac{\delta^{2}}{\delta
h_{ij}\delta
h_{kl}}-\dfrac{\ell_{P}^{2}}{2c\kappa}h^{1/2}\left({{}^{(3)}R}-2\Lambda-2\kappa\ell_{P}^{2}\varrho[h]\right)\right\\}\Psi(h)=0.$
(7.115)
In analogy to the generic cosmology [221] the conjecture (7.114) describes
isotropic spacetimes, and is related to the strata of the Wheeler superspace,
called midisuperspace, in which wave functionals are functions of a one
variable.
###### Step 2: Reduction of Quantum Geometrodynamics
Let us consider the Jacobi formula for determinant of the space-time metric
$\delta g=gg^{\mu\nu}\delta g_{\mu\nu},$ (7.116)
which can be rewritten in components
$\delta g=g\left(g^{00}\delta g_{00}+g^{ij}\delta g_{ij}+g^{0j}\delta
g_{0j}+g^{i0}\delta g_{i0}\right).$ (7.117)
The $3+1$ splitting (6.21) allows determine the partial variations
$\displaystyle\delta g_{00}$ $\displaystyle=$ $\displaystyle-\delta
N^{2}+N^{i}N^{j}\delta h_{ij}+h_{ij}N^{i}\delta N^{j}+h_{ij}N^{j}\delta
N^{i},$ (7.118) $\displaystyle\delta g_{ij}$ $\displaystyle=$
$\displaystyle\delta h_{ij},$ (7.119) $\displaystyle\delta g_{0j}$
$\displaystyle=$ $\displaystyle h_{ij}\delta N^{i}+N^{i}\delta h_{ij},$
(7.120) $\displaystyle\delta g_{i0}$ $\displaystyle=$ $\displaystyle
h_{ij}\delta N^{j}+N^{j}\delta h_{ij},$ (7.121)
as well as the total variation
$\displaystyle\delta g=N^{2}\delta h+h\delta N^{2}.$ (7.122)
Collecting all one obtains the result relevant for an induced metric
$N^{2}\delta h=N^{2}hh^{ij}\delta h_{ij},$ (7.123)
which allows to establish the Jacobian matrix for transformation of variables
$h_{ij}\rightarrow h$
$\displaystyle\mathcal{J}\left(h_{ij},h\right)=\dfrac{\delta(h)}{\delta(h_{ij})}=\dfrac{\delta
h}{\delta h_{ij}}\equiv hh^{ij}.$ (7.124)
Because of the approximation (7.114) the functional derivative
$\dfrac{\delta}{\delta h_{ij}}$ acts on a wave functional depending only on
$h$. It allows us to express the functional derivative with respect $h_{ij}$
through the functional derivative $\dfrac{\delta}{\delta h}$. Therefore one
has
$\dfrac{\delta\Psi[h]}{\delta h_{ij}}=hh^{ij}\dfrac{\delta\Psi[h]}{\delta h}.$
(7.125)
Consequently, application of (7.125) within the differential operator of the
Wheeler–DeWitt equation (7.115) leads to
$\displaystyle G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=G_{ijkl}h^{ij}h^{kl}h^{2}\dfrac{\delta^{2}}{\delta h^{2}}.$ (7.126)
So that the reduction is given by the double projection of the DeWitt
supermetric onto an induced metric
$\displaystyle G_{ijkl}h^{ij}h^{kl}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)h^{ij}h^{kl}=$
(7.127) $\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{h}}\left(h_{ik}h^{kl}h^{ij}h_{jl}+h_{il}h^{ij}h_{jk}h^{kl}-h_{ij}h^{ij}h_{kl}h^{kl}\right)=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{h}}\left(\delta^{l}_{i}\delta^{i}_{l}+\delta^{j}_{l}\delta^{l}_{j}-\delta^{i}_{i}\delta^{k}_{k}\right)=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{h}}\left(\delta^{i}_{i}+\delta^{j}_{j}-(\delta^{i}_{i})^{2}\right)=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{h}}\left(2\delta^{i}_{i}-(\delta^{i}_{i})^{2}\right)=$
$\displaystyle=$ $\displaystyle\dfrac{1}{2\sqrt{h}}\left(2\cdot
3-(3)^{2}\right)=-\dfrac{3}{2}h^{-1/2},$
where we have used the relations for three-dimensional embedded space
$h^{ab}h_{bc}=h^{a}_{c}$, $h^{a}_{a}=\delta^{a}_{a}=\mathrm{Tr}h_{ab}=3$.
Jointing (7.126) and (7.127) one obtains finally the transformation
$\displaystyle G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=-\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta h^{2}},$ (7.128)
which leads to the quantum geometrodynamics
$\left[2c\kappa\dfrac{\hslash^{2}}{\ell_{P}^{2}}\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta
h^{2}}-\dfrac{\ell_{P}^{2}}{2c\kappa}h^{1/2}\left({{}^{(3)}R}-2\Lambda-2\kappa\ell_{P}^{2}\varrho[h]\right)\right]\Psi(h)=0.$
(7.129)
Because the relation (7.124) arises due to $3+1$ approximation, so (7.127) is
an approximation within the ansatz.
###### Step 3: Dimensional Reduction
The quantum geometrodynamics (7.129) can be rewritten in the form of the
Klein–Gordon equation
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+\omega^{2}\right)\Psi=0,$ (7.130)
where $\omega^{2}$ is squared _gravitational dimensionless frequency_ of the
field $\Psi$
$\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{6(8\pi)^{2}}\dfrac{1}{h}\left({}^{(3)}R-2\Lambda-2\kappa\ell_{P}^{2}\varrho\right)=$
(7.131) $\displaystyle=$
$\displaystyle-\dfrac{1}{6(8\pi)^{2}}\dfrac{1}{h}(K_{ij}K^{ij}-K^{2}),$
(7.132)
where the Hamiltonian constraint was involved in the second line. In general
the squared mass can be positive, negative or even vanishing identically. The
equation (7.130) can be treated as the classical-field-theoretical
Euler–Lagrange equations of motion arising from stationarity of the action
functional
$S[\Psi]=\int\delta hL\left(\Psi,\dfrac{\delta\Psi}{\delta h}\right),$ (7.133)
where $L=L\left(\Psi,\dfrac{\delta\Psi}{\delta h}\right)$ is the field-
theoretic Lagrange function
$\displaystyle L$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\dfrac{\delta\Psi}{\delta
h}\right)^{2}-\dfrac{\omega^{2}}{2}\Psi^{2}=$ (7.134) $\displaystyle=$
$\displaystyle\dfrac{1}{2}\Pi_{\Psi}^{2}-\dfrac{\omega^{2}}{2}\Psi^{2},$
(7.135)
where $\Pi_{\Psi}$ is the momentum conjugated to the classical scalar field
$\Psi$
$\Pi_{\Psi}=\dfrac{\partial L}{\partial\left(\dfrac{\delta\Psi}{\delta
h}\right)}=\dfrac{\delta\Psi}{\delta h}.$ (7.136)
The action $S[\Psi]$ is a field-theoretic action functional in the classical
field $\Psi$, and therefore arbitrary dependence on the variable $h$ of the
mass $m=m[h]$ does not play a role, i.e. behaves as a coefficient, in
derivation of the Euler–Lagrange equations of motion
$\delta S[\Psi]=\int\delta h\left[\dfrac{\partial
L}{\partial\Psi}-\dfrac{\delta}{\delta h}\dfrac{\partial
L}{\partial\left(\dfrac{\delta\Psi}{\delta
h}\right)}\right]\delta\Psi+\int\delta h\dfrac{\delta}{\delta
h}\left(\dfrac{\partial L}{\partial\Psi}\delta\Psi\right)=0,$ (7.137)
what gives the result
$\dfrac{\partial L}{\partial\Psi}-\dfrac{\delta}{\delta h}\dfrac{\partial
L}{\partial\left(\dfrac{\delta\Psi}{\delta h}\right)}=0,$ (7.138)
where we have taken _ad hoc_ the field theoretical condition of vanishing of
the boundary term
$\int\delta h\dfrac{\delta}{\delta h}\left(\dfrac{\partial
L}{\partial\Psi}\delta\Psi\right)=\int\delta\left(\dfrac{\partial
L}{\partial\Psi}\delta\Psi\right)=\left.\dfrac{\partial
L}{\partial\Psi}\delta\Psi\right|_{0}=0.$ (7.139)
It can be seen by straightforward computation that the equation (7.138)
coincides with (7.130).
By application of the conjugate momentum $\Pi_{\Psi}$ one rewrites the
equation (7.130) in the following form
$\dfrac{\delta\Pi_{\Psi}}{\delta h}+\omega^{2}\Psi=0,$ (7.140)
and therefore the equations (7.136) and (7.140) are the system of canonical
Hamilton equations of motion
$\displaystyle\dfrac{\delta}{\delta h}\Psi$ $\displaystyle=$
$\displaystyle\dfrac{\delta}{\delta\Pi_{\Psi}}H\left(\Psi,\Pi_{\Psi}\right),$
(7.141) $\displaystyle\dfrac{\delta}{\delta h}\Pi_{\Psi}$ $\displaystyle=$
$\displaystyle-\dfrac{\delta}{\delta\Psi}H\left(\Psi,\Pi_{\Psi}\right),$
(7.142)
where the Hamilton function $H\left(\Psi,\Pi_{\Psi}\right)$ is obtained from
the Lagrange function (7.135) via the Legendre transformation
$\displaystyle H\left(\Psi,\Pi_{\Psi}\right)$ $\displaystyle=$
$\displaystyle\Pi_{\Psi}\dfrac{\delta\Psi}{\delta
h}-L\left(\Psi,\dfrac{\delta\Psi}{\delta h}\right)=$ (7.143) $\displaystyle=$
$\displaystyle\dfrac{1}{2}\Pi_{\Psi}^{2}-\dfrac{\omega^{2}}{2}\Psi^{2}.$
(7.144)
If one recognizes the kinetic $T$ and the potential $V$ energies as
$\displaystyle T$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}\Pi_{\Psi}^{2},$
(7.145) $\displaystyle V$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\omega^{2}\Psi^{2},$ (7.146)
then the Hamilton function (7.144) is $H=T-V$ and the Lagrange function
(7.135) is $L=T+V$, what means that the field theory presented above is the
Euclidean field theory of a simple harmonic oscillator of the mass $1$ and
frequency $\omega$. In this context the classical scalar field - the wave
function $\Psi$ \- becomes the generalized coordinate.
Let us introduce the two-component field
$\Phi=\left[\begin{array}[]{c}\Pi_{\Psi}\\\ \Psi\end{array}\right],$ (7.147)
which components obey the equations (7.136)-(7.140). The system of the
Hamilton canonical equations of motion (7.136)-(7.140) can be rewritten in the
form of the vector equation
$\left(-i\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right]\dfrac{\delta}{\delta
h}-\left[\begin{array}[]{cc}-\dfrac{1}{\Pi_{\Psi}}\dfrac{\delta}{\delta\Pi_{\Psi}}&0\\\
0&-\dfrac{1}{\Psi}\dfrac{\delta}{\delta\Psi}\end{array}\right]H\left(\Psi,\Pi_{\Psi}\right)\right)\Phi=0,$
(7.148)
which for the situation given by the Hamiltonian (7.144) leads the appropriate
one-dimensional Dirac equation for the classical two-component field $\Phi$
$\left(-i\gamma\dfrac{\delta}{\delta h}-M\right)\Phi=0,$ (7.149)
where $M$ is the mass matrix of the field $\Phi$
$M=\left[\begin{array}[]{cc}-1&0\\\ 0&-\omega^{2}\end{array}\right],$ (7.150)
and the $\gamma$ matrix is the Pauli matrix $\sigma_{y}$
$\gamma=\sigma_{y}=\left[\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right],$
(7.151)
obeying the following algebra
$\gamma^{2}=\mathbf{I}_{2}\quad,\quad\left\\{\gamma,\gamma\right\\}=2\mathbf{I}_{2}\quad,\quad\mathbf{I}_{2}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right].$ (7.152)
The algebra (7.152) is the four-dimensional Clifford algebra over the complex
vector space $\mathbb{C}^{2}$ (For basics and advances in Clifford algebras
see e.g. the Ref. [601])
$\mathcal{C}\ell_{2}(\mathbb{C})=\mathcal{C}\ell_{0}(\mathbb{C})\otimes\mathrm{M}_{2}(\mathbb{C})\cong\mathrm{M}_{2}(\mathbb{C})=\mathbb{C}\oplus\mathbb{C},$
(7.153)
where $\mathcal{C}\ell_{n}\equiv\mathcal{C}\ell_{n,0}$, and
$\mathrm{M}_{2}(\mathbb{C})$ denotes algebra of all $2\times 2$ matrices over
$\mathbb{C}$. The Clifford algebra $\mathcal{C}\ell_{2,0}(\mathbb{C})$
possesses a two-dimensional complex representation. Restriction to the pinor
group $\textrm{Pin}_{2,0}(\mathbb{R})$ yields a complex representation of two-
dimensional pinor group, i.e. the two-dimensional spinor representation,
whereas restriction to the spinor group $\textrm{Spin}_{2,0}(\mathbb{R})$
splits $\mathcal{C}\ell_{1,1}(\mathbb{R})$ onto a sum of two half spin
representations of dimension 1, i.e. the one dimensional Weyl representations.
There is the isomorphism
$\textrm{Spin}_{2,0}(\mathbb{R})\cong\mathrm{U}(1)\cong\mathrm{SO}(2),$
(7.154)
and the spinor group $\textrm{Spin}_{2,0}(\mathbb{R})$ acts on a 1-sphere
$S^{1}$ in such a way that one has a fibre bundle with fibre
$\textrm{Spin}_{1,0}(\mathbb{R})$
$\textrm{Spin}_{1,0}(\mathbb{R})\longrightarrow\textrm{Spin}_{2,0}(\mathbb{R})\longrightarrow
S^{1},$ (7.155)
and the homotopy sequence is
$\pi_{1}\left(\textrm{Spin}_{1,0}(\mathbb{R})\right)\longrightarrow\pi_{1}\left(\textrm{Spin}_{2,0}(\mathbb{R})\right)\longrightarrow\pi_{1}\left(S^{1}\right).$
(7.156)
The Clifford algebra $\mathcal{C}\ell_{2}(\mathbb{C})$ can be generated by
complexification
$\mathcal{C}\ell_{2}(\mathbb{C})\cong\mathcal{C}\ell_{1,1}(\mathbb{R})\otimes\mathcal{C}\ell_{0}(\mathbb{C}),$
(7.157)
where $\mathcal{C}\ell_{1,1}(\mathbb{R})$ is the four-dimensional Clifford
algebra over the real vector space $\mathbb{R}^{2,0}$
$\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathrm{M}_{2}(\mathbb{R})\otimes\mathcal{C}\ell_{0}(\mathbb{R})\cong{\mathrm{M}_{2}(\mathbb{R})},$
(7.158)
with $\mathrm{M}_{2}(\mathbb{R})$ being algebra of $2\times 2$ matrices over
$\mathbb{R}$, and
$\displaystyle\mathcal{C}\ell_{0}(\mathbb{R})=\mathbb{R},$ (7.159)
$\displaystyle\mathcal{C}\ell_{0}(\mathbb{C})=\mathbb{C}.$ (7.160)
The Clifford algebra (7.158) can be decomposed into a direct sum of central
simple algebras isomorphic to matrix algebra over $\mathbb{R}$
$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})$ $\displaystyle=$
$\displaystyle\mathcal{C}\ell^{+}_{1,1}(\mathbb{R})\oplus\mathcal{C}\ell^{-}_{1,1}(\mathbb{R}),$
(7.161) $\displaystyle\mathcal{C}\ell^{\pm}_{1,1}(\mathbb{R})$
$\displaystyle=$
$\displaystyle\dfrac{1\pm\gamma}{2}\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathbb{R},$
(7.162)
as well as into a tensor product
$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})$ $\displaystyle=$
$\displaystyle\mathcal{C}\ell_{2,0}(\mathbb{R})\otimes\mathcal{C}\ell_{0,0}(\mathbb{R}),$
(7.163) $\displaystyle\mathcal{C}\ell_{2,0}(\mathbb{R})$ $\displaystyle=$
$\displaystyle\mathrm{M}_{2}(\mathbb{R})\otimes\mathcal{C}\ell_{0,0}(\mathbb{R})\cong{\mathrm{M}_{2}(\mathbb{R})}.$
(7.164)
#### D Field Quantization in Static Fock Space
The one-dimensional Dirac equation (7.149) can be canonically quantized
$\left(-i\gamma\dfrac{\delta}{\delta h}-M\right)\hat{\Phi}=0,$ (7.165)
according to the canonical commutation relations (CCR) characteristic for
bosonic fields
$\displaystyle i\left[\hat{\Pi}_{\Psi}[h^{\prime}],\hat{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle\delta(h^{\prime}-h),$ (7.166) $\displaystyle
i\left[\hat{\Pi}_{\Psi}[h^{\prime}],\hat{\Pi}_{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle 0,$ (7.167) $\displaystyle
i\left[\hat{\Psi}[h^{\prime}],\hat{\Psi}[h]\right]$ $\displaystyle=$
$\displaystyle 0,$ (7.168)
where the choice of the bosonic CCR follows form the fact that one has the
one-dimensional situation in which there is no difference between bosons and
fermions. Particles obeying one-dimensional quantum evolutions are called
_axions_ , and in this manner the second quantized one-dimensional Dirac
equation (7.165) describes axions obeying the Bose–Einstein statistics, which
are _gravitons_ in our understanding.
Let us apply the Fock space formalism, which allows to write out explicitly
the decomposition of the solution
$\hat{\Phi}=\mathbf{Q}\mathfrak{B},$ (7.169)
where $\mathbf{Q}$ is the matrix of secondary quantization
$\mathbf{Q}=\left[\begin{array}[]{cc}\sqrt{\dfrac{1}{2\omega}}&\sqrt{\dfrac{1}{2\omega}}\\\
-i\sqrt{\dfrac{\omega}{2}}&i\sqrt{\dfrac{\omega}{2}}\end{array}\right],$
(7.170)
and $\mathfrak{B}=\mathfrak{B}[h]$ is a dynamical repère
$\mathfrak{B}=\left\\{\left[\begin{array}[]{c}\textsf{G}[h]\\\
\textsf{G}^{\dagger}[h]\end{array}\right]:\left[\textsf{G}[h^{\prime}],\textsf{G}^{\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\textsf{G}[h^{\prime}],\textsf{G}[h]\right]=0\right\\},$
(7.171)
on the Fock space of creation and annihilation operators
$\mathcal{F}=\left(\textsf{G},\textsf{G}^{\dagger}\right).$ (7.172)
Application of the decomposition (7.169) yields the Heisenberg equations of
motion modified by the non-diagonal components
$\dfrac{\delta\mathfrak{B}}{\delta h}=\mathbf{X}\mathfrak{B},$ (7.173)
where $\mathbf{X}$ is the matrix
$\mathbf{X}=\left[\begin{array}[]{cc}-i\omega&\dfrac{1}{2\omega}\dfrac{\delta\omega}{\delta
h}\\\ \dfrac{1}{2\omega}\dfrac{\delta\omega}{\delta
h}&i\omega\end{array}\right].$ (7.174)
Let us suppose that there is another repère $\mathfrak{F}$ determined by the
Bogoliubov transformation
$\displaystyle\mathfrak{F}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}u&v\\\
v^{\ast}&u^{\ast}\end{array}\right]\mathfrak{B},$ (7.177)
where the Bogoliubov coefficients $u$ and $v$ forms the
Gauss–Lobachevsky–Bolyai hyperbolic space and obey the constraint
$|u|^{2}-|v|^{2}=1,$ (7.178)
and together with the frequency $\Omega$ are functionals of $h$. As the second
requirement let us suppose also that dynamics of the repère is governed by the
Heisenberg equations of motion
$\dfrac{\delta\mathfrak{F}}{\delta h}=\left[\begin{array}[]{cc}-i\Omega&0\\\
0&i\Omega\end{array}\right]\mathfrak{F}.$ (7.179)
Application of the system of equations (7.177)- (7.179) to the equations
(7.173) leads to the equation for the vector of the Bogoliubov coefficients
$\mathbf{b}=\left[\begin{array}[]{c}u\\\ v\end{array}\right],$ (7.180)
which is given by the following vector equation
$\dfrac{\delta\mathbf{b}}{\delta h}=\mathbf{X}\mathbf{b},$ (7.181)
and gives trivial value of the unknown frequency
$\Omega\equiv 0.$ (7.182)
Therefore, the conjectured repère $\mathfrak{F}$ becomes the static Fock
repère with respect to initial data ($I$)
$\mathfrak{F}=\left\\{\left[\begin{array}[]{c}\textsf{G}_{I}\\\
\textsf{G}^{\dagger}_{I}\end{array}\right]:\left[\textsf{G}_{I},\textsf{G}^{\dagger}_{I}\right]=1,\left[\textsf{G}_{I},\textsf{G}_{I}\right]=0\right\\},$
(7.183)
and the vacuum state $\left|\textrm{0}\right\rangle$ is correctly defined
$\displaystyle\textsf{G}_{I}\left|\textrm{0}\right\rangle$ $\displaystyle=$
$\displaystyle 0,$ (7.184)
$\displaystyle\left\langle\textrm{0}\right|\textsf{G}_{I}^{\dagger}$
$\displaystyle=$ $\displaystyle 0.$ (7.185)
Integrability of the system of equations (7.181) is the crucial element of the
scheme presented above. The Bogoliubov transformation (7.177), however,
suggests application of the superfluid parametrization
$\displaystyle u$ $\displaystyle=$ $\displaystyle e^{i\theta}\cosh\phi,$
(7.186) $\displaystyle v$ $\displaystyle=$ $\displaystyle
e^{i\theta}\sinh\phi,$ (7.187)
where $\theta$ and $\phi$ are the angles which for the present situation are
$\displaystyle\theta$ $\displaystyle=$ $\displaystyle\pm
i\int_{h_{I}}^{h}\omega^{\prime}\delta h^{\prime},$ (7.188)
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\ln{\sqrt{\left|\dfrac{\omega_{I}}{\omega}\right|}},$ (7.189)
where $\omega^{\prime}=\omega(h^{\prime})$ and $\omega_{I}$ is the initial
datum of gravitational dimensionless frequency
$\omega_{I}=-\dfrac{1}{8\pi\sqrt{6}},$ (7.190)
which yield the Bogoliubov coefficients
$\displaystyle u$ $\displaystyle=$
$\displaystyle\dfrac{\mu+1}{2\sqrt{\mu}}\exp\left\\{i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\},$ (7.191) $\displaystyle v$ $\displaystyle=$
$\displaystyle\dfrac{\mu-1}{2\sqrt{\mu}}\exp\left\\{-i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\},$ (7.192)
where $\mu=\dfrac{\omega}{\omega_{I}}$ measures the relative gravitational
dimensionless frequency. For convenience one can apply also the reciprocal of
$\mu$, i.e. the parameter $\lambda=\dfrac{\omega_{I}}{\omega}=\dfrac{1}{\mu}$
$\lambda=\sqrt{\left|\dfrac{h}{{}^{(3)}R-2\Lambda-2\kappa\ell_{P}^{2}\varrho}\right|}=\sqrt{\left|\dfrac{h}{K_{ij}K^{ij}-K^{2}}\right|},$
(7.193)
and we understand $\lambda\equiv\lambda[h]$,
$\lambda^{\prime}=\lambda[h^{\prime}]$.
Consequently, the integrability problem is solved by the equation
$\hat{\Phi}=\mathbf{Q}\mathbf{G}\mathfrak{F},$ (7.194)
where $\mathbf{G}$ is the monodromy matrix
$\mathbf{G}=\left[\begin{array}[]{cc}\dfrac{1+\mu}{2\sqrt{\mu}}\exp\left\\{-i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}\vspace*{10pt}&\dfrac{1-\mu}{2\sqrt{\mu}}\exp\left\\{i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}\\\
\dfrac{1-\mu}{2\sqrt{\mu}}\exp\left\\{-i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}&\dfrac{1+\mu}{2\sqrt{\mu}}\exp\left\\{i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}\end{array}\right].$ (7.195)
Now it can be seen straightforwardly that the presented version of quantum
geometrodynamics formulates quantum gravity as a quantum field theory of
gravity, where the quantum gravitational field is associated with
configuration of embedded space and given by the decomposition (7.194) in the
static Fock space. In this manner one can write out straightforwardly
conclusions following form the global one-dimensional model of quantum
gravity.
It must be noticed that the functional measure $\delta h$ in any integrals of
the form $\int\delta h^{\prime}f[h^{\prime}]$ for the case of a fixed
configuration of space, i.e. $h=constant$, becomes the Riemann–Lebesgue
measure $dh$. However, because of $h$ in general is a smooth function of
space-time coordinates and free parameters, the measure $\delta h$ as a total
variation over space-time coordinates is the Lebesgue–Stieltjes measure which
can be rewritten as the Riemann–Lebesgue measure on space-time. In the most
general case $h=h(x_{0},x_{1},x_{2},x_{3})$ one can use the transformation
$\delta h=\dfrac{\partial^{4}h(x_{0},x_{1},x_{2},x_{3})}{\partial
x_{0}\partial x_{1}\partial x_{2}\partial x_{3}}d^{4}x,$ (7.196)
where $d^{4}x=dx_{0}dx_{1}dx_{2}dx_{3}$, and compute the integral
$\int\delta h^{\prime}f[h^{\prime}]=\int
d^{4}x^{\prime}\dfrac{\partial^{4}h(x_{0}^{\prime},x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime})}{\partial
x_{0}^{\prime}\partial x_{1}^{\prime}\partial x_{2}^{\prime}\partial
x_{3}^{\prime}}f(x_{0}^{\prime},x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime}).$
(7.197)
In this manner the transformation (7.196) establishes the relation between the
Wheeler superspace and space-time.
The initial data condition $m=m_{I}$ generates the equation for the initial
manifold
${{}^{(3)}}R^{(I)}-2\Lambda-2\kappa\ell_{P}^{2}\varrho_{I}=h_{I},$ (7.198)
or equivalently
$K^{(I)}_{ij}K^{(I)ij}-K^{(I)2}=h^{I},$ (7.199)
where the superscript $I$ means initial value of given quantity. The quantum
evolution (7.130) in such a situation takes the form
$\left(\dfrac{\delta^{2}}{\delta
h_{I}^{2}}-\dfrac{1}{6(8\pi)^{2}}\right)\Psi(h_{I})=0,$ (7.200)
and after taking into account the suitable boundary conditions
$\displaystyle\Psi(h_{I}=h_{0})=\Psi_{0},$ (7.201)
$\displaystyle\left.\dfrac{\delta\Psi(h_{I})}{\delta
h_{I}}\right|_{h_{I}=h_{0}}=\Pi_{\Psi}^{0},$ (7.202)
can be solved straightforwardly
$\Psi(h_{I})=\Psi_{0}\cosh\left\\{\dfrac{h_{I}-h_{0}}{8\pi\sqrt{6}}\right\\}+8\pi\sqrt{6}\ell_{P}^{2}\Pi_{\Psi}^{0}\sinh\left\\{\dfrac{h_{I}-h_{0}}{8\pi\sqrt{6}}\right\\}.$
(7.203)
#### E Several Implications
The quantum field-theoretic geometrodynamics just was formulated. However,
still we do not know what it the role of an one-dimensional wave function
which solves the equation (7.115). The same problem is to define any geometric
quantities related to the midisuperspace quantum geometrodynamics. The quantum
field theory of gravity (7.194) has also unprecise significance. Let us
present now several conclusions arising from the previous section, which shall
clarify our doubts in some detail.
##### E1 The Global 1D Wave Function
The one-dimensional Dirac equation (8.20) can be rewritten in the form of
Schrödinger equation
$\dfrac{\delta\Phi}{\delta h}=H\Phi,$ (7.204)
where $H$ is the hermitian Hamiltonian
$H=i\gamma{M}=\left[\begin{array}[]{cc}0&-\omega^{2}\\\
1&0\end{array}\right]=\left[\begin{array}[]{cc}0&-\dfrac{\omega_{I}^{2}}{\lambda^{2}}\\\
1&0\end{array}\right],$ (7.205)
yielding the evolution operator
$U[h,h_{I}]=\exp\int_{h_{I}}^{h}\delta
h^{\prime}H[h^{\prime}]=\exp\left[\begin{array}[]{cc}0&-\int_{h_{I}}^{h}\delta{h^{\prime}}\omega^{2}[h^{\prime}]\\\
h-h_{I}&0\end{array}\right],$ (7.206)
where $h\geqslant h_{I}$, which is explicitly
$U[h,h_{I}]=\left[\begin{array}[]{cc}\cos
f[h,h_{I}]&-\left(\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)\dfrac{\sin f[h,h_{I}]}{f[h,h_{I}]}\\\
(h-h_{I})\dfrac{\sin f[h,h_{I}]}{f[h,h_{I}]}&\cos
f[h,h_{I}]\end{array}\right],$ (7.207)
where $f[h,h_{I}]$ is the functional
$f[h,h_{I}]=\sqrt{{(h-h_{I})\int_{h_{I}}^{h}{\omega^{\prime}}^{2}\delta
h^{\prime}}},$ (7.208)
so that the solution of the equation (8.30) is
$\Phi[h,h_{I}]=U[h,h_{I}]\Phi[h_{I}].$ (7.209)
Straightforward elementary algebraic manipulations allow to determine the
global one-dimensional wave function as
$\Psi[h,h_{I}]]=\Psi^{I}\cos f[h,h_{I}]+\Pi_{\Psi}^{I}(h-h_{I})\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]},$ (7.210)
and similarly the canonical conjugate momentum is
$\Pi_{\Psi}[h,h_{I}]=\Pi_{\Psi}^{I}\cos
f[h,h_{I}]-\Psi^{I}\left(\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)\dfrac{\sin f[h,h_{I}]}{f[h,h_{I}]},$
(7.211)
where $\Psi^{I}$ and $\Pi_{\Psi}^{I}$ are initial data
$\displaystyle\Psi^{I}$ $\displaystyle=$ $\displaystyle\Psi[h_{I}],$ (7.212)
$\displaystyle\Pi_{\Psi}^{I}$ $\displaystyle=$
$\displaystyle\Pi_{\Psi}[h_{I}]=\left.\dfrac{\delta\Psi}{\delta
h}\right|_{h=h_{I}}.$ (7.213)
Because of $U^{\dagger}[h,h_{I}]=U^{T}[h,h_{I}]$ one sees that
$\displaystyle\Psi^{\star}[h,h_{I}]$ $\displaystyle=$
$\displaystyle\Psi[h,h_{I}],$ (7.214)
$\displaystyle\Pi_{\Psi}^{\star}[h,h_{I}]$ $\displaystyle=$
$\displaystyle\Pi_{\Psi}[h,h_{I}],$ (7.215)
and for consistency also must be $(\Psi^{I})^{\star}=\Psi^{I}$,
$(\Pi_{\Psi}^{I})^{\star}=\Pi_{\Psi}^{I}$. The probability density in the
quantum mechanics is
$\Omega[h,h_{I}]=\Phi^{\dagger}[h,h_{I}]\Phi[h,h_{I}]=\Phi^{\dagger}[h_{I}]U^{\dagger}[h,h_{I}]U[h,h_{I}]\Phi[h_{I}].$
(7.216)
Computing the matrix $U^{\dagger}[h,h_{I}]U[h,h_{I}]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!U^{\dagger}[h,h_{I}]U[h,h_{I}]$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\cos^{2}f[h,h_{I}]+\left[(h-h_{I})\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]}\right]^{2}\\\ \left(h-h_{I}-\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)\dfrac{\sin
2f[h,h_{I}]}{2f[h,h_{I}]}\end{array}\right.$ (7.222)
$\displaystyle\left.\begin{array}[]{cc}\left(h-h_{I}-\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)\dfrac{\sin 2f[h,h_{I}]}{2f[h,h_{I}]}\\\
\cos^{2}f[h,h_{I}]+\left[\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]}\right]^{2}\end{array}\right],$
and taking into account that
$\Phi^{\dagger}[h_{I}]=\left[\begin{array}[]{c}\Pi_{\Psi}^{I}\\\
\Psi^{I}\end{array}\right]^{\dagger}=\left[\begin{array}[]{c}(\Pi_{\Psi}^{I})^{\star}\\\
(\Psi^{I})^{\star}\end{array}\right]^{T}=\left[\begin{array}[]{c}\Pi_{\Psi}^{I}\\\
\Psi^{I}\end{array}\right]^{T}=\left[\Pi_{\Psi}^{I},\Psi^{I}\right],$ (7.223)
one obtains finally
$\Omega[h,h_{I}]=A[h,h_{I}]\left(\Psi^{I}\right)^{2}+2B[h,h_{I}]\Psi^{I}\Pi_{\Psi}^{I}+C[h,h_{I}]\left(\Pi_{\Psi}^{I}\right)^{2},$
(7.224)
where the functional coefficients $A[h,h_{I}]$, $B[h,h_{I}]$, $C[h,h_{I}]$ in
(7.224) are
$\displaystyle A[h,h_{I}]$ $\displaystyle=$
$\displaystyle\cos^{2}f[h,h_{I}]+\left(\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)^{2}\left(\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]}\right)^{2},$ (7.225) $\displaystyle B[h,h_{I}]$
$\displaystyle=$ $\displaystyle\left(h-h_{I}-\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)\dfrac{\sin 2f[h,h_{I}]}{2f[h,h_{I}]},$
(7.226) $\displaystyle C[h,h_{I}]$ $\displaystyle=$
$\displaystyle\cos^{2}f[h,h_{I}]+(h-h_{I})^{2}\left(\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]}\right)^{2},$ (7.227)
The initial data $\Psi^{I}$ and $\Pi_{\Psi}^{I}$ are not arbitrary, but
constrained by the normalization condition for the probability density
$\int_{h_{I}}^{h_{F}}\Omega[h^{\prime}]\delta h^{\prime}=1,$ (7.228)
where $h_{F}$ is some maximal value of $h$, which in explicit form leads to
the algebraic equation
$C[h_{I}](\Pi_{\Psi}^{I})^{2}+2B[h_{I}]\Psi^{I}\Pi_{\Psi}^{I}+A[h_{I}](\Psi^{I})^{2}-1=0,$
(7.229)
with the coefficients $A[h_{I}]$, $B[h_{I}]$, $C[h_{I}]$ given by the
integrals
$\displaystyle A[h_{I}]$ $\displaystyle=$
$\displaystyle\int_{h_{I}}^{h_{F}}\delta h^{\prime}A[h^{\prime},h_{I}],$
(7.230) $\displaystyle B[h_{I}]$ $\displaystyle=$
$\displaystyle\int_{h_{I}}^{h_{F}}\delta h^{\prime}B[h^{\prime},h_{I}],$
(7.231) $\displaystyle C[h_{I}]$ $\displaystyle=$
$\displaystyle\int_{h_{I}}^{h_{F}}\delta h^{\prime}C[h^{\prime},h_{I}].$
(7.232)
The equation (7.229) can be solved straightforwardly. In result one obtains
$\displaystyle\Pi_{\Psi}^{I}=-\dfrac{B[h_{I}]}{C[h_{I}]}\Psi^{I}\pm\sqrt{{\dfrac{B^{2}[h_{I}]-A[h_{I}]C[h_{I}]}{C^{2}[h_{I}]}(\Psi^{I})^{2}+\dfrac{1}{C[h_{I}]}}}.$
(7.233)
Application of the explicit form of initial data of the conjugate momentum
$\Pi_{\Psi}^{I}=\dfrac{\delta\Psi^{I}}{\delta h_{I}}$ to the equation (7.229)
yields the differential equation for initial data of the classical scalar
field $\Psi^{I}$
$C[h_{I}]\left(\dfrac{\delta\Psi^{I}}{\delta
h_{I}}\right)^{2}+2B[h_{I}]\Psi^{I}\dfrac{\delta\Psi^{I}}{\delta
h_{I}}+A[h_{I}](\Psi^{I})^{2}-1=0,$ (7.234)
which in general is very hard to solve. However, there is the case defined by
the values of the coefficients $A[h_{I}]=A$, $B[h_{I}]=B$, $C[h_{I}]=C$ which
are independent on $h_{I}$. In such a situation the equation (7.234) possesses
solutions which are easy to extract
$\Psi^{I}_{\mp}=f_{\pm}^{(-1)}\left(\mp\dfrac{h_{I}}{C}+C_{1}\right),$ (7.235)
where $C_{1}$ is an integration constant, and $f_{\pm}(x)$ are the functions
$\displaystyle
f_{\pm}(x)=\pm\dfrac{B}{AC}\Bigg{\\{}\dfrac{1}{2}\ln\left|Ax^{2}-1\right|\pm\mathrm{artanh}\left[\dfrac{Bx}{\sqrt{C+\left(B^{2}-AC\right)x^{2}}}\right]\mp$
$\displaystyle\mp\dfrac{\sqrt{B^{2}-AC}}{B}\ln\left|2\left(B^{2}-AC\right)x+2\sqrt{B^{2}-AC}\sqrt{C+\left(B^{2}-AC\right)x^{2}}\right|\Bigg{\\}},$
(7.236)
The solution (7.235) has been received computationally, and by using it one
can construct the analytical solutions straightforwardly. Let us introduce the
following parameter
$x=\pm\dfrac{h_{I}-h_{0}}{C},$ (7.237)
where for consistency we have taken $C_{1}=\mp\dfrac{h_{0}}{C}$.
Differentiating the solutions (7.235) with respect to $x$ one obtains
$\dfrac{\delta\Psi^{I}_{\mp}}{\delta\left(\pm\dfrac{h_{I}}{C}+C_{1}\right)}=\pm
C\dfrac{\delta\Psi^{I}_{\mp}}{\delta h_{I}}=-\dfrac{1}{f^{\prime}_{\mp}(x)},$
(7.238)
where $f^{\prime}(x)=\dfrac{df(x)}{dx}$, and by this reason
$\dfrac{\delta\Psi^{I}_{\mp}}{\delta
h_{I}}=\mp\dfrac{1}{Cf^{\prime}_{\mp}(x)}.$ (7.239)
Substitution of the derivative (7.239) to the equation (7.234) gives
$\dfrac{1}{C(f^{\prime}_{\mp}(x))^{2}}\mp\dfrac{2B}{Cf^{\prime}_{\mp}(x)}\Psi^{I}_{\mp}+A(\Psi^{I}_{\mp})^{2}-1=0,$
(7.240)
or equivalently one obtains quadratic equation for $\Psi^{I}$
$AC(f^{\prime}_{\mp}(x))^{2}(\Psi^{I}_{\mp})^{2}\mp
2Bf^{\prime}_{\mp}(x)\Psi^{I}_{\mp}+1-C(f^{\prime}(x))^{2}=0.$ (7.241)
The equation (7.241) possesses two solutions
$\Psi^{I}_{\mp}=\pm\dfrac{B}{ACf^{\prime}_{\mp}(x)}\left(1+\dfrac{1}{ABC}\sqrt{B^{2}-AC+4AC^{2}(f^{\prime}_{\mp}(x))^{2}}\right),$
(7.242)
where the derivative $f^{\prime}_{\mp}(x)$ can be established
straightforwardly
$f^{\prime}_{\mp}(x)=\pm\dfrac{B}{C}\left[\dfrac{x}{Ax^{2}-1}\mp\dfrac{B}{\sqrt{C+(B^{2}-AC)x^{2}}}\left(\dfrac{x^{2}}{Ax^{2}-1}-\dfrac{C}{B^{2}}\right)\right].$
(7.243)
Substitution of the derivative (7.243) to the formula (7.242) gives
$\displaystyle\Psi^{I}_{\mp}=\left(\dfrac{Ax}{Ax^{2}-1}\mp\dfrac{AB}{\sqrt{C+(B^{2}-AC)x^{2}}}\left(\dfrac{x^{2}}{Ax^{2}-1}-\dfrac{C}{B^{2}}\right)\right)^{-1/2}\times$
$\displaystyle\Bigg{\\{}1+\dfrac{1}{AC}\Bigg{[}1-\dfrac{AC}{B^{2}}+\dfrac{4Ax^{2}}{(Ax^{2}-1)^{2}}+\dfrac{4AB}{C+(B^{2}-AC)x^{2}}\left(\dfrac{x^{2}}{Ax^{2}-1}-\dfrac{C}{B^{2}}\right)^{2}\pm$
$\displaystyle\dfrac{ABx}{(Ax^{2}-1)\sqrt{C+(B^{2}-AC)x^{2}}}\left(\dfrac{x^{2}}{Ax^{2}-1}-\dfrac{C}{B^{2}}\right)\Bigg{]}^{1/2}\Bigg{\\}}.$
(7.244)
##### E2 The Unitary Three-Manifolds
The evolution operator (7.207) is in general non unitary, i.e. it can be shown
by straightforward calculation that the condition
$U^{\dagger}U=UU^{\dagger}=\mathbf{1}_{2},$ (7.245)
is broken. However, there are situations within the theory for which the
evolution operator (7.207) is unitary. It can be proved easy that for the
unitarity of $U$ the necessary and sufficient conditions are
$\displaystyle\left(h-h_{I}\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)\dfrac{\sin 2f[h,h_{I}]}{2f[h,h_{I}]}$
$\displaystyle=$ $\displaystyle 0,$ (7.246)
$\displaystyle\cos^{2}f[h,h_{I}]+\left(\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)^{2}\left(\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]}\right)^{2}$ $\displaystyle=$ $\displaystyle 1,$
(7.247) $\displaystyle\left[\left(\int_{h_{I}}^{h}\delta
h^{\prime}{\omega^{\prime}}^{2}\right)^{2}-\left(h-h_{I}\right)^{2}\right]\left(\dfrac{\sin
f[h,h_{I}]}{f[h,h_{I}]}\right)^{2}$ $\displaystyle=$ $\displaystyle 0.$
(7.248)
These equations possess two solutions. The first solution is trivial
$f[h,h_{I}]=0,$ (7.249)
and corresponds to the initial data point
$h=h_{I}.$ (7.250)
In this situation the evolution operator is trivially equal to the unit
$2\times 2$ matrix, what in fact means that there is no evolution. The second
solution, however, is non trivial
$f^{2}[h,h_{I}]=-(h-h_{I})^{2},$ (7.251)
and corresponds to the equation
$\omega^{2}=-1,$ (7.252)
which is associated with purely imaginary frequency and therefore also energy,
i.e. tachyon. With using of the definition (7.131) the condition (7.252)
generates the equation for the embedded three-dimensional space
${{}^{(3)}}R=6(8\pi)^{2}h+2\Lambda+2\kappa\ell_{P}^{2}\varrho,$ (7.253)
or equivalently with using of the second definition (7.132)
$K_{ij}K^{ij}-K^{2}=6(8\pi)^{2}h.$ (7.254)
In this case the unitary evolution operator has the form
$U[h,h_{I}]=\left[\begin{array}[]{cc}\cosh(h-h_{I})&\sinh(h-h_{I})\\\
-\sinh(h-h_{I})&\cosh(h-h_{I})\end{array}\right],$ (7.255)
and is the rotation matrix of the unitary Lie group $U(1)\cong SO(2)$, where
the angle of the rotation is $i(h-h_{I})$. Hence in such a situation the
classical scalar field $\Psi$ and its conjugate momentum field $\Pi_{\Psi}$
are
$\displaystyle\Psi[h,h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{I}\cosh(h-h_{I})-\Pi_{\Psi}^{I}\sinh(h-h_{I}),$ (7.256)
$\displaystyle\Pi_{\psi}[h,h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{I}\sinh(h-h_{I})+\Pi_{\Psi}^{I}\cosh(h-h_{I}),$ (7.257)
and the corresponding probability density equals to
$\displaystyle\Omega[h,h_{I}]$ $\displaystyle=$
$\displaystyle\left(\Psi^{I}\right)^{2}\cosh^{2}(h-h_{I})+\left(\Pi_{\Psi}^{I}\right)^{2}\sinh^{2}(h-h_{I})-$
(7.258) $\displaystyle-$ $\displaystyle
2\Psi^{I}\Pi^{I}\sinh(h-h_{I})\cosh(h-h_{I}).$ (7.259)
In this manner the normalization condition (7.228) generates the equation for
the initial data
$\alpha(h_{F}-h_{I})\left(\Pi_{\Psi}^{I}\right)^{2}-2\beta(h_{F}-h_{I})\Psi^{I}\Pi^{I}+\gamma(h_{F}-h_{I})\left(\Psi^{I}\right)^{2}=\dfrac{1}{h_{F}-h_{I}},$
(7.260)
where the coefficients-functions of $h_{I}$ are
$\displaystyle\alpha(h_{F}-h_{I})$ $\displaystyle=$
$\displaystyle\dfrac{\sinh(2(h_{F}-h_{I}))-2(h_{F}-h_{I})}{4},$ (7.261)
$\displaystyle\beta(h_{F}-h_{I})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\sinh^{2}(2(h_{F}-h_{I})),$ (7.262)
$\displaystyle\gamma(h_{F}-h_{I})$ $\displaystyle=$
$\displaystyle\dfrac{\sinh(2(h_{F}-h_{I}))+2(h_{F}-h_{I})}{4}.$ (7.263)
Application of the basic definition of initial data of the conjugate momentum
$\Pi_{\Psi}^{I}=\dfrac{\delta\Psi_{I}}{\delta h_{I}}$ to the equation (7.264)
gives the differential equation for the initial data do the scalar field
$\Psi_{I}$
$\alpha(h_{F}-h_{I})\left(\dfrac{\delta\Psi_{I}}{\delta
h_{I}}\right)^{2}-2\beta(h_{F}-h_{I})\Psi^{I}\dfrac{\delta\Psi_{I}}{\delta
h_{I}}+\gamma(h_{F}-h_{I})\left(\Psi^{I}\right)^{2}-\dfrac{1}{h_{F}-h_{I}}=0.$
(7.264)
for finite values of $h_{F}$ this equation is difficult to solve
straightforwardly. When the upper limit is infinite, i.e.
$h_{F}\rightarrow\infty$, then the third term in (7.264) vanishes identically
whereas the coefficients $\alpha$, $\beta$, and $\gamma$ tends to infinity.
Then however, one can divide both sides of the equation (7.264) by
$\dfrac{1}{2}\sinh(2(h_{F}-h_{I}))$ and obtain the finite limit
$\left(\dfrac{\delta\Psi_{I}}{\delta
h_{I}}\right)^{2}-2\Psi^{I}\dfrac{\delta\Psi_{I}}{\delta
h_{I}}+\left(\Psi^{I}\right)^{2}=\left(\dfrac{\delta\Psi_{I}}{\delta
h_{I}}-\Psi^{I}\right)^{2}=0.$ (7.265)
With using of the boundary condition $\Psi_{I}(h_{0})=\Psi_{0}$, this equation
can be solved immediately
$\Psi_{I}=\Psi_{0}\exp\left\\{h_{I}-h_{0}\right\\}.$ (7.266)
Because of the equations (7.253) and (7.254) are related to the unitary
operator of evolution (7.255) we shall call their solutions _the unitary
three-manifolds_. Interestingly, the equation (7.253) suggests that in general
the unitary three-manifolds are deformations of the three-manifolds defined by
the Ricci scalar-curvature proportional to determinant of an induced metric,
i.e. to the global dimension
${{}^{(3)}}R=6(8\pi)^{2}h,$ (7.267)
and deformation is due to the cosmological constant $\Lambda$ and the energy
density $\varrho$ of Matter fields. Because of the three-dimensional manifolds
defined by (7.267) are also the unitary three-manifolds we shall call them
_the global unitary three-manifolds_.
##### E3 The Fourier Analysis
The quantum gravity given by one-dimensional the Klein–Gordon equation (7.130)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta
h^{2}}+\omega^{2}[h]\right)\Psi(h)=0,$ (7.268)
$\displaystyle\omega^{2}[h]=-\dfrac{1}{6(8\pi)^{2}}\dfrac{1}{h}\left({{}^{(3)}}\\!R-2\Lambda-2\kappa\ell_{P}^{2}\varrho\right),$
(7.269)
can be considered as the equation for the 3-dimensional scalar curvature
${{}^{(3)}\\!R}$
$^{(3)}\\!R=2\left(\Lambda+\kappa\ell_{P}^{2}\varrho\right)+6(8\pi)^{2}\varphi(h)h,$
(7.270)
where we have introduced the function
$\varphi(h)=\dfrac{1}{\Psi(h)}\dfrac{\delta^{2}\Psi(h)}{\delta{h^{2}}}.$
(7.271)
In the stationary case
$\varrho\equiv 0\cap\Lambda\equiv
0\quad\textrm{or}\quad\varrho=-\dfrac{\Lambda}{\kappa\ell_{P}^{2}},$ (7.272)
one obtains from (7.270) that
$^{(3)}\\!R=6(8\pi)^{2}\varphi_{n}h,$ (7.273)
where $\varphi_{n}$ are eigenvalues determined by the equation
$\dfrac{\delta^{2}\Psi}{\delta{h^{2}}}=\varphi_{n}\Psi.$ (7.274)
If one wishes to consider the non-stationary situation then it is easy to see
that
$\varphi(h)=\varphi_{n}-\dfrac{2}{6(8\pi)^{2}}\dfrac{\Lambda+\kappa\ell_{P}^{2}\varrho}{h}.$
(7.275)
One can apply the analytical form of the classical scalar field $\Psi$
$\Psi=\sum_{n=-\infty}^{\infty}a_{n}(h-h_{I})^{n},$ (7.276)
to the eigenequation (7.274)
$\sum_{n=-\infty}^{\infty}(n+1)(n+2)a_{n+2}(h-h_{I})^{n}=\varphi_{n}\sum_{n=-\infty}^{\infty}a_{n}(h-h_{I})^{n},$
(7.277)
and express eigenvalues $\varphi_{n}$ by the series coefficients $a_{n}$
$\varphi_{n}=(n+1)(n+2)\dfrac{a_{n+2}}{a_{n}}.$ (7.278)
Because, however, the series coefficients are defined as
$a_{n}=\left.\dfrac{\delta^{n}\Psi}{\delta h^{n}}\right|_{h=h_{I}},$ (7.279)
one obtains from (7.278)
$\varphi_{n}=(n+1)(n+2)\left.\dfrac{\dfrac{\delta^{n+2}}{\delta
h^{n+2}}\Psi(h)}{\dfrac{\delta^{n}}{\delta h^{n}}\Psi(h)}\right|_{h=h_{I}}.$
(7.280)
In the light of the fact
$\dfrac{\delta^{n+2}\Psi}{\delta h^{n+2}}=\dfrac{\delta^{n}}{\delta
h^{n}}\left(\dfrac{\delta^{2}\Psi}{\delta
h^{2}}\right)=-\dfrac{\delta^{n}}{\delta
h^{n}}\left(\omega^{2}[h]\Psi(h)\right),$ (7.281)
where we have applied the equation (7.268), one has
$\varphi_{n}=-(n+1)(n+2)\left.\dfrac{\dfrac{\delta^{n}}{\delta
h^{n}}\left(\omega^{2}[h]\Psi(h)\right)}{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}\right|_{h=h_{I}}.$ (7.282)
Let us assume that there are generalized Fourier transforms
$\displaystyle\widetilde{\Psi}(s)$ $\displaystyle=$ $\displaystyle\int\delta
he^{-2i\pi sh}\Psi(h),$ (7.283) $\displaystyle\widetilde{m^{2}}(s)$
$\displaystyle=$ $\displaystyle\int\delta he^{-2i\pi sh}m^{2}[h],$ (7.284)
as well as the inverted Fourier transforms
$\displaystyle\Psi(h)$ $\displaystyle=$ $\displaystyle\int\delta se^{2i\pi
sh}\widetilde{\Psi}(s),$ (7.285) $\displaystyle m^{2}[h]$ $\displaystyle=$
$\displaystyle\int\delta se^{2i\pi sh}\widetilde{m^{2}}(s).$ (7.286)
Applying the generalized Leibniz product formula
$\dfrac{\delta^{n}}{\delta
h^{n}}\left(\omega^{2}[h]\Psi(h)\right)=\sum_{r=0}^{n}\binom{n}{r}\left(\dfrac{\delta^{r}}{\delta
h^{r}}\omega^{2}[h]\right)\left(\dfrac{\delta^{n-r}}{\delta
h^{n-r}}\Psi(h)\right),$ (7.287)
one obtains
$\displaystyle\dfrac{\dfrac{\delta^{n}}{\delta
h^{n}}\left(\omega^{2}[h]\Psi(h)\right)}{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}\sum_{r=0}^{n}\binom{n}{r}\left(\dfrac{\delta^{r}}{\delta
h^{r}}\omega^{2}[h]\right)\left(\dfrac{\delta^{n-r}}{\delta
h^{n-r}}\Psi(h)\right)=$ (7.288) $\displaystyle=$
$\displaystyle\dfrac{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}\sum_{r=0}^{n}\binom{n}{r}\left(\dfrac{\delta^{r}}{\delta
h^{r}}\omega^{2}[h]\right)\left(\dfrac{\delta^{-r}}{\delta
h^{-r}}\Psi(h)\right)=$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{n}\binom{n}{r}\left(\dfrac{\delta^{r}}{\delta
h^{r}}\omega^{2}[h]\right)\left(\dfrac{\delta^{-r}}{\delta
h^{-r}}\Psi(h)\right).$
In this manner using of the inverted Fourier transforms (7.285) and (7.286)
gives
$\displaystyle\dfrac{\delta^{r}}{\delta h^{r}}\omega^{2}[h]$ $\displaystyle=$
$\displaystyle\int\delta se^{2i\pi sh}(2i\pi s)^{r}\widetilde{\omega^{2}}(s),$
(7.289) $\displaystyle\dfrac{\delta^{-r}}{\delta h^{-r}}\Psi(h)$
$\displaystyle=$ $\displaystyle\int\delta se^{2i\pi sh}(2i\pi
s)^{-r}\widetilde{\Psi}(s),$ (7.290)
and by this reason the formula (7.288) becomes
$\dfrac{\dfrac{\delta^{n}}{\delta
h^{n}}\left(\omega^{2}[h]\Psi(h)\right)}{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}=\int\int\delta s\delta
s^{\prime}e^{2i\pi(^{\prime}+s^{\prime})h}\left[\sum_{r=0}^{n}\binom{n}{r}\left(\dfrac{s}{s^{\prime}}\right)^{r}\right]\widetilde{\omega^{2}}(s)\widetilde{\Psi}(s^{\prime}).$
(7.291)
Applying the standard summation procedure
$\sum_{r=0}^{n}\binom{n}{r}x^{r}=(1+x)^{n},$ (7.292)
one obtains
$\dfrac{\dfrac{\delta^{n}}{\delta
h^{n}}\left(\omega^{2}[h]\Psi(h)\right)}{{\dfrac{\delta^{n}}{\delta
h^{n}}\Psi(h)}}=\int\int\delta{s}\delta{s^{\prime}}e^{2i\pi(s^{\prime}+s)h}\left(1+\dfrac{s}{s^{\prime}}\right)^{n}\widetilde{\omega^{2}}(s)\widetilde{\Psi}(s^{\prime}).$
(7.293)
By this reason the eigenvalues (7.282) are
$\varphi_{n}=-(n+1)(n+2)\left.\int\int\delta{s}\delta{s^{\prime}}e^{2i\pi(s+s^{\prime})h}\left(1+\dfrac{s}{s^{\prime}}\right)^{n}\widetilde{\omega^{2}}(s)\widetilde{\Psi}(s^{\prime})\right|_{h=h_{I}}.$
(7.294)
Applying the Fourier transforms (7.283) and (7.284) one receives
$\widetilde{\omega^{2}}(s)\widetilde{\Psi}(s^{\prime})=\int\int\delta h\delta
h^{\prime}e^{-2i\pi(sh+s^{\prime}h^{\prime})}\omega^{2}[h]\Psi(h^{\prime}),$
(7.295)
and by this reason one obtains finally
$\displaystyle\varphi_{n}$ $\displaystyle=$
$\displaystyle-\left.\int\int\delta h\delta
h^{\prime}\mathcal{G}_{n}(h-h^{\prime})\omega^{2}[h]\Psi(h^{\prime})\right|_{h=h_{I},h^{\prime}=h_{I}}=$
(7.296) $\displaystyle=$ $\displaystyle-\left.\int\int\delta h\delta
h^{\prime}\mathcal{G}_{n}(h-h^{\prime})\dfrac{\delta^{2}\Psi(h)}{\delta
h^{2}}\dfrac{\Psi(h^{\prime})}{\Psi(h)}\right|_{h=h_{I},h^{\prime}=h_{I}}=$
(7.297) $\displaystyle=$ $\displaystyle-\left.\int\int\delta h\delta
h^{\prime}\mathcal{G}_{n}(h-h^{\prime})\dfrac{\delta\Pi_{\Psi}(h)}{\delta
h}\dfrac{\Psi(h^{\prime})}{\Psi(h)}\right|_{h=h_{I},h^{\prime}=h_{I}},$
(7.298)
where in the second line we have used equations of motion, and in the third
line we have applied definition of the conjugate momentum field. The kernel
$\mathcal{G}(h-h^{\prime})$ is given by the relation
$\mathcal{G}_{n}(h-h^{\prime})=(n+1)(n+2)\int\int\delta s\delta
s^{\prime}e^{2i\pi
s^{\prime}(h-h^{\prime})}\left(1+\dfrac{s^{\prime}}{s}\right)^{n}.$ (7.299)
Estimation of the kernel (7.299) is the crucial element of the proposed
analysis. To make it consistently let us determine the range of $s$ and
$s^{\prime}$. In quantum mechanics the Fourier analysis transforms theory to
the momentum space representation. Therefore, $s$ and $s^{\prime}$ are the
momenta conjugated to $h$ and $h^{\prime}$, respectively.
The double integral in the kernel (7.299) can be transformed as
$\int\delta s^{\prime}e^{2i\pi s^{\prime}(h-h^{\prime})}\left[\int\delta
s\left(1+\dfrac{s}{s^{\prime}}\right)^{n}\right],$ (7.300)
and therefore one sees that the variable $s$ is _internal_ , while the
variable $s^{\prime}$ is _external_. Let us take _ad hoc_ the finite range of
the internal variable $s\in[0,S]$ and the infinite range of the external
variable $s^{\prime}\in[0,\infty]$. The internal integral can be easy computed
$\displaystyle\int\delta s\left(1+\dfrac{s}{s^{\prime}}\right)^{n}$
$\displaystyle=$ $\displaystyle
s^{\prime}\int\delta\left(1+\dfrac{s}{s^{\prime}}\right)\left(1+\dfrac{s}{s^{\prime}}\right)^{n}=$
(7.301) $\displaystyle=$ $\displaystyle
s^{\prime}\int_{t=1+\frac{s}{s^{\prime}}}\delta
tt^{n}=\dfrac{s^{\prime}}{n+1}\left(1+\dfrac{S}{s^{\prime}}\right)^{n+1},$
(7.302)
and by this reason the double integral (7.300) becomes
$\dfrac{1}{n+1}\int\delta s^{\prime}e^{2i\pi
s^{\prime}(h-h^{\prime})}s^{\prime}\left(1+\dfrac{S}{s^{\prime}}\right)^{n+1}.$
(7.303)
In the range $s^{\prime}\in[0,\infty]$ the integral in (7.303) converges for
$n<1$
$\int\delta s^{\prime}e^{2i\pi
s^{\prime}(h-h^{\prime})}s^{\prime}\left(1+\dfrac{S}{s^{\prime}}\right)^{n+1}=-\dfrac{\Gamma(1-n)}{4\pi^{2}(h-h^{\prime})^{2}}U\left(-1-n,-1,-2i\pi
S\left(h-h^{\prime}\right)\right),$ (7.304)
where $\Gamma(z)$ is the Euler gamma-function
$\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt,$ (7.305)
and $U\left(a,b,z\right)$ is the Tricomi confluent hypergeometric function
(See e.g. the Ref. [602] for basic knowledge about special functions)
$U(a,b,z)=\dfrac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)+\dfrac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z),$
(7.306)
where $M(a,b,z)$ is the Kummer confluent hypergeometric function
$M(a,b,z)=\sum_{n=0}^{\infty}\dfrac{(a)_{n}}{(b)_{n}}\dfrac{z^{n}}{n!}=1+\dfrac{\Gamma(b)}{\Gamma(a)}\sum_{n=1}^{\infty}\dfrac{\Gamma(a+n)}{\Gamma(b+n)}\dfrac{z^{n}}{n!},$
(7.307)
where $(\alpha)_{n}$ are the Pochhammer symbols
$(\alpha)_{n}=\dfrac{\Gamma(\alpha+n)}{\Gamma(\alpha)}\quad,\quad(\alpha)_{0}=1,$
(7.308)
and for $\Re a>0$ there is the integral relation
$\dfrac{(a)_{n}}{(b)_{n}}=\dfrac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_{0}^{1}t^{a-1+n}(1-t)^{b-a-1}dt.$
(7.309)
Applying the values $a=1-n>0$, $b=3$, $z=-2i\pi S(h-h^{\prime})$ to the Kummer
transformation
$U(a,b,z)=z^{1-b}U(1+a-b,2-b,z),$ (7.310)
one receives
$U\left(-1-n,-1,-2i\pi
S\left(h-h^{\prime}\right)\right)=-4\pi^{2}S^{2}(h-h^{\prime})^{2}U\left(1-n,3,-2i\pi
S\left(h-h^{\prime}\right)\right),$ (7.311)
and by this reason the integral (7.304) becomes
$\int\delta s^{\prime}e^{2i\pi
s^{\prime}(h-h^{\prime})}s^{\prime}\left(1+\dfrac{S}{s^{\prime}}\right)^{n+1}=S^{2}\Gamma(1-n)U\left(1-n,3,-2i\pi
S\left(h-h^{\prime}\right)\right).$ (7.312)
Because of now $1-n>0$ the Tricomi confluent hypergeometric functions can be
computed in the integral representation
$U(a,b,z)=\dfrac{1}{\Gamma(a)}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}dt.$
(7.313)
In this manner the kernel (7.299) (for $n<1$) is given by the formula
$\mathcal{G}_{n}(h-h^{\prime})=S^{2}(n+2)\Gamma(1-n)U\left(1-n,3,-2i\pi
S\left(h-h^{\prime}\right)\right),$ (7.314)
and consequently the eigenvalue (7.278) can be evaluated as
$\varphi_{n}=-S^{2}(n+2)\Gamma(1-n)\left.\int\int\delta h\delta
h^{\prime}U\left(1-n,3,-2i\pi
S\left(h-h^{\prime}\right)\right)\Upsilon(h,h^{\prime})\right|_{h=h_{I},h^{\prime}=h_{I}},$
(7.315)
where we have introduced the symbol $\Upsilon(h,h^{\prime})$
$\Upsilon(h,h^{\prime})=\dfrac{\Psi(h^{\prime})}{\Psi(h)}\dfrac{\delta^{2}\Psi(h)}{\delta
h^{2}}=\dfrac{\Psi(h^{\prime})}{\Psi(h)}\dfrac{\delta\Pi_{\Psi}(h)}{\delta
h},$ (7.316)
which can be straightforwardly established when both wave function $\Psi(h)$
and its second derivative $\dfrac{\delta^{2}\Psi(h)}{\delta h^{2}}$, or
equivalently derivative of its conjugate momentum field
$\dfrac{\delta\Pi_{\Psi}(h)}{\delta h}$, are given explicitly.
##### E4 Quantum Correlations
With using of the matrices (7.195) and (7.169), and the relation (7.194) one
derives the quantum field
$\hat{\Psi}(h)=\sqrt{\dfrac{\omega_{I}}{8}}\dfrac{1}{\omega}\left(\exp\left\\{-i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}\textsf{G}_{I}+\exp\left\\{i\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}\textsf{G}_{I}^{\dagger}\right).$ (7.317)
Let us take into account the $n$-particle one-point quantum states determined
as
$|h,n\rangle\equiv\hat{\Psi}^{n}\left|\textrm{0}\right\rangle=\left(\sqrt{\dfrac{\omega_{I}}{8}}\dfrac{1}{\omega}\exp\left\\{\int_{h_{I}}^{h}\omega^{\prime}\delta
h^{\prime}\right\\}\right)^{n}\textsf{G}^{\dagger
n}_{I}\left|\textrm{0}\right\rangle,$ (7.318)
yields two-point correlators
$\mathrm{Cor}_{n^{\prime}n}(h^{\prime},h)\equiv\langle
n^{\prime},h^{\prime}|h,n\rangle$ or explicitly
$\displaystyle\mathrm{Cor}_{n^{\prime}n}(h^{\prime},h)$ $\displaystyle=$
$\displaystyle\left(\dfrac{\omega_{I}}{8}\right)^{(n^{\prime}+n)/2}\exp\left\\{i\left(n^{\prime}\int_{h^{\prime}}^{h_{I}}+n\int_{h_{I}}^{h}\right)\omega^{\prime\prime}\delta
h^{\prime\prime}\right\\}\times$ (7.319) $\displaystyle\times$
$\displaystyle\dfrac{\left\langle\textrm{0}\right|\textsf{G}_{I}^{n^{\prime}}\textsf{G}^{\dagger
n}_{I}\left|\textrm{0}\right\rangle}{{{\omega^{\prime}}^{n^{\prime}}\omega^{n}}}.$
Basically one obtains
$\displaystyle\mathrm{Cor}_{00}(h,h)=\mathrm{Cor}_{00}(h^{\prime},h)=\mathrm{Cor}_{00}(h_{I},h_{I})=\left\langle{\textrm{0}}|{\textrm{0}}\right\rangle,$
(7.320) $\displaystyle\mathrm{Cor}_{11}(h_{I},h_{I})=\dfrac{1}{8\omega_{I}},$
(7.321)
$\displaystyle\dfrac{\mathrm{Cor}_{n^{\prime}n}(h_{I},h_{I})}{\left[\mathrm{Cor}_{11}(h_{I},h_{I})\right]^{(n^{\prime}+n)/2}}=\left\langle\textrm{0}\right|\textsf{G}_{I}^{n^{\prime}}\textsf{G}^{\dagger
n}_{I}\left|\textrm{0}\right\rangle,$ (7.322)
and by elementary algebraic manipulations one receives
$\displaystyle\mathrm{Cor}_{11}(h^{\prime},h)=\dfrac{\sqrt{{\mathrm{Cor}_{11}(h^{\prime},h^{\prime})\mathrm{Cor}_{11}(h,h)}}}{\mathrm{Cor}_{11}(h_{I},h_{I})}\exp\left\\{i\int_{h^{\prime}}^{h}\omega^{\prime\prime}\delta
h^{\prime\prime}\right\\},$ (7.323)
$\displaystyle\dfrac{\mathrm{Cor}_{nn}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}=\left[\dfrac{\mathrm{Cor}_{11}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}\right]^{n},$
(7.324)
$\displaystyle\dfrac{\mathrm{Cor}_{11}(h,h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}=\left(\dfrac{m_{I}}{m}\right)^{2}\mathrm{Cor}_{11}(h_{I},h_{I}).$
(7.325)
Straightforwardly from (7.325) one can relate a size scale with quantum
correlations
$\displaystyle\lambda=\dfrac{\omega_{I}}{\omega}=\sqrt{{\dfrac{\mathrm{Cor}_{11}(h,h)}{\mathrm{Cor}_{11}(h_{I},h_{I})\mathrm{Cor}_{00}(h_{I},h_{I})}}},$
(7.326)
and consequently one receives the formulas
$\displaystyle\dfrac{\mathrm{Cor}_{n^{\prime}n}(h,h)}{\mathrm{Cor}_{n^{\prime}n}(h_{I},h_{I})}=\lambda^{n^{\prime}+n}\exp\left\\{-i(n^{\prime}-n)\int_{h_{I}}^{h}\omega^{\prime\prime}\delta
h^{\prime\prime}\right\\},$ (7.327)
$\displaystyle\dfrac{\mathrm{Cor}_{11}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})\mathrm{Cor}_{11}(h_{I},h_{I})}=\lambda^{\prime}\lambda\exp\left\\{i\int_{h^{\prime}}^{h}\omega^{\prime\prime}\delta
h^{\prime\prime}\right\\},$ (7.328)
$\displaystyle\dfrac{\mathrm{Cor}_{nn}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}=\lambda^{\prime
n}\lambda^{n}[\mathrm{Cor}_{11}(h_{I},h_{I})]^{n}\exp\left\\{in\int_{h^{\prime}}^{h}\omega^{\prime\prime}\delta
h^{\prime\prime}\right\\}.$ (7.329)
A whole information about the quantum gravity is contained in the parameters
of the theory, i.e. $m$, $\lambda$, and the initial data $m_{I}$. It is
evident that the quantum correlations are strictly determined by these
fundamental quantities only. In other words measurement of quantum
correlations can be used for deduction of values of the fundamental parameters
of the theory.
The conclusions presented in this section have purely formal character,
however, they show manifestly a general feature of the proposed theory of
quantum gravity. These conclusions are partial, but they show a non trivial
both physical and mathematical implications following from the theory of
quantum gravity. Let us see more consequences of the theory.
### Chapter 8 The Invariant Global Dimension
#### A The Invariant Global Quantum Gravity
Let us included explicitly presence of Matter fields, and extend the global
one-dimensional wave function $\Psi(h)$ to a functional $\Psi[h,\phi]$, which
we shall call _extended global wave function_. Still we take the global
dimension $h=\det h_{ij}$.
The theory of quantum gravity proposed in the previous chapter can be
rewritten as
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{eff}[h,\phi]\right)\Psi[h,\phi]=0.$
(8.1)
where $V_{eff}$ is _the (effective) gravitational potential_
$V_{eff}\equiv\dfrac{1}{6(8\pi)^{2}}\left(-\dfrac{{{}^{(3)}\\!R}}{h}+\dfrac{2\Lambda}{h}+\dfrac{2\kappa\ell_{P}^{2}}{h}\varrho[\phi]\right).$
(8.2)
The manifestly singular behavior $\sim 1/h$ of the potential (8.2), however,
can be regularized by the suitable change of variables
$h\rightarrow\xi=\xi[h],$ (8.3)
which generates the adequate Jacobi formula
$\delta\xi=\left(\dfrac{\delta\xi}{\delta h}\right)hh^{ij}\delta h_{ij}.$
(8.4)
The dimension $\xi[h]$, as a functional of the global dimension $h$, is also
diffeoinvariant. Applying the change of variables (8.3) within the equation
(8.1) one obtains
$\left\\{\left(\dfrac{\delta\xi}{\delta
h}\right)^{2}\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V_{eff}\left[\xi,\phi\right]\right\\}\Psi\left[\xi,\phi\right]=0,$
(8.5)
and therefore for all nonsingular situations $\dfrac{\delta\xi}{\delta h}\neq
0$ it can be rewritten in more convenient form
$\left\\{\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V[\xi,\phi]\right\\}\Psi\left[\xi,\phi\right]=0,$
(8.6)
where the potential $V[\xi,\phi]$ is
$V[\xi,\phi]=\left(\dfrac{\delta\xi}{\delta
h}\right)^{-2}V_{eff}\left[\xi,\phi\right].$ (8.7)
We shall call it _the generalized gravitational potential_. In fact, the
choice of the invariant dimension $\xi$ is a kind of the choice of a gauge for
the theory of quantum gravity.
Naturally, the generic gauge is the global dimension $\xi[h]\equiv h$. Another
situations can be generated straightforwardly from this fundamental case.
There is a lot of possible choices of gauge for quantum gravity. However, note
that the particular choice
$\xi=\dfrac{1}{4\pi}\sqrt{{\dfrac{h}{6}}},$ (8.8)
which is associated to the measure
$\delta\xi=\dfrac{1}{8\pi}\sqrt{{\dfrac{h}{6}}}h^{ij}\delta h_{ij},$ (8.9)
removes the singularity $1/h$ present in the gravitational potential
$V_{eff}\left[h,\phi\right]$ (8.2), and consequently the equation (8.6) reads
$\left\\{\dfrac{\delta^{2}}{\delta{\xi^{2}}}-\left({{}^{(3)}\\!R[\xi]}-2\Lambda-2\kappa\ell_{P}^{2}\varrho[\phi]\right)\right\\}\Psi\left[\xi,\phi\right]=0.$
(8.10)
The appropriate normalization condition should be chosen as
$\int\left|\Psi\left[\xi,\phi\right]\right|^{2}\delta\mu(\xi,\phi)=1,$ (8.11)
where $\delta\mu(\xi,\phi)=\delta\xi\delta\phi.$ is the invariant product
functional measure. Note that similarly as $\delta h$ the measure
$\delta\sqrt{h}$ is also the Lebesgue–Stieltjes (Radon) type integral measure,
which can be transformed to the Riemann–Lebesgue measure over space-time. In
the general case $h=h(x_{0},x_{1},x_{2},x_{3})$
$\delta\sqrt{h}=\dfrac{\partial^{4}\sqrt{h}}{\partial x_{0}\partial
x_{1}\partial x_{2}\partial x_{3}}d^{4}x.$ (8.12)
By the special role of the change of variables (8.8) we shall call this
dimension _invariant global dimension_ , and the theory of quantum gravity
(8.10) will be called _the invariant global quantum gravity_.
#### B The One-Dimensional Dirac Equation
The equation (8.6) can be derived as the Euler–Lagrange equations of motion by
the field theoretical variational principle $\delta S[\Psi]=0$ applied to the
action functional
$\displaystyle S[\Psi]$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\int\delta\xi\delta\phi\Psi[\xi,\phi]\left(\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V[\xi,\phi]\right)\Psi[\xi,\phi]=$
(8.13) $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\int\delta\phi\Psi[\xi,\phi]\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}+$
$\displaystyle+$
$\displaystyle\dfrac{1}{2}\int\delta\xi\delta\phi\left\\{\left(\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right)^{2}+V[\xi,\phi]\Psi^{2}[\xi,\phi]\right\\},$
where the integration by parts was applied. One can choose the coordinate
system in the Wheeler superspace by the condition of vanishing of the material
surface term
$-\dfrac{1}{2}\int\delta\phi\Psi[\xi,\phi]\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}=0,$
(8.14)
which we shall call _the material coordinate system_. In the material
coordinates the action functional (8.13) is reduced to the action of the
Euclidean field theory
$S[\Psi]\equiv\int\delta\xi\delta\phi
L\left[\Psi[\xi,\phi],\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right],$ (8.15)
where the Euclidean Lagrangian has the form
$L\left[\Psi[\xi,\phi],\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right]=\dfrac{1}{2}\left(\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right)^{2}+\dfrac{V[\xi,\phi]}{2}\Psi^{2}[\xi,\phi],$
(8.16)
and the corresponding canonical momentum conjugated to the classical scalar-
valued field $\Psi^{[}\xi,\phi]$ is
$\Pi_{\Psi}[\xi,\phi]=\dfrac{\partial
L}{\partial\left(\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right)}=\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}.$
(8.17)
Therefore the choice of the coordinate system (8.14) actually means the choice
of the orthogonal coordinates in the space field-theoretic phase space
$(\Psi[\xi,\phi],\Pi_{\Psi}[\xi,\phi])$
$\forall\xi,\phi\in\Sigma(\xi,\phi):\int\delta\phi\Psi[\xi,\phi]\Pi_{\Psi}[\xi,\phi]=0,$
(8.18)
where by $\Sigma(\xi,\phi)$ we denoted the midisuperspace strata of the
Wheeler superspace. With using of the conjugate momentum (8.17) the equation
(8.6) can be rewritten in the form
$\dfrac{\delta\Pi_{\Psi}[\xi,\phi]}{\delta\xi}+V[\xi,\phi]\Psi[\xi,\phi]=0,$
(8.19)
and together with the equation (8.17) creates the Hamilton canonical equations
of motion, yielding the appropriate one-dimensional Dirac equation
$\left(-i\gamma\dfrac{\delta}{\delta\xi}-M[\xi,\phi]\right)\Phi[\xi,\phi]=0,$
(8.20)
where $\Phi[\xi,\phi]$ is the two-component classical field
$\Phi[\xi,\phi]=\left[\begin{array}[]{c}\Pi_{\Psi}[\xi,\phi]\\\
\Psi[\xi,\phi]\end{array}\right],$ (8.21)
$M[\xi,\phi]$ is the mass matrix of this field
$M[\xi,\phi]=\left[\begin{array}[]{cc}1&0\\\ 0&V[\xi,\phi]\end{array}\right],$
(8.22)
and the $\gamma$-matrices algebra
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right]\equiv\sigma_{y},$ (8.25) $\displaystyle\gamma^{2}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right]\equiv\mathbf{I}_{2},$ (8.28)
that in itself creates the Clifford algebra $\mathcal{C}\ell_{2}(\mathbb{C})$
$\displaystyle\left\\{\gamma,\gamma\right\\}$ $\displaystyle=$ $\displaystyle
2\mathbf{I}_{2}.$ (8.29)
#### C The Cauchy-Like Wave Functionals
The one-dimensional Dirac equation (8.20) can be rewritten in the form of the
Schrödinger equation
$i\dfrac{\delta\Phi[\xi,\phi]}{\delta\xi}=H[\xi,\phi]\Phi[\xi,\phi],$ (8.30)
where the Hamiltonian is
$H[\xi,\phi]=i\left[\begin{array}[]{cc}0&-V[\xi,\phi]\\\
1&0\end{array}\right].$ (8.31)
Solution of the $\xi$-evolution (8.30) can be written out straightforwardly
$\Phi[\xi,\phi]=U[\xi,\phi]\Phi[\xi^{I},\phi],$ (8.32)
where $\Phi[\xi^{I},\phi]$ is an initial data vector with respect to $\xi$
only, and $U[\xi,\phi]$ is the operator of $\xi$-evolution
$\displaystyle
U=\exp\left\\{-i\int_{\Sigma(\xi)}\delta\xi^{\prime}H[\xi^{\prime},\phi]\right\\}=\exp\left\\{-i\Omega(\xi,\phi)\langle
H\rangle(\xi,\phi)\right\\},$ (8.33)
where $\Sigma(\xi)$ is the finite integration region in the subset of
midisuperspace, which we shall call _$\xi$ -space_,
$\Omega=V\left(\Sigma(\phi,\xi)\right)$ is the volume of full configuration
space, and $\langle H\rangle(\phi)$ is the Hamiltonian averaged on
midisuperspace
$\displaystyle\Omega(\xi,\phi)$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.34) $\displaystyle\langle H\rangle(\xi,\phi)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi,\phi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}H[\xi^{\prime},\phi],$
(8.35)
where $\Sigma(\xi,\phi)$ is the midisuperspace
$\Sigma(\xi,\phi)=\Sigma(\xi)\times\Sigma(\phi),$ (8.36)
which is assumed to be finite integration region. Explicitly one obtains
$\displaystyle
U[\xi,\phi]=\mathbf{I}_{2}\cos\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]+$
$\displaystyle+\left[\begin{array}[]{cc}0&-\sqrt{{\langle
V\rangle(\xi,\phi)}}\\\ \dfrac{1}{\sqrt{{\langle
V\rangle(\xi,\phi)}}}&0\end{array}\right]\sin\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$ (8.39)
where we have introduced averaged generalized gravitational potential
$\langle
V\rangle(\xi,\phi)=\dfrac{1}{\Omega(\xi,\phi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}V[\xi^{\prime},\phi].$
(8.40)
Elementary algebraic manipulations yield the extended global wave functional
$\displaystyle\Psi[\xi,\phi]$ $\displaystyle=$
$\displaystyle\Psi[\xi^{I},\phi]\cos\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]+$ (8.41) $\displaystyle+$
$\displaystyle\Pi_{\Psi}[\xi^{I},\phi]\dfrac{\sin\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle V\rangle(\xi,\phi)}}},$
and the canonical conjugate momentum as the solution is
$\displaystyle\Pi_{\Psi}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\Pi_{\Psi}[\xi^{I},\phi]\cosh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]+$ (8.42) $\displaystyle-$
$\displaystyle\Psi[\xi^{I},\phi]\sqrt{{\langle
V\rangle(\xi,\phi)}}\sinh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$
where $\Psi[\xi^{I},\phi]$ and $\Pi_{\Psi}[\xi^{I},\phi]$ are initial data
with respect to $\xi$ only. Applying, however, the equation (8.17) to (8.42)
one obtains the relation
$\displaystyle\Pi_{\Psi}[\xi,\phi]=\dfrac{\Pi_{\Psi}[\xi^{I},\phi]}{\sqrt{\langle
V\rangle(\xi,\phi)}}\dfrac{\delta}{\delta\xi}\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]\cos\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]$
$\displaystyle-\Bigg{\\{}\Psi[\xi^{I},\phi]\dfrac{\delta}{\delta\xi}\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]-\Pi_{\Psi}[\xi^{I},\phi]\dfrac{\delta}{\delta\xi}\left[\dfrac{1}{\sqrt{\langle
V\rangle(\xi,\phi)}}\right]\Bigg{\\}}$
$\displaystyle\times\sin\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right],$ (8.43)
which after calculation of the functional derivatives
$\dfrac{\delta}{\delta\xi}\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]=\dfrac{\sqrt{\langle
V\rangle(\xi,\phi)}}{2}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}+1\right),$
(8.44)
and
$\dfrac{\delta}{\delta\xi}\left[\dfrac{1}{\sqrt{\langle
V\rangle(\xi,\phi)}}\right]=\dfrac{1}{2}\dfrac{1}{\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}-1\right)$
(8.45)
and using them within the formula (8.43) yields
$\displaystyle\Pi_{\Psi}[\xi,\phi]=\Pi_{\Psi}[\xi^{I},\phi]\dfrac{1}{2}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}+1\right)\cos\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]$
$\displaystyle-\Bigg{[}\Psi[\xi^{I},\phi]\dfrac{\sqrt{\langle
V\rangle(\xi,\phi)}}{2}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}+1\right)$
$\displaystyle-\dfrac{\Pi_{\Psi}[\xi^{I},\phi]}{2\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}-1\right)\Bigg{]}\sin\left[\Omega(\xi,\phi)\sqrt{\langle
V\rangle(\xi,\phi)}\right].$ (8.46)
After comparison with (8.42) one obtains the equations
$\dfrac{1}{2}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}+1\right)=1,$
(8.47)
and
$\Psi[\xi^{I},\phi]\dfrac{1}{2}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}+1\right)-\dfrac{\Pi_{\Psi}[\xi^{I},\phi]}{\Omega(\xi,\phi)\langle
V\rangle(\xi,\phi)}\dfrac{1}{2}\left(\dfrac{\delta\Omega(\xi,\phi)}{\delta\xi}-1\right)=\Psi[\xi^{I},\phi],$
(8.48)
The equation (8.47) yields the relation
$\dfrac{\delta\Omega}{\delta\xi}=1=\int_{\Sigma(\phi)}\delta\phi^{\prime},$
(8.49)
where the last integral arises by the formula (8.34). Application of the
result (8.49) to the equation (8.48) leads to the self-consistent identity
$\Psi[\xi^{I},\phi]=\Psi[\xi^{I},\phi]$. Such a situation means that the
volume of midisuperspace $\Omega(\xi,\phi)$ is $\phi$-invariant
$\Omega(\xi,\phi)=\int_{\Sigma(\xi,\phi)}\delta\xi^{\prime}\delta\phi^{\prime}=\int_{\Sigma(\phi)}\delta\phi^{\prime}\int_{\Sigma(\xi)}\delta\xi^{\prime}=\int_{\Sigma(\xi)}\delta\xi^{\prime}=\Omega(\xi),$
(8.50)
i.e. does not depend on Matter fields.
Directly from (8.41) the probability density can be deduced easily as
$\displaystyle|\Psi[\xi,\phi]|^{2}$ $\displaystyle=$
$\displaystyle(\Psi[\xi^{I},\phi])^{2}\cos^{2}\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]$ (8.51) $\displaystyle+$
$\displaystyle(\Pi_{\Psi}[\xi^{I},\phi])^{2}\left(\dfrac{\sin\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]}{\sqrt{\langle V\rangle(\xi,\phi)}}\right)^{2}$
$\displaystyle+$
$\displaystyle\Psi[\xi^{I},\phi]\Pi_{\Psi}[\xi^{I},\phi]\dfrac{\sin\left[2\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]}{\sqrt{\langle V\rangle(\xi,\phi)}},$
and in the light of the relation (8.18) it simplifies to
$\displaystyle|\Psi[\xi,\phi]|^{2}$ $\displaystyle=$
$\displaystyle(\Psi[\xi^{I},\phi])^{2}\cos^{2}\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]$ (8.52) $\displaystyle+$
$\displaystyle(\Pi_{\Psi}[\xi^{I},\phi])^{2}\left(\dfrac{\sin\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]}{\sqrt{\langle V\rangle(\xi,\phi)}}\right)^{2}.$
Let us take into account _ad hoc_ the following separation conditions
$\displaystyle\Psi[\xi^{I},\phi]$ $\displaystyle=$
$\displaystyle\Psi[\xi^{I}]\Gamma_{\Psi}[\phi],$ (8.53)
$\displaystyle\Pi_{\Psi}[\xi^{I},\phi]$ $\displaystyle=$
$\displaystyle\Pi_{\Psi}[\xi^{I}]\Gamma_{\Pi}[\phi],$ (8.54)
where $\Gamma_{\Psi}$ and $\Gamma_{\Pi}$ are functionals of $\phi$ only and
$\Psi[\xi^{I}]$, and $\Pi_{\Psi}[\xi^{I}]$ are constant functionals. Applying
the usual normalization condition
$\int_{\Sigma(\xi,\phi)}|\Psi[\xi^{\prime},\phi^{\prime}]|^{2}\delta\xi^{\prime}\delta\phi^{\prime}=1,$
(8.55)
one obtains the simple constraint for the initial data
$A(\Pi_{\Psi}[\xi^{I}])^{2}+B(\Psi[\xi^{I}])^{2}-1=0,$ (8.56)
where the constants $A$ and $B$ are given by the integrals
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\Gamma_{\Pi}[\phi^{\prime}]\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.57) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\Gamma_{\Psi}[\phi^{\prime}]\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V(\xi^{\prime},\phi^{\prime})\rangle}\right]\delta\xi^{\prime}\delta\phi^{\prime},$
(8.58)
which in our assumption are convergent, finite, and independent on the initial
data $\xi^{I}$. The equation (8.56), however, can be solved straightforwardly.
In result one obtains
$\displaystyle\Pi_{\Psi}[\xi^{I}]=\pm\sqrt{{\dfrac{1}{A}-\dfrac{B}{A}(\Psi[\xi^{I}])^{2}}},$
(8.59)
which together with the definition
$\Pi_{\Psi}[\xi^{I},\phi]=\dfrac{\delta\Psi[\xi^{I},\phi]}{\delta\xi^{I}},$
(8.60)
and the separability conditions (8.53)-(8.54) yields the differential equation
for the initial data of the field $\Psi[\xi]$
$\dfrac{1}{\Gamma[\phi]}\dfrac{\delta\Psi[\xi^{I}]}{\delta\xi^{I}}=\pm\sqrt{{\dfrac{1}{A}-\dfrac{B}{A}(\Psi[\xi^{I}])^{2}}},$
(8.61)
where $\Gamma[\phi]$ is the coefficient dependent on Matter fields only
$\Gamma[\phi]\equiv\dfrac{\Gamma_{\Pi}[\phi]}{\Gamma_{\Psi}[\phi]},$ (8.62)
which can be integrated straightforwardly
$\sqrt{A}\int\dfrac{\delta\Psi[\xi^{I}]}{\sqrt{{1-B(\Psi[\xi^{I}])^{2}}}}=\pm\Gamma[\phi]\xi^{I}+C,$
(8.63)
where $C$ is a constant of integration, with the result
$\sqrt{{\dfrac{A}{B}}}\arcsin\left\\{\sqrt{{\dfrac{B}{A}}}\Psi[\xi^{I}]\right\\}=\pm\Gamma[\phi]\xi^{I}+C,$
(8.64)
so that after elementary algebraic manipulations one obtains
$\Psi[\xi^{I}]=\sqrt{{\dfrac{A}{B}}}\sin\theta(\xi^{I},\phi),$ (8.65)
where $\theta(\xi^{I},\phi)$ is the phase
$\theta(\xi^{I},\phi)=\sqrt{{\dfrac{B}{A}}}\left(\pm\Gamma[\phi]\xi^{I}+C\right),$
(8.66)
Albeit, because of $\Psi[\xi^{I}]$ must be a functional of $\xi^{I}$ only,
must hold $\Gamma[\phi]=\Gamma_{0}$, where $\Gamma_{0}$ is a constant
independent on $\phi$ and $\xi^{I}$, for which the phase
$\theta(\xi^{I},\phi)$ is reduced to
$\theta(\xi^{I})=\sqrt{{\dfrac{B}{A}}}\left(\pm\Gamma_{0}\xi^{I}+C\right).$
(8.67)
Taking into account the relation (8.59) one obtains finally
$\displaystyle\Psi[\xi^{I}]$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{A}{B}}\sin\theta(\xi^{I}),$ (8.68)
$\displaystyle\Pi_{\Psi}[\xi^{I}]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{A}-\sin^{2}\theta(\xi^{I})}}.$ (8.69)
In the light of the equation (8.18), however, one of the relations
$\displaystyle\sin\theta(\xi^{I})$ $\displaystyle=$ $\displaystyle 0,$ (8.70)
$\displaystyle\sin\theta(\xi^{I})$ $\displaystyle=$
$\displaystyle\pm\sqrt{\dfrac{1}{A}},$ (8.71)
is always true. One sees that both these conditions define discrete values of
the initial data of the invariant global dimension $\xi_{I}$. Namely, the
first relation (8.70) leads to the solution
$\sqrt{{\dfrac{B}{A}}}\left(\pm\Gamma_{0}\xi^{I}+C\right)=k\pi,$ (8.72)
where $k\in\mathbb{Z}$ is an integer, what leads to
$\xi^{I}=\pm\dfrac{1}{\Gamma_{0}}\left(\sqrt{{\dfrac{A}{B}}}k\pi-C\right).$
(8.73)
Similarly the second relation (8.71) can be solved immediately
$\xi^{I}=\pm\dfrac{1}{\Gamma_{0}}\left(\pm\sqrt{{\dfrac{A}{B}}}\arcsin\sqrt{{\dfrac{1}{A}}}-C\right).$
(8.74)
For the first case one has
$\displaystyle\Psi[\xi^{I}]$ $\displaystyle=$ $\displaystyle 0,$ (8.75)
$\displaystyle\Pi_{\Psi}[\xi^{I}]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{A}}},$ (8.76)
and for the second one hold
$\displaystyle\Psi[\xi^{I}]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{B}}},$ (8.77)
$\displaystyle\Pi_{\Psi}[\xi^{I}]$ $\displaystyle=$ $\displaystyle 0.$ (8.78)
Finally one sees that the invariant global wave functional (8.41) is
$\Psi[\xi,\phi]=\pm\Gamma_{\Psi}[\phi]\Gamma_{0}\sqrt{{\dfrac{1}{A}}}\dfrac{\sin\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle V\rangle(\xi,\phi)}}},$ (8.79)
in the first case (8.70), and
$\Psi[\xi,\phi]=\pm\Gamma_{\Psi}[\phi]\sqrt{{\dfrac{1}{B}}}\cos\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$ (8.80)
for the second one (8.71).
Applying the normalization condition (8.55) to the solutions (10.150) and
(10.157) one receives the equations
$\displaystyle|\Gamma_{\Psi}[\phi]\Gamma_{0}|^{2}$ $\displaystyle=$
$\displaystyle 1,$ (8.81) $\displaystyle\Gamma_{\Psi}[\phi]\Gamma_{0}$
$\displaystyle=$ $\displaystyle 1,$ (8.82)
which can be solved easy and lead to the relation
$\Gamma_{\Psi}[\phi]=\dfrac{1}{\Gamma_{0}}.$ (8.83)
Therefore, one obtains finally
$\displaystyle\Psi_{1}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|A|}}}\dfrac{\sin\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle V\rangle(\xi,\phi)}}},$ (8.84)
$\displaystyle\Psi_{2}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|B|}}}\cos\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (8.85)
where now the constants $A$ and $B$ are equal to
$\displaystyle
A=\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.86) $\displaystyle
B=\dfrac{1}{\Gamma_{0}}\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]\delta\xi^{\prime}\delta\phi^{\prime}.$
(8.87)
In this manner the general solutions of the theory of quantum gravity can be
now constructed straightforwardly by using of the solutions (10.150) and
(10.157), in which the integrals must be putted
$\displaystyle\Omega(\xi)$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi)}\delta\xi^{\prime}=\xi,$ (8.88)
$\displaystyle\langle V\rangle(\xi,\phi)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}V[\xi^{\prime},\phi],$
(8.89)
where the $\xi$-measure is
$\displaystyle\delta\xi=\dfrac{1}{8\pi}\dfrac{\delta h}{\sqrt{6h}}.$ (8.90)
Because, however, the generalized gravitational potential $V[\xi,\phi]$ has
the form of an algebraic sum
$V[\xi,\phi]=-{{}^{(3)}\\!R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho,$ (8.91)
one has very convenient separability
$\langle{V}\rangle(\xi,\phi)=-\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}\leavevmode\nobreak\
{{}^{(3)}\\!R}+2\Lambda+\dfrac{2\kappa\ell_{P}^{2}}{\Omega(\xi)}\int_{\Sigma(\phi)}\delta\phi^{\prime}\int_{\Sigma(\xi)}\delta\xi^{\prime}\rho.$
(8.92)
Therefore, for a concretely given geometry of a three-dimensional space-like
embedded space one should estimate the functionally averaged three-dimensional
Ricci scalar
$\displaystyle\langle{{}^{(3)}\\!R}\rangle=\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}\leavevmode\nobreak\
{{}^{(3)}\\!R},$ (8.93)
and the functionally averaged energy density of Matter fields
$\langle\rho\rangle=\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\phi)}\delta\phi^{\prime}\int_{\Sigma(\xi)}\delta\xi^{\prime}\rho,$
(8.94)
and using these quantities construct the functionally averaged generalized
gravitational potential
$\langle
V\rangle(\xi,\phi)=-\langle{{}^{(3)}\\!R}\rangle+2\Lambda+2\kappa\ell_{P}^{2}\langle\rho\rangle.$
(8.95)
Applying this averaging method one can construct the solutions (10.150) and
(10.157) straightforwardly. If the Ricci scalar curvature ${{}^{(3)}\\!R}$ of
an embedded space and/or the energy density of Matter fields $\varrho$ are
functions of a space-time point $x^{\mu}=(x^{0},x^{1},x^{2},x^{3})$ or any one
space-time coordinate the integration procedure is understood as the procedure
of performing of the Lebesgue–Stieltjes integral. For instance in the case
when the integrand is a function of $x^{\mu}$ then the Lebesgue–Stieltjes
measure has the following form
$\delta\xi=\dfrac{\partial^{4}\xi}{\partial x_{0}\partial x_{1}\partial
x_{2}\partial x_{3}}d^{4}x,$ (8.96)
or in terms of determinant of induced metric $h$
$\delta\xi=\dfrac{1}{8\pi}\dfrac{\partial^{4}h}{\partial x_{0}\partial
x_{1}\partial x_{2}\partial x_{3}}\dfrac{d^{4}x}{\sqrt{6h}}.$ (8.97)
#### D Problem I: Inverted Transformation
There is the problem of inverted transformation
$\xi\rightarrow h_{ij},$ (8.98)
within the obtained solutions. In fact, this problem is strictly related to
the problem of finding of a induced metric if one knows its determinant. Such
a procedure can not be performed analytically, but there are functional
methods which enable performing of this step.
The solutions (10.150) and (10.157) expressed via $h_{ij}$ are
$\displaystyle\Psi_{1}[h_{ij},\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|A|}}}\dfrac{\sin\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right]}{\sqrt{{\langle V\rangle(h_{ij},\phi)}}},$
(8.99) $\displaystyle\Psi_{2}[h_{ij},\phi]$ $\displaystyle=$
$\displaystyle\pm\sqrt{{\dfrac{1}{|B|}}}\cos\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right],$ (8.100)
where now the constants $A$ and $B$ are equal to
$\displaystyle
A=\dfrac{1}{8\pi}\int_{\Sigma(h_{ij},\phi)}\left(\dfrac{\sin\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right)^{2}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},$ (8.101) $\displaystyle
B=\dfrac{1}{8\pi\Gamma_{0}}\int_{\Sigma(h_{ij},\phi)}\cos^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},$ (8.102)
and assumed to be convergent and finite. The solutions (8.99) and (8.100) are
two independent states transformed to the standard quantum geometrodynamics,
but in general they are not solutions of the Wheeler–DeWitt equation.
Interestingly, in such a situation the normalization condition (8.55) becomes
$\dfrac{1}{8\pi}\int_{\Sigma(h_{ij},\phi)}|\Psi[h_{ij}^{\prime},\phi^{\prime}]|^{2}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}=1,$ (8.103)
and can be used to construct the $\pi$ number definition
$\pi=\dfrac{1}{8}\int_{\Sigma(h_{ij},\phi)}|\Psi[h_{ij}^{\prime},\phi^{\prime}]|^{2}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}.$ (8.104)
#### E Problem II: The Hilbert Space and Superposition
Because, however, both the equations of quantum geometrodynamics (6.101) and
(8.10) are linear in the field $\Psi$, in general the superposition
$\Psi=\sum_{i=1,2}\alpha_{i}\Psi_{i},$ (8.105)
where $\alpha_{i}$ are arbitrary constants, and $\Psi_{i}$ are (8.99) and
(8.100), should be also a solution. The normalization (8.103) of (8.105) gives
$|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+(\alpha^{\star}_{1}\alpha_{2}+\alpha_{1}\alpha^{\star}_{2})I=1,$
(8.106)
where $I$ is the integral
$I=\dfrac{1}{\sqrt{|A||B|}}\dfrac{1}{8\pi}\int_{\Sigma(h_{ij},\phi)}\dfrac{\sin\left[2\Omega(h_{ij}^{\prime})\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}}\sqrt{{\dfrac{h^{\prime}}{6}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}.$ (8.107)
For vanishing $I=0$ one obtains form (8.106) simply
$|\alpha_{2}|=\sqrt{{1-|\alpha_{1}|^{2}}}\quad,\quad|\alpha_{1}|\geqslant 1.$
(8.108)
The case of $I\neq 0$ is much more complicated. Note that the equation (8.106)
can be rewritten in form
$(\alpha_{1}+\alpha_{2}I)\alpha^{\star}_{1}+(\alpha_{2}+\alpha_{1}I)\alpha_{2}^{\star}=0,$
(8.109)
what leads to the result
$\dfrac{\alpha^{\star}_{1}}{\alpha_{2}^{\star}}=\dfrac{-\alpha_{1}I+\alpha_{2}}{\alpha_{1}+\alpha_{2}I},$
(8.110)
or equivalently
$\displaystyle C\alpha^{\star}_{1}$ $\displaystyle=$
$\displaystyle-\alpha_{1}I+\alpha_{2},$ (8.111) $\displaystyle
C\alpha_{2}^{\star}$ $\displaystyle=$ $\displaystyle\alpha_{1}+\alpha_{2}I,$
(8.112)
where $0\neq C\in\mathbb{R}$ is a constant. The relations (8.111)-(8.112) lead
to
$\displaystyle C|\alpha_{1}|^{2}$ $\displaystyle=$
$\displaystyle-\alpha^{2}_{1}I+\alpha_{2}\alpha_{1},$ (8.113) $\displaystyle
C|\alpha_{2}|^{2}$ $\displaystyle=$
$\displaystyle\alpha_{1}\alpha_{2}+\alpha_{2}^{2}I.$ (8.114)
Mutual addition and application of (8.106) yields
$CI[(\alpha^{\star}_{1}-\alpha_{2})\alpha_{2}+(\alpha_{2}^{\star}+\alpha_{1})\alpha_{1}]=\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{1},$
(8.115)
which generates the equations
$\displaystyle CI(\alpha^{\star}_{1}-\alpha_{2})$ $\displaystyle=$
$\displaystyle\alpha_{1},$ (8.116) $\displaystyle
CI(\alpha_{2}^{\star}+\alpha_{1})$ $\displaystyle=$ $\displaystyle\alpha_{2}.$
(8.117)
Complex decomposition of (8.116)-(8.117) leads to
$\displaystyle\Re\alpha_{2}$ $\displaystyle=$
$\displaystyle(CI-1)\Re\alpha_{1},$ (8.118) $\displaystyle\Im\alpha_{2}$
$\displaystyle=$ $\displaystyle(CI-1)\Im\alpha_{1},$ (8.119)
or equivalently
$\displaystyle\alpha_{2}$ $\displaystyle=$ $\displaystyle(CI-1)\alpha_{1},$
(8.120) $\displaystyle|\alpha_{2}|^{2}$ $\displaystyle=$
$\displaystyle(CI-1)^{2}|\alpha_{1}|^{2}.$ (8.121)
Employing (8.120)-(8.121) within the constraint (8.106) yields to
$\dfrac{1}{|\alpha_{1}|^{2}}=IC^{2}+(I^{2}-2I)C-I+2.$ (8.122)
Because of the natural condition $\dfrac{1}{|\alpha_{1}|^{2}}>0$ one has the
inequality for $C$
$IC^{2}+I(I-2)C-(I-2)>0,$ (8.123)
where $I\geqslant 0$. One sees that the case $I=0$ gives $2>0$ what is true.
The inequality (8.123) has two different solutions which are dependent on
$\mathrm{sgn}(I-2)$. When $0\leqslant I<2$ then the solution is
$C\in\left(C_{-},C_{+}\right).$ (8.124)
When $I>2$ the solution is somewhat different
$C\in(-\infty,C_{-})\cup(C_{+},\infty).$ (8.125)
Here are the constants
$C_{\mp}=\dfrac{2-I}{2}\mp\dfrac{1}{2}\sqrt{\dfrac{I^{3}-4I^{2}+8I-8}{I}}.$
(8.126)
For consistency the constants $C_{\mp}$ must be real numbers. For this it is
necessary and sufficient to satisfy the condition
$I^{3}-4I^{2}+8I-8\geqslant 0,$ (8.127)
what is satisfied for $I\geqslant 2$ or $I<0$. Because of $I>0$ the first case
is true, and therefore (8.125) is true region of validity of $C$. Applying the
definition of $I$, $A$, and $B$ one receives the following inequality
$\displaystyle\int_{\Sigma(h_{ij},\phi)}\dfrac{\sin\left[2\Omega(h_{ij}^{\prime})\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}}\sqrt{h^{\prime}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}\geqslant$
$\displaystyle\dfrac{2}{\sqrt{|\Gamma_{0}|}}\left(\int_{\Sigma(h_{ij},\phi)}\left(\dfrac{\sin\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right)^{2}\sqrt{h^{\prime}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}\right)^{1/2}\times$
$\displaystyle\left(\int_{\Sigma(h_{ij},\phi)}\cos^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]\sqrt{h^{\prime}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}\right)^{1/2},$ (8.128)
or in terms of the invariant global dimension
$\displaystyle\int_{\Sigma(\xi,\phi)}\dfrac{\sin\left[2\Omega(\xi^{\prime})\sqrt{{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}}\delta\xi^{\prime}\delta\phi^{\prime}\geqslant$
$\displaystyle\dfrac{2}{\sqrt{|\Gamma_{0}|}}\left(\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2}\times$
$\displaystyle\left(\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2}.$
(8.129)
Let us consider the concept of an inner product space, called also pre-Hilbert
space. Such a space is a vector space equipped with an additional structure
called an inner product
$\left<\bullet,\bullet\right>:V\times V\rightarrow\mathbb{F},$ (8.130)
where $V$ is a vector space over the field of scalars $\mathbb{F}$. A pre-
Hilbert space is called a Hilbert space if is complete as a normed space under
the induced norm $||\bullet||$. Every Hilbert space is the Banach space. The
structure of an inner product allows to build intuitive geometrical notions
such as length of a vector or the angle between two vectors, and deduce
orthogonality between vectors by vanishing of an inner product. Pre-Hilbert
spaces generalize Euclidean spaces to vector spaces of any, including
infinite, dimension and are the theme of functional analysis (For more
detailed discussion of functional analysis see e.g. the Ref. [603]). The
Lebesgue space $L^{2}\left(\Sigma,\mu\right)$ where $\Sigma$ is the
configurational space equipped with the Lebesgue measure $\mu$, called also
the space of square-integrable functions, which is the special case of
$L^{p}$-spaces (For more details see e.g. the Ref. [604]), and moreover is a
Hilbert space. This particular Hilbert space is the fundament of quantum
mechanics, in which wave functions belong to $L^{2}$.
Let us construct the Lebesgue space $L^{2}$ for the theory of quantum gravity
_in accordance_ with the superposition principle. In such a situation the
configurational space is the midisuperspace $\Sigma(\xi,\phi)$ with the
measure $\mu=\delta\xi\delta\phi$. The Lebesgue space
$L^{2}\left(\Sigma(\xi,\phi),\delta\xi\delta\phi\right)$ is the Hilbert space
with the inner product
$\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle=\int_{\Sigma(\xi,\phi)}f^{\star}(\xi^{\prime},\phi^{\prime})g(\xi^{\prime},\phi^{\prime})\delta\xi^{\prime}\delta\phi^{\prime},$
(8.131)
satisfying the conditions
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$
$\displaystyle\overline{\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle},$
(8.132)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{af(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$ $\displaystyle
a\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle,$ (8.133)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{f(\xi,\phi)+h(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle+\left\langle{h(\xi,\phi),g(\xi,\phi)}\right\rangle.$
(8.134)
The induced norm
$\left|\left|f(\xi,\phi)\right|\right|=\sqrt{\left\langle{f(\xi,\phi),f(\xi,\phi)}\right\rangle}=\left(\int_{\Sigma(\xi,\phi)}|f(\xi^{\prime},\phi^{\prime})|^{2}\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2}<\infty,$
(8.135)
is homogeneous
$\left|\left|af(\xi,\phi)\right|\right|=|a|\left|\left|f(\xi,\phi)\right|\right|,$
(8.136)
and satisfies the triangle inequality
$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|\leqslant\left|\left|f(\xi,\phi)\right|\right|+\left|\left|g(\xi,\phi)\right|\right|.$
(8.137)
For the orthogonal situation
$\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle=0,$ (8.138)
the Pythagoras theorem holds
$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}=\left|\left|f(\xi,\phi)\right|\right|^{2}+\left|\left|g(\xi,\phi)\right|\right|^{2}.$
(8.139)
The parallelogram law
$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}+\left|\left|f(\xi,\phi)-g(\xi,\phi)\right|\right|^{2}=2\left|\left|f(\xi,\phi)\right|\right|^{2}+2\left|\left|g(\xi,\phi)\right|\right|^{2},$
(8.140)
is a necessary and sufficient condition for the existence of a scalar product
corresponding to a given norm. This scalar product follows from the
polarization identity
$\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}=\left|\left|f(\xi,\phi)\right|\right|^{2}+\left|\left|g(\xi,\phi)\right|\right|^{2}+\Re\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle,$
(8.141)
called also the law of cosinuses, and equals to
$\displaystyle\left(f(\xi,\phi),g(\xi,\phi)\right)$ $\displaystyle=$
$\displaystyle\dfrac{\left|\left|f(\xi,\phi)+g(\xi,\phi)\right|\right|^{2}-\left|\left|f(\xi,\phi)-g(\xi,\phi)\right|\right|^{2}}{2}=$
(8.142) $\displaystyle=$
$\displaystyle\Re\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle.$
When $f_{i}(\xi,\phi)$, $i=1,\ldots,N$, are orthogonal vectors then holds the
relation
$\sum_{i=1}^{N}\left|\left|f_{i}(\xi,\phi)\right|\right|=\left|\left|\sum_{i=1}^{N}f_{i}(\xi,\phi)\right|\right|.$
(8.143)
When $V$ is a complete pre-Hilbert space, i.e. is a Hilbert space, then the
equation (8.143) becomes the Parseval identity
$\sum_{i=1}^{\infty}\left|\left|f_{i}(\xi,\phi)\right|\right|=\left|\left|\sum_{i=1}^{\infty}f_{i}(\xi,\phi)\right|\right|,$
(8.144)
provided that the infinite series on the LHS is convergent. Recall that
completeness of the space is necessary for convergence of the sequence of
partial sums on the space
$s_{k}=\sum_{i=1}^{k}\left|\left|f_{i}(\xi,\phi)\right|\right|,$ (8.145)
which is a Cauchy sequence.
The inner product and the norm can be used for definition of the angle
$\alpha$ between two vectors $f(\xi,\phi)$ and $g(\xi,\phi)$
$\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)=\arccos\dfrac{\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle}{\left|\left|f(\xi,\phi)\right|\right|\left|\left|g(\xi,\phi)\right|\right|},$
(8.146)
and it can be shown by straightforward computation that the
Cauchy–Bunyakovsky–Schwarz inequality holds
$\left|\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle\right|\leqslant\left|\left|f(\xi,\phi)\right|\right|\left|\left|g(\xi,\phi)\right|\right|.$
(8.147)
Taking into account
$\displaystyle f(\xi,\phi)$ $\displaystyle=$
$\displaystyle\dfrac{\sin\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]}{\sqrt{\langle V\rangle(\xi,\phi)}},$ (8.148)
$\displaystyle g(\xi,\phi)$ $\displaystyle=$
$\displaystyle\cos\left[\Omega(\xi)\sqrt{\langle V\rangle(\xi,\phi)}\right],$
(8.149)
one obtains
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\dfrac{\sin\left[2\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{2\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.150)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\left|f(\xi,\phi)\right|\right|^{2}$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime},$
(8.151)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left|\left|g(\xi,\phi)\right|\right|^{2}$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi)\sqrt{\langle
V\rangle(\xi,\phi)}\right]\delta\xi^{\prime}\delta\phi^{\prime}.$ (8.152)
By this reason one can rewrite the integral (8.107) as
$I=\dfrac{\left\langle{f(\xi,\phi),g(\xi,\phi)}\right\rangle}{\left|\left|f(\xi,\phi)\right|\right|\left|\left|g(\xi,\phi)\right|\right|},$
(8.153)
or with using of the definition (8.146)
$I=\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right).$ (8.154)
In this manner the inequality (8.129) can be rewritten as
$\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\geqslant\dfrac{2}{\sqrt{|\Gamma_{0}|}}$
(8.155)
whereas the Cauchy–Bunyakovsky–Schwarz inequality (8.147) says that
$\left|\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\right|\leqslant 1.$
(8.156)
Hence the superposition principle requires
$\dfrac{2}{\sqrt{|\Gamma_{0}|}}\leqslant\left|\cos\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\right|\leqslant
1,$ (8.157)
or equivalently
$\alpha\left(f(\xi,\phi),g(\xi,\phi)\right)\in\left[\arccos\dfrac{2}{\sqrt{|\Gamma_{0}|}}+2k\pi,2\pi+2k\pi\right],\quad
k\in\mathbb{Z},$ (8.158)
what is consistent if and only if
$\sqrt{|\Gamma_{0}|}\geqslant 2.$ (8.159)
Let us consider the wave functions (8.99) and (8.100) expressed via the
invariant global dimension
$\displaystyle\Psi_{1}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]}{\sqrt{{\langle
V\rangle(\xi,\phi)}}}=\dfrac{1}{\sqrt{|A|}}f(\xi,\phi),$ (8.160)
$\displaystyle\Psi_{2}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]=\dfrac{1}{\sqrt{|B|}}g(\xi,\phi),$ (8.161)
with the constants $A$ and $B$ given by
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A$
$\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\left(\dfrac{\sin\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]}{\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}}\right)^{2}\delta\xi^{\prime}\delta\phi^{\prime}\equiv\left|\left|f(\xi,\phi)\right|\right|^{2},$
(8.162)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!B$
$\displaystyle=$
$\displaystyle\dfrac{1}{|\Gamma_{0}|}\int_{\Sigma(\xi,\phi)}\cos^{2}\left[\Omega(\xi^{\prime})\sqrt{\langle
V\rangle(\xi^{\prime},\phi^{\prime})}\right]\delta\xi^{\prime}\delta\phi^{\prime}\equiv\dfrac{1}{|\Gamma_{0}|}\left|\left|g(\xi,\phi)\right|\right|^{2}.$
(8.163)
In other words
$\displaystyle\Psi_{1}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\dfrac{f(\xi,\phi)}{\left|\left|f(\xi,\phi)\right|\right|},$
(8.164) $\displaystyle\Psi_{2}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\dfrac{g(\xi,\phi)}{\left|\left|g(\xi,\phi)\right|\right|},$
(8.165) $\displaystyle I$ $\displaystyle=$
$\displaystyle\cos\alpha\left(\Psi_{1}[\xi,\phi],\Psi_{2}[\xi,\phi]\right).$
(8.166)
The wave functions (8.164) and (8.165) are elements of the Lebesgue space
$L^{2}\left(\Sigma(\xi,\phi),\delta\xi\delta\phi\right)$ with the scalar
product
$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)=\Re\left\langle{\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right\rangle,$
(8.167)
where the inner product is
$\left\langle{\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right\rangle=\int_{\Sigma(\xi,\phi)}\Psi_{1}^{\star}(\xi^{\prime},\phi^{\prime})\Psi_{2}(\xi^{\prime},\phi^{\prime})\delta\xi^{\prime}\delta\phi^{\prime},$
(8.168)
and the induced norm
$\left|\left|\Psi_{i}(\xi,\phi)\right|\right|=\left(\int_{\Sigma(\xi,\phi)}|\Psi_{i}(\xi^{\prime},\phi^{\prime})|^{2}\delta\xi^{\prime}\delta\phi^{\prime}\right)^{1/2},$
(8.169)
where $i=1,2$. It is easy to see that
$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)=\sqrt{|\Gamma_{0}|}I,$
(8.170)
and because of $I\geqslant 2$ the scalar product satisfies the inequality
$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)\geqslant
2\sqrt{|\Gamma_{0}|}.$ (8.171)
In the light of the inequality (8.159) one has
$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)\geqslant 4.$ (8.172)
It can be seen also that
$\displaystyle||\Psi_{1}||$ $\displaystyle=$ $\displaystyle 1,$ (8.173)
$\displaystyle||\Psi_{2}||$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\geqslant 2,$ (8.174)
i.e. the state $\Psi_{1}$ is a ray whereas the state $\Psi_{2}$ is not a ray
in the Hilbert space. Because, however the superposition is given by
$\Psi=\alpha_{1}\Psi_{1}+\alpha_{2}\Psi_{2},$ (8.175)
one can selected the constants $\alpha_{1}$ and $\alpha_{2}$ in such a way
that the wave function $\Psi_{2}$ will be normalized to unity. The choice is
easy to deduce. Because of $\Psi_{1}$ is normalized to unity the coefficient
$\alpha_{1}$ is an arbitrary constant, while the coefficient $\alpha_{2}$
should be exchanged on
$\alpha_{2}^{\prime}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}\alpha_{2},$ (8.176)
where $\alpha_{2}$ is an arbitrary constant. Still, however, in general the
problem of normalization of the state $\Psi_{2}$ is unsolved. Redefinition of
the scalar product
$\left({\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right)=\dfrac{\left\langle{\Psi_{1}(\xi,\phi),\Psi_{2}(\xi,\phi)}\right\rangle}{\left|\left|\Psi_{1}(\xi,\phi)\right|\right|\left|\left|\Psi_{2}(\xi,\phi)\right|\right|}=I\geqslant
2,$ (8.177)
does not work consistently in the light of (8.154). The only solution is to
apply _ad hoc_ the scaling
$\Psi_{2}^{\prime}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}\Psi_{2}\leqslant\dfrac{1}{\sqrt{2}}\Psi_{2},$
(8.178)
which, however, is consistent with the choice (8.176).
Another alternative can be constructed as follows. Let us assume that the
superposition state
$\Psi^{\prime}=\alpha_{1}^{\prime}\Psi_{1}^{\prime}+\alpha_{2}^{\prime}\Psi_{2}^{\prime},$
(8.179)
is such that the states $\Psi_{1}^{\prime}$ and $\Psi_{2}^{\prime}$ are
orthonormal, i.e. are orthogonal rays in the Hilbert space
$\displaystyle||\Psi_{1}^{\prime}||$ $\displaystyle=$ $\displaystyle 1,$
(8.180) $\displaystyle||\Psi_{2}^{\prime}||$ $\displaystyle=$ $\displaystyle
1,$ (8.181) $\displaystyle\left(\Psi_{1}^{\prime},\Psi_{2}^{\prime}\right)$
$\displaystyle=$ $\displaystyle 0.$ (8.182)
The problem is to construct such constants $\alpha_{1}^{\prime}$,
$\alpha_{2}^{\prime}$, and states $\Psi_{1}^{\prime}$ and $\Psi_{2}^{\prime}$
which satisfy these requirements. This problem can be solved with help of the
Gram–Schmidt algorithm, in which the orthonormal states $\Psi_{i}^{\prime}$
(here $i=1,2$) are expressed via the non-orthonormal states $\Psi_{i}$ via
using of the Gram determinants
$\displaystyle G_{0}$ $\displaystyle=$ $\displaystyle 1,$ (8.183)
$\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left|(\Psi_{1},\Psi_{1})\right|,$ (8.184) $\displaystyle G_{2}$
$\displaystyle=$
$\displaystyle\det\left[\begin{array}[]{cc}(\Psi_{1},\Psi_{1})&(\Psi_{1},\Psi_{2})\\\
(\Psi_{1},\Psi_{2})&(\Psi_{2},\Psi_{2})\end{array}\right],$ (8.187)
as follows
$\displaystyle\Psi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{G_{0}G_{1}}}\Psi_{1},$ (8.188)
$\displaystyle\Psi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{G_{1}G_{2}}}\det\left[\begin{array}[]{cc}(\Psi_{1},\Psi_{1})&(\Psi_{1},\Psi_{2})\\\
\Psi_{1}&\Psi_{2}\end{array}\right].$ (8.191)
It can be seen by straightforward computation that
$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle 1,$ (8.192)
$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle|\Gamma_{0}|\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right),$
(8.193)
and consequently
$\displaystyle\Psi_{1}^{\prime}$ $\displaystyle=$ $\displaystyle\Psi_{1},$
(8.194) $\displaystyle\Psi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{-1/2}\left(-I\Psi_{1}+\dfrac{1}{\sqrt{|\Gamma_{0}|}}\Psi_{2}\right).$
(8.195)
One sees now that the second term in (8.195) converges with the proposed
scaled state (8.178). In this manner the superposition state (8.179) can be
rewritten as
$\Psi^{\prime}=\alpha_{1}^{\prime}\Psi_{1}+\alpha_{2}^{\prime}\Psi_{2}^{\prime},$
(8.196)
where now $\Psi_{2}^{\prime}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}\Psi_{2}$ is the
scaled state (8.178) and
$\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\alpha_{1}-\alpha_{2}I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{-1/2},$
(8.197) $\displaystyle\alpha_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\alpha_{2}\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{-1/2}.$
(8.198)
The states $\Psi_{1}$ and $\Psi_{2}$ can be expressed via the orthonormal
states $\Psi_{1}^{\prime}$ and $\Psi_{2}^{\prime}$
$\displaystyle\Psi_{1}$ $\displaystyle=$ $\displaystyle\Psi_{1}^{\prime},$
(8.199) $\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\left(\Psi_{2}^{\prime}+I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}\Psi_{1}^{\prime}\right),$
(8.200)
and by this reason
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\Psi_{1},\Psi_{2})$
$\displaystyle=$
$\displaystyle\left(\Psi_{1}^{\prime},\sqrt{|\Gamma_{0}|}\left(\Psi_{2}^{\prime}+I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}\Psi_{1}^{\prime}\right)\right)=$
(8.201) $\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}\left[(\Psi_{1}^{\prime},\Psi_{2}^{\prime})+I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}(\Psi_{1}^{\prime},\Psi_{1}^{\prime})\right]=$
$\displaystyle=$
$\displaystyle\sqrt{|\Gamma_{0}|}I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}.$
In the light of the relation (8.170) one has
$\sqrt{|\Gamma_{0}|}I\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}=\sqrt{|\Gamma_{0}|}I,$
(8.202)
which, because of $I\geqslant 2$, gives uniquely and unambiguously
$\left(\dfrac{1}{\sqrt{|\Gamma_{0}|}}-I^{2}\right)^{1/2}=1,$ (8.203)
or equivalently
$I^{2}=\dfrac{1}{\sqrt{|\Gamma_{0}|}}-1.$ (8.204)
In the light of the fact $I\geqslant 2$ one has
$\sqrt{|\Gamma_{0}|}\leqslant\dfrac{1}{5},$ (8.205)
what however is inconsistent in the light of the fact (8.159). In this manner
the problem of superposition in the theory of quantum gravity can not be
solved by application of the Gram–Schmidt algorithm of orthonormalization to
the Hilbert space constructed in this subsection.
#### F Problem III: The Problem of Time
It is easy to see that within the invariant global one-dimensional quantum
gravity the problem of time is naturally solved. Namely the role of physical
time plays the invariant global dimension
$\xi=\dfrac{1}{4\pi}\sqrt{\dfrac{h}{6}}\equiv\dfrac{t_{\textrm{physical}}}{\tau},$
(8.206)
where $\tau$ is a reference constant, which can be taken _ad hoc_ as the
Planck time, i.e. $\tau=t_{P}$. Such an identification is justified by the
fact that $\xi$ is invariant with respect to action of the spatial
diffeomorphisms group, because of the volume form $\sqrt{h}$ is such a
diffeoinvariant. In this manner, in the sense of Kuchař $\xi$ is an
observable.
Such a situation is analogous to the quantum cosmology presented in the
chapter 5, in which the role of time plays the cosmic scale factor parameter
$a$. This parameter is a function of the conformal or cosmological time. By
this reason the proposed model of quantum gravity can be rewritten in more
fashionable form
$\left(\dfrac{d^{2}}{dt^{2}}+V[t]\right)\Psi(t,\phi)=0.$ (8.207)
This global one-dimensional evolutionary equation defines in itself non-
trivial dynamics. Namely, this is the time-dependent $0+1$ Schrödinger wave
equation.
### Chapter 9 Examples of Invariant 1D Wave Functions
Let us present with no detailed computations the wave functionals associated
to the three classical solutions of the Einstein field equations.
#### A The Minkowski Space-time
Let us consider first empty space with no cosmological constant, i.e. the
Minkowski space-time. In such a case, in the Cartesian coordinates, the
spatial metric coincides with the metric of the Euclidean space
$h_{ij}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right],$
(9.1)
and therefore $h=1$ so that the invariant global dimension is
$\xi=\dfrac{1}{4\pi}\dfrac{1}{\sqrt{6}}.$ (9.2)
The Minkowski space-time is characterized by
${{}^{(3)}}R=0\quad,\quad\varrho=0\quad,\quad\Lambda=0,$ (9.3)
and by this reason the averaged generalized gravitational potential is
$\langle{V}\rangle(\xi,\phi)=0.$ (9.4)
In this manner the wave functions for this case are
$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\xi,$ (9.5) $\displaystyle\Psi_{2}$
$\displaystyle=$ $\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.6)
where the integration constants $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{1}{18(4\pi)^{3}\sqrt{6}},$ (9.7) $\displaystyle B$
$\displaystyle=$ $\displaystyle\dfrac{1}{4\Gamma_{0}\pi\sqrt{6}}.$ (9.8)
In other words the wave functions
$\displaystyle\Psi_{1}$ $\displaystyle=$ $\displaystyle\sqrt{12\pi\sqrt{6}},$
(9.9) $\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\sqrt{4|\Gamma_{0}|\pi\sqrt{6}},$ (9.10)
are constant on the midisuperspace. This case is singular because of constant
wave functions can not be normalized to unity separately. However, taking the
superposed state
$\Psi=\alpha_{1}\Psi_{1}+\alpha_{2}\Psi_{2},$ (9.11)
one obtains the normalization condition
$3|\alpha_{1}|^{2}+|\Gamma_{0}||\alpha_{2}|^{2}+\left(\alpha_{1}^{\star}\alpha_{2}+\alpha_{1}\alpha_{2}^{\star}\right)\sqrt{3|\Gamma_{0}|}=1,$
(9.12)
having the following solutions
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\dfrac{\beta_{1}}{\sqrt{3}}\exp(i\alpha),$ (9.13)
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\dfrac{\beta_{2}}{\sqrt{|\Gamma_{0}|}}\exp(i\alpha),$ (9.14)
where $\alpha$ is an arbitrary real phase, and the constants $\beta_{1}$ and
$\beta_{2}$ are constrained by
$\beta_{2}=\pm 1-\beta_{1}.$ (9.15)
In other words the superposed states
$\Psi_{\pm}^{M}=\exp(i\alpha)\left(\dfrac{\beta}{\sqrt{3}}\Psi_{1}+\dfrac{\pm
1-\beta}{\sqrt{|\Gamma_{0}|}}\Psi_{2}\right)=\pm\exp(i\alpha)\sqrt{4\pi\sqrt{6}},$
(9.16)
are consistent wave functions of the Minkowski space-time. Because of the
Minkowski vacuum is nonlinearly stable, the wave functions (9.16) are the
reference states for another $\pi$ number definition
$\pi:=\dfrac{\left|\Psi_{\pm}^{M}\right|^{2}}{4\sqrt{6}}.$ (9.17)
#### B The Kasner Space-time
Let us consider the simple solution of the Einstein field equations describing
an anisotropic universe without matter, called the Kasner metric. The spatial
part of this metric is
$h_{ij}=\left[\begin{array}[]{ccc}t^{2p_{1}}&0&0\\\ 0&t^{2p_{2}}&0\\\
0&0&t^{2p_{3}}\end{array}\right],$ (9.18)
where $t$ is time coordinate, and
$\displaystyle\sum_{i}p_{i}$ $\displaystyle=$ $\displaystyle 1,$ (9.19)
$\displaystyle\sum_{i}p_{i}^{2}$ $\displaystyle=$ $\displaystyle 1.$ (9.20)
In this case the global dimension is
$h=t^{2},$ (9.21)
while the invariant global dimension is
$\xi=\dfrac{t}{4\pi\sqrt{6}}.$ (9.22)
Because of the cosmological constant and the Matter fields are absent here,
and the three-dimensional Ricci scalar curvature vanishes one has
$\langle{V}\rangle=0$. In this manner the wave functions of the Kasner space-
time are
$\displaystyle\Psi_{1}^{K}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{t}{4\pi\sqrt{6}},$ (9.23)
$\displaystyle\Psi_{2}^{K}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.24)
where the constants of integration are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{1}{3}\dfrac{T^{3}}{6(4\pi)^{3}\sqrt{6}},$ (9.25)
$\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{T}{4\pi\sqrt{6}},$ (9.26)
where $T$ is some reference value of time $t$.
#### C The Schwarzschild Space-time
The second situation which we shall present here is the Schwarzschild space-
time. This metric is a spherically symmetric vacuum solution of the Einstein
field equations with no cosmological constant, i.e. a situation
${{}^{(3)}\\!R[h]}=0$, $\Lambda=0$, $\varrho=0$. It means that in this case
also $\langle V\rangle(h_{ij},\phi)=0$. The spatial part of the metric has the
form
$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{r_{S}}{r}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right],$ (9.27)
where $r_{S}$ is the Schwarzschild radius of the spherically symmetric non-
rotating object of mass $M$
$r_{S}=\dfrac{2GM}{c^{2}}=\dfrac{\kappa}{4\pi}Mc^{2}.$ (9.28)
Therefore in this case the global dimension is
$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{r_{S}}{r}},$ (9.29)
so that the invariant global dimension is
$\xi=\dfrac{1}{4\pi}\dfrac{r^{2}\sin\theta}{\sqrt{6\left(1-\dfrac{r_{S}}{r}\right)}},$
(9.30)
and the total volume of the Schwarzschild midisuperspace is $\Omega(\xi)=\xi$.
The integration constants $A$ and $B$ can be established easy
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\Omega^{2}(\xi)\delta\xi=\dfrac{\Xi^{3}}{3}=\dfrac{1}{(4\pi)^{3}}\dfrac{R^{6}\sin^{3}\Theta}{\left(6\left(1-\dfrac{r_{S}}{R}\right)\right)^{3/2}},$
(9.31) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\delta\xi=\dfrac{\Xi}{\Gamma_{0}}=\dfrac{1}{4\Gamma_{0}\pi}\dfrac{R^{2}\sin\Theta}{\sqrt{6\left(1-\dfrac{r_{S}}{R}\right)}},$
(9.32)
where $\Xi=\xi(R,\Theta)$ is
$\Xi=\dfrac{1}{4\pi}\dfrac{R^{2}\sin\Theta}{\sqrt{6\left(1-\dfrac{r_{S}}{R}\right)}}.$
(9.33)
In this manner the wave functions of the Schwarzschild space-time are
$\displaystyle\Psi_{1}^{S}$ $\displaystyle=$
$\displaystyle\sqrt{4\pi\sqrt{6}}\sqrt{\dfrac{1}{R^{2}\sin\Theta}\sqrt{1-\dfrac{r_{S}}{R}}}\sqrt{\dfrac{1-\dfrac{r_{S}}{R}}{1-\dfrac{r_{S}}{r}}}\dfrac{r^{2}\sin\theta}{R^{2}\sin\Theta},$
(9.34) $\displaystyle\Psi_{2}^{S}$ $\displaystyle=$
$\displaystyle\sqrt{4\pi|\Gamma_{0}|\sqrt{6}}\sqrt{\dfrac{1}{R^{2}\sin\Theta}\sqrt{1-\dfrac{r_{S}}{R}}}.$
(9.35)
#### D The (Anti-) De Sitter Space-time
Let us consider now the spherically symmetric solution of vacuum Einstein’s
field equations in presence of the cosmological constant, i.e. lambdavacuum
Einstein’s field equations
$R_{\mu\nu}=\Lambda g_{\mu\nu},$ (9.36)
called the (Anti-) De Sitter space-time. Such a space-time is a submanifold of
the Minkowski space-time of one higher dimension described by the hyperboloid
of one sheet
$-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=\pm\alpha^{2},$ (9.37)
where the sign plus describes De Sitter space-time, whereas the sign minus is
related to the Anti-De Sitter space. The (Anti-) De Sitter space-time is an
Einstein manifold
$R_{\mu\nu}=\pm\dfrac{3}{\alpha^{2}}g_{\mu\nu},$ (9.38)
what in the light of the lambdavacuum Einstein field equations (9.36) the
cosmological constant is
$\Lambda=\pm\dfrac{3}{\alpha^{2}}.$ (9.39)
The three-dimensional Ricci scalar curvature of the (Anti-) De Sitter space-
time can be established easy
${{}^{(3)}}R=h^{ij}R_{ij}=3\Lambda,$ (9.40)
and because of there is no Matter fields $\varrho=0$, so that the averaged
generalized gravitational potential has the value
$\langle{V}\rangle(\xi,\phi)=-3\Lambda+2\Lambda=-\Lambda.$ (9.41)
In other words for the De Sitter space-time
$\langle{V}\rangle(\xi,\phi)=-\dfrac{3}{\alpha^{2}},$ (9.42)
while for the Anti-De Sitter space-time
$\langle{V}\rangle(\xi,\phi)=\dfrac{3}{\alpha^{2}}.$ (9.43)
The spatial part of the (Anti-) De Sitter metric is
$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{\Lambda}{3}r^{2}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right],$ (9.44)
and by this reason the global dimension is
$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{\Lambda}{3}r^{2}},$ (9.45)
whereas the invariant global dimension is
$\xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{\Lambda}{3}r^{2}}},$
(9.46)
and of course the volume of the related midisuperspace is $\Omega(\xi)=\xi$.
Let us consider first the case of the De Sitter space-time, i.e.
$\Lambda=\dfrac{3}{\alpha^{2}}$. The constants of integration can be evaluated
as follows
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sinh\left(\sqrt{\Lambda}\xi\right)}{\sqrt{\Lambda}}\right)^{2}\delta\xi=\dfrac{1}{\Lambda}\left(-\dfrac{\Xi}{2}+\dfrac{\sinh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.47) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cosh^{2}\left(\sqrt{\Lambda}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cosh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.48)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{\Lambda}{3}R^{2}}}.$
(9.49)
By this reason the wave functions of the De Sitter space-time are
$\displaystyle\Psi_{1}^{DS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{\Lambda}}\sinh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{\Lambda}{3}r^{2}}}\right),$
(9.50) $\displaystyle\Psi_{2}^{DS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{\Lambda}{3}r^{2}}}\right).$
(9.51)
Similarly for the case of Anti-De Sitter space-time one obtains the following
wave functions
$\displaystyle\Psi_{1}^{ADS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{|\Lambda|}}\sin\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.52) $\displaystyle\Psi_{2}^{ADS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.53)
where $|\Lambda|=\dfrac{3}{\alpha^{2}}$, and the constants of integration $A$
and $B$ are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sin\left(\sqrt{|\Lambda|}\xi\right)}{\sqrt{|\Lambda|}}\right)^{2}\delta\xi=\dfrac{1}{|\Lambda|}\left(\dfrac{\Xi}{2}-\dfrac{\sin\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.54) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cos^{2}\left(\sqrt{|\Lambda|}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cos\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.55)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1+\dfrac{|\Lambda|}{3}R^{2}}}.$
(9.56)
#### E The (Anti-) De Sitter–Schwarzschild Space-time
The (Anti-) De Sitter–Schwarzschild space-time is the case jointing the
(Anti-) De Sitter and Schwarzschild space-times. This is the spherically
symmetric solution of lambdavacuum Einstein’s field equations with suitable
boundary conditions. The spatial part of space-time metric has the form
$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right].$ (9.57)
In this manner the global dimension is
$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}},$
(9.58)
while the invariant global dimension is
$\xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}}},$
(9.59)
and of course the volume of the related midisuperspace is $\Omega(\xi)=\xi$.
The wave functions of the De Sitter–Schwarzschild space-time, called also the
Kottler space-time, can be written in the form
$\displaystyle\Psi_{1}^{DS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{\Lambda}}\sinh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}}}\right),$
(9.60) $\displaystyle\Psi_{2}^{DS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\dfrac{\sqrt{\Lambda}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}-\dfrac{\Lambda}{3}r^{2}}}\right).$
(9.61)
where the constants of integration $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sinh\left(\sqrt{\Lambda}\xi\right)}{\sqrt{\Lambda}}\right)^{2}\delta\xi=\dfrac{1}{\Lambda}\left(-\dfrac{\Xi}{2}+\dfrac{\sinh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.62) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cosh^{2}\left(\sqrt{\Lambda}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cosh\left(2\sqrt{\Lambda}\Xi\right)}{4\sqrt{\Lambda}}\right),$
(9.63)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{r_{S}}{R}-\dfrac{\Lambda}{3}R^{2}}}.$
(9.64)
Similarly, the wave functions of the Anti-De Sitter–Schwarzschild space-time
are
$\displaystyle\Psi_{1}^{ADS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{\sqrt{|\Lambda|}}\sin\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.65) $\displaystyle\Psi_{2}^{ADS-S}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\dfrac{\sqrt{|\Lambda|}}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{|\Lambda|}{3}r^{2}}}\right),$
(9.66)
where $|\Lambda|=\dfrac{3}{\alpha^{2}}$, and the constants of integration $A$
and $B$ are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left(\dfrac{\sin\left(\sqrt{|\Lambda|}\xi\right)}{\sqrt{|\Lambda|}}\right)^{2}\delta\xi=\dfrac{1}{|\Lambda|}\left(\dfrac{\Xi}{2}-\dfrac{\sin\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.67) $\displaystyle B$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\int\cos^{2}\left(\sqrt{|\Lambda|}\xi\right)\delta\xi=\dfrac{1}{\Gamma_{0}}\left(\dfrac{\Xi}{2}+\dfrac{\cos\left(2\sqrt{|\Lambda|}\Xi\right)}{4\sqrt{|\Lambda|}}\right),$
(9.68)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{r_{S}}{R}+\dfrac{|\Lambda|}{3}R^{2}}}.$
(9.69)
#### F The Kerr Space-Time
Kerr’s space-time is a solution of vacuum Einstein’s field equations for
spherical body rotating with the angular momentum $J$. Because of the spatial
Ricci scalar curvature is zero, cosmological constant is not present, and
Matter fields are absent, in such a situation the averaged generalized
gravitational potential identically vanishes. The spatial part of the Kerr
metric has the form
$h_{ij}=\left[\begin{array}[]{ccc}\dfrac{r^{2}+\alpha^{2}\cos^{2}\theta}{r^{2}-r_{s}r+\alpha^{2}}&0&0\\\
0&r^{2}+\alpha^{2}\cos^{2}\theta&0\\\
0&0&r^{2}+\alpha^{2}+\dfrac{\alpha^{2}r_{S}r\sin^{2}\theta}{r^{2}+\alpha^{2}\cos^{2}\theta}\end{array}\right],$
(9.70)
where $\alpha$ is the coefficient related to the angular momentum and the mass
of the rotating object
$\alpha=\dfrac{J}{Mc}.$ (9.71)
In this manner the global dimension is
$h=\dfrac{\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)^{2}}{r^{2}-r_{s}r+\alpha^{2}}\left(r^{2}+\alpha^{2}+\dfrac{\alpha^{2}r_{S}r\sin^{2}\theta}{r^{2}+\alpha^{2}\cos^{2}\theta}\right),$
(9.72)
while the invariant global dimension is
$\xi=\dfrac{1}{4\pi\sqrt{6}}\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\sqrt{\dfrac{\left(r^{2}+\alpha^{2}\right)^{2}-\alpha^{2}\left(r^{2}-r_{S}r+\alpha^{2}\right)\sin^{2}\theta}{\left(r^{2}-r_{s}r+\alpha^{2}\right)\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)}},$
(9.73)
and of course the volume of the Kerr midisuperspace is $\Omega(\xi)=\xi$.
Introducing the reference parameter
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\left(R^{2}+\alpha^{2}\cos^{2}\Theta\right)\sqrt{\dfrac{\left(R^{2}+\alpha^{2}\right)^{2}-\alpha^{2}\left(R^{2}-r_{S}R+\alpha^{2}\right)\sin^{2}\Theta}{\left(R^{2}-r_{s}R+\alpha^{2}\right)\left(R^{2}+\alpha^{2}\cos^{2}\Theta\right)}},$
(9.74)
one can write out the constants of integration $A$ and $B$
$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.75)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.76)
so that the wave functions of the Kerr space-time have the form
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Psi_{1}^{Kerr}$
$\displaystyle=$
$\displaystyle\dfrac{r^{2}+\alpha^{2}\cos^{2}\theta}{4\pi\sqrt{|A|}\sqrt{6}}\sqrt{\dfrac{\left(r^{2}+\alpha^{2}\right)^{2}-\alpha^{2}\left(r^{2}-r_{S}r+\alpha^{2}\right)\sin^{2}\theta}{\left(r^{2}-r_{s}r+\alpha^{2}\right)\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)}},$
(9.77)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Psi_{2}^{Kerr}$
$\displaystyle=$ $\displaystyle\dfrac{1}{\sqrt{|B|}}.$ (9.78)
#### G The Kerr–Newman Space-time
The Kerr–Newman space-time is a solution of the electrovacuum Einstein field
equations called also Einstein–Maxwell equations, which are combination of the
Einstein field equations with the stress-energy tensor of electromagnetic
field and no cosmological constant, and the source-free Maxwell equations
$\displaystyle R_{\mu\nu}=\kappa\ell_{P}^{2}{T}_{\mu\nu}^{em},$ (9.79)
$\displaystyle F_{\mu\nu;\kappa}+F_{\nu\kappa;\mu}+F_{\kappa\mu;\nu}=0,$
(9.80) $\displaystyle F^{\mu\nu}_{;\nu}=0,$ (9.81)
where $T_{\mu\nu}^{em}$ is the stress-energy tensor of electromagnetic field,
which can be given _ad hoc_ in analogy with the case of flat space-time
$T_{\mu\nu}^{em}=\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{\mu}{F}_{\alpha\nu}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.82)
and $F_{\mu\nu}$ is the electromagnetic field tensor
$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&\dfrac{E_{x}}{c}&\dfrac{E_{y}}{c}&\dfrac{E_{z}}{c}\\\
-\dfrac{E_{x}}{c}&0&-B_{z}&-B_{y}\\\ -\dfrac{E_{y}}{c}&B_{z}&0&-B_{x}\\\
-\dfrac{E_{z}}{c}&-B_{y}&B_{x}&0\end{array}\right],$ (9.87) $\displaystyle
F^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&-\dfrac{E_{x}}{c}&-\dfrac{E_{y}}{c}&-\dfrac{E_{z}}{c}\\\
\dfrac{E_{x}}{c}&0&-B_{z}&-B_{y}\\\ \dfrac{E_{y}}{c}&B_{z}&0&-B_{x}\\\
\dfrac{E_{z}}{c}&-B_{y}&B_{x}&0\end{array}\right],$ (9.92)
where $\vec{E}=[E_{x},E_{y},E_{z}]$ and $\vec{B}=[B_{x},B_{y},B_{z}]$ are
three-vectors of the electric and the magnetic field, respectively. The speed
of light $c$ can be expressed in terms of the vacuum permeability $\mu_{0}$
and the vacuum permittivity $\epsilon_{0}$ via the formula
$\epsilon_{0}\mu_{0}=1/c^{2}$.
There is the question how to derive the stress-energy tensor of
electromagnetic field (9.82) via using of the definition
$T_{\mu\nu}=-\dfrac{2}{\sqrt{-g}}\dfrac{\delta(\sqrt{-g}\mathcal{L})}{\delta
g^{\mu\nu}},$ (9.93)
following from the Hilbert–Palatini action principle, and describing Matter
fields characterized by Lagrangian $\mathcal{L}$ in a four-dimensional
Riemannian space-time with a metric $g^{\mu\nu}$. The Lagrangian of the
Maxwell electromagnetic field has the form (For more general approach see e.g.
the Ref. [605])
$\mathcal{L}=-\dfrac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}.$ (9.94)
With using of the Jacobi formula
$\delta\sqrt{-g}=-\dfrac{1}{2}\sqrt{-g}g_{\mu\nu}\delta{g}^{\mu\nu}$ the
stress-energy tensor (9.93) can be rewritten in more convenient form
$T_{\mu\nu}=-2\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L},$ (9.95)
so that the problem is the establish the functional derivative
$\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}=-\dfrac{1}{4\mu_{0}}\dfrac{\delta}{\delta
g^{\mu\nu}}\left(F_{\alpha\beta}F^{\alpha\beta}\right)=-\dfrac{1}{4\mu_{0}}\left(\dfrac{\delta{F}_{\alpha\beta}}{\delta
g^{\mu\nu}}F^{\alpha\beta}+F_{\alpha\beta}\dfrac{\delta{F}^{\alpha\beta}}{\delta
g^{\mu\nu}}\right).$ (9.96)
Let us notice that the tensor $F_{\alpha\beta}$ can be presented in more
complex form
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!F_{\alpha\beta}$
$\displaystyle=$ $\displaystyle
g^{\gamma}_{\alpha}{g}^{\delta}_{\beta}{F}_{\gamma\delta}=\left(g^{\gamma}_{\mu}{g}^{\mu\nu}g_{\nu\alpha}\right)\left(g^{\delta}_{\nu}{g}^{\nu\mu}g_{\mu\beta}\right){F}_{\gamma\delta}=$
(9.97) $\displaystyle=$
$\displaystyle\left(g^{\gamma}_{\mu}{g}^{\mu\nu}g_{\nu\alpha}\right)\left(g^{\delta}_{\nu}{g}^{\mu\nu}g_{\mu\beta}\right){F}_{\gamma\delta}={g}^{\mu\nu}{g}^{\mu\nu}\left(g^{\gamma}_{\mu}{g}_{\nu\alpha}g^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\gamma\delta}.$
In this manner one has
$\displaystyle\dfrac{\delta{F}_{\alpha\beta}}{\delta{g}^{\mu\nu}}$
$\displaystyle=$ $\displaystyle
2{g}^{\mu\nu}\left(g^{\gamma}_{\mu}{g}_{\nu\alpha}g^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\gamma\delta}=2g^{\gamma}_{\mu}{g}^{\delta}_{\nu}\left({g}^{\mu\nu}g_{\nu\alpha}\right){g}_{\mu\beta}{F}_{\gamma\delta}=$
(9.98) $\displaystyle=$ $\displaystyle
2\left(g^{\gamma}_{\mu}{g}^{\delta}_{\nu}{g}^{\mu}_{\alpha}{g}_{\mu\beta}\right){F}_{\gamma\delta}=2\left(g^{\gamma}_{\mu}{g}^{\mu}_{\alpha}\right){g}^{\delta}_{\nu}{g}_{\mu\beta}{F}_{\gamma\delta}=$
$\displaystyle=$ $\displaystyle
2\left(g^{\gamma}_{\alpha}{g}^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\gamma\delta}=2\left({g}^{\delta}_{\nu}{g}_{\mu\beta}\right)g^{\gamma}_{\alpha}{F}_{\gamma\delta}=2\left({g}^{\delta}_{\nu}{g}_{\mu\beta}\right){F}_{\alpha\delta}=$
$\displaystyle=$
$\displaystyle-2{g}_{\mu\beta}\left({g}^{\delta}_{\nu}{F}_{\delta\alpha}\right)=-2{g}_{\mu\beta}F_{\nu\alpha},$
and consequently
$\dfrac{\delta{F}_{\alpha\beta}}{\delta
g^{\mu\nu}}F^{\alpha\beta}=-2{g}_{\mu\beta}F_{\nu\alpha}F^{\alpha\beta}=2F_{\nu\alpha}\left({g}_{\mu\beta}F^{\beta\alpha}\right)=2F_{\nu\alpha}F_{\mu}^{\alpha}=2F_{\mu}^{\alpha}{F}_{\alpha\nu}.$
(9.99)
Similarly, one can rewrite ${F}^{\alpha\beta}$ in the form
$F^{\alpha\beta}=g^{\alpha\kappa}g^{\beta\lambda}F_{\kappa\lambda},$ (9.100)
and consequently one has
$\displaystyle\dfrac{\delta{F}^{\alpha\beta}}{\delta{g}^{\mu\nu}}$
$\displaystyle=$
$\displaystyle\dfrac{\delta{g}^{\alpha\kappa}}{\delta{g}^{\mu\nu}}g^{\beta\lambda}F_{\kappa\lambda}+{g}^{\alpha\kappa}\dfrac{\delta{g}^{\beta\lambda}}{\delta{g}^{\mu\nu}}F_{\kappa\lambda}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
(9.101) $\displaystyle=$ $\displaystyle
g^{\alpha}_{\mu}{g}^{\kappa}_{\nu}{g}^{\beta\lambda}F_{\kappa\lambda}+{g}^{\alpha\kappa}g^{\beta}_{\mu}{g}^{\lambda}_{\nu}{F}_{\kappa\lambda}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle-g^{\alpha}_{\mu}{g}^{\kappa}_{\nu}{g}^{\beta\lambda}{F}_{\lambda\kappa}-g^{\alpha\kappa}g^{\beta}_{\mu}{g}^{\lambda}_{\nu}{F}_{\lambda\kappa}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle-g^{\alpha}_{\mu}{g}^{\kappa}_{\nu}{F}^{\beta}_{\kappa}-g^{\alpha\kappa}g^{\beta}_{\mu}{F}_{\nu\kappa}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle-g^{\alpha}_{\mu}{F}^{\beta}_{\nu}-g^{\beta}_{\mu}{g}^{\alpha\kappa}{F}_{\nu\kappa}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle
g^{\alpha}_{\mu}{F}^{\beta}_{\nu}+g^{\beta}_{\mu}{g}^{\alpha\kappa}{F}_{\kappa\nu}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle
g^{\alpha}_{\mu}{F}^{\beta}_{\nu}+g^{\beta}_{\mu}{F}^{\alpha}_{\nu}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}.$
By this reason one obtains
$\displaystyle
F_{\alpha\beta}\dfrac{\delta{F}^{\alpha\beta}}{\delta{g}^{\mu\nu}}$
$\displaystyle=$ $\displaystyle
F_{\alpha\beta}\left[g^{\alpha}_{\mu}{F}^{\beta}_{\nu}+g^{\beta}_{\mu}{F}^{\alpha}_{\nu}+{g}^{\alpha\kappa}{g}^{\beta\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}\right]=$
(9.102) $\displaystyle=$
$\displaystyle\left(F_{\alpha\beta}g^{\alpha}_{\mu}\right){F}^{\beta}_{\nu}+\left(F_{\alpha\beta}g^{\beta}_{\mu}\right){F}^{\alpha}_{\nu}+\left(F_{\alpha\beta}{g}^{\alpha\kappa}{g}^{\beta\lambda}\right)\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle\left(-F_{\beta\alpha}g^{\alpha}_{\mu}\right){F}^{\beta}_{\nu}-F_{\alpha\mu}{F}^{\alpha}_{\nu}-\left(F_{\beta\alpha}{g}^{\alpha\kappa}{g}^{\beta\lambda}\right)\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle-
F_{\beta\mu}{F}^{\beta}_{\nu}+{F}^{\alpha}_{\nu}{F}_{\alpha\mu}-\left(F_{\beta}^{\kappa}{g}^{\beta\lambda}\right)\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$
$\displaystyle{F}^{\beta}_{\nu}{F}_{\beta\mu}+{F}^{\alpha}_{\nu}{F}_{\alpha\mu}-F^{\kappa\lambda}\dfrac{\delta{F}_{\kappa\lambda}}{\delta{g}^{\mu\nu}}=$
$\displaystyle=$ $\displaystyle
2{F}^{\alpha}_{\nu}{F}_{\alpha\mu}-2F_{\mu}^{\alpha}{F}_{\alpha\nu}=0,$
where we have applied the identity
${F}^{\alpha}_{\nu}{F}_{\alpha\mu}=g^{\alpha\beta}{F}_{\beta\nu}{F}_{\alpha\mu}=g^{\beta\alpha}{F}_{\alpha\mu}{F}_{\beta\nu}=F_{\mu}^{\beta}{F}_{\beta\nu}=F_{\mu}^{\alpha}{F}_{\alpha\nu}.$
(9.103)
Taking into account (9.99) and (9.102) one receives
$\dfrac{\delta{F}_{\alpha\beta}}{\delta{g}^{\mu\nu}}F^{\alpha\beta}+F_{\alpha\beta}\dfrac{\delta{F}^{\alpha\beta}}{\delta{g}^{\mu\nu}}=-2{F}^{\alpha}_{\mu}{F}_{\alpha\nu}.$
(9.104)
By this reason one obtains
$\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}=-\dfrac{1}{4\mu_{0}}\left(-2{F}^{\alpha}_{\mu}{F}_{\alpha\nu}\right)=\dfrac{1}{2\mu_{0}}{F}^{\alpha}_{\mu}{F}_{\alpha\nu}=-\dfrac{1}{2\mu_{0}}{F}^{\alpha}_{\mu}{F}_{\alpha\nu},$
(9.105)
where in the last step we have changed order of indexes in
${F}^{\alpha}_{\mu}$, which is invisible by the only inconvenient notation. In
this manner one receives finally
$T_{\mu\nu}^{em}=-2\dfrac{\delta\mathcal{L}}{\delta
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}=\dfrac{1}{\mu_{0}}\left({F}^{\alpha}_{\mu}{F}_{\alpha\nu}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.106)
what coincides with the stress-energy tensor of electromagnetic field (9.82)
given _ad hoc_ in analogy with the case of flat space-time.
For the case of the Kerr–Newman metric in general the spatial Ricci scalar
curvature and energy density of Matter fields are manifestly non zero. Let us
calculate them straightforwardly. The spatial Ricci curvature tensor can be
taken from (9.79) as $R_{ij}=\kappa{T}^{em}_{ij}$, where
$T_{ij}^{em}=\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{1}{4}h_{ij}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.107)
is the spatial part of the stress-energy tensor for electromagnetic field.
When the space-time metric is the Minkowski metric $\eta_{\mu\nu}$ then the
spatial part of the stress-energy tensor of electromagnetic field is
$\left.T_{ij}^{em}\right|_{g_{\mu\nu}=\eta_{\mu\nu}}=-\sigma^{f\/lat}_{ij},$
(9.108)
where $\sigma_{ij}^{f\/lat}$ is the Maxwell stress tensor of electromagnetic
field
$\sigma_{ij}^{f\/lat}=\epsilon_{0}E_{i}E_{j}+\dfrac{1}{\mu_{0}}B_{i}B_{j}-\dfrac{1}{2}\left(\epsilon_{0}\vec{E}^{2}+\dfrac{1}{\mu_{0}}\vec{B}^{2}\right)\delta_{ij}.$
(9.109)
In this manner the natural generalization of the Maxwell stress tensor of
electromagnetic field to the case of non-flat space-time is the tensor
$\sigma_{ij}=-\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{1}{4}h_{ij}F_{\alpha\beta}F^{\alpha\beta}\right),$
(9.110)
which we shall call the the curved-space Maxwell stress tensor of
electromagnetic field. Therefore the three-dimensional Ricci scalar curvature
${{}^{(3)}}R=h^{ij}R_{ij}$ up to the minus sign becomes the curved-space
Maxwell stress tensor of electromagnetic field projected onto the induced
metric
$\displaystyle{{}^{(3)}}R$ $\displaystyle=$
$\displaystyle-\kappa\ell_{P}^{2}{h}^{ij}\sigma_{ij}=\kappa\ell_{P}^{2}{h}^{ij}\dfrac{1}{\mu_{0}}\left(F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{1}{4}h_{ij}F_{\alpha\beta}F^{\alpha\beta}\right)=$
(9.111) $\displaystyle=$
$\displaystyle\dfrac{\kappa\ell_{P}^{2}}{\mu_{0}}\left({h}^{ij}F^{\alpha}_{i}{F}_{\alpha{j}}-\dfrac{3}{4}F_{\alpha\beta}F^{\alpha\beta}\right).$
where we have used the identity $h^{ij}h_{ij}=3$. Using of the transformation
${h}^{ij}F^{\alpha}_{i}{F}_{\alpha{j}}={h}^{ij}h_{ik}F^{k\alpha}{F}_{\alpha{j}}=F^{j\alpha}{F}_{\alpha{j}}=F^{\alpha{j}}{F}_{{j}\alpha},$
(9.112)
and the definition
$F_{\alpha\beta}F^{\alpha\beta}=F^{\alpha{j}}{F}_{\alpha{j}}+F^{\alpha{0}}{F}_{\alpha{0}}=-F^{\alpha{j}}{F}_{{j}\alpha}-F^{\alpha{0}}{F}_{{0}\alpha},$
(9.113)
together with the properties of the electromagnetic field tensor
$\displaystyle{F}_{\alpha\beta}F^{\alpha\beta}$ $\displaystyle=$
$\displaystyle 2\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right),$ (9.114)
$\displaystyle{F}_{\alpha 0}F^{0\alpha}$ $\displaystyle=$
$\displaystyle\dfrac{\vec{E}^{2}}{c^{2}},$ (9.115)
one obtains
${h}^{ij}F^{\alpha}_{i}{F}_{\alpha{j}}=-F_{\alpha\beta}F^{\alpha\beta}-F^{\alpha{0}}{F}_{{0}\alpha}=-2\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right)-\dfrac{\vec{E}^{2}}{c^{2}}=2\vec{B}^{2}+\dfrac{\vec{E}^{2}}{c^{2}}.$
(9.116)
Collecting all together one receives finally
${{}^{(3)}}R=\dfrac{\kappa\ell_{P}^{2}}{\mu_{0}}\left[2\vec{B}^{2}+\dfrac{\vec{E}^{2}}{c^{2}}-\dfrac{3}{2}\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right)\right]=\dfrac{\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}+5\epsilon_{0}\vec{E}^{2}\right).$
(9.117)
Similarly one can calculate the energy density of electromagnetic field.
Applying the definition $n^{\mu}={n}^{\beta}{g}^{\mu}_{\beta}$, and the
identity $g_{\mu\nu}n^{\mu}{n}^{\nu}=n^{\mu}{n}_{\mu}=-1$ one obtains
$\displaystyle\varrho$ $\displaystyle=$ $\displaystyle
T_{\mu\nu}n^{\mu}{n}^{\nu}=\dfrac{1}{\mu_{0}}\left(F_{\mu\alpha}g^{\alpha\beta}F_{\nu\beta}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)n^{\mu}{n}^{\nu}=$
(9.118) $\displaystyle=$
$\displaystyle\dfrac{1}{\mu_{0}}\left({F}_{\mu\alpha}({g}^{\alpha\beta}n^{\mu}{n}^{\nu})F_{\nu\beta}-\dfrac{1}{4}g_{\mu\nu}n^{\mu}{n}^{\nu}{F}_{\alpha\beta}F^{\alpha\beta}\right).$
The second term in the formula (9.118) can be easy transformed with using of
the identity $g_{\mu\nu}n_{\mu}{n}_{\nu}=n^{\mu}{n}_{\mu}=-1$. The first term,
however, is not so easy to transform. Let us notice that
${g}^{\alpha\beta}n^{\mu}{n}^{\nu}=g^{\alpha\mu}g^{\nu\beta}(g_{\mu\nu}n^{\mu}{n}^{\nu})=-g^{\alpha\mu}g^{\nu\beta},$
(9.119)
where we have applied $g_{\mu\nu}n^{\mu}{n}^{\nu}=n_{\nu}n^{\nu}=-1$. In this
manner
$\displaystyle{F}_{\mu\alpha}({g}^{\alpha\beta}n^{\mu}{n}^{\nu})F_{\nu\beta}$
$\displaystyle=$
$\displaystyle-{F}_{\mu\alpha}(g^{\alpha\mu}g^{\nu\beta})F_{\nu\beta}={F}_{\mu\alpha}(g^{\alpha\mu}g^{\nu\beta})F_{\beta\nu}=$
(9.120) $\displaystyle=$
$\displaystyle{F}^{\mu}_{\mu}{F}^{\nu}_{\nu}=F_{\mu\nu}(g^{\nu\mu}g_{\nu\mu})F^{\mu\nu}=$
$\displaystyle=$ $\displaystyle
F_{\mu\nu}(g^{\nu\mu}g_{\mu\nu})F^{\mu\nu}=4F_{\mu\nu}F^{\mu\nu},$
where we have applied the identities $F^{\mu}_{\mu}=F_{\mu\nu}g^{\nu\mu}$,
$F^{\nu}_{\nu}=g_{\nu\mu}F^{\mu\nu}$, and $g^{\nu\mu}g_{\mu\nu}=4$. In this
manner one receives finally the energy density
$\displaystyle\varrho$ $\displaystyle=$
$\displaystyle\dfrac{1}{\mu_{0}}\left(4{F}_{\alpha\beta}{F}^{\alpha\beta}+\dfrac{1}{4}{F}_{\alpha\beta}F^{\alpha\beta}\right)=\dfrac{17}{4\mu_{0}}{F}_{\alpha\beta}F^{\alpha\beta}=$
(9.121) $\displaystyle=$
$\displaystyle\dfrac{17}{2\mu_{0}}\left(\vec{B}^{2}-\dfrac{\vec{E}^{2}}{c^{2}}\right)=\dfrac{17}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}-\epsilon_{0}\vec{E}^{2}\right),$
where we have applied the property (9.114) of electromagnetic field tensor.
Consequently the generalized gravitational potential is
$\displaystyle
V=-{{}^{(3)}}R+2\kappa\ell_{P}^{2}\varrho=-\dfrac{\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}+5\epsilon_{0}\vec{E}^{2}\right)+17\kappa\ell_{P}^{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}-\epsilon_{0}\vec{E}^{2}\right)=$
$\displaystyle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\vec{B}^{2}-\dfrac{13}{11}\epsilon_{0}\vec{E}^{2}\right).$
(9.122)
Now it is easy to see that $V$ averaged on midisuperspace is
$\langle{V}\rangle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right),$
(9.123)
where $\langle\vec{B}^{2}\rangle$ and $\langle\vec{E}^{2}\rangle$ are the
midisuperspace means of the squared fields $\vec{B}$ and $\vec{E}$,
respectively. One has explicitly
$\displaystyle\langle\vec{B}^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi)}\int\vec{B}^{2}\delta\xi,$ (9.124)
$\displaystyle\langle\vec{E}^{2}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Omega(\xi)}\int\vec{E}^{2}\delta\xi.$ (9.125)
Because of the fields $\vec{B}$ and $\vec{E}$ are in general functions on
space-time, it is evident that both the averages (9.124) and (9.125) must be
treated as functions on space-time, of course after performing in all the
functional integrals the suitable transformation from midisuperspace to space-
time $\xi\rightarrow\xi(x)$. In other words one has
$\langle{V}\rangle(\xi,\phi)=\langle{V}\rangle(x).$ (9.126)
Because of the considerations presented above are independent on the concrete
form of the electrovacuum solution, they can be applied to any solution of the
Einstein–Maxwell equations. Interestingly, one can _ad hoc_ add the
cosmological constant
$\langle{V}\rangle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)+2\Lambda,$
(9.127)
so that the averaged generalized gravitational potential can be ordered by two
equivalent ways
$\displaystyle\langle{V}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\epsilon_{0}\langle\vec{E}^{2}\rangle\right)+2\Lambda-3\kappa\ell_{P}^{2}\epsilon_{0}\langle\vec{E}^{2}\rangle=$
(9.128) $\displaystyle=$
$\displaystyle\dfrac{39\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\epsilon_{0}\langle\vec{E}^{2}\rangle\right)+2\Lambda-\dfrac{3\kappa\ell_{P}^{2}}{\mu_{0}}\langle\vec{B}^{2}\rangle.$
(9.129)
If the averages $\langle\vec{B}^{2}\rangle=B_{\Lambda}^{2}$ and
$\langle\vec{E}^{2}\rangle=E_{\Lambda}^{2}$ are constant then one can take
$2\Lambda=3\kappa\ell_{P}^{2}\epsilon_{0}E^{2}_{\Lambda}=\dfrac{3\kappa\ell_{P}^{2}}{\mu_{0}}B^{2}_{\Lambda},$
(9.130)
and one the averaged generalized gravitational potential vanishes
$\langle{V}\rangle=0.$ (9.131)
The condition (9.130, however, leads to the relation
$E_{\Lambda}=cB_{\Lambda},$ (9.132)
which can be used to define the speed of light
$c:=\dfrac{E_{\Lambda}}{B_{\Lambda}},$ (9.133)
if one knows $E_{\Lambda}$ and $B_{\Lambda}$.
Let us return to the Kerr–Newman space-time. The spatial part of the
Kerr–Newman metric has the form
$h_{ij}=\left[\begin{array}[]{ccc}\dfrac{r^{2}+\alpha^{2}\cos^{2}\theta}{\Delta}&0&0\\\
0&r^{2}+\alpha^{2}\cos^{2}\theta&0\\\
0&0&\dfrac{(r^{2}+\alpha^{2})^{2}-\alpha^{2}\Delta\sin^{2}\theta}{r^{2}+\alpha^{2}\cos^{2}\theta}\sin^{2}\theta\end{array}\right],$
(9.134)
where $\Delta=r^{2}\alpha^{2}-r_{S}r+r_{Q}^{2}$, and by this reason the global
dimension can be deduced easy
$h=\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\left[r^{2}+\alpha^{2}\cos^{2}\theta+\dfrac{(r^{2}+\alpha^{2})(r_{S}r-r_{Q}^{2})}{r^{2}+\alpha^{2}-r_{S}r+r_{Q}^{2}}\right]\sin^{2}\theta,$
(9.135)
so that the invariant global dimension is
$\xi=\dfrac{1}{4\pi\sqrt{6}}\sqrt{\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\left[r^{2}+\alpha^{2}\cos^{2}\theta+\dfrac{(r^{2}+\alpha^{2})(r_{S}r-r_{Q}^{2})}{r^{2}+\alpha^{2}-r_{S}r+r_{Q}^{2}}\right]}\sin\theta.$
(9.136)
The volume of the Kerr-Newman midisuperspace is $\Omega(\xi)=\xi$. Here
$r_{S}$ is the Schwarzschild radius and $r_{Q}$ is a length-scale
corresponding to the electric charge Q of the mass
$r_{Q}^{2}=\dfrac{Q^{2}G}{4\pi\epsilon_{0}c^{4}}=\dfrac{Q^{2}}{4\pi\epsilon_{0}}\dfrac{\kappa}{8\pi}.$
(9.137)
There are three possible situations. Namely, when the averaged generalized
gravitational potential (9.123) is $1^{\circ}$ vanishing, $2^{\circ}$
positive, $3^{\circ}$ negative. The first case is rather trivial.
$\langle{V}\rangle$ vanishes if and only if
$\langle\vec{B}^{2}\rangle=\dfrac{13}{11}\dfrac{\langle\vec{E}^{2}\rangle}{c^{2}},$
(9.138)
what in fact means that
$|\vec{B}|=\sqrt{\dfrac{13}{11}}\dfrac{|\vec{E}|}{c}\approx
1.087\dfrac{|\vec{E}|}{c}.$ (9.139)
In such a situation one can construct straightforwardly and easy the wave
functions of the Kerr–Newman space-time
$\Psi_{1}^{KN}=\dfrac{\sin\theta}{4\pi\sqrt{|A|}\sqrt{6}}\sqrt{\left(r^{2}+\alpha^{2}\cos^{2}\theta\right)\left[r^{2}+\alpha^{2}\cos^{2}\theta+\dfrac{(r^{2}+\alpha^{2})(r_{S}r-r_{Q}^{2})}{r^{2}+\alpha^{2}-r_{s}r+r_{Q}^{2}}\right]},$
(9.140) $\Psi_{2}^{KN}=\dfrac{1}{\sqrt{|B|}},$ (9.141)
where the constants of integration $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.142)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.143)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\sqrt{\left(R^{2}+\alpha^{2}\cos^{2}\Theta\right)\left[R^{2}+\alpha^{2}\cos^{2}\Theta+\dfrac{(R^{2}+\alpha^{2})(r_{S}R-r_{Q}^{2})}{R^{2}+\alpha^{2}-r_{s}R+r_{Q}^{2}}\right]}\sin\Theta.$
(9.144)
The second situation, i.e. positive $\langle{V}\rangle$, is defined for
$\langle\vec{B}^{2}\rangle>\dfrac{13}{11}\dfrac{\langle\vec{E}^{2}\rangle}{c^{2}},$
(9.145)
what can be presented in the form
$|\vec{B}|>\sqrt{\dfrac{13}{11}}\dfrac{|\vec{E}|}{c}.$ (9.146)
In such a situation the constants of integration $A$ and $B$ are not easy to
calculate for general fields $\vec{B}$ and $\vec{E}$, but can be presented in
the compact form
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}}\right]^{2}\delta\xi,$
(9.147) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cos^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)\delta\xi,$
(9.148)
and the wave functions of the Kerr–Newman space-time are
$\displaystyle\Psi_{1}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}},$
(9.149) $\displaystyle\Psi_{2}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right).$
(9.150)
The case of negative averaged generalized gravitational potential can be
considered analogously. In this case
$|\vec{B}|<\sqrt{\dfrac{13}{11}}\dfrac{|\vec{E}|}{c}.$ (9.151)
So that the wave functions of the Kerr–Newman space-time are
$\displaystyle\Psi_{1}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}},$
(9.152) $\displaystyle\Psi_{2}^{KN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right).$
(9.153)
where the integration constants $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}}\right]^{2}\delta\xi,$
(9.154) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cosh^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)\delta\xi,$
(9.155)
#### H The Reissner–Nordström Space-time
Another solution of the electrovacuum Einstein field equations is the
Reissner–Nordström metric describing spherically symmetric static massive
charged object. Therefore in this case one has also
$\langle{V}\rangle=\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right),$
(9.156)
and the only change is the metric. The spatial part of Reissner–Nordström
space-time metric is
$h_{ij}=\left[\begin{array}[]{ccc}\left(1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}\right)^{-1}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{array}\right],$ (9.157)
so that the global dimension is
$h=\dfrac{r^{4}\sin^{2}\theta}{1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}}$
(9.158)
while the invariant global dimension has the form
$\xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}}}$
(9.159)
and of the volume of the Reissner–Nordström midisuperspace is
$\Omega(\xi)=\xi$. There are three possible situations defined by absolute
values of the electric and magnetic fields. For the case of negative
$\langle{V}\rangle$ the wave functions of the Reissner–Nordström space-time
are
$\displaystyle\Psi_{1}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}},$
(9.160) $\displaystyle\Psi_{2}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cosh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right).$
(9.161)
where the integration constants $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sinh\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}}\right]^{2}\delta\xi,$
(9.162) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cosh^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left|\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right|}\right)\delta\xi.$
(9.163)
When $\langle{V}\rangle$ is positive then
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int\left[\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}}\right]^{2}\delta\xi,$
(9.164) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int\cos^{2}\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)\delta\xi,$
(9.165)
and the wave functions of the Reissner–Nordström space-time are
$\displaystyle\Psi_{1}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right)}{\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}},$
(9.166) $\displaystyle\Psi_{2}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\xi\sqrt{\dfrac{33\kappa\ell_{P}^{2}}{2}\left(\dfrac{1}{\mu_{0}}\langle\vec{B}^{2}\rangle-\dfrac{13}{11}\epsilon_{0}\langle\vec{E}^{2}\rangle\right)}\right).$
(9.167)
For vanishing $\langle{V}\rangle$ one has
$\displaystyle\Psi_{1}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi\sqrt{|A|}\sqrt{6}}\dfrac{r^{2}\sin\theta}{\sqrt{1-\dfrac{r_{S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}}},$
(9.168) $\displaystyle\Psi_{2}^{RN}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.169)
where the constants of integration $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.170)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.171)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}\dfrac{R^{2}\sin\Theta}{\sqrt{1-\dfrac{r_{S}}{R}+\dfrac{r_{Q}^{2}}{R^{2}}}}.$
(9.172)
#### I The Gödel Space-time
Let us consider the solution of the Einstein field equations with presence of
the cosmological constant and the stress-energy tensor of dust
$\displaystyle
R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}{{}^{(4)}}R+\Lambda{g}_{\mu\nu}=\kappa\ell_{P}^{2}{T}_{\mu\nu},$
(9.173) $\displaystyle T_{\mu\nu}=\varepsilon{u}_{\mu}{u}_{\nu},$ (9.174)
which is given by the Gödel space-time
$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}-a^{2}&0&0&-a^{2}e^{x}\\\
0&a^{2}&0&0\\\ 0&0&a^{2}&0\\\
-a^{2}e^{x}&0&0&-\dfrac{1}{2}a^{2}e^{2x}\end{array}\right],$ (9.179)
$\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}\dfrac{1}{a^{2}}&0&0&-\dfrac{2e^{-x}}{a^{2}}\\\
0&\dfrac{1}{a^{2}}&0&0\\\ 0&0&\dfrac{1}{a^{2}}&0\\\
-\dfrac{2e^{-x}}{a^{2}}&0&0&\dfrac{2e^{-2x}}{a^{2}}\end{array}\right].$
(9.184)
having the following spatial part
$h_{ij}=\left[\begin{array}[]{ccc}a^{2}&0&0\\\ 0&a^{2}&0\\\
0&0&-\dfrac{1}{2}a^{2}e^{2x}\end{array}\right]\quad,\quad
h^{ij}=\left[\begin{array}[]{ccc}\dfrac{1}{a^{2}}&0&0\\\
0&\dfrac{1}{a^{2}}&0\\\ 0&0&\dfrac{2e^{-2x}}{a^{2}}\end{array}\right],$
(9.185)
where $a=a(x,y,z)$. In this case the global dimension is
$|h|=\dfrac{a^{6}}{2}e^{2x},$ (9.186)
so the invariant global dimension has the form
$\xi=\dfrac{1}{8\pi\sqrt{3}}a^{3}e^{x},$ (9.187)
and therefore the volume of the Gödel midisuperspace is $\Omega(\xi)=\xi$. The
Gödel metric satisfies the Einstein field equations for which the cosmological
constant is related to the parameter $a$ by the relation
$\Lambda=-\dfrac{1}{2a^{2}}.$ (9.188)
The energy density $\epsilon$ of the dust is
$\epsilon=\dfrac{1}{\kappa\ell_{P}^{2}a^{2}}.$ (9.189)
The Gödel metric can be decomposed in the ADM $3+1$ form. It is easy to see
that such an _ad hoc_ decomposition generates the equations
$\displaystyle-N^{2}+N_{i}N^{i}$ $\displaystyle=$ $\displaystyle-a^{2},$
(9.190) $\displaystyle N_{i}$ $\displaystyle=$
$\displaystyle\left[0,0,-a^{2}e^{x}\right].$ (9.191)
Taking into account the spatial metric (9.185) one receives
$N^{i}=h^{ij}N_{j}=\left[0,0,-2e^{-x}\right],$ (9.192)
what gives $N_{i}N^{i}=2a^{2}$ and by this reason
$N=\sqrt{3}a.$ (9.193)
In this manner one can establish the normal unit vector field
$\displaystyle n^{\mu}$ $\displaystyle=$
$\displaystyle\left[\dfrac{1}{N},-\dfrac{N^{i}}{N}\right]=\dfrac{1}{\sqrt{3}}\left[\dfrac{1}{a},0,0,\dfrac{2e^{-x}}{a}\right],$
(9.194) $\displaystyle n_{\mu}$ $\displaystyle=$
$\displaystyle\left[-N,0_{i}\right]=\sqrt{3}\left[-a,0,0,0\right].$ (9.195)
The problem is to choose the velocity vector $u_{\mu}$ according to the
general rules
$\displaystyle u_{\mu}{u}^{\mu}$ $\displaystyle=$ $\displaystyle-1,$ (9.196)
$\displaystyle u_{\mu}{n}^{\mu}$ $\displaystyle=$ $\displaystyle-1,$ (9.197)
$\displaystyle u_{\mu}{n}^{\mu}{u}_{\nu}{n}^{\nu}$ $\displaystyle=$
$\displaystyle 1.$ (9.198)
However, it can be seen by straightforward calculation that the following
choice
$\displaystyle u^{\mu}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{3}}\left[\dfrac{1}{a},0,0,0\right],$ (9.199)
$\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}u^{\nu}=\sqrt{3}\left[-a,0,0,-ae^{x}\right],$ (9.200)
satisfies the conditions (9.196)-(9.198). The contraction of the Einstein
field equations (9.173)-(9.174) with metric $g^{\mu\nu}$ leads to
${{}^{(4)}}R=4\Lambda+\kappa\ell_{P}^{2}\epsilon,$ (9.201)
so that the equations (9.173)-(9.174) can be presented in the form
$R_{\mu\nu}=\Lambda{g}_{\mu\nu}+\kappa\ell_{P}^{2}\epsilon\left({u}_{\mu}{u}_{\nu}+\dfrac{1}{2}g_{\mu\nu}\right),$
(9.202)
what after including the fact that for the Gödel Universe one has
$\kappa\ell_{P}^{2}\epsilon=-2\Lambda,$ (9.203)
one receives the four-dimensional Ricci curvature tensor
$R_{\mu\nu}=\kappa\ell_{P}^{2}\epsilon{u}_{\mu}{u}_{\nu}=-2\Lambda{u}_{\mu}{u}_{\nu}.$
(9.204)
In this manner the three-dimensional Ricci curvature tensor has the form
$R_{ij}=-2\Lambda{u}_{i}{u}_{j},$ (9.205)
and consequently the three-dimensional Ricci scalar curvature is
${{}^{(3)}}R=h^{ij}R_{ij}=-2\Lambda{u}_{i}{u}^{i}=0.$ (9.206)
Similarly one can establish the energy density of Matter fields
$\varrho=T_{\mu\nu}n^{\mu}{n}^{\nu}=\epsilon{u}_{\mu}{n}^{\mu}{u}_{\nu}{n}^{\nu}=\epsilon=-2\dfrac{\Lambda}{\kappa\ell_{P}^{2}}.$
(9.207)
In this manner the generalized gravitational potential for the Gödel Universe
has the following form
$V=-{{}^{(3)}}R+2\Lambda+2\kappa\ell_{P}^{2}\varrho=-2\Lambda=\kappa\ell_{P}^{2}\epsilon=\dfrac{1}{a^{2}}>0,$
(9.208)
so that
$\displaystyle\langle{V}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{a^{3}e^{x}}\int\dfrac{d({a}^{3}e^{x})}{{a}^{2}}=\dfrac{1}{a^{3}e^{x}}\int\dfrac{3a^{2}dae^{x}+a^{3}e^{x}dx}{{a}^{2}}=$
(9.209) $\displaystyle=$
$\displaystyle\dfrac{1}{a^{3}e^{x}}\int\left(3e^{x}da+ae^{x}dx\right),$
(9.210)
and by this reason
$\sqrt{\langle{V}\rangle}\left(\xi,\phi\right)=\dfrac{1}{a^{3/2}e^{x/2}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}.$
(9.211)
In this manner the wave functions of the Gödel dust Universe are
$\displaystyle\Psi_{1}^{G}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{\sin\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right)}{\dfrac{1}{a^{3/2}e^{x/2}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}},$
(9.212) $\displaystyle\Psi_{2}^{G}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}}\cos\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right),$
(9.213)
where $\mathrm{sgn}(a)\neq 0$, and the constants of integration $A$ and $B$
are
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\sqrt{3}S_{P}}\int\left(\dfrac{\sin\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right)}{\dfrac{1}{a^{3/2}e^{x/2}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}}\right)^{2}d(a^{3}e^{x}),$
(9.214)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!B$
$\displaystyle=$
$\displaystyle\dfrac{1}{\Gamma_{0}}\dfrac{1}{2\sqrt{3}S_{P}}\int\cos^{2}\left(\dfrac{a^{3/2}e^{x/2}}{8\pi\sqrt{3}}\sqrt{\int\left(3e^{x}da+ae^{x}dx\right)}\right)d(a^{3}e^{x}).$
(9.215)
Of course when $a=constans$ the situation is much more simpler, but we will
not derive these particular results.
#### J The Einstein–Rosen Gravitational Waves
The example of vacuum solution are also the Einstein–Rosen cylindrical
gravitational waves. The spatial part of the metric has the form
$h_{ij}=\left[\begin{array}[]{ccc}e^{2\gamma-2\psi}&0&0\\\ 0&e^{2\psi}&0\\\
0&0&r^{2}e^{-2\psi}\end{array}\right],$ (9.216)
where $r$ is radial distance from the $z$ axis, and $\psi=\psi(t,r)$ and
$\gamma=\gamma(t,r)$ are functions satisfying the equations
$\displaystyle\ddot{\psi}$ $\displaystyle=$
$\displaystyle\dfrac{\psi^{\prime}}{r}+\psi^{\prime\prime},$ (9.217)
$\displaystyle\dfrac{\gamma^{\prime}}{r}$ $\displaystyle=$
$\displaystyle\psi^{\prime 2}+\dot{\psi}^{2},$ (9.218)
$\displaystyle\dfrac{\dot{\gamma}}{r}$ $\displaystyle=$ $\displaystyle
2\dot{\psi}\psi^{\prime}.$ (9.219)
In other words when one solves the equation (9.217) then the function $\gamma$
can be established by
$\gamma=\int\left[r\left(\psi^{\prime
2}+\dot{\psi}^{2}\right)dr+2r\dot{\psi}\psi^{\prime}dt\right].$ (9.220)
In this manner the global dimension for the Einstein–Rosen waves is
$h=r^{2}e^{2\gamma-2\psi},$ (9.221)
while the invariant global dimension has the form
$\xi=\dfrac{1}{\sqrt{6}S_{P}}re^{\gamma-\psi}.$ (9.222)
The generalized gravitational potential vanishes in this case, so that the
wave functions are
$\displaystyle\Psi_{1}^{ER}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{4\pi\sqrt{6}}re^{\gamma-\psi},$
(9.223) $\displaystyle\Psi_{2}^{ER}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.224)
where the constants of integration $A$ and $B$ are determined by the relations
$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.225)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.226)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}Re^{\gamma(T,R)-\psi(T,R)},$ (9.227)
where $T$ and $R$ are reference values of $t$ and $R$, respectively.
#### K The Taub–Newman–Unti–Tamburino Space-time
Let us consider the generalized axisymmetric solution of the vacuum Einstein
field equations presented in the Weyl canonical coordinates
$h_{ij}=\left[\begin{array}[]{ccc}e^{2\gamma-2\psi}&0&0\\\
0&e^{2\gamma-2\psi}&0\\\
0&0&r^{2}e^{-2\psi}-A^{2}e^{2\gamma}\end{array}\right],$ (9.228)
for which the field equations are
$\displaystyle\bigtriangleup\psi$ $\displaystyle=$ $\displaystyle 0,$ (9.229)
$\displaystyle\dfrac{\gamma^{\prime}}{r}$ $\displaystyle=$
$\displaystyle\psi^{\prime 2}+\dot{\psi}^{2},$ (9.230)
$\displaystyle\dfrac{\gamma^{\prime}_{z}}{r}$ $\displaystyle=$ $\displaystyle
2\dot{\psi}\psi^{\prime}.$ (9.231)
Here $r$ is the radial distance from the axis of symmetry $z$. In such a
situation the global dimension is
$h=e^{4\gamma-4\psi}\left(r^{2}e^{-2\psi}-A^{2}e^{2\gamma}\right),$ (9.232)
so that the invariant global dimension is
$\xi=\dfrac{1}{4\pi\sqrt{6}}e^{2\gamma-2\psi}\sqrt{r^{2}e^{-2\psi}-A^{2}e^{2\gamma}}.$
(9.233)
For this case the cosmological constant as well as Matter fields are absent.
Therefore the three-dimensional Ricci scalar curvature, and the generalized
gravitational potential are trivial. The wave functions of any static
axisymmetric vacuum solution can be established as
$\displaystyle\Psi_{1}^{SAVS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|A|}}\dfrac{1}{4\pi\sqrt{6}}e^{2\gamma-2\psi}\sqrt{r^{2}e^{-2\psi}-A^{2}e^{2\gamma}},$
(9.234) $\displaystyle\Psi_{2}^{SAVS}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{|B|}},$ (9.235)
where the constants of integration $A$ and $B$ are
$\displaystyle A$ $\displaystyle=$ $\displaystyle\dfrac{\Xi^{3}}{3},$ (9.236)
$\displaystyle B$ $\displaystyle=$ $\displaystyle\dfrac{1}{\Gamma_{0}}\Xi,$
(9.237)
where $\Xi$ is the reference constant
$\Xi=\dfrac{1}{4\pi\sqrt{6}}e^{2\gamma(R,Z)-2\psi(R,Z)}\sqrt{R^{2}e^{-2\psi(R,Z)}-A^{2}(R,Z)e^{2\gamma(R,Z)}},$
(9.238)
where $R$ and $Z$ are reference values of $r$ and $z$. Let us see the wave
functions of two particular cases.
The Taub–Newman–Unti–Tamburino space-time is the particular case the static
axisymmetric vacuum space-time characterized by
$\displaystyle e^{2\psi}$ $\displaystyle=$
$\displaystyle\dfrac{(r_{+}+r_{-})^{2}-(r_{S}^{2}+4l^{2})}{(r_{+}+r_{-}+r_{S})^{2}+4l^{2}},$
(9.239) $\displaystyle e^{2\gamma}$ $\displaystyle=$
$\displaystyle\dfrac{(r_{+}+r_{-})^{2}-(r_{S}^{2}+4l^{2})}{4r_{+}r_{-}},$
(9.240) $\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{2l(r_{+}-r_{-})}{\sqrt{r_{S}^{2}+4l^{2}}},$ (9.241)
$\displaystyle r_{\pm}^{2}$ $\displaystyle=$ $\displaystyle
r^{2}+\left(z\pm\dfrac{1}{2}\sqrt{r_{S}^{2}+4l^{2}}\right)^{2}.$ (9.242)
### Chapter 10 The Functional Objective Geometry
#### A Effective Scalar Curvature
Let us assume that the concrete form of the gravitational potential $V_{eff}$
(8.2) is fixed _ad hoc_ as functional or function of the global dimension $h$.
In such a situation one can express the Ricci scalar of a three-dimensional
embedded space as follows
${{}^{(3)}}R=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}hV_{eff},$ (10.1)
whereas the global one-dimensional quantum gravity is given by the
evolutionary equation
$\left(\dfrac{\delta^{2}}{\delta h^{2}}+V_{eff}\right)\Psi[h]=0.$ (10.2)
In this manner the quantum gravity is the system describing geometry (10.1),
and quantum mechanics (10.2) of an embedded space.
The quantum gravity (10.1)-(10.2) is in itself non trivial. In fact, this can
be expressed in more general notation
$\displaystyle{{}^{(3)}}R$ $\displaystyle=$ $\displaystyle f[h_{ij}],$ (10.3)
$\displaystyle\dfrac{\delta^{2}\Psi[h]}{\delta h^{2}}$ $\displaystyle=$
$\displaystyle-V_{eff}[h_{ij}]\Psi[h],$ (10.4)
where both $f[h_{ij}]$ as well as $V_{eff}[h_{ij}]$ are scalar-valued
functionals of an induced three-dimensional metric $h_{ij}$, i.e. are
_objective functionals_
$\displaystyle f[h_{ij}]$ $\displaystyle=$ $\displaystyle
f[I_{\mathbf{h}},II_{\mathbf{h}},III_{\mathbf{h}}],$ (10.5)
$\displaystyle\Psi[h_{ij}]$ $\displaystyle=$
$\displaystyle\Psi[I_{\mathbf{h}},II_{\mathbf{h}},III_{\mathbf{h}}],$ (10.6)
where $I_{\mathbf{h}}$, $II_{\mathbf{h}}$, and $III_{\mathbf{h}}$ are the
$3\times 3$ matrix invariants of an induced metric $h_{ij}$
$I_{\mathbf{h}}=\mathrm{Tr}\mathbf{h}\quad,\quad{II}_{\mathbf{h}}=\dfrac{\left(\mathrm{Tr}\mathbf{h}\right)^{2}-\mathrm{Tr}\mathbf{h}^{2}}{2}\quad,\quad{III}_{\mathbf{h}}=\det\mathbf{h},$
(10.7)
which according to the Cayley–Hamilton theorem are the coefficients of the
characteristic polynomial of the matrix $h_{ij}$
$\mathbf{h}^{3}-I_{\mathbf{h}}\mathbf{h}^{2}+II_{\mathbf{h}}\mathbf{h}-III_{\mathbf{h}}\mathbf{I}_{3\times
3}=0.$ (10.8)
In this manner we shall call _the functional objective geometry_ the quantum
gravity given by system of equations (10.3)- (10.4).
We shall call the Ricci scalar curvature (10.1) describing the three-geometry
of an embedded space _the effective scalar curvature_ and study its meaning in
this section. For convenience let us present $V_{eff}$ as an algebraic sum of
three elementary energetic constituents
$V_{eff}=V_{G}+V_{C}+V_{M},$ (10.9)
where $V_{G}$, $V_{C}$, and $V_{M}$ are the geometric and the cosmological,
and the material contributions
$\displaystyle V_{G}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{6(8\pi)^{2}}\dfrac{{{}^{(3)}\\!R}}{h},$ (10.10)
$\displaystyle V_{C}$ $\displaystyle=$
$\displaystyle\dfrac{1}{6(8\pi)^{2}}\dfrac{2\Lambda}{h},$ (10.11)
$\displaystyle V_{M}$ $\displaystyle=$
$\displaystyle\dfrac{1}{6(8\pi)^{2}}\dfrac{2\kappa}{h}\varrho.$ (10.12)
One can list several examples of physical scenarios within the global one-
dimensional quantum gravity, with respect to the choice of the form of the
potential $V_{eff}$.
1. 1.
The case of constant non vanishing effective gravitational potential
$V_{eff}=V_{c}\neq 0$. In such a situation the Ricci scalar curvature of an
embedded space and the global one-dimensional quantum gravity are
$\displaystyle{{}^{(3)}}R=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}hV_{c},$
(10.13)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{c}\right)\Psi_{c}[h]=0,$
(10.14)
where $\Psi_{c}[h]$ is a wave functional related to $V_{eff}=V_{c}$.
2. 2.
The case of trivial effective gravitational potential $V_{eff}=0$. In such a
situation the three-dimensional Ricci scalar curvature and the global one-
dimensional quantum gravity are
$\displaystyle{{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho,$ (10.15)
$\displaystyle\dfrac{\delta^{2}}{\delta{h^{2}}}\Psi_{0}[h]=0,$ (10.16)
where $\Psi_{0}$ is a ”free” wave functional related to $V_{eff}=0$.
3. 3.
The case when a sum of geometric and cosmological contributions is trivial
$V_{G}+V_{C}=0$, but the effective potential does not vanish identically
$V_{eff}\neq 0$. In such a situation the three-dimensional Ricci scalar
curvature and the global one-dimensional quantum gravity are
$\displaystyle{{}^{(3)}}R=2\Lambda,$ (10.17)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-\dfrac{1}{6(8\pi)^{2}}\dfrac{2\kappa\ell_{P}^{2}}{h}\varrho[h]\right)\Psi_{M}[h]=0,$
(10.18)
where $\Psi_{M}$ is a ”material” wave functional related to $V_{M}\neq 0$.
4. 4.
The case when a sum of geometric and material contributions vanishes
$V_{G}+V_{M}=0$, but the gravitational potential is in general non trivial
$V_{eff}\neq 0$. In such a situation the Ricci scalar curvature of an embedded
space and the global one-dimensional quantum gravity are
$\displaystyle{{}^{(3)}\\!R}=2\kappa\ell_{P}^{2}\varrho,$ (10.19)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+\dfrac{1}{6(8\pi)^{2}}\dfrac{2\Lambda}{h}\right)\Psi_{C}[h]=0.$
(10.20)
Here $\Psi_{C}$ is the ”cosmological” wave functional related to $V_{C}\neq
0$.
5. 5.
The case when a sum of cosmological and material contributions is trivial
$V_{C}+V_{M}=0$, but the effective gravitational potential is non zero
$V_{eff}\neq 0$. In such a situation the energy density of Matter fields and
the global one-dimensional quantum gravity are
$\displaystyle\varrho=-\dfrac{\Lambda}{\kappa\ell_{P}^{2}},$ (10.21)
$\displaystyle\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-\dfrac{1}{6(8\pi)^{2}}\dfrac{{{}^{(3)}\\!R}}{h}\right)\Psi_{G}[h]=0,$
(10.22)
where $\Psi_{G}$ is the ”geometric” wave functional related to $V_{G}\neq 0$.
6. 6.
More general approach can be based on complex analysis (For basics advances
see e.g. [606]). Let us pu _ad hoc_ the functional Laurent series expansion in
the global dimension $h$ of the effective gravitational potential $V_{eff}[h]$
in an infinitesimal neighborhood, i.e. a 1-sphere (circle) of a radius
$h_{\epsilon}$, of any fixed initial value $h_{0}$
$V_{eff}[h]=\sum_{-\infty}^{\infty}a_{n}\left(h-h_{0}\right)^{n}\quad\mathrm{in}\quad
C(h_{\epsilon})=\left\\{h:|h-h_{0}|<h_{\epsilon}\right\\},$ (10.23)
where $a_{n}$ are the series coefficients given by the classical functional
integral
$a_{n}=\dfrac{1}{2\pi
i}\int_{C(h_{\epsilon})}\dfrac{V_{eff}[h]}{\left(h-h_{0}\right)^{n+1}}\delta
h,$ (10.24)
which is the Cauchy integral with the Radon/Lebesgue–Stieltjes measure $\delta
h$. Let us take into considerations $h_{0}=0$. Then the Ricci scalar curvature
of a three-dimensional embedded space is
${{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}\sum_{-\infty}^{\infty}b_{n}(h-h_{0})^{n},$
(10.25)
where $b_{n}$ is the series coefficient
$b_{n}=a_{n-1}+h_{0}a_{n}=\dfrac{1}{2\pi
i}\int_{C(h_{\epsilon})}\dfrac{h}{\left(h-h_{0}\right)^{n+1}}V_{eff}[h]\delta
h,$ (10.26)
and the global one-dimensional quantum gravity yields
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+\sum_{-\infty}^{\infty}a_{n}(h-h_{0})^{n}\right)\Psi[h]=0.$
(10.27)
By the triangle inequality one has
$|b_{n}|\leqslant|a_{n-1}|+|h_{0}||a_{n}|$ (10.28)
so it is easy to see that
$\dfrac{|b_{n}|}{|a_{n}|}\leqslant\dfrac{|a_{n-1}|}{|a_{n}|}+|h_{0}|.$ (10.29)
Applying the inequality
$\left|\int{f}\right|\leqslant\int|f|,$ (10.30)
where $f$ is considered as Riemann-integrable function and integral is
defined, to the coefficients $a_{n}$ and $b_{n}$ one has
$\displaystyle|a_{n}|$ $\displaystyle\leqslant$
$\displaystyle\dfrac{1}{h_{\epsilon}^{n+1}}\dfrac{1}{2\pi}\int_{C(h_{\epsilon})}\left|V_{eff}\right|\delta{h}\leqslant\dfrac{1}{h_{\epsilon}^{n+1}}|a_{-1}|,$
(10.31)
where $a_{-1}$ is the residue of the effective gravitational potential in the
point $h=h_{0}$ given by the Cauchy integral formula
$a_{-1}=\mathrm{Res}(V_{eff},h_{0})=\dfrac{1}{2\pi
i}\int_{C(h_{\epsilon})}{V}_{eff}\delta{h},$ (10.32)
where $C(h_{\epsilon})$ traces out a circle around $h_{0}$ in a
counterclockwise manner on the punctured disk
$D=\left\\{z:0<|h-h_{0}|<R\right\\}$. If the point $h=h_{0}$ is a pole of
order $n$, then
$\mathrm{Res}(V_{eff},h_{0})=\dfrac{1}{\Gamma(n)}\lim_{h\rightarrow
h_{0}}\dfrac{\delta^{n-1}}{\delta{h}^{n-1}}\left((h-h_{0})V_{eff}\right).$
(10.33)
It can be seen straightforwardly that
$\dfrac{|a_{n-1}|}{|a_{n}|}\geqslant{h}_{\epsilon},$ (10.34)
and hence the inequality (10.29) gives
$\dfrac{|b_{n}|}{|a_{n}|}\geqslant{h}_{\epsilon}+|h_{0}|.$ (10.35)
Because by the triangle inequality
$|b_{n+1}|=|a_{n}+h_{0}a_{n+1}|\leqslant|a_{n}|+|h_{0}||a_{n+1}|,$ (10.36)
and
$\dfrac{|a_{n+1}|}{|a_{n}|}\leqslant\dfrac{1}{h_{\epsilon}},$ (10.37)
one obtains
$\dfrac{|b_{n+1}|}{|a_{n}|}\leqslant 1+\dfrac{|h_{0}|}{h_{\epsilon}}.$ (10.38)
applying the inequality (10.35) in the equivalent form
$\dfrac{|a_{n}|}{|b_{n}|}\leqslant\dfrac{1}{{h}_{\epsilon}+|h_{0}|}.$ (10.39)
one receives the upper bound
$\dfrac{|b_{n+1}|}{|a_{n}|}\dfrac{|a_{n}|}{|b_{n}|}=\dfrac{|b_{n+1}|}{|b_{n}|}\leqslant\dfrac{1}{{h}_{\epsilon}+|h_{0}|}\left(1+\dfrac{|h_{0}|}{h_{\epsilon}}\right).$
(10.40)
Another bound for $\dfrac{|b_{n+1}|}{|b_{n}|}$ can be obtained as follows.
Because of
$a_{n}=\dfrac{b_{n}-a_{n-1}}{h_{0}},$ (10.41)
one has
$b_{n+1}=\dfrac{b_{n}-a_{n-1}}{h_{0}}+h_{0}a_{n+1},$ (10.42)
or after small algebraic manipulations
$h_{0}b_{n+1}+a_{n-1}=h_{0}a_{n+1}+b_{n}.$ (10.43)
This equation can be rewritten in the form
$1=\left|\dfrac{h_{0}b_{n+1}}{h_{0}a_{n+1}+b_{n}}+\dfrac{a_{n-1}}{h_{0}a_{n+1}+b_{n}}\right|,$
(10.44)
which after taking into account the triangle inequality
$\left|\dfrac{h_{0}b_{n+1}}{h_{0}a_{n+1}+b_{n}}+\dfrac{a_{n-1}}{h_{0}a_{n+1}+b_{n}}\right|\leqslant\left|\dfrac{h_{0}b_{n+1}}{h_{0}a_{n+1}+b_{n}}\right|+\left|\dfrac{a_{n-1}}{h_{0}a_{n+1}+b_{n}}\right|,$
(10.45)
leads to
$|h_{0}a_{n+1}+b_{n}|\leqslant|h_{0}||b_{n+1}|+|a_{n-1}|.$ (10.46)
Applying again the triangle inequality
$|h_{0}a_{n+1}+b_{n}|\leqslant|h_{0}||a_{n+1}|+|b_{n}|,$ (10.47)
one receives
$|h_{0}||b_{n+1}|-|b_{n}|\leqslant|h_{0}||a_{n+1}|-|a_{n-1}|.$ (10.48)
This inequality can be rewritten as
$|h_{0}|\dfrac{|b_{n+1}|}{|a_{n}|}-\dfrac{|b_{n}|}{|a_{n}|}\leqslant|h_{0}|\dfrac{|a_{n+1}|}{|a_{n}|}-\dfrac{|a_{n-1}|}{|a_{n}|},$
(10.49)
or equivalently
$|h_{0}|\dfrac{|b_{n+1}|}{|b_{n}|}-1\leqslant\dfrac{|a_{n}|}{|b_{n}|}\left(|h_{0}|\dfrac{|a_{n+1}|}{|a_{n}|}-\dfrac{|a_{n-1}|}{|a_{n}|}\right).$
(10.50)
In the light of the inequality (10.39) and the relation
$|h_{0}|\dfrac{|a_{n+1}|}{|a_{n}|}-\dfrac{|a_{n-1}|}{|a_{n}|}\leqslant\dfrac{|h_{0}|}{{h}_{\epsilon}}-{h}_{\epsilon},$
(10.51)
one obtains the bound
$\dfrac{|b_{n+1}|}{|b_{n}|}\leqslant\dfrac{1}{|h_{0}|}\left(1+\dfrac{1}{h_{\epsilon}+|h_{0}|}\left(\dfrac{|h_{0}|}{h_{\epsilon}}-h_{\epsilon}\right)\right).$
(10.52)
Comparing this bound to the previous one (10.40) one receives
$|h_{0}|(|h_{0}|-1)\geqslant 0,$ (10.53)
what gives the condition for $h_{0}$
$|h_{0}|\in\\{0\\}\cup[1,\infty).$ (10.54)
Naturally, there is many other opportunities for selection of a form of the
effective gravitational potential $V_{eff}[h]$. However, in this section we
shall discuss the only a particular case.
#### B The Newton–Coulomb Potential
Let us consider _the residual approximation_ in which the series coefficient
of the effective gravitational potential are
$a_{n}=\left\\{\begin{array}[]{cc}a_{-1}=const&\mathrm{for}\leavevmode\nobreak\
n=-1\\\ 0&\mathrm{for}\leavevmode\nobreak\ n\neq-1\end{array}\right.,$ (10.55)
i.e. the effective gravitational potential (8.2) has the form of the
Newton–Coulomb potential
$V_{eff}=\dfrac{a_{-1}}{h-h_{0}}.$ (10.56)
In such a situation the coefficients $b_{n}$ are
$b_{n}=\left\\{\begin{array}[]{cc}b_{-1}=h_{0}a_{-1}&\mathrm{for}\leavevmode\nobreak\
n=-1\\\ b_{0}=a_{-1}&\mathrm{for}\leavevmode\nobreak\ n=0\\\
0&\mathrm{for}\leavevmode\nobreak\ n\neq-1,0\end{array}\right.,$ (10.57)
so that the Ricci scalar curvature of a three-dimensional space is
${{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}a_{-1}\left(1+\dfrac{h_{0}}{h-h_{0}}\right),$
(10.58)
whereas the Klein–Gordon equation (8.1) is
$\left(\dfrac{\delta^{2}}{\delta h^{2}}+\dfrac{a_{-1}}{h-h_{0}}\right)\Psi=0.$
(10.59)
The equation (10.64) defines some three-geometries, but even in the vacuum
situation, i.e. $\varrho=0$ and $\Lambda=0$, it is difficult to establish an
induced geometry which Ricci scalar curvature behaves like
${{}^{(3)}\\!R}\sim 1+\dfrac{h_{0}}{h-h_{0}}.$ (10.60)
Interestingly, in general the residue of the three-dimensional Ricci scalar
curvature in the point $h_{0}$ is
$\mathrm{Res}({{}^{(3)}}R,h_{0})=2\kappa\ell_{P}^{2}\mathrm{Res}(\varrho,h_{0})-6(8\pi)^{2}a_{-1}h_{0},$
(10.61)
i.e. it can be taken equal to zero if and only if the residue of energy
density of Matter fields is
$\mathrm{Res}(\varrho,h_{0})=\dfrac{3(8\pi)^{2}}{\kappa\ell_{P}^{2}}a_{-1}h_{0}.$
(10.62)
If one takes _ad hoc_ the relation
$a_{-1}=\dfrac{\Lambda}{3(8\pi)^{2}},$ (10.63)
then the Ricci scalar curvature of induced three-geometry has the form
${{}^{(3)}\\!R}=2\kappa\ell_{P}^{2}\varrho-\dfrac{2\Lambda{h_{0}}}{h-h_{0}},$
(10.64)
and its residue
$\mathrm{Res}({{}^{(3)}}R,h_{0})=2\kappa\ell_{P}^{2}\mathrm{Res}(\varrho,h_{0})-2\Lambda{h_{0}},$
(10.65)
vanishes if and only if
$\mathrm{Res}(\varrho,h_{0})=\dfrac{\Lambda}{\kappa\ell_{P}^{2}}h_{0}.$
(10.66)
Then also the geometry of an embedded three-manifold is Ricci-flat if and only
if the energy density of Matter fields has the following form
$\varrho=\dfrac{\Lambda}{\kappa\ell_{P}^{2}}\dfrac{{h_{0}}}{h-h_{0}}.$ (10.67)
Another possible Ricci-flat three-manifold is obtained for $\Lambda=0$ and
$\displaystyle\varrho=\dfrac{3(8\pi)^{2}}{\kappa\ell_{P}^{2}}a_{-1}\left(1+\dfrac{h_{0}}{h-h_{0}}\right).$
(10.68)
In general three-spaces having induced metrics satisfying the Ricci scalar
curvature (10.60) are not known yet. However, it is evidently seen that in the
particular case $h_{0}=0$, which is in full accordance with the general
condition (10.54), the situation is much more simpler, i.e.
$\displaystyle{{}^{(3)}\\!R}=2\Lambda+2\kappa\ell_{P}^{2}\varrho-6(8\pi)^{2}a_{-1},$
(10.69) $\displaystyle\left(\dfrac{\delta^{2}}{\delta
h^{2}}+\dfrac{a_{-1}}{h}\right)\Psi=0.$ (10.70)
Let us consider this particular case as the basic situation. We shall call the
global one-dimensional quantum gravity described by the system of equations
(10.69)-(10.70) _the Newton–Coulomb quantum gravity_.
As the example we shall consider vanishing of the energy density
$\varrho\equiv 0,$ (10.71)
i.e. stationarity of Matter fields. We shall call this case _the
Newton–Coulomb stationary quantum gravity_. In such a situation the Ricci
scalar curvature of three-dimensional embedded space becomes
${{}^{(3)}}R=2\Lambda-6(8\pi)^{2}a_{-1}=constant.$ (10.72)
Therefore the Ricci curvature tensor of the three-dimensional manifolds
describes the three-dimensional Einstein manifolds [597]
$R_{ij}=\lambda h_{ij},$ (10.73)
where the sign $\lambda$ of the Einstein manifolds is defined by the
Newton–Coulomb stationary quantum gravity as follows
$\dfrac{2}{3}\Lambda-2(8\pi)^{2}a_{-1}=\lambda.$ (10.74)
Because, however, energy density of Matter fields vanishes therefore the
Einstein manifolds described by the sign (10.74) possess _maximal symmetry_.
By this reason the Newton–Coulomb stationary quantum gravity geometrically
corresponds to three-dimensional _the maximally symmetric Einstein manifolds_.
In this manner in general one can consider the classification of the three-
dimensional spaces, which are maximally symmetric three-dimensional Einstein
manifolds (10.73), with respect to the value of the sign $\lambda$ (10.74) of
a manifold. The following conclusion can be deduced straightforwardly.
###### Conclusion.
The Newton–Coulomb stationary quantum gravity, defined by the effective
gravitational potential $V_{eff}[h]=\dfrac{a_{-1}}{h}$, determines the three-
dimensional embedded spaces which are the maximally symmetric three-
dimensional Einstein manifolds, characterized by the sign (10.74). There are
particular situations:
1. 1.
When the sign is non zero $\lambda\neq 0$ and the residue of the effective
gravitational potential is negative $a_{-1}=-|\alpha|$, then the effective
potential $V_{eff}[h]$ becomes the Newtonian attractive potential energy
$V_{eff}=-\dfrac{|\alpha|}{h}=-\dfrac{Gm_{1}m_{2}}{\ell_{P}h}.$ (10.75)
When the cosmological constant is positive $\Lambda=+|\Lambda|$ then the
maximally symmetric Einstein three-manifolds are characterized by positive
Ricci scalar curvature
${{}^{(3)}}R=\dfrac{2}{3}|\Lambda|+2(8\pi)^{2}|\alpha|.$ (10.76)
When the cosmological constant is negative $\Lambda=-|\Lambda|$ the maximally
symmetric Einstein three-manifolds are characterized by the Ricci scalar
curvature
${{}^{(3)}}R=-\dfrac{2}{3}|\Lambda|+2(8\pi)^{2}|\alpha|,$ (10.77)
which is negative if $|\Lambda|>3(8\pi)^{2}|\alpha|$, and positive if
$|\Lambda|<3(8\pi)^{2}|\alpha|$.
2. 2.
When the sign is non zero $\lambda\neq 0$ and the residue of the effective
gravitational potential is positive $a_{-1}=+|\alpha|$, then the effective
potential $V_{eff}[h]$ becomes the Coulomb repulsive potential energy
$V_{eff}=\dfrac{|\alpha|}{h}=\dfrac{q_{1}q_{2}}{4\pi\epsilon_{0}\ell_{P}h}.$
(10.78)
When the cosmological constant is negative $\Lambda=-|\Lambda|$ then the
maximally symmetric Einstein three-manifolds are characterized by negative
Ricci scalar curvature
${{}^{(3)}}R=-\dfrac{2}{3}|\Lambda|-2(8\pi)^{2}|\alpha|.$ (10.79)
When the cosmological constant is positive $\Lambda=+|\Lambda|$ then the
maximally symmetric Einstein three-manifolds are characterized by the Ricci
scalar curvature
${{}^{(3)}}R=\dfrac{2}{3}|\Lambda|-2(8\pi)^{2}|\alpha|,$ (10.80)
which is negative if $|\Lambda|<3(8\pi)^{2}|\alpha|$, an positive if
$|\Lambda|>3(8\pi)^{2}|\alpha|$.
3. 3.
When the sign is vanishing $\lambda=0$, i.e. the maximally symmetric Einstein
three-manifolds are Ricci-flat, one determines uniquely the value of the
reside of the effective gravitational potential as
$a_{-1}=\pm\dfrac{|\Lambda|}{3(8\pi)^{2}},$ (10.81)
In such a case one obtains the values of cosmological constant
$|\Lambda|=\left\\{\begin{array}[]{rl}\dfrac{3(8\pi)^{3}E_{P}}{4\ell_{P}^{2}}r_{g}(m_{1})r_{g}(m_{2})&\mathrm{for\leavevmode\nobreak\
the\leavevmode\nobreak\ Newton\leavevmode\nobreak\ law}\vspace*{10pt}\\\
\dfrac{3(8\pi)^{3}E_{P}}{\ell_{P}^{2}}r_{e}(q_{1})r_{e}(q_{2})&\mathrm{for\leavevmode\nobreak\
the\leavevmode\nobreak\ Coulomb\leavevmode\nobreak\ law}\end{array}\right.$
(10.82)
where $m$ is mass of a body generating Newtonian gravitational field in vacuum
and $r_{g}(m)=\dfrac{2Gm}{c^{2}}=\kappa{c^{2}}\dfrac{m}{4\pi}$ is its
gravitational radius, $q$ is charge generating Coulombic electrical field in
vacuum and
$r_{e}(q)=\dfrac{q}{\sqrt{4\pi\epsilon_{0}}}\sqrt{\dfrac{G}{c^{4}}}=\sqrt{\dfrac{\kappa}{2\epsilon_{0}}}\dfrac{q}{4\pi}$
is its electrical radius.
Note that in fact, by assuming the relation for the series coefficients
(10.24), the residue $a_{-1}$ is the Cauchy integral of the effective
potential $V_{eff}$ in the fixed point $h_{0}=0$
$a_{-1}=\textrm{Res}\left[\dfrac{1}{6(8\pi)^{2}}\dfrac{1}{h}\left(-{{}^{(3)}\\!R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho\right),h=0\right],$
(10.83)
and its value can be straightforwardly established as
$a_{-1}=\dfrac{1}{6(8\pi)^{2}}\left.\left(-{{}^{(3)}\\!R}+2\Lambda+2\kappa\ell_{P}^{2}\varrho\right)\right|_{h=0}=-\dfrac{{{}^{(3)}}R_{0}}{6(8\pi)^{2}}+\dfrac{\Lambda}{3(8\pi)^{2}}+\dfrac{\kappa\ell_{P}^{2}\varrho_{0}}{3(8\pi)^{2}}\quad,$
(10.84)
where subscript ”$0$” on the LHS means value of a quantity in $h=0$.
When one associates the effective gravitational potential
$V_{eff}=\dfrac{a_{-1}}{h}$ the Newton or the Coulomb potential energy, then
the global dimension becomes a spatial distance $r=\sqrt{x^{2}+y^{2}+z^{2}}$
$h\equiv r\quad,$ (10.85)
so that the evolution (10.59) becomes radial wave equation
$\left(\dfrac{d^{2}}{d{r^{2}}}+\dfrac{\mp|\alpha|}{r}\right)\Psi(r)=0,$
(10.86)
describing a quantum Kepler problem. There is a lot of metrics $h_{ij}$
possessing the same determinant, for instance
$h_{ij}=r^{1/3}r_{ij},$ (10.87)
where $r_{ij}$ is $SO(3)$ group rotation matrix, which can be expressed via
the Euler angles $(\theta,\varphi,\phi)$
$r_{ij}(\theta,\varphi,\phi)\equiv
r_{il}^{(3)}(\theta)r_{lk}^{(2)}(\varphi)r_{kj}^{(3)}(\phi)\quad,$ (10.88)
where $r_{ij}^{(p)}(\vartheta)$ are rotation matrices around a $p$-axis
$\displaystyle r_{ij}^{(3)}(\vartheta)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\cos\vartheta&-\sin\vartheta&0\\\
\sin\vartheta&\cos\vartheta&0\\\ 0&0&1\end{array}\right],$ (10.92)
$\displaystyle r_{ij}^{(2)}(\vartheta)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\cos\vartheta&0&\sin\vartheta\\\
0&1&0\\\ -\sin\vartheta&0&\cos\vartheta\end{array}\right].$ (10.96)
Interestingly, one can connect the radial wave equation (10.86) with the
radial Schrödinger equation for a particle equipped with energy $E$, mass $m$,
and potential energy $V$ (For more detailed discussion of the radial
Schrödinger equation see e.g. the Ref. [607]). Applying the substitution
$\Psi(r)=rR(r)$ where $R(r)$ is the radial part of the wave function of a
particle, one can present the radial Schrödinger equation in the following
form
$\left[\dfrac{d^{2}}{dr^{2}}-\left(\dfrac{2m}{\hslash^{2}}V+\dfrac{l(l+1)}{r^{2}}\right)\right]\Psi(r)=\dfrac{2mE}{\hslash^{2}}\psi(r),$
(10.97)
where $l$ is one of subscripts of the spherical harmonics
$Y_{l}^{m}(\theta,\phi)$ of degree $l$ and order $m$
$Y_{l}^{m}(\theta,\phi)=\sqrt{\dfrac{2l+1}{4\pi}\dfrac{(l-m)!}{(l+m)!}}P_{lm}(\cos\theta)e^{im\phi},$
(10.98)
which are the angular part of the wave function of a particle. Here
$P_{lm}(x)$ are associated Legendre polynomials
$P_{lm}(x)=\dfrac{1}{2^{l}l!}(1-x^{2})^{m/2}\dfrac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}.$
(10.99)
Putting _ad hoc_ the values $E=0$, $l=0$ and
$-\dfrac{2m}{\hslash^{2}}V=V_{eff}$ one receives excellent coincidence with
the equation (10.86). This nice property allows to construct the physical
interpretation of the wave functional $\Psi(r)$. Namely, it is strictly
related to the wave function $\psi(r,\theta,\phi)=R(r)Y_{0}^{m}(\theta,\phi)$
of a particle equipped with mass $m$ and having the total energy $E=0$ and the
potential energy $V=-\dfrac{\hslash^{2}}{2m}\dfrac{\pm|\alpha|}{r}$. The
identification is as follows
$\Psi(r)=r\dfrac{\psi(r,\theta,\phi)}{Y_{0}^{0}(\theta,\phi)},$ (10.100)
where $Y_{0}^{0}(\theta,\phi)=\dfrac{1}{2\sqrt{\pi}}$ is the only non trivial
value of $Y_{0}^{m}(\theta,\phi)$.
Another linkage to the radial Schrödinger equation is also possible. Let the
wave functional $\Psi(r)$ is the radial part $R(r)$ of the wave function of a
particle, i.e. $\psi(r,\theta,\phi)=\Psi(r)Y_{l}^{m}(\theta,\phi)$, equipped
with mass $m$ and total energy $E$, and moving in _arbitrary_ potential
$V(r)$. Then $\Psi(r)$ satisfies the radial Schrödinger equation
$-\dfrac{1}{r^{2}}\dfrac{d}{dr}\left(r\dfrac{d\Psi}{dr}\right)+\left(\dfrac{2m}{\hslash^{2}}V+\dfrac{l(l+1)}{r^{2}}\right)\Psi=\dfrac{2mE}{\hslash^{2}}\Psi,$
(10.101)
or equivalently
$\dfrac{d^{2}\Psi}{dr^{2}}+\dfrac{1}{r}\dfrac{d\Psi}{dr}-\left(\dfrac{2m}{\hslash^{2}}rV+\dfrac{l(l+1)}{r}\right)\Psi=-\dfrac{2mE}{\hslash^{2}}r\Psi.$
(10.102)
This equations can be separated on the system of two equations
$\displaystyle\dfrac{d^{2}\Psi}{dr^{2}}-\dfrac{2m}{\hslash^{2}}rV\Psi$
$\displaystyle=$ $\displaystyle 0,$ (10.103)
$\displaystyle\dfrac{d\Psi}{dr}-l(l+1)\Psi+\dfrac{2mE}{\hslash^{2}}r^{2}\Psi$
$\displaystyle=$ $\displaystyle 0.$ (10.104)
The equation (10.103) is exactly the wave equation (10.86) if and only if the
equation holds
$V(r)=-\dfrac{\hslash^{2}}{2mr}V_{eff}.$ (10.105)
The equation (10.104) must be solved. It is easy to see that this equation can
be integrated straightforwardly
$\int_{\Psi(r_{0})}^{\Psi(r)}\dfrac{d\Psi}{\Psi}=\int_{r_{0}}^{r}\left(l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right)dr,$
(10.106)
where $r_{0}$ is some reference value of $r$, what gives
$\ln\left|\dfrac{\Psi(r)}{\Psi(r_{0})}\right|=l(l+1)(r-r_{0})-\dfrac{2mE}{\hslash^{2}}\dfrac{r^{3}-r_{0}^{3}}{3},$
(10.107)
and results in the solution
$\Psi(r)=\Psi(r_{0})\exp\left\\{-l(l+1)r_{0}+\dfrac{2mE}{3\hslash^{2}}r_{0}^{3}\right\\}\exp\left\\{l(l+1)r-\dfrac{2mE}{3\hslash^{2}}r^{3}\right\\}.$
(10.108)
Now one can differentiate the equation (10.104) with respect to $r$ and
obtains
$\displaystyle\dfrac{d^{2}\Psi}{dr^{2}}$ $\displaystyle=$ $\displaystyle
l(l+1)\dfrac{d\Psi}{dr}-\dfrac{2mE}{\hslash^{2}}\left(2r\Psi+r^{2}\dfrac{d\Psi}{dr}\right)=$
(10.109) $\displaystyle=$
$\displaystyle\left[l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right]\dfrac{d\Psi}{dr}-\dfrac{4mE}{\hslash^{2}}r\Psi=$
$\displaystyle=$
$\displaystyle\left\\{\left[l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right]^{2}-\dfrac{4mE}{\hslash^{2}}r\right\\}\Psi.$
Application of the equation (10.103) to the result (10.109) leads to
$\dfrac{2m}{\hslash^{2}}rV=\left\\{\left[l(l+1)-\dfrac{2mE}{\hslash^{2}}r^{2}\right]^{2}-\dfrac{4mE}{\hslash^{2}}r\right\\},$
(10.110)
what allows to establish the potential $V$
$V(r)=\dfrac{\hslash^{2}}{2m}\dfrac{l^{2}(l+1)^{2}}{r}-2E-2El(l+1)r+\dfrac{2mE^{2}}{\hslash^{2}}r^{3}.$
(10.111)
Using of the identification (10.105) and the explicit form of
$V_{eff}=\dfrac{\pm|\alpha|}{r}$, one receives the equation for $r$
$\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}r^{5}-\dfrac{4mE}{\hslash^{2}}l(l+1)r^{3}-\dfrac{4mE}{\hslash^{2}}r^{2}+l^{2}(l+1)^{2}r\pm|\alpha|=0,$
(10.112)
which is very difficult to solve, but simplifies for suggested value $l=0$
$\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}r^{5}-\dfrac{4mE}{\hslash^{2}}r^{2}\pm|\alpha|=0.$
(10.113)
The equation (10.113) can be solved easy. Let us focus on the real solution
which for the Coulombic case is
$\displaystyle r$ $\displaystyle=$
$\displaystyle\Bigg{[}\dfrac{\hslash^{2}}{2mE}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}-1\right)^{1/3}+$
(10.114) $\displaystyle+$
$\displaystyle\dfrac{2}{3}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{-1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}-1\right)^{-1/3}\Bigg{]}^{1/2},$
while for the Newtonian case one obtains
$\displaystyle r$ $\displaystyle=$
$\displaystyle\Bigg{[}\dfrac{\hslash^{2}}{2mE}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}+1\right)^{1/3}+$
(10.115) $\displaystyle+$
$\displaystyle\dfrac{2}{3}\left(\dfrac{mE}{\hslash^{2}}|\alpha|\right)^{-1/3}\left(\sqrt{1-\dfrac{32}{27}\dfrac{2mE}{\hslash^{2}|\alpha|^{2}}}+1\right)^{-1/3}\Bigg{]}^{1/2},$
Recall that in our theory $h=r$, i.e. in fact the results (10.114)-(10.115)
fix value of $h$. The distance must be real number. For this must be
$E\leqslant\dfrac{27}{32}\dfrac{\hslash^{2}}{2m}|\alpha|^{2}.$ (10.116)
In general solutions of the algebraic equation (10.112) can be found by
methods of the Galois group. Despite the real and positive solution is
difficult to extract, some simpler solution can be constructed. Let us
substitute to the equation (10.112) as the unknown $r=x+iy$, where the
imaginary part $y$ will be $y\rightarrow 0$ in a certain stage of the
construction. In such a situation the equation (10.112) is equivalent to the
statement that its both real and imaginary part vanish. Such a method
generates the system of equations
$\displaystyle\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}y^{4}+2\left(\dfrac{2mE}{\hslash^{2}}\right)\left(l(l+1)-5\left(\dfrac{2mE}{\hslash^{2}}\right)x^{2}\right)y^{2}-$
$\displaystyle-\left(\dfrac{2mE}{\hslash^{2}}\right)x\left(4+6l(l+1)x-5\left(\dfrac{2mE}{\hslash^{2}}\right)x^{3}\right)+l^{2}(l+1)^{2}=0,$
(10.117) $\displaystyle
5\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}xy^{4}+2\left(\dfrac{2mE}{\hslash^{2}}\right)\left(1+3l(l+1)x-5\left(\dfrac{2mE}{\hslash^{2}}\right)x^{3}\right)y^{2}+$
$\displaystyle+x\left(-2\left(\dfrac{2mE}{\hslash^{2}}\right)x+\left(l(l+1)-\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}\right)^{2}\right)\pm|\alpha|=0,$
(10.118)
where we introduced the shortened notation $L=l(l+1)$. Now one can put $y=0$,
so that $x=r$, and obtain the system of equations
$\displaystyle-\left(\dfrac{2mE}{\hslash^{2}}\right)r\left(4+6l(l+1)r-5\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)+l^{2}(l+1)^{2}=0,$
(10.119) $\displaystyle
l^{2}(l+1)^{2}r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{2}\left(2+2l(l+1)r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)\pm|\alpha|=0,$
(10.120)
which can be presented in the form
$\displaystyle\left(\dfrac{2mE}{\hslash^{2}}\right)r^{2}$ $\displaystyle=$
$\displaystyle\dfrac{l^{2}(l+1)^{2}r}{\left(4+6l(l+1)r-5\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)},$
(10.121) $\displaystyle\left(\dfrac{2mE}{\hslash^{2}}\right)r^{2}$
$\displaystyle=$
$\displaystyle\dfrac{l^{2}(l+1)^{2}r\pm|\alpha|}{\left(2+2l(l+1)r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)},$
(10.122)
and lead to the relation
$\dfrac{l^{2}(l+1)^{2}r}{\left(4+6l(l+1)r-5\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)}=\dfrac{l^{2}(l+1)^{2}r\pm|\alpha|}{\left(2+2l^{2}(l+1)^{2}r-\left(\dfrac{2mE}{\hslash^{2}}\right)r^{3}\right)}.$
(10.123)
The relation (10.123), however, can be rewritten in much more convenient form
of the 4th order algebraic equation
$\displaystyle 4\left(\dfrac{2mE}{\hslash^{2}}\right)l^{2}(l+1)^{2}r^{4}\pm
5\left(\dfrac{2mE}{\hslash^{2}}\right)|\alpha|r^{3}-4l^{3}(l+1)^{3}r^{2}-$
$\displaystyle 2l(l+1)\left(l(l+1)\pm 3|\alpha|\right)r-(\pm 4|\alpha|)=0.$
(10.124)
In other words the 5th order algebraic equation very difficult to solve has
been reduced to the 4th order algebraic equation which in general can solved
straightforwardly. In our interest is the real and positive solution of the
equation (B) which is given by the formula
$\displaystyle r=-\dfrac{\pm
5|\alpha|}{16l^{2}(l+1)^{2}}+\dfrac{1}{2}\Bigg{(}\dfrac{1}{2^{2/3}3A_{l}}\left(45\dfrac{|\alpha|^{2}}{l(l+1)}-81(\pm|\alpha|)+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}\right)+$
$\displaystyle\dfrac{25|\alpha|^{2}}{64l^{4}(l+1)^{4}}+\dfrac{\hslash^{2}l(l+1)}{3mE}+\dfrac{\hslash^{2}A_{l}}{{2}^{10/3}3mEl^{2}(l+1)^{2}}\Bigg{)}^{1/2}+\dfrac{1}{2}\Bigg{(}\dfrac{25|\alpha|^{2}}{32l^{4}(l+1)^{4}}+$
$\displaystyle\dfrac{2\hslash^{2}l(l+1)}{3mE}-\dfrac{1}{2^{2/3}3A_{l}}\left(45\dfrac{|\alpha|^{2}}{l(l+1)}-81(\pm|\alpha|)+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}\right)-$
$\displaystyle\dfrac{\hslash^{2}A_{l}}{2^{10/3}3mEl^{2}(l+1)^{2}}+\dfrac{1}{4l(l+1)}\left(\dfrac{2\hslash^{2}l(l+1)}{mE}-\dfrac{\pm
125|\alpha|^{3}}{64l^{5}(l+1)^{5}}+\dfrac{\pm
7\hslash^{2}|\alpha|}{2mE}\right)\times$
$\displaystyle\Bigg{(}\dfrac{25|\alpha|^{2}}{64l^{4}(l+1)^{4}}+\dfrac{\hslash^{2}l(l+1)}{3mE}+\dfrac{\hslash^{2}A_{l}}{{2}^{10/3}3mEl^{2}(l+1)^{2}}+$
$\displaystyle\dfrac{1}{2^{2/3}3A_{l}}\left(45\dfrac{|\alpha|^{2}}{l(l+1)}-81(\pm|\alpha|)+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}\right)\Bigg{)}^{-1/2}\Bigg{)}^{1/2},$
(10.125)
where
$\displaystyle A_{l}=\dfrac{2mE}{\hslash^{2}}\Bigg{(}-\dfrac{\pm
675\hslash^{2}|\alpha|^{3}}{2mE}+\dfrac{\hslash^{4}l^{6}(l+1)^{6}}{2(mE)^{2}}\bigg{(}\dfrac{351|\alpha|^{2}}{l(l+1)}-\dfrac{\pm
297|\alpha|}{l(l+1)}-$
$\displaystyle\dfrac{8\hslash^{2}l^{3}(l+1)^{3}}{mE}+54\bigg{)}+\dfrac{\hslash^{2}}{2mE}\Bigg{(}-\dfrac{\hslash^{2}l^{6}(l+1)^{6}}{mE}\bigg{(}\dfrac{45|\alpha|^{2}}{l(l+1)}+\dfrac{4\hslash^{2}l^{4}(l+1)^{4}}{mE}-$
$\displaystyle 81(\pm|\alpha|)\bigg{)}^{3}+\bigg{(}\pm
675|\alpha|^{3}-\dfrac{\hslash^{2}l^{8}(l+1)^{8}}{mE}\bigg{(}\dfrac{351|\alpha|^{2}}{l(l+1)}-\dfrac{\pm
297|\alpha|}{l(l+1)}-$
$\displaystyle\dfrac{8\hslash^{2}l^{3}(l+1)^{3}}{mE}\bigg{)}\bigg{)}^{2}+54\Bigg{)}^{1/2}\Bigg{)}^{1/3}.$
(10.126)
Another separation of the radial Schrödinger equation (10.102) is possible.
Namely, one can rewrite this equation as the system
$\displaystyle\dfrac{d^{2}\Psi}{dr^{2}}-\left(\dfrac{2m}{\hslash^{2}}rV+\dfrac{l(l+1)}{r}\right)\Psi$
$\displaystyle=$ $\displaystyle 0,$ (10.127)
$\displaystyle\dfrac{d\Psi}{dr}+\dfrac{2mE}{\hslash^{2}}r^{2}\Psi$
$\displaystyle=$ $\displaystyle 0.$ (10.128)
The procedure analogous to the previous separation gives the solution
$\Psi(r)=\Psi(r_{0})\exp\left\\{\dfrac{2mE}{3\hslash^{2}}r_{0}^{3}\right\\}\exp\left\\{-\dfrac{2mE}{3\hslash^{2}}r^{3}\right\\},$
(10.129)
the potential
$V(r)=-\dfrac{\hslash^{2}}{2m}\dfrac{l(l+1)}{r^{2}}-2E+\dfrac{2mE^{2}}{\hslash^{2}}r^{3},$
(10.130)
and the equation for $r$
$\left(\dfrac{2mE}{\hslash^{2}}\right)^{2}r^{4}-\dfrac{4mE}{\hslash^{2}}r-\dfrac{\hslash^{2}}{2m}l^{2}(l+1)^{2}\pm|\alpha|=0.$
(10.131)
This is equation can be solved for arbitrary $l$ with no problems. There are
real and positive solutions if and only if
$\pm|\alpha|\leqslant\dfrac{\hslash^{2}}{2m}l^{2}(l+1)^{2}.$ (10.132)
Then there is the only one real and positive solution given by
$\displaystyle
r=\dfrac{1}{2}\Bigg{\\{}\dfrac{2\hslash^{2}}{mE}\Bigg{[}\dfrac{2^{-1/3}}{(mE/\hslash^{2})^{2/3}}\beta_{l}^{1/3}+\dfrac{2^{2/3}\alpha_{l}}{3(mE/\hslash^{2})^{4/3}}\beta_{l}^{-1/3}\Bigg{]}^{-1/2}$
$\displaystyle-\dfrac{2^{-1/3}}{(mE/\hslash^{2})^{2/3}}\beta_{l}^{1/3}-\dfrac{2^{2/3}\alpha_{l}}{3(mE/\hslash^{2})^{4/3}}\beta_{l}^{-1/3}\Bigg{\\}}^{1/2}$
$\displaystyle+\dfrac{1}{2}\left(\dfrac{2^{-1/3}}{(mE/\hslash^{2})^{2/3}}\beta_{l}^{1/3}+\dfrac{2^{2/3}\alpha_{l}}{3(mE/\hslash^{2})^{4/3}}\beta_{l}^{-1/3}\right)^{1/2}$
(10.133)
where we have introduced the shortened notation
$\displaystyle\alpha_{l}$ $\displaystyle=$
$\displaystyle-\dfrac{\hslash^{2}}{2m}l^{2}(l+1)^{2}\pm|\alpha|,$ (10.134)
$\displaystyle\beta_{l}$ $\displaystyle=$ $\displaystyle
1+\sqrt{1-\dfrac{4}{27}\dfrac{\hslash^{4}}{m^{2}E^{2}}\alpha_{l}^{3}}.$
(10.135)
It is easy to deduce the solution for $l=0$.
#### C Boundary Conditions for The Wave Functionals
In this subsection we shall consider certain solutions of the global one-
dimensional quantum gravity (8.1) for the approximation of the effective
gravitational potential $V_{eff}$ discussed in the previous subsection. For
the considered situation the $h$-evolution
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}\mp\dfrac{|\alpha|}{h}\right)\Psi^{\mp}[h]=0,$
(10.136)
is solved by two type of wave functions $\Psi^{\mp}$ where the attractive wave
functions $\Psi_{G}^{-}[h]$ are associated with the the Newton-like effective
gravitational potential, and the repulsive ones $\Psi^{+}[h]$ are associated
with the Coulomb-like effective gravitational potential. Because of manifest
one-dimensionality of the functional evolutionary equation (10.136) one can
solve this equation in frames of the theory of ordinary differential equations
by treatment of the functional derivative as the ordinary one, i.e.
$\dfrac{\delta}{\delta h}=\dfrac{d}{dh}$, and the functional and a function
$\Psi[h]=\Psi(h)$ with no loss of the generality.
In this manner we shall consider here the equation
$\left(\dfrac{d^{2}}{dh^{2}}\mp\dfrac{|\alpha|}{h}\right)\Psi^{\mp}(h)=0.$
(10.137)
The general solution of this differential equation can be constructed
straightforwardly by application of the Bessel functions $J_{n}$ and $Y_{n}$
for the case of the attractive potential
$\Psi^{-}[h]=\sqrt{|\alpha|h}\left[C_{1}^{-}J_{1}\left(2\sqrt{|\alpha|h}\right)+2iC_{2}^{-}Y_{1}\left(2\sqrt{|\alpha|h}\right)\right],$
(10.138)
and in terms of the modified Bessel $I_{n}$ and $K_{n}$ for the case of the
repulsive potential
$\Psi^{+}[h]=-\sqrt{|\alpha|h}\left[C_{1}^{+}I_{1}\left(2\sqrt{|\alpha|h}\right)+2C_{2}^{+}K_{1}\left(2\sqrt{|\alpha|h}\right)\right],$
(10.139)
where $C_{1}^{\pm}$ and $C_{2}^{\pm}$ are constants of integration. In
concrete calculations one can take the standard definitions of the Bessel
functions of first and second kind [602], $J_{\alpha}(x)$ and $Y_{\alpha}(x)$,
which are
$\displaystyle J_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}dt\cos\left(x\cos t-\alpha
t\right),$ (10.140) $\displaystyle Y_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\dfrac{J_{\alpha}(x)\cos\left(\alpha\pi\right)-J_{-\alpha}(x)}{\sin\left(\alpha\pi\right)},$
(10.141)
as well as the modified Bessel functions of the first and the second kind,
$I_{\alpha}(x)$ and $K_{\alpha}(x)$, which are
$\displaystyle I_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}dt\exp\left(x\cos
t\right)\cos\left(\alpha t\right),$ (10.142) $\displaystyle K_{\alpha}(x)$
$\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\dfrac{I_{-\alpha}(x)-I_{\alpha}(x)}{\sin\left(\alpha\pi\right)}.$
(10.143)
Recall that standardly the values of the Bessel functions of second kind and
the modified Bessel functions of second kind for any integers $n$ can be
received by application of the limiting procedure
$\displaystyle Y_{n}(x)$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow n}Y_{\alpha}(x),$ (10.144)
$\displaystyle{K}_{n}(x)$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow n}K_{\alpha}(x).$ (10.145)
In further part of this section we shall present solutions to the one-
dimensional quantum mechanics (10.137) with respect to several selected
boundary conditions for the general solutions (10.138) and (10.139).
###### Boundary Conditions I
Let us consider the global one-dimensional quantum mechanics (8.1) with the
boundary conditions for some selected initial value of the dimension
$h=h_{I}$:
$\displaystyle\Psi[h_{I}]$ $\displaystyle=$ $\displaystyle\Psi_{I},$ (10.146)
$\displaystyle\dfrac{\delta\Psi}{\delta h}[h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{\prime}_{I}.$ (10.147)
For construction of the solution one can use the regularized hypergeometric
functions ${{}_{p}}\tilde{F}_{q}$
${{}_{p}}\tilde{F}_{q}\left(\begin{array}[]{c}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{array};x\right)=\dfrac{{{}_{p}}F_{q}\left(\begin{array}[]{c}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{array};x\right)}{\Gamma(b_{1})\ldots\Gamma(b_{q})},$
(10.148)
where ${{}_{p}}F_{q}$ is the confluent hypergeometric function
${{}_{p}}F_{q}\left(\begin{array}[]{c}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{array};x\right)=\sum_{r=0}^{\infty}\dfrac{(a_{1})_{r}\ldots(a_{p})_{r}}{(b_{1})_{r}\ldots(b_{q})_{r}}\dfrac{x^{r}}{r!},$
(10.149)
where $(a)_{r}=\dfrac{\Gamma(a+r)}{\Gamma(a)}$ are Pochhammer symbols. The
general solutions (10.138) and (10.139) with respect to the boundary
conditions (10.146)-(10.147) can be written out in the form
$\Psi^{-}=C^{-}_{1}\left(2\sqrt{{|\alpha|h}}\right)K_{1}\left(2\sqrt{{|\alpha|h}}\right)+C^{-}_{2}\left(2\sqrt{{|\alpha|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h\right),$ (10.150)
where constants of integration are
$\displaystyle C^{-}_{1}$ $\displaystyle=$
$\displaystyle\Psi_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
1\end{array};|\alpha|h_{I}\right)-\Psi^{\prime}_{I}h_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right),$ (10.155) $\displaystyle C^{-}_{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\Psi_{I}K_{0}\left(2\sqrt{{|\alpha|h_{I}}}\right)+\Psi^{\prime}_{I}\sqrt{{\dfrac{h_{I}}{|\alpha|}}}K_{1}\left(2\sqrt{{|\alpha|h_{I}}}\right)\right),$
(10.156)
for the Newton-like case, and
$\Psi^{+}=C^{+}_{1}\left(2\sqrt{{|\alpha|h}}\right)Y_{1}\left(2\sqrt{{|\alpha|h}}\right)+C^{+}_{2}\left(2\sqrt{{|\alpha|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h\right),$ (10.157)
where constants of integration are
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C^{+}_{1}$ $\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\left(\Psi^{\prime}_{I}h_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right)-\Psi_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
1\end{array};-|\alpha|h_{I}\right)\right),$ (10.162)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C^{+}_{2}$ $\displaystyle=$
$\displaystyle\dfrac{\pi}{2}\left(\Psi_{I}Y_{0}\left(2\sqrt{{|\alpha|h_{I}}}\right)-\Psi^{\prime}_{I}\sqrt{{\dfrac{h_{I}}{|\alpha|}}}Y_{1}\left(2\sqrt{{|\alpha|h_{I}}}\right)\right),$
(10.163)
for the Coulomb-like case.
###### Boundary Conditions II
The second case which we want to present in this paper, are the boundary
conditions for 1st and 2nd functional derivatives
$\displaystyle\dfrac{\delta\Psi}{\delta h}[h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{\prime}_{I},$ (10.164)
$\displaystyle\dfrac{\delta^{2}\Psi}{\delta h^{2}}[h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{\prime\prime}_{I}.$ (10.165)
By using of the hypergeometric functions, one can express the solution for
attractive case as follows
$\Psi^{-}=C^{-}_{1}\left(2\sqrt{{|\alpha|h}}\right)K_{1}\left(2\sqrt{{|\alpha|h}}\right)+C^{-}_{2}\left(2\sqrt{{|\alpha|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h\right),$ (10.166)
where $C^{-}_{1}$ and $C^{-}_{2}$ are constants defined as
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C^{-}_{1}$
$\displaystyle=$ $\displaystyle-
h_{I}\left(\Psi^{\prime}_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right)-\dfrac{\Psi^{\prime\prime}_{I}}{|\alpha|}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
1\end{array};|\alpha|h_{I}\right)\right),$ (10.171)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C^{-}_{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\sqrt{\dfrac{h_{I}}{|\alpha|}}\left(\Psi^{\prime\prime}_{I}\sqrt{\dfrac{h_{I}}{|\alpha|}}K_{0}\left(2\sqrt{{|\alpha|h_{I}}}\right)+\Psi^{\prime}_{I}K_{1}\left(2\sqrt{{|\alpha|h_{I}}}\right)\right).$
(10.172)
Similarly for the repulsive case one obtains easily
$\Psi^{+}=C^{+}_{1}\left(2\sqrt{{|\alpha|h}}\right)Y_{1}\left(2\sqrt{{|\alpha|h}}\right)+C^{+}_{2}\left(2\sqrt{{|\alpha|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h\right),$ (10.173)
where the constants of integration are
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C^{+}_{1}$
$\displaystyle=$ $\displaystyle\dfrac{\pi
h_{I}}{2}\left(\Psi^{\prime}_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right)+\dfrac{\Psi^{\prime\prime}_{I}}{|\alpha|}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
1\end{array};-|\alpha|h_{I}\right)\right),$ (10.178)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!C^{+}_{2}$
$\displaystyle=$
$\displaystyle\dfrac{\pi}{4}\sqrt{\dfrac{h_{I}}{|\alpha|}}\left(\Psi^{\prime\prime}_{I}\sqrt{\dfrac{h_{I}}{|\alpha|}}Y_{0}\left(2\sqrt{{|\alpha|h_{I}}}\right)+\Psi^{\prime}_{I}Y_{1}\left(2\sqrt{{|\alpha|h_{I}}}\right)\right).$
(10.179)
###### Boundary Conditions III
The third possible choice of the boundary conditions for the considered
problem has the following form
$\displaystyle\Psi[h_{I}]$ $\displaystyle=$ $\displaystyle\Psi_{I},$ (10.180)
$\displaystyle\dfrac{\delta^{2}\Psi}{\delta h^{2}}[h_{I}]$ $\displaystyle=$
$\displaystyle\Psi^{\prime\prime}_{I}.$ (10.181)
These conditions are formally improper for the problem, because of they lead
to manifestly singular solutions. In such a situation, however, one can
present the solutions in the form in which constants of integration are
formally singular. For the case of the attractive Newton-like potential one
has
$\Psi_{G}^{-}=C^{-}_{1}\left(2\sqrt{|\alpha|h}\right)K_{1}\left(2\sqrt{\left|\alpha\right|h}\right)+C^{-}_{2}\left(2\sqrt{|\alpha|h}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h\right),$ (10.182)
where constants of integration are ($\epsilon\rightarrow 0$)
$\displaystyle C^{-}_{1}$ $\displaystyle=$
$\displaystyle\dfrac{2}{\epsilon}\sqrt{|\alpha|h_{I}}\left(\Psi_{I}-\dfrac{h_{I}}{\left|\alpha\right|}\Psi^{\prime\prime}_{I}\right){{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right),$ (10.185) $\displaystyle C^{-}_{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\epsilon}\left(\Psi_{I}-\dfrac{h_{I}}{|\alpha|}\Psi^{\prime\prime}_{I}\right)K_{1}\left(2\sqrt{|\alpha|h_{I}}\right).$
(10.186)
Similarly for case of the repulsive Coulomb-like potential one obtains
$\Psi_{G}^{+}=C^{+}_{1}\left(2\sqrt{|\alpha|h}\right)Y_{1}\left(2\sqrt{\left|\alpha\right|h}\right)+C^{+}_{2}\left(2\sqrt{|\alpha|h}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h\right),$ (10.187)
where constants of integration are ($\epsilon\rightarrow 0$)
$\displaystyle C^{+}_{1}$ $\displaystyle=$
$\displaystyle\dfrac{2}{\epsilon}\sqrt{|\alpha|h_{I}}\left(\Psi_{I}+\dfrac{h_{I}}{\left|\alpha\right|}\Psi^{\prime\prime}_{I}\right){{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right),$ (10.190) $\displaystyle C^{+}_{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\epsilon}\left(\Psi_{I}+\dfrac{h_{I}}{|\alpha|}\Psi^{\prime\prime}_{I}\right)Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right).$
(10.191)
There is the question how to regularize these solutions to obtain non-singular
constants of integration. There is no general method for such an productive
procedure. Let us propose some constructive regularization. For regularization
of the solution let us take into account the following _the ansatz for
boundary conditions_
$\pm\dfrac{h_{I}}{\left|\alpha\right|}{\Psi^{\pm}_{I}}^{\prime\prime}+\Psi^{\pm}_{I}\equiv\epsilon
f_{\pm}[h_{I},|\alpha|],$ (10.192)
where $f_{\pm}[h_{I},|\alpha|]\neq 0$ are some nonsingular functionals of
$h_{I}$ and $|\alpha|$, which are presently unknown and arbitrary. For formal
correctness and consistency of the method we shall put the limiting procedure
$\epsilon\rightarrow 0$ in some step of the regularization process. The sign
$+$ is related to the Newton-like case, and the sign $-$ to the Coulomb-like
case. It can be seen by straightforward calculation that in such a situation
the singularity of the solutions (10.182) and (10.187) can be removed. We are
going to show now that the ansatz (10.192) allows to express the problem of
choice of the initial data via integral equations for the functionals
$f_{\pm}[h_{I},|\alpha|]$. For basics and applications of the theory of
integral equations we suggest to see e.g. the books in the Ref. [608].
For the attractive Newton-like case the initial value $\Psi_{I}$ can be
obtained as follows
$\displaystyle\Psi^{-}_{I}=-|\alpha|h_{I}\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right)\left[c^{-}_{1}+2\epsilon\sqrt{|\alpha|}\int_{1}^{h_{I}}\dfrac{dt}{\sqrt{t}}f_{-}[t,|\alpha|]K_{1}\left(2\sqrt{|\alpha|t}\right)\right]$
(10.195)
$\displaystyle+\,2\sqrt{|\alpha|h_{I}}K_{1}\left(2\sqrt{|\alpha|h_{I}}\right)\left[c^{-}_{2}+\epsilon|\alpha|\int_{1}^{h_{I}}dtf_{-}[t,|\alpha|]\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|t\right)\right],$ (10.198) (10.199)
and similarly for the repulsive Coulomb-like case one receives
$\displaystyle\Psi^{+}_{I}=|\alpha|h_{I}\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right)\left[c^{+}_{1}-\epsilon\pi\sqrt{|\alpha|}\int_{1}^{h_{I}}\dfrac{dt}{\sqrt{t}}f_{+}[t,|\alpha|]Y_{1}\left(2\sqrt{|\alpha|t}\right)\right]$
(10.202)
$\displaystyle+\,2i\sqrt{|\alpha|h_{I}}Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right)\left[c^{+}_{2}-\epsilon\dfrac{i\pi}{2}|\alpha|\int_{1}^{h_{I}}dtf_{+}[t,|\alpha|]\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|t\right)\right],$ (10.205) (10.206)
where $c^{\pm}_{1,2}$ are now non-singular constants of integration. The
functionals $f_{\pm}[h_{I},|\alpha|]$ can be established by straightforward
application of the ansatz (10.192) within the general solutions (10.182) and
(10.187). Such an application yields the following results
$\displaystyle\Psi_{I}^{-}$ $\displaystyle=$ $\displaystyle
8|\alpha|h_{I}K_{1}\left(2\sqrt{|\alpha|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right)f_{-}[h_{I},|\alpha|],$ (10.209)
$\displaystyle\Psi_{I}^{+}$ $\displaystyle=$ $\displaystyle
8|\alpha|h_{I}Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right)f_{+}[h_{I},|\alpha|].$ (10.212)
Employing straightforwardly these results within the equations (10.199) and
(10.206) one obtains the integral equations for the functionals $f_{\pm}$. For
the Coulomb-like situation one receives the following Volterra integral
equation of the second kind
$f_{-}[h_{I},|\alpha|]=g^{-}[h_{I},|\alpha|]+\epsilon\int_{1}^{h_{I}}dt\mathcal{K}^{-}(t,|\alpha|,h_{I})f_{-}[t,|\alpha|],$
(10.213)
where the function $g^{-}[h_{I},|\alpha|]$ causing the non-homogeneity is
$g^{-}[h_{I},|\alpha|]=-\dfrac{c^{-}_{1}}{8K_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}+\dfrac{c^{-}_{2}}{2\left(2\sqrt{|\alpha|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right)},$ (10.214)
and the kernel $\mathcal{K}^{-}(t,|\alpha|,h_{I})$ has the form
$\displaystyle\mathcal{K}^{-}(t,|\alpha|,h_{I})$ $\displaystyle=$
$\displaystyle\dfrac{|\alpha|}{\sqrt{t}}\Bigg{[}\dfrac{\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|t\right)}{2\left(2\sqrt{|\alpha|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};|\alpha|h_{I}\right)}-$ (10.219) $\displaystyle-$
$\displaystyle\dfrac{\sqrt{h_{I}}K_{1}\left(2\sqrt{|\alpha|t}\right)}{2\left(2\sqrt{|\alpha|h_{I}}\right)K_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}\Bigg{]}.$
(10.220)
Similar procedure can be performed for the Newton-like situation. In this case
the Volterra integral equation of the second kind is
$f_{+}[h_{I},|\alpha|]=g^{+}[h_{I},|\alpha|]+\epsilon\int_{1}^{h_{I}}dt\mathcal{K}^{+}(t,|\alpha|,h_{I})f_{+}[t,|\alpha|]$
(10.221)
where the function $g^{-}[h_{I},|\alpha|]$ causing the non-homogeneity is
$g^{+}[h_{I},|\alpha|]=\dfrac{c^{+}_{1}}{8Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}+\dfrac{ic^{+}_{2}}{2\left(2\sqrt{|\alpha|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right)},$ (10.222)
and the kernel $\mathcal{K}^{+}(t,|\alpha|,h_{I})$ has the form
$\displaystyle\mathcal{K}^{+}(t,|\alpha|,h_{I})$ $\displaystyle=$
$\displaystyle\dfrac{\pi|\alpha|}{2\sqrt{t}}\Bigg{[}-\dfrac{\sqrt{h_{I}}Y_{1}\left(2\sqrt{|\alpha|t}\right)}{2\left(2\sqrt{|\alpha|h_{I}}\right)Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}-$
(10.227) $\displaystyle-$
$\displaystyle\dfrac{\,i\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|t\right)}{2\left(2\sqrt{|\alpha|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\
2\end{array};-|\alpha|h_{I}\right)}\Bigg{]}.$
The number $\epsilon$ plays the role analogous to the eigenvalue in linear
algebra. It is evidently seen that the integral operators acting on the
functionals $f_{\pm}$ are non-singular when the limiting procedure
$\epsilon\rightarrow 0$ is performed. Such a property guarantees stability and
effectiveness of the regularization method given by the ansatz (10.192). By
this reason one can perform the limit $\epsilon\rightarrow 0$
straightforwardly within the Volterra integral equations of the second kind
(10.213) and (10.221), and extract the unknown functionals
$f_{\pm}[h_{I},|\alpha|]=g^{\pm}[h_{I},|\alpha|].$ (10.228)
Taking into account elementary properties of the hypergeometric function the
final resultscan be presented in the following form
$\displaystyle f_{-}[h_{I},|\alpha|]$ $\displaystyle=$
$\displaystyle\dfrac{-c^{-}_{1}}{8K_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}+\dfrac{c^{-}_{2}}{4I_{1}\left(2\sqrt{|\alpha|h_{I}}\right)},$
(10.229) $\displaystyle f_{+}[h_{I},|\alpha|]$ $\displaystyle=$
$\displaystyle\dfrac{c^{+}_{1}}{8Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}+\dfrac{ic^{+}_{2}}{4J_{1}\left(2\sqrt{|\alpha|h_{I}}\right)}.$
(10.230)
In this manner the conditions for the initial data in the considered situation
given by the improper boundary conditions (10.180)-(10.181) and in the light
to the ansatz (10.192) can not be chosen arbitrary, but according to the
following selection rules
$\displaystyle\Psi_{I}^{-}$ $\displaystyle=$
$\displaystyle\sqrt{|\alpha|h_{I}}\left[-c^{-}_{1}I_{1}\left(2\sqrt{|\alpha|h_{I}}\right)+2c^{-}_{2}K_{1}\left(2\sqrt{|\alpha|h_{I}}\right)\right],$
(10.231) $\displaystyle\Psi_{I}^{+}$ $\displaystyle=$
$\displaystyle\sqrt{|\alpha|h_{I}}\left[c^{+}_{1}J_{1}\left(2\sqrt{|\alpha|h_{I}}\right)+2ic^{+}_{2}Y_{1}\left(2\sqrt{|\alpha|h_{I}}\right)\right].$
(10.232)
It must be emphasized that the proposed ansatz for the boundary conditions
(10.192) is not unique, and can be replaced by another proposal. However, as
we have mentioned earlier, this type of regularization method assures
stability and effectiveness of the limiting procedure $\epsilon\rightarrow 0$,
and therefore in general guarantees consistency of the method. The ansatz
(10.192) allowed to formulate the regularization in terms of the Volterra
integral equation of the second kind which, however, was not solved because of
the limit $\epsilon\rightarrow 0$ was performed. Albeit, it must be emphasized
also that according to the theory of integral equations the obtained Volterra
integral equations of the second kind should be solved for arbitrary
$\epsilon$ and then the limit $\epsilon\rightarrow 0$ should be performed.
Such a solution can be constructed straightforwardly by application of the
Neumann series called also the Born series. If one rewrites the Volterra
integral equations (10.213) and (10.221) in more convenient symbolic form
$f_{\pm}[h_{I},|\alpha|]=g^{\pm}[h_{I},|\alpha|]+\epsilon\textsf{K}^{\pm}f_{\pm}[t,|\alpha|],$
(10.233)
where the integral operators $\textsf{K}^{\pm}$ are defined as
$\textsf{K}^{\pm}=\int_{1}^{h_{I}}dt\mathcal{K}^{\pm}(t,|\alpha|,h_{I}),$
(10.234)
then the solution can be written out straightforwardly via using of the
Neumann series
$f_{\pm}[h_{I},|\alpha|]=g^{\pm}[h_{I},|\alpha|]+\sum_{n=1}^{\infty}\epsilon^{n}{\textsf{K}^{\pm}}^{n}g^{\pm}[h_{I},|\alpha|].$
(10.235)
It is easy to see now that in the limit $\epsilon\rightarrow 0$ one obtains
the result $f_{\pm}[h_{I},|\alpha|]=g^{\pm}[h_{I},|\alpha|]$ received earlier.
The problem is, however, convergence of the series as well as definiteness of
the integral operators $\textsf{K}^{\pm}$ because of in the presented
situation the integral kernels $\textsf{K}^{\pm}(t,|\alpha|,h_{I})$ are
special functions. The presented method in general reflects the typical
problems arising from application of the improper boundary conditions.
### Chapter 11 _Ab Initio_ Thermodynamics of Space Quanta Æther
The quantum field theory formulated in terms of the static Fock repère is the
most natural approach to formulation of the state of thermodynamic
equilibrium. The effectiveness of such a method is emphasized by the fact that
despite that the system is manifestly nonequilibrium (For details see e.g.
[609]) application of the Fock space formulation leads to the properly defined
equilibrium. In the particular situation related to the global one-dimensional
quantum gravity, the equilibrium is strictly related to the ensemble of the
quanta of space which in itself create the Æther. Such a situation gives the
most natural conditions for straightforward application of the first
principles of statistical mechanics [610], what will be resulting in _ab
initio_ formulation of the thermodynamics of such an Æther of space quanta. In
this section we shall present the constructive approach which uses the
simplest possible approximation, but in itself is the most fundamental
contribution to the thermodynamics. Namely, as the example of the
thermodynamic strategy we shall apply so called _one-particle approximation_ ,
based on the corresponding method of _one-particle density matrix method_. The
simplicity of the one-particle approximation is the fact that in such a
situation the density operator D is equivalent to an occupation number
operator. Then thermodynamic equilibrium is determined with respect to the
static Fock repère in the stable Bogoliubov vacuum, and therefore the one-
particle density matrix in equilibrium $\mathbb{D}$ is established according
to the von Neumann–Heisenberg picture.
#### A Entropy I: The Analytic Approach
The non-equilibrium density operator D, which possesses a dynamical nature by
its existence in a dynamical Fock repère, is expressed in the static Fock
repère $\mathfrak{F}$ via the equilibrium density matrix $\mathbb{D}$ in the
following way
$\textsf{D}={\textsf{G}}^{\dagger}{\textsf{G}}=\mathfrak{F}^{\dagger}\mathbb{D}\mathfrak{F},$
(11.1)
where in the case of global one-dimensional quantum gravity one the density
matrix has the form
$\mathbb{D}=\left[\begin{array}[]{cc}\dfrac{(\mu+1)^{2}}{{4\mu}}&\dfrac{1-\mu^{2}}{{4\mu}}\\\
\dfrac{1-\mu^{2}}{{4\mu}}&\dfrac{(\mu-1)^{2}}{{4\mu}}\end{array}\right],$
(11.2)
here $\mu=\dfrac{\omega}{\omega_{I}}$ is the mass scale of the system of space
quanta. Note that in the present situation
$\displaystyle\mathbb{D}^{2}$ $\displaystyle=$
$\displaystyle(\mathrm{Tr}\mathbb{D})\mathbb{D},$ (11.3)
$\displaystyle\det\mathbb{D}$ $\displaystyle=$ $\displaystyle 0.$ (11.4)
Vanishing of the determinant means that in the one-particle approximation the
corresponding thermodynamics is _irreversible_ , or in other words that the
thermodynamic processes in the Æther of space quanta are irreversible.
Employing the density matrix (11.90) one can establish the value of the
occupation number
$N=\dfrac{\mathrm{Tr}\left(\mathbb{D}^{2}\right)}{\mathrm{Tr}\mathbb{D}}=\dfrac{\mathrm{Tr}\left((\mathrm{Tr}\mathbb{D})\mathbb{D}\right)}{\mathrm{Tr}\mathbb{D}}=\dfrac{(\mathrm{Tr}\mathbb{D})^{2}}{\mathrm{Tr}\mathbb{D}}=\mathrm{Tr}\mathbb{D}=\dfrac{\mu^{2}+1}{2\mu},$
(11.5)
and the entropy can be derived from its basic definition
$\displaystyle
S=\dfrac{\mathrm{Tr}(\mathbb{D}\ln\mathbb{D})}{\mathrm{Tr}\mathbb{D}}.$ (11.6)
The problem is to derive $\ln\mathbb{D}$. It can be performed effectively by
application of the matrix Taylor series
$\ln\mathbb{D}=\ln[(\mathbb{D}-\mathbb{I})+\mathbb{I}]=\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n}(\mathbb{D}-\mathbb{I})^{n},$
(11.7)
and the matrix Newton binomial series
$\displaystyle(\mathbb{D}-\mathbb{I})^{n}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\binom{n}{k}\mathbb{D}^{k}(-\mathbb{I})^{n-k}=\sum_{k=1}^{n}(-1)^{n-k}\binom{n}{k}\mathbb{D}^{k}=$
(11.8) $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}(-1)^{n-k}\binom{n}{k}(\mathrm{Tr}\mathbb{D})^{k-1}\mathbb{D},$
(11.9)
where
$\binom{n}{m}=\dfrac{n!}{m!(n-m)!},$ (11.10)
is the Newton binomial symbol, which allows to write out the formula
$S=\sum_{n=1}^{\infty}\sum_{k=1}^{n}\dfrac{(-1)^{k-1}}{n}\binom{n}{k-1}S_{k},$
(11.11)
where $S_{k}$ are _cluster entropies_
$S_{k}=\dfrac{\mathrm{Tr}(\mathbb{D}^{k})}{\mathrm{Tr}\mathbb{D}}=\dfrac{\mathrm{Tr}\left((\mathrm{Tr}\mathbb{D})^{k-1}\mathbb{D}\right)}{\mathrm{Tr}\mathbb{D}}=\dfrac{(\mathrm{Tr}\mathbb{D})^{k-1}\mathrm{Tr}\mathbb{D}}{\mathrm{Tr}\mathbb{D}}=N^{k-1}.$
(11.12)
We shall call the series expansion (11.11) _the cluster series_. Its summation
is the method for obtaining entropy in the analytical way.
According to a certain mathematical tradition the series expansion (11.6)
converges if and only if the spectral radius $\rho$ of the difference
$\mathbb{D}-\mathbb{I}$, where $\mathbb{I}$ is $2\times 2$ unit matrix, is
$\rho(\mathbb{D}-\mathbb{I})<1.$ (11.13)
Straightforward calculation leads to the condition for the mass scale
$\mu\in[1;2+\sqrt{3}),$ (11.14)
and the result of the summation procedure is
$S=\dfrac{\zeta(1)}{2}\left(\dfrac{\mu^{2}-1}{\mu^{2}+1}\right)^{2}+\dfrac{\mu^{4}+6\mu^{2}+1}{(\mu^{2}+1)^{2}}\ln\dfrac{(\mu-1)^{2}}{2\mu},$
(11.15)
where $\zeta(s)$ is the Riemann zeta function
$\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^{s}},$ (11.16)
and $\zeta(1)$ is manifestly divergent.
Note that by straightforward application of the Hagedorn hadronization formula
$\omega\sim T_{H}$ [611], where $m$ is the mass of the system, one can
establish the hadronized temperature as
$\dfrac{T_{H}}{T_{I}}=\mu.$ (11.17)
By the relation $\Delta\omega\sim\Delta T_{H}$ one obtains the hadronized
temperature normalized to $T_{I}$ value
$\dfrac{\Delta T_{H}}{T_{I}}=\mu_{\max}-\mu_{\min}=1+\sqrt{3}\approx 2.732,$
(11.18)
so that one can establish the ratio
$\dfrac{T_{H}}{\Delta
T_{H}}\in\left[\dfrac{\sqrt{3}-1}{2},\dfrac{\sqrt{3}+1}{2}\right),$ (11.19)
what allows to determine the reciprocal
$\dfrac{\Delta T_{H}}{T_{H}}\in\left(-(\sqrt{3}+1),\sqrt{3}+1\right].$ (11.20)
Defining anisotropy as $\delta T_{H}=\Delta T_{H}-T_{H}$ one derives
$\dfrac{\delta T_{H}}{T_{H}}\in\left(-(\sqrt{3}+2),\sqrt{3}\right],$ (11.21)
so that half of the difference anisotropy is exactly
$\Delta\left(\dfrac{\delta T_{H}}{T_{H}}\right)=\left(\dfrac{\Delta
T_{H}}{T_{H}}\right)_{\max}-\left(\dfrac{\Delta
T_{H}}{T_{H}}\right)_{\min}=2(\sqrt{3}+1)=2\dfrac{\Delta T_{H}}{T_{I}}.$
(11.22)
In this manner one obtains
$\Delta T_{H}=\dfrac{T_{I}}{2}\Delta\left(\dfrac{\delta
T_{H}}{T_{H}}\right)\approx 2.732T_{I}.$ (11.23)
The difference (11.23) can be identified with a background temperature, i.e.
$\Delta T_{H}\equiv T_{B}$. For the initial datum $T_{I}\sim 1\mathrm{K}$ it
is very close to the averaged cosmic microwave background radiation
temperature, $T_{CMB}\approx 2.725\mathrm{K}$. Because of the quantity
$|T_{CMB}-T_{B}|$ is small $7\cdot 10^{-3}\mathrm{K}$, one can deduce that
next approximations will be resulting in successive contributions to
$T_{CMB}$. In other words
$T_{CMB}=T_{B}+\ldots.$ (11.24)
In the dynamical Fock repère Hamiltonian operator is
$\textsf{H}=\dfrac{m}{2}\left(\textsf{G}^{\dagger}\textsf{G}+\textsf{G}\textsf{G}^{\dagger}\right)=\mathfrak{F}^{\dagger}\mathbb{H}\mathfrak{F},$
(11.25)
where $\mathbb{H}$ is the Hamiltonian matrix in the static Fock repère
$\mathbb{H}=\left[\begin{array}[]{cc}\dfrac{\omega_{I}}{4}\left(1+\mu^{2}\right)&\dfrac{\omega_{I}}{4}\left(1-\mu^{2}\right)\vspace*{5pt}\\\
\dfrac{\omega_{I}}{4}\left(1-\mu^{2}\right)&\dfrac{\omega_{I}}{4}\left(1+\mu^{2}\right)\end{array}\right],$
(11.26)
which for fixed mass scale has discrete spectrum
$\mathrm{Spec}\leavevmode\nobreak\
\mathbb{H}=\left\\{\dfrac{\omega_{I}}{2}\mu^{2},\dfrac{\omega_{I}}{2}\right\\}.$
(11.27)
The internal energy calculated from the Hamiltonian matrix (11.26) is
$U=\dfrac{\mathrm{Tr}(\mathbb{D}\mathbb{H})}{\mathrm{Tr}\mathbb{D}}=\dfrac{\omega_{I}}{4}(\mu^{2}+1).$
(11.28)
The Hamiltonian matrix $\mathbb{H}$, however, consists constant term
$\mathbb{H}_{I}$
$\mathbb{H}_{I}=\left[\begin{array}[]{cc}\dfrac{\omega_{I}}{4}&\dfrac{\omega_{I}}{4}\vspace*{5pt}\\\
\dfrac{\omega_{I}}{4}&\dfrac{\omega_{I}}{4}\end{array}\right]$ (11.29)
which can be eliminated by simple renormalization
$\mathbb{H}\rightarrow\mathbb{H}^{\prime}=\mathbb{H}-\mathbb{H}_{I}=\left[\begin{array}[]{cc}\dfrac{\omega_{I}}{4}\mu^{2}&-\dfrac{\omega_{I}}{4}\mu^{2}\vspace*{5pt}\\\
-\dfrac{\omega_{I}}{4}\mu^{2}&\dfrac{\omega_{I}}{4}\mu^{2}\end{array}\right].$
(11.30)
The spectrum of the renormalized Hamiltonian matrix is
$\mathrm{Spec}\leavevmode\nobreak\
\mathbb{H}^{\prime}=\left\\{\dfrac{\omega_{I}}{2}\mu^{2},0\right\\},$ (11.31)
and straightforward computation of the renormalized internal energy yields the
following result
$U^{\prime}=\dfrac{\mathrm{Tr}(\mathbb{D}\mathbb{H}^{\prime})}{\mathrm{Tr}\mathbb{D}}=\dfrac{\omega_{I}}{4}\mu^{2}\equiv
U-U_{I},$ (11.32)
where $U_{I}=\dfrac{\omega_{I}}{4}$ is the constant term, which possesses the
property of the Eulerian homogeneity of degree $2$, i.e.
$U^{\prime}[\alpha\mu]=\alpha^{2}U^{\prime}[\mu].$ (11.33)
In this manner the thermodynamics describing space quanta Æther can be
formulated in the standard way of the Eulerian systems.
The three elementary and fundamental physical characteristics, i.e. the
occupation number $N$, the internal energy $U$, and the entropy $S$, just were
derived, and therefore one can conclude the thermodynamics by straightforward
application of the first principles. Actually the entropy (11.91) is
manifestly divergent due to the presence of formal infinity $\zeta(1)$.
Straightforward calculation shows that the temperature $T=\dfrac{\delta
U}{\delta S}$ arising from the entropy (11.91) contains the term with
$\zeta(1)$. However, it is also visible that such a temperature possesses
finite limit if and only if one re-scale initial data mass to formal infinity,
i.e. $\omega_{I}\rightarrow\omega_{I}\zeta(1)$. Because of the mass $m$ is
related to length $l$ like $m\sim 1/l$, performing of the limiting procedure
$\omega_{I}\rightarrow\infty$ corresponds with introduction to the theory
_purely point object_ $l_{I}\rightarrow 0$.
Scaling of initial data is not good procedure, however, because of it has no
well-defined physical meaning. There is another possibility for reorganization
of the troublesome divergence. Namely, it can be seen by straightforward
calculation that the entropy renormalization $S\rightarrow\dfrac{S}{\zeta(1)}$
with performing the formal limit $\zeta(1)\rightarrow\infty$ leads to the
equivalent result for the thermodynamics with no necessity of application of
unclear scaling in initial data. Such an _entropy renormalization_ corresponds
to an initial quantum state of an embedded three-dimensional space being a
purely point object, and yields perfect accordance with the second law of
thermodynamics
$S\longrightarrow
S^{\prime}=\lim_{\zeta(1)\rightarrow\infty}\dfrac{S}{\zeta(1)}=\dfrac{1}{2}\left(\dfrac{\mu^{2}-1}{\mu^{2}+1}\right)^{2}\geqslant
0.$ (11.34)
Calculating the temperature $T^{\prime}$ of space quanta one obtains the
formula
$T^{\prime}=\dfrac{\delta U^{\prime}}{\delta
S^{\prime}}=\omega_{I}\dfrac{(\mu^{2}+1)^{3}}{8(\mu^{2}-1)},$ (11.35)
and one sees that initially, _i.e._ for $\mu=1$, temperature is infinite. Such
a situation describes the _Hot Big Bang (HBB) phenomenon_. It can be seen that
after the HBB point the system is cooled right up until mass scale reaches the
value
$\mu_{PT}=\sqrt{2}\approx 1.414,$ (11.36)
and then is warmed, what means that the value $\mu_{PT}$ is the phase
transition point.
This phenomenon is better visible when one computes the energetic heat
capacity $C_{U}$
$C_{U}=T\dfrac{\delta S^{\prime}}{\delta T}=\dfrac{\delta U^{\prime}}{\delta
T}=\dfrac{(\mu^{2}-1)^{2}}{(\mu^{2}-2)(\mu^{2}+1)^{2}},$ (11.37)
which possesses the singularity in the point $\mu_{PT}$. Application of the
generalized law of equipartition
$\dfrac{\delta U^{\prime}}{\delta T^{\prime}}=\dfrac{f}{2},$ (11.38)
allows to establish the variability of the number of degrees of freedom
$f=2C_{U}.$ (11.39)
The Helmholtz free energy $F=U^{\prime}-T^{\prime}S^{\prime}$ that is
$F=-\dfrac{\omega_{I}}{16}(\mu^{4}-4\mu^{2}-1),$ (11.40)
is finite for finite value of initial data $\omega_{I}$, increases since the
initial point $\mu=1$ until the phase transition $\mu_{PT}$, and then
decreases. Therefore, the _thermal equilibrium point_ is the HBB point related
to the initial point $\mu_{eq}=1$. In the region of mass scales for which
$1\leqslant\mu<\mu_{PT}$ the mechanical isolation is absent, but it is present
after the phase transition, i.e. in the region $\mu>\mu_{PT}$.
Calculating the chemical potential
$\varpi=\dfrac{\delta F}{\delta
N}=-\omega_{I}\dfrac{\mu^{3}(\mu^{2}-2)}{2(\mu^{2}-1)},$ (11.41)
one can see straightforwardly that in the HBB point $\mu_{eq}=1$ this
potential diverges and in the phase transition point $\mu_{PT}$ it vanishes.
Applying the chemical potential (11.41) together with the occupation number
$N$ and the Helmholtz free energy $F$ yields the appropriate free energy
defined by the Landau grand potential $\Omega$
$\Omega=F-\varpi
N=\omega_{I}\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{16(\mu^{2}-1)},$ (11.42)
and therefore the corresponding Massieu–Planck free entropy $\Xi$ can be also
derived straightforwardly
$\Xi=-\dfrac{\Omega}{T}=-\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{2(\mu^{2}+1)^{3}}.$
(11.43)
Consequently the grand partition function $Z$ is established as
$\displaystyle
Z=e^{\Xi}=\exp\left\\{-\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{2(\mu^{2}+1)^{3}}\right\\}.$
(11.44)
The 2nd order Eulerian homogeneity of the system of space quanta yields the
equation of state $\dfrac{PV}{T}=\ln Z=\Xi$ and determines the product of
pressure $P$ and volume $V$ as
$PV=-\Omega,$ (11.45)
and together with the appropriate Gibbs–Duhem equation
$V\delta P=S^{\prime}\delta T+N\delta\varpi$ (11.46)
allows to establish the value of the pressure
$|P|=\exp\left\\{-\int\left(\dfrac{S}{\Omega}\delta
T+\dfrac{N}{\Omega}\delta\varpi\right)\right\\}.$ (11.47)
Similarly, the first law of thermodynamics
$-\delta\Omega=S^{\prime}\delta T+P\delta V+N\delta\varpi,$ (11.48)
together with the equation of state (11.45) determine the volume
$V=\dfrac{|\Omega|}{|P|},$ (11.49)
which is positive by definition. Regarding the equation of state (11.45) for
$\Omega=-|\Omega|<0$ the pressure is $P=|P|$, whereas for $\Omega=|\Omega|>0$
the pressure has the value $P=-|P|$. In this manner one receives
$\displaystyle
P=P(\mu_{0})\exp\left[\int_{\mu_{0}}^{\mu}\dfrac{8t\left(t^{6}-3t^{2}+4\right)}{3t^{8}-2t^{6}-12t^{4}+10t^{2}+1}dt\right],$
(11.50)
where $\mu_{0}$ is some reference value of $\mu$. Regarding the relation
(11.49), $V$ is a fixed parameter and its value can be established as follows
$V=\dfrac{\omega_{I}}{P(\mu_{0})}\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{16(\mu^{2}-1)}\exp\left[-\int_{\mu_{0}}^{\mu}\dfrac{8t\left(t^{6}-3t^{2}+4\right)}{3t^{8}-2t^{6}-12t^{4}+10t^{2}+1}dt\right].$
(11.51)
One can also determine two another thermodynamical potentials: the Gibbs free
energy $G=U-TS+PV$ and enthalpy $H=U+PV$. The results are as follows
$\displaystyle G$ $\displaystyle=$
$\displaystyle-\dfrac{\omega_{I}\mu^{2}\left(\mu^{4}-\mu^{2}-2\right)}{4\left(\mu^{2}-1\right)},$
(11.52) $\displaystyle H$ $\displaystyle=$
$\displaystyle-\dfrac{\omega_{I}\left(3\mu^{6}-3\mu^{4}-7\mu^{2}-1\right)}{16\left(\mu^{2}-1\right)}.$
(11.53)
Equivalently, the thermodynamics of space quanta Æther can be expressed by the
size scale $\lambda=\dfrac{1}{\mu}$. There are the relations relating both the
scales with an occupation number
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
N\left(1\mp\sqrt{{1-\dfrac{1}{N^{2}}}}\right),$ (11.54) $\displaystyle\mu$
$\displaystyle=$ $\displaystyle
N\left(1\pm\sqrt{{1-\dfrac{1}{N^{2}}}}\right),$ (11.55)
that in the limit of infinite $N$ are equal
$\displaystyle\lambda(N=\infty)$ $\displaystyle=$
$\displaystyle\left\\{0,\infty\right\\},$ (11.56) $\displaystyle\mu(N=\infty)$
$\displaystyle=$ $\displaystyle\left\\{\infty,0\right\\}.$ (11.57)
Therefore, there are two possible asymptotic behaviors. The first situation is
$\lambda=0$, $\mu=\infty$ which can be interpreted as a _black hole_ as well
as with HBB. The second situation is $\lambda=\infty$ which defined stable
classical physical object - the classical space-time.
One can establish the number $n$ of space quanta generated from the stable
Bogoliubov vacuum determined by the static Fock space related to initial data.
We shall call such class of states _vacuum space quanta_. By definition this
is the vacuum expectation value of the one-particle density operator. In other
words
$n=\dfrac{\left\langle\textrm{0}\right|\textsf{D}\left|\textrm{0}\right\rangle}{\left\langle{\textrm{0}}|{\textrm{0}}\right\rangle}=\dfrac{\left\langle\textrm{0}\right|{\textsf{G}}^{\dagger}{\textsf{G}}\left|\textrm{0}\right\rangle}{\left\langle{\textrm{0}}|{\textrm{0}}\right\rangle}.$
(11.58)
Straightforward application of the Bogoliubov transformation, the canonical
commutation relations of the static Fock space leads to
$\displaystyle{\textsf{G}}^{\dagger}{\textsf{G}}$ $\displaystyle=$
$\displaystyle\left(u\textsf{G}_{I}^{\dagger}-v\textsf{G}_{I}\right)\left(-v^{\ast}\textsf{G}_{I}^{\dagger}+u^{\ast}\textsf{G}_{I}\right)=$
(11.59) $\displaystyle=$
$\displaystyle|v|^{2}\textsf{G}_{I}\textsf{G}_{I}^{\dagger}+|u|^{2}\textsf{G}_{I}^{\dagger}\textsf{G}_{I}-v^{\ast}{u}\textsf{G}_{I}^{\dagger}\textsf{G}_{I}^{\dagger}-vu^{\ast}\textsf{G}_{I}\textsf{G}_{I}=$
$\displaystyle=$
$\displaystyle|v|^{2}+\left(|u|^{2}+|v|^{2}\right)\textsf{G}_{I}^{\dagger}\textsf{G}_{I}-v^{\ast}{u}\textsf{G}_{I}^{\dagger}\textsf{G}_{I}^{\dagger}-vu^{\ast}\textsf{G}_{I}\textsf{G}_{I}.$
and using of the properties of the stable Bogoliubov vacuum gives
$\displaystyle\left\langle\textrm{0}\right|{\textsf{G}}^{\dagger}{\textsf{G}}\left|\textrm{0}\right\rangle$
$\displaystyle=$
$\displaystyle|v|^{2}\left\langle{\textrm{0}}|{\textrm{0}}\right\rangle+\left(|u|^{2}+|v|^{2}\right)\left\langle\textrm{0}\right|\textsf{G}_{I}^{\dagger}\textsf{G}_{I}\left|\textrm{0}\right\rangle-$
(11.60) $\displaystyle-$ $\displaystyle
v^{\ast}{u}\left\langle\textrm{0}\right|\textsf{G}_{I}^{\dagger}\textsf{G}_{I}^{\dagger}\left|\textrm{0}\right\rangle-
vu^{\ast}\left\langle\textrm{0}\right|\textsf{G}_{I}\textsf{G}_{I}\left|\textrm{0}\right\rangle=$
(11.61) $\displaystyle=$
$\displaystyle|v|^{2}\left\langle{\textrm{0}}|{\textrm{0}}\right\rangle.$
(11.62)
Therefore the number of vacuum space quanta is
$n=|v|^{2}=\dfrac{(\mu-1)^{2}}{4\mu}.$ (11.63)
In this manner the mass scale has two values
$\mu_{\pm}(n)=\left(\sqrt{n}\pm\sqrt{n+1}\right)^{2},$ (11.64)
corresponding to two independent phases of the space quanta Æther. For
convenience we shall call the phase described by the sign $+$ _the positive
phase of the Æther_ , and the phase described by the sign $-$ _the negative
phase of the Æther_. The phases have completely different asymptotic
behaviour. Namely,
$\lim_{n\rightarrow\infty}\mu_{\pm}(n)=\left\\{\begin{array}[]{cc}\infty&\textrm{for\leavevmode\nobreak\
the\leavevmode\nobreak\ positive\leavevmode\nobreak\ phase}\\\
0&\leavevmode\nobreak\ \textrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
negative\leavevmode\nobreak\ phase}\end{array}\right..$ (11.65)
Let us consider the asymptotic $n\rightarrow\infty$ thermodynamics of the
Æther. The case of positive phase is
$\displaystyle T$ $\displaystyle\rightarrow$ $\displaystyle\infty,$ (11.66)
$\displaystyle C_{U}$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.67)
$\displaystyle f$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.68)
$\displaystyle F$ $\displaystyle\rightarrow$ $\displaystyle\infty,$ (11.69)
$\displaystyle\varpi$ $\displaystyle\rightarrow$ $\displaystyle\infty,$
(11.70) $\displaystyle\Omega$ $\displaystyle\rightarrow$
$\displaystyle\infty,$ (11.71) $\displaystyle\Xi$ $\displaystyle\rightarrow$
$\displaystyle-\dfrac{3}{2},$ (11.72) $\displaystyle Z$
$\displaystyle\rightarrow$ $\displaystyle e^{-3/2},$ (11.73) $\displaystyle G$
$\displaystyle\rightarrow$ $\displaystyle\infty,$ (11.74) $\displaystyle H$
$\displaystyle\rightarrow$ $\displaystyle\infty,$ (11.75) $\displaystyle P$
$\displaystyle\rightarrow$ $\displaystyle P_{\infty},$ (11.76) $\displaystyle
V$ $\displaystyle\rightarrow$ $\displaystyle\infty,$ (11.77)
where $P_{\infty}$ is the (constant) value of the pressure in the asymptotic
value $\mu=\infty$ which can be assessed numerically. For instance in the
trivial case $\mu_{0}=0$ one obtains $P_{\infty}\approx P(0)\exp(23.9527)$,
whereas for $\mu_{0}=1$ one receives $P_{\infty}\approx P(1)\exp(510959)$.
Similarly one can analyse the negative phase
$\displaystyle T$ $\displaystyle\rightarrow-\dfrac{\omega_{I}}{8},$ (11.78)
$\displaystyle C_{U}$ $\displaystyle\rightarrow$ $\displaystyle-\dfrac{1}{2},$
(11.79) $\displaystyle f$ $\displaystyle\rightarrow$ $\displaystyle-1,$
(11.80) $\displaystyle F$ $\displaystyle\rightarrow$
$\displaystyle\dfrac{\omega_{I}}{16},$ (11.81) $\displaystyle\varpi$
$\displaystyle\rightarrow$ $\displaystyle 0,$ (11.82) $\displaystyle\Omega$
$\displaystyle\rightarrow$ $\displaystyle\dfrac{\omega_{I}}{16},$ (11.83)
$\displaystyle\Xi$ $\displaystyle\rightarrow$ $\displaystyle\dfrac{1}{2},$
(11.84) $\displaystyle Z$ $\displaystyle\rightarrow$ $\displaystyle e^{1/2},$
(11.85) $\displaystyle G$ $\displaystyle\rightarrow$ $\displaystyle 0,$
(11.86) $\displaystyle H$ $\displaystyle\rightarrow$
$\displaystyle-\dfrac{\omega_{I}}{16},$ (11.87) $\displaystyle P$
$\displaystyle\rightarrow$ $\displaystyle P_{0},$ (11.88) $\displaystyle V$
$\displaystyle\rightarrow$ $\displaystyle\dfrac{\omega_{I}}{16P_{0}},$ (11.89)
where $P_{0}$ is the (constant) value of the pressure in $\mu=0$ which can be
assessed numerically. For instance when $\mu_{0}=0$ one has $P_{0}=P(0)$. For
$\mu_{0}=1$ one obtains $P_{0}\approx P(1)\exp(-151.536)$.
In this section we have presented the next implication of the global one-
dimensional quantum gravity. It was shown that this algorithm yields
constructive, consistent, and plausible phenomenology, that is thermodynamics
of Æther, in the discussed situation describing space quanta behavior. The
theory of quantum gravity as well as the thermodynamics can be applied to any
general relativistic space-times which metrics can be presented in the form of
the $3+1$ splitting. Such space-time satisfy the Mach principle, _i.e._ are
isotropic. Their importance for elementary particle physics, cosmology and
high energy astrophysics is experimentally confirmed; one can say that these
are _phenomenological space-times_.
As the example of _ab initio_ formulation of thermodynamics we have employed
the one-particle approximation of the density matrix. Application of the
renormalization method to the entropy and to the Hamiltonian matrix resulted
in the second order Eulerian homogeneity property. The Landau grand potential
$\Omega$ and the Massieu–Planck free entropy $\Xi$ were employed to the
consistent description. The grand partition function $Z$ and thermodynamic
volume $V$ were determined constructively. Another thermodynamical potentials
were derived in frames of the _entropic formalism_ , which accords with the
first and the second principles of thermodynamics. Physical information
following from the thermodynamics of space quanta Æther is the crucial point
of the construction presented in this section. Actually the proposed approach
differ from another ones (Cf. _e.g._ [612, 613, 614, 615]) by _ab initio_
treatment of the quantum gravity phenomenology.
Studying of particular physical situations in frames of the proposed approach
seems to be the most important prospective arising from the thermodynamics of
space quanta Æther. From experimental point of view the presented
considerations possess evident usefulness, because of bosonic systems are
common in high energy physics.
#### B Entropy II: The Algebraic Approach
Let us consider the space quanta Æther in the grand canonical ensemble. First
of all let us express the static one-particle density matrix via the
Bogoliubov coefficients
$\mathbb{D}=\left[\begin{array}[]{cc}|u|^{2}&-uv\\\
-u^{\ast}v^{\ast}&|v|^{2}\end{array}\right].$ (11.90)
The basic quantity is an entropy, which for an arbitrary quantum system is
defined by the standard Boltzmann–von Neumann formula
$S=\dfrac{\mathrm{Tr}\left(\mathbb{D}\ln\mathbb{D}\right)}{\mathrm{Tr}\mathbb{D}},$
(11.91)
and in the present situation can be immediately computed from the density
matrix (11.90).
The problem is to establish the logarithm of the one-particle density matrix
$\ln\mathbb{D}$. It can be performed by application of of the algebraic
methods, particularly polynomial long division algorithm, the characteristic
polynomial, and the Cayley–Hamilton theorem (For some details of basic and
advanced algebra see e.g. books in the Ref. [616]). Let us present the method
in detail for any analytical function of a matrix $\mathbb{D}$. Let us
consider the characteristic polynomial $ch_{\mathbb{D}}(\lambda)$ of the
matrix $\mathbb{D}$, where $\lambda$ is eigenvalue of $\mathbb{D}$. If
$p(\lambda)$ is an analytical function, i.e. possesses power series expansion,
then via using of the division transformation one can present $p(\lambda)$ as
the dividend which divisor is the characteristic polynomial
$p(\lambda)=q(\lambda)ch_{\mathbb{D}}(\lambda)+r(\lambda).$ (11.92)
In other words the problem is to establish the remainder polynomial
$r(\lambda)$ and the quotient polynomial $q(\lambda)$. Recall that according
to the Cayley–Hamilton theorem the characteristic polynomial of a matrix
$\mathbb{D}$ evaluated on this matrix vanishes identically
$ch_{\mathbb{D}}(\mathbb{D})=0,$ (11.93)
so that consequently the evaluation $p(\mathbb{D})$ is exactly equal to the
reminder polynomial evaluated on the matrix $\mathbb{D}$
$p(\mathbb{D})=r(\mathbb{D}).$ (11.94)
If a matrix $\mathbb{D}$ is a matrix of dimension $n\times n$ then its
characteristic polynomial $ch_{\mathbb{D}}(\lambda)$ is a polynomial of degree
$n$ which coefficients are invariants of a matrix $\mathbb{D}$. Therefore the
remainder polynomial $r(\lambda)$ must of the order $n-1$ at most. There is
some problem when there are eigenvalues of $\mathbb{D}$ for which the function
$p(\lambda)$ has singularity. Then, however, we shall not include such
eigenvalues, and for determination of the quotient $q(\lambda)$ and the
remainder polynomial $r(\lambda)$ we shall differentiate the relation (11.92)
$n$ times and evaluate all $n+1$ relations on the non-singular eigenvalues of
$\mathbb{D}$. Such a procedure generates the system of $n+1$ equations which
allows to establish the coefficients of the remainder polynomial $r(\lambda)$
as well as leads to the quotient $q(\lambda)$ as the result of solving an
appropriate differential equation of degree $n$ at most. It must be emphasized
that the quotient $q(\lambda)$ as a solution of ordinary differential equation
is assured to be an analytical function.
Let us apply such a method to the situation given by the $2\times 2$ matrix
$\mathbb{D}$ having zero determinant $\det\mathbb{D}=0$, and the function
$p(x)=\ln{x}$. The characteristic polynomial of the matrix $\mathbb{D}$ is
$ch_{\mathbb{D}}(\lambda)=\det\left(\mathbb{D}-\lambda\mathbb{I}\right)=\lambda^{2}-(\mathrm{Tr}\mathbb{D})\lambda,$
(11.95)
and its eigenvalues are $\lambda=0$ and $\lambda=\mathrm{Tr}\mathbb{D}$. By
the Cayley–Hamilton theorem one has
$\mathbb{D}^{2}-(\mathrm{Tr}\mathbb{D})\mathbb{D}=0,$ (11.96)
i.e. $\mathbb{D}^{2}=(\mathrm{Tr}\mathbb{D})\mathbb{D}$. Let us write out the
relation (11.92) for this case
$\ln\lambda=q(\lambda)\left(\lambda^{2}-(\mathrm{Tr}\mathbb{D})\lambda\right)+a_{0}+a_{1}\lambda,$
(11.97)
where $a_{0}$ and $a_{1}$ are the coefficients of the polynomial $r(\lambda)$.
The problem is to establish $q(\lambda)$ and the coefficients $a_{0}$ and
$a_{1}$. First of all let us note that $\lambda=0$ is the singularity of
$\ln\lambda$, what means that we shall not consider this eigenvalue. For
determination of the three unknown quantities let us differentiate the
relation (11.97) two times. The results are as follows
$\displaystyle\dfrac{1}{\lambda}$ $\displaystyle=$ $\displaystyle
q^{\prime}(\lambda)\left(\lambda^{2}-(\mathrm{Tr}\mathbb{D})\lambda\right)+q(\lambda)\left(2\lambda-\mathrm{Tr}\mathbb{D}\right)+a_{1},$
(11.98) $\displaystyle-\dfrac{1}{\lambda^{2}}$ $\displaystyle=$ $\displaystyle
q^{\prime\prime}(\lambda)\left(\lambda^{2}-(\mathrm{Tr}\mathbb{D})\lambda\right)+2q^{\prime}(\lambda)\left(2\lambda-\mathrm{Tr}\mathbb{D}\right)+2q(\lambda).$
(11.99)
Taking into account the fact that evaluation of the characteristic polynomial
on an eigenvalue is zero. Application of this fundamental fact in the case of
the eigenvalue $\lambda=\mathrm{Tr}\mathbb{D}$ leads to significant
simplification of the equations (11.97), (11.98) and (11.99)
$\displaystyle\ln\lambda$ $\displaystyle=$ $\displaystyle a_{0}+a_{1}\lambda,$
(11.100) $\displaystyle\dfrac{1}{\lambda}$ $\displaystyle=$ $\displaystyle
q(\lambda)\lambda+a_{1},$ (11.101) $\displaystyle-\dfrac{1}{\lambda^{2}}$
$\displaystyle=$ $\displaystyle 2q^{\prime}(\lambda)\lambda+2q(\lambda),$
(11.102)
which can be presented in the following form
$\displaystyle a_{0}=\ln\lambda-a_{1}\lambda,$ (11.103) $\displaystyle
a_{1}=\dfrac{1}{\lambda}-q(\lambda)\lambda,$ (11.104) $\displaystyle\lambda
q^{\prime}(\lambda)+q(\lambda)+\dfrac{1}{2\lambda^{2}}=0.$ (11.105)
The equation (11.105) is the differential equation for the quotient
$q(\lambda)$ and can be solved straightforwardly
$q(\lambda)=\dfrac{1}{2\lambda^{2}}+\dfrac{C}{\lambda},$ (11.106)
where $C$ is constant of integration. Because of we are still interested in
the concrete eigenvalue $\lambda=\mathrm{Tr}\mathbb{D}$ one receives
$q(\mathrm{Tr}\mathbb{D})=\dfrac{1}{2(\mathrm{Tr}\mathbb{D})^{2}}+\dfrac{C}{\mathrm{Tr}\mathbb{D}},$
(11.107)
and by this reason one receives
$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\mathrm{Tr}\mathbb{D}}-q(\mathrm{Tr}\mathbb{D})\mathrm{Tr}\mathbb{D}=\dfrac{1}{2\mathrm{Tr}\mathbb{D}}-C,$
(11.108) $\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\ln\mathrm{Tr}\mathbb{D}-a_{1}\mathrm{Tr}\mathbb{D}=\ln\mathrm{Tr}\mathbb{D}-\dfrac{1}{2}+C\mathrm{Tr}\mathbb{D}.$
(11.109)
In this manner one can compute the function $\ln\mathbb{D}$ as follows
$\ln\mathbb{D}=a_{0}\mathbb{I}+a_{1}\mathbb{D}=\left(\ln\mathrm{Tr}\mathbb{D}-\dfrac{1}{2}+C\mathrm{Tr}\mathbb{D}\right)\mathbb{I}+\left(\dfrac{1}{2\mathrm{Tr}\mathbb{D}}-C\right)\mathbb{D},$
(11.110)
and consequently one obtains
$\displaystyle\mathbb{D}\ln\mathbb{D}$ $\displaystyle=$
$\displaystyle\left(\ln\mathrm{Tr}\mathbb{D}-\dfrac{1}{2}+C\mathrm{Tr}\mathbb{D}\right)\mathbb{D}+\left(\dfrac{1}{2\mathrm{Tr}\mathbb{D}}-C\right)\mathbb{D}^{2}=$
(11.111) $\displaystyle=$
$\displaystyle\left(\ln\mathrm{Tr}\mathbb{D}-\dfrac{1}{2}+C\mathrm{Tr}\mathbb{D}\right)\mathbb{D}+\left(\dfrac{1}{2\mathrm{Tr}\mathbb{D}}-C\right)(\mathrm{Tr}\mathbb{D})\mathbb{D}=$
(11.112) $\displaystyle=$
$\displaystyle\left(\ln\mathrm{Tr}\mathbb{D}-\dfrac{1}{2}+C\mathrm{Tr}\mathbb{D}\right)\mathbb{D}+\left(\dfrac{1}{2}-C\mathrm{Tr}\mathbb{D}\right)\mathbb{D}=$
(11.113) $\displaystyle=$
$\displaystyle\left(\ln\mathrm{Tr}\mathbb{D}-\dfrac{1}{2}+C\mathrm{Tr}\mathbb{D}+\dfrac{1}{2}-C\mathrm{Tr}\mathbb{D}\right)\mathbb{D}=$
(11.114) $\displaystyle=$
$\displaystyle\left(\ln\mathrm{Tr}\mathbb{D}\right)\mathbb{D}.$ (11.115)
Now it is easy to establish the entropy
$S=\dfrac{\mathrm{Tr}(\mathbb{D}\ln\mathbb{D})}{\mathrm{Tr}\mathbb{D}}=\dfrac{\mathrm{Tr}\left((\ln\mathrm{Tr}\mathbb{D})\mathbb{D}\right)}{\mathrm{Tr}\mathbb{D}}=\dfrac{(\ln\mathrm{Tr}\mathbb{D})\mathrm{Tr}\mathbb{D}}{\mathrm{Tr}\mathbb{D}}=\ln\mathrm{Tr}\mathbb{D}.$
(11.116)
In our situation the trace of the density matrix is
$\mathrm{Tr}\mathbb{D}=|u|^{2}+|v|^{2}=2|v|^{2}+1$ (11.117)
and by this reason one obtains
$S=\ln\left(2|v|^{2}+1\right)=-\ln\Sigma,$ (11.118)
where $\Sigma$ is the quantum statistics of the system of space quanta
$\Sigma=\dfrac{1}{2|v|^{2}+1}=\dfrac{1}{2n+1},$ (11.119)
where $n$ is the number of vacuum space quanta (11.62).
Let us compute the thermodynamical potentials for the entropy received entropy
(11.118) and the same values of the internal energy $U$, and the occupation
number $N$ established in the previous section, i.e.
$\displaystyle S$ $\displaystyle=$
$\displaystyle\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right],$ (11.120)
$\displaystyle U$ $\displaystyle=$
$\displaystyle\dfrac{\omega_{I}}{4}\mu^{2},$ (11.121) $\displaystyle N$
$\displaystyle=$ $\displaystyle\dfrac{\mu^{2}+1}{2\mu}.$ (11.122)
The most important is of course the temperature of the system
$T=\dfrac{\omega_{I}\mu^{2}}{2}\dfrac{\mu^{2}+1}{\mu^{2}-1},$ (11.123)
which for $\mu>1$ is manifestly negative. The heat capacity has the form
$C_{U}=\dfrac{1}{2}+\dfrac{1}{\mu^{4}-2\mu^{2}-1},$ (11.124)
and the number of degrees of freedom is
$f=1+\dfrac{2}{\mu^{4}-2\mu^{2}-1}.$ (11.125)
The Helmholtz free energy can be also derived straightforwardly
$F=\dfrac{\omega_{I}\mu^{2}}{4}\left(1-2\dfrac{\mu^{2}+1}{\mu^{2}-1}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right]\right),$
(11.126)
and the chemical potential has the form
$\varpi=-\dfrac{2\omega_{I}\mu^{3}\left(\mu^{4}-2\mu^{2}-1\right)}{\left(\mu^{2}-1\right)^{3}}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right].$
(11.127)
The Landau grand potential and the Massieu–Planck free entropy are
respectively
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\dfrac{\omega_{I}\mu^{2}}{4}\left(1+2\dfrac{\left(\mu^{2}-3\right)\left(\mu^{2}+1\right)^{2}}{\left(\mu^{2}-1\right)^{3}}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right]\right),$
(11.128) $\displaystyle\Xi$ $\displaystyle=$
$\displaystyle-\dfrac{\mu^{2}-1}{2\left(\mu^{2}+1\right)}-\dfrac{\left(\mu^{2}-3\right)\left(\mu^{2}+1\right)}{\left(\mu^{2}-1\right)^{2}}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right].$
(11.129)
The grand partition function can be established as
$Z=\exp\left[-\dfrac{\mu^{2}-1}{2\left(\mu^{2}+1\right)}\right]\left(\dfrac{\mu^{2}+1}{2\mu}\right)^{-\dfrac{\left(\mu^{2}-3\right)\left(\mu^{2}+1\right)}{\left(\mu^{2}-1\right)^{2}}}.$
(11.130)
The pressure is
$P=P(\mu_{0})\exp\left[-4\int_{\mu_{0}}^{\mu}\dfrac{(t^{2}-1)^{2}\left(t^{4}-2t^{2}-1\right)+4\left(t^{4}+4t^{2}+1\right)\ln\left[\dfrac{t^{2}+1}{2t}\right]}{t\left(t^{2}-1\right)\left(\left(t^{2}-1\right)^{3}+2\left(t^{2}-3\right)\left(t^{2}+1\right)^{2}\ln\left[\dfrac{t^{2}+1}{2t}\right]\right)}dt\right],$
(11.131)
where $\mu_{0}$ is some reference value of $\mu$, and the thermodynamical
volume has the form
$\displaystyle
V=\dfrac{\omega_{I}\mu^{2}}{4P(\mu_{0})}\left|1+2\dfrac{\left(\mu^{2}-3\right)\left(\mu^{2}+1\right)^{2}}{\left(\mu^{2}-1\right)^{3}}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right]\right|\times$
(11.132)
$\displaystyle\times\exp\left[4\int_{\mu_{0}}^{\mu}\dfrac{(t^{2}-1)^{2}\left(t^{4}-2t^{2}-1\right)+4\left(t^{4}+4t^{2}+1\right)\ln\left[\dfrac{t^{2}+1}{2t}\right]}{t\left(t^{2}-1\right)\left(\left(t^{2}-1\right)^{3}+2\left(t^{2}-3\right)\left(t^{2}+1\right)^{2}\ln\left[\dfrac{t^{2}+1}{2t}\right]\right)}dt\right].$
The Gibbs free energy and enthalpy are respectively
$\displaystyle G$ $\displaystyle=$
$\displaystyle\dfrac{\omega_{I}\mu^{2}\left(\mu^{2}+1\right)\left(\mu^{4}-2\mu^{2}-1\right)}{\left(\mu^{2}-1\right)^{3}}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right],$
(11.133) $\displaystyle H$ $\displaystyle=$
$\displaystyle-\dfrac{\omega_{I}\mu^{2}\left(\mu^{2}-3\right)\left(\mu^{2}+1\right)^{2}}{2\left(\mu^{2}-1\right)^{3}}\ln\left[\dfrac{\mu^{2}+1}{2\mu}\right].$
(11.134)
There is the question about asymptotic $n\rightarrow\infty$ thermodynamics of
the Æther. The positive phase is then
$\displaystyle T$ $\displaystyle\rightarrow\infty,$ (11.135) $\displaystyle
C_{U}$ $\displaystyle\rightarrow$ $\displaystyle\dfrac{1}{2},$ (11.136)
$\displaystyle f$ $\displaystyle\rightarrow$ $\displaystyle 1,$ (11.137)
$\displaystyle F$ $\displaystyle\rightarrow$ $\displaystyle-\infty,$ (11.138)
$\displaystyle\varpi$ $\displaystyle\rightarrow$ $\displaystyle-\infty,$
(11.139) $\displaystyle\Omega$ $\displaystyle\rightarrow$
$\displaystyle\infty,$ (11.140) $\displaystyle\Xi$ $\displaystyle\rightarrow$
$\displaystyle-\infty,$ (11.141) $\displaystyle Z$ $\displaystyle\rightarrow$
$\displaystyle 0,$ (11.142) $\displaystyle G$ $\displaystyle\rightarrow$
$\displaystyle-\infty,$ (11.143) $\displaystyle H$ $\displaystyle\rightarrow$
$\displaystyle-\infty,$ (11.144) $\displaystyle P$ $\displaystyle\rightarrow$
$\displaystyle P_{\infty},$ (11.145) $\displaystyle V$
$\displaystyle\rightarrow$ $\displaystyle\infty,$ (11.146)
where $P_{\infty}$ is the numerical value of the pressure in the asymptotic
value $\mu=\infty$. For example in the trivial situation $\mu_{0}=0$ one
obtains the value $P_{\infty}\approx{P}(0)\exp(473.822)$, whereas when
$\mu_{0}=1$ one receives another result $P_{\infty}\approx
P(1)\exp(0.208615)\approx 1.23197P(1)$.
Similarly, the asymptotic thermodynamics of the negative phase of the space
quanta Æther can be analyzed. The results are as follows
$\displaystyle T$ $\displaystyle\rightarrow 0,$ (11.147) $\displaystyle C_{U}$
$\displaystyle\rightarrow$ $\displaystyle-\dfrac{1}{2},$ (11.148)
$\displaystyle f$ $\displaystyle\rightarrow$ $\displaystyle-1,$ (11.149)
$\displaystyle F$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.150)
$\displaystyle\varpi$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.151)
$\displaystyle\Omega$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.152)
$\displaystyle\Xi$ $\displaystyle\rightarrow$ $\displaystyle-\infty,$ (11.153)
$\displaystyle Z$ $\displaystyle\rightarrow$ $\displaystyle-\infty,$ (11.154)
$\displaystyle G$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.155)
$\displaystyle H$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.156)
$\displaystyle P$ $\displaystyle\rightarrow$ $\displaystyle P_{0},$ (11.157)
$\displaystyle V$ $\displaystyle\rightarrow$ $\displaystyle 0,$ (11.158)
where $P_{0}$ is the numerical value of the pressure for the value $\mu=0$.
For example in the trivial situation $\mu_{0}=0$ one has $P_{0}=P(0)$, while
when $\mu_{0}=1$ one obtains $P_{0}\approx P(1)\exp(-510956)$.
### Epilogue
Science is a differential equation. Religion is a boundary condition.
Alan Turing
This book presented the constructive model of physical Reality based on the
realization of the fusion of two fundamental concepts of Antiquity:
Aristotelian Æther and the Epicurean–Islamic Multiverse. The theory in itself
creates the unified point of view which I proposed to call _Æthereal
Multiverse_. There is evident opportunity and necessity to apply the proposed
approach straightforwardly to the concrete problems which theoretical results
could be compared with experimental and observational data. The only such a
treatment guarantees physical consistency of _Æthereal Multiverse_.
I believe that _Æthereal Multiverse_ governs Nature at comparatively small
scales. In my opinion the Planck scale, i.e. the scale in which quantum
physics meets classical physics, is the good candidate for such effects.
However, this is the only my personal belief, and therefore there is justified
necessity to verify its consequences empirically. Possibly, the scope of
applicability is much more wide than I think presently, but this is also
possible that the region of applicability is completely different then I have
suggested. Another possible candidates for the new physics having a place at
comparatively small scales include the Compton scale or introduced in this
book the Compton–Planck scale.
Anyway, there is a certain knowledge which we have established in this book.
Namely, we have proved that straightforward philosophical reasoning involving
both the concepts of Æther and Multiverse can be productively performed, and
applied to consistent construction of new physics. The philosophical spirit of
the modal realism, which was the ideological fundament of our consideration,
resulted in fruitful and fashionable development of two fundamental ideas of
Antiquity, which at the first glance look like in an old-fashioned manner or
manifestly fossilized than like the foundations of the new physics. The
crucial point of the presented deductions was constructiveness of the applied
theoretical approach based on the method of analogy. The central
methodological background of our deductions was the principle of simplicity,
which enabled to receive new physical description from well-established
knowledge of abstract mathematics and mathematical physics.
From the philosophical point of view we have proven obvious non triviality.
Namely, we showed that it is possible to joint manifestly the systems of
Aristotelian and Epicurean–Islamic philosophy in the one unified and
productive picture of physical Reality. This is in itself a paradoxical
situation because of these philosophies have been understood as completely
different ideological systems which are impossible to unify. This creates the
strong belief that new physics can be effectively formulated via involving of
the old and well-established philosophy and knowledge to the new applications.
In other words, it is my deep conviction that the new physics is the only
sophisticated structure hidden in a philosophical interpretation of a
mathematical formalism of the old physics. The identification method, which we
frequently applied in our studies, is the only straightforward purely logical
consequence of a philosophical interpretation which play the fundamental role
in all natural sciences.
In this book a number of results obtained in my earlier research work was
definitely updated, improved or even rejected from the general physical
scenario. However, I have a deep conviction that the mathematical truth was
established in a great detail. This level of discussion allows to think that
the presented scheme of philosophical reasoning is the most productive way for
new constructive deductions and, moreover, in itself creates the new logical
system of physics. At the first glance this logic can be difficult to accept
or even impossible to practical applications. However, it has been shown in
this book that when the application is realized in a constructive way it
results in the elegant picture of the physical Reality.
The physical scenario presented in this book is essentially novel, because of
in a whole is a certain application of both the methods and the philosophical
background of theory of relativity and quantum theory, which are fundamentally
confirmed as the fundamental theories of physics. Their possible deformations
and modifications follow from the recent deductions of high energy physics and
astroparticle physics. I will be satisfied when _Æthereal Multiverse_ will be
turned out a productive approach to theoretical physics.
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See pages - of index.pdf
|
arxiv-papers
| 2011-02-24T14:28:46 |
2024-09-04T02:49:17.213755
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lukasz Andrzej Glinka",
"submitter": "Lukasz Glinka",
"url": "https://arxiv.org/abs/1102.5002"
}
|
1102.5040
|
ROM2F/2011/02
Supersymmetry Breaking in a Minimal Anomalous Extension of the MSSM
A. Lionetto111Andrea.Lionetto@roma2.infn.it♮ and A.
Racioppi222Antonio.Racioppi@kbfi.ee♭
♮ Dipartimento di Fisica dell’Università di Roma , “Tor Vergata” and
I.N.F.N. - Sezione di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - 00133 Roma, ITALY
♭ National Institute of Chemical Physics and Biophysics,
Ravala 10, Tallinn 10143, Estonia
We study a supersymmetry breaking mechanism in the context of a minimal
anomalous extension of the MSSM. The anomaly cancellation mechanism is
achieved through suitable counterterms in the effective action, i.e. Green-
Schwarz terms. We assume that the standard MSSM superpotential is
perturbatively realized, i.e. all terms allowed by gauge symmetries, except
for the $\mu$-term which has a non-perturbative origin. The presence of this
term is expected in many intersecting D-brane models which can be considered
as the ultraviolet completion of our model. We show how soft supersymmetry
breaking terms arise in this framework and we study the effect of some
phenomenological constraints on this scenario.
## 1 Introduction
The LHC era has begun and the high energy physics community is analyzing and
discussing the first results. One of the key goals of LHC, besides shedding
light on the electroweak (EW) symmetry breaking sector of the Standard Model
(SM), is to find some signature of physics beyond the SM. Supersymmetric
particles and extra neutral gauge bosons $Z^{\prime}$ are widely studied
examples of such signatures. A large class of phenomenological and string
models aiming to describe the low energy physics accessible to LHC predict the
existence of additional abelian $U(1)$ gauge groups as well as $N=1$
supersymmetry softly broken roughly at the TeV scale. In particular in string
theory the presence of extra anomalous $U(1)$’s seems ubiquitous. D-brane
models in orientifold vacua contain several abelian factors and they are
typically anomalous [1]-[16]. In [17] we studied a string inspired extension
of the (Minimal Supersymmetric SM) MSSM with an additional anomalous $U(1)$
(see [18] for other anomalous $U(1)$ extensions of the SM and see [19] for
extensions of the MSSM). The term anomalous refers to the peculiar mechanism
of gauge anomaly cancellation [20] which does not rely on the fermion charges
but rather on the presence of suitable counterterms in the effective action.
These terms are usually dubbed as Green-Schwarz (GS) [18, 21] and Generalized
Chern-Simons (GCS) [22]-[26]. They can be considered as the low energy
remnants of the higher dimensional anomaly cancellation mechanism in string
theory. In our model we assumed the usual MSSM superpotential and soft
supersymmetry breaking terms allowed by the symmetries (the well known result
[27]). In this paper we address the question of the origin of the latter in
the context of a global supersymmetry breaking mechanism. This means that we
do not rely on a supergravity origin of the soft terms but rather on a local
setup based for example on intersecting D-brane constructions in superstring
theory in which gravity is essentially decoupled (see for instance [28] for a
recent attempt in this direction). Moreover in [17] we made the assumption
that all the MSSM superpotential terms were perturbatively realized, i.e.
allowed by the extra abelian $U(1)$ symmetries. In the following we assume
instead that the $\mu$-term is perturbatively forbidden. The origin of this
term is rather non-perturbative and can be associated to an exotic instanton
contribution which naturally arises from euclidean D-brane in the framework of
a type IIA intersecting brane model (see [29] and references therein).
The paper is organized as follows: in Sec. 2 we describe the basic setup of
the model and we discuss the perturbative and non-perturbative origin of the
superpotential terms. We argue how the latter can naturally come from an
intersecting D-brane model considered as the ultraviolet (UV) completion of
our model. In Sec. 3 we describe the (global) supersymmetry breaking mechanism
that gives mass to all the soft terms. In Sec. 5 we compute the gauge vector
boson masses while in Sec. 4 we study the scalar potential of the theory in
the neutral sector. In Sec. 6 we describe the neutralino sector while in Sec.
7 we describe the sfermion mass matrices. In Sec. 8 we study the phenomenology
of our model and the bounds that can be put by some experimental constraints.
Finally in Sec. 9 we draw our conclusions.
## 2 Model Setup
The model is an extension of the MSSM with two extra abelian gauge groups,
$U(1)_{A}$ and $U(1)_{B}$. The first one is anomalous while the second one is
anomaly free. This assumption is quite generic since in models with several
anomalous $U(1)$ symmetries there exists a unique linear combination which is
anomalous while the other combinations are anomaly free. The charge assignment
for the chiral superfields is shown in Table 1.
| $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{A}$ | $U(1)_{B}$
---|---|---|---|---|---
$Q_{i}$ | ${\bf 3}$ | ${\bf 2}$ | $1/6$ | $q_{Q}$ | 0
$U^{c}_{i}$ | $\bar{\bf 3}$ | ${\bf 1}$ | $-2/3$ | $q_{U^{c}}$ | 0
$D^{c}_{i}$ | $\bar{\bf 3}$ | ${\bf 1}$ | $1/3$ | $q_{D^{c}}$ | 0
$L_{i}$ | ${\bf 1}$ | ${\bf 2}$ | $-1/2$ | $q_{L}$ | 0
$E^{c}_{i}$ | ${\bf 1}$ | ${\bf 1}$ | $1$ | $q_{E^{c}}$ | 0
$H_{u}$ | ${\bf 1}$ | ${\bf 2}$ | $1/2$ | $q_{H_{u}}$ | 0
$H_{d}$ | ${\bf 1}$ | ${\bf 2}$ | $-1/2$ | $q_{H_{d}}$ | 0
$\Phi^{+}$ | ${\bf 1}$ | ${\bf 1}$ | 0 | 1 | 1
$\Phi^{-}$ | ${\bf 1}$ | ${\bf 1}$ | 0 | -1 | -1
Table 1: Charge assignment.
The vector and matter chiral multiplets undergo the usual gauge
transformations
$\displaystyle V$ $\displaystyle\to$ $\displaystyle
V+i\left(\Lambda-\Lambda^{\dagger}\right)$ $\displaystyle\Phi$
$\displaystyle\to$ $\displaystyle e^{-iq\Lambda}\Phi$ (1)
The anomaly cancellation of the $U(1)_{A}$ gauge group is achieved by the four
dimensional analogue of the higher dimensional GS mechanism which involves the
Stückelberg superfield $S=s+2\theta\psi_{S}+\theta^{2}F_{S}$ transforming as a
shift
$S\to S-2iM_{V_{A}}\Lambda$ (2)
where $M_{V_{A}}$ is a mass parameter related to the anomalous $U(1)_{A}$
gauge boson mass. It turns out that not all the anomalies can be cancelled in
this way. In particular the so called mixed anomalies between anomalous and
non anomalous $U(1)$’s require the presence of trilinear GCS counterterms. For
further details about the anomaly cancellation mechanism see Appendix A (see
also for instance [17] and [24]). The effective superpotential of our model at
the scale $E=M_{V_{A}}$ is given by
$W=W_{MSSM}+\lambda e^{-kS}H_{u}H_{d}+m\Phi^{+}\Phi^{-}$ (3)
where $W_{MSSM}$ is given by
$W_{MSSM}=y_{u}^{ij}Q_{i}U^{c}_{j}H_{u}-y_{d}^{ij}Q_{i}D^{c}_{j}H_{d}-y_{e}^{ij}L_{i}E^{c}_{j}H_{d}$
(4)
which is the usual MSSM superpotential without the $\mu$-term which is
forbidden for a generic choice of the charges $q_{H_{u}}$ and $q_{H_{d}}$. The
second term in (3) is the only gauge invariant coupling allowed between the
Stückelberg superfield and the two Higgs fields. This is the only allowed
coupling with matter fields for a field transforming as (2). We will argue
later about how non perturbative effects can generate such a term. The last
term in (3) is a mass term for $\Phi^{\pm}$ which are charged under both
$U(1)_{A}$ and $U(1)_{B}$. These fields have been considered as supersymmetry
breaking mediators in the context of anomalous models by Dvali and Pomarol
[30]. They play a key role in generating gaugino masses. In the effective
lagrangian, besides the usual kinetic terms (they are charged under both
$U(1)_{A}$ and $U(1)_{B}$), the two $U(1)_{B}$ fields $\Phi^{\pm}$ couple to
the gauge field strength $W_{a}^{\alpha}$ through the dimension six effective
operator
${\cal{L}}_{g}=c_{a}\frac{\Phi^{+}\Phi^{-}}{\Lambda^{2}}W_{a}^{\alpha}W_{\alpha\,a}$
(5)
where $a=A,B,Y,2,3$, $\Lambda$ is the cut-off scale of the theory while
$c_{a}$ are constants that have to be computed in the UV completion of the
theory.
The non perturbative term in (3) is expected to be generated in the effective
action of intersecting D-brane models which can be considered as the UV
completion of our model. This is the leading order term when the coupling
$H_{u}H_{d}$ is not allowed by gauge invariance. In string theory there are
many axions related to the GS mechanism of anomaly cancellation which are
charged under some Ramond-Ramond (RR) form. For example in type IIA
orientifold model with D6-branes, axion fields are associated to the $C_{3}$
RR-form (see for a recent review [31]). Instantons charged under this RR-form,
such as euclidean E2-branes wrapping some $\gamma_{3}$ 3-cycle in the Calabi-
Yau (CY) compactification manifold, give a contribution to the holomorphic
couplings in the $N=1$ superpotential. Our analysis does not rely on any
concrete intersecting brane model but rather on the generic appearance of such
instanton induced terms. The exponential suppression factor of the classical
instanton action is
$e^{-{\rm Vol_{E2}}/g_{s}}$ (6)
where ${\rm Vol_{E2}}$ is the volume of the 3-cycle in the CY wrapped by a
$E2$-brane measured in string units while $g_{s}$ is the string coupling. Such
exponential factor is independent from the $d=4$ gauge coupling and thus this
instanton is usually termed as stringy or exotic instanton (see [29] and [32]
and references therein). Moreover the instanton contribution can be sizable
even in the case $g_{s}\ll 1$ if ${\rm Vol_{E2}}\ll 1$ measured in string
units.
In type IIA orientifold models with intersecting branes the complexified
moduli, whose imaginary part are the generalized axion fields (depending on
the cycle $\gamma_{3}^{i}$), can be written as
$U_{i}=e^{-\varphi}\int_{\gamma_{3}^{i}}\Omega_{3}+i\int_{\gamma_{3}^{i}}C_{3}$
(7)
where $\varphi$ is the dilaton, $\Omega_{3}$ is the CY volume 3-form (which is
a complex form) and $C_{3}$ is the RR-form. The integral of this form is dual
to the axion whose shift symmetry is gauged in the GS mechanism. The generic
contribution of an $E2$ instanton is formally given by
$W\sim\prod_{i=1}^{n}\Phi_{a_{i},\,b_{i}}e^{-S_{E2}}$ (8)
where $\Phi_{a_{i},\,b_{i}}$ are chiral superfields localized at the
intersection of two D6-branes described by open strings while $S_{E2}$ denotes
the instanton classical action
$e^{-S_{E2}}=\exp\left[-\frac{2\pi}{l_{s}^{3}}\left(\frac{1}{g_{s}}\int_{\gamma}Re(\Omega_{3})-i\int_{\gamma}C_{3}\right)\right]$
(9)
This result can be immediately extended to the supersymmetric case which
involves the complete Stückelberg multiplet. The appearance of the exponential
suppression factor is dictated by the fact that the superpotential is a
holomorphic quantity. Thus the only allowed functional dependence on the
string coupling $g_{s}=e^{<\varphi>}$ and the axionic field is an exponential.
Any other dependence can be excluded due to the shift transformation (2).
## 3 Supersymmetry Breaking
The D-term contribution of the $U(1)_{A}$ vector multiplet $V_{A}$ relevant to
supersymmetry breaking is given, in the limit of vanishing kinetic mixing
$\delta_{YA},\delta_{AB}=0$, by the following lagrangian:
$\mathcal{L}=\frac{1}{2}D_{A}D_{A}+\sum_{i}g_{A}q_{i}\phi_{i}^{\dagger}D_{A}\phi_{i}+\xi
D_{A}$ (10)
where the sum is extended to all the scalars charged under the $U(1)_{A}$.
There is no D-term contribution related to the $U(1)_{B}$ except that of
$\phi^{\pm}$ since all the MSSM chiral fields are uncharged under $U(1)_{B}$
(see Table 1). The last term in (10) is a tree-level field dependent Fayet-
Iliopoulos (FI) term which comes from the supersymmetrized Stückelberg
lagrangian
$\displaystyle\mathcal{L}_{axion}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left.\left(S+S^{\dagger}+2M_{V_{A}}V_{A}\right)^{2}\right|_{\theta^{2}\bar{\theta}^{2}}+\ldots$
(11) $\displaystyle=$ $\displaystyle
M_{V_{A}}\left.(S+S^{\dagger})V_{A}\right|_{\theta^{2}\bar{\theta}^{2}}+\ldots$
$\displaystyle=$ $\displaystyle M_{V_{A}}\alpha D_{A}+\ldots$
where in the last line $\alpha$ denotes the real part of the lowest component
of the Stückelberg chiral multiplet $s=\alpha+i\varphi$. The fields $\alpha$
and $\varphi$ are called the saxion and the axion respectively333with a slight
abuse of notation with respect to the previous section where we denoted the
dilaton with $\varphi$.. We assume that the real part $\alpha$ gets an
expectation value. This gives a contribution to the gauge coupling constants
which can be absorbed in the following redefinition
$\frac{1}{16g_{a}^{2}\tau_{a}}=\frac{1}{16\tilde{g}_{a}^{2}\tau_{a}}-\frac{1}{2}b^{aa}\langle\alpha\rangle$
(12)
where the gauge factors $\tau_{a}$ take the values $1,1,1,1/2,1/2$ and the
$b^{aa}$ constants are given in (105). The tree-level FI term is then given by
$\xi=M_{V_{A}}\left<\alpha\right>$ (13)
Moreover in the following we assume that 1-loop FI terms are absent (see the
discussion in [33]). The FI term induces a mass term for the scalars. This can
be seen by solving the equations of motion for $D_{A}$
$D_{A}+\sum_{i}g_{A}q_{i}\phi_{i}^{\dagger}\phi_{i}+\xi=0$ (14)
where the index $i$ runs over all chiral superfields. The D-term contribution
to the scalar potential is given by
$V(\phi_{i},\phi_{i}^{\dagger})=\frac{1}{2}\left(\xi+g_{A}\sum_{i}q_{i}\left|\phi_{i}\right|^{2}\right)^{2}$
(15)
The quadratic part gives the scalar mass term
$\sum_{i}\xi
g_{A}q_{i}\left|\phi_{i}\right|^{2}=\sum_{i}m_{i}^{2}\left|\phi_{i}\right|^{2}$
(16)
where we have defined
$m_{i}^{2}=\xi
g_{A}q_{i}=\left<\alpha\right>g_{A}M_{V_{A}}q_{i}=q_{i}m_{\xi}^{2}$ (17)
with
$m_{\xi}^{2}=\left<\alpha\right>g_{A}M_{V_{A}}=g_{A}\xi$ (18)
The typical scale for the mass $m_{\xi}$ is of the order of few hundreds of
GeV if $M_{V_{A}}\sim\left<\alpha\right>\sim 1$ TeV and $g_{A}\sim 0.1$. It is
interesting to note that in this scenario a low subTeV supersymmetry breaking
scale $m_{\xi}$ is due to the Stückelberg mechanism which gives mass to
$V_{A}$. This is the most important difference with the scenario proposed in
[30], where the scale $m_{\xi}$ is dynamically generated by some dynamics in a
strong coupling regime.
Mass terms for the gauginos, i.e. $\lambda_{a}\lambda_{a}$, are generated by
the dimension six effective operator (5) in the broken phase where
$\phi^{\pm}$ get vacuum expectation value (vev). The contribution coming from
this mechanism is
$M_{a}=c_{a}\frac{\langle F^{+}\phi^{-}\rangle+\langle
F^{-}\phi^{+}\rangle}{\Lambda^{2}}=c_{a}\frac{m\left(v_{+}^{2}+v_{-}^{2}\right)}{2\Lambda^{2}}$
(19)
where $v_{\pm}/\sqrt{2}=\left<\phi_{\pm}\right>$ and where in the right hand
side we have used the F-term equations of motion for $F^{\pm}$
$F^{\pm}=-\frac{\partial W^{*}}{\partial\phi^{\pm*}}=-m\phi^{\mp*}$ (20)
having assumed $m$ real without any loss in generality. We assume $c_{a}=c$
for each $a$. This is an assumption of universality as a boundary condition at
the cutoff scale $\Lambda$ which does not affect in a crucial way our
analysis. In section 4 we study the scalar potential of our model and we
derive the conditions for having a vev for $\phi^{\pm}$ different from zero.
Since we are breaking supersymmetry in the global limit in which the Planck
mass $M_{P}\to\infty$ the F-term induced contribution to the scalar masses
$m^{2}_{i}\sim\frac{\left<F_{\pm}\right>}{M_{P}^{2}}$ (21)
vanishes leaving (17) as the leading contribution.
The requirement of gauge invariance of the superpotential implies the
following constraints on the $U(1)_{A}$ charges
$\displaystyle q_{U^{c}}$ $\displaystyle=$ $\displaystyle-q_{Q}-q_{H_{u}}$
$\displaystyle q_{D^{c}}$ $\displaystyle=$ $\displaystyle-q_{Q}-q_{H_{d}}$
$\displaystyle q_{E^{c}}$ $\displaystyle=$ $\displaystyle-q_{L}-q_{H_{d}}$
(22)
and
$k=\frac{{q_{H_{u}}}+{q_{H_{d}}}}{2M_{V_{A}}}$ (23)
As we said at the beginning of this section we assume that the net kinetic
mixing between $U(1)_{Y}$ and $U(1)_{A}$ vanishes 444We postpone the
discussion about the kinetic mixing between $U(1)_{A}$ and $U(1)_{B}$ to the
next section.. There are two contributions for the $U(1)_{Y}-U(1)_{A}$ kinetic
mixing: the 1-loop mixing $\delta_{YA}$ and $b^{YA}$ coming from the GS
coupling $SW_{Y}W_{A}$ (see eq. (A)). The following conditions imply a bound
on the charges
$\displaystyle\delta_{YA}=0$ $\displaystyle\Rightarrow$
$\displaystyle\sum_{f}q_{f}Y_{f}=0$ $\displaystyle b^{YA}=0$
$\displaystyle\Rightarrow$ $\displaystyle\sum_{f}q_{f}^{2}Y_{f}=0$ (24)
where the sum is extended over all the chiral fermions in the theory. The
constraints (24) can be solved in terms of $q_{Q}$ and $q_{L}$. By using the
conditions (22) we get
$\displaystyle{q_{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(3{q_{H_{u}}}-4{q_{H_{d}}}\right)$
$\displaystyle{q_{Q}}$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\left(5{q_{H_{u}}}-2{q_{H_{d}}}\right)$ (25)
The positive squared mass condition for the sfermions
$m^{2}_{\tilde{f}}=g_{A}q_{f}M_{V_{A}}\langle\alpha\rangle>0$ (26)
implies $q_{f}>0$ for all the sfermions having assumed without loss of
generality $\langle\alpha\rangle>0$. Using the constraints (22) and (25) we
get the allowed parameter space
${q_{H_{u}}}<0\,,\quad\frac{5}{2}{q_{H_{u}}}<{q_{H_{d}}}<\frac{3}{4}{q_{H_{u}}}$
(27)
## 4 Scalar Potential
The key ingredient in our model is the instanton induced term in (3) which
couples the Stückelberg field to the Higgs fields. The $\theta^{2}$ component
of this superpotential term gives the following contribution to the lagrangian
$\displaystyle W_{inst}|_{\theta^{2}}$ $\displaystyle=$ $\displaystyle\lambda
e^{-kS}H_{u}H_{d}|_{\theta^{2}}$ (28) $\displaystyle=$ $\displaystyle\lambda
e^{-ks}h_{u}F_{d}+\lambda e^{-ks}F_{u}h_{d}-\lambda ke^{-ks}F_{S}h_{u}h_{d}+$
$\displaystyle\sqrt{2}\lambda
e^{-ks}k\left(h_{u}\psi_{S}\tilde{h}_{d}+h_{d}\psi_{S}\tilde{h}_{u}\right)-\lambda
e^{-ks}k^{2}h_{u}h_{d}\psi_{S}\psi_{S}$
where $F_{u,d}$ are the F-terms of $H_{u,d}$. Solving the F-terms equations
for $H_{u}$ and $H_{d}$ we get the following contributions for the instanton
induced term in the scalar potential
$V_{inst}=2\lambda^{2}e^{-2k\alpha}h_{u}^{\dagger}h_{u}+2\lambda^{2}e^{-2k\alpha}h_{d}^{\dagger}h_{d}+\lambda
ke^{-k\alpha}\left(e^{-ik\varphi}F_{S}h_{u}h_{d}+h.c.\right)$ (29)
In the following we assume that $\alpha$ gets a vev different from zero and
that the mass of this field is much higher than $\Lambda$ so that its dynamics
is not described by the low energy effective action. From the point of view of
the UV completion (for example a type IIA intersecting brane model) this
amounts to saying that the closed string modulus related to $\alpha$ is
stabilized. Moreover we made the assumption that the same dynamics that
stabilizes $\alpha$ also fixes $F_{S}$. By supersymmetry the saxion field
$\alpha$, being part of the Stuckelberg multiplet, has a tree-level mass
$M_{V_{A}}$. Thus if we want to consider a frozen dynamics for $\alpha$ at the
TeV scale we have to assume a mass parameter for the anomalous $U(1)_{A}$ just
slightly above the TeV scale, i.e. $M_{V_{A}}>1$ TeV. In this way the
effective instanton induced potential at a scale $E\simeq 1$ TeV is thus given
by
$V_{inst}=2\lambda^{2}e^{-2k\langle\alpha\rangle}h_{u}^{\dagger}h_{u}+2\lambda^{2}e^{-2k\langle\alpha\rangle}h_{d}^{\dagger}h_{d}+\lambda
ke^{-k\langle\alpha\rangle}\left(\langle F_{S}\rangle
e^{-ik\varphi}h_{u}h_{d}+h.c.\right)$ (30)
The first two terms are $\mu$-terms while the third one is a b-term. The
complete effective scalar potential is given by
$\displaystyle V$ $\displaystyle=$
$\displaystyle(|\mu|^{2}+m^{2}_{h_{u}})\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}\right)+(|\mu|^{2}+m^{2}_{h_{d}})\left(|h_{d}^{0}|^{2}+|h_{d}^{-}|^{2}\right)$
(31)
$\displaystyle+(|m|^{2}+m^{2}_{\phi^{+}})|\phi^{+}|^{2}+(|m|^{2}+m^{2}_{\phi^{-}})|\phi^{-}|^{2})$
$\displaystyle+\left[be^{-ik\varphi}\left(h_{u}^{+}h_{d}^{-}-h_{u}^{0}h_{d}^{0}\right)+h.c.\right]$
$\displaystyle+\frac{1}{8}(g_{2}^{2}+g_{Y}^{2})\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}-|h_{d}^{0}|^{2}-|h_{d}^{-}|^{2}\right)^{2}+\frac{1}{2}g_{2}^{2}\left|h_{u}^{+}h_{d}^{0*}+h_{u}^{0}h_{d}^{-*}\right|^{2}$
$\displaystyle+\frac{1}{2}g_{A}^{2}\left[{q_{H_{u}}}\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}\right)+{q_{H_{d}}}\left(|h_{d}^{0}|^{2}+|h_{d}^{-}|^{2}\right)+|\phi^{+}|^{2}-|\phi^{-}|^{2}\right]^{2}$
$\displaystyle+\frac{1}{2}g_{B}^{2}\left[|\phi^{+}|^{2}-|\phi^{-}|^{2}\right]^{2}$
where
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\sqrt{2}\lambda
e^{-k\langle\alpha\rangle}$ (32) $\displaystyle b$ $\displaystyle=$
$\displaystyle\lambda ke^{-k\langle\alpha\rangle}\langle F_{S}\rangle$ (33)
These relations give a solution of the well known $\mu$-problem since both
terms have a common origin (see the analysis in Sec. (8.2)). The soft squared
masses are generated by the FI $U(1)_{A}$ term
$\displaystyle m^{2}_{h_{u}}$ $\displaystyle=$ $\displaystyle
q_{H_{u}}m_{\xi}^{2}$ (34) $\displaystyle m^{2}_{h_{d}}$ $\displaystyle=$
$\displaystyle q_{H_{d}}m_{\xi}^{2}$ (35) $\displaystyle m^{2}_{\phi^{+}}$
$\displaystyle=$ $\displaystyle m_{\xi}^{2}$ (36) $\displaystyle
m^{2}_{\phi^{-}}$ $\displaystyle=$ $\displaystyle-m_{\xi}^{2}$ (37)
with $m_{\xi}^{2}$ given by (17). The scalar potential depends on the
following new parameters: $\langle\alpha\rangle$, $\langle F_{S}\rangle$,
$\lambda$, $m$, $g_{A,B}$, $q_{H_{u,d}}$, $M_{V_{A}}$.
In order to have a vacuum preserving the electromagnetism the charged field
vevs must vanish. Thus we are left with the problem of finding a minimum for
the neutral scalar potential
$\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle(|\mu|^{2}+m^{2}_{h_{u}})|h_{u}^{0}|^{2}+(|\mu|^{2}+m^{2}_{h_{d}})|h_{d}^{0}|^{2}-(b\,e^{-ik\varphi}\,h_{u}^{0}h_{d}^{0}+h.c.)$
$\displaystyle+(|m|^{2}+m^{2}_{\phi^{+}})|\phi^{+}|^{2}+(|m|^{2}+m^{2}_{\phi^{-}})|\phi^{-}|^{2}$
$\displaystyle+\frac{1}{8}(g_{2}^{2}+g_{Y}^{2})\left(|h_{u}^{0}|^{2}-|h_{d}^{0}|^{2}\right)^{2}$
$\displaystyle+\frac{1}{2}g_{A}^{2}\left(q_{H_{u}}|h_{u}^{0}|^{2}+q_{H_{d}}|h_{d}^{0}|^{2}+|\phi^{+}|^{2}-|\phi^{-}|^{2}\right)^{2}$
$\displaystyle+\frac{1}{2}g_{B}^{2}\left[|\phi^{+}|^{2}-|\phi^{-}|^{2}\right]^{2}$
Since there are no D-flat directions along which the quartic part vanishes,
the potential is always bounded from below. To find the minimum we solve
$\partial V_{0}/\partial z^{i}=0$ where the scalar field $z^{i}$ runs over
$\\{\varphi,h_{u}^{0},h_{d}^{0},\phi^{+},\phi^{-}\\}$. The conditions for
having a non-trivial minimum boils down to the same condition of the MSSM
$b^{2}>(|\mu|^{2}+m^{2}_{h_{u}})(|\mu|^{2}+m^{2}_{h_{d}})$ (39)
Moreover in order to generate a mass term for the gauginos (see eq. (19)) the
condition $v_{-}\neq 0$ must hold since $v_{+}=0$ due to the positive sign of
the coefficient of the $\phi^{+}$ quadratic term in (36). This implies the
following condition for the coefficient of the $\phi^{-}$ quadratic term
$|m|^{2}+m^{2}_{\phi^{-}}<0$ (40)
The minimum is attained at $\varphi=\phi^{+}=0$. Actually since the potential
for the axion $\varphi$ is periodic the minimum condition holds for
$\varphi=2n\pi/k$ with $n\in\mathbb{Z}$. All these minima are physically
equivalent and thus we arbitrarily choose $n=0$. The remaining three
conditions imply the following constraints on the parameters
$\displaystyle
m_{h_{d}}^{2}+\mu^{2}-b\,t_{\beta}+\frac{1}{8}(g_{Y}^{2}+g_{2}^{2})v^{2}c_{2\beta}+\frac{1}{2}g_{A}^{2}{q_{H_{d}}}\left[v^{2}\left({q_{H_{d}}}c_{\beta}^{2}+{q_{H_{u}}}s_{\beta}^{2}\right)-v_{-}^{2}\right]$
$\displaystyle=$ $\displaystyle 0$ (41) $\displaystyle
m_{h_{u}}^{2}+\mu^{2}-b\,t_{\beta}^{-1}-\frac{1}{8}(g_{Y}^{2}+g_{2}^{2})v^{2}c_{2\beta}+\frac{1}{2}g_{A}^{2}{q_{H_{u}}}\left[v^{2}\left({q_{H_{d}}}c_{\beta}^{2}+{q_{H_{u}}}s_{\beta}^{2}\right)-v_{-}^{2}\right]$
$\displaystyle=$ $\displaystyle 0$ (42)
$\displaystyle\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}-g_{A}^{2}v^{2}\left({q_{H_{d}}}c_{\beta}^{2}+{q_{H_{u}}}s_{\beta}^{2}\right)+2\left(|m|^{2}+m^{2}_{\phi^{-}}\right)$
$\displaystyle=$ $\displaystyle 0$ (43)
where we have defined in order to keep a compact notation
$c_{\beta}=\cos\beta,\quad s_{\beta}=\sin\beta,\quad t_{\beta}=\tan\beta,\quad
c_{2\beta}=\cos(2\beta),\quad s_{2\beta}=\sin(2\beta)$ (44)
and as usual as $\tan\beta=v_{u}/v_{d}$.
In the previous discussion we treated the scalar potential in an exact way. In
the following we want to introduce some useful approximation in order to
compute the mass eigenstates. Let us go back to the minima equations (41-43).
Supposing
$v\ll v_{-}$ (45)
we can neglect all the $g_{A}v$ terms. With this approximation the minima
equations read
$\displaystyle\tilde{m}_{h_{d}}^{2}+\mu^{2}-b\,t_{\beta}+\frac{1}{8}(g_{Y}^{2}+g_{2}^{2})v^{2}c_{2\beta}$
$\displaystyle=$ $\displaystyle 0$ (46)
$\displaystyle\tilde{m}_{h_{u}}^{2}+\mu^{2}-b\,t_{\beta}^{-1}-\frac{1}{8}(g_{Y}^{2}+g_{2}^{2})v^{2}c_{2\beta}$
$\displaystyle=$ $\displaystyle 0$ (47)
$\displaystyle\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}+2\left(|m|^{2}+m^{2}_{\phi^{-}}\right)$
$\displaystyle=$ $\displaystyle 0$ (48)
where we have defined
$\displaystyle\tilde{m}_{h_{d}}^{2}$ $\displaystyle=$ $\displaystyle
m_{h_{d}}^{2}-\frac{1}{2}g_{A}^{2}{q_{H_{d}}}v_{-}^{2}$ (49)
$\displaystyle\tilde{m}_{h_{u}}^{2}$ $\displaystyle=$ $\displaystyle
m_{h_{u}}^{2}-\frac{1}{2}g_{A}^{2}{q_{H_{u}}}v_{-}^{2}$ (50)
Equations (46) and (47) have the same functional form as in the MSSM case.
Moreover $v_{-}$ does not depend on any parameter of the visible sector.
Within this approximation the dynamics of the fields $\phi^{\pm}$ is decoupled
from that of the Higgs sector and thus the Higgs potential can be studied by
fixing $\phi^{\pm}$ at their vevs. We get
$\displaystyle V$ $\displaystyle\simeq$
$\displaystyle(|\mu|^{2}+m^{2}_{h_{u}})\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}\right)+(|\mu|^{2}+m^{2}_{h_{d}})\left(|h_{d}^{0}|^{2}+|h_{d}^{-}|^{2}\right)$
(51)
$\displaystyle+\left[be^{-ik\varphi}\left(h_{u}^{+}h_{d}^{-}-h_{u}^{0}h_{d}^{0}\right)+h.c.\right]$
$\displaystyle+{1\over
8}(g_{2}^{2}+g_{Y}^{2})\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}-|h_{d}^{0}|^{2}-|h_{d}^{-}|^{2}\right)^{2}+{1\over
2}g_{2}^{2}\left|h_{u}^{+}h_{d}^{0*}+h_{u}^{0}h_{d}^{-*}\right|^{2}$
$\displaystyle+{1\over
2}g_{A}^{2}\left[{q_{H_{u}}}\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}\right)+{q_{H_{d}}}\left(|h_{d}^{0}|^{2}+|h_{d}^{-}|^{2}\right)-\frac{1}{2}v_{-}^{2}\right]^{2}$
neglecting further constant terms in $v_{-}$. Close to the minima the relevant
term in the last line of eq. (51) is the double product of the Higgs part with
the $v_{-}^{2}$ term. Hence by using (45) we finally get
$\displaystyle V_{h,\varphi}$ $\displaystyle\simeq$
$\displaystyle(|\mu|^{2}+\tilde{m}^{2}_{h_{u}})\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}\right)+(|\mu|^{2}+\tilde{m}^{2}_{h_{d}})\left(|h_{d}^{0}|^{2}+|h_{d}^{-}|^{2}\right)$
$\displaystyle+\left[be^{-ik\varphi}\left(h_{u}^{+}h_{d}^{-}-h_{u}^{0}h_{d}^{0}\right)+h.c.\right]$
$\displaystyle+{1\over
8}(g_{2}^{2}+g_{Y}^{2})\left(|h_{u}^{0}|^{2}+|h_{u}^{+}|^{2}-|h_{d}^{0}|^{2}-|h_{d}^{-}|^{2}\right)^{2}+{1\over
2}g_{2}^{2}\left|h_{u}^{+}h_{d}^{0*}+h_{u}^{0}h_{d}^{-*}\right|^{2}$
This potential has the same form (except for the contribution of the
exponential term in $\varphi$) of the MSSM potential and the corresponding
minima equations are exactly given in eqs (46) and (47). Thus all the well
known MSSM results apply here [36].
In particular one of the constraints is $t_{\beta}\gtrsim 1.2$ [36] which
implies555The presence of the extra field $\varphi$ does not affect this
result since the minima conditions are the same as the MSSM.
$\tilde{m}^{2}_{h_{u}}<\tilde{m}^{2}_{h_{d}}$. By using the equations (49) and
(50) we get
$g_{A}{q_{H_{u}}}\left(M_{V_{A}}\langle\alpha\rangle-\frac{1}{2}g_{A}v_{-}^{2}\right)<g_{A}{q_{H_{d}}}\left(M_{V_{A}}\langle\alpha\rangle-\frac{1}{2}g_{A}v_{-}^{2}\right)$
(53)
By assuming $M_{V_{A}}>1$ TeV, $v_{-}$ in the TeV range, $g_{A}\sim O(0.1)$
the term between brackets is positive and we get the following constraint
${q_{H_{u}}}<{q_{H_{d}}}$ (54)
for the $U(1)_{A}$ Higgs charges.
### 4.1 Higgs mass matrices
We discuss the mass eigenvalues starting from the exact form of the scalar
potential (31), switching to the approximated expression (4) when needed. In
the neutral sector the singlet scalar $\phi^{+}$ does not mix with any other
scalar so it is a mass eigenstate with square mass
$M_{\phi^{+}}^{2}=2|m|^{2}$ (55)
The same holds for the imaginary part of $\phi^{-}$ which becomes the
longitudinal mode of the gauge vector $Z_{2}$. The mass matrix for the real
scalar fields $\\{\varphi,Im(h_{u}^{0}),Im(h_{d}^{0})\\}$ is given by
${\cal M}_{S}^{(Im)}=\left(\begin{array}[]{ccc}b\,t_{\beta}&\dots&\dots\\\
b&b\,t_{\beta}^{-1}&\dots\\\
-b\,k\,v\,s_{\beta}&-b\,k\,v\,c_{\beta}&b\,k^{2}\,v^{2}\,c_{\beta}s_{\beta}\end{array}\right)$
(56)
The determinant of this matrix is zero. Two eigenvalues are zero which
correspond to the Goldstone modes of $Z_{0}$ and $Z_{1}$. The physical massive
state is an axi-higgs state with mass given by
$M_{A^{0}}^{2}=\frac{2b}{s_{2\beta}}\left[1-\frac{1}{16}\frac{\left({q_{H_{u}}}+{q_{H_{d}}}\right)^{2}v^{2}}{M_{V_{A}}^{2}}s_{2\beta}^{2}\right]$
(57)
where we used the relation (23). The mass matrix for the real scalar fields
$\\{Re(h_{u}^{0})$, $Re(h_{d}^{0})$, $\phi_{R}^{-}\equiv Re(\phi^{-})\\}$
reads as
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal M}_{S}^{(Re)}=$ (58)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!{\left(\begin{array}[]{ccc}\left(\frac{1}{4}g_{EW}^{2}+g_{A}^{2}{q_{H_{d}}}^{2}\right)v^{2}c_{\beta}^{2}+bt_{\beta}&\dots&\dots\\\
\\!\\!-b-\left(\frac{1}{4}g_{EW}^{2}-g_{A}^{2}{q_{H_{d}}}{q_{H_{u}}}\right)v^{2}c_{\beta}s_{\beta}&\left(\frac{1}{4}g_{EW}^{2}+g_{A}^{2}{q_{H_{u}}}^{2}\right)v^{2}s_{\beta}^{2}+bt_{\beta}^{-1}&\dots\\\
-g_{A}^{2}{q_{H_{d}}}v\,v_{-}c_{\beta}&-g_{A}^{2}{q_{H_{u}}}v\,v_{-}s_{\beta}&\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}\\\
\end{array}\right)}$ (62)
where $g_{EW}^{2}=(g_{Y}^{2}+g_{2}^{2})$. The matrix can be diagonalized
exactly but the results are cumbersome and difficult to read. It is much more
convenient starting from the approximated potential (4) neglecting the mixing
between Higgses and $\phi^{-}$. In this case we can apply the MSSM equations
and get the following mass eigenvalues
$\displaystyle\\!\\!\\!\\!\\!M^{2}_{h^{0},H^{0}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\left(\frac{2b}{s_{2\beta}}\mp\sqrt{\left(\frac{2b}{s_{2\beta}}-\frac{1}{4}(g_{Y}^{2}+g_{2}^{2})v^{2}\right)^{2}+2b(g_{Y}^{2}+g_{2}^{2})v^{2}s_{2\beta}}\right)$
(63) $\displaystyle\\!\\!\\!\\!\\!M_{\phi^{-}_{R}}^{2}$ $\displaystyle\simeq$
$\displaystyle\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}$ (64)
The charged sector is unchanged with respect to the MSSM, so
$M^{2}_{H^{\pm}}=\frac{2b}{s_{2\beta}}+M_{W}^{2}$ (65)
As in the standard MSSM case the mass of the lightest Higgs $M_{h^{0}}$ has a
theoretical bound. It is a well known problem in the MSSM that the upper bound
[37] is not compatible with the LEP bound [38]. In our case the bound is
increased due to the presence of $D_{A}$-term corrections
$M_{h^{0}}^{2}<\frac{1}{4}(g_{Y}^{2}+g_{2}^{2})v^{2}c^{2}_{2\beta}+\frac{1}{4}g_{A}^{2}v^{2}\left[{q_{H_{d}}}+{q_{H_{u}}}+\left({q_{H_{d}}}-{q_{H_{u}}}\right)c_{2\beta}\right]^{2}$
(66)
where the first term is the MSSM bound. In principle, for arbitrary high
values of $g_{A}{q_{H_{d}}}$, $g_{A}{q_{H_{u}}}$ we get an increasing upper
bound. However, as in the standard MSSM case, $M_{h^{0}}^{2}$ undergoes to
relatively drastic quantum corrections [36]. Hence in Section 8 we consider
tree-level masses for all the particles except for $h_{0}$ for which we use
the 1-loop corrected expression (see eq. (103)).
## 5 Vector mass matrix
We now discuss the vector mass matrix. All the neutral scalars could in
principle take a vev different from zero, hence we assume
$\displaystyle\langle\phi_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\frac{v_{\pm}}{\sqrt{2}}$ (67) $\displaystyle\langle
h^{0}_{u,d}\rangle$ $\displaystyle=$ $\displaystyle\frac{v_{u,d}}{\sqrt{2}}$
(68)
The neutral vector square mass matrix in the base
$(V_{B},V_{A},V_{Y},V_{2}^{3})$ is
${\cal
M}_{V}=\left({\begin{array}[]{cccc}g_{B}^{2}v_{\phi}^{2}&\dots&\dots&\dots\\\
g_{A}g_{B}v_{\phi}^{2}&g_{A}^{2}\left[\left(c_{\beta}^{2}{q_{H_{d}}}^{2}+s_{\beta}^{2}{q_{H_{u}}}^{2}\right)v^{2}+v_{\phi}^{2}\right]+M_{V_{A}}^{2}&\dots&\dots\\\
0&\frac{1}{2}g_{A}g_{Y}q_{H}(\beta)v^{2}&\frac{1}{4}g_{Y}^{2}v^{2}&\dots\\\
0&-\frac{1}{2}g_{A}g_{2}q_{H}(\beta)v^{2}&-\frac{1}{4}g_{Y}g_{2}v^{2}&\frac{1}{4}g_{2}^{2}v^{2}\end{array}}\right)$
(69)
where
$\displaystyle v_{\phi}^{2}$ $\displaystyle=$ $\displaystyle
v_{+}^{2}+v_{-}^{2}$ (70) $\displaystyle v^{2}$ $\displaystyle=$
$\displaystyle v_{u}^{2}+v_{d}^{2}$ (71) $\displaystyle q_{H}(\beta)$
$\displaystyle=$
$\displaystyle\left(s_{\beta}^{2}{q_{H_{u}}}-c_{\beta}^{2}{q_{H_{d}}}\right)$
(72)
By taking $M_{V_{A}}>1$ TeV (see Sec. 4), $V_{A}$ can be considered as
decoupled from the low energy gauge sector (namely $E\lesssim 1$ TeV), and we
can ignore with very good approximation any mixing term666 The kinetic mixing
between $U(1)_{A}$ and $U(1)_{B}$ deserves some comment, in particular if we
relax the $M_{V_{A}}>1$ TeV assumption. Actually the presence of this mixing
turns out to be irrelevant for the phenomenology of the visible sector. Anyway
one has to take into account that for
$\textnormal{Tr}\left(q_{A}q_{B}\right)\neq 0$ such a mixing arises at the
1-loop level. In such a case it can be assumed that the two $U(1)$’s are in
the kinetic diagonalized basis with $\textnormal{Tr}\left(q_{A}q_{B}\right)=0$
thanks to some additional heavy chiral multiplet charged under both $U(1)_{A}$
and $U(1)_{B}$. These multiplets generate a counterterm in the effective
theory that cancels against $\delta_{AB}$ making the net kinetic mixing term
equal to zero. This mechanism is analogous to the anomaly cancellation one
where the GS mechanism can be generated by an anomaly free theory with some
heavy chiral fermion integrated out of the mass spectrum [24]. involving
$V_{A}$. From now on we will apply this approximation.
Since $V_{B}$ is a hidden gauge boson, it is decoupled from the SM sector. The
charged vector sector is unchanged with respect to the MSSM, so
$\displaystyle W^{\pm}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{V_{2}^{1\mu}\mp iV_{2}^{2\mu}}{\sqrt{2}}$ (73)
$\displaystyle M^{2}_{W}$ $\displaystyle=$
$\displaystyle\frac{1}{4}g_{2}^{2}v^{2}$ (74)
## 6 Neutralinos
In comparison with the standard MSSM we now have five new neutral fermionic
fields: $\psi_{S}$, $\lambda_{A}$, $\lambda_{B}$, $\tilde{\phi}^{\pm}$.
However under the assumption $M_{V_{A}}>1$ TeV, $\psi_{S}$ and $\lambda_{A}$
are not in the low energy sector because of the $M_{V_{A}}$ mass term777 We
stress that the $\psi_{S}-\lambda_{A}$ sector presents a different parameters
choice with respect to [39]-[41], where we realized a scenario in which the
mixing between $\psi_{S}$ and $\lambda_{A}$ was suppressed.. Thus we have
$\mathcal{L}_{\mbox{neutralino mass}}=-\frac{1}{2}(\psi^{0})^{T}{\cal
M}_{\tilde{N}}\psi^{0}+h.c.$ (75)
where
$(\psi^{0})^{T}=(\lambda_{B},\ \tilde{\phi}^{-},\ \tilde{\phi}^{+},\
\lambda_{Y},\ \lambda_{2}^{0},\ \tilde{h}_{d}^{0},\ \tilde{h}_{u}^{0})$ (76)
In this basis the neutralino mass matrix ${\cal M}_{\tilde{N}}$ is written as
${\cal
M}_{\tilde{N}}={\left(\begin{array}[]{ccccccccc}M_{B}&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\
-g_{B}v_{-}&0&\ldots&\ldots&\ldots&\ldots&\ldots\\\
0&-m&0&\ldots&\ldots&\ldots&\ldots\\\ 0&0&0&M_{1}&\ldots&\ldots&\ldots\\\
0&0&0&0&M_{2}&\ldots&\ldots\\\
0&0&0&-\frac{g_{1}v_{d}}{2}&\frac{g_{2}v_{d}}{2}&0&\ldots\\\
0&0&0&\frac{g_{1}v_{u}}{2}&-\frac{g_{2}v_{u}}{2}&-\mu&0\end{array}\right)}$
(77)
where $\mu$ is given in eq. (32). We remind that gaugino masses arise from the
Dvali-Pomarol term (5).
${\cal M}_{\tilde{N}}$ factorizes in a $4\times 4$ MSSM block in the lower
right corner, and in a $3\times 3$ new sector block in the upper left corner.
The new sector block is given by the $\lambda_{B}$ and $\tilde{\phi}^{\pm}$
contributions. This last block has a MSSM-like structure that can be easily
understood just considering the superpotential (3), the gaugino masses (19)
and by reminding that $\phi^{-}$ gets a vev $v_{-}$ different from zero, while
$v_{+}=0$.
Finally there are also corrections coming from the anomalous axino couplings:
F-term couplings of the type $b^{aa}\langle
F_{S}\rangle\lambda_{a}\lambda_{a}$ and D-term couplings of the type
$b^{aa}\psi_{S}\lambda_{a}\langle D_{a}\rangle$, and corrections coming from
the superpotential term $e^{-kS}H_{u}H_{d}+h.c.$. However such corrections are
always subdominant and thus we neglect them with very good approximation.
We assume the lightest supersymmetric particle (LSP) in our model comes from
the neutralino sector. In Sec. 8 we show the parameter regions in which this
holds true. In order to ensure that the neutralino is the LSP we keep fixed
the gravitino mass $m_{3/2}\sim{\rm O(TeV)}$ in the limit $M_{P}\to\infty$.
## 7 Sfermion masses
The sfermion masses receive several contributions. As we have seen in Sec. 3
the leading contribution comes from the induced soft masses (17). But there
are further contributions. We have MSSM-like contributions: F-term corrections
proportional to the Yukawa couplings, $D_{Y}$ and $D_{2}$ term correction from
the Higgs sector. Moreover there are $D_{A}$ term corrections from the Higgs
and $\phi^{-}$ sector. As an aside, the appearance of such terms in the low-
energy action, given our assumption $M_{V_{A}}>1$ TeV, can be understood in
terms of quantum corrections to Kahler potential [34]. Considering the first
two families we neglect the corresponding Yukawa couplings (the so called
third family approximation). In this approximation the mass eigenvalues are
given by
$\displaystyle m^{2}_{\tilde{u}_{L}}\simeq m^{2}_{\tilde{c}_{L}}$
$\displaystyle=$ $\displaystyle
m^{2}_{\tilde{Q}}+\left(\frac{1}{3}g_{Y}^{2}-g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{Q}}\tilde{m}^{2}_{D_{A}}$ (78) $\displaystyle
m^{2}_{\tilde{u}_{R}}\simeq m^{2}_{\tilde{c}_{R}}$ $\displaystyle=$
$\displaystyle m^{2}_{\tilde{U}^{c}}-g_{Y}^{2}\frac{\Delta
v^{2}}{6}+{q_{U^{c}}}\tilde{m}^{2}_{D_{A}}$ (79) $\displaystyle
m^{2}_{\tilde{d}_{L}}\simeq m^{2}_{\tilde{s}_{L}}$ $\displaystyle=$
$\displaystyle
m^{2}_{\tilde{Q}}+\left(\frac{1}{3}g_{Y}^{2}+g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{Q}}\tilde{m}^{2}_{D_{A}}$ (80) $\displaystyle
m^{2}_{\tilde{d}_{R}}\simeq m^{2}_{\tilde{s}_{R}}$ $\displaystyle=$
$\displaystyle m^{2}_{\tilde{D}^{c}}+g_{Y}^{2}\frac{\Delta
v^{2}}{12}+{q_{D^{c}}}\tilde{m}^{2}_{D_{A}}$ (81) $\displaystyle
m^{2}_{\tilde{\nu}_{e}}=m^{2}_{\tilde{\nu}_{\mu}}$ $\displaystyle=$
$\displaystyle m^{2}_{\tilde{L}}-\left(g_{Y}^{2}+g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{L}}\tilde{m}^{2}_{D_{A}}$ (82) $\displaystyle
m^{2}_{\tilde{e}_{L}}\simeq m^{2}_{\tilde{\mu}_{L}}$ $\displaystyle=$
$\displaystyle m^{2}_{\tilde{L}}-\left(g_{Y}^{2}-g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{L}}\tilde{m}^{2}_{D_{A}}$ (83) $\displaystyle
m^{2}_{\tilde{e}_{R}}\simeq m^{2}_{\tilde{\mu}_{R}}$ $\displaystyle=$
$\displaystyle m^{2}_{\tilde{E}^{c}}+g_{Y}^{2}\frac{\Delta
v^{2}}{4}+{q_{E^{c}}}\tilde{m}^{2}_{D_{A}}$ (84)
The first terms on the right hand side
$m^{2}_{{\tilde{Q}},{\tilde{U}^{c}},{\tilde{D}^{c}},{\tilde{L}},{\tilde{E}^{c}}}$
are the corresponding soft masses (17), the second terms are the $D_{Y,2}$
contributions with $\Delta v^{2}=v_{u}^{2}-v_{d}^{2}=-v^{2}c_{2\beta}$, while
the last terms are the $D_{A}$ corrections given by
$\tilde{m}^{2}_{D_{A}}=\frac{1}{2}\left({q_{H_{u}}}v_{u}^{2}+{q_{H_{d}}}v_{d}^{2}-v_{-}^{2}\right)$
(85)
There is an approximated degeneracy between the sfermions with the same
charges.
The mass matrix for the third family sfermions is parametrized as
${\cal
M}_{\tilde{f}}^{2}=\left(\begin{array}[]{cc}{M^{\tilde{f}}_{LL}}^{2}\,\,{M^{\tilde{f}}_{LR}}^{2}\\\
{M^{\tilde{f}}_{LR}}^{2}\,\,{M^{\tilde{f}}_{RR}}^{2}\end{array}\right)$ (86)
where the off-diagonal terms are generated by F-term corrections proportional
to the Yukawa couplings. The stop mass matrix elements are
$\displaystyle{M^{\tilde{t}}_{LL}}^{2}=m_{t}^{2}+m^{2}_{\tilde{Q}}+\left(\frac{1}{3}g_{Y}^{2}-g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{Q}}\tilde{m}^{2}_{D_{A}}$
$\displaystyle{M^{\tilde{t}}_{RR}}^{2}=m_{t}^{2}+m^{2}_{\tilde{U}^{c}}-g_{Y}^{2}\frac{\Delta
v^{2}}{6}+{q_{U^{c}}}\tilde{m}^{2}_{D_{A}}$
$\displaystyle{M^{\tilde{t}}_{LR}}^{2}=-\mu\,m_{t}\,t_{\beta}^{-1}$ (87)
The sbottom mass matrix elements are
$\displaystyle{M^{\tilde{b}}_{LL}}^{2}=m_{b}^{2}+m^{2}_{\tilde{Q}}+\left(\frac{1}{3}g_{Y}^{2}+g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{Q}}\tilde{m}^{2}_{D_{A}}$
$\displaystyle{M^{\tilde{b}}_{RR}}^{2}=m_{b}^{2}+m^{2}_{\tilde{D}^{c}}+g_{Y}^{2}\frac{\Delta
v^{2}}{12}+{q_{D^{c}}}\tilde{m}^{2}_{D_{A}}$
$\displaystyle{M^{\tilde{b}}_{LR}}^{2}=-\mu\,m_{b}\,t_{\beta}$ (88)
The stau mass matrix elements are
$\displaystyle{M^{\tilde{\tau}}_{LL}}^{2}=m_{\tau}^{2}+m^{2}_{\tilde{L}}-\left(g_{Y}^{2}-g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{L}}\tilde{m}^{2}_{D_{A}}$
$\displaystyle{M^{\tilde{\tau}}_{RR}}^{2}=m_{\tau}^{2}+m^{2}_{\tilde{E}^{c}}+g_{Y}^{2}\frac{\Delta
v^{2}}{4}+{q_{E^{c}}}\tilde{m}^{2}_{D_{A}}$
$\displaystyle{M^{\tilde{\tau}}_{LR}}^{2}=-\mu\,m_{\tau}\,t_{\beta}$ (89)
The tau sneutrino mass is
$m^{2}_{\tilde{\nu}_{\tau}}=m^{2}_{\tilde{L}}-\left(g_{Y}^{2}+g_{2}^{2}\right)\frac{\Delta
v^{2}}{8}+{q_{L}}\tilde{m}^{2}_{D_{A}}$ (90)
where $m_{t}$, $m_{b}$ and $m_{\tau}$ are the masses of the corresponding
standard fermions (i.e. further F-term contributions proportional to the
Yukawa couplings). The structure of the diagonal terms of (86) is the same as
in eq. (78)-(84): soft masses, MSSM D-term contribution and $D_{A}$ term
correction. Furthermore we stress that there is a mass degeneracy between the
three sneutrinos $\tilde{\nu}_{e,\mu,\tau}$ since the soft masses (17) are
flavor blind.
## 8 Phenomenology
In the following we derive the phenomenological consequences of our scenario.
Following our assumption of having a mass parameter for the anomalous
$U(1)_{A}$ just slightly above the TeV scale, we fix $M_{V_{A}}=10$ TeV. The
mass scale in the gaugino sector $\Lambda$ is set to be $O(M_{V_{A}})$.
### 8.1 Charge Bounds
The model parameter space can in principle be constrained by precision EW
measurements [35].
Figure 1: Higgs couplings bounds. The yellow spot represents our charge
choice.
However, since $M_{V_{A}}=10$ TeV every value of $g_{A}{q_{H_{u}}}$ and
$g_{A}{q_{H_{d}}}$ is allowed by EW precision data if
$|g_{A}{q_{H_{u}}}|,|g_{A}{q_{H_{d}}}|\lesssim 0.1$. So the only relevant
constraints are (27) and (54), that are plotted respectively with a red and a
blue region, in Fig. 1 in the plane ($g_{A}{q_{H_{u}}}$, $g_{A}{q_{H_{d}}}$).
### 8.2 Free Parameters
Here we discuss which parameters remain free in our model after all the
constraints discussed in the previous sections are imposed. Our choice for the
Higgs $U(1)_{A}$ charges corresponds to the yellow spot in Fig. 1
$\displaystyle g_{A}=0.1\qquad M_{V_{A}}=10\text{ TeV}$
$\displaystyle{q_{H_{d}}}=-(1/3)\qquad{q_{H_{u}}}=-(2/5)$ (91)
In order to fix the remaining parameters ($\langle\alpha\rangle$, $\langle
F_{S}\rangle$, $\lambda$, $m$, $g_{B}$) we assume $v\simeq 246$ GeV and then
we choose some benchmark value for $g_{B}$ and $v_{-}$ in the $U(1)_{B}$
sector888We remind that $v_{+}=0$ (see Section (4)).:
$\displaystyle A)\quad g_{B}=0.4\qquad v_{-}=5\text{ TeV}$ (92) $\displaystyle
B)\quad g_{B}=0.1\qquad v_{-}=4\text{ TeV}$ (93)
The next step is to solve the minima conditions (41)-(43) determining $\langle
F_{S}\rangle$, $\lambda$, $m$ as function of $\langle\alpha\rangle$. In the
limit in which $v^{2}\ll M_{V_{A}}\langle\alpha\rangle,v_{-}^{2}$, we get
$\displaystyle\lambda^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{8}\,e^{\frac{2\langle\alpha\rangle
g_{A}({q_{H_{d}}}+{q_{H_{u}}})}{M_{V_{A}}}}\Big{[}g_{A}\left(g_{A}v_{-}^{2}-2\langle\alpha\rangle
M_{V_{A}}\right)(\sec(2\beta)({q_{H_{d}}}-{q_{H_{u}}})+{q_{H_{d}}}+{q_{H_{u}}})\Big{]}$
$\displaystyle\langle F_{S}\rangle$ $\displaystyle\simeq$
$\displaystyle-\,e^{\frac{\langle\alpha\rangle
g_{A}({q_{H_{d}}}+{q_{H_{u}}})}{M_{V_{A}}}}\frac{M_{V_{A}}\tan(2\beta)}{4({q_{H_{d}}}+{q_{H_{u}}})\lambda}({q_{H_{d}}}-{q_{H_{u}}})\left(2\langle\alpha\rangle
M_{V_{A}}-g_{A}v_{-}^{2}\right)$ $\displaystyle|m|^{2}$ $\displaystyle\simeq$
$\displaystyle g_{A}\langle\alpha\rangle
M_{V_{A}}-\frac{1}{2}\left[\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}\right]$
(94)
In Appendix B we report the exact formulae. Thus the only remaining free
parameters are $t_{\beta}$ and $\langle\alpha\rangle$ and we perform the
following analysis of the mass spectrum as a function of $t_{\beta}$ and
$\langle\alpha\rangle$. A lower bound on $\langle\alpha\rangle$ as a function
of $v_{-}$ can be obtained, given the approximation (45), from the equation
(48)
$\langle\alpha\rangle\simeq\frac{|m|^{2}+\left(1/2\right)\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}}{g_{A}M_{V_{A}}}$
(95)
where we used the relation
$m_{\phi^{-}}^{2}=-m_{\xi}^{2}=-\langle\alpha\rangle g_{A}M_{V_{A}}$ (96)
Thus the lower bound on $\langle\alpha\rangle$ is obtained simply by setting
$|m|=0$,
$\langle\alpha\rangle_{m}\simeq\frac{\left(1/2\right)\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}}{g_{A}M_{V_{A}}}$
(97)
The condition $\left<\alpha\right>>\langle\alpha\rangle_{m}$ must hold since
otherwise we would have a massless scalar field in the spectrum (see eq.
(55)). Another lower bound, $\langle\alpha\rangle_{b}$, can be obtained from
the condition (39), by solving the minima conditions (41)-(43) and by
substituting the corresponding $\langle F_{S}\rangle$, $\lambda$ and $m$
values (94). The resulting lower bound can be expressed as
$\langle\alpha\rangle>\max\left[\langle\alpha\rangle_{m},\langle\alpha\rangle_{b}\right]$
(98)
No upper bound can be imposed, hence we decide to perform our analysis by
considering $\langle\alpha\rangle\lesssim 100$ TeV.
The parameters $\lambda$ and $\langle F_{S}\rangle$ are of a particular
phenomenological importance since they appear in the $\mu$ and $b$ terms (see
eqs. (32) and (33)). In the case A, $\mu$ is in the range $(900,6000)$ GeV and
$\sqrt{b}$ is in the range $(50,1200)$ GeV while in the case B, $\mu$ is in
the range $(500,6000)$ GeV and $\sqrt{b}$ is in the range $(25,1200)$ GeV.
These values are in the right range to solve the $\mu$-problem.
### 8.3 Mass spectrum
#### A)
With such choice the gauge vector sector is completely fixed up to a
$t_{\beta}$ dependence. Anyway even such a dependence can be safely ignored
with a very good approximation in the new gauge sector since the mixing is
strongly suppressed. So for each $t_{\beta}$ value we have
$\displaystyle M_{Z_{1}}$ $\displaystyle\simeq$ $\displaystyle 10\text{ TeV}$
(99) $\displaystyle M_{Z_{2}}$ $\displaystyle\simeq$ $\displaystyle 2\text{
TeV}$ (100)
where with $Z_{1}$ we denote the $V_{A}$-like vector .
#### B)
As in the previous case, we just give the $Z_{1,2}$ masses
$\displaystyle M_{Z_{1}}$ $\displaystyle\simeq$ $\displaystyle 10\text{ TeV}$
(101) $\displaystyle M_{Z_{2}}$ $\displaystyle\simeq$ $\displaystyle 400\text{
GeV}$ (102)
where as in the previous case $Z_{1}$ is $V_{A}$-like.
We will not give the exact values of the $Z_{0}$ mass. It is enough for our
purposes to know that they are compatible with the bounds of Section 8.1. Both
case A and B are compatible with CDF bounds about $Z^{\prime}$ direct
production [42].
Figure 2: Allowed $\langle\alpha\rangle$ and $\tan\beta$ values for case A
(up) and case B (down), $\Lambda_{c}=5$ (left) and $\Lambda_{c}=10$ (right).
The red region is the one in which
$\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[124,126]\text{ GeV}$, the
magenta region is the one in which
$\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[114.5,131]\text{ GeV}$ and the
blue region satisfies all the mass bounds on the sparticles (from PDG) and
requires a neutralino LSP. The yellow dots are our benchmark points.
Figure 3: Allowed $\langle\alpha\rangle$ and $\tan\beta$ values for case A
(up) and case B (down), $\Lambda_{c}=5$ (left) and $\Lambda_{c}=10$ (right).
The red region is the one in which
$\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[124,126]\text{ GeV}$, the
magenta region is the one in which
$\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[114.5,131]\text{ GeV}$ and the
blue region satisfies all the mass bounds on the sparticles (from preliminary
LHC data) and requires a neutralino LSP. The yellow dots are our benchmark
points.
Figure 4: Mass spectrum, case A, $\Lambda_{c}=5$, $\langle\alpha\rangle=0.3$
TeV and $t_{\beta}=50$ (left), $\Lambda_{c}=10$, $\langle\alpha\rangle=0.5$
TeV and $t_{\beta}=10$ (right).
Figure 5: Mass spectrum, case B, $\Lambda_{c}=10$, $\langle\alpha\rangle=0.45$
TeV and $t_{\beta}=50$ (left), $\langle\alpha\rangle=8$ TeV and
$t_{\beta}=2.5$ (right).
Recent LHC data have restricted the most probable range for the Higgs particle
mass to be $[115.5,131]$ GeV (ATLAS) [47] and $[114.5,127]$ (CMS) [46].
Moreover, there are hints observed by both CMS and ATLAS of an excess of
events that might correspond to decays of a Higgs particle with a mass in a
range close to 125 GeV. So, in Fig. 2 and 3 we give region plots showing the
allowed values of $\langle\alpha\rangle$ and $t_{\beta}$ for case A (B) and
$\Lambda/\sqrt{c}=(5)10\text{ TeV}$. The red region is the one in which
$\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[124,126]\text{ GeV}$ where the
$h^{0}$ mass is computed considering 1-loop corrections. Since it turns out
that the top squarks have small mixing angle and considering the limit
$M_{A^{0}}\gg M_{Z_{0}}$, we have [36]
$\displaystyle\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}$
$\displaystyle\simeq$
$\displaystyle\left.M_{h^{0}}^{2}\right|_{\text{tree}}+\frac{3}{4\pi^{2}}s_{\beta}^{2}y_{t}^{2}m_{t}^{2}\ln\left(m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}/m_{t}^{2}\right)$
(103) $\displaystyle\simeq$
$\displaystyle\left.M_{h^{0}}^{2}\right|_{\text{tree}}+\frac{3}{2\pi^{2}}\frac{m_{t}^{4}}{v^{2}}\ln\left(m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}/m_{t}^{2}\right)$
where $\left.M_{h^{0}}\right|_{\text{tree}}$ is the tree-level $h^{0}$ mass
and we used $m_{t}=y_{t}v_{u}/2=y_{t}vs_{\beta}/2$. There is an approximated
inverse correlation between $\langle\alpha\rangle$ and $t_{\beta}$ in the
$h_{0}$ mass allowed region because the 1-loop correction in (103) increases
for increasing values of $\langle\alpha\rangle$ or $t_{\beta}$. The $h_{0}$
mass allowed region is almost the same for case A and B because of two reasons
* i.
the mixing with $\phi^{\pm}$ is suppressed
* ii.
the parameters $\tilde{m}_{h_{u}},\tilde{m}_{h_{d}},\mu,b$ in the scalar
potential (4) are ruled by the square mass parameters
$g_{A}\langle\alpha\rangle M_{V_{A}}$ and $\left(g_{A}v_{-}\right)^{2}$ and
the first one turns out to be dominant.
The magenta region satisfies a milder constraint on the light Higgs boson:
$\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[114.5,131]$ GeV. In order to be
more conservative we imposed the joint constraints of ATLAS and CMS.
The blue region satisfies all the mass bounds on the sparticles and requires a
neutralino LSP. We considered two possibilities: one more optimistic (Fig. 2)
using the PDG bounds [43, 44] and one more conservative (Fig. 3) using recent
LHC data [45]. The combination of the gluino mass bound with a neutralino LSP
is a strong constraint that reduces drastically the allowed parameter space.
In some cases there is not even a blue region, which means that we cannot
satisfy simultaneously all the mass bounds and have a neutralino LSP, so they
are completely ruled out. When the gluino mass bound is from PDG, Case A is
allowed, otherwise it is completely ruled out, and only case B for
$\Lambda/\sqrt{c}=10\text{ TeV}$ presents allowed regions. We notice that case
A favors low $\langle\alpha\rangle$ values, while case B favors big
$\langle\alpha\rangle$ values. For every allowed case we choose a benchmark
point (yellow spots in Fig. 2 and Fig. 3)
* i.
case A, $\Lambda_{c}=5$, $\langle\alpha\rangle=3$ TeV and $t_{\beta}=50$ so
that $\left.M_{h^{0}}\right|_{\text{1-loop}}\simeq 121.6$ GeV
* ii.
case A, $\Lambda_{c}=10$, $\langle\alpha\rangle=5$ TeV and $t_{\beta}=10$ so
that $\left.M_{h^{0}}\right|_{\text{1-loop}}\simeq 124.7$ GeV.
* iii.
case B, $\Lambda_{c}=10$, $\langle\alpha\rangle=4.5$ TeV and $t_{\beta}=50$ so
that $\left.M_{h^{0}}\right|_{\text{1-loop}}\simeq 125.1$ GeV.
* iv.
case B, $\Lambda_{c}=10$, $\langle\alpha\rangle=50$ TeV and $t_{\beta}=2.5$ so
that $\left.M_{h^{0}}\right|_{\text{1-loop}}\simeq 130.1$ GeV.
and we give the full mass spectrum in Fig. 4 and in Fig. 5.
All the benchmark points share some common features
* •
the LSP is the lightest neutralino of the new sector: in case A it is a
combination of $\tilde{\phi}^{\pm}$ and $\lambda_{B}$ while in case B is
almost a pure $\lambda_{B}$.
* •
an approximated mass degeneracy of $H^{0}$, $A^{0}$ and $H^{\pm}$ holds, and
their masses satisfy the bounds of [38, 48].
* •
the lightest sleptons is a sneutrino, except for $t_{\beta}=50$ when it is
$\tilde{\tau}_{1}$
* •
the lightest squark is $\tilde{u}_{L}$, except for $t_{\beta}=50$ when it is
$\tilde{b}_{1}$
* •
the first and second family left-handed squarks/sleptons are likely to be
lighter than their right-handed counterparts. This is at odds with the usual
MSSM cases [36].
* •
$\tilde{C}_{1(2)}$ is close in mass with $\tilde{N}^{\text{MSSM}}_{1(4)}$.
$\tilde{C}_{2}$ and $\tilde{N}^{\text{MSSM}}_{4}$ are heavier than all
sfermions.
* •
the gluino is close in mass to $\tilde{C}_{1}$ and
$\tilde{N}^{\text{MSSM}}_{1}$ which are gaugino-like. Moreover it is lighter
than all the squarks except for point i). So it turns out to be long lived,
specially in case B where the approximated mass degeneracy involves also the
LSP. Long lived gluinos bind with SM quarks and gluons from the vacuum during
the hadronisation process, and produce R-hadrons. R-hadrons are among the most
interesting searches at LHC. Anyway we will come back to this point with a
more detailed study in a forthcoming paper.
* •
there is an approximated mass degeneracy between $\tilde{e}_{R}$ and
$\tilde{u}_{R}$ because using the charge constraints (25) and (91) we get
${q_{E^{c}}}=3$ and ${q_{U^{c}}}\simeq 2.9$.
* •
$m_{\phi^{-}_{R}}<m_{\phi^{+}}$ except for point i)
Case B points deserve some more comments.
$\phi^{+}$ and $\tilde{N}^{\text{new}}_{2,3}$ are out of the plot of point iv)
because they are heavier than 6 TeV. $Z_{2}$ is among the lightest not SM
particle, so it can decay only into SM particles, because of energy and
R-parity conservation. So $Z_{2}$ is long lived, because SM particles are
coupled to $Z_{2}$ only through the suppressed $V_{A,B}$ mixing or through the
Higgs scalars which present a tiny mixing with $\phi^{\pm}$.
It is not an easy task to compare the resulting spectrum we get for our model
with those related to the rich zoology of supersymmetry breaking scenarios. It
is worth to stress anyway that the two representative spectrums showed in Fig.
5 which encode the key features of our scenarios listed above are not
reproduced in any of the benchmark points showed in [49].
## 9 Conclusions
In this paper we presented a viable mechanism to generate soft supersymmetry
breaking terms in the framework of a minimal supersymmetric anomalous
extension of the SM. The crucial ingredient is a non perturbative term in the
superpotential (3) which couples the Stückelberg field $S$ to the Higgs
sector. This term is related to the generation of a suitable $\mu$ and $b$
terms (see Eq. (32) and (33)) in the low energy effective action when the
Stückelberg gets vev. We argued about the origin of this term from an exotic
instanton in an intersecting D-brane setup. We computed the spectrum of our
model as a function of the saxion vev $\left<\alpha\right>$ and for different
choices of the remaining free parameters. We checked our results against known
phenomenological bounds, namely current lower bounds on the mass of the scalar
and fermionic superpartners. We analyzed a scenario in which the anomalous
sector is the source of the soft supersymmetry breaking terms while the
corresponding vector and Stückelberg multiplets are not present in the low
energy effective action. For what concern the non anomalous sector we took
into account two different cases (dubbed case A and case B).
As we stated in Sec. 8, by applying some phenomenological constraints we were
able to derive some bounds on the saxion vev $\langle\alpha\rangle$, which is
the relevant parameter setting the mass scale of the scalars. The strongest
constraints on $\langle\alpha\rangle$ and $t_{\beta}$ comes from the combined
requirement of $\left.M_{h^{0}}^{2}\right|_{\text{1-loop}}\in[124,126]\text{
GeV}$ or ($[114.5,131]\text{ GeV}$), a neutralino LSP and that all mass bounds
(specially the gluino one) are fulfilled. In Fig. 2 (pre-LHC bounds) and 3
(preliminary LHC bounds) we summarize the allowed regions for
$\langle\alpha\rangle$. In the first case, by requiring a phenomenological
appealing neutralino LSP, we get an allowed $\langle\alpha\rangle$ of few TeV
up to $10$ TeV for the A and B scenarios respectively. In the second case
(preliminary LHC bounds) we get that only the B scenario is allowed with
$\langle\alpha\rangle\gtrsim 5$ TeV. These results can be seen as a bound that
a concrete D-brane model has to satisfy. We deserve this analysis for future
work.
In Fig. 5 we explicitly showed two benchmark mass spectrums for our model with
$\langle\alpha\rangle$ and $t_{\beta}$ which fulfill the above bounds. The
cases shared different peculiar features: the LSP is the lightest neutralino
of the new sector, there is a near mass degeneracy between $H^{0}$, $A^{0}$
and $H^{\pm}$, and between $\tilde{e}_{R}$ and $\tilde{u}_{R}$, the lightest
sleptons is a sneutrino except for $t_{\beta}=50$ when it is stau, the
lightest squark is a $\tilde{u}_{L}$ except for $t_{\beta}=50$ when it is a
sbottom, the first and second family left-handed squarks/sleptons are
typically lighter than their right-handed counterparts. Moreover in case B the
gluino is long lived and can produce R-hadrons. It turns out that these
features are not reproduced in any of the widely studied benchmark points
presented in [49].
Acknowledgments
A.L. acknowledges M. Bianchi, E. Kiritsis and R. Richter for useful
discussions and comments. A. R. acknowledges M. Raidal for discussions and the
ESF JD164 contract for financial support.
## Appendix A Anomalous Lagrangians
The Lagrangian involved in the anomaly cancellation procedure is
$\displaystyle\mathcal{L}_{S}$ $\displaystyle=$
$\displaystyle{\frac{1}{4}}\left.\left(S+S^{\dagger}+2M_{V_{A}}V_{A}\right)^{2}\right|_{\theta^{2}\bar{\theta}^{2}}$
$\displaystyle-2\left\\{\left[\sum_{a}g_{a}^{2}b^{aa}S\textnormal{Tr}\left(W_{a}W_{a}\right)+g_{Y}g_{A}b^{YA}SW_{Y}W_{A}\right]_{\theta^{2}}+h.c.\right\\}$
where the index $a=A,B,Y,2,3$ runs over the $U(1)_{A}$, $U(1)_{B}$,
$U(1)_{Y}$, $SU(2)$ and $SU(3)$ gauge groups respectively, and the constants
$b^{ab}$ are fixed by the anomaly cancellation.
Since we have only one anomalous $U(1)$ we can avoid the use of GCS terms,
distributing the anomalies only on the $U(1)_{A}$ vertices. So we have
$\displaystyle b^{AA}=-\frac{g_{A}\mathcal{A}_{AA}}{96\pi^{2}M_{V_{A}}}\qquad
b^{YY}=-\frac{g_{A}\mathcal{A}_{YY}}{32\pi^{2}M_{V_{A}}}\qquad
b^{22}=-\frac{g_{A}\mathcal{A}_{22}}{16\pi^{2}M_{V_{A}}}$ $\displaystyle
b^{33}\ =-\frac{g_{A}\mathcal{A}_{33}}{16\pi^{2}M_{V_{A}}}\qquad
b^{YA}=-\frac{g_{A}\mathcal{A}_{YA}}{32\pi^{2}M_{V_{A}}}$ (105)
where the $\mathcal{A}$’s are the corresponding anomalies
$\displaystyle\mathcal{A}_{AA}$ $\displaystyle=$
$\displaystyle-10{q_{H_{d}}}^{3}-9{q_{H_{d}}}^{2}({q_{L}}+3{q_{Q}})-9{q_{H_{d}}}\left({q_{L}}^{2}+3{q_{Q}}^{2}\right)$
(106)
$\displaystyle-7{q_{H_{u}}}^{3}-27{q_{H_{u}}}^{2}{q_{Q}}-27{q_{H_{u}}}{q_{Q}}^{2}+3{q_{L}}^{3}$
$\displaystyle\mathcal{A}_{YY}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(7{q_{H_{d}}}+7{q_{H_{u}}}+3{q_{L}}+9{q_{Q}})$ (107)
$\displaystyle\mathcal{A}_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2}({q_{H_{d}}}+{q_{H_{u}}}+3{q_{L}}+9{q_{Q}})$ (108)
$\displaystyle\mathcal{A}_{33}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}({q_{H_{d}}}+{q_{H_{u}}})$ (109)
$\displaystyle\mathcal{A}_{YA}$ $\displaystyle=$ $\displaystyle
5{q_{H_{d}}}^{2}+6{q_{H_{d}}}({q_{L}}+{q_{Q}})-{q_{H_{u}}}(5{q_{H_{u}}}+12{q_{Q}})$
(110)
where we used the constraints (22). Imposing the conditions (25) we get
$\displaystyle\mathcal{A}_{AA}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\left(-1168{q_{H_{d}}}^{3}+1776{q_{H_{d}}}^{2}{q_{H_{u}}}-996{q_{H_{d}}}{q_{H_{u}}}^{2}+53{q_{H_{u}}}^{3}\right)$
(111) $\displaystyle\mathcal{A}_{YY}$ $\displaystyle=$
$\displaystyle-\frac{11}{4}({q_{H_{d}}}+{q_{H_{u}}})$ (112)
$\displaystyle\mathcal{A}_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}({q_{H_{d}}}+{q_{H_{u}}})$ (113)
$\displaystyle\mathcal{A}_{33}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}({q_{H_{d}}}+{q_{H_{u}}})$ (114)
$\displaystyle\mathcal{A}_{YA}$ $\displaystyle=$ $\displaystyle 0$ (115)
We remind that (115) is not a consequence of (25), but rather (25) is a
consequence of imposing (115) in order to cancel the $U(1)_{Y}-U(1)_{A}$
kinetic mixing.
## Appendix B Exact fixed parameters
In this Appendix we give the exact values for the $\langle F_{S}\rangle$,
$\lambda$, $m$ parameters determined in section 8.2. Solving the minima
conditions (41)-(43), we get
$\displaystyle\lambda^{2}=\frac{e^{\,2\langle\alpha\rangle
g_{A}({q_{H_{d}}}+{q_{H_{u}}})/M_{V_{A}}}}{32}\times$
$\displaystyle\times\Big{[}-g_{A}\sec(2\beta)({q_{H_{d}}}-{q_{H_{u}}})\left(8\langle\alpha\rangle
M_{V_{A}}+g_{A}v^{2}({q_{H_{d}}}+{q_{H_{u}}})\left(\cos(4\beta)+3\right)-4g_{A}v_{-}^{2}\right)$
$\displaystyle-8\langle\alpha\rangle
g_{A}M_{V_{A}}({q_{H_{d}}}+{q_{H_{u}}})-2v^{2}\left(2g_{A}^{2}\left({q_{H_{d}}}^{2}+{q_{H_{u}}}^{2}\right)+g_{Y}^{2}+g_{2}^{2}\right)+4g_{A}^{2}v_{-}^{2}({q_{H_{d}}}+{q_{H_{u}}})\Big{]}$
$\displaystyle\langle F_{S}\rangle=-e^{\langle\alpha\rangle
g_{A}({q_{H_{d}}}+{q_{H_{u}}})/M_{V_{A}}}\times\frac{M_{V_{A}}\tan(2\beta)}{8g_{A}({q_{H_{d}}}+{q_{H_{u}}})\lambda}\times$
$\displaystyle\times\Big{[}g_{A}({q_{H_{d}}}-{q_{H_{u}}})\left(4\langle\alpha\rangle
M_{V_{A}}+g_{A}v^{2}({q_{H_{d}}}+{q_{H_{u}}})-2g_{A}v_{-}^{2}\right)+$
$\displaystyle
v^{2}\cos(2\beta)\left(g_{A}^{2}({q_{H_{d}}}-{q_{H_{u}}})^{2}+g_{Y}^{2}+g_{2}^{2}\right)\Big{]}$
$\displaystyle|m|^{2}=g_{A}\langle\alpha\rangle
M_{V_{A}}-\frac{1}{2}\left[\left(g_{A}^{2}+g_{B}^{2}\right)v_{-}^{2}+g_{A}^{2}v^{2}\left({q_{H_{d}}}c_{\beta}^{2}+{q_{H_{u}}}s_{\beta}^{2}\right)\right]$
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|
arxiv-papers
| 2011-02-24T17:18:52 |
2024-09-04T02:49:17.269457
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Lionetto and A. Racioppi",
"submitter": "Antonio Racioppi",
"url": "https://arxiv.org/abs/1102.5040"
}
|
1102.5088
|
# Estimation of the relative risk following group sequential procedure based
upon the weighted log-rank statistic
Grant Izmirlianlabel=e1]izmirlig@mail.nih.gov [ National Cancer Institute;
Executive Plaza North, Suite 3131
6130 Executive Blvd, MSC 7354; Bethesda, MD 20892-7354
###### Abstract
In this paper we consider a group sequentially monitored trial on a survival
endpoint, monitored using a weighted log-rank (WLR) statistic with
deterministic weight function. We introduce a summary statistic in the form of
a weighted average logged relative risk and show that if there is no sign
change in the instantaneous logged relative risk, there always exists a
bijection between the WLR statistic and the weighted average logged relative
risk. We show that this bijection can be consistently estimated at each
analysis under a suitable shape assumption, for which we have listed two
possibilities. We indicate how to derive a design-adjusted p-value and
confidence interval and suggest how to apply the bias-correction method.
Finally, we document several decisions made in the design of the NLST interim
analysis plan and in reporting its results on the primary endpoint.
62L12,
62L12,
62N022,
Weighted Logrank Statistic,
Group Sequential,
Interim Analysis,
Estimation,
###### keywords:
[class=AMS]
###### keywords:
t1This article is a U.S. Government work and is in the public domain in the
U.S.A.
## 1 Introduction
Time to event, e.g. disease specific mortality, is the primary endpoint in
many clinical trials. The use of group sequential boundaries in monitoring the
trial is not only commonplace, but ethically mandated in all trials of human
subjects. The logrank statistic is often the monitoring statistic of choice
due to its natural connection with the relative risk, which is often the
parameter of inference. This natural connection, which is based upon the
assumption of proportional hazards, admits a one-to-one correspondence between
the inferential procedure based upon the usual standard normal scale and that
based on the scale of the natural parameter. However, the assumption of
proportional hazards is not always a reasonable assumption. In many subject
areas, e.g. in disease-prevention trials, one expects that the hazard ratio
will not be constant. Much of the prior work on the use of the weighted
logrank statistic in a sequential design is confined to the use a weighting
function from the $G^{\rho,\gamma}(t)=S^{\rho}(t)(1-S(t))^{\gamma}$ family, of
Fleming and Harrington, [2]. They suggest two major types of problems which
can arise. First, they argue that use of the weighted logrank statistic does
not reproduce the single point analysis in the way that is desired. Most
notably, they argue, there is no clinically meaningful parameter that allows
the values of the monitoring statistic and sequential boundaries to be cast
into a clinically meaningful scale. They believe that this problem is further
aggrivated when the range of the weighting function over the duration of the
trial is quite large, such as is the case with the $G^{0,1}$ weight function
(Gillen and Emerson, [4]) and suggest a re-weighting scheme whereby the most
weight is given to the most recent data collected at each analysis. Secondly,
they argue that if the chosen weighting function is non-deterministic or
trial-specific then it is impossible to compare results from different
clinical trials, (Gillen and Emerson, [3, 5]). While the bulk of these
cautious remarks are useful to know in their own right, several important
points have been omitted from the discussion. Firstly, as we will show, there
is a natural, clinically meaningful parameter, the weighted average logged
relative risk, that is connected bijectively to the weighted logrank statistic
when there is no change in sign in the instantaneous logged relative risk.
Under suitable shape assumptions, the bijection can be estimated at each
analysis. We will show that the asymptotic distribution of the WLR statistic,
suitably normalized is a Brownian motion plus drift under nothing but
boundeness conditions. In two corollaries, we demonstrate how each of two
presented shape assumptions translates into a form of the drift function and
consequently, into an estimator of the weighted average logged relative risk.
We then demonstrate how the usual results concerning monitoring and end of
trial estimation follow. Finally, we note that this bijection between the
weighted logrank statistic and the weighted average logged relative risk
allows the values of the monitoring statistic, efficacy and futility
boundaries, and reported point estimate and confidence interval to be cast
into a clinically meaningful scale.
## 2 Terminology and framework
We consider a two armed randomized trial of the effect of an intervention upon
a time to event that is run until time $\tau$. Let $\tilde{T}_{i}$ be the
possibly unobserved time to event and let $C_{i}$ a right censoring time. We
assume non-informative censoring for simplicity. Let
$T_{i}=\tilde{T}_{i}\wedge C_{i}$ be the observed time on study and let
$\delta_{i}=I(\tilde{T}_{i}\leq C_{i})$ be the event indicator. Let $X_{i}$
indicates membership in the intervention arm ($X_{i}=1$) or control arm
($X_{i}=0$). We assume, conditional upon $X_{i}$, that individuals,
$i=1,\ldots,n$ are distributed independently and identically. Let $dH_{0}(t)$
and $dH_{1}(t)$ be the trial arm specific cumulative hazard increments. We
assume throughout that $H_{0}(t)$ is finite for all $t$ on $[0,\tau]$. For the
instantaneous logged hazard ratio, we write
$\beta(t)=\log\left\\{\frac{dH_{1}(t)}{dH_{0}(t)}\right\\}\,.$ (2.1)
Let $N_{i}(t)=I(T_{i}\leq t,\delta_{i}=1)$ and $dN_{i}(t)=N_{i}(t)-N_{i}(t-)$
be the subject level counting process and its increments, respectively. Let
$N_{n}(t)=\sum_{i}N_{i}(t)$ and $dN_{n}(t)=N_{n}(t)-N_{n}(t-)$ be the
aggregated counting process and its increments, respectively. Note that the
following difference is a compensated counting process martingale:
$dM_{i}(t)=dN_{i}(t)-I(T_{i}\geq t)\exp(X_{i}\beta(t))dH_{0}(t)$ (2.2)
Let $E_{n}(t,0)=\sum_{i}X_{i}I(T_{i}\geq t)/\sum_{i}I(T_{i}\geq t)$ denote the
proportion of the population at risk at time $t$ in the intervention arm, and
let $e(t,0)=\mathop{{\mathrm{lim}}_{a.s.}}_{n\rightarrow\infty}E_{n}(t,0)$ and
let $G(t)=\mathop{{\mathrm{lim}}_{a.s.}}dN_{n}(t)/n$. Let
${I}\kern-3.00003pt{F}_{n}(t)=\int_{0}^{t}E_{n}(\xi,0)(1-E_{n}(\xi,0))\,dN_{n}(\xi)/n$
and let ${I}\kern-3.00003pt{F}(t)=\int_{0}^{t}e(\xi,0)(1-e(\xi,0))\,dG(\xi)$.
We introduce the following notation for cross moment integrals against
$d{I}\kern-3.00003pt{F}$ over $(0,t)$:
$\langle\psi_{1}|{I}\kern-3.00003pt{F}|\psi_{2}\rangle_{t}=\int_{0}^{t}\psi_{1}(\xi)\,\psi_{2}(\xi)d{I}\kern-3.00003pt{F}(\xi)\,.$
(2.3)
For reasons that will become clear below, we consider the target of our
investigation to be the following weighted average logged relative risk:
$\beta^{\star}=\frac{\langle
Q|{I}\kern-3.00003pt{F}|\beta\rangle_{\tau}}{\langle
Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}}\,.$ (2.4)
Let $q(t)=\beta(t)/\beta^{\star}$. This provides a representaton of the
instantaneous logged relative risk function, $\beta(t)=\beta^{\star}\,q(t)$ as
the product of its weighted average value, $\beta^{\star}$ times a shape
function, $q$. Note it follows that the shape function has weighted average
value equal to 1:
$1=\frac{\langle Q|{I}\kern-3.00003pt{F}|q\rangle_{\tau}}{\langle
Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}}\,.$ (2.5)
At follow-up time $t$, the $\sqrt{n}$ normalized score statistic with
weighting function $Q$ is:
$U_{n}(t)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{t}Q(\xi)\left\\{X_{i}-E_{n}(\xi,0)\right\\}dN_{i}(\xi)\,.$
(2.6)
Its estimated variance is:
$V_{n}(t)=\frac{1}{n}\int_{0}^{t}Q^{2}(\xi)E_{n}(\xi,0)\left(1-E_{n}(\xi,0)\right)dN_{n}(\xi)=\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{t}\,.$ (2.7)
Let $v(t)=\mathop{{\mathrm{lim}}_{a.s.}}V_{n}(t)$. Note that $v(t)=\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{t}$. Let
$f_{n}(t;\tau)=V_{n}(t)/V_{n}(\tau)$ and $f(t;\tau)=v(t)/v(\tau)$. We will on
occasion use the shorthand $f_{n,j}$ and $f_{j}$ for $f_{n}(t;\tau)$ and
$f(t;\tau)$, respectively. Also, let $m_{n}(t)=\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{t}$ and $m(t)=\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{t}$. We consider the weighted log-rank (WLR)
statistic at time $t$ on several “scales”
* (i)
The standard normal scale: $Z_{n}(t)=U_{n}(t)/\sqrt{V_{n}(t)}$
* (ii)
The “Brownian scale”: $X_{n}(t)=U_{n}(t)/\sqrt{V_{n}(\tau)}$
## 3 Main Result
###### Condition 3.1.
The instantaneous logged relative risk function, $\beta$, is bounded on
$[0,\tau]$.
###### Condition 3.2.
The chosen weighting function, $Q$, is bounded on $[0,\tau]$ and
deterministic.
Recall that a weighting functions is always non-negative. The stipulated
boundedness in conditions 3.1 and 3.2 above can be relaxed to being of class
$L^{2}$ with respect to the measure $d{I}\kern-3.00003pt{F}$, as this is all
that is really required.
While the context will involve monitoring the statistic at a sequence of
interim analyses, for the time being, we suppress this aspect and consider
instead the following more general and generic result which holds under the
weakest set of assumptions:
###### Theorem 3.1.
Under conditions 3.1 and 3.2, then under the family of local alternatives,
$\beta_{n}^{\star}=b^{\star}/\sqrt{n}$, the score statistic, normalized to the
“Brownian scale” is asymptotically a Brownian motion on $[0,1]$ plus a drift.
$X_{n}(t)\buildrel\cal D\over{\longrightarrow}W(f(t;\tau))+\mu(t)\,$ (3.1)
where the “time scale” for the Brownian motion is the variance ratio or
information fraction, $f(t;\tau)=v(t)/v(\tau)$, and the drift, parameterized
by $t$ is
$\mu(t)=\frac{\langle Q|{I}\kern-3.00003pt{F}|q\rangle_{t}}{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}}}\,b^{\star}\,.$ (3.2)
The proof of 3.1 is given in appendix 8.1. Notice, first, that from equations
2.5 and 3.2, it follows that the value of the drift function at the scheduled
end of the trial is
$\mu(\tau)=\frac{\langle
Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}}{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}}}\,b^{\star}\,.$ (3.3)
Thus, without any additional assumptions on the shape function, $q$, we have
the following corollary:
###### Corallary 3.1.
At the planned conclusion of the trial, $\tau$, an estimate of $\beta^{\star}$
is given by the following:
${\widehat{\beta}^{\star}}=X_{n}(\tau)\frac{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}}{\sqrt{n}\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}}\,.$ (3.4)
* (i)
${\widehat{\beta}^{\star}}$ is unbiased
* (ii)
An estimate of its variance is given by
${\mathrm{var\left[\widehat{\beta}^{\star}\right]}}=\frac{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}{n\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}^{2}}\,.$ (3.5)
## 4 Estimates of $\beta^{\star}$ in a Trial Stopped Early
Obtaining an estimate of $\beta^{\star}$ at a trial stopped early due to an
efficacy boundary crossing will require more assumptions on the shape
function, $q$. At a minimum in order to have a monotone drift function which
is necessary for propper monitoring, we require the following.
###### Condition 4.1.
The shape function, $q$, is non-negative.
Since the drift’s function’s dependence on $t$ is through an integral of a
non-negative function, we have the following corollary:
###### Corallary 4.1.
If conditions 3.1, 3.2 and 4.1 are true then the conclusion of theorem 3.1
holds and the drift function is monotone increasing or decreasing in $t$,
depending upon the sign of $b^{\star}$.
Note also that as the inverse of an increasing function is also increasing,
the drift function can also be considered a monotone function of the
information fraction. This would, of course, lead to a natural estimate of
$\beta^{\star}$ in a trial stopped early except for the fact that we have no
knowledge of $q$. In order to have a more useful estimator for $\beta^{\star}$
in trials stopped early, we opt for a semi-parametric model. In the following,
we list two possibilities. The most natural shape condition to impose is true
if our choice of weight function was the optimal one among all possible
choices.
###### Condition 4.2.
The shape function, $q$, is proportional to our chosen weighting function,
$q(t)=K\,Q(t)$.
Note that as the weighted average of the shape function must equal 1 as in
equation 2.5 it follows that the constant of proportionality, $K$, must be
$K=\frac{\langle Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}}{\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}}\,.$ (4.1)
###### Corallary 4.2.
If conditions 3.1, 3.2 and 4.2 are true then
* (i)
$X_{n}$ is asymptotically a Brownian motion with a drift that is linear in the
information fraction:
$\mu(t)=\frac{\langle Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}}{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}}}f(t;\tau)\,b^{\star}\,.$ (4.2)
* (ii)
If the trial is stopped at an analysis number $J$ at calender time $t_{J}$ due
to an effacacy boundary crossing, then we have the following estimate of
$\beta^{\star}$
${\widehat{\beta}^{\star}}=\frac{X_{n}(t_{J})}{f_{n}(t_{J};\tau)}\,\frac{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}}{\sqrt{n}\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}}$ (4.3)
* (iii)
An estimate of the mean-squared error is given by:
${\mathrm{mse\left[\widehat{\beta}^{\star}\right]}}=\frac{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}{n\,f_{n}(t_{J};\tau)\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}^{2}}$ (4.4)
Another natural shape condition is true when we have opted for a weighted
statistic but the true shape is constant.
###### Condition 4.3.
The shape function, $q$, is identically 1.
###### Corallary 4.3.
If conditions 3.1, 3.2 and 4.3 are true then
* (i)
$X_{n}$ is asymptotically a Brownian motion the following drift:
$\mu(t)=\frac{\langle Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}}{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}}}r(t,\tau)\,b^{\star}\,,$ (4.5)
where $r(t;\tau)=\langle Q|{I}\kern-3.00003pt{F}|1\rangle_{t}/\langle
Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}$, which is an increasing function of
$t$ and takes the values $0$ at $t=0$ and $1$ at $t=\tau$.
* (ii)
If the trial is stopped at an analysis number $J$ at calender at time $t_{J}$
due to an effacacy boundary crossing, then we have the following estimate of
$\beta^{\star}$
${\widehat{\beta}^{\star}}=\frac{X_{n}(t_{J})}{r_{n}(t_{J};\tau)}\,\frac{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}}{\sqrt{n}\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}}\,,$ (4.6)
where $r_{n}(t;\tau)=\langle Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{t}/\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}$
* (iii)
An estimate of the mean-squared error is given by:
${\mathrm{mse\left[\widehat{\beta}^{\star}\right]}}=\frac{f_{n}(t_{J};\tau)\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}{n\,r_{n}(t_{J};\tau)^{2}\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}^{2}}$ (4.7)
## 5 Application to Monitoring and Final Reporting in a Clinical Trial
The relationship between the drift of the WLR statistic and the weighted
average logged relative risk parameter provided by theorem 3.1 and its
corallaries can be used in the monitoring and final reporting of a clinical
trial.
### 5.1 Futility Boundary
Our comments regarding monitoring a trial are made within the context of
boundaries constructed using the Lan-Demets procedure, [6]. Construction of
the efficacy boundary is done under the null hypothesis that the drift
function is identically zero and can be done without appealing to the results
presented here. If a futility boundary is specified in the design then under
either of the shape assumptions, one can apply the corresponding corollary 4.2
or corollary 4.3 to calculate the drift function at each interim analysis
which is required to compute the futility boundary under the Lan-Demets
approach [6]. Note that the shape assumption being made must be part of the
interim analysis plan design. In the following discussion we will assume that
the optimal weighting shape condition 4.2 was specified in the design so that
the discussion focuses on the application of corollary 4.2. In this case,
$\beta^{\star}$ is the weighted average logged relative risk for which the
study is powered to detect and must also be specified in the interim analysis
plan design. The values of $v(\tau)=\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}$ and $m(\tau)=\langle
Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}$ at the planned termination of the
study, $\tau$, must also be specified in the interim analysis plan design. We
demonstrate in appendix 8.2 when the only source of censoring is
administrative censoring or other cause mortality, how these functionals can
be projected for a specific choice of weighting function, $Q$, based upon
projected values of the cross-arm pooled cumulative hazard function at several
landmark times on study. We remark here that following consensus, we recommend
using a non-binding futility boundary which is constructed after construction
of an efficacy boundary which ignores the existence of the futility boundary.
This is preferred to the joint construction of efficacy and futility
boundaries as that approach results in a discounted efficacy criterion.
### 5.2 Prediction at End of Trial
When the trial is stopped at an efficacy or futility boundary crossing, or at
the scheduled end of the trial, and if the optimal weighting shape assumption
4.2 was specified in the design, then corollary 4.2 can be used to convert the
value of the WLR statistic on the Brownian scale, $X_{n}(t_{j})$, to an
estimate of the weighted average logged relative risk,
$\widehat{\beta}^{\star}$. Therefore, our point estimate is
${\widehat{\beta}^{\star}}=\frac{X_{n}(t_{j})}{f_{n,j}}\,\frac{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}}{\sqrt{n}\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}}$ (5.1)
We use the values of $v(\tau)=\langle Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}$
and $m(\tau)=\langle Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}$ which are
specified in the interim analysis plan design. As mentioned above, when it is
obtained at an efficacy boundary crossing, these type of estimates are known
to be biased away from the null (see e.g. Liu and Hall, [7]). The construction
of a design-adjusted confidence interval and adjustment of this estimate for
the above mentioned bias are standard results, especially under the optimal
weighting shape condition 4.2 which leads, in corollary 4.2, to a drift that
is linear in the information fraction. For sake of completeness, we outline
below how to compute a design adjusted p-value, construct a design-adjusted
confidence interval and how to calculate the bias adjusted estimate of the
weighted average logged relative risk. All three of these tasks involve the
sampling density under the null hypothesis of the sufficient statistic,
$(J,X_{n}(t_{J}))$, where $J$ and $X_{n}(t_{J})$ are the analysis number and
the value of the weighted logrank statistic at an efficacy crossing. The
sampling density of $(J,X_{n}(t_{J}))$ takes the following form. First, for
$j=1$, $\pi((1,x))={{\rm I}\kern-1.79993pt{\rm P}}\\{X_{n}(t_{1})=x\\}$. For
$j>1$,
$\displaystyle\pi((j,x)\kern-7.5pt$ ;
$\displaystyle\kern-7.5pt\mathbf{b}_{1:(j-1)},\mathbf{f}_{1:j})$
$\displaystyle=$ $\displaystyle\frac{d}{dx}{{\rm I}\kern-1.79993pt{\rm
P}}_{H_{0}}\\{J=j\mathrm{~{}and~{}}X_{n}(t_{\ell})<\sqrt{f_{\ell}}b_{\ell}\,,\,\ell=1,\ldots,j-1,X_{n}(t_{j})=x\\}$
Here $\mathbf{b}_{1:(j-1)}$ is the sequence of efficacy boundary points at all
prior analyses and $\mathbf{f}_{1:j}$ is the sequence of information fractions
at all analyses prior and current. In the following $\mathbf{b}_{1:{\ell}}$
and $\mathbf{f}_{1:{\ell}}$ for $\ell<1$ denote the empty sequence. The
construction and form of this density is reviewed in appendix 8.3. Let
$\bar{\Pi}((j,x);\mathbf{b}_{1:(j-1)},\mathbf{f}_{1:j})=\int_{x}^{\infty}\pi((j,\xi);\mathbf{b}_{1:(j-1)},\mathbf{f}_{1:j})d\xi$
(5.3)
be the joint probability under $\pi$ that $J=j$ and $X_{n}(t_{j})$ is in the
right tail $(x,\infty)$. In order to calculate a p-value and construct a
confidence interval which account for the sequential design, we must choose an
ordering of the sample space for the statistic $(J,X_{n}(t_{J}))$. Here we
prefer to use the following ordering: $(j,x)>(k,y)$ if and only if ($j=k$ and
$x>y$) or $j<k$. This ordering is applicable when the rejection region is
convex, as is the case with Lan-Demets boundaries constructed using a smooth
spending function. The discussion of the p-value and of the confidence
interval is in the setting of symmetric 2-sided boundaries and when sign of
the alternative hypothesis is positive as it is a simple matter to apply these
results to the case where the sign of the alternative hypothesise is negative.
P-value Under the ordering given above, the region further away from the null
than $(J,X_{n}(t_{J}))$ is the union of all prior rejection regions with the
right tail at $X_{n}(t_{J})$. Thus the design-adjusted or sequential p-value
is:
$\bar{\Pi}((J,X_{n}(t_{J}));\mathbf{b}_{1:(J-1)},\mathbf{f}_{1:J})+\sum_{\ell=1}^{J-1}\bar{\Pi}((\ell,b_{\ell});\mathbf{b}_{1:{\ell-1}},\mathbf{f}_{1:\ell})\,,$
(5.4)
Confidence Interval If the probability of type one error that remained prior
to analysis $J$ is $\alpha_{tot}-\alpha_{J-1}$ then a two sided design-
adjusted confidence interval for $\widehat{\beta}^{\star}$ is derived as
follows. If we denote by $x_{u}$ the solution in $x$ of the equation
$\alpha_{tot}-\alpha_{J-1}=\bar{\Pi}((J,x);\mathbf{b}_{1:(J-1)},\mathbf{f}_{1:J})+\sum_{\ell=1}^{J-1}\bar{\Pi}((\ell,b_{\ell});\mathbf{b}_{1:{\ell-1}},\mathbf{f}_{1:\ell})\,,$
(5.5)
then the design-adjusted confidence interval is
$\widehat{\beta}^{\star}\pm\frac{x_{u}}{\sqrt{f_{n,J}}}\sqrt{{\mathrm{mse\left[\widehat{\beta}^{\star}\right]}}}\,,$
(5.6)
where ${\mathrm{mse\left[\widehat{\beta}^{\star}\right]}}$ is the estimated
mean-squared error of $\widehat{\beta}^{\star}$ as given in part (iii) of
corollary 4.2. Note that when the efficacy boundary is one-sided one can still
construct a 2-sided confidence interval by replacing
$\alpha_{tot}-\alpha_{J-1}$ above with 1/2 its value.
Bias Adjustment As in Liu and Hall, [7], bias adjustment is done recursively
as follows. First,
$\widetilde{\zeta}(1,x)=\frac{x}{f_{1}}$ (5.7)
Continuing,
$\widetilde{\zeta}(j,x)=\int_{-\infty}^{\sqrt{f_{j}}b_{j}}\widetilde{\zeta}(j-1,\xi)\,\pi((j-1,\xi);\mathbf{b}_{1:(j-1)},\mathbf{f}_{1:(j-1)})\,\phi_{{}_{\Delta_{j}}}(x-\xi)\,d\xi$
(5.8)
The bias adjusted estimate, $\widetilde{\beta}^{\star}$, of the weighted
average logged relative risk, $\beta^{\star}$, is obtained by replacing
$X_{n}(t_{J})/f_{n,J}$ in part (ii) of corollary 4.2 with
$\widetilde{\zeta}(J,X_{n}(t_{J}))$ to obtain the following:
$\widetilde{\beta}^{\star}=\widetilde{\zeta}(J,X_{n}(t_{J}))\,\frac{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}}{\sqrt{n}\,\langle
Q|{I}\kern-3.00003pt{F}_{n}|1\rangle_{\tau}}$ (5.9)
The design-adjusted confidence interval is the same as given above, but now
centered about $\widetilde{\beta}^{\star}$
$\widetilde{\beta}^{\star}\pm\frac{x_{u}}{\sqrt{f_{n,J}}}\sqrt{{\mathrm{mse\left[\widehat{\beta}^{\star}\right]}}}\,,$
(5.10)
## 6 The NLST
The design of the National Lung Screening Trial (NLST) [8] interim analysis
plan stipulated a one-sided efficacy boundary constructed using the Lan-Demets
procedure with a total probability of type one error set to 0.05. The trial
had 90% power to detect a relative risk of 0.79 at a sample size of 25,000 per
arm, accounting for contamination and non-compliance that could attenuate this
effect to 0.85. The trial began randomization on August 5th, 2002 and
concluded randomization on April 26th, 2004. A non-binding futility boundary
was used. The drift was derived under the optimal weighting shape assumption,
4.2, and incorporated the design alternative $\beta^{\star}=\log(0.85)$.
Initial estimates of $v(\tau)$ and $m(\tau)$ were posed in the design. These
were updated by using a least squares quadratic curve to project required
future values of $H$ as data accumulated. During the run of the trial,
projected values of the end of trial functionals $v(\tau)$ and $m(\tau)$ did
not vary more than $\pm 5\%$. Interim analyses occured starting in Spring of
2006 and continued annually until the 5th analysis. The 6th analysis occured 6
months after the 5th. Data on the primary endpoint was backdated roughly 18
months to allow more complete ascertainment by the endpoint verification team.
The efficacy boundary was crossed at the sixth interim analysis, using data
backdated to January 15th 2009. Data on the primary endpoint was collected
only for events occurring through December 31, 2009 so this was used as the
scheduled termination date. The raw estimated weighted logged relative risk
and its design-adjusted confidence interval were derived. The bias adjusted
weighted logged relative risk was compared to the raw estimate. As the raw
estimate is asymptotically unbiased, and since the crude risk ratio is the
most straightforward and tangible summary of the trial results, the trial
leadership decided to report the crude risk ratio together with the
exponentiated raw estimate’s design-adjusted confidence interval.
## 7 Discussion
We have shown that there is a natural clinically meaningful parameter, the
weighted average logged relative risk, that is connected the weighted logrank
statistic. When $\beta(t)$ does not change sign, the connection is a
bijection. We have shown that under suitable shape assumptions, this bijection
can be estimated at each analysis. We have shown how this bijection between
the weighted logrank statistic and the weighted average logged relative risk
allows the values of the monitoring statistic, efficacy and futility
boundaries, and reported point estimate and confidence interval to be cast
into a clinically meaningful scale. We have indicated how to derive a design-
adjusted p-value and confidence interval and how bias adjustment of the
estimate may be done using known methods. Finally, we have documented several
decisions made in the design of the NLST interim analysis plan and in
reporting its results on the primary endpoint.
## References
* [1] [author] Armitage, P.P., McPherson, C. K.C. K. and Rowe, B. C.B. C. (1969). Repeated significnce tests on accumulating data. Journal of the Royal Statistical Society, Series A 132 235–244.
* [2] [author] Fleming, Thomas R.T. R. and Harrington, David P.D. P. (1991). Counting processes and survival analysis. Wiley, New York.
* [3] [author] Gillen, DavidD. and Emerson, ScottS. (2005). Information growth in a family of weighted logrank statistics under interim analyses. Sequential Analysis 24 1–22.
* [4] [author] Gillen, DavidD. and Emerson, ScottS. (2005). A note on P-values under group sequential testing and nonproportional hazards. Biometrics 61 546–551.
* [5] [author] Gillen, DavidD. and Emerson, ScottS. (2007). Non-transitivity in a class of weighted logrank statistics under nonproportional hazards. Statistics and Probability Letters 77 123–130.
* [6] [author] Lan, K. K. G.K. K. G. and DeMets, David L.D. L. (1983). Discrete sequential boundaries for clinical trials. Biometrika 70 659–663.
* [7] [author] Liu, AiyiA. and Hall, W. J.W. J. (1999). Unbiased estimation following a group sequential test. Biometrika 86 71–78.
* [8] [author] The National Lung Screening Trial Research Team, (2011). The National Lung Screening Trial: Overview and Study Design. Radiology 258 243–253.
## 8 Appendices
### 8.1 Proof of Theorem 3.1
We follow the usual method of adding and subtracting the differential of the
compensator, and thereby express $U_{n}$ as a sum of a term that is
asymptotically mean zero Gaussian process and a drift function which grows as
$\sqrt{n}$.
$\displaystyle U_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{t}Q(\xi)\left\\{X_{i}-E_{n}(\xi,0)\right\\}dM_{i}(\xi)$
(8.1)
$\displaystyle+\;\;\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{t}Q(\xi)\left\\{X_{i}-E_{n}(\xi,0)\right\\}I(T_{i}\geq\xi)\exp(X_{i}q(\xi)\beta^{\star})dH_{0}(\xi)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{t}Q(\xi)\left\\{X_{i}-E_{n}(\xi,0)\right\\}dM_{i}(\xi)$
$\displaystyle+\;\;\sqrt{n}\int_{0}^{t}Q(\xi)\left\\{E_{n}(\xi,\beta^{\star})-E_{n}(\xi,0)\right\\}R_{n}(\xi,\beta^{\star})dH_{0}(\xi)\,,$
where in the above,
$R_{n}(\xi,\beta^{\star})=1/n\sum_{i}I(T_{i}\geq\xi)\exp(X_{i}q(\xi)\beta^{\star})$,
and
$E_{n}(\xi,\beta^{\star})=1/(nR_{n}(\xi,\beta^{\star}))\sum_{i}X_{i}I(T_{i}\geq\xi)\exp(X_{i}q(\xi)\beta^{\star})$.
By linearizing the difference, $E_{n}(\xi,\beta^{\star})-E_{n}(\xi,0)$ about
$\beta^{\star}=0$ we obtain
$\displaystyle U_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{t}Q(\xi)\left\\{X_{i}-E_{n}(\xi,0)\right\\}dM_{i}(\xi)$
$\displaystyle+\;\;\sqrt{n}\beta^{\star}\int_{0}^{t}Q(\xi)q(\xi)E_{n}(\xi,0)\left\\{1-E_{n}(\xi,0)\right\\}R_{n}(\xi,\beta^{\star})dH_{0}(\xi)\,.$
We normalize by $\sqrt{V_{n}(\tau)}$ and replace the differential
$R_{n}(\xi,\beta^{\star})dH_{0}(\xi)$ with $dN_{n}(\xi)/n$. The latter is
possible because integrals of bounded functions against the difference of the
differentials are consistent to zero.
$\displaystyle X_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n\,V_{n}(\tau)}}\sum_{i=1}^{n}\int_{0}^{t}Q(\xi)\left\\{X_{i}-E_{n}(\xi,0)\right\\}dM_{i}(\xi)$
(8.3)
$\displaystyle+\;\;\sqrt{\frac{n}{V_{n}(\tau)}}\beta^{\star}\int_{0}^{t}Q(\xi)q(\xi)E_{n}(\xi,0)\left\\{1-E_{n}(\xi,0)\right\\}\frac{dN_{n}(\xi)}{n}$
$\displaystyle=$ $\displaystyle W_{n}(f_{n}(t;\tau))\;+\;\frac{\langle
Q|{I}\kern-3.00003pt{F}_{n}|q\rangle_{t}}{\sqrt{\langle
Q|{I}\kern-3.00003pt{F}_{n}|Q\rangle_{\tau}}}\,\sqrt{n}\beta^{\star}\,.$
The first term is easily recognized to be asymptotic in distribution to a
standard Brownian motion. The reader can either directly apply Robolledo’s
martingale central limit theorem, verifying that in the case that integrands
and intensities are bounded all conditions are satisfied, or apply a more
direct result, such as theorem (6.2.1) in Fleming and Harrington [2]. Under
the family of local alternatives, $\beta^{\star}_{n}=b^{\star}/\sqrt{n}$, then
by the comments following expression LABEL:eqn:U, the second term is easily
seen to be consistent to the drift function listed in expression 3.2.
Therefore the result follows by Slutzky’s theorem.
### 8.2 End of Trial Functionals
In this section we demonstrate how to project values of the variance
$v(\tau)=\langle Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}$, and the “first
moment” $m(\tau)=\langle Q|{I}\kern-3.00003pt{F}|1\rangle_{\tau}$ at the
scheduled end of study, $\tau$. This is done in the specific case of the “ramp
plateau” weighting function which was used for interim monitoring and
reporting in the NLST. This is the function which takes the value 0 at $t=0$,
has linear increase to the value 1 at $t=t_{c}$ and then maintains this
constant value forward.
$Q(t)=\frac{t}{t_{c}}\wedge 1$ (8.4)
In the NLST, the value of $t_{c}=4$ years was used. Next, by imposing some
mild assumptions we will be able to express all quantities in the integrands
in terms of the cross-arm pooled cancer mortality cumulative hazard function,
$H$ and thereby solve the integrals via a simple change of variables. The
resulting expressions require only values of $H(t)$ at $t=t_{c}$, $t=\tau-
t_{er}$ and $t=\tau$, where $t_{er}$ is the calender time at which
randomization was concluded. First we shall list the required assumptions. In
the following discussion, $S$, $S_{lr}$ and $S_{oth}$ are survival functions
corresponding to the cross-arm pooled cancer mortality, administrative
censoring or “live removal” and other cause mortality. The latter two were the
only sources of censoring in the NLST because complete ascertainment with
respect to mortality was possibly through the use of the matching death
certificates through the national death index.
###### Condition 8.1.
Other cause mortality is proportional to cancer mortality, i.e. that
$\theta=-dlog(S_{oth})/dH$ is constant.
###### Condition 8.2.
Proportional allocation: $e(\xi,0)\equiv e(0,0)$.
###### Condition 8.3.
Accrual is uniform on the scale of $H$, so that
$S_{lr}(\xi)=\frac{H(\tau)-H(\xi)}{H(\tau)-H(\tau-t_{er})}\wedge 1,$ (8.5)
where $\tau$ is the time at which the required number of events are obtained,
and $t_{er}$ is the time at which randomization is completed.
###### Condition 8.4.
$Q(\xi)=\frac{\xi}{t_{c}}\wedge 1\equiv\frac{1-\exp(-H(\xi)\wedge
H(t_{c}))}{1-\exp(-H(t_{c}))}.$ (8.6)
The other cause versus cancer proportionality assumption is perhaps the most
arguable. However, the extent to which it is violated in practice has little
impact upon our results as other cause mortality enters our results only
through its survival function which maintains a value in excess of 0.95
throughout the trial. The proportional allocation assumption approximates what
we see in practice quite closely, especially in the case of a large trial of a
rare event. In the NLST there was 1 to 1 randomization so that $e(0,0)=1/2$.
The extent to which the latter two assumptions 8.3 and 8.4 hold both depend
upon the extent to which pooled cancer specific mortality grows at a constant
rate. In the case of the NLST, the pooled cancer mortality cumulative hazard
function did grow at an approximately linear rate.
Variance at Planned Termination
$\displaystyle v(\tau)$ $\displaystyle=$ $\displaystyle\langle
Q|{I}\kern-3.00003pt{F}|Q\rangle_{\tau}=\int_{0}^{\tau}Q^{2}(\xi)e(\xi,0)\left(1-e(\xi,0)\right)dG(\xi)$
(8.7) $\displaystyle=$
$\displaystyle\int_{0}^{\tau}Q^{2}(\xi)e(\xi,0)\left(1-e(\xi,0)\right)S_{oth}(\xi)S_{lr}(\xi)S(\xi)dH(\xi)\,.$
Here, $S$, $S_{lr}$ and $S_{oth}$ are survival functions corresponding to the
cross-arm pooled cancer mortality, administrative censoring or “live removal”
and other cause mortality. The latter two were the only sources of censoring
in the NLST because complete ascertainment with respect to mortality was
possibly through the use of the matching death certificates through the
national death index. Therefore, we can express the differential, $dG$, in
this way. Under assumptions 8.1, 8.2, 8.3, and 8.4, we apply the change of
variables, $\eta=H(\xi)$, to obtain
$\displaystyle v(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{H(\tau)}\left(1-{\mathrm{e}}^{-(\eta\wedge
H(t_{c}))}\right)^{2}\,{\mathrm{e}}^{-\theta\eta}\left\\{\frac{H(\tau)-\eta}{H(\tau)-H(\tau-
t_{er})}\wedge 1\right\\}\,{\mathrm{e}}^{-\eta}d\eta$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{H(t_{c})\wedge H(\tau-
t_{er})}\left(1-2{\mathrm{e}}^{-\eta}+{\mathrm{e}}^{-2\eta}\right)\,{\mathrm{e}}^{-(\theta+1)\eta}d\eta$
$\displaystyle\;\;+\frac{I\left(t_{c}<\tau-
t_{er}\right)}{4}\,\left(1-{\mathrm{e}}^{-H(t_{c})}\right)^{2}\,\int_{H(t_{c})}^{H(\tau-
t_{er})}{\mathrm{e}}^{-(\theta+1)\eta}d\eta$ $\displaystyle\;\;+\frac{I(\tau-
t_{er}<t_{c})}{4\left(H(\tau)-H(\tau-t_{er})\right)}\,\int_{H(\tau-
t_{er})}^{H(t_{c})}\,\left(1-2{\mathrm{e}}^{-\eta}+{\mathrm{e}}^{-2\eta}\right)\,{\mathrm{e}}^{-(\theta+1)\eta}\,\left(H(\tau)-\eta\right)d\eta$
$\displaystyle\;\;+\frac{\left(1-{\mathrm{e}}^{-H(t_{c})}\right)^{2}}{4\left(H(\tau)-H(\tau-
t_{er})\right)}\,\int_{H(\tau-t_{er})\vee
H(t_{c})}^{H(\tau)}{\mathrm{e}}^{-(\theta+1)\eta}\,\left(H(\tau)-\eta\right)d\eta$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}+I_{3}+I_{4}\,.$
These evaluate to:
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\frac{1-{\mathrm{e}}^{-(\theta+1)H_{m}}}{\theta+1}\;-\;2\,\frac{1-{\mathrm{e}}^{-(\theta+2)H_{m}}}{\theta+2}\;+\;\frac{1-{\mathrm{e}}^{-(\theta+3)H_{m}}}{\theta+3}\right\\}\;\;{\rm
where~{}}H_{m}=H(t_{c})\wedge H(\tau-t_{er})\,,$ $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle I(t_{c}<\tau-
t_{er})\,\left(1-{\mathrm{e}}^{-H(t_{c})}\right)^{2}\,\frac{{\mathrm{e}}^{(\theta+1)H(t_{c})}-{\mathrm{e}}^{-(\theta+1)H(\tau-
t_{er})}}{4(\theta+1)}\,,$ $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\frac{I(\tau-t_{er}<t_{c})}{4(H(\tau)-H(\tau-t_{er}))}$
$\displaystyle\;\;\times\;\left\\{\left(\frac{{\mathrm{e}}^{-(\theta+1)H(\tau-
t_{er})}}{\theta+1}-2\frac{{\mathrm{e}}^{-(\theta+2)H(\tau-
t_{er})}}{\theta+2}+\frac{{\mathrm{e}}^{-(\theta+3)H(\tau-
t_{er})}}{\theta+3}\right)\left(H(\tau)-H(\tau-t_{er})\right)\right.$
$\displaystyle\qquad-\;\left(\frac{{\mathrm{e}}^{-(\theta+1)H(t_{c})}}{\theta+1}-2\frac{{\mathrm{e}}^{-(\theta+2)H(t_{c})}}{\theta+2}+\frac{{\mathrm{e}}^{-(\theta+3)H(t_{c})}}{\theta+3}\right)\left(H(\tau)-H(t_{c})\right)$
$\displaystyle\qquad-\;\left(\frac{{\mathrm{e}}^{-(\theta+1)H(\tau-
t_{er})}-{\mathrm{e}}^{-(\theta+1)H(t_{c})}}{(\theta+1)^{2}}-2\frac{{\mathrm{e}}^{-(\theta+2)H(\tau-
t_{er})}-{\mathrm{e}}^{-(\theta+2)H(t_{c})}}{(\theta+2)^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\;\left.\left.\frac{{\mathrm{e}}^{-(\theta+3)H(\tau-
t_{er})}-{\mathrm{e}}^{-(\theta+3)H(t_{c})}}{(\theta+3)^{2}}\right)\right\\}\,,$
$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\frac{\left(1-{\mathrm{e}}^{-H(t_{c})}\right)^{2}}{4(\theta+1)}$
$\displaystyle\kern-10.00002pt\times\;\left\\{\frac{H(\tau)-(H(\tau-
t_{er})\vee H(t_{c}))}{H(\tau)-H(\tau-
t_{er})}\,{\mathrm{e}}^{-(\theta+1)\left(H(\tau-t_{er})\vee
H(t_{c})\right)}\;-\;\frac{{\mathrm{e}}^{-(\theta+1)(H(\tau-t_{er})\vee
H(t_{c}))}-{\mathrm{e}}^{-(\theta+1)H(\tau)}}{(\theta+1)(H(\tau)-H(\tau-
t_{er})}\right\\}$
respectively.
First Moment at Planned Termination
$\displaystyle m(\tau)$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}Q(\xi)e(\xi,0)\left(1-e(\xi,0)\right)dG(\xi)$
(8.8) $\displaystyle=$
$\displaystyle\int_{0}^{\tau}Q(\xi)e(\xi,0)\left(1-e(\xi,0)\right)S_{oth}(\xi)S_{lr}(\xi)S(\xi)dH(\xi)\,.$
Under assumptions 8.1, 8.2, 8.3, and 8.4, we again apply the change of
variables, $\eta=H(\xi)$, to obtain
$\displaystyle m(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{H(\tau)}\left(1-{\mathrm{e}}^{-\eta\wedge
H(t_{c})}\right)\,{\mathrm{e}}^{-\theta\eta}\,\left\\{\frac{H(\tau)-\eta}{H(\tau)-H(\tau-
t_{er})}\wedge 1\right\\}\,{\mathrm{e}}^{-\eta}d\eta$ $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{H(t_{c})\wedge H(\tau-
t_{er})}\left(1-{\mathrm{e}}^{-\eta}\right)\,{\mathrm{e}}^{-\theta\eta}\,{\mathrm{e}}^{-\eta}d\eta$
$\displaystyle\;+\;\frac{1}{4}\,I(t_{c}<\tau-
t_{er})\,\left(1-{\mathrm{e}}^{-H(t_{c})}\right)\,\int_{H(t_{c})}^{H(\tau-
t_{er})}\,{\mathrm{e}}^{-\theta\eta}\,{\mathrm{e}}^{-\eta}d\eta$
$\displaystyle\;+\;\frac{1}{4}\,I(t_{c}>\tau-t_{er})\,\int_{H(\tau-
t_{er})}^{H(t_{c})}\left(1-{\mathrm{e}}^{-\eta}\right)\,{\mathrm{e}}^{-\theta\eta}\,\frac{H(\tau)-\eta}{H(\tau)-H(\tau-
t_{er})}\,{\mathrm{e}}^{-\eta}d\eta$
$\displaystyle\;+\;\frac{1}{4}\,I(t_{c}<\tau)\,\left(1-{\mathrm{e}}^{-H(t_{c})}\right)\,\int_{H(t_{c})\vee
H(\tau-
t_{er})}^{H(\tau)}\,{\mathrm{e}}^{-\theta\eta}\,\frac{H(\tau)-\eta}{H(\tau)-H(\tau-
t_{er})}\,{\mathrm{e}}^{-\eta}d\eta$ $\displaystyle=$ $\displaystyle
J_{1}+J_{2}+J_{3}+J_{4}$
These evaluate to
$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\frac{1-{\mathrm{e}}^{-(\theta+1)(H(t_{c})\wedge
H(\tau-t_{er}))}}{\theta+1}-\frac{1-{\mathrm{e}}^{-(\theta+2)(H(t_{c})\wedge
H(\tau-t_{er}))}}{\theta+2}\right\\}\,,$ $\displaystyle J_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4}\,I(t_{c}<\tau-
t_{er})\,\left(1-{\mathrm{e}}^{-H(t_{c})}\right)\,\frac{{\mathrm{e}}^{-(\theta+1)H(t_{c})}-{\mathrm{e}}^{-(\theta+1)H(\tau-
t_{er})}}{\theta+1}\,,$ $\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\frac{I(t_{c}>\tau-t_{er})}{4\left(H(\tau)-H(\tau-
t_{er})\right)}$
$\displaystyle\qquad\qquad\times\;\left\\{\left(\frac{\left(H(\tau)-H(\tau-
t_{er})\right)\,{\mathrm{e}}^{-(\theta+1)H(\tau-
t_{er})}-\left(H(\tau)-H(t_{c})\right)\,{\mathrm{e}}^{-(\theta+1)H(t_{c})}}{\theta+1}\right.\right.$
$\displaystyle\qquad\qquad-\left.\frac{\left(H(\tau)-H(\tau-
t_{er})\right)\,{\mathrm{e}}^{-(\theta+2)H(\tau-
t_{er})}-\left(H(\tau)-H(t_{c})\right)\,{\mathrm{e}}^{-(\theta+2)H(t_{c})}}{\theta+2}\right)$
$\displaystyle\qquad\qquad-\left.\left(\frac{{\mathrm{e}}^{-(\theta+1)H(\tau-
t_{er})}-{\mathrm{e}}^{-(\theta+1)H(t_{c})}}{(\theta+1)^{2}}\;-\;\frac{{\mathrm{e}}^{-(\theta+2)H(\tau-
t_{er})}-{\mathrm{e}}^{-(\theta+2)H(t_{c})}}{(\theta+2)^{2}}\right)\right\\}$
$\displaystyle J_{4}$ $\displaystyle=$
$\displaystyle\frac{I(t_{c}<\tau)\,\left(1-{\mathrm{e}}^{-H(t_{c})}\right)}{4\left(H(\tau)-H(\tau-
t_{er})\right)}$
$\displaystyle\quad\times\;\left\\{\frac{\left(H(\tau)-H(t_{c}\vee(\tau-
t_{er}))\right)\,{\mathrm{e}}^{-(\theta+1)H(t_{c}\vee(\tau-
t_{er}))}}{\theta+1}\;-\;\frac{{\mathrm{e}}^{-(\theta+1)H(t_{c}\vee(\tau-
t_{er}))}-{\mathrm{e}}^{-(\theta+1)H(\tau)}}{(\theta+1)^{2}}\right\\}$
respectively.
Duration of Trial The duration the NLST was part of the design. In other
situations in which the design stipulates that the trial should run until
required number of events is attained, the above change of variables technique
can be used to find a closed form expression for
$G(\tau)=\int_{0}^{\tau}S_{oth}(\xi)S_{lr}(\xi)S(\xi)dH(\xi)\,.$ (8.9)
in terms of the projected values of $H$ at $t=\tau$ and $t=\tau-t_{er}$. Then
using the plug-in estimate ${{\rm I}\kern-1.79993pt{\rm E}}N_{n}(\tau)/n$ for
$G(\tau)$ this expression can be inverted to solve for $\tau$, the duration of
the trial.
### 8.3 Sampling density of $(J,X_{n}(t_{J}))$
As in Armitage, McPherson and Rowe, [1], the sampling density of
$(J,X_{n}(t_{J}))$ can be derived recursively as follows. Let
$\Delta_{j}=f_{n,j}-f_{n,j-1}$ and let $\phi_{v}(x)=\phi(x/\sqrt{v})/\sqrt{v}$
where $\phi$ is the density of the standard normal. First,
$\displaystyle\pi((1,x))=\phi_{{}_{f_{1}}}(x)\,.$ (8.10)
Next, for all $j>1$,
$\displaystyle\pi((j,x)\kern-10.00002pt$ ;
$\displaystyle\kern-10.00002pt\mathbf{b}_{1:(j-1)},\mathbf{f}_{1:j})$
$\displaystyle=$
$\displaystyle\kern-10.00002pt\int_{-\infty}^{\sqrt{f_{j-1}}b_{j-1}}\pi((j-1,\xi);\mathbf{b}_{1:(j-2)},\mathbf{f}_{1:(j-1)})\,\phi_{{}_{\Delta_{j}}}(x-\xi)\,d\xi$
|
arxiv-papers
| 2011-02-24T20:58:53 |
2024-09-04T02:49:17.278024
|
{
"license": "Public Domain",
"authors": "Grant Izmirlian",
"submitter": "Grant Izmirlian",
"url": "https://arxiv.org/abs/1102.5088"
}
|
1102.5353
|
# Regularization Schemes and Higher Order Corrections
William B. Kilgore Physics Department, Brookhaven National Laboratory, Upton,
New York 11973, USA.
[kilgore@bnl.gov]
###### Abstract
I apply commonly used regularization schemes to a multiloop calculation to
examine the properties of the schemes at higher orders. I find complete
consistency between the conventional dimensional regularization scheme and
dimensional reduction, but I find that the four-dimensional helicity scheme
produces incorrect results at next-to-next-to-leading order and singular
results at next-to-next-to-next-to-leading order. It is not, therefore, a
unitary regularization scheme.
## I Introduction
Dimensional regularization ’t Hooft and Veltman (1972) is an elegant and
efficient means of handling the divergences that arise in perturbation theory
beyond the tree level. Among its many favorable qualities it respects gauge
and Lorentz invariance and allows one to handle both ultraviolet and infrared
divergences in the same manner. The application of dimensional regularization
to different kinds of problems has led to the development of a variety of
regularization schemes, which share the dimensional regularization of momentum
integrals, but differ in their handling of external (or observed) states and
of spin degrees of freedom.
The original formulation of dimensional regularization ’t Hooft and Veltman
(1972), known as the ’t Hooft-Veltman (HV) scheme, specifies that observed
states are to be treated as four-dimensional, while internal states are to be
treated as $D_{m}=4-2\,{\varepsilon}$ dimensional. That is, both their momenta
and spin degrees of freedom were to be continued from four to $D_{m}$
dimensions. It turns out that one has the freedom to choose the value of the
trace of the Dirac unit matrix to take its canonical value of four, so
fermions continue to have two spin degrees of freedom, even though their
momenta are continued to $D_{m}$ dimensions. Internal gauge bosons, however,
have $D_{m}-2$ spin degrees of freedom (internal massive gauge bosons have
$D_{m}-1$ degrees of freedom).
A slight variation on the HV scheme has come to be called conventional
dimensional regularization (CDR) Collins (1984). In this variation, all
particles and momenta are taken to be $D_{m}$ dimensional. This often turns
out to be computationally more convenient, since one set of rules governs all
interactions. This is particularly so when computing higher order corrections
to theories subject to infrared sensitivities, like QCD. In the HV scheme, if
two external states have infrared sensitive overlaps, they must be treated as
internal, or $D_{m}$ dimensional states. In the CDR scheme, all states are
already treated as $D_{m}$ dimensional, so there is no possibility of failing
to properly account for infrared overlaps.
A third variation, called dimensional reduction (DRED) Siegel (1979), was
devised for application to supersymmetric theories. In supersymmetry, it is
essential that the number of bosonic degrees of freedom is exactly equal to
the number of fermionic degrees of freedom. This requirement is violated in
the HV and CDR schemes. In the DRED scheme, the continuation to $D_{m}$
dimensions is taken as a compactification from four dimensions. Thus, while
space-time is taken to be four-dimensional and particles have the standard
number of degrees of freedom, momenta span a $D_{m}$ dimensional vector space
and momentum integrals are regularized dimensionally.
A fourth variation, the four-dimensional helicity (FDH) scheme Bern and
Kosower (1992); Bern et al. (2002), was developed primarily for use in
constructing one-loop amplitudes from unitarity cuts. The most efficient
building blocks for such calculations are tree-level helicity amplitudes,
which necessarily have two spin degrees of freedom for both fermions and gauge
bosons. The FDH scheme resembles the DRED scheme in that it regularizes
momentum integrals dimensionally while maintaining the spin degrees of freedom
of a four-dimensional theory (and therefore appears to be a valid
supersymmetric regularization scheme Bern et al. (2002)), but there are
crucial differences, which I will discuss in detail.
The fact that the HV scheme respects the unitarity of the $S$-matrix was
proven at its introduction ’t Hooft and Veltman (1972). The arguments which
establish the validity of the HV scheme carry over to the CDR scheme and
establish that it too is a valid regularization scheme. After some initial
confusion over the proper renormalization procedure van Damme and ’t Hooft
(1985); Capper et al. (1980); Jack et al. (1994a) for the DRED scheme, it was
established that it too is a proper, unitary regularization scheme Jack et al.
(1994a) and that it is indeed equivalent to the CDR scheme Jack et al.
(1994b). The FDH scheme has never been subjected to such stringent
examination. It has been used successfully in a number of landmark next-to-
leading order (NLO) calculations, but it has never been established whether it
is a proper, unitary regularization scheme, or merely a set of shortcuts that
allow expert users to obtain correct results.
In this paper, I will perform a well-known multiloop calculation in the
various regularization schemes. I will show that while the HV and CDR scheme
calculations yield the correct result and the DRED scheme calculation, while
far more complicated is completely equivalent, the FDH scheme calculation
yields incorrect results which inevitably violate unitarity at sufficiently
high order. A detailed comparison of the various calculations identifies the
source of the unitarity violations in the FDH scheme.
The plan of this paper is as follows: in section two, I will describe the test
calculation to be performed and present the result to be obtained. In sections
three, four and five, I will describe in detail the calculation to next-to-
next-to-leading order (NNLO) as it is performed in the CDR, DRED and FDH
schemes, respectively. In section six, I present partial results at N3LO which
solidify the conclusion that the CDR and DRED schemes are equivalent and
correct, but that the FDH scheme violates unitarity. In section seven, I will
discuss my results and draw my conclusions.
## II The Test Environment
To test the regularization schemes, I will calculate two quantities: the
massless nonsinglet contributions to
1. 1.
the hadronic decay width of a fictitious neutral vector boson $V$, of mass
$M_{V}$;
2. 2.
the single photon approximation to the total hadronic annihilation cross
section for an electron – positron pair.
I will perform these calculations by means of the optical theorem, taking the
imaginary part of the forward scattering amplitudes. In both cases, this means
taking the imaginary part of the vacuum polarization tensor sandwiched between
external states. Since the optical theorem is a direct consequence of the
unitarity of the $S$-matrix, any unitary regularization scheme must give the
same result, once one expands in terms of a standard coupling. To avoid
complications involving prescriptions for handling $\gamma_{5}$ and the Levi-
Civita tensor, I will take $V$ to have only vectorlike couplings. In this way,
the vacuum polarization tensor for the $V$ boson will be identical to that of
the off shell photon, up to coupling constants and so the QCD expansion of the
two results will differ only by constant numerical factors.
Each regularization scheme will start from the same four-dimensional
Lagrangian,
$\begin{split}{\cal
L}=&-\frac{1}{2}A^{a}_{\mu}\left(\partial^{\mu}\partial^{\nu}(1-\xi^{-1})-g^{\mu\nu}\Box\right)A^{a}_{\nu}-g\,f^{abc}(\partial^{\mu}\,A^{a\,\nu})A^{b}_{\mu}\,A^{c}_{\nu}-\frac{g^{2}}{4}f^{abc}\,f^{ade}\,A^{b\,\mu}\,A^{c\,\nu}\,A^{d}_{\mu}\,A^{e}_{\nu}\\\
&+i\sum_{f}\,\overline{\psi}_{f}^{i}\left(\delta_{ij}\not{\partial}-i\,g\,t^{a}_{ij}\not{A}^{a}-i\,g_{V}\,Q_{f}\not{V}\right)\,\psi_{f}^{j}-\overline{c}^{a}\Box\,c^{a}+g\,f^{abc}\left(\partial_{\mu}\,\overline{c}^{a}\right)\,A^{b\,\mu}\,c^{c}\,,\end{split}$
(1)
where $A^{a\,\mu}$ is the QCD gauge field, $V^{\mu}$ is the massive vector
boson, $\psi_{f}$ is the quark field of flavor $f$, $\overline{c}^{a}$ and
$c^{a}$ are the Faddeev-Popov ghost fields, $g$ is the QCD coupling, $g_{V}$
is the $V$ gauge coupling and $Q_{f}$ represents the charge of the quark
flavor $f$ under the $V$ symmetry. I will not be computing nontrivial
corrections in $g_{V}$, so there is no need to specify the $V$-self
interaction parts of the Lagrangian.
Figure 1: Sample diagrams of one-, two- and three-loop contributions to the
vacuum polarization of $V$.
The result to N3LO is well known Chetyrkin et al. (1979); Dine and Sapirstein
(1979); Celmaster and Gonsalves (1980); Gorishnii et al. (1988, 1991),
$\begin{split}\Gamma^{V}_{had}=&\Gamma^{V}_{0,\,{\rm had}}\,{\cal
F}({\alpha_{s}^{\overline{\rm{MS}}}},Q^{2}=M_{V}^{2})\hskip
70.0pt\Gamma^{V}_{0,\,{\rm
had}}=\frac{\alpha_{V}\,M_{V}}{3}\,N_{c}\sum_{f}\,Q_{f}^{2}\\\
\sigma^{e^{+}\,e^{-}\to\ {\rm had}}(Q^{2})=&\sigma_{0}^{e^{+}\,e^{-}\to\ {\rm
had}}(Q^{2})\,{\cal F}({\alpha_{s}^{\overline{\rm{MS}}}},Q^{2})\hskip
40.0pt\sigma_{0}^{e^{+}\,e^{-}\to\ {\rm
had}}(Q^{2})=\frac{4\,\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\sum_{f}\,Q_{f}^{2}\\\
\end{split}$ (2)
and
$\begin{split}{\cal
F}({\alpha_{s}^{\overline{\rm{MS}}}},Q^{2})=\hskip-12.0pt&\hskip
12.0pt\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\left({\beta_{1}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}+{\beta_{0}^{\overline{\rm{MS}}}}^{\,2}\,\ln^{2}\frac{\mu^{2}}{Q^{2}}\right)\right]\right.\\\
&+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\left[\left(-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right)\right.\\\
&\qquad\qquad\times\left.\left(1+2{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right)\right]\\\
&+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\left[-C_{F}^{3}\frac{69}{128}+C_{F}^{2}\,C_{A}\left(-\frac{127}{64}-\frac{143}{16}\zeta_{3}+\frac{55}{4}\,\zeta_{5}\right)\right.\\\
&\qquad\qquad+C_{F}\,C_{A}^{2}\left(\frac{90445}{3456}-\frac{2737}{144}\,\zeta_{3}-\frac{55}{24}\,\zeta_{5}\right)\\\
&\qquad\qquad+C_{F}^{2}\,N_{f}\left(-\frac{29}{128}+\frac{19}{8}\,\zeta_{3}-\frac{5}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\,N_{f}\left(-\frac{485}{54}+\frac{56}{9}\,\zeta_{3}+\frac{5}{12}\,\zeta_{5}\right)\\\
&\left.\left.\qquad\qquad+C_{F}\,N_{f}^{2}\left(\frac{151}{216}-\frac{19}{36}\,\zeta_{3}\right)-\frac{1}{4}\,\pi^{2}\,C_{F}\,{\beta_{0}^{\overline{\rm{MS}}}}^{2}\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{4}\right)\right\\}\,.\end{split}$
(3)
To obtain the hadronic decay width at LO, NLO and NNLO, I need to compute the
QCD corrections to the vacuum polarization of the $V$ (photon) at $1$, $2$ and
$3$ loops, respectively. Sample diagrams are shown in Fig. (1).
### II.1 Methods
In each scheme, I will need to compute the vacuum polarization of $V$ and the
necessary coupling renormalization constants. As a cross-check on the
reliability of my calculational framework, I reproduce known results on the
QCD $\beta$-functions and mass anomalous dimensions to three-loop order, as
well as the three-loop QCD contributions to the $\beta$-function of $V$ (where
needed).
In all calculations, I generate the contributing diagrams using QGRAF Nogueira
(1993). The symbolic algebra program FORM Vermaseren (2000) is used to
implement the Feynman rules and perform algebraic manipulations to reduce the
result to a set of Feynman integrals to be performed and their coefficients.
The set of Feynman integrals are then reduced to master integrals using the
program REDUZE Studerus (2010). Using the method of Ref. Davydychev et al.
(1998), the vertex corrections can be expressed in terms of the same
propagator integrals used to compute the vacuum polarization and wave function
renormalizations. The complete set of master integrals at one, two and three
loops are shown in Fig. (2).
a) b)
c)
Figure 2: Master integrals for the evaluation of vacuum polarization at a) one
loop, b) two loops and c) three loops.
Most of the master integrals are trivial iterated-bubble diagrams and the
others were evaluated long ago Chetyrkin et al. (1980); Kazakov (1984). As an
additional cross-check, the integral reduction and evaluation is also
performed using the program MINCERGorishnii et al. (1989); Larin et al.
(1991).
### II.2 Notation
The various schemes that I will consider span a variety of vector spaces, each
with their own metric tensor. To establish some level of consistency, I will
denote the metric tensor of classical four-dimensional space-time as
$\eta^{\mu\nu}$; the metric tensor of the $D_{m}$ dimensional vector space in
which momentum integrals are regularized will be denoted as
$\hat{g}^{\mu\nu}$; and the metric tensor of the largest vector space will be
denoted $g^{\mu\nu}$. Where it does not vanish, the complement of
$\hat{g}^{\mu\nu}$ will be denoted as
$\delta^{\mu\nu}=g^{\mu\nu}-\hat{g}^{\mu\nu}$. Similarly, the Dirac matrices
$\gamma^{\mu}$, will be denoted $\gamma_{(4)}^{\mu}$ when they are strictly
four-dimensional, $\hat{\gamma}^{\mu}$ when they span the $D_{m}$ dimensional
space and $\bar{\gamma}^{\mu}$ in the space spanned by $\delta^{\mu\nu}$.
I will now present the details of the calculation in the CDR, DRED and FDH
schemes.
## III Conventional Dimensional Regularization
In the CDR scheme, the calculation is quite straightforward. The Lagrangian
and Feynman rules are just the same as for a four-dimensional calculation,
except that the Dirac matrices $\gamma^{\mu}$ and the metric tensor
$g^{\mu\nu}$ have been extended to span a $D_{m}$ dimensional vector space.
That is,
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\,g^{\mu\nu}\,,\qquad
g^{\mu\nu}\,g_{\mu\nu}=D_{m}\,,\qquad\gamma^{\mu}\,\gamma_{\mu}=D_{m}\,,\qquad
g^{\mu\nu}\equiv\hat{g}^{\mu\nu}\,.$ (4)
The Dirac trace, $\mathop{\rm Tr\left[{1}\right]}\nolimits=4$, retains its
standard normalization.
Although $D_{m}$ is given the representation $D_{m}=4-2\,{\varepsilon}$, the
sign of ${\varepsilon}$ is not determined. If it is taken to be positive, so
that $D_{m}<4$, then the Feynman integrals that one encounters are convergent
under the rules of ultraviolet power counting. On the other hand, infrared
power counting would prefer ${\varepsilon}<0\Rightarrow D_{m}>4$. In practice,
the sign of ${\varepsilon}$ does not matter and it can be used to regularize
both infrared and ultraviolet divergences. Regardless of the sign of
${\varepsilon}$, it is important that the vector space in which momenta take
values is larger than the standard $3+1$ dimensional space-time. This means
that the standard four-dimensional metric tensor $\eta^{\mu\nu}$ spans a
smaller space than the $D_{m}$ dimensional metric tensor, and the four-
dimensional Dirac matrices $\gamma^{0,1,2,3}$ form a subset of the full
$\gamma^{\mu}$,
$g^{\mu\nu}\,g_{\mu}^{\rho}=g^{\nu\rho}\,,\qquad\qquad
g^{\mu\nu}\,\eta_{\mu}^{\rho}=\eta^{\nu\rho}\,,\qquad\qquad\eta^{\mu\nu}\,\eta_{\mu}^{\rho}=\eta^{\nu\rho}\,.$
(5)
These considerations are of particular importance when considering chiral
objects involving $\gamma_{5}$ and the Levi-Civita tensor, but will play a
role in our discussion below.
Because the Dirac trace is unchanged, fermions still have exactly two degrees
of freedom in the CDR scheme. Gauge bosons, however, acquire extra spin
degrees of freedom in the $D_{m}$ dimensional vector space. The spin sum over
polarization vectors in a physical (axial) gauge takes the form
$-g_{\mu\nu}\,\sum_{\lambda}\epsilon^{*\,\mu}(k,\lambda)\,\epsilon^{\nu}(k,\lambda)=g_{\mu\nu}\,\left(g^{\mu\nu}-\frac{k^{\mu}\,n^{\nu}+n^{\mu}\,k^{\nu}}{k\cdot
n}\right)=D_{m}-2=2-2\,{\varepsilon}\,,$ (6)
where $n$ is the axial gauge reference vector. For massive vector bosons, the
spin sum becomes
$-g_{\mu\nu}\,\sum_{\lambda}\epsilon^{*\,\mu}(k,\lambda)\,\epsilon^{\nu}(k,\lambda)=g_{\mu\nu}\,\left(g^{\mu\nu}-\frac{k^{\mu}\,k^{\nu}}{M^{2}}\right)=D_{m}-1=3-2\,{\varepsilon}\,,$
(7)
### III.1 Renormalization
The renormalization constants in the CDR scheme are defined as
$\begin{split}\Gamma^{(B)}_{AAA}&=Z_{1}\Gamma_{AAA}\,,\qquad\psi^{(B)\,i}_{f}=Z^{\frac{1}{2}}_{2}\,\psi^{i}_{f}\,,\qquad
A^{(B)\,a}_{\mu}=Z^{\frac{1}{2}}_{3}\,A^{a}_{\mu}\\\
\Gamma^{(B)}_{c\overline{c}A}&=\widetilde{Z}_{1}\Gamma_{q\overline{q}A}\,,\qquad\
c^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,c^{a}\,,\qquad\ \
\overline{c}^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,\overline{c}^{a}\,,\\\
\Gamma^{(B)}_{q\overline{q}A}&=Z_{1\,F}\Gamma_{q\overline{q}A}\,,\qquad\xi^{(B)}=\xi\,Z_{3}\,,\end{split}$
(8)
where $\Gamma_{abc}$ represents the vertex function involving fields $a$, $b$
and $c$.
Although we treat the quark fields as massless, we can compute the mass
anomalous dimension by introducing a fictitious scalar particle $\phi$ and
computing the $\beta$-function of its Yukawa coupling to the quarks. The
equivalence is clear from the standard model, where the Higgs Yukawa coupling
and the fermion mass are proportional at leading electroweak order and must
behave the same under QCD renormalization. For this purpose, I introduce one
more renormalization constant,
$\Gamma^{(B)}_{q\overline{q}\phi}=Z_{1\,\phi}\Gamma_{q\overline{q}\phi}$. One
can introduce a wave function renormalization for $\phi$, $Z_{3\,\phi}$, but
it will not contribute because $Z_{3\,\phi}=1+{\cal O}(\alpha_{\phi})$. Note
also that I do not need to compute the QCD corrections to the $\beta$-function
for $\alpha_{V}$, which will start at order $\alpha_{V}^{2}$ because of the
Ward Identity.
In the $\overline{\rm MS}$ scheme, the couplings renormalize as
$\begin{split}{\alpha_{s}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}\,{\alpha_{s}^{\overline{\rm{MS}}}}\,,\qquad
Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}=\frac{Z_{1}^{2}}{Z_{3}^{3}}=\frac{Z_{1\,F}^{2}}{Z_{2}^{2}\,Z_{3}}=\frac{\widetilde{Z}_{1}^{2}}{\widetilde{Z}_{3}^{2}\,Z_{3}}\\\
{\alpha_{\phi}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}\,{\alpha_{\phi}^{\overline{\rm{MS}}}}\,,\qquad
Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}=\frac{Z_{1\,\phi}^{2}}{Z_{2}^{2}\,Z_{3\,\phi}}\end{split}$
(9)
The structure of the renormalization constants
$Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}$ and
$Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}$ is determined entirely by their
lowest order ($1/{\varepsilon}$) poles, which in turn define the
$\beta$-functions.
$\begin{split}{\beta^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{\pi}&=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{\pi}\left(1+\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}}{\partial{\alpha_{s}^{\overline{\rm{MS}}}}}\right)^{-1}\\\
&=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{\pi}-\sum_{n=0}^{\infty}\,{\beta_{n}^{\overline{\rm{MS}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{n+2}\\\
{\beta_{\phi}^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{\pi}+\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}{\partial{\alpha_{s}^{\overline{\rm{MS}}}}}\,{\beta^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})\right)\left(1+\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}{\partial{\alpha_{\phi}^{\overline{\rm{MS}}}}}\right)^{-1}\\\
&=-\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{\pi}\left({\varepsilon}+\sum_{n=0}^{\infty}\,{\beta_{\phi\,,n}^{\overline{\rm{MS}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{n+1}\right)\\\
\end{split}$ (10)
The mass anomalous dimension,
${\gamma^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})=\frac{\mu^{2}}{m^{{\overline{\rm
MS}}}}\frac{d}{d\mu^{2}}m^{{\overline{\rm
MS}}}=\sum_{n=0}^{\infty}-{\gamma_{n}^{\overline{\rm{MS}}}}{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{n+1}$
(11)
is defined in terms of $m$, rather than $m^{2}$, with the result that
${\gamma_{n}^{\overline{\rm{MS}}}}=\frac{1}{2}{\beta_{\phi\,,n}^{\overline{\rm{MS}}}}$.
The results for ${\beta_{n}^{\overline{\rm{MS}}}}$ and
${\gamma_{n}^{\overline{\rm{MS}}}}$ through three loops are given in Appendix
A.
### III.2 Vacuum polarization in the CDR scheme
The imaginary part of the unrenormalized vacuum polarization tensor in the CDR
scheme is
$\begin{split}\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{CDR}}\right]&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt1+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{4}+{\varepsilon}\left(\frac{55}{8}-6\,\zeta_{3}\right)+{\varepsilon}^{2}\,\left(\frac{1711}{48}-\frac{15}{4}\,\zeta_{2}-19\,\zeta_{3}-9\,\zeta_{4}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\,\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)\right.\\\
&\hskip-30.0pt-\frac{3}{32}\,C_{F}^{2}+C_{F}\,C_{A}\left(\frac{487}{48}-\frac{33}{4}\zeta_{3}\right)+C_{F}\,N_{f}\left(-\frac{11}{6}+\frac{3}{2}\,\zeta_{3}\right)\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{143}{32}-\frac{111}{8}\,\zeta_{3}+\frac{45}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\left(\frac{50339}{576}-\frac{231}{32}\,\zeta_{2}-\frac{109}{2}\,\zeta_{3}-\frac{99}{8}\,\zeta_{4}-\frac{15}{4}\,\zeta_{5}\right)\right.\\\
&\hskip-30.0pt\left.\left.\left.+C_{F}\,N_{f}\left(-\frac{4417}{288}+\frac{21}{16}\,\zeta_{2}+\frac{19}{2}\,\zeta_{3}+\frac{9}{4}\,\zeta_{4}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(12)
Upon renormalizing the QCD coupling according to Eq. (9), setting
${\alpha_{V}^{B}}\to{\alpha_{V}}\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}$,
and dropping terms of order $({\varepsilon})$, I obtain
$\begin{split}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR}}\right]&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(13)
In this way of performing the calculation, all of the QCD states that appear
are internal states, so the HV scheme gives exactly the same result.
### III.3 Total Decay rate and annihilation cross section in the CDR scheme
The decay rate and the annihilation cross section are determined by computing
the imaginary part of the forward scattering amplitude. For the decay rate,
this means attaching the polarization vector ${\varepsilon}^{\mu}(Q,\lambda)$
and its conjugate ${\varepsilon}^{\nu}(Q,\lambda)^{*}$ ($Q^{2}=M_{V}^{2}$) and
averaging over the spins,
$\Gamma^{{CDR}}_{V\to\ {\rm hadrons}}=\frac{1}{M_{V}}\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR}}\right]\,{\varepsilon}^{\nu}(Q,\lambda)^{*}\,,$
(14)
where
$\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,{\varepsilon}^{\nu}(Q,\lambda)^{*}=\frac{1}{N_{\rm
spins}}\left(-g^{\mu\nu}+\frac{Q^{\mu}\,Q^{\nu}}{M_{V}^{2}}\right)\,.$ (15)
Notice that because the imaginary part of the vacuum polarization tensor is
finite, it does not matter whether the spin sum is taken in
$D_{m}=4-2\,{\varepsilon}$ dimensions as in the CDR scheme or in four
dimensions as in the HV scheme as the difference is of order ${\varepsilon}$.
The result is
$\begin{split}\Gamma^{{CDR}}_{V\to\ {\rm
hadrons}}=&\frac{{\alpha_{V}}\,M_{V}}{3}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(16)
in agreement with Eqs. (LABEL:eqn:knownresult-3).
For the annihilation cross section $\sigma_{e^{+}\,e^{-}\to\ {\rm hadrons}}$,
one attaches fermion bilinears to each end of the vacuum polarization tensor
and averages over the spins.
$\sigma^{{CDR}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=\frac{2}{Q^{2}}\frac{e^{2}}{4}\sum_{\lambda\,\lambda^{{}^{\prime}}}\frac{{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\gamma^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}}{Q^{2}}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR},\,{\alpha_{V}}\to\alpha}\right]\frac{{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\gamma^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}}{Q^{2}}\,.$
(17)
Because this is a forward scattering amplitude, the spinor bilinears can be
combined into a trace,
$\frac{1}{2}\sum_{\lambda\,\lambda^{{}^{\prime}}}{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\gamma^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\gamma^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}=\frac{1}{2}\mathop{\rm
Tr\left[{\not{p}_{e^{+}}\,\gamma^{\mu}\not{p}_{e^{-}}\,\gamma^{\nu}}\right]}\nolimits=\left(-Q^{2}\,g^{\mu\,\nu}+Q^{\mu}\,Q^{\nu}\right)\,,$
(18)
where the last identification results from the fact that
$Q^{\mu}=p_{e^{-}}^{\mu}+p_{e^{+}}^{\mu}$,
$p_{e^{-}}\cdot\,Q=p_{e^{+}}\cdot\,Q=Q^{2}/2$. The result is
$\begin{split}\sigma^{{CDR}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=&\frac{4\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(19)
again in agreement with Eqs. (LABEL:eqn:knownresult-3).
Thus, I have established that I can reproduce the known results in the CDR
scheme through three-loop order, which is a strong check on my computational
framework.
## IV Dimensional Reduction
In dimensional reduction, one starts from standard four-dimensional space-time
and compactifies to a smaller vector space of dimension
$D_{m}=4-2\,{\varepsilon}<4$ in which momenta take values. The particles in
the spectrum, however, retain the spin degrees of freedom of four dimensions.
That is, both fermions and gauge bosons still have two degrees of freedom.
This is by design, of course, since it is required by supersymmetry. All Dirac
algebra can be treated as four-dimensional. However, now the four-dimensional
metric tensor $\eta^{\mu\nu}$ spans a larger space than the $D_{m}$
dimensional metric $\hat{g}^{\mu\nu}$ that might arise from tensor momentum
integrals,
$\hat{g}^{\mu\nu}\,\eta_{\mu}^{\rho}=\hat{g}^{\nu\rho}\,.$ (20)
There is also a very serious consequence of the fact that the $D_{m}$
dimensional vector space is smaller than four-dimensional space-time. The Ward
Identity only applies to the $D_{m}$ dimensional vector space. This means that
the $2\,{\varepsilon}$ spin degrees of freedom that are not protected by the
Ward Identity must renormalize differently than the $2-2\,{\varepsilon}$
degrees of freedom that are protected. In supersymmetric theories, the
supersymmetry provides the missing Ward Identity which demands that the
$2\,{\varepsilon}$ spin degrees of freedom be treated as gauge bosons. In
nonsupersymmetric theories, however, they must be considered to be distinct
particles, with distinct couplings and renormalization properties. It is
common to refer to these extra degrees of freedom as “${\varepsilon}$-scalars”
or as “evanescent” degrees of freedom.
Once the evanescent degrees of freedom (which I will label
$A_{e}^{a\,\tilde{\mu}}$, to distinguish them from the gluons, $A^{a\,\mu}$)
are recognized as independent particles, it is apparent that their couplings
are also independent, not only of the QCD coupling, but of one another. That
is, the coupling $g_{e}$ of the evanescent gluons to the quarks is not only
distinct from $g$, the coupling of QCD, but is also distinct from
$\lambda_{i}$, the quartic couplings of the evanescent gluons to themselves.
(The quartic gauge coupling of QCD splits into three independent quartic
couplings of the evanescent gluons.) Note that the massive vector boson
$V^{\mu}$ also has evanescent degrees of freedom, $V_{e}^{\tilde{\mu}}$, which
couple to quarks with strength $g_{Ve}$.
Thus, the Lagrangian in the DRED scheme becomes:
$\begin{split}{\cal
L}=&-\frac{1}{2}A^{a}_{\mu}\left(\partial^{\mu}\partial^{\nu}(1-\xi^{-1})-\hat{g}^{\mu\nu}\Box\right)A^{a}_{\nu}-g\,f^{abc}(\partial^{\mu}\,A^{a\,\nu})A^{b}_{\mu}\,A^{c}_{\nu}-\frac{g^{2}}{4}f^{abc}\,f^{ade}\,A^{b\,\mu}\,A^{c\,\nu}\,A^{d}_{\mu}\,A^{e}_{\nu}\\\
&+i\sum_{f}\,\overline{\psi}_{f}^{i}\left(\delta_{ij}\not{\partial}-i\,g\,t^{a}_{ij}\not{A}^{a}-i\,g_{V}\,Q_{f}\not{V}\right)\,\psi_{f}^{j}-\overline{c}^{a}\Box\,c^{a}+g\,f^{abc}\left(\partial_{\mu}\,\overline{c}^{a}\right)\,A^{b\,\mu}\,c^{c}\\\
&+\frac{1}{2}A_{e\,\tilde{\mu}}^{a}\Box\
A_{e}^{a\,\tilde{\mu}}-g\,f^{abc}(\partial^{\mu}\,A_{e}^{a\,\tilde{\nu}})A^{b}_{\mu}\,A^{c}_{e\,\tilde{\nu}}+\frac{g^{2}}{2}f^{abc}\,f^{adf}\,A^{b\,\mu}\,A_{e}^{c\,\tilde{\nu}}\,A^{d}_{\mu}\,A_{e\,\tilde{\nu}}^{f}-\frac{1}{4}\sum_{i}\lambda_{i}\,H_{i}^{bcdf}\,A_{e}^{b\,\tilde{\mu}}\,A_{e}^{c\,\tilde{\nu}}\,A_{e\,\tilde{\mu}}^{d}\,A_{e\,\tilde{\nu}}^{f}\\\
&+\sum_{f}\,\overline{\psi}_{f}^{i}\left(g_{e}\,t^{a}_{ij}\not{A}_{e}^{a}+g_{Ve}\,Q_{f}\not{V}_{e}\right)\,\psi_{f}^{j}\,.\end{split}$
(21)
As mentioned above, the quartic coupling of the evanescent gluons splits into
three terms, which mix under renormalization. One can choose the tensors
$H_{i}^{bcde}$ to be Harlander et al. (2006a)
$\begin{split}H_{1}^{bcde}=&\frac{1}{2}\left(f^{abc}\,f^{ade}+f^{abe}\,f^{adc}\right)\\\
H_{2}^{bcde}=&\delta^{bc}\delta^{de}+\delta^{bd}\delta^{ce}+\delta^{be}\delta^{cd}\\\
H_{3}^{bcde}=&\frac{1}{2}\left(\delta^{bc}\delta^{de}+\delta^{be}\delta^{cd}\right)-\delta^{bd}\delta^{ce}\,,\end{split}$
(22)
Although the quartic couplings enter the $\beta$-functions and anomalous
dimension at three loops and are essential to the renormalization program,
they do not explicitly contribute to the calculation at hand.
Now that the correct spectrum has been identified, one must carefully consider
the renormalization program. The naïve application of the principle of minimal
subtraction leads to the violation of unitarity van Damme and ’t Hooft (1985).
Because the contributions of evanescent states and couplings to scattering
amplitudes are weighted by a factor ${\varepsilon}$, the leading one-loop
contribution is finite and therefore not subtracted. As one proceeds to higher
orders, there is a mismatch among the counterterms such that the
renormalization program fails to remove all of the ultraviolet singularities.
A successful renormalization program for the DRED scheme Jack et al. (1994a,
b) applies the principle of minimal subtraction to the evanescent Green
functions (that is, Green functions with external evanescent states)
themselves. At each order, the renormalization scheme renders the evanescent
Green functions finite. Since evanescent Green functions enter into the
scattering amplitudes of physical particles at order ${\varepsilon}$ and they
are rendered finite by renormalization, they never contribute to physical
scattering amplitudes.
The evanescent coupling still contributes to Green functions with only
physical external states, but the contribution is rendered finite by the
prescribed renormalization program Jack et al. (1994a, b); Harlander et al.
(2006b, a). Because the evanescent coupling, $\alpha_{e}$ renormalizes
differently than the gauge coupling $\alpha_{s}$, the two cannot be
identified, even at the end of the calculation. One can choose a
renormalization point where the two coincide, but they evolve differently
under renormalization group transformations and their values will diverge as
one moves away from the renormalization point.
Still, the evanescent coupling is essentially a fictitious quantity and one
finds that if one computes a physical quantity in the DRED scheme and then
converts the running couplings of the DRED scheme to those of a scheme such as
CDR that has no evanescent couplings, the factors of $\alpha_{e}$ drop out
Harlander et al. (2006b, a).
### IV.1 Renormalization
The renormalization constants in the DRED scheme are defined as
$\begin{split}\Gamma^{(B)}_{AAA}&=Z_{1}\Gamma_{AAA}\,,\qquad\psi^{(B)\,i}_{f}=Z^{\frac{1}{2}}_{2}\,\psi^{i}_{f}\,,\qquad
A^{(B)\,a}_{\mu}=Z^{\frac{1}{2}}_{3}\,A^{a}_{\mu}\\\
\Gamma^{(B)}_{c\overline{c}A}&=\widetilde{Z}_{1}\Gamma_{q\overline{q}A}\,,\qquad\
c^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,c^{a}\,,\qquad\ \
\overline{c}^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,\overline{c}^{a}\,,\\\
\Gamma^{(B)}_{q\overline{q}A}&=Z_{1\,F}\Gamma_{q\overline{q}A}\,,\qquad\xi^{(B)}=\xi\,Z_{3}\,,\\\
\Gamma^{(B)}_{q\overline{q}e}&=Z_{1\,e}\Gamma_{q\overline{q}e}\,,\qquad
A^{(B)\,a}_{e\,\mu}=Z^{\frac{1}{2}}_{3\,e}\,A^{a}_{e\,\mu}\,,\qquad\Gamma^{(B)\,i}_{eeee}=Z^{i}_{1\,eeee}\,\Gamma^{i}_{eeee}\,,\\\
\Gamma^{(B)}_{q\overline{q}V_{e}}&=Z_{1\,Ve}\Gamma_{q\overline{q}V_{e}}\,,\qquad
V^{(B)}_{e\,\mu}=Z^{\frac{1}{2}}_{3\,Ve}\,V_{e\,\mu}\,.\end{split}$ (23)
In addition, I again introduce the fictitious scalar that allows me to compute
the mass anomalous dimension for massless quarks. Note that while the Ward
Identity protects $\alpha_{V}$ from leading QCD corrections, it does not
protect $\alpha_{Ve}$. That is why I need to introduce renormalization
constants for the vertex and wave-function and why I need to compute the
$\beta$-function of $\alpha_{Ve}$.
In the ${\overline{\rm DR}}$ scheme (modified minimal subtraction in the DRED
scheme), the couplings renormalize as
$\begin{split}{\alpha_{s}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\alpha_{s}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}=\frac{Z_{1}^{2}}{Z_{3}^{3}}=\frac{Z_{1\,F}^{2}}{Z_{2}^{2}\,Z_{3}}=\frac{\widetilde{Z}_{1}^{2}}{\widetilde{Z}_{3}^{2}\,Z_{3}}\,,\\\
{\alpha_{e}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\alpha_{e}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}=\frac{Z_{1\,e}^{2}}{Z_{2}^{2}\,Z_{3\,e}}\,,\\\
{\alpha_{Ve}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}\,{\alpha_{Ve}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}=\frac{Z_{1\,Ve}^{2}}{Z_{2}^{2}\,Z_{3\,Ve}}\,,\\\
\
{\alpha_{\phi}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}\,{\alpha_{\phi}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}=\frac{Z_{1\,\phi}^{2}}{Z_{2}^{2}\,Z_{3\,\phi}}\,.\end{split}$
(24)
and the $\beta$-functions are given by
$\begin{split}{\beta^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\beta_{e}^{\overline{\rm{DR}}}}+\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\left(1+\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}-\sum_{i,j,k,l,m}\,{\beta_{ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\\\
{\beta_{e}^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\beta^{\overline{\rm{DR}}}}+\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\left(1+\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-{\varepsilon}\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{\pi}-\sum_{i,j,k,l,m}\,{\beta_{e,\,ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\\\
{\beta_{Ve}^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\beta^{\overline{\rm{DR}}}}+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\beta_{e}^{\overline{\rm{DR}}}}+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\\\ &\hskip
100.0pt\times\left(1+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{Ve}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{\pi}\left({\varepsilon}+\sum_{i,j,k,l,m}\,{\beta_{Ve,\,ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\right)\\\
{\beta_{\phi}^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\beta^{\overline{\rm{DR}}}}+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\beta_{e}^{\overline{\rm{DR}}}}+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\\\ &\hskip
100.0pt\times\left(1+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{\phi}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{\pi}\left({\varepsilon}+\sum_{i,j,k,l,m}\,{\beta_{\phi,\,ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\right)\\\ \end{split}$ (25)
Through three-loop order, the $\eta_{i}$ do not contribute to the QCD
$\beta$-function, ${\beta^{\overline{\rm{DR}}}}$, nor to the vacuum
polarization of $V$ (or $V_{e}$). To three-loop order, I find agreement with
known results Harlander et al. (2006b, a) and derive new results for the
$\beta$-function of $\alpha_{Ve}$. The coefficients of the $\beta$-functions
and anomalous dimensions are given in Appendix B.
By comparing ${\beta_{Ve,\,\,}^{\overline{\rm{DR}}}}$ and
${\gamma^{\overline{\rm{DR}}}}$ in Eqs. (61-62), we see that the term
“${\varepsilon}$-scalar” is a misnomer. If the evanescent part of $V$ were a
true scalar, its $\beta$-function would coincide (but for a factor of $2$)
with the mass anomalous dimension. The pure
${\alpha_{s}^{\overline{\rm{DR}}}}$ terms do coincide, because there is no
nonvanishing contraction of the Lorentz indices of the evanescent $V$ and
those of the gluons. Because there are contractions between the Lorentz
indices of the evanescent $V$ and those of the evanescent gluons, however,
terms involving ${\alpha_{e}^{\overline{\rm{DR}}}}$ do not agree.
Calculations in the DRED scheme naturally produce results in terms of
${\alpha_{s}^{\overline{\rm{DR}}}}$ while the standard result has been
expressed in terms of ${\alpha_{s}^{\overline{\rm{MS}}}}$. One can always
convert one renormalized coupling to another. The rule for converting
${\alpha_{s}^{\overline{\rm{DR}}}}\to{\alpha_{s}^{\overline{\rm{MS}}}}$,
derived in Refs. Kunszt et al. (1994); Harlander et al. (2006b), is
${\alpha_{s}^{\overline{\rm{DR}}}}={\alpha_{s}^{\overline{\rm{MS}}}}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\frac{C_{A}}{12}+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\frac{11}{72}C_{A}^{2}-{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}\frac{C_{F}\,N_{f}}{16}+\ldots\right]$
(26)
When the result is expressed in terms of ${\alpha_{s}^{\overline{\rm{MS}}}}$,
all ${\alpha_{e}^{\overline{\rm{DR}}}}$ terms drop out.
### IV.2 Vacuum polarization in the DRED scheme
In the DRED scheme, there are two independent transverse vacuum polarization
tensors,
$\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{DRED}}\right]=\frac{-Q^{2}\,\hat{g}_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}\,\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{DRED}}\right]-Q^{2}\,\frac{\delta_{\mu\nu}}{2\,{\varepsilon}}\,\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{DRED}}\right]\,,$
(27)
where
$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{DRED}}\right]&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt1+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{4}+{\varepsilon}\left(\frac{51}{8}-6\,\zeta_{3}\right)+{\varepsilon}^{2}\,\left(\frac{497}{16}-\frac{15}{4}\zeta_{2}-15\,\zeta_{3}-9\,\zeta_{4}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[-{\varepsilon}\,\frac{3}{4}-{\varepsilon}^{2}\,\frac{29}{8}+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)-\frac{3}{32}C_{F}^{2}+\left(\frac{77}{8}-\frac{33}{4}\zeta_{3}\right)\,C_{F}\,C_{A}-\left(\frac{7}{4}-\frac{3}{2}\zeta_{3}\right)\,C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{141}{32}-\frac{111}{8}\,\zeta_{3}+\frac{45}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\left(\frac{15301}{192}-\frac{231}{32}\,\zeta_{2}-\frac{193}{4}\,\zeta_{3}-\frac{99}{8}\,\zeta_{4}-\frac{15}{4}\,\zeta_{5}\right)\right.\\\
&\hskip-20.0pt\left.\left.+C_{F}\,N_{f}\left(-\frac{1355}{96}+\frac{21}{16}\,\zeta_{2}+\frac{17}{2}\,\zeta_{3}+\frac{9}{4}\,\zeta_{4}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{3}{4}C_{F}^{2}-\frac{3}{8}C_{F}\,C_{A}+\frac{3}{16}C_{F}\,N_{f}-{\varepsilon}\left(\frac{47}{8}C_{F}^{2}-\frac{11}{4}C_{F}\,C_{A}+\frac{7}{4}C_{F}\,N_{f}\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}\left.+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[-\frac{9}{8}C_{F}^{2}-{\varepsilon}\left(\frac{141}{16}C_{F}^{2}+\frac{21}{16}C_{F}\,C_{A}\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)\right\\}\,,\end{split}$
(28)
and
$\begin{split}\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{DRED}}\right]&={\alpha_{Ve}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{{\varepsilon}+2\,{\varepsilon}^{2}+\left(4-\frac{3}{2}\zeta_{2}\right){\varepsilon}^{3}+{\cal
O}({\varepsilon}^{4})\right.\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{2}+{\varepsilon}\frac{29}{4}+{\varepsilon}^{2}\,\left(\frac{227}{8}-\frac{15}{2}\zeta_{2}-6\,\zeta_{3}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[-1-4\,{\varepsilon}-{\varepsilon}^{2}\left(\frac{27}{2}-5\,\zeta_{2}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\frac{9}{8}C_{F}^{2}+\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)+\frac{279}{32}C_{F}^{2}+\frac{199}{32}C_{F}\,C_{A}-\frac{17}{16}C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(\frac{3139}{64}-\frac{189}{16}\,\zeta_{2}-\frac{45}{4}\,\zeta_{3}\right)+C_{F}\,C_{A}\left(\frac{2473}{64}-\frac{231}{32}\,\zeta_{2}-\frac{75}{8}\,\zeta_{3}\right)\right.\\\
&\hskip-20.0pt\left.\left.+C_{F}\,N_{f}\left(-\frac{207}{32}+\frac{21}{16}\,\zeta_{2}+\frac{3}{2}\,\zeta_{3}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[-\frac{1}{{\varepsilon}}\frac{9}{4}C_{F}^{2}-\frac{129}{8}C_{F}^{2}-\frac{3}{8}C_{F}\,C_{A}\right.\\\
&\hskip-30.0pt\phantom{1}\left.-{\varepsilon}\left(\left(\frac{671}{8}-\frac{189}{8}\zeta_{2}-9\,\zeta_{3}\right)\,C_{F}^{2}+\frac{53}{16}C_{F}\,C_{A}\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(C_{F}^{2}-\frac{1}{4}C_{F}\,C_{A}+\frac{1}{8}C_{F}\,N_{f}\right)+\frac{13}{2}C_{F}^{2}-\frac{3}{2}C_{F}\,C_{A}+\frac{15}{16}C_{F}\,N_{f}\right.\\\
&\hskip-50.0pt\phantom{1}\left.+{\varepsilon}\left(\left(31-\frac{21}{2}\zeta_{2}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}-\left(\frac{53}{8}-\frac{21}{8}\zeta_{2}-\frac{3}{8}\zeta_{3}\right)\,C_{F}\,C_{A}+\left(\frac{157}{32}-\frac{21}{16}\zeta_{2}\right)\,C_{F}\,N_{f}\right)+{\cal
O}({\varepsilon}^{2})\right]\\\ &\hskip-50.0pt\phantom{1}\left.+{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)\right\\}\,,\\\
\end{split}$ (29)
where $\displaystyle{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)$
denotes terms for which the sum of the powers of
$\displaystyle\left(\frac{{\alpha_{s}^{B}}}{\pi}\right)$ and
$\displaystyle\left(\frac{{\alpha_{e}^{B}}}{\pi}\right)$ is at least three.
Upon renormalization according to Eq. (24) and expanding in terms of
${\alpha_{s}^{\overline{\rm{MS}}}}$ according to Eq. (26), I find that
$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{DRED}}\right]&={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}\frac{3}{4}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}{\beta_{20}^{\overline{\rm{DR}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{2}\left[-C_{F}^{2}\frac{3}{32}+C_{F}\,C_{A}\left(\frac{121}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\left(-\frac{11}{16}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)\right\\}\\\
&\hskip-45.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{DRED}}\right]&={\cal
O}({\varepsilon})\,.\end{split}$ (30)
### IV.3 Total Decay rate and annihilation cross section in the DRED scheme
As in the CDR scheme, the decay rate and annihilation cross section are
determined from the imaginary part of the forward scattering amplitude.
$\Gamma^{{DRED}}_{V\to\ {\rm hadrons}}=\frac{1}{M_{V}}\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{DRED}}\right]\,{\varepsilon}^{\nu}(Q,\lambda)^{*}\,,$
(31)
where
$\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,{\varepsilon}^{\nu}(Q,\lambda)^{*}=\frac{1}{3}\left(-\hat{g}^{\mu\nu}+\frac{Q^{\mu}\,Q^{\nu}}{M_{V}^{2}}-\delta^{\mu\nu}\right)\,.$
(32)
The evanescent part of the spin average contracts only with the $\Pi_{B}(Q)$
term, which has been renormalized to be of order $({\varepsilon})$, so that
the result is
$\begin{split}\Gamma^{{DRED}}_{V\to\ {\rm
hadrons}}=&\frac{{\alpha_{V}}\,M_{V}}{3}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(33)
just like in the CDR calculation.
For the annihilation cross section $\sigma_{e^{+}\,e^{-}\to\ {\rm hadrons}}$,
one attaches fermion bilinears to each end of the vacuum polarization tensor
and averages over the spins.
$\begin{split}\sigma^{{DRED}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}&=\frac{2}{Q^{2}}\frac{e^{2}}{4}\sum_{\lambda\,\lambda^{{}^{\prime}}}\frac{{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\hat{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}}{Q^{2}}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{DRED},\,{\alpha_{V}}\to\alpha}\right]\frac{{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\hat{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}}{Q^{2}}\\\
&+\frac{2}{Q^{2}}\frac{e_{\ell\,e}^{2}}{4}\sum_{\lambda\,\lambda^{{}^{\prime}}}\frac{{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\bar{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}}{Q^{2}}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{DRED},\,{\alpha_{V}}\to\alpha}\right]\frac{{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\bar{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}}{Q^{2}}\,,\end{split}$
(34)
where $e_{\ell\,e}$ represents the coupling of the evanescent photon to the
electron. Combining the spinor bilinears into traces,
$\begin{split}\frac{1}{2}&\sum_{\lambda\,\lambda^{{}^{\prime}}}{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\hat{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\hat{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}=\frac{1}{2}\mathop{\rm
Tr\left[{\not{p}_{e^{+}}\,\gamma^{\mu}\not{p}_{e^{-}}\,\gamma^{\nu}}\right]}\nolimits=\left(-Q^{2}\,\hat{g}^{\mu\,\nu}+Q^{\mu}\,Q^{\nu}\right)\\\
\frac{1}{2}&\sum_{\lambda\,\lambda^{{}^{\prime}}}{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\bar{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\bar{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}=\frac{1}{2}\mathop{\rm
Tr\left[{\not{p}_{e^{+}}\,\bar{\gamma}^{\mu}\not{p}_{e^{-}}\,\bar{\gamma}^{\nu}}\right]}\nolimits=\left(-Q^{2}\,\delta^{\mu\,\nu}\right)\,\end{split}$
(35)
The final result is
$\begin{split}\sigma^{{DRED}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=&\frac{4\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(36)
again in agreement with Eqs. (LABEL:eqn:knownresult-3). As promised, under the
DRED scheme renormalization program, evanescent Green functions are rendered
finite by renormalization and contribute to scattering amplitudes at order
$({\varepsilon})$. Also as promised, the results are completely equivalent to
those of the CDR scheme.
## V The Four-Dimensional Helicity Scheme
In the four-dimensional helicity scheme, one defines an enlarged vector space
of dimensionality $D_{m}=4-2\,{\varepsilon}$, in which loop momenta take
values, as in the CDR scheme. In addition, one defines a still larger vector
space, of dimensionality $D_{s}=4$, in which internal spin degrees of freedom
take values. The precise rules for the FDH scheme are given in Ref. Bern et
al. (2002). They are:
1. 1.
As in ordinary dimensional regularization, all momentum integrals are
integrated over $D_{m}$ dimensional momenta. Metric tensors resulting from
tensor integrals are $D_{m}$ dimensional.
2. 2.
All “observed” external states are taken to be four-dimensional, as are their
momenta and polarization vectors. This facilitates the use of helicity states
for observed particles.
3. 3.
All “unobserved” or internal states are treated as $D_{s}$ dimensional, and
the $D_{s}$ dimensional vector space is taken to be larger than the $D_{m}$
dimensional vector space. Unobserved states include virtual states inside of
loops, virtual states inside of trees as well as external states that have
infrared sensitive overlaps with other external states.
4. 4.
Both the $D_{s}$ and $D_{m}$ dimensional vector spaces are larger than the
standard four-dimensional space-time, so that contraction of four-dimensional
objects with $D_{m}$ or $D_{s}$ dimensional objects yields only four-
dimensional components.
To keep track of the many vector spaces and their overlapping domains, I give
the result of the contractions of the various metric tensors with one another,
$\begin{split}g^{\mu\nu}\,g_{\mu\nu}&=D_{s}\,,\qquad\
\hat{g}^{\mu\nu}\,\hat{g}_{\mu\nu}=D_{m}\,,\qquad\eta^{\mu\nu}\,\eta_{\mu\nu}=4\,,\qquad\delta^{\mu\nu}\,\delta_{\mu\nu}=D_{x}=D_{s}-D_{m}\\\
g^{\mu\nu}\hat{g}^{\rho}_{\nu}&=\hat{g}^{\mu\rho}\,,\qquad
g^{\mu\nu}\eta^{\rho}_{\nu}=\eta^{\mu\rho}\,,\qquad\hat{g}^{\mu\nu}\eta^{\rho}_{\nu}=\eta^{\mu\rho}\,,\\\
g^{\mu\nu}\delta^{\rho}_{\nu}&=\delta^{\mu\rho}\,,\qquad\hat{g}^{\mu\nu}\delta^{\rho}_{\nu}=0\,,\qquad\quad\
\eta^{\mu\nu}\delta^{\rho}_{\nu}=0\,.\\\ \end{split}$ (37)
Like the HV scheme, the FDH scheme treats observed states as four-dimensional.
In inclusive calculations, however, where there are infrared overlaps among
external states, the external states are taken to be $D_{s}$ dimensional in
the infrared regions.
As in the DRED scheme, spin degrees of freedom take values in a vector space
that is larger than that in which momenta take values. It would seem,
therefore, that the same remarks regarding the Ward Identity and the
conclusion that the $D_{x}=D_{s}-D_{m}$ dimensional components of the gauge
fields and their couplings must be considered as distinct from the $D_{m}$
dimensional gauge fields and couplings would apply. That is not, however, how
the FDH scheme is used. All field components in the $D_{s}$ dimensional space
are treated as gauge fields and no distinction is made between the couplings.
It is common, however, to define an interpolating scheme, the “$\delta_{R}$”
scheme, in which $D_{s}=4-2\,{\varepsilon}\,\delta_{R}$. The parameter
$\delta_{R}$ interpolates between the HV scheme ($\delta_{R}=1$) and the FDH
scheme ($\delta_{R}=0$). Using this scheme gives one a handle on the impact of
the evanescent degrees of freedom on the result, but not on the impact of a
distinct evanescent coupling.
It is claimed Bern et al. (2002) that the essential difference between the FDH
and DRED schemes is that in the former $D_{m}>4$, while in the latter
$D_{m}<4$. It must be this difference, then, that allows for the very
different handling of the evanescent couplings and degrees of freedom. We
shall see what impact this choice has in the calculation and discussion below.
### V.1 Renormalization
I will not give detailed results for the renormalization parameters of the FDH
scheme. There is no point in doing so because, as I will show, the rules of
the FDH scheme enumerated in the previous section are not consistent with a
successful renormalization program. The first sign that there is a problem
with the renormalization program comes in the computation of the one-loop
renormalization constants. In particular, the gluon vacuum polarization tensor
splits into two independent components,
$\Pi_{A}^{\mu\nu}=\Pi_{A}(Q^{2})\,\left((-Q^{2}\hat{g}^{\mu\nu}+Q^{\mu}\,Q^{\nu}\right)$
and $\Pi_{B}^{\mu\nu}=\Pi_{B}(Q^{2})\,\delta^{\mu\nu}$, both of which are
singular. This is a clear warning that what the FDH scheme calls the gluon is
in fact two distinct sets of degrees of freedom. If I ignore $\Pi_{B}$ and
just renormalize $\Pi_{A}$, I find the usual result that
${\beta_{0}^{\overline{\rm{FDH}}}}=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}\,.$
(38)
Note that I also get this result if I take the spin average (trace) of the
full vacuum polarization tensor. Because $\Pi_{B}$ is weighted by a factor of
$2\,{\varepsilon}$, its contribution to the spin average is not singular.
Because the leading order term in the quantities being calculated is of order
one, and the NLO term of order $\alpha_{s}$, this result for the one-loop
$\beta$-function is all that is needed to compute the renormalized cross
section at NNLO. Furthermore, the many NLO results that have been obtained
using the FDH scheme have all renormalized using the above result for
${\beta_{0}^{\overline{\rm{FDH}}}}$.
When I try to proceed to the two-loop beta function, I find that both
$\Pi_{A}$ and $\Pi_{B}$ contribute singular terms to the spin-averaged vacuum
polarization, while if I again ignore $\Pi_{B}$ and renormalize $\Pi_{A}$, I
obtain the usual value for $\beta_{1}$,
${\beta_{1}^{\overline{\rm{FDH}}}}=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}\,.$
(39)
This seems to be the choice made in Ref. Bern et al. (2002) as they quote only
the result for terms proportional to $Q^{\mu}Q^{\nu}$, which would be part of
my $\Pi_{A}$. Since the standard lore has been that
${\alpha_{s}^{\overline{\rm{FDH}}}}$ and ${\alpha_{s}^{\overline{\rm{DR}}}}$
coincide, at least through second order corrections, this seems to be the most
reasonable choice. Furthermore, it means that the conversion to
${\alpha_{s}^{\overline{\rm{MS}}}}$ will be Kunszt et al. (1994); Bern et al.
(2002)
${\alpha_{s}^{\overline{\rm{FDH}}}}={\alpha_{s}^{\overline{\rm{MS}}}}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\frac{C_{A}}{12}+\ldots\right]$
(40)
As it turns out, it does not matter what choice one makes as even the one-loop
result for ${\beta_{0}^{\overline{\rm{FDH}}}}$, which seems safe if only
because it is familiar, leads to the violation of unitarity.
### V.2 Vacuum polarization in the FDH scheme
Leaving aside the question of renormalization beyond one-loop, I will proceed
with the calculation of the $V$-boson vacuum polarization. In performing
calculations in the FDH scheme, it becomes apparent that the results are
identical, term-by-term. to the calculation in the DRED scheme, except that
the evanescent gluons are identified as gluons and the coupling $\alpha_{e}$
is set to $\alpha_{s}$. Therefore I find that
$\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{FDH}}\right]=\frac{-Q^{2}\,\hat{g}_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}\,\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{FDH}}\right]-Q^{2}\,\frac{\delta_{\mu\nu}}{2\,{\varepsilon}}\,\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{FDH}}\right]\,,$
(41)
where
$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{FDH}}\right]&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt1+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{4}+{\varepsilon}\left(\frac{45}{8}-6\,\zeta_{3}\right)+{\varepsilon}^{2}\,\left(\frac{439}{16}-\frac{15}{4}\zeta_{2}-15\,\zeta_{3}-9\,\zeta_{4}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)-\frac{15}{32}C_{F}^{2}+\left(\frac{37}{4}-\frac{33}{4}\zeta_{3}\right)\,C_{F}\,C_{A}-\left(\frac{25}{16}-\frac{3}{2}\zeta_{3}\right)\,C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{235}{32}-\frac{111}{8}\,\zeta_{3}+\frac{45}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\left(\frac{14521}{192}-\frac{231}{32}\,\zeta_{2}-\frac{193}{4}\,\zeta_{3}-\frac{99}{8}\,\zeta_{4}-\frac{15}{4}\,\zeta_{5}\right)\right.\\\
&\hskip-20.0pt\left.\left.\left.+C_{F}\,N_{f}\left(-\frac{1187}{96}+\frac{21}{16}\,\zeta_{2}+\frac{17}{2}\,\zeta_{3}+\frac{9}{4}\,\zeta_{4}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(42)
and
$\begin{split}\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{FDH}}\right]&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt{\varepsilon}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{1}{2}+{\varepsilon}\frac{13}{4}+{\varepsilon}^{2}\,\left(\frac{119}{8}-\frac{5}{2}\zeta_{2}-6\,\zeta_{3}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(-\frac{1}{8}C_{F}^{2}+\frac{7}{16}C_{F}\,C_{A}\right)-\frac{29}{32}C_{F}^{2}+\frac{139}{32}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{245}{64}+\frac{21}{16}\,\zeta_{2}-3\,\zeta_{3}\right)+C_{F}\,C_{A}\left(\frac{1837}{64}-\frac{147}{32}\,\zeta_{2}-9\,\zeta_{3}\right)\right.\\\
&\hskip-20.0pt\left.\left.\left.+C_{F}\,N_{f}\left(-\frac{25}{16}+\frac{3}{2}\,\zeta_{3}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\right)\right\\}\,.\\\
\end{split}$ (43)
Upon renormalizing such that
${\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\to{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{-{\varepsilon}}\left(1-\frac{{\beta_{0}^{\overline{\rm{FDH}}}}}{{\varepsilon}}{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\right)\,,\qquad\qquad{\alpha_{V}^{B}}\to{\alpha_{V}}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{-{\varepsilon}}\,,$
(44)
I find that
$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{FDH}}\right]&={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\frac{3}{4}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}{\beta_{0}^{\overline{\rm{FDH}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{2}\left[-C_{F}^{2}\frac{15}{32}+C_{F}\,C_{A}\left(\frac{131}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\left(-\frac{5}{8}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\right)\right\\}\\\
&\hskip-45.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{15}{32}+C_{F}\,C_{A}\,\left(\frac{133}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{5}{8}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{FDH}}\right]&={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\frac{1}{2}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}{\beta_{0}^{\overline{\rm{FDH}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{2}\left[\frac{1}{{\varepsilon}}\left(-C_{F}^{2}\frac{1}{8}-C_{F}\,C_{A}\frac{1}{48}+C_{F}\,N_{f}\frac{1}{12}\right)\left(1+3{\varepsilon}\ln\frac{\mu^{2}}{Q^{2}}\right)\right.\\\
&\left.\left.-C_{F}^{2}\frac{29}{32}+C_{F}\,C_{A}\frac{131}{96}+C_{F}\,N_{f}\frac{5}{12}\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\right)\right\\}\\\
&\hskip-45.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\frac{1}{2}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}{\beta_{0}^{\overline{\rm{FDH}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\left[\frac{1}{{\varepsilon}}\left(-C_{F}^{2}\frac{1}{8}-C_{F}\,C_{A}\frac{1}{48}+C_{F}\,N_{f}\frac{1}{12}\right)\left(1+3{\varepsilon}\ln\frac{\mu^{2}}{Q^{2}}\right)\right.\\\
&\left.\left.-C_{F}^{2}\frac{29}{32}+C_{F}\,C_{A}\frac{45}{32}+C_{F}\,N_{f}\frac{5}{12}\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(45)
### V.3 Total Decay rate and annihilation cross section in the FDH scheme
The results of the vacuum polarization calculation look to be disastrous as
$\Pi_{B}$ is singular at order $\alpha_{s}^{2}$. However, the rules of the FDH
scheme, enumerated above, specify that external states are taken to be four-
dimensional. This means that the spin average of the vector polarizations is
$\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,{\varepsilon}^{\nu}(Q,\lambda)^{*}=\frac{1}{3}\left(-\eta^{\mu\nu}+\frac{Q^{\mu}\,Q^{\nu}}{M_{V}^{2}}\right)\,,$
(46)
which annihilates $\left.\Pi_{B}^{\mu\nu}\right|_{FDH}$. For the annihilation
rate, the rules are a bit ambiguous, as they could be read to mean that the
lepton spinors are four-dimensional but the vertex ($\gamma^{\mu}$) connecting
them to the loop part of the amplitude is $D_{s}$ dimensional. This would
bring $\left.\Pi_{B}^{\mu\nu}\right|_{FDH}$ into the calculation and lead to a
singular result at order $\alpha_{s}^{2}$. However, Rule $4$ could also be
read to mean that the vertex sandwiched between four-dimensional states is
also reduced to being four-dimensional.
Assuming this interpretation, I find that
$\begin{split}\Gamma^{{FDH}}_{V\to\ {\rm
hadrons}}=&\frac{{\alpha_{V}}\,M_{V}}{3}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{15}{32}+C_{F}\,C_{A}\,\left(\frac{133}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{5}{8}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(47)
and
$\begin{split}\sigma^{{FDH}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=&\frac{4\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{15}{32}+C_{F}\,C_{A}\,\left(\frac{133}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{5}{8}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(48)
The results agree with one another, are correct through NLO and are finite
through NNLO. Unfortunately, the NNLO terms are not correct! Because the
discrepancy is finite, there remains the possibility that the conversion from
${\alpha_{s}^{\overline{\rm{FDH}}}}$ to ${\alpha_{s}^{\overline{\rm{MS}}}}$
given in Eq. (40) is incorrect, although this would contradict previous
results Kunszt et al. (1994); Bern et al. (2002). If this were the case, then
one would expect that the N3LO result would also be finite but incorrect. If,
instead, the finite discrepancy at NNLO is the result of a failure of the
renormalization program, the N3LO result should be singular.
## VI Partial results at N3LO
Although first computed some time ago, the vacuum polarization at four loops
Gorishnii et al. (1988, 1991) remains a formidable calculation. It is only
necessary, however, to look at a small part of the calculation: the terms
proportional to the square of the number of fermion flavors, $N_{f}^{2}$. This
is fortunate for a couple of reasons: 1) there are only three four-loop
diagrams to be computed, see Fig. (3), (plus three more in the DRED scheme,
where the gluons are replaced by evanescent gluons); and 2) the contributions
from renormalization in the CDR and FDH schemes come only from the leading
term in the QCD $\beta$-function ($\beta_{0}$ and $\beta_{0}^{2}$). Thus, my
result will not depend on how the higher order terms of the $\beta$-function
are chosen in the FDH scheme.
Figure 3: Four-loop diagrams that contribute to the $N_{f}^{2}$ term at N3LO.
### VI.1 The CDR scheme
In the CDR scheme, there are only three four-loop diagrams that need to be
calculated. The first two are simply iterated-bubble diagrams and are
essentially trivial. The third is slightly nontrivial, so I again use my
QGRAF-FORM-REDUZE suite of programs to address the problem. All of the four-
loop master integrals can be found in Ref. Baikov and Chetyrkin (2010). I find
the result of the four-loop calculation to be
$\begin{split}\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{CDR}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}\\\
&\qquad\times{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,C_{F}\,N_{f}^{2}\left[\frac{1}{48\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\left(\frac{121}{288}-\frac{1}{3}\zeta_{3}\right)+\frac{2777}{576}-\frac{3}{8}\zeta_{2}-\frac{19}{6}\zeta_{3}-\frac{1}{2}\zeta_{4}\right]\end{split}$
(49)
Renormalizing, I find
$\begin{split}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\,C_{F}\,N_{f}^{2}\\\
&\times\left[\frac{151}{216}-\frac{1}{24}\zeta_{2}-\frac{19}{36}\zeta_{3}+\left(\frac{11}{48}-\frac{1}{6}\zeta_{3}\right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{1}{48}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\end{split}$
(50)
Using this term to compute the $\alpha_{s}^{3}\,N_{f}^{2}$ contribution to the
decay rate and annihilation cross section as in Eqs. (14,17), I find the
result expected from Eqs. (LABEL:eqn:knownresult-3).
### VI.2 The DRED scheme
In the DRED scheme, there are three extra four-loop diagrams to compute,
obtained by replacing gluon propagators with evanescent gluon propagators. I
find
$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\,\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\phantom{+}{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[\frac{1}{48\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\left(\frac{13}{32}-\frac{1}{3}\zeta_{3}\right)+\frac{7847}{1728}-\frac{3}{8}\zeta_{2}-\frac{53}{18}\zeta_{3}-\frac{1}{2}\zeta_{4}\right]\\\
&\left.+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{3}\,\left[-\frac{1}{{\varepsilon}}\frac{3}{64}-\frac{83}{128}\right]\right\\}\\\
\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{Ve}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\,\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\phantom{+}{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[\frac{1}{72\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\frac{73}{432}+\frac{3595}{2592}-\frac{1}{4}\zeta_{2}-\frac{1}{3}\zeta_{3}\right]\\\
&\left.+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{3}\,\left[-\frac{1}{48\,{\varepsilon}^{2}}-\frac{1}{{\varepsilon}}\frac{11}{48}-\frac{155}{96}+\frac{3}{8}\zeta_{2}\right]\right\\}\end{split}$
(51)
Upon renormalizing according to Eq. (24) and converting the coupling to
${\alpha_{s}^{\overline{\rm{MS}}}}$, I obtain
$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&\\\
&\hskip-55.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,C_{F}\,N_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\,\left[\frac{151}{216}-\frac{1}{24}\zeta_{2}-\frac{19}{36}\zeta_{3}+\left(\frac{11}{48}-\frac{1}{6}\zeta_{3}\right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{1}{48}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\cal
O}({\varepsilon})\,.\end{split}$ (52)
As for the CDR scheme, this leads to the expected result for the decay rate
and annihilation cross section.
### VI.3 The FDH scheme
In the FDH scheme, however, I find that
$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\\\
&\times{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[\frac{1}{48\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\left(\frac{23}{64}-\frac{1}{3}\zeta_{3}\right)+\frac{13453}{3456}-\frac{3}{8}\zeta_{2}-\frac{53}{18}\zeta_{3}-\frac{1}{2}\zeta_{4}\right]\,,\\\
\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\\\
&\times{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[-\frac{1}{144\,{\varepsilon}^{2}}-\frac{1}{{\varepsilon}}\frac{13}{216}-\frac{295}{1296}+\frac{1}{8}\zeta_{2}-\frac{1}{3}\zeta_{3}\right]\,.\end{split}$
(53)
I renormalize according to
${\alpha_{s}^{B}}=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,{\alpha_{s}^{\overline{\rm{FDH}}}}\,\left[1-{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\frac{{\beta_{0}^{\overline{\rm{FDH}}}}}{{\varepsilon}}+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{2}\left(\frac{{\beta_{0}^{\overline{\rm{FDH}}}}^{2}}{{\varepsilon}^{2}}-\frac{1}{2}\frac{{\beta_{1}^{\overline{\rm{FDH}}}}}{{\varepsilon}}\right)\right]\,,$
(54)
keeping only terms proportional to
${\alpha_{s}^{\overline{\rm{FDH}}}}^{3}\,N_{f}^{2}$. Such terms can only come
from the ${\beta_{0}^{\overline{\rm{FDH}}}}$ and
${\beta_{0}^{\overline{\rm{FDH}}}}^{2}$ terms, so any uncertainty about
${\beta_{1}^{\overline{\rm{FDH}}}}$ has no effect here. The renormalized
result is
$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&\\\
&\hskip-75.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,C_{F}\,N_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\,\left[-\frac{1}{192\,{\varepsilon}}+\frac{1843}{3456}-\frac{1}{24}\zeta_{2}-\frac{19}{36}\zeta_{3}+\left(\frac{3}{16}-\frac{1}{6}\zeta_{3}\right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{1}{48}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&\\\
&\hskip-75.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,C_{F}\,N_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\,\left[\frac{1}{144\,{\varepsilon}^{2}}-\frac{5}{432\,{\varepsilon}}-\frac{869}{2592}+\frac{1}{18}\zeta_{2}-\frac{5}{27}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{1}{36}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\,.\end{split}$
(55)
The demand that external states be four-dimensional removes the $\Pi_{B}$
term, but there is also a pole in $\Pi_{A}$ and no finite renormalization to
put the result in terms of ${\alpha_{s}^{\overline{\rm{MS}}}}$ can remove it.
I must therefore conclude that the FDH scheme is not consistent with
unitarity.
## VII Discussion
In this paper, I have performed a high-order calculation in each of three
regularization schemes: the conventional dimensional regularization (CDR)
scheme; the dimensional reduction (DRED) scheme; and the four-dimensional
helicity (FDH) scheme. Of these, the CDR scheme is by far the most widely
used, and was, in fact, used to compute the original results that I use as my
test basis. The FDH scheme has primarily been used to produce one-loop
helicity amplitudes, although it has been used in a few cases in two-loop
calculations and also as a supersymmetric regulator. The primary purpose of
this paper was to put the FDH scheme to a stringent test and determine its
reliability in a high-order calculation. The DRED scheme is primarily used as
a supersymmetric regulator and is quite cumbersome for nonsupersymmetric
calculations. It is, however, closely related to the FDH scheme and has been
demonstrated Jack et al. (1994a, b); Harlander et al. (2006b, a) to be
equivalent to the CDR scheme through four loops. A close comparison of the
details of the calculations in the FDH and DRED schemes helps to identify
where and when things go wrong with the former.
In the cases of the CDR and DRED schemes, I have reproduced the known result
for the hadronic decay width of a massive vector boson (or equivalently, the
$e^{+}e^{-}$ annihilation rate to hadrons) through NNLO, and a few terms at
N3LO. This represents computing the QCD corrections to the vacuum polarization
of the photon ($V$ boson) through three loops, with partial results at four
loops. In addition, I have reproduced the renormalization parameters of QCD
($\beta$-function(s), mass anomalous dimension) through three-loop order. This
establishes that I have theoretical control over all of the needed
calculations through three-loop order. In order to obtain the partial N3LO
result in the DRED scheme, I also needed the three-loop QCD corrections to the
$\beta$-function of the evanescent photon ($V$ boson).
The calculation of the $V$ boson decay rate provides another instance of the
equivalence the CDR and DRED schemes at the four-loop level Harlander et al.
(2006a). The ability to obtain the correct result using the DRED scheme
required a delicate balance of the many extra couplings and their
renormalization effects upon one another. Indeed, given the complexity needed
to make the DRED scheme work, it seems that there should be little surprise
that the FDH scheme, with its greater simplicity, should fail.
Perhaps, it is worth considering how it is that the FDH scheme has been used
successfully in so many calculations. Its most common use has been in the
construction of one-loop scattering amplitudes via unitarity cuts, using four-
dimensional helicity amplitudes as the primary building blocks. Thus, it is
natural that it restricts observed (external) states to be four-dimensional.
Because the FDH scheme defines that $D_{s}>D_{m}>4$, this restriction excludes
evanescent fields from appearing as external states. This is very important
because, as one can see from comparing Eqs. (30) and (55), terms involving
external evanescent states are the most dangerous. Even though it does not
renormalize evanescent states and couplings properly the FDH is able to get
the nonevanescent part of the vacuum polarization tensor correct at NLO, while
the evanescent part is ready to contribute a finite error at NLO. Because the
DRED scheme defines $4>D_{m}$, the evanescent states are parts of the
classical four-dimensional states. It would not seem natural to exclude them
from appearing as external states. Instead, they are handled through the
renormalization program so that their effects are removed from physical
scattering amplitudes. In the FDH scheme, the evanescent states are instead
additions to the four-dimensional states (as are the extra degrees of freedom
that come from regularizing momentum integrals) and there is no barrier to
excluding them as observed states.
In an FDH scheme calculation, a tree-level term is strictly four-dimensional
and is free from evanescent contributions. (Depending on interpretation, this
may be a stronger condition than is given in the rules of Ref Bern et al.
(2002), but it is the actual condition imposed if one defines the tree-level
amplitude as being a four-dimensional helicity amplitude.) Because evanescent
terms are absent at tree-level, they cannot generate ultraviolet poles at one
loop. Even if one were to renormalize them properly, as in the DRED scheme,
there would be nowhere to make the counter-term insertion! In fact, the one-
loop contributions are not even finite, as the counting over the number of
states ($2{\varepsilon}$) makes the result of order ${\varepsilon}$. This is
clearly illustrated in Eq. (28). Neither $\alpha_{s}$, nor $\alpha_{e}$ appear
at LO. Therefore, the contributions at NLO are finite for $\alpha_{s}$ and of
order ${\varepsilon}$ (because of the counting over the number of states) for
$\alpha_{e}$. In more complicated QCD calculations, $\alpha_{s}$ will appear
at LO and will therefore contribute an ultraviolet pole at one-loop, which
will be removed by renormalization. $\alpha_{e}$, however, will still make its
first appearance at NLO and that contribution will be of order
${\varepsilon}$. Thus, one can expect that the FDH scheme, used as above,
should be reliable for computing NLO corrections through finite order
(${\varepsilon}^{0}$). The error from improperly identifying evanescent
quantities should be of order ${\varepsilon}$. At NNLO and beyond however, the
failure to properly identify and renormalize the evanescent parameters leads
to incorrect results and the violation of unitarity.
So, as suggested Bern et al. (2002), one of the FDH scheme’s most important
assets is that it defines $D_{s}>D_{m}>4$. This feature is also the scheme’s
undoing, though not of necessity. Because the effects of external evanescent
states can be removed (or indeed never seen) by imposing a four-dimensionality
restriction, and because the effects of internal evanescent states therefore
contribute at order ${\varepsilon}$ at one loop, it appears that one can
simply ignore the distinction between gauge and evanescent terms. In contrast,
because the DRED scheme must deal with external evanescent terms from the
beginning, its advocates were forced to develop a successful renormalization
program Jack et al. (1994a, b). Extensive testing Jack et al. (1994a, b);
Harlander et al. (2006b, a) has shown that this program works to at least the
fourth order and that it handles the effects of both internal and external
evanescent contributions. As I remarked earlier, calculations in the DRED and
FDH schemes are term-by-term identical, except for the identification of the
couplings and propagating states. Thus, one could make the FDH scheme a
unitary regularization scheme for nonsupersymmetric calculations by
recognizing the distinction between gauge and evanescent terms and adopting
the DRED scheme’s renormalization program. This would, of course, do away with
any notion of the FDH scheme being simple, but it would at least be correct.
The FDH scheme would still be distinguished from the DRED scheme by the fact
that $D_{s}>D_{m}>4$, which facilitates helicity amplitude calculations and,
in chiral theories, improves its situation with regard to $\gamma_{5}$ and the
Levi-Civita tensor Siegel (1980); Stockinger (2005). Furthermore, with a valid
renormalization program, the requirement of four-dimensional observed states
could be made optional. This would lead to two linked, slightly different,
schemes, just like the HV and CDR schemes. This suggestion has already been
made by Signer and Stöckinger Signer and Stockinger (2009) who in fact define
their version of the DRED scheme to have precisely the $D_{s}>D_{m}>4$
hierarchy of the FDH scheme.
Thus, in conclusion, the CDR and DRED schemes are correct and equivalent ways
of performing QCD calculations through N3LO. The FDH scheme, however, has been
shown to be incorrect and to violate unitarity beyond NLO when applied to
nonsupersymmetric theories. It must therefore be viewed as a shortcut for
performing NLO calculations and should only be used for such calculations with
great caution.
#### Acknowledgments:
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.
## Appendix A Renormalization parameters for the CDR scheme
To three-loop order, I find the coefficients of the $\beta$-function to be
$\begin{split}{\beta_{0}^{\overline{\rm{MS}}}}&=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}\,,\qquad\qquad{\beta_{1}^{\overline{\rm{MS}}}}=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}\,,\\\
{\beta_{2}^{\overline{\rm{MS}}}}&=\frac{2857}{3456}C_{A}^{3}-\frac{1415}{3456}C_{A}^{2}\,N_{f}-\frac{205}{1152}C_{A}\,C_{F}\,N_{f}+\frac{1}{64}C_{F}^{2}\,N_{f}+\frac{79}{3456}C_{A}\,N_{f}^{2}+\frac{11}{576}C_{F}\,N_{f}^{2}\,,\end{split}$
(56)
while the coefficients of the mass anomalous dimension are
$\begin{split}{\gamma_{0}^{\overline{\rm{MS}}}}&=\frac{3}{4}C_{F}\,,\hskip
80.0pt{\gamma_{1}^{\overline{\rm{MS}}}}=\frac{3}{32}C_{F}^{2}+\frac{97}{96}C_{F}\,C_{A}-\frac{5}{48}C_{F}\,N_{f}\,,\\\
{\gamma_{2}^{\overline{\rm{MS}}}}&=\frac{129}{128}C_{F}^{3}-\frac{129}{256}C_{F}^{2}\,C_{A}+\frac{11413}{6912}C_{F}\,C_{A}^{2}-\left(\frac{23}{64}-\frac{3}{8}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\left(\frac{139}{864}+\frac{3}{8}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\frac{35}{1728}C_{F}\,N_{f}^{2}\,,\end{split}$
(57)
in agreement with known results Tarasov et al. (1980); Larin and Vermaseren
(1993); Chetyrkin (1997); Vermaseren et al. (1997).
## Appendix B Renormalization parameters for the DRED scheme
The coefficients of the QCD $\beta$-function,
${\beta^{\overline{\rm{DR}}}}({\alpha_{s}^{\overline{\rm{DR}}}})$ through
three loops are:
$\begin{split}{\beta_{20}^{\overline{\rm{DR}}}}&=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}\,,\hskip
60.0pt{\beta_{30}^{\overline{\rm{DR}}}}=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}\,,\\\
{\beta_{40}^{\overline{\rm{DR}}}}&=\frac{3115}{3456}C_{A}^{3}-\frac{1439}{3456}C_{A}^{2}\,N_{f}-\frac{193}{1152}C_{A}\,C_{F}\,N_{f}+\frac{1}{64}C_{F}^{2}\,N_{f}+\frac{79}{3456}C_{A}\,N_{f}^{2}+\frac{11}{576}C_{F}\,N_{f}^{2}\,,\\\
{\beta_{31}^{\overline{\rm{DR}}}}&=-\frac{1}{16}C_{F}\,N_{f}\left(\frac{3}{2}C_{F}\right)\,,\qquad\quad{\beta_{22}^{\overline{\rm{DR}}}}=-\frac{1}{16}C_{F}\,N_{f}\left(\frac{1}{2}C_{A}-C_{F}-\frac{1}{4}N_{f}\right)\,,\end{split}$
(58)
where the notation is that
${\beta^{\overline{\rm{DR}}}}({\alpha_{s}^{\overline{\rm{DR}}}})=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}-\sum_{i,j,k,l,m}\,{\beta_{ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\,.$ (59)
The last three indices of ${\beta_{ijklm}^{\overline{\rm{DR}}}}$ are omitted
when they are all equal to $0$.
The $\beta$-function of evanescent QCD coupling,
${\beta_{e,\,\,}^{\overline{\rm{DR}}}}({\alpha_{e}^{\overline{\rm{DR}}}})$ is
$\begin{split}{\beta_{e,\,0\,2}^{\overline{\rm{DR}}}}&=\frac{1}{2}C_{A}-C_{F}-\frac{1}{4}N_{f}\,,\qquad{\beta_{e,\,1\,1}^{\overline{\rm{DR}}}}=\frac{3}{2}C_{F}\,,\\\\[5.0pt]
{\beta_{e,\,0\,3}^{\overline{\rm{DR}}}}&=\frac{3}{8}C_{A}^{2}-\frac{5}{4}C_{A}\,C_{F}+C_{F}^{2}-\frac{3}{16}C_{A}\,N_{f}+\frac{3}{8}C_{F}\,N_{f}\,,\qquad{\beta_{e,\,1\,2}^{\overline{\rm{DR}}}}=-\frac{3}{8}C_{A}^{2}+\frac{5}{2}C_{A}\,C_{F}-\frac{11}{4}C_{F}^{2}-\frac{5}{16}C_{F}\,N_{f}\,,\\\
{\beta_{e,\,2\,1}^{\overline{\rm{DR}}}}&=-\frac{7}{64}C_{A}^{2}+\frac{55}{48}C_{A}\,C_{F}+\frac{3}{16}C_{F}^{2}+\frac{1}{16}C_{A}\,N_{f}-\frac{5}{24}C_{F}\,N_{f}\\\\[5.0pt]
{\beta_{e,\,0\,2100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{8}\qquad{\beta_{e,\,0\,2010}^{\overline{\rm{DR}}}}=\frac{5}{4}\qquad{\beta_{e,\,0\,2001}^{\overline{\rm{DR}}}}=\frac{3}{4}\\\
{\beta_{e,\,0\,1200}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{27}{64}\qquad{\beta_{e,\,0\,1020}^{\overline{\rm{DR}}}}=-\frac{15}{4}\qquad{\beta_{e,\,0\,1002}^{\overline{\rm{DR}}}}=\frac{21}{32}\qquad{\beta_{e,\,0\,1101}^{\overline{\rm{DR}}}}=-\frac{9}{16}\\\
{\beta_{e,\,0\,4}^{\overline{\rm{DR}}}}&=-\left(\frac{7}{4}+\frac{9}{4}\zeta_{3}\right)\,C_{F}^{3}+\left(\frac{17}{8}+\frac{15}{2}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}-\left(\frac{3}{4}+\frac{69}{16}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}+\left(\frac{1}{16}+\frac{9}{16}\zeta_{3}\right)\,C_{A}^{3}\\\
&+\left(\frac{13}{32}-\frac{33}{16}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\left(\frac{1}{32}+\frac{51}{32}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\left(\frac{21}{128}+\frac{9}{32}\zeta_{3}\right)\,C_{A}^{2}\,N_{f}-\left(\frac{1}{128}C_{F}-\frac{7}{256}C_{A}\right)\,N_{f}^{2}\\\
{\beta_{e,\,1\,3}^{\overline{\rm{DR}}}}&=\left(\frac{13}{2}-3\,\zeta_{3}\right)\,C_{F}^{3}-\left(10-6\,\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\left(\frac{133}{32}-\frac{15}{4}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}-\left(\frac{25}{64}-\frac{3}{4}\zeta_{3}\right)\,C_{A}^{3}\\\
&+\left(\frac{13}{16}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\frac{9}{8}\left(1-\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}+\left(\frac{7}{32}-\frac{3}{8}\zeta_{3}\right)\,C_{A}^{2}\,N_{f}+\frac{3}{64}\,C_{A}\,N_{f}^{2}\\\
{\beta_{e,\,2\,2}^{\overline{\rm{DR}}}}&=-\left(\frac{139}{64}-\frac{27}{4}\zeta_{3}\right)\,C_{F}^{3}-\left(\frac{793}{128}+18\,\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\left(\frac{1587}{256}+\frac{207}{16}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}-\left(\frac{427}{512}+\frac{45}{16}\zeta_{3}\right)\,C_{A}^{3}\\\
&-\left(\frac{569}{256}-\frac{99}{16}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\left(\frac{31}{16}-\frac{171}{32}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\left(\frac{871}{1024}-\frac{45}{32}\zeta_{3}\right)\,C_{A}^{2}\,N_{f}+\left(\frac{1}{16}C_{F}-\frac{1}{256}C_{A}\right)\,N_{f}^{2}\\\
{\beta_{e,\,3\,1}^{\overline{\rm{DR}}}}&=\frac{129}{64}C_{F}^{3}-\frac{457}{128}C_{F}^{2}\,C_{A}+\frac{11875}{3456}C_{F}\,C_{A}^{2}-\frac{3073}{4608}C_{A}^{3}\\\
&-\left(\frac{23}{32}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\left(\frac{157}{1728}+\frac{3}{4}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}+\frac{463}{2304}C_{A}^{2}\,N_{f}-\left(\frac{35}{864}C_{F}+\frac{5}{576}C_{A}\right)\,N_{f}^{2}\\\
{\beta_{e,\,0\,3100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{64}+\frac{243}{128}N_{f}\qquad{\beta_{e,\,0\,3010}^{\overline{\rm{DR}}}}=\frac{5}{8}-\frac{45}{64}N_{f}\qquad{\beta_{e,\,0\,3001}^{\overline{\rm{DR}}}}=\frac{3}{32}-\frac{81}{64}N_{f}\\\
{\beta_{e,\,1\,2100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{219}{16}\qquad{\beta_{e,\,1\,2010}^{\overline{\rm{DR}}}}=\frac{145}{48}\qquad{\beta_{e,\,1\,2001}^{\overline{\rm{DR}}}}=\frac{73}{8}\\\
{\beta_{e,\,2\,1100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{1125}{1024}\qquad{\beta_{e,\,2\,1010}^{\overline{\rm{DR}}}}=\frac{105}{128}\qquad{\beta_{e,\,2\,1001}^{\overline{\rm{DR}}}}=\frac{615}{512}\\\
{\beta_{e,\,0\,2200}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{1413}{512}-\frac{729}{1024}N_{f}\qquad{\beta_{e,\,0\,2020}^{\overline{\rm{DR}}}}=-\frac{115}{32}+\frac{135}{64}N_{f}\qquad{\beta_{e,\,0\,2002}^{\overline{\rm{DR}}}}=-\frac{161}{256}-\frac{567}{512}N_{f}\\\
{\beta_{e,\,0\,2110}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{75}{8}\qquad{\beta_{e,\,0\,2101}^{\overline{\rm{DR}}}}=-\frac{471}{128}+\frac{243}{256}N_{f}\qquad{\beta_{e,\,0\,2011}^{\overline{\rm{DR}}}}=-\frac{85}{8}\\\
{\beta_{e,\,0\,1300}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{1701}{1024}\qquad{\beta_{e,\,0\,1210}^{\overline{\rm{DR}}}}=-\frac{405}{128}\qquad{\beta_{e,\,0\,1201}^{\overline{\rm{DR}}}}=\frac{1701}{512}\\\
{\beta_{e,\,0\,1120}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{135}{32}\qquad{\beta_{e,\,0\,1111}^{\overline{\rm{DR}}}}=\frac{135}{16}\qquad{\beta_{e,\,0\,1102}^{\overline{\rm{DR}}}}=-\frac{81}{128}\\\
{\beta_{e,\,0\,1021}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{315}{32}\qquad{\beta_{e,\,0\,1012}^{\overline{\rm{DR}}}}=-\frac{315}{32}\qquad{\beta_{e,\,0\,1003}^{\overline{\rm{DR}}}}=\frac{63}{128}\qquad\end{split}$
(60)
The mass anomalous dimension in the DRED scheme is
$\begin{split}{\gamma_{10}^{\overline{\rm{DR}}}}&=\frac{3}{4}C_{F}\\\\[5.0pt]
{\gamma_{20}^{\overline{\rm{DR}}}}&=\frac{3}{32}C_{F}^{2}+\frac{91}{96}C_{A}\,C_{F}-\frac{5}{48}C_{F}\,N_{f}\qquad{\gamma_{11}^{\overline{\rm{DR}}}}=-\frac{3}{8}C_{F}^{2}\qquad{\gamma_{02}^{\overline{\rm{DR}}}}=\frac{1}{4}C_{F}^{2}-\frac{1}{8}C_{A}\,C_{F}+\frac{1}{16}C_{F}\,N_{f}\\\\[5.0pt]
{\gamma_{30}^{\overline{\rm{DR}}}}&=\frac{129}{128}C_{F}^{3}-\frac{133}{256}C_{F}^{2}\,C_{A}+\frac{10255}{6912}C_{F}\,C_{A}^{2}-\left(\frac{23}{64}-\frac{3}{8}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\left(\frac{281}{1728}+\frac{3}{8}\zeta_{3}\right)\,C_{A}\,C_{F}\,N_{f}-\frac{35}{1728}C_{F}\,N_{f}^{2}\\\
{\gamma_{21}^{\overline{\rm{DR}}}}&=-\frac{27}{64}C_{F}^{3}-\frac{21}{32}C_{F}^{2}\,C_{A}-\frac{15}{256}C_{F}\,C_{A}^{2}+\frac{9}{64}C_{F}^{2}\,N_{f}\\\
{\gamma_{12}^{\overline{\rm{DR}}}}&=\frac{9}{8}C_{F}^{3}-\frac{21}{32}C_{F}^{2}\,C_{A}+\frac{3}{64}C_{F}\,C_{A}^{2}+\frac{3}{128}C_{F}\,C_{A}\,N_{f}+\frac{3}{16}C_{F}^{2}\,N_{f}\\\
{\gamma_{03}^{\overline{\rm{DR}}}}&=-\frac{3}{8}C_{F}^{3}+\frac{3}{8}C_{F}^{2}\,C_{A}-\frac{3}{32}C_{F}\,C_{A}^{2}+\frac{1}{16}C_{F}\,C_{A}\,N_{f}-\frac{5}{32}C_{F}^{2}\,N_{f}-\frac{1}{128}C_{F}\,N_{f}^{2}\\\\[5.0pt]
{\gamma_{02100}^{\overline{\rm{DR}}}}\hskip-10.0pt&\hskip
10.0pt=\frac{3}{8}\qquad{\gamma_{02010}^{\overline{\rm{DR}}}}=-\frac{5}{12}\qquad{\gamma_{02001}^{\overline{\rm{DR}}}}=-\frac{1}{4}\qquad\\\
{\gamma_{01200}^{\overline{\rm{DR}}}}\hskip-10.0pt&\hskip
10.0pt=-\frac{9}{64}\qquad{\gamma_{01101}^{\overline{\rm{DR}}}}=\frac{3}{16}\qquad{\gamma_{01020}^{\overline{\rm{DR}}}}=\frac{5}{4}\qquad{\gamma_{01002}^{\overline{\rm{DR}}}}=-\frac{7}{32}\end{split}$
(61)
The above results for ${\beta^{\overline{\rm{DR}}}}$,
${\beta_{e,\,\,}^{\overline{\rm{DR}}}}$ and ${\gamma^{\overline{\rm{DR}}}}$
all agree with the results of Refs. Harlander et al. (2006b, a)
The QCD contributions to the $\beta$-function of the evanescent part of a non-
QCD gauge coupling is a new result. I find
$\begin{split}{\beta_{Ve,\,1\,0}^{\overline{\rm{DR}}}}&=\frac{3}{2}C_{F}\qquad{\beta_{Ve,\,0\,1}^{\overline{\rm{DR}}}}=-C_{F}\\\\[5.0pt]
{\beta_{Ve,\,2\,0}^{\overline{\rm{DR}}}}&=\frac{3}{16}C_{F}^{2}+\frac{91}{48}C_{F}\,C_{A}-\frac{5}{24}C_{F}\,N_{f}\qquad{\beta_{Ve,\,1\,1}^{\overline{\rm{DR}}}}=-\frac{11}{4}C_{F}^{2}-\frac{3}{4}C_{F}\,C_{A}\qquad{\beta_{Ve,\,0\,2}^{\overline{\rm{DR}}}}=C_{F}^{2}+\frac{3}{8}C_{F}\,N_{f}\\\\[5.0pt]
{\beta_{Ve,\,3\,0}^{\overline{\rm{DR}}}}&=\frac{129}{64}C_{F}^{3}-\frac{133}{128}C_{F}^{2}\,C_{A}-\left(\frac{23}{32}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\frac{10255}{3456}C_{F}\,C_{A}^{2}-\left(\frac{281}{864}+\frac{3}{4}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\frac{35}{864}C_{F}\,N_{f}^{2}\\\
{\beta_{Ve,\,2\,1}^{\overline{\rm{DR}}}}&=-\left(\frac{139}{64}-\frac{27}{4}\zeta_{3}\right)\,C_{F}^{3}-\left(\frac{331}{64}+\frac{81}{8}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\frac{11}{16}C_{F}^{2}\,N_{f}-\left(\frac{195}{256}-\frac{27}{8}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}+\frac{5}{64}C_{F}\,C_{A}\,N_{f}\\\
{\beta_{Ve,\,1\,2}^{\overline{\rm{DR}}}}&=\left(\frac{13}{2}-3\,\zeta_{3}\right)\,C_{F}^{3}-\left(\frac{7}{8}-\frac{9}{2}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\left(\frac{63}{64}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\left(\frac{7}{16}-\frac{3}{2}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}\\\
&-\left(\frac{3}{64}-\frac{3}{4}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}\\\\[5.0pt]
{\beta_{Ve,\,0\,3}^{\overline{\rm{DR}}}}&=-\left(\frac{7}{4}+\frac{9}{4}\zeta_{3}\right)\,C_{F}^{3}+\left(\frac{1}{8}+\frac{27}{8}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}-\frac{27}{32}C_{F}^{2}\,N_{f}+\left(\frac{1}{16}-\frac{9}{8}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}+\frac{3}{64}C_{F}\,C_{A}\,N_{f}+\frac{3}{64}C_{F}\,N_{f}^{2}\\\
{\beta_{Ve,\,0\,2100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{3}{8}\qquad{\beta_{Ve,\,0\,2010}^{\overline{\rm{DR}}}}=-\frac{25}{6}\qquad{\beta_{Ve,\,0\,2001}^{\overline{\rm{DR}}}}=-\frac{1}{4}\\\
{\beta_{Ve,\,0\,1200}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{63}{64}\qquad{\beta_{Ve,\,0\,1101}^{\overline{\rm{DR}}}}=\frac{21}{16}\qquad{\beta_{Ve,\,0\,1020}^{\overline{\rm{DR}}}}=\frac{65}{4}\qquad{\beta_{Ve,\,0\,1002}^{\overline{\rm{DR}}}}=-\frac{49}{32}\end{split}$
(62)
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|
arxiv-papers
| 2011-02-25T21:25:37 |
2024-09-04T02:49:17.290424
|
{
"license": "Public Domain",
"authors": "William B. Kilgore",
"submitter": "William Kilgore",
"url": "https://arxiv.org/abs/1102.5353"
}
|
1102.5398
|
# A proof of the classification theorem of overtwisted contact structures via
convex surface theory
Yang Huang University of Southern California, Los Angeles, CA 90089
huangyan@usc.edu
###### Abstract.
In [2], Y. Eliashberg proved that two overtwisted contact structures on a
closed oriented 3-manifold are isotopic through contact structures if and only
if they are homotopic as 2-plane fields. We provide an alternative proof of
this theorem using the convex surface theory and bypasses.
###### Contents
1. 1 Preliminaries
2. 2 Outline of the proof
3. 3 Local properties of bypass attachments
4. 4 Isotoping contact structures up to the 2-skeleton
5. 5 Bypass triangle attachments
6. 6 Overtwisted contact structures on $S^{2}\times[0,1]$ induced by isotopies.
7. 7 Classification of overtwisted contact structures on $S^{2}\times[0,1]$
8. 8 Proof of the main theorem
A contact manifold $(M,\xi)$ is a smooth manifold with a contact structure
$\xi$, i.e., a maximally non-integrable codimension 1 tangent distribution. In
particular, if the dimension of the manifold is three, it was realized through
the work of D. Bennequin and Y. Eliashberg in [1], [3] that contact structures
fall into two classes: tight or overtwisted. Since then, dynamical systems and
foliation theory of surfaces embedded in contact 3-manifolds have been studied
extensively to analyze this dichotomy. Based on these developments, Eliashberg
gave a classification of overtwisted contact structures in [2], which we now
explain.
Let $M$ be a closed oriented manifold and $\triangle\subset M$ be an oriented
embedded disk. Furthermore, we fix a point $p\in\triangle$. We denote by
$Cont^{ot}(M,\triangle)$ the space of cooriented, positive, overtwisted
contact structures on $M$ which are overtwisted along $\triangle$, i.e., the
contact distribution is tangent to $\triangle$ along $\partial\triangle$. Let
$Distr(M,\triangle)$ be the space of cooriented 2-plane distributions on $M$
which are tangent to $\triangle$ at $p$. Both spaces are equipped with the
$C^{\infty}$-topology.
###### Theorem 0.1 (Eliashberg).
Let $M$ be a closed, oriented 3-manifold. Then the inclusion
$j:Cont^{ot}(M,\triangle)\to Distr(M,\triangle)$ is a homotopy equivalence.
In particular, we have:
###### Theorem 0.2.
Let $M$ be a closed, oriented 3-manifold. If $\xi$ and $\xi^{\prime}$ are two
positive overtwisted contact structures on $M$, then they are isotopic if and
only if they are homotopic as 2-plane fields.
Consequently, overtwisted contact structures are completely determined by the
homotopy classes of the underlying 2-plane fields. On the other hand, the
classification of tight contact structures is much more subtle and contains
more topological information about the ambient 3-manifold.
The goal of this paper is to provide an alternative proof of Theorem 0.2 based
on convex surface theory. Convex surface theory was introduced by E. Giroux in
[8] building on the work of Eliashberg-Gromov [4]. Given a closed oriented
surface $\Sigma$, we consider contact structures on $\Sigma\times[0,1]$ such
that $\Sigma\times\\{0,1\\}$ is convex. By studying the “film picture” of the
characteristic foliations on $\Sigma\times\\{t\\}$ as $t$ goes from 0 to 1,
Giroux showed in [9] that, up to an isotopy, there are only finitely many
levels $\Sigma\times\\{t_{i}\\}$, $0<t_{1}<\cdots<t_{n}<1$, which are not
convex. Moreover, for small $\epsilon>0$, the characteristic foliations on
$\Sigma\times\\{t_{i}-\epsilon\\}$ and $\Sigma\times\\{t_{i}+\epsilon\\}$,
$i=1,2,\cdots,n$, change by a bifurcation. In [10], K. Honda gave an
alternative description of the bifurcation of characteristic foliations in
terms of dividing sets. Namely, he defined an operation, called the bypass
attachment, which combinatorially acts on the dividing set. It turns out that
a bypass attachment is equivalent to a bifurcation on the level of
characteristic foliations. Hence, in order to study contact structures on
$\Sigma\times[0,1]$ with convex boundary, it suffices to consider the isotopy
classes of contact structures given by sequences of bypass attachments. In
particular, we will study sequences of (overtwisted) bypass attachments on
$S^{2}\times[0,1]$, which is the main ingredient in our proof of Theorem 0.2.
This paper is organized as follows. In Section 1 we recall some basic
knowledge in contact geometry, in particular, convex surface theory and the
definition of a bypass. Section 2 gives an outline of our approach to the
classification problem. Section 3 is devoted to establishing some necessary
local properties of the bypass attachment. Using techniques from previous
sections, we show in Section 4 that how to isotop homotopic overtwisted
contact structures so that they agree in a neighborhood of the 2-skeleton.
Section 5, 6 and 7 are devoted to studying overtwisted contact structures on
$S^{2}\times[0,1]$ which is the technical part of this paper. We finally
finish the proof of Theorem 0.2 in Section 8.
## 1\. Preliminaries
Let $M$ be a closed, oriented 3-manifold. Throughout this paper, we only
consider cooriented, positive contact structures $\xi$ on $M$, i.e., those
that satisfy the following conditions:
1. (1)
there exists a global 1-form $\alpha$ such that $\xi=\ker(\alpha)$.
2. (2)
$\alpha\wedge d\alpha>0$, i.e., the orientation induced by the contact form
$\alpha$ agrees with the orientation on $M$.
A contact structure $\xi$ is overtwisted if there exists an embedded disk
$D^{2}\subset M$ such that $\xi$ is tangent to $D^{2}$ on $\partial D^{2}$.
Otherwise, $\xi$ is said to be tight. We will focus on overtwisted contact
structures for the rest of this paper.
Let $\Sigma\subset M$ be a closed, embedded, oriented surface in $M$. The
characteristic foliation $\Sigma_{\xi}$ on $\Sigma$ is by definition the
integral of the singular line field $\Sigma_{\xi}(x)\coloneqq\xi_{x}\cap
T_{x}\Sigma$. One way to describe the contact structure near $\Sigma$ is to
look at its characteristic foliation.
###### Proposition 1.1 (Giroux).
Let $\xi_{0}$ and $\xi_{1}$ be two contact structures which induce the same
characteristic foliation on $\Sigma$. Then there exists an isotopy
$\phi_{t}:M\to M$, $t\in[0,1]$ fixing $\Sigma$ such that $\phi_{0}=id$ and
$(\phi_{1})_{*}\xi_{0}=\xi_{1}$.
Possibly after a $C^{\infty}$-small perturbation, we can always assume that
$\Sigma\subset M$ is convex, i.e., there exists a vector field $v$ transverse
to $\Sigma$ such that the flow of $v$ preserves the contact structure. Using
this transverse contact vector field $v$, we define the dividing set on
$\Sigma$ to be
$\Gamma_{\Sigma}\coloneqq\\{x\in\Sigma~{}|~{}v_{x}\in\xi_{x}\\}$. Note that
the isotopy class of $\Gamma_{\Sigma}$ does not depend on the choice of $v$.
We refer to [8] for a more detailed treatment of basic properties of convex
surfaces. The significance of dividing sets in contact geometry is made clear
by Giroux’s flexibility theorem.
###### Theorem 1.2 (Giroux).
Assume $\Sigma$ is convex with characteristic foliation $\Sigma_{\xi}$,
contact vector field $v$, and dividing set $\Gamma_{\Sigma}$. Let
$\mathscr{F}$ be another singular foliation on $\Sigma$ divided by
$\Gamma_{\Sigma}$. Then there exists an isotopy $\phi_{t}:M\to M,t\in[0,1]$
such that
1. (1)
$\phi_{0}=id$ and $\phi_{t}|_{\Gamma_{\Sigma}}=id$ for all $t$.
2. (2)
$v$ is transverse to $\phi_{t}(\Sigma)$ for all $t$.
3. (3)
$\phi_{1}(\Sigma)$ has characteristic foliation $\mathscr{F}$.
We now look at contact structures on $\Sigma\times[0,1]$ with convex boundary.
The first important result relating to this problem is the following theorem
due to Giroux.
###### Theorem 1.3 (Giroux).
Let $\xi$ be a contact structure on $W=\Sigma\times[0,1]$ so that
$\Sigma\times\\{0,1\\}$ is convex. There exists an isotopy relative to the
boundary $\phi_{s}:W\to W$, $s\in[0,1]$, such that the surfaces
$\phi_{1}(\Sigma\times\\{t\\})$ are convex for all but finitely many
$t\in[0,1]$ where the characteristic foliations satisfy the following
properties:
1. (1)
The singularities and closed orbits are all non-degenerate.
2. (2)
The limit set of any half-orbit is either a singularity or a closed orbit.
3. (3)
There exists a single “retrogradient” saddle-saddle connection, i.e., an orbit
from a negative hyperbolic point to a positive hyperbolic point.
In the light of Giroux’s flexibility theorem, one should expect a
corresponding “film picture” of dividing sets on convex surfaces. It turns out
that the correct notion corresponding to a bifurcation is the bypass
attachment, which we now describe.
###### Definition 1.4.
Let $\Sigma$ be a convex surface and $\alpha$ be a Legendrian arc in $\Sigma$
which intersects $\Gamma_{\Sigma}$ in three points, two of which are endpoints
of $\alpha$. A bypass is a convex half-disk $D$ with Legendrian boundary,
where $D\cap\Sigma=\alpha$, $D\pitchfork\Sigma$, and $tb(\partial D)=-1$. We
call $\alpha$ an admissible arc, and $D$ a bypass along $\alpha$ on $\Sigma$.
###### Remark 1.5.
The admissible arc $\alpha$ in the above definition is also known as the arc
of attachment for a bypass in literature.
###### Remark 1.6.
We do not distinguish isotopic admissible arcs $\alpha_{0}$ and $\alpha_{1}$,
i.e., if there exists a path of admissible arcs $\alpha_{t}$, $t\in[0,1]$
connecting them.
The following lemma shows how a bypass attachment combinatorially acts on the
dividing set.
###### Lemma 1.7 (Honda).
Following the terminology from Definition 1.4, let $D$ be a bypass along
$\alpha$ on $\Sigma$. There exists a neighborhood of $\Sigma\cup D\subset M$
diffeomorphic to $\Sigma\times[0,1]$, such that $\Sigma\times\\{0,1\\}$ are
convex, and $\Gamma_{\Sigma\times\\{1\\}}$ is obtained from
$\Gamma_{\Sigma\times\\{0\\}}$ by performing the bypass attachment operation
as depicted in Figure 1 in a neighborhood of $\alpha$.
$(a)$$(b)$
Figure 1. A bypass attachment along $\alpha$. (a) The dividing set on
$\Sigma\times\\{0\\}$ before the bypass is attached. (b) The dividing set on
$\Sigma\times\\{1\\}$ after the bypass is attached.
It is worthwhile to mention that there are two distinguished bypasses, namely,
the trivial bypass and the overtwisted bypass as depicted in Figure 2. The
effect of a trivial bypass attachment is isotopic to an $I$-invariant contact
structure where no bypass is attached, while the overtwisted bypass attachment
immediately introduces an overtwisted disk in the local model, hence, for
example, is disallowed in tight contact manifolds.
\begin{overpic}[scale={.4}]{TrivialOT.eps} \put(16.0,3.0){$\longrightarrow$}
\put(76.0,3.0){$\longrightarrow$} \put(17.0,-5.0){$(a)$}
\put(77.0,-5.0){$(b)$} \end{overpic}
Figure 2. (a) The trivial bypass attachment. (b) The overtwisted bypass
attachment.
## 2\. Outline of the proof
Let $\xi$ and $\xi^{\prime}$ be two overtwisted contact structures on $M$,
homotopic to each other as 2-plane field distributions. Our approach to
Theorem 0.2 has three main steps.
Step 1. Fix a triangulation $T$ of $M$. Isotop $\xi$ and $\xi^{\prime}$
through contact structures such that $T$ becomes an overtwisted contact
triangulation in the sense that the 1-skeleton $T^{(1)}$ is a Legendrian
graph, the 2-skeleton $T^{(2)}$ is convex and each 3-cell is an overtwisted
ball with respect to both contact structures. We first show that if
$e(\xi)=e(\xi^{\prime})\in H^{2}(M;\mathbb{Z})$, then one can isotop $\xi$ and
$\xi^{\prime}$ so that they agree in a neighborhood of $T^{(2)}$.
Step 2. We can assume that there exists a ball $B^{3}\subset M$ such that
$\xi$ and $\xi^{\prime}$ agree on $M\setminus B^{3}$. Taking a small Darboux
ball $B^{3}_{std}\subset B^{3}$, observe that $\xi|_{B^{3}}$ and
$\xi^{\prime}|_{B^{3}}$ can both be realized as attaching sequences of
bypasses to $B^{3}_{std}$. In section 5, we will define the notion of a stable
isotopy. Then we show that both of sequences of bypass are stably isotopic to
some power of the bypass triangle attachment. Moreover, the boundary relative
homotopy classes of $\xi|_{B^{3}}$ and $\xi^{\prime}|_{B^{3}}$, measured by
the Hopf invariant, are uniquely determined by the number of bypass triangles
attached according to [11].
Step 3. By elementary obstruction theory, the Hopf invariants of
$\xi|_{B^{3}}$ and $\xi^{\prime}|_{B^{3}}$ are not necessarily the same, but
they can at most differ by an integral multiple of the divisibility of the
Euler class of either $\xi$ or $\xi^{\prime}$. See Section 8 for the
definition of the divisibility. We show that this ambiguity can be resolved by
further isotoping the contact structures in a neighborhood of $T^{(2)}$. This
finishes the proof of Theorem 0.2.
## 3\. Local properties of bypass attachments
Let $M$ be an overtwisted contact 3-manifold. Let $\Sigma\subset M$ be a
closed convex surface with dividing set $\Gamma_{\Sigma}$. For convenience, we
choose a metric on $M$ and denote $M\setminus\Sigma$ the metric closure of the
open manifold $M-\Sigma$. In this paper, we restrict ourself to the case that
each connected component of $M\setminus\Sigma$ is overtwisted111In general it
is possible that all components of $M\setminus\Sigma$ are tight even if $M$ is
overtwisted.. In order to isotop convex surfaces through bypasses freely, we
must show that there are enough bypasses. In fact, bypasses exist along any
admissible Legendrian arc on $\Sigma$ provided that the contact structure is
overtwisted. This is the content of the following lemma.
###### Lemma 3.1.
Suppose that $M\setminus\Sigma$ is overtwisted. For any admissible arc
$\alpha\subset\Sigma$, there exists a bypass along $\alpha$ in
$M\setminus\Sigma$. If $\Sigma$ separates $M$ into two overtwisted components,
then there exists such a bypass in each component.
###### Proof.
The technique for proving this lemma is essentially due to Etnyre and Honda
[5], and independently Torisu [12]. We construct a bypass $D$ along $\alpha$
as follows. Let $\tilde{D}\subset M\setminus\Sigma$ be an overtwisted disk.
First we push the interior of $\alpha$ slightly into $M\setminus\Sigma$ with
the endpoints of $\alpha$ fixed to obtain another Legendrian arc
$\tilde{\alpha}$, such that $\alpha$ and $\tilde{\alpha}$ cobound a convex
bigon $B$ with $tb(\partial B)=-2$. Next, take a Legendrian arc $\gamma$
connecting $\tilde{\alpha}$ and $\partial\tilde{D}$ in the complement of
$\Sigma\cup\tilde{D}\cup B$, namely, the two endpoints of $\gamma$ are
contained in $\tilde{\alpha}$ and $\partial\tilde{D}$ respectively and the
interior of $\gamma$ is disjoint from $\Sigma\cup\tilde{D}\cup B$ as depicted
in Figure 3.
\begin{overpic}[scale={.33}]{legarc.eps} \put(48.0,17.5){$B$}
\put(72.0,78.0){$\tilde{D}$} \put(54.0,49.0){$\gamma$}
\put(54.0,9.0){$\alpha$} \end{overpic} Figure 3. The Legendrian arc $\gamma$
connecting $\partial B$ and $\partial\tilde{D}$.
Suppose $N(\gamma)\cong\gamma\times[-\epsilon,\epsilon]$ is a band with the
core $\gamma\times\\{0\\}$ identified with $\gamma$, such that the
characteristic foliation is non-singular and is given by
$\gamma\times\\{t\\}$, $t\in[-\epsilon,\epsilon]$. In particular
$\gamma\times\\{-\epsilon\\}$ and $\gamma\times\\{\epsilon\\}$ are both
Legendrian. We want to glue $N(\gamma)$ to $\tilde{D}$ and $B$ so that the
characteristic foliations match along the common boundary. In order to do so,
we recall the following lemma first observed by Fraser [6].
###### Lemma 3.2.
Let $S$ be an embedded disk in a contact manifold $(M,\xi)$ with a
characteristic foliation $\xi|_{S}$ which consists only of one positive
elliptic singularity p and unstable orbits from p which exit transversely from
$\partial S$. If $\delta_{1}$, $\delta_{2}$ are two unstable orbits meeting at
$p$, and $\delta_{i}\cap S=p_{i}$, then, after a $C^{\infty}$-small
perturbation of $S$ fixing $\partial S$, we obtain $S^{\prime}$ whose
characteristic foliation has exactly one positive elliptic singularity
$p^{\prime}$ and unstable orbits from $p^{\prime}$ exiting transversely from
$\partial S$, and for which the orbits passing through $p_{1}$, $p_{2}$ meet
tangentially at $p^{\prime}$.
We first glue $N(\gamma)$ to $\tilde{D}$ as follows. Let
$p_{1}=\gamma\cap\partial\tilde{D}$. By the Flexibility Theorem we may suppose
that $p_{1}$ is a half-elliptic singular point of the characteristic foliation
$\xi|_{\tilde{D}}$ on $\tilde{D}$. Consider a slightly larger disk
$\tilde{D}^{\prime}\supset\tilde{D}$ such that $p_{1}$ is an elliptic
singularity of $\xi|_{\tilde{D}^{\prime}}$. Let $S\subset\tilde{D}^{\prime}$
be a small disk neighbothood of $p_{1}$, which satisfies the conditions in
Lemma 3.2. Applying Lemma 3.2, we can perturb $S$ to get a disk $\hat{D}$ on
which the characteristic foliation (in a neighbothood of $p_{1}$) looks like
the one depicted in Figure 4.
\begin{overpic}[scale={.5}]{neck.eps} \put(51.5,32.5){\small{$p_{1}$}}
\put(83.0,8.0){\small{$\hat{D}$}} \end{overpic} Figure 4.
Now we can glue $N(\gamma)$ to $\hat{D}$ in the obvious way such that the
characteristic foliations match along the common boundary. We can apply the
same trick to glue $N(\gamma)$ to $B$. In the end we obtain a half disk, which
we denote by $\tilde{D}\cup N(\gamma)\cup B$ by abuse of notation, on which
the characteristic foliation is as depicted in Figure 5.
\begin{overpic}[scale={.5}]{charbypass.eps} \put(101.0,7.0){$\alpha$}
\put(34.0,18.2){\small{$p_{1}$}} \put(78.8,18.2){\small{$p_{2}$}}
\put(19.7,13.8){$+$} \put(34.0,14.0){$-$} \put(79.7,13.0){$+$}
\put(101.0,14.0){$+$} \put(101.0,-3.0){$-$} \put(101.0,33.0){$-$}
\put(-4.0,14.0){$-$} \put(2.0,2.0){$-$} \put(2.0,28.5){$-$}
\put(15.0,-3.0){$-$} \put(15.0,34.0){$-$} \put(28.2,2.0){$-$}
\put(28.2,28.5){$-$} \end{overpic} Figure 5. The preferred characteristic
foliation on $\tilde{D}\cup N(\gamma)\cup B$.
Note that since the characteristic foliation contains a flowline from the
negative half-elliptic-half-hyperbolic singularity to the positive half-
elliptic-half-hyperbolic singularity, the half disk $\tilde{D}\cup
N(\gamma)\cup B$ is not convex. However we can perform a $C^{\infty}$-small
perturbation in a neighborhood of $p_{1}$ and $p_{2}$ to obtain a new half
disk $D$ such that the singularities $p_{1}$ and $p_{2}$ are eliminated. The
characteristic foliation on $D$ is given by Figure 6, which is easily seen to
be of Morse-Smale type. Therefore $D$ is convex with Legendrian boundary. The
dividing set $\Gamma$ on $D$ has to separate the positive and negative
singularities and to be transverse to the characteristic foliation. So
$\Gamma$ is, up to isotopy, the half-circle as depicted in Figure 6 as
desired, and therefore $D$ is a bypass along $\alpha$.
\begin{overpic}[scale={.5}]{charbypassfinal.eps} \put(101.0,7.0){$\alpha$}
\put(19.7,13.8){$+$} \put(101.0,14.0){$+$} \put(101.0,-3.0){$-$}
\put(101.0,33.0){$-$} \put(-4.0,14.0){$-$} \put(2.0,2.0){$-$}
\put(2.0,28.5){$-$} \put(15.0,-3.0){$-$} \put(15.0,34.0){$-$}
\put(28.2,2.0){$-$} \put(28.2,28.5){$-$} \end{overpic} Figure 6. The bypass
$D$ along $\alpha$.
∎
We then show the triviality of the trivial bypass, i.e., attaching a trivial
bypass does not change the isotopy class of the contact structure in a
neighborhood of the convex surface. The proof essentially follows the lines of
the proof of Proposition 4.9.7 in Geiges [7]. Here the contact structure may
be either overtwisted or tight.
###### Lemma 3.3.
Let $(\Sigma\times[0,1],\xi)$ be a contact manifold with the contact structure
$\xi$ obtained by attaching a trivial bypass on
$(\Sigma\times\\{0\\},\xi|_{\Sigma\times\\{0\\}})$. Then there exists another
contact structure $\tilde{\xi}$, which is isotopic to $\xi$ relative to the
boundary, such that $\Sigma\times\\{t\\}$ is convex with respect to
$\tilde{\xi}$ for all $t\in[0,1]$.
###### Proof.
Since this is a local problem, we may assume that $\Sigma\times[0,1]$ is a
neighborhood of the trivial bypass attachment. By Theorem 1.2, any Morse-Smale
type characteristic foliation adapted to $\Gamma_{\Sigma\times\\{0\\}}$ can be
realized as the characteristic foliation of a contact structure isotopic to
$\xi$ in a neighborhood of $\Sigma\times\\{0\\}$. In particular, we can assume
that the characteristic foliation on $\Sigma\times\\{0\\}$ looks exactly the
same as in Figure 7(a) such that $e_{-}$ does not connect to any negative
hyperbolic point other than $h_{-}$ along the flow line.
\begin{overpic}[scale={.3}]{Bifurcation.eps} \put(4.4,12.9){\tiny{$e_{-}$}}
\put(15.4,12.7){\tiny{$h_{-}$}} \put(33.7,12.7){\tiny{$h_{+}$}}
\put(62.5,12.9){\tiny{$e_{-}$}} \put(73.3,12.7){\tiny{$h_{-}$}}
\put(91.8,12.7){\tiny{$h_{+}$}} \put(20.0,-4.0){(a)} \put(80.0,-4.0){(b)}
\end{overpic}
Figure 7. (a) The characteristic foliation on $\Sigma\times\\{0\\}$. The
trivial bypass is attached along the Legendrian arc in dash line. (b) The
characteristic foliation on $\Sigma\times\\{1\\}$ after attaching the trivial
bypass. Here $e_{\pm}$ (resp. $h_{\pm}$) denote the $\pm$-elliptic (resp.
$\pm$-hyperbolic) singular points of the foliation.
Look at the characteristic foliations on $\Sigma\times\\{t\\}$ as $t$ goes
from 0 to 1. Generically we can assume that the Morse-Smale condition fails at
one single level, say, $\Sigma\times\\{1/2\\}$, where an unstable saddle-
saddle connection has to appear as shown in Figure 8(a).
\begin{overpic}[scale={.35}]{SScon.eps} \put(3.4,15.8){\tiny{$e_{-}$}}
\put(17.0,15.5){\tiny{$h_{-}$}} \put(30.5,15.5){\tiny{$h_{+}$}}
\put(-1.0,23.0){$\Omega$} \put(66.0,23.0){$\Omega$} \put(20.0,-5.0){(a)}
\put(80.0,-5.0){(b)} \end{overpic}
Figure 8. (a) The characteristic foliation on $\Sigma\times\\{1/2\\}$, where a
saddle-saddle connect from $h_{-}$ to $h_{+}$ exists. The region $\Omega$
contains exactly two singular points $\\{e_{-},h_{-}\\}$ which are in
elimination position. (b) The nonsingular characteristic foliation on $\Omega$
after the elimination.
Let $\Omega\subset\Sigma\times\\{1/2\\}$ be an open neighborhood of the flow
line from $h_{-}$ to $e_{-}$ as depicted in Figure 8(a). Observe that the
characteristic foliation inside $\Omega$ is of Morse-Smale type, and therefore
stable in the $t$-direction. According to the proof of Proposition
4.9.7222This is a stronger version of the usual Elimination Lemma. in Geiges
[7], for a small $\delta>0$, there exists an isotopy
$\phi_{s}:\Sigma\times[0,1]\to\Sigma\times[0,1]$, $s\in[0,1]$, compactly
supported in $\Omega\times(1/2-2\delta,1/2+2\delta)\subset\Sigma\times[0,1]$
and $\phi_{0}=id$, such that $\tilde{\xi}=(\phi_{1})_{*}\xi$ satisfies the
following:
1. (1)
The characteristic foliation on $\Omega\times\\{t\\}$ with respect to
$\tilde{\xi}$ is isotopic to the one in Figure 8(b) for
$t\in[1/2-\delta,1/2+\delta]$. In particular, it is nonsingular.
2. (2)
For $t\in(1/2-2\delta,1/2-\delta)\cup(1/2+\delta,1/2+2\delta)$, The
characteristic foliation on $\Omega\times\\{t\\}$ with respect to
$\tilde{\xi}$ is almost Morse-Smale except that there may exist a half-
elliptic-half-hyperbolic point.
We remark here that the above conditions are achieved in [7] by isotoping
surfaces $\Sigma\times\\{t\\}$, $t\in[1/2-2\delta,1/2+2\delta]$ while fixing
the contact structure $\xi$, but this is equivalent to isotoping $\xi$ while
fixing $\Sigma\times\\{t\\}$. We will switch between these two equivalent
point of view again in the proof of Proposition 4.3.
Now we can make $\Sigma\times\\{t\\}$ convex for $t\in[1/2-\delta,1/2+\delta]$
because the only unstable saddle-saddle connection is eliminated and therefore
the characteristic foliation becomes Morse-Smale. For
$t\notin[1/2-\delta,1/2+\delta]$, there may exist half-elliptic-half-
hyperbolic singular points, but we can as well construct a contact structure
realizing this type of singularity so that each $\Omega\times\\{t\\}$ stays
convex. Hence $\tilde{\xi}$ constructed above is as required. ∎
###### Remark 3.4.
Let $(\Sigma\times[0,1],\xi)$ be a contact manifold such that
$\xi|_{\Sigma_{0}}=\xi|_{\Sigma_{1}}$ and $\Sigma\times\\{t\\}$ is convex for
all $t\in[0,1]$. If $\Sigma\neq S^{1}\times S^{1}$ and $\xi$ is tight, then it
is a standard fact that $\xi$ is isotopic to an $I$-invariant contact
structure relative to the boundary. However, if either $\Sigma=S^{1}\times
S^{1}$ or $\xi$ is overtwisted, then the above fact is not true anymore. We
will study this phenomenon in detail in the case when $\Sigma=S^{2}$ and $\xi$
is overtwisted in Section 6.
## 4\. Isotoping contact structures up to the 2-skeleton
We are now ready to take the first main step towards the proof of Theorem 0.2.
Since we will isotop contact structures skeleton by skeleton, we start with
the following definition.
###### Definition 4.1.
Let $(M,\xi)$ be an overtwisted contact manifold, and $T$ be a triangulation
of $M$. The triangulation $T$ is called an overtwisted contact triangulation
if the following conditions hold:
1. (1)
The 1-skeleton is a Legendrian graph.
2. (2)
Each 2-simplex is convex with Legendrian boundary.
3. (3)
Each 3-simplex is an overtwisted ball.
###### Remark 4.2.
The overtwisted contact triangulation defined above is different from the
usual contact triangulation where the 3-simplexes are assumed to be tight.
The goal for this section is to prove the following Proposition.
###### Proposition 4.3.
Let $M$ be a closed, oriented 3-manifold with a fixed triangulation $T$. Let
$\xi$ and $\xi^{\prime}$ be homotopic overtwisted contact structures on $M$.
Then they are isotopic up to the 2-skeleton, i.e., there exists an isotopy
$\phi_{t}:M\to M$, $t\in[0,1]$, $\phi_{0}=id$ such that
$(\phi_{1})_{*}\xi=\xi^{\prime}$ in a neighborhood of $T^{(2)}$.
###### Proof.
Before we go into details of the proof, observe that if $\phi_{t}:M\to M$,
$t\in[0,1]$, $\phi_{0}=id$ is an isotopy, then $(M,\phi_{1}(\xi),T)$ and
$(M,\xi,\phi_{1}^{-1}(T))$ carries the same contact information. In fact, we
will isotop the skeletons of the triangulation $T$ and think of them as
isotopies of contact structures.
By a $C^{0}$-small perturbation of the 1-skeleton $T^{(1)}$, we can assume
that $T^{(1)}$ is a Legendrian graph with respect to $\xi$ and $\xi^{\prime}$.
Performing stabilizations to edges of $T^{(1)}$ if necessary, we can further
assume that $\xi=\xi^{\prime}$ in a neighborhood of $T^{(1)}$. For each
2-simplex $\sigma^{2}$ in $T^{(2)}$, we can always stabilize the Legendrian
unknot $\partial\sigma^{2}$ sufficiently many times so that
$tb(\partial\sigma^{2})<0$. Therefore a $C^{\infty}$-small perturbation of
$\sigma^{2}$ relative to $\partial\sigma^{2}$ makes it convex with respect to
$\xi$ (resp. ${\xi^{\prime}}$) with dividing set $\Gamma_{\sigma^{2}}^{\xi}$
(resp. $\Gamma_{\sigma^{2}}^{\xi^{\prime}}$). Both $\Gamma_{\sigma^{2}}^{\xi}$
and $\Gamma_{\sigma^{2}}^{\xi^{\prime}}$ are proper 1-submanifolds of
$\sigma^{2}$ and generically the endpoints are contained in the interior of
the 1-simplexes. See Figure 9 for an example.
In order to make $T$ an overtwisted contact triangulation for $\xi$ and
$\xi^{\prime}$, we still need to make sure that all 3-simplexes are
overtwisted. We do this for $\xi$, and the same argument applies to
$\xi^{\prime}$. Take an overtwisted disc $D$ in $(M,\xi)$. We can assume that
$D$ is contained in a 3-simplex $\sigma^{3}_{1}$. Let $\sigma^{3}_{2}$ be
another 3-simplex which shares a 2-face with $\sigma^{3}_{1}$, i.e.,
$\sigma^{3}_{1}\cap\sigma^{3}_{2}=\sigma^{2}$ is a 2-simplex. We claim that by
isotoping $\sigma^{2}$ relative to $\partial\sigma^{2}$ if necessary, we can
make both $\sigma^{3}_{1}$ and $\sigma^{3}_{2}$ overtwisted. The fact that $M$
is closed immediately implies that a finite steps of such isotopies will make
$T$ an overtwisted contact triangulation. To prove the claim, we first take a
parallel copy of the overtwisted disk $D$ in an $I$-invariant neighborhood of
$D$, denoted by $D^{\prime}$. Pick an arc $\gamma$ connecting $D^{\prime}$ to
$\sigma^{2}$ inside $\sigma^{3}_{1}$. Let $\tilde{\sigma}^{2}$ be another
2-simplex obtained by isotoping $\sigma^{2}$ across $D^{\prime}$ along
$\gamma$, i.e., $\tilde{\sigma}^{2}$ satisfying the following conditions:
1. (1)
$\partial\tilde{\sigma}^{2}=\partial\sigma^{2}$.
2. (2)
$\sigma^{2}\cup\tilde{\sigma}^{2}$ bounds a neighborhood of
$D^{\prime}\cup\gamma$.
3. (3)
$\tilde{\sigma}^{2}$ is convex.
By replacing $\sigma^{2}$ with $\tilde{\sigma}^{2}$, we obtain two new
3-simplexes, each of which contains an overtwisted disk in the interior as
claimed.
\begin{overpic}[scale={.2}]{DivSet.eps} \end{overpic} Figure 9. An example of
the dividing set on a 2-simplex.
Now by Giroux’s flexibility theorem, it suffices to isotop $\xi$ and
${\xi^{\prime}}$ so that they induce isotopic dividing sets on each 2-simplex
relative to $T^{(1)}$. To achieve this goal, we define the difference
2-cocycle $\delta$ by assigning to each oriented 2-simplex $\sigma^{2}$ an
integer
$\chi(R_{+}(\Gamma^{\xi^{\prime}}_{\sigma^{2}}))-\chi(R_{-}(\Gamma^{\xi^{\prime}}_{\sigma^{2}}))-\chi(R_{+}(\Gamma^{\xi}_{\sigma^{2}}))+\chi(R_{-}(\Gamma^{\xi}_{\sigma^{2}}))$.
Since $\xi$ is homotopic to ${\xi^{\prime}}$ as 2-plane fields,
$[\delta]=e(\xi)-e(\xi^{\prime})=0\in H^{2}(M,\mathbb{Z})$. Hence there exists
an integral 1-cocycle $\theta$ so that $2d\theta=\delta$ since the Euler class
is always even.333More precisely, if we fix a trivialization of $TM$ and
consider the Gauss map associated to the contact distribution, then the Euler
class of the contact distribution is exactly twice the Poincaré dual of the
Pontryagin submanifold of the Gauss map. One should think of $\theta$ as an
element in $Hom(C_{1}(M),\mathbb{Z})$.
Let $\sigma^{2}\in T^{(2)}$ be an oriented convex 2-simplex and
$\sigma^{1}\subset\partial\sigma^{2}$ be an oriented 1-simplex with the
induced orientation. We study the effect of stabilizing the 1-simplex
$\sigma^{1}$ to the overtwisted contact triangulation. If we positively
stabilize $\sigma^{1}$ once and isotop $\sigma^{2}$ accordingly to obtain a
new 2-simplex $\tilde{\sigma}^{2}$, then the dividing set
$\Gamma^{\xi}_{\tilde{\sigma}^{2}}$ on $\tilde{\sigma}^{2}$ is obtained from
$\Gamma^{\xi}_{\sigma^{2}}$ by adding a properly embedded arc contained in the
negative region with both endpoints on the interior of $\sigma^{1}$ as
depicted in Figure 10. Similarly, if we negatively stabilize $\sigma^{1}$ once
and isotop $\sigma^{2}$ accordingly as before, then the dividing set on the
isotoped $\sigma^{2}$ is obtained from $\Gamma^{\xi}_{\sigma^{2}}$ by adding a
properly embedded arc contained in the positive region and with both endpoints
on the interior of $\sigma^{1}$.
\begin{overpic}[scale={.3}]{PosStab.eps} \put(3.5,2.0){\tiny{$-$}}
\put(10.0,6.0){\tiny{$+$}} \put(19.5,6.0){\tiny{$-$}}
\put(26.0,2.0){\tiny{$+$}} \put(11.7,12.5){\tiny{$-$}}
\put(72.0,2.0){\tiny{$-$}} \put(78.0,6.0){\tiny{$+$}}
\put(88.5,8.0){\tiny{$-$}} \put(86.5,1.5){\tiny{$+$}}
\put(94.6,2.0){\tiny{$+$}} \put(80.3,12.5){\tiny{$-$}} \put(12.0,-6.0){(a)}
\put(82.0,-6.0){(b)} \end{overpic}
Figure 10. (a) The dividing set on $\sigma^{2}$ divides it into $\pm$-regions.
The bottom edge is $\sigma^{1}$. (b) One possible dividing set on
$\tilde{\sigma}^{2}$ after positively stabilizing $\sigma^{1}$ once.
Note that in general, the new overtwisted contact triangulation obtained by
$\pm$-stabilizing a 1-simplex $\sigma^{1}$ is not unique. In fact, different
choices may give non-isotopic dividing sets on the isotoped $\sigma^{2}$ in
the new triangulation. However, for our purpose, we only care about the
quantity $\chi(R_{+})-\chi(R_{-})$ on each 2-simplex and it is easy to see
that different choices give the same value to this quantity. Thus we will
ignore this ambiguity by arbitrarily choosing an isotopy of the 2-simplex.
We denote the overtwisted contact triangulation obtained by $\pm$-stabilizing
$\sigma^{1}$ once in $(M,\xi)$ by $S^{\pm}_{\sigma^{1}}(\xi)$. As remarked at
the beginning of the proof, one should think of $S^{\pm}_{\sigma^{1}}(\xi)$ as
isotopies of $\xi$. It is easy to see that $S^{\pm}_{\sigma^{1}}(\xi)$ changes
$\chi(R_{+}(\Gamma^{\xi}_{\sigma^{2}}))-\chi(R_{-}(\Gamma^{\xi}_{\sigma^{2}}))$
by $\pm 1$ for any 2-simplex $\sigma^{2}\in T^{(2)}$ so that
$\sigma^{1}\subset\partial\sigma^{2}$ as an oriented boundary edge. The same
holds for $\xi^{\prime}$ as well.
Now we argue that one can isotop $\xi$ and ${\xi^{\prime}}$ so that
$\chi(R_{+}(\Gamma^{\xi}_{\sigma^{2}}))-\chi(R_{-}(\Gamma^{\xi}_{\sigma^{2}}))=\chi(R_{+}(\Gamma^{\xi^{\prime}}_{\sigma^{2}}))-\chi(R_{-}(\Gamma^{\xi^{\prime}}_{\sigma^{2}}))$
on each 2-simplex $\sigma^{2}$. This can be done as follows. For each oriented
1-simplex $\sigma^{1}\in T^{(1)}$, the 1-cocycle $\theta$ sends it to an
integer $n=\theta(\sigma^{1})$. We perform $n$ times the isotopy
$S^{+}_{\sigma^{1}}(\xi)$ to $\xi$ and $n$ times the isotopy
$S^{-}_{\sigma^{1}}({\xi^{\prime}})$ to ${\xi^{\prime}}$ at the same time. If
we perform such operation to every 1-simplex in $T$, it is easy to see that
the following properties are satisfied:
1. (1)
$\xi={\xi^{\prime}}$ in a neighborhood of $T^{(1)}$.
2. (2)
$\chi(R_{+}(\Gamma^{\xi}_{\sigma^{2}}))-\chi(R_{-}(\Gamma^{\xi}_{\sigma^{2}}))=\chi(R_{+}(\Gamma^{\xi^{\prime}}_{\sigma^{2}}))-\chi(R_{-}(\Gamma^{\xi^{\prime}}_{\sigma^{2}}))$,
$\forall\sigma^{2}\in T^{(2)}$.
The second property implies that $\Gamma^{\xi^{\prime}}_{\sigma^{2}}$ can be
obtained from $\Gamma^{\xi}_{\sigma^{2}}$ by attaching a sequence of bypasses
for each 2-simplex $\sigma^{2}$. Recall that $T$ is an overtwisted contact
triangulation and in particular each 3-simplex is an overtwisted ball. Hence
bypasses exist along any admissible arc in $\sigma^{2}$ inside any 3-simplex
with $\sigma^{2}$ as a 2-face by Lemma 3.1. Therefore by isotoping 2-simplexes
through bypasses, we can assume that $\xi$ and $\xi^{\prime}$ induce isotopic
dividing sets on each 2-simplex relative to its boundary. The conclusion now
follows immediately from Giroux’s flexibility theorem. ∎
## 5\. Bypass triangle attachments
In this section we study the effect of attaching a bypass triangle to the
contact structure, in particular, we give an alternative definition of the
bypass triangle attachment. We start with the definition of the bypass
triangle attachment.
Notation: Let $\Sigma$ be a convex surface and $\alpha\subset\Sigma$ be an
admissible arc. We denote the bypass attachment along $\alpha$ on $\Sigma$ by
$\sigma_{\alpha}$. Let $\beta$ be another admissible arc on the convex surface
obtained by attaching the bypass along $\alpha$ on $\Sigma$. We denote the
composition of bypass attachments by $\sigma_{\alpha}\ast\sigma_{\beta}$,
where the composition rule is to attach the bypass along $\alpha$ first, then
attach the bypass along $\beta$ in the same direction. If $(M,\xi)$ is a
contact manifold with convex boundary, then $\xi\ast\sigma_{\alpha}$ denotes
the contact structure obtained by attaching a bypass along $\alpha$ to
$(M,\xi)$.
###### Remark 5.1.
In general, bypass attachments are not commutative unless the attaching arcs
are disjoint.
###### Definition 5.2.
Let $\Sigma$ be a convex surface and $\alpha\subset\Sigma$ be an admissible
arc. A bypass triangle attachment along $\alpha$ is the composition of three
bypass attachments along admissible arcs $\alpha$, $\alpha^{\prime}$ and
$\alpha^{\prime\prime}$ in a neighborhood of $\alpha$ as depicted in Figure
11. We denote the bypass triangle attachment along $\alpha$ by
$\triangle_{\alpha}=\sigma_{\alpha}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$.
###### Remark 5.3.
The second admissible arc $\alpha^{\prime}$ in the bypass bypass triangle is
also known as the arc of anti-bypass attachment to $\sigma_{\alpha}$.
\begin{overpic}[scale={.32}]{BypassTriangle.eps} \put(13.5,46.0){(a)}
\put(81.0,46.0){(b)} \put(46.0,-7.0){(c)} \put(17.5,69.5){\tiny{$\alpha$}}
\put(87.5,64.5){\tiny{$\alpha^{\prime}$}}
\put(49.5,9.5){\tiny{$\alpha^{\prime\prime}$}}
\put(48.0,70.0){\small{$\sigma_{\alpha}$}}
\put(72.0,37.0){\small{$\sigma_{\alpha^{\prime}}$}}
\put(21.0,37.0){\small{$\sigma_{\alpha^{\prime\prime}}$}} \end{overpic}
Figure 11. (a) A neighborhood of $\alpha$ on $\Sigma$, along which the first
bypass $\sigma_{\alpha}$ is attached. (b) The second bypass
$\sigma_{\alpha^{\prime}}$ is attached along the dotted arc $\alpha^{\prime}$.
(c) The third bypass $\sigma_{\alpha^{\prime\prime}}$ is attached along the
dotted arc $\alpha^{\prime\prime}$ and finishes the bypass triangle.
Warning: When we define a bypass attachment $\sigma_{\alpha}$ along $\alpha$
on $(\Sigma,\Gamma_{\Sigma})$, there are several choices involved. Namely, we
need to choose a multicurve, i.e., a 1-submanifold of $\Sigma$, representing
the isotopy class of $\Gamma_{\Sigma}$, an admissible arc representing the
isotopy class of $\alpha$, a neighborhood of $\alpha$ where $\sigma_{\alpha}$
is supported. Since the space of choices of $\alpha$ and its neighborhood is
contractible according to Theorem 1.2, we can neglect this ambiguity. However
the space of choices of multicurves representing $\Gamma_{\Sigma}$ is not
necessarily contractible. This point will be made clear in the next section.
For the rest of this paper, $\Gamma_{\Sigma}$ always means a multicurve on
$\Sigma$ rather than its isotopy class.
###### Remark 5.4.
If $\Sigma=S^{2}$ and $\Gamma_{\Sigma}=S^{1}$, then the space of choices of
multicurve is simply-connected since there is a unique tight contact structure
in a neighborhood of $S^{2}$ up to isotopy.
Observe that, up to an isotopy supported in a neighborhood of the admissible
arc $\alpha$, the bypass triangle attachment does not change
$\Gamma_{\Sigma}$.
In what follows we look at bypass triangle attachments along different
admissible arcs, which leads to our alternative definition of the bypass
triangle attachment.
###### Lemma 5.5.
Let $\xi_{\alpha}$ and $\xi_{\beta}$ be two (overtwisted) contact structures
on $S^{2}\times[0,1]$, where $\alpha$ and $\beta$ are admissible arcs on
$S^{2}\times\\{0\\}$, such that
1. (1)
$S^{2}\times\\{0,1\\}$ is convex with respect to both $\xi_{\alpha}$ and
$\xi_{\beta}$.
2. (2)
$\xi_{\alpha}=\xi_{\beta}$ in a neighborhood of $S^{2}\times\\{0\\}$ and
$\\#\Gamma^{\xi_{\alpha}}_{S^{2}\times\\{0\\}}=\\#\Gamma^{\xi_{\beta}}_{S^{2}\times\\{0\\}}=1$.
3. (3)
$\xi_{\alpha}$ is obtained by attaching a bypass triangle $\triangle_{\alpha}$
to $\xi_{\alpha}|_{S^{2}\times\\{0\\}}$, and $\xi_{\beta}$ is obtained by
attaching a bypass triangle $\triangle_{\beta}$ to
$\xi_{\beta}|_{S^{2}\times\\{0\\}}$.
Then $\xi_{\alpha}$ is isotopic to $\xi_{\beta}$ relative to the boundary.
###### Proof.
Up to isotopy, there are only two different admissible arcs on
$(S^{2}\times\\{0\\},\xi_{\alpha}|_{S^{2}\times\\{0\\}})$ (or,
$(S^{2}\times\\{0\\},\xi_{\beta}|_{S^{2}\times\\{0\\}})$). Namely, one gives
the trivial bypass and the other gives the overtwisted bypass. We may assume
without loss of generality that $\alpha$ is not isotopic to $\beta$, and
$\sigma_{\alpha}$ is the trivial bypass and $\sigma_{\beta}$ is the
overtwisted bypass. We complete the bypass triangles $\triangle_{\alpha}$ and
$\triangle_{\beta}$ as depicted in Figure 12.
\begin{overpic}[scale={.3}]{BTonS2.eps} \put(7.0,32.0){\tiny{$\alpha$}}
\put(32.8,29.8){\tiny{$\alpha^{\prime}$}}
\put(62.8,31.8){\tiny{$\alpha^{\prime\prime}$}} \put(11.5,7.0){\tiny{$\beta$}}
\put(35.2,6.4){\tiny{$\beta^{\prime}$}}
\put(65.0,4.2){\tiny{$\beta^{\prime\prime}$}}
\put(21.0,34.5){\tiny{$\sigma_{\alpha}$}}
\put(48.5,34.5){\tiny{$\sigma_{\alpha^{\prime}}$}}
\put(76.5,34.5){\tiny{$\sigma_{\alpha^{\prime\prime}}$}}
\put(21.0,9.5){\tiny{$\sigma_{\beta}$}}
\put(48.5,9.5){\tiny{$\sigma_{\beta^{\prime}}$}}
\put(76.5,9.5){\tiny{$\sigma_{\beta^{\prime\prime}}$}} \end{overpic}
Figure 12.
Observe that $\alpha^{\prime}$ is isotopic to $\beta$, $\alpha^{\prime\prime}$
is isotopic to $\beta^{\prime}$ and bypass attachments along $\alpha$ and
$\beta^{\prime\prime}$ are trivial according to Lemma 3.3, we have the
following isotopies:
$\displaystyle\triangle_{\alpha}$
$\displaystyle=\sigma_{\alpha}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\sigma_{\beta}\ast\sigma_{\beta^{\prime}}$
$\displaystyle\simeq\sigma_{\beta}\ast\sigma_{\beta^{\prime}}\ast\sigma_{\beta^{\prime\prime}}=\triangle_{\beta}.$
Since $S^{2}\times\\{0,1\\}$ are convex, we can make sure that the isotopies
above are supported in the interior of $S^{2}\times[0,1]$. ∎
###### Definition 5.6.
A minimal overtwisted ball $(B^{3},\xi_{ot})$ is an overtwisted ball where
$\partial B^{3}$ has a tight neighborhood, and the contact structure
$\xi_{ot}$ is obtained by attaching a bypass triangle to the standard tight
ball $(B^{3},\xi_{std})$.
###### Remark 5.7.
By Lemma 5.5, the minimal overtwisted ball is well-defined even if we do not
specify the admissible arc along which the bypass triangle is attached.
With the above preparation, we can now redefine the bypass triangle attachment
which is more convenient for our purpose. Let $(M,\xi)$ be a contact
3-manifold with convex boundary $\partial M=\Sigma$. Identify a collar
neighborhood of $\partial M$ with $\Sigma\times[-1,0]$ such that $\partial
M=\Sigma\times\\{0\\}$ and the contact vector field transverse to $\partial M$
is identified with the $[-1,0]$-direction. Let $\alpha\subset\partial M$ be an
admissible arc along which the bypass triangle is attached. Push $\alpha$ into
the interior of $M$ to obtain another admissible arc, parallel to $\alpha$,
contained in $\Sigma\times\\{-1/2\\}$, which we still denote by $\alpha$. Let
$N$ be a neighborhood of $\alpha$ in $\Sigma\times\\{-1/2\\}$. Consider the
ball with corners $N\times[-2/3,-1/3]\subset M$. By rounding the corners, we
get a smoothly embedded tight ball
$(B^{3}_{1},\xi|_{B^{3}_{1}})\subset(M,\xi)$, in particular, $\partial
B^{3}_{1}$ has a tight neighborhood in $(M,\xi)$. Let $(B^{3}_{2},\xi_{ot})$
be a minimal overtwisted ball. We construct a new contact manifold
$(M,\tilde{\xi})=(M\setminus B^{3}_{1},\xi)\cup_{\phi}(B^{3}_{2},\xi_{ot})$,
where $\phi$ is an orientation-reversing diffeomorphism identifying the
standard tight neighborhoods of $\partial B^{3}_{1}$ and $\partial B^{3}_{2}$.
It is easy to see that $\tilde{\xi}$ is isotopic to the contact structure
obtained by attaching a bypass triangle to $(M,\xi)$ along $\alpha$.
###### Remark 5.8.
The uniqueness of the tight contact structure on 3-ball, due to Eliashberg,
guarantees that the bypass triangle attachment described above is well-
defined.
Using the above alternative description of the bypass triangle attachment, we
prove the following generalization of Lemma 5.5.
###### Lemma 5.9.
Let $(M,\xi)$ be a contact 3-manifold with convex boundary, and let
$\alpha,\beta$ be two admissible arcs on $\partial M$. Let $\xi_{\alpha}$
(resp. $\xi_{\beta}$) be the contact structure on $M$ obtained by attaching a
bypass triangle $\triangle_{\alpha}$ (resp. $\triangle_{\beta}$) along
$\alpha$ (resp. $\beta$) to $(M,\xi)$. Then $\xi_{\alpha}$ is isotopic to
$\xi_{\beta}$ relative to the boundary.
###### Proof.
Without loss of generality, we can assume that $\alpha$ and $\beta$ are
disjoint. If not, we take another admissible arc $\gamma$ which is disjoint
from $\alpha$ and $\beta$. We then show that $\xi_{\alpha}\simeq\xi_{\gamma}$
and $\xi_{\beta}\simeq\xi_{\gamma}$, which implies
$\xi_{\alpha}\simeq\xi_{\beta}$.
As before, since $\partial M$ is convex, we can push $\alpha$ and $\beta$
slightly into the manifold $M$, which we still denote by $\alpha$ and $\beta$.
Now let $B^{3}_{\alpha}\subset M$ and $B^{3}_{\beta}\subset M$ be smoothly
embedded tight balls containing $\alpha$ and $\beta$ respectively. Take a
Legendrian arc $\tau$ connecting $B^{3}_{\alpha}$ and $B^{3}_{\beta}$, i.e.,
the endpoints of $\tau$ are contained in $\partial B^{3}_{\alpha}$ and
$\partial B^{3}_{\beta}$, respectively, and the interior of $\tau$ is disjoint
from $B^{3}_{\alpha}$ and $B^{3}_{\beta}$. Moreover, we can assume that
$\tau\cap\partial B^{3}_{\alpha}\in\Gamma_{\partial B^{3}_{\alpha}}$ and
$\tau\cap\partial B^{3}_{\beta}\in\Gamma_{\partial B^{3}_{\beta}}$. Let
$N(\tau)$ be a closed tubular neighborhood of $\tau$. By rounding the corners
of $B^{3}_{\alpha}\cup B^{3}_{\beta}\cup N(\tau)$, we get a smoothly embedded
ball $B^{3}\subset M$ with tight convex boundary. Using our cut-and-paste
definition of the bypass triangle attachment, it is easy to see that
$(B^{3},\xi_{\alpha}|_{B^{3}})$ and $(B^{3},\xi_{\beta}|_{B^{3}})$ are
isotopic, relative to the boundary, to the contact boundary sums
$(B^{3},\xi_{ot})\\#_{b}(B^{3},\xi_{std})$ and
$(B^{3},\xi_{std})\\#_{b}(B^{3},\xi_{ot})$, respectively. Hence both are
isotopic to the minimal overtwisted ball. One simply extends the isotopy by
identity to the rest of $M$ to conclude that $\xi_{\alpha}\simeq\xi_{\beta}$
on $M$. ∎
According to Lemma 5.9, the isotopy class of the contact structure obtained by
attaching a bypass triangle does not depend on the choice of the attaching
arcs. We shall write $\triangle$ for a bypass triangle attachment along an
arbitrary admissible arc. An immediate consequence of this fact is that the
bypass triangle attachment commutes with any bypass attachment. This is the
content of the following corollary:
###### Corollary 5.10.
Let $(M,\xi)$ be contact 3-manifold with convex boundary, and $\alpha$ be an
admissible arc on $\partial M$. Then
$\xi\ast\sigma_{\alpha}\ast\triangle\simeq\xi\ast\triangle\ast\sigma_{\alpha}$.
###### Proof.
By Lemma 5.9, we can arbitrarily choose an admissible arc
$\beta\subset\partial M$ along which the bypass triangle $\triangle$ is
attached. In particular, we require that $\beta$ is disjoint from $\alpha$.
Hence a neighborhood of $\beta$ where $\triangle_{\beta}$ is supported in is
also disjoint from $\alpha$. Thus we have the following isotopies:
$\displaystyle\xi\ast\sigma_{\alpha}\ast\triangle$
$\displaystyle\simeq\xi\ast\sigma_{\alpha}\ast\triangle_{\beta}$
$\displaystyle\simeq\xi\ast\triangle_{\beta}\ast\sigma_{\alpha}$
$\displaystyle\simeq\xi\ast\triangle\ast\sigma_{\alpha}.$
which proves the commutativity. ∎
###### Corollary 5.11.
Let $(S^{2}\times[0,1],\xi)$ be a contact manifold with convex boundary, where
$\xi$ is isotopic to a sequence of bypass attachments
$\sigma_{1}\ast\sigma_{2}\ast\cdots\ast\sigma_{n}$, i.e., there exists
$0=t_{0}<t_{1}<\cdots<t_{n}=1$ such that $S^{2}\times\\{t_{i}\\}$ are convex
for $0\leq i\leq n$ and $S^{2}\times[t_{i-1},t_{i}]$ with the restricted
contact structure is isotopic to the bypass attachment $\sigma_{i}$. Then
$\xi\ast\triangle$ is isotopic to $\xi_{k}$ for $0\leq k\leq n$, where
$\xi_{k}$ is the contact structure isotopic to a sequence of bypass
attachments
$\sigma_{1}\ast\cdots\ast\sigma_{k}\ast\triangle\ast\sigma_{k+1}\cdots\ast\sigma_{n}$.
###### Proof.
This is an iterated application of Corollary 5.10. ∎
However, observe that subtracting a bypass triangle is in general not well-
defined. So we need the following definition.
###### Definition 5.12.
Two contact structures $\xi$ and $\xi^{\prime}$ on $S^{2}\times[0,1]$ are
stably isotopic, denoted by $\xi\sim\xi^{\prime}$, if they become isotopic
after attaching finitely many bypass triangles to $S^{2}\times\\{1\\}$
simultaneously, i.e.,
$\xi\ast\triangle^{n}\simeq\xi^{\prime}\ast\triangle^{n}$ for some
$n\in\mathbb{N}$.
## 6\. Overtwisted contact structures on $S^{2}\times[0,1]$ induced by
isotopies.
Let $\xi$ be an overtwisted contact structure on $S^{2}\times[0,1]$ such that
$S^{2}\times\\{0\\}$ and $S^{2}\times\\{1\\}$ are convex spheres. In general,
any such $\xi$ can be represented by a sequence of bypass attachments. More
precisely, by Theorem 1.3, there exists an increasing sequence
$0=t_{0}<t_{1}<\cdots<t_{n}=1$ such that $S^{2}\times\\{t_{i}\\}$ is convex
and $\xi|_{S^{2}\times[t_{i-1},t_{i}]}$ is isotopic to a bypass attachment
$\sigma_{i}$ for $i=1,\cdots,n$. In this section, we consider a special class
of overtwisted contact structures on $S^{2}\times[0,1]$ such that
$S^{2}\times\\{t\\}$ is convex for $t\in[0,1]$, in other words, there is no
bypass attached.
Let $\xi_{0}$ be an $I$-invariant contact structure on $S^{2}\times[0,1]$ with
dividing set $\Gamma_{0}$ on $S^{2}\times\\{0\\}$. Let $\phi_{t}:S^{2}\to
S^{2}$, $t\in[0,1]$, be an isotopy such that $\phi_{0}=id$. We define a new
contact structure $\xi_{\Gamma_{0},\Phi}=\Phi_{*}(\xi_{0})$ on
$S^{2}\times[0,1]$, where $\Phi:S^{2}\times[0,1]\to S^{2}\times[0,1]$ is
defined by $(x,t)\mapsto(\phi_{t}(x),t)$. Observe that $S^{2}\times\\{t\\}$ is
convex with respect to $\xi_{\Gamma_{0},\Phi}$ for all $t\in[0,1]$ by
construction. Hence we get a smooth family of dividing sets
$\Gamma_{S^{2}\times\\{t\\}}$ for $t\in[0,1]$. Conversely, a smooth family of
dividing sets $\Gamma_{S^{2}\times\\{t\\}}$, $t\in[0,1]$ defines a unique
contact structure on $S^{2}\times[0,1]$, which is isotopic to
$\xi_{\Gamma_{0},\Phi}$ constructed above for some isotopy $\phi_{t}$,
$t\in[0,1]$. In practice, it is usually easier to keep track of the dividing
sets rather than the isotopy.
###### Definition 6.1.
A contact structure $\xi$ on $S^{2}\times[0,1]$ is induced by an isotopy if
$S^{2}\times\\{t\\}$ is convex for all $t\in[0,1]$, or, equivalently, there
exists an isotopy $\Phi:S^{2}\times[0,1]\to S^{2}\times[0,1]$ such that $\xi$
is isotopic to $\xi_{\Gamma_{0},\Phi}$ as constructed above.
It is convenient to have the following lemma.
###### Lemma 6.2.
Let $\xi$, $\xi^{\prime}$ be two contact structures on $S^{2}\times[0,1]$
induced by isotopies and let $\Gamma_{t}$, $\Gamma^{\prime}_{t}$ be dividing
sets on $S^{2}\times\\{t\\}$, $0\leq t\leq 1$, with respect to $\xi$,
$\xi^{\prime}$ respectively. If $\Gamma_{0}=\Gamma^{\prime}_{0}$,
$\Gamma_{1}=\Gamma^{\prime}_{1}$ and there exists a path of smooth families of
multicurves $\Gamma^{s}_{t}$, $0\leq s\leq 1$ satisfying the following:
1. (1)
$\Gamma^{s}_{t}$ is a multicurve, i.e., a finite disjoint union of simple
closed curves, contained in $S^{2}\times\\{t\\}$ for $0\leq s\leq 1$, $0\leq
t\leq 1$.
2. (2)
$\Gamma^{0}_{t}=\Gamma_{t}$, $\Gamma^{1}_{t}=\Gamma^{\prime}_{t}$ for $0\leq
t\leq 1$,
3. (3)
$\Gamma^{s}_{0}=\Gamma_{0}$, $\Gamma^{s}_{1}=\Gamma_{1}$ for $0\leq s\leq 1$.
then $\xi$ is isotopic to $\xi^{\prime}$ relative to the boundary.
###### Proof.
By Giroux’s flexibility theorem, the path $\Gamma^{s}_{t}$, $0\leq s\leq 1$ of
multicurves determines a path of contact structures $\xi^{s}$ on
$S^{2}\times[0,1]$ such that $\xi^{0}=\xi$, $\xi^{1}=\xi^{\prime}$. Hence
$\xi$ is isotopic to $\xi^{\prime}$ relative to the boundary by Gray’s
stability theorem. ∎
We first consider a bypass attachment to the contact structures on
$S^{2}\times[0,1]$ induced by an isotopy.
###### Lemma 6.3.
Let $\xi_{\Gamma_{0},\Phi}$ be a contact structure on $S^{2}\times[0,1/2]$
induced by an isotopy $\phi_{t}:S^{2}\to S^{2}$, $t\in[0,1/2]$, and
$(S^{2}\times[1/2,1],\sigma_{\alpha})$ be a bypass attachment along an
admissible arc $\alpha\subset S^{2}\times\\{1/2\\}$. Then there exists an
admissible arc $\tilde{\alpha}\subset S^{2}\times\\{0\\}$ such that
$(S^{2}\times[0,1],\xi_{\Gamma_{0},\Phi}\ast\sigma_{\alpha})$ is isotopic,
relative to the boundary, to
$(S^{2}\times[0,1],\sigma_{\tilde{\alpha}}\ast\xi_{\Gamma^{\prime}_{0},\Phi})$,
where $\Gamma^{\prime}_{0}$ is the dividing set obtained by attaching a bypass
along $\alpha$ to $\Gamma_{0}$.
###### Proof.
We basically re-foliate the contact manifold
$(S^{2}\times[0,1],\xi_{\Gamma_{0},\Phi}\ast\sigma_{\alpha})$. Recall that
$\sigma_{\alpha}$ attaches a bypass $D$ on $S^{2}\times\\{1/2\\}$ so that
$\partial D=\alpha\cup\beta$ is the union of two Legendrian arcs, where
$tb(\alpha)=-1$, $tb(\beta)=0$. We extend $D$ to a new bypass $\tilde{D}$ on
$S^{2}\times\\{0\\}$ through the isotopy $\phi_{t}:S^{2}\to S^{2}$,
$t\in[0,1/2]$, by defining $\tilde{D}=D\cup\Phi(\tilde{\alpha}\times[0,1/2])$,
where $\tilde{\alpha}=\phi_{1/2}^{-1}(\alpha)\subset S^{2}\times\\{0\\}$ is
the new admissible arc along which $\tilde{D}$ is attached, and
$\Phi:S^{2}\times[0,1/2]\to S^{2}\times[0,1/2]$ is defined by
$(x,t)\mapsto(\phi_{t}(x),t)$. By attaching the new bypass $\tilde{D}$ on
$S^{2}\times\\{0\\}$, observe that the rest of $S^{2}\times[0,1]$ can be
foliated by convex surfaces, and the contact structure is also induced by
$\Phi$. Hence $\xi_{\Gamma_{0},\Phi}\ast\sigma_{\alpha}$ is isotopic to
$\sigma_{\tilde{\alpha}}\ast\xi_{\Gamma^{\prime}_{0},\Phi}$ as desired. ∎
###### Definition 6.4.
The admissible arc $\tilde{\alpha}$ constructed in Lemma 6.3 is called a push-
down of $\alpha$. Conversely, we call $\alpha$ a pull-up of $\tilde{\alpha}$.
The rest of this section is rather technical and can be skipped at the first
time reading. The only result needed for our proof of Theorem 0.2 is
Proposition 6.15.
We consider a subclass of the contact structures on $S^{2}\times[0,1]$ induced
by isotopies which we will be mainly interested in. Fix a metric on $S^{2}$.
Without loss of generality, we assume that there exists a small disk
$D^{2}_{\epsilon}(y)\subset S^{2}$ centered at $y$ of radius $\epsilon$ and a
codimension 0 submanifold $\tilde{\Gamma}_{S^{2}\times\\{0\\}}$ of
$\Gamma_{S^{2}\times\\{0\\}}$ such that
$\tilde{\Gamma}_{S^{2}\times\\{0\\}}\subset D^{2}_{\epsilon}(y)$ and
$D^{2}_{\epsilon}(y)\cap\Gamma_{S^{2}\times\\{0\\}}=\tilde{\Gamma}_{S^{2}\times\\{0\\}}$.
Let $\gamma(s)\subset S^{2}\times\\{0\\}$, $s\in[0,1]$ be an embedded oriented
loop such that $\gamma(0)=\gamma(1)=y$. Let $A(\gamma)$ be an annulus
neighborhood of $\gamma$ containing $D^{2}_{\epsilon}(y)$ and disjoint from
other components of the dividing set as depicted in Figure 13. We define an
isotopy $\phi_{t}:S^{2}\to S^{2}$, $t\in[0,1]$, supported in $A(\gamma)$ which
parallel transports $D^{2}_{\epsilon}(y)$ along $\gamma$ in $A(\gamma)$. More
precisely, by applying the stereographic projection map, we can identify
$A(\gamma)$ with an annulus in $\mathbb{R}^{2}$. Then the parallel
transportation is given by an affine map $\phi_{t}:x\mapsto
x+\gamma(t)-\gamma(0)$ for any $x\in D^{2}_{\epsilon}(y)$ and $t\in[0,1]$.
\begin{overpic}[scale={.3}]{Permutation.eps}
\put(7.5,24.0){{\color[rgb]{1,0,0}\small{$\tilde{\Gamma}$}}}
\put(46.0,24.0){{\color[rgb]{1,0,0}\small{$\Gamma\setminus{\tilde{\Gamma}}$}}}
\put(110.0,24.0){{\color[rgb]{1,0,0}\small{$\Gamma\setminus{\tilde{\Gamma}}$}}}
\put(90.0,33.0){\small{$\gamma$}} \put(91.0,45.0){\small{$A(\gamma)$}}
\end{overpic} Figure 13.
###### Definition 6.5.
With the small disk
$D^{2}_{\epsilon}(y)\supset\tilde{\Gamma}_{S^{2}\times\\{0\\}}$ such that
$\tilde{\Gamma}_{S^{2}\times\\{0\\}}\cap\partial
D^{2}_{\epsilon}(y)=\emptyset$, the annulus $A(\gamma)\supset\gamma$ and the
isotopy $\phi_{t}:S^{2}\to S^{2}$ chosen as above, we say that the contact
structure $\xi_{\Gamma_{S^{2}\times\\{0\\}},\Phi}$ on $S^{2}\times[0,1]$ is
induced by a pure braid of the dividing set, where $\Phi:S^{2}\times[0,1]\to
S^{2}\times[0,1]$ is induced by $\phi_{t}$ as before. We denote such contact
structures by $\xi_{\Gamma,\Phi(\tilde{\Gamma},D^{2}_{\epsilon}(y),\gamma)}$.
When there is no confusion, we also abbreviate it by
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$.
###### Remark 6.6.
For any simply connected region $D\subset S^{2}\times\\{0\\}$ containing
$\tilde{\Gamma}_{S^{2}\times\\{0\\}}$, one can isotop so that $D$ becomes a
round disk with small radius as required in Definition 6.5. The isotopy class
of the contact structure on $S^{2}\times[0,1]$ induced by a pure braid of the
dividing set only depends on the choice of
$D\supset\tilde{\Gamma}_{S^{2}\times\\{0\\}}$ and the isotopy class of
$\gamma$.
###### Remark 6.7.
If $\xi$ is a contact structure on $S^{2}\times[0,1]$ induced by a pure braid
of the dividing set, then
$\Gamma_{S^{2}\times\\{0\\}}=\Gamma_{S^{2}\times\\{1\\}}$.
Before we give a complete classification of contact structures on
$S^{2}\times[0,1]$ induced by pure braids of the dividing set, we make a
digression into the study of its homotopy classes using a generalized version
of the Pontryagin-Thom construction for manifolds with boundary. See [11] for
more discussions on the generalized Pontryagin-Thom construction.
We can always assume that the isotopy
$\phi_{t}(\tilde{\Gamma},D^{2}_{\epsilon}(y),\gamma):S^{2}\to S^{2}$,
$t\in[0,1]$, discussed in Definition 6.5 is supported in a disk $D^{2}\subset
S^{2}$. Trivialize the tangent bundle of $D^{2}\times[0,1]$ by embedding it
into $\mathbb{R}^{3}$ so that $D^{2}$ is contained in the $xy$-plane. Consider
the Gauss map
$G:(D^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})\to S^{2}$.
By Lemma 6.2, we can assume without loss of generality that the dividing set
is a disjoint union of round circles in $D^{2}\times\\{t\\}$ for all $0\leq
t\leq 1$, and $p=(1,0,0)\in S^{2}\subset\mathbb{R}^{3}$ is a regular value.
Suppose the number of connected components $\\#\Gamma_{D^{2}\times\\{0\\}}=m$,
then the Pontryagin submanifold $\mathcal{B}=G^{-1}(p)$ is an oriented framed
monotone braid in the sense that $\mathcal{B}$ transversely intersects
$D^{2}\times\\{t\\}$ in $m$ points for any $0\leq t\leq 1$, and each connected
component of the dividing set contains exactly one point. It is easy to check
that the pull-back framing is the blackboard framing, and consequently the
self-linking number of $\mathcal{B}$ is exactly $writhe(\mathcal{B})$. It
follows from the generalized Pontryagin-Thom construction that the homotopy
class of a contact structure on $D^{2}\times[0,1]$ relative to the boundary is
uniquely determined by the relative framed cobordism class of its Pontryagin
submanifold $\mathcal{B}$, and hence is uniquely determined by
$writhe(\mathcal{B})$ since
$H_{1}(D^{2}\times[0,1],\partial(D^{2}\times[0,1]);\mathbb{Z})=0$. One may
think of $writhe(\mathcal{B})$ as a relative version of the Hopf invariant
associated with boundary relative homotopy classes of maps
$D^{2}\times[0,1]\simeq B^{3}\to S^{2}$.
###### Example 6.8.
If $\Gamma_{D^{2}\times\\{0\\}}$ is the disjoint union of two isolated
circles, and $\tilde{\Gamma}_{D^{2}\times\\{0\\}}=S^{1}\subset
D^{2}_{\epsilon}(y)$ is the circle on the left as depicted in Figure 14. The
isotopy $\phi_{t}$ parallel transports $D^{2}_{\epsilon}(y)$ along the
oriented loop $\gamma$. We compute the homotopy class of the contact structure
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$.
\begin{overpic}[scale={.3}]{braidEx1.eps} \put(15.0,-8.0){(a)}
\put(81.0,-8.0){(b)} \put(14.0,6.7){\tiny{$p_{1}$}}
\put(27.0,6.7){\tiny{$p_{2}$}} \put(14.0,31.0){\tiny{$p_{1}$}}
\put(27.2,31.0){\tiny{$p_{2}$}} \put(74.0,-2.5){\tiny{$p_{1}$}}
\put(90.0,-2.5){\tiny{$p_{2}$}} \put(74.0,39.5){\tiny{$p_{1}$}}
\put(90.0,39.5){\tiny{$p_{2}$}} \put(4.0,7.0){\tiny{$+$}}
\put(4.0,31.7){\tiny{$+$}} \put(10.0,7.0){\tiny{$-$}}
\put(23.0,7.0){\tiny{$-$}} \put(10.0,31.7){\tiny{$-$}}
\put(23.0,31.7){\tiny{$-$}} \put(11.0,-2.3){\tiny{$D^{2}\times[0,1]$}}
\put(16.0,12.2){\tiny{$\gamma$}} \end{overpic}
Figure 14. (a) The contact structure on $S^{2}\times[0,1]$ induced by a full
twist of the dividing circles, where $\\{p_{1},p_{2}\\}$ are pre-images of the
regular value $p=(1,0,0)\in S^{2}$. (b) The oriented braid with the blackboard
framing $\mathcal{B}$ as the Pontryagin submanifold.
According to the Pontryagin-Thom construction, since $writhe(\mathcal{B})=-2$,
the homotopy class of $\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ is in
general different from the $I$-invariant contact structure, and the difference
is measured by decreasing the Hopf invariant by 2.444However, if the
divisibility of the Euler class is 2, then $\phi_{t}$ gives a contact
structure which is homotopic to the $I$-invariant contact structure. We will
discuss the divisibility of the Euler class in detail in Section 8.
###### Example 6.9.
If $\Gamma_{D^{2}\times\\{0\\}}$ is the disjoint union of three circles, and
$\tilde{\Gamma}_{D^{2}\times\\{0\\}}=S^{1}\subset D^{2}_{\epsilon}(y)$ is the
circle on the left as depicted in Figure 15. The isotopy $\phi_{t}$ parallel
transports $D^{2}_{\epsilon}(y)$ along the oriented loop $\gamma$. We compute
the homotopy class of the contact structure
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$.
\begin{overpic}[scale={.35}]{braidEx2.eps} \put(14.5,5.2){\tiny{$p_{1}$}}
\put(20.5,5.2){\tiny{$p_{2}$}} \put(31.3,5.2){\tiny{$p_{3}$}}
\put(14.5,33.2){\tiny{$p_{1}$}} \put(20.8,33.2){\tiny{$p_{2}$}}
\put(31.3,33.2){\tiny{$p_{3}$}} \put(15.0,-8.0){(a)} \put(81.0,-8.0){(b)}
\put(12.0,-3.3){\tiny{$D^{2}\times[0,1]$}} \put(73.0,-2.0){\tiny{$p_{1}$}}
\put(82.0,-2.0){\tiny{$p_{2}$}} \put(92.0,-2.0){\tiny{$p_{3}$}}
\put(73.0,42.0){\tiny{$p_{1}$}} \put(82.0,42.0){\tiny{$p_{2}$}}
\put(92.0,42.0){\tiny{$p_{3}$}} \put(3.5,5.5){\tiny{$+$}}
\put(10.0,5.5){\tiny{$-$}} \put(24.8,5.66){\tiny{$+$}}
\put(28.0,5.5){\tiny{$-$}} \put(3.5,34.3){\tiny{$+$}}
\put(10.0,34.3){\tiny{$-$}} \put(24.8,34.3){\tiny{$+$}}
\put(28.0,34.3){\tiny{$-$}} \put(17.0,11.5){\tiny{$\gamma$}} \end{overpic}
Figure 15. (a) A braiding by a full twist of the left-hand side dividing
circle along $\gamma$, where $\\{p_{1},p_{2},p_{3}\\}=G^{-1}(p)$ is the pre-
image of the regular value $p=(1,0,0)\in S^{2}$. (b) The oriented framed braid
$\mathcal{B}$ as the Pontryagin submanifold.
In this case, one computes that $writhe(\mathcal{B})=0$, hence
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ is homotopic to the
$I$-invariant contact structure.
Now we are ready to classify the contact structures induced by pure braids of
the dividing set up to stable isotopy in the sense of Definition 6.5. One goal
is to establish an isotopy equivalence relation between a pure braid of the
dividing set and the bypass triangle attachment. To start with, we consider
the contact structures induced by two special pure braids of the dividing set
as depicted in Figure 16. In Figure 16(a), the dividing set
$\tilde{\Gamma}\subset D^{2}_{\epsilon}(y)$ is a single circle, and the
dividing set contained in the disk bounded by $\gamma$ and disjoint from
$\tilde{\Gamma}$ is also a single circle. In Figure 16(b), the dividing set
$\tilde{\Gamma}\subset D^{2}_{\epsilon}(y)$ consists of $m$ isolated circles
nested in another circle, and the dividing set contained in the disk bounded
by $\gamma$ and disjoint from $\tilde{\Gamma}$ consists of $n$ isolated
circles nested in another circle. We also assume that either $m$ or $n$ is not
zero. For technical reasons, it is convenient to have the following
definitions.
###### Definition 6.10.
Given two disjoint embedded circles $\gamma,\gamma^{\prime}\subset D^{2}$, we
say $\gamma<\gamma^{\prime}$ if and only if $\gamma$ is contained in the disk
bounded by $\gamma^{\prime}$.
###### Definition 6.11.
Let $\Gamma\subset D^{2}$ be a finite disjoint union of embedded circles. The
depth of $\Gamma$ is the maximum length of chains
$\gamma_{1}<\gamma_{2}<\cdots<\gamma_{r}$, where $\gamma_{i}\subset\Gamma$ is
a single circle for any $i\in\\{1,2,\cdots,r\\}$.
Observe that the depth of the dividing set in Figure 16(a) is 1, and the depth
of the dividing set in Figure 16(b) is 2. It turns out that to study the
contact structure induced by an arbitrary pure braid of the dividing set, it
suffices to consider a finite composition of these two special cases.
\begin{overpic}[scale={.25}]{Perm1.eps} \put(10.0,-4.5){(a)}
\put(77.0,-4.5){(b)} \put(22.5,10.0){\small{$\gamma$}}
\put(100.0,10.0){\small{$\gamma$}} \put(-2.2,2.0){\small{$\Gamma^{\prime}$}}
\put(51.5,2.0){\small{$\Gamma^{\prime}$}} \put(60.0,4.5){$\underbrace{}$}
\put(84.7,4.5){$\underbrace{}$} \put(63.2,0.8){\tiny{$m$}}
\put(88.5,0.8){\tiny{$n$}} \end{overpic}
Figure 16.
###### Lemma 6.12.
If $(S^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})$ is a
contact manifold with contact structure induced by a pure braid of the
dividing set where $\tilde{\Gamma}\subset D^{2}_{\epsilon}$ and $\gamma$ are
chosen as in Figure 16(a), then
$(S^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})$ is isotopic
relative to the boundary to $(S^{2}\times[0,1],\triangle^{2})$, where
$\triangle^{2}$ denotes the contact structure obtained by attaching two bypass
triangles on
$(S^{2}\times\\{0\\},\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}|_{S^{2}\times\\{0\\}})$.
###### Proof.
Let $\alpha$ be an admissible arc as depicted in Figure 17(b). Suppose that
both bypass triangles are attached along $\alpha$.
\begin{overpic}[scale={.2}]{braidLem1.eps} \put(14.0,-7.0){(a)}
\put(80.0,-7.0){(b)} \put(16.0,0.5){\tiny{$\gamma$}}
\put(82.0,7.0){\tiny{$\alpha$}} \put(82.0,21.5){\tiny{$\alpha$}}
\put(38.0,20.0){$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$}
\put(102.0,11.0){$\triangle_{\alpha}$} \put(102.0,24.0){$\triangle_{\alpha}$}
\end{overpic}
Figure 17. (a) The contact structure is induced by parallel transporting
$\tilde{\Gamma}\subset D^{2}_{\epsilon}$ along $\gamma$. (b) Attaching two
bypass triangles along the admissible arc $\alpha$.
Observe that
$\triangle_{\alpha}=\sigma_{\alpha}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$,
where $\sigma_{\alpha}$, $\sigma_{\alpha^{\prime}}$ and
$\sigma_{\alpha^{\prime\prime}}$ are all trivial bypass attachments. Hence the
contact manifold $(S^{2}\times[0,1],\triangle_{\alpha}^{2})$ can be foliated
by convex surfaces by Lemma 3.3. In other words, it is induced by an isotopy.
By Theorem 0.5555The 3-dimensional obstruction class $o_{3}$ used in Theorem
0.5 in [11] is by definition the relative version of the Hopf invariant which
we have discussed above. in [11], we know that attaching two bypass triangles
$\triangle_{\alpha}^{2}$ decreases the Hopf invariant by 2. In Example 6.8, we
checked by Pontryagin-Thom construction that
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ also decreases the Hopf
invariant by 2. Observe that the isotopy class relative to the boundary of a
2-strand oriented monotone braid with blackboard framing is uniquely
determined by its self-linking number, which is equal to the Hopf invariant.
Hence $\triangle^{2}_{\alpha}$ is isotopic
$\Phi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ in the region where both
operations are supported. By extending the isotopy by identity to the rest of
$S^{2}$, we conclude that
$(S^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})$ is isotopic
relative to the boundary to $(S^{2}\times[0,1],\triangle^{2})$. ∎
###### Lemma 6.13.
If $(S^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})$ is a
contact manifold with contact structure induced by a pure braid of the
dividing set where $\tilde{\Gamma}\subset D^{2}_{\epsilon}$ and $\gamma$ are
chosen as in Figure 16(b), then
$(S^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})$ is stably
isotopic to $(S^{2}\times[0,1],\triangle^{2(m-1)(n-1)})$.
###### Proof.
Let $\alpha\subset S^{2}\times\\{1\\}$ be an admissible arc as depicted in the
left-hand side of Figure 18(a). By Lemma 6.3, if $\tilde{\alpha}$ is the push-
down of $\alpha$, then
$\xi_{\Gamma,\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\alpha}\simeq\sigma_{\tilde{\alpha}}\ast\xi_{\Gamma^{\prime},\Phi}$,
where $\Gamma^{\prime}$ is obtained from $\Gamma$ by attaching a bypass along
$\alpha$. We remark here that
$\xi_{\Gamma,\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}$ and
$\xi_{\Gamma^{\prime},\Phi}$ are contact structures induced by the same
isotopy, but are push-forwards of different contact structures on
$S^{2}\times[0,1]$. Choose $\tilde{\Gamma}^{\prime}\subset
D^{2^{\prime}}_{\epsilon}$ to be the $m$ isolated circles on the left and
$\gamma^{\prime}$ be an oriented loop as depicted in the right-hand side of
Figure 18(a). Let
$\xi_{\tilde{\Gamma}^{\prime},D^{2^{\prime}}_{\epsilon},\gamma^{\prime}}$ be
the contact structure induced by an isotopy which parallel transports
$\tilde{\Gamma}^{\prime}\subset D^{2^{\prime}}_{\epsilon}$ along
$\gamma^{\prime}$. Then Lemma 6.2 implies that $\xi_{\Gamma^{\prime},\Phi}$ is
isotopic, relative to the boundary, to
$\xi_{\tilde{\Phi}}\ast\xi_{\tilde{\Gamma}^{\prime},D^{2^{\prime}}_{\epsilon},\gamma^{\prime}}$,
where $\tilde{\Phi}$ is induced by an isotopy that rounds the outmost dividing
circle. An iterated application of Lemma 6.12 implies that
$\xi_{\tilde{\Gamma}^{\prime},D^{2^{\prime}}_{\epsilon},\gamma^{\prime}}\simeq\triangle^{2m(n-1)}$.
We next isotop the contact structure
$\sigma_{\tilde{\alpha}}\ast\xi_{\tilde{\Phi}}$. Consider the $n$ isolated
circles nested in a larger circle. Let $\tilde{\Gamma}^{\prime\prime}\subset
D^{2^{\prime\prime}}_{\epsilon}$ be the leftmost circle among the $n$ circles
and $\gamma^{\prime\prime}$ be an oriented loop as depicted in the right-hand
side of Figure 18(b). We pull up $\tilde{\alpha}$ through an isotopy which
parallel transports $\tilde{\Gamma}^{\prime\prime}\subset
D^{2^{\prime\prime}}_{\epsilon}$ along $\gamma^{\prime\prime}$, and observe
that the pull-up of $\tilde{\alpha}$ is isotopic to $\alpha$. By using Lemma
6.3 one more time, we get the isotopy of contact structures
$\sigma_{\tilde{\alpha}}\ast\xi_{\tilde{\Phi}}\simeq\xi_{\tilde{\Gamma}^{\prime\prime},D^{2^{\prime\prime}}_{\epsilon},\gamma^{\prime\prime}}\ast\sigma_{\alpha}$.
It is left to determine the isotopy class of the contact structure
$\xi_{\tilde{\Gamma}^{\prime\prime},D^{2^{\prime\prime}}_{\epsilon},\gamma^{\prime\prime}}$.
Since $\gamma^{\prime\prime}$ is oriented counterclockwise, by applying Lemma
6.12 $(n-1)$ times, we get a stable isotopy
$\xi_{\tilde{\Gamma}^{\prime\prime},D^{2^{\prime\prime}}_{\epsilon},\gamma^{\prime\prime}}\sim\triangle^{2(1-n)}$,
i.e.,
$\xi_{\tilde{\Gamma}^{\prime\prime},D^{2^{\prime\prime}}_{\epsilon},\gamma^{\prime\prime}}\ast\triangle^{2(n-1)}$
is isotopic to the $I$-invariant contact structure.
\begin{overpic}[scale={.2}]{braidLem2.eps} \put(38.0,47.0){(a)}
\put(38.0,-4.0){(b)} \put(38.0,18.0){$\simeq$} \put(38.0,73.0){$\simeq$}
\put(32.0,13.0){$\sigma_{\tilde{\alpha}}$}
\put(80.0,10.0){$\xi_{\tilde{\Gamma}^{\prime\prime},D^{2^{\prime\prime}}_{\epsilon},\delta^{\prime\prime}}$}
\put(80.0,24.0){$\sigma_{\alpha}$} \put(19.0,4.0){\tiny{$\tilde{\alpha}$}}
\put(69.0,2.0){\tiny{$\gamma^{\prime\prime}$}}
\put(62.5,19.0){\tiny{$\alpha$}}
\put(32.0,64.0){$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$}
\put(32.0,78.0){$\triangle_{\alpha}$} \put(15.0,56.2){\tiny{$\gamma$}}
\put(15.0,74.5){\tiny{$\alpha$}} \put(80.0,62.0){$\sigma_{\tilde{\alpha}}$}
\put(80.0,74.0){$\xi_{\tilde{\Gamma}^{\prime},D^{2^{\prime}}_{\epsilon},\gamma^{\prime}}$}
\put(80.0,86.0){$\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$}
\put(68.0,52.8){\tiny{$\tilde{\alpha}$}}
\put(62.5,65.8){\tiny{$\gamma^{\prime}$}}
\put(8.0,55.3){\tiny{$\tilde{\Gamma}$}}
\put(57.6,66.5){\tiny{$\tilde{\Gamma}^{\prime}$}}
\put(65.7,2.5){\tiny{$\tilde{\Gamma}^{\prime\prime}$}} \end{overpic}
Figure 18. (a) Pushing down the bypass attachment $\sigma_{\alpha}$. (b)
Pulling up the bypass attachment $\sigma_{\tilde{\alpha}}$.
To summarize what we have done so far, we have the following (stable)
isotopies of contact structures:
$\displaystyle\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}\ast\triangle_{\alpha}$
$\displaystyle=\xi_{\Gamma,\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\alpha}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\sigma_{\tilde{\alpha}}\ast\xi_{\Gamma^{\prime},\Phi}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\sigma_{\tilde{\alpha}}\ast\xi_{\tilde{\Phi}}\ast\xi_{\tilde{\Gamma}^{\prime},D^{2^{\prime}}_{\epsilon},\gamma^{\prime}}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\sigma_{\tilde{\alpha}}\ast\xi_{\tilde{\Phi}}\ast\triangle^{2m(n-1)}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\xi_{\tilde{\Gamma}^{\prime\prime},D^{2^{\prime\prime}}_{\epsilon},\gamma^{\prime\prime}}\ast\sigma_{\alpha}\ast\triangle^{2m(n-1)}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\sim\triangle^{2(1-n)}\ast\sigma_{\alpha}\ast\triangle^{2m(n-1)}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle\simeq\triangle^{2(m-1)(n-1)}\ast\sigma_{\alpha}\ast\sigma_{\alpha^{\prime}}\ast\sigma_{\alpha^{\prime\prime}}$
$\displaystyle=\triangle^{2(m-1)(n-1)}\ast\triangle_{\alpha}.$
Note that the third equation from the bottom is only a stable isotopy so that
the (possibly) negative power of the bypass triangle attachment makes sense.
See Definition 5.12. We will use the same trick in the proof of the following
Proposition 6.14 without further mentioning. Hence by definition,
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ is stably isotopic to
$\triangle^{2(m-1)(n-1)}$ as desired. ∎
We now completely classify contact structures on $S^{2}\times[0,1]$ induced by
pure braids of the dividing set.
###### Proposition 6.14.
If $(S^{2}\times[0,1],\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma})$ is a
contact manifold with contact structure induced by a pure braid of the
dividing set, then $\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ is stably
isotopic to $(S^{2}\times[0,1],\triangle^{l})$ for some $l\in\mathbb{N}$.
###### Proof.
Recall that $\tilde{\Gamma}\subset D^{2}_{\epsilon}$ is a codimension 0
submanifold of $\Gamma_{S^{2}\times\\{0\\}}$, and $\gamma$ is an oriented loop
in the complement of $\Gamma_{S^{2}\times\\{0\\}}$ as in Definition 6.5. Let
$\tilde{\Gamma}^{\prime}$ be the union of components of
$\Gamma_{S^{2}\times\\{0\\}}$ contained in a disk bounded by $\gamma$ and
outside of $A(\gamma)$. We may choose the disk so that $-\gamma$ is the
oriented boundary. Since the contact structure
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ is induced by a pure braid of
the dividing set, we have
$\Gamma_{S^{2}\times\\{0\\}}=\Gamma_{S^{2}\times\\{1\\}}$. Hence we also view
$\tilde{\Gamma}$ and $\tilde{\Gamma}^{\prime}$ as dividing sets on
$S^{2}\times\\{1\\}$. Choose pairwise disjoint admissible arcs
$\alpha_{1},\alpha_{2},\cdots,\alpha_{r},\alpha_{r+1},\cdots,\alpha_{k}$ on
$S^{2}\times\\{1\\}$ such that the following conditions hold:
1. (1)
$\alpha_{1},\alpha_{2},\cdots,\alpha_{r-1}$ are admissible arcs contained in
$D^{2}_{\epsilon}$ such that by attaching bypasses along these arcs, the depth
of $\tilde{\Gamma}$ becomes at most 2.
2. (2)
$\alpha_{r},\alpha_{r+1},\cdots,\alpha_{k}$ are admissible arcs contained in
the disk bounded by $\gamma$ and outside of $A(\gamma)$ such that by attaching
bypasses along these arcs, the depth of $\tilde{\Gamma}^{\prime}$ becomes at
most 2.
Observe that we choose $\alpha_{1},\alpha_{2},\cdots,\alpha_{k}$ such that the
isotopy class of each $\alpha_{i}$ is invariant under the time-1 map
$\phi_{1}$ which is supported in $A(\gamma)\setminus D^{2}_{\epsilon}$. Hence,
by abuse of notation, we do not distinguish $\alpha_{i}$ and its push-down
through $\phi_{t}(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)$. By Lemma 6.3, we
have the isotopy of contact structures
$\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}\ast\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{k}}\simeq\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{k}}\ast\xi_{\Phi}$,
where $\xi_{\Phi}$ is the contact structure induced by a finite composition of
special pure braids of the dividing set considered in Lemma 6.12 and Lemma
6.13, Therefore $\xi_{\Phi}$ is stable isotopic to a power of the bypass
triangle attachment, say $\triangle^{l}$ for some $l\in\mathbb{N}$. To
summarize, we have the following (stable) isotopies of contact structures,
relative to the boundary.
$\displaystyle\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}\ast\triangle^{k}$
$\displaystyle\simeq\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}\ast\triangle_{\alpha_{1}}\ast\cdots\ast\triangle_{\alpha_{k}}$
$\displaystyle=\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}\ast(\sigma_{\alpha_{1}}\ast\sigma_{\alpha^{\prime}_{1}}\ast\sigma_{\alpha^{\prime\prime}_{1}})\ast\cdots\ast(\sigma_{\alpha_{k}}\ast\sigma_{\alpha^{\prime}_{k}}\ast\sigma_{\alpha^{\prime\prime}_{k}})$
$\displaystyle\simeq(\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}\ast\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{k}})\ast(\sigma_{\alpha^{\prime}_{1}}\ast\sigma_{\alpha^{\prime\prime}_{1}})\ast\cdots\ast(\sigma_{\alpha^{\prime}_{k}}\ast\sigma_{\alpha^{\prime\prime}_{k}})$
$\displaystyle\simeq(\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{k}}\ast\xi_{\Phi})\ast(\sigma_{\alpha^{\prime}_{1}}\ast\sigma_{\alpha^{\prime\prime}_{1}})\ast\cdots\ast(\sigma_{\alpha^{\prime}_{k}}\ast\sigma_{\alpha^{\prime\prime}_{k}})$
$\displaystyle\sim(\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{k}}\ast\triangle^{l})\ast(\sigma_{\alpha^{\prime}_{1}}\ast\sigma_{\alpha^{\prime\prime}_{1}})\ast\cdots\ast(\sigma_{\alpha^{\prime}_{k}}\ast\sigma_{\alpha^{\prime\prime}_{k}})$
$\displaystyle\simeq\triangle^{l}\ast(\sigma_{\alpha_{1}}\ast\sigma_{\alpha^{\prime}_{1}}\ast\sigma_{\alpha^{\prime\prime}_{1}})\ast\cdots\ast(\sigma_{\alpha_{k}}\ast\sigma_{\alpha^{\prime}_{k}}\ast\sigma_{\alpha^{\prime\prime}_{k}})$
$\displaystyle=\triangle^{l}\ast\triangle^{k}.\qed$
Hence $\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ is stably isotopic to
$\triangle^{l}$ by definition.
To conclude this section, we prove the following technical result which
asserts that under certain assumptions and up to possible bypass triangle
attachments, one can separate two bypasses.
###### Proposition 6.15.
Let $(S^{2},\Gamma)$ be a convex sphere with dividing set $\Gamma$ and
$\alpha\subset(S^{2},\Gamma)$ be an admissible arc such that the bypass
attachment $\sigma_{\alpha}$ increases $\\#\Gamma$ by 2. Suppose that
$(S^{2},\Gamma^{\prime})$ is the new convex sphere obtained by attaching
$\sigma_{\alpha}$ to $(S^{2},\Gamma)$ and suppose
$\beta\subset(S^{2},\Gamma^{\prime})$ is another admissible arc such that the
bypass attachment $\sigma_{\beta}$ decreases $\\#\Gamma^{\prime}$ by 2. Then
there exists an admissible arc $\tilde{\beta}\subset(S^{2},\Gamma)$ disjoint
from $\alpha$, a map $\Phi:S^{2}\times[0,1]\to S^{2}\times[0,1]$ induced by an
isotopy, and an integer $l\in\mathbb{N}$ such that
$\sigma_{\alpha}\ast\sigma_{\beta}\sim\sigma_{\alpha}\ast\sigma_{\tilde{\beta}}\ast\triangle^{l}\ast\xi_{\Phi}$
relative to the boundary.
###### Proof.
Let $\delta$ be the arc of anti-bypass attachment to $\sigma_{\alpha}$
contained in $(S^{2},\Gamma^{\prime})$ as discussed in Remark 5.3. Then
$\delta$ intersects $\Gamma^{\prime}$ in three points
$\\{p_{1},p_{2},p_{3}\\}$ as depicted in Figure 19(b). Let $\delta_{1}$ and
$\delta_{2}$ be subarcs of $\delta$ from $p_{1}$ to $p_{2}$ and from $p_{2}$
to $p_{3}$ respectively. Observe that, in order to find an admissible arc
$\tilde{\beta}\subset(S^{2},\Gamma)$ which is disjoint from $\alpha$ and
satisfy all the conditions in the lemma, it suffices to find an admissible arc
on $(S^{2},\Gamma^{\prime})$, which we still denote by $\tilde{\beta}$, and
which is disjoint from $\delta$ and also satisfies the conditions in the
lemma. In fact, by symmetry, we only need $\tilde{\beta}$ to be disjoint from
$\delta_{1}$. Without loss of generality, we can assume that $\beta$
intersects $\delta$ transversely and the intersection points are different
from $p_{1}$, $p_{2}$ and $p_{3}$.
\begin{overpic}[scale={.35}]{antibypass.eps} \put(34.3,31.0){\tiny{$\alpha$}}
\put(85.0,23.0){\tiny{$\delta$}} \put(73.7,27.0){\tiny{$p_{1}$}}
\put(80.0,22.0){\tiny{$p_{2}$}} \put(88.0,17.9){\tiny{$p_{3}$}}
\put(11.0,-5.0){(a)} \put(83.0,-5.0){(b)} \end{overpic}
Figure 19. (a) The convex sphere $(S^{2},\Gamma)$ with an admissible arc
$\alpha$. (b) The convex sphere $(S^{2},\Gamma^{\prime})$ obtained by
attaching a bypass along $\alpha$, where $\delta$ is the arc of the anti-
bypass attachment.
_Claim: Up to isotopy and possibly a finite number of bypass triangle
attachments, one can arrange so that $\beta$ and $\delta_{1}$ do not cobound a
bigon $B$ on $S^{2}$ as depicted in Figure 20(a)_.
\begin{overpic}[scale={.23}]{cancelbigon.eps}
\put(8.5,11.5){\tiny{$\delta_{1}$}} \put(13.5,16.5){\tiny{$\beta$}}
\put(38.5,16.0){\tiny{$\gamma$}} \put(79.3,13.0){\tiny{$\tilde{\beta}$}}
\put(9.0,-5.0){(a)} \put(48.0,-5.0){(b)} \put(88.0,-5.0){(c)} \end{overpic}
Figure 20. (a) The admissible arc $\beta$ together with $\delta_{1}$ bound a
minimal bigon, which contains other components of the dividing set in the
interior. (b) Choose a disk $D^{2}_{\epsilon}$ containing all the dividing
sets $\tilde{\Gamma}$ in the bigon and an oriented loop $\gamma$ so that it
intersects $\beta$ in exactly one point. (c) The pull-up of $\beta$ through
the contact structure $\xi_{\tilde{\Gamma},D^{2}_{\epsilon},\gamma}$ bounds a
trivial bigon with $\delta_{1}$.
To verify the claim, note that if $B$ is a trivial bigon, i.e., it contains no
component of the dividing set in the interior, then we can easily isotop
$\beta$ to eliminate $B$. If otherwise, we consider a minimal bigon bounded by
$\beta$ and $\delta_{1}$ in the sense that the interior of the bigon does not
intersect with $\beta$. Take a disk $D^{2}_{\epsilon}\subset B$ containing all
components of the dividing set $\tilde{\Gamma}$ in $B$, namely,
$\Gamma^{\prime}\cap D^{2}_{\epsilon}=\tilde{\Gamma}$ and
$\Gamma^{\prime}\cap(B\setminus D^{2}_{\epsilon})=\emptyset$. By our
assumption, the bypass attachment $\sigma_{\beta}$ decreases
$\\#\Gamma^{\prime}$ by 2, so $\beta$ must intersect $\Gamma^{\prime}$ in
three points which are contained in three different connected components of
$\Gamma^{\prime}$ respectively. One can find an oriented loop $\gamma:[0,1]\to
S^{2}\setminus\Gamma^{\prime}$ with $\gamma(0)=\gamma(1)\in D^{2}_{\epsilon}$
such that $\gamma$ intersects $\beta$ in one point. Orient $\gamma$ in such a
way that it goes from $\gamma\cap\beta$ to $\gamma(1)$ in the interior of $B$
as depicted in Figure 20(b). Suppose that $\Phi:S^{2}\times[0,1]\to
S^{2}\times[0,1]$ is induced by an isotopy $\phi_{t}$ which parallel
transports $D^{2}_{\epsilon}$ along $\gamma$. By pulling up the the bypass
attachment $\sigma_{\beta}$ through $\xi_{\Gamma^{\prime},\Phi}$, we get the
following isotopy of contact structures (cf. proof of Lemma 6.13):
$\displaystyle\sigma_{\beta}\ast\xi_{\Gamma^{\prime\prime},\Phi(D^{2}_{\epsilon},\gamma)}\simeq\xi_{\Gamma^{\prime},\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\tilde{\beta}}$
where $\Gamma^{\prime\prime}$ is obtained from $\Gamma^{\prime}$ by attaching
a bypass along $\beta$, and $\tilde{\beta}$ is the pull-up of $\beta$ which is
isotopic to the one depicted in Figure 20(c).
Since $\tilde{\beta}$ and $\delta_{1}$ cobound a trivial bigon, a further
isotopy of $\tilde{\beta}$ will eliminate the bigon so that $\beta^{\prime}$
does not intersect $\delta_{1}$ in this local picture. By Proposition 6.14,
the contact structure
$\xi_{\Gamma^{\prime},\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}$ is stably
isotopic to $\triangle^{n}$ for some $n\in\mathbb{N}$. Define
$\Phi^{-1}:S^{2}\times[0,1]\to S^{2}\times[0,1]$ by
$(x,t)\mapsto(\phi^{-1}_{t}(x),t)$, then it is easy to see that
$\xi_{\Gamma^{\prime\prime},\Phi(D^{2}_{\epsilon},\gamma)}\ast\xi_{\Gamma^{\prime\prime},\Phi^{-1}(D^{2}_{\epsilon},\gamma)}$
is isotopic, relative to the boundary, to an $I$-invariant contact structure.
Since we will use this trick many times, we simply write $\xi_{\Phi^{-1}}$ for
$\xi_{\Gamma^{\prime\prime},\Phi^{-1}(D^{2}_{\epsilon},\gamma)}$ when there is
no confusion. To summarize, we have
$\displaystyle\sigma_{\beta}$
$\displaystyle\simeq\xi_{\Gamma^{\prime},\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Gamma^{\prime\prime},\Phi^{-1}(D^{2}_{\epsilon},\gamma)}$
$\displaystyle\sim\triangle^{n}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Gamma^{\prime\prime},\Phi^{-1}(D^{2}_{\epsilon},\gamma)}$
$\displaystyle\simeq\sigma_{\tilde{\beta}}\ast\triangle^{n}\ast\xi_{\Gamma^{\prime\prime},\Phi^{-1}(D^{2}_{\epsilon},\gamma)}$
By applying the above argument finitely many times, we can eliminate all
bigons bounded by $\beta$ and $\delta_{1}$. Hence the claim is proved.
Let us assume that $\beta$ intersects $\delta_{1}$ nontrivially, and $\beta$
and $\delta_{1}$ do not cobound any bigon on $S^{2}$. We consider the
following two cases separately.
_Case 1._ Suppose $\beta$ does not intersect any of the three components of
the dividing set generated by the bypass attachment $\sigma_{\alpha}$. Let
$\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ be the three dividing circles
which intersect with $\beta$.
\begin{overpic}[scale={.26}]{cancelintersection1.eps}
\put(-1.5,12.0){\tiny{$\Gamma_{1}$}} \put(17.1,19.0){\tiny{$\Gamma_{2}$}}
\put(19.0,21.0){\tiny{$\Gamma_{3}$}} \put(11.0,11.5){\tiny{$\beta$}}
\put(43.0,20.0){\tiny{$\gamma$}} \put(79.0,20.0){\tiny{$\tilde{\beta}$}}
\put(11.0,-4.5){(a)} \put(49.0,-4.5){(b)} \put(87.0,-4.5){(c)} \end{overpic}
Figure 21. (a) The convex sphere $(S^{2},\Gamma^{\prime})$ with an admissible
arc $\beta$ intersecting $\delta_{1}$ in exactly one point. (b) Choose a disk
$D^{2}_{\epsilon}$ containing $\Gamma_{1}$ and an oriented loop $\gamma$,
along which we apply the isotopy. (c) The pull-up of $\beta$ through the
contact structure $\xi_{\Gamma_{1},D^{2}_{\epsilon},\gamma}$ bounds a trivial
bigon with $\delta_{1}$.
If $\beta$ intersects $\delta_{1}$ in exactly one point as depicted in Figure
21(a), then we choose a disk $D^{2}_{\epsilon}\supset\Gamma_{1}$ and an
oriented loop $\gamma$ in the complement of the dividing set as depicted in
Figure 21(b) such that
$\sigma_{\beta}\simeq\xi_{\Gamma^{\prime},\Phi(\Gamma_{1},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Phi^{-1}}\sim\triangle^{m}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Phi^{-1}}$
by arguments as before for some $m\in\mathbb{N}$, where $\tilde{\beta}$
intersects $\delta^{1}$ in exactly two points and cobound a trivial bigon as
depicted in Figure 21(c). Hence an obvious further isotopy of $\tilde{\beta}$
makes it disjoint from $\delta_{1}$ as desired.
If $\beta$ intersects $\delta_{1}$ in more than one point, we orient $\beta$
so that it starts from the point $q=\beta\cap\Gamma_{1}$ as depicted in Figure
22(a). Let $q_{1}$ and $q_{2}$ be the first and the second intersection points
of $\beta$ with $\delta_{1}$ respectively. Note that since we assume $\beta$
and $\delta_{1}$ do not cobound any bigon, there is no more intersection point
$\beta\cap\delta_{1}$ along $\delta_{1}$ between $q_{1}$ and $q_{2}$. Let
$\overrightarrow{qq_{1}}$, $\overrightarrow{q_{1}q}$ and
$\overrightarrow{q_{1}q_{2}}$ be oriented subarcs of $\beta$ and
$\overrightarrow{q_{2}q_{1}}$ be an oriented subarc of $\delta_{1}$. We obtain
a closed, oriented (but not embedded) loop
$\gamma=\overrightarrow{qq_{1}}\cup\overrightarrow{q_{1}q_{2}}\cup\overrightarrow{q_{2}q_{1}}\cup\overrightarrow{q_{1}q}$
by gluing the arcs together. To apply Proposition 6.14 in this case, we take
an embedded loop close to $\gamma$ as depicted in Figure 22(b), which we still
denote by $\gamma$. Let $D^{2}_{\epsilon}$ be a small disk containing
$\Gamma_{1}$ as usual. Again by pulling up the bypass attachment
$\sigma_{\beta}$ through
$\xi_{\Gamma^{\prime},\Phi(\Gamma_{1},D^{2}_{\epsilon},\gamma)}$, we have
(stable) isotopies of contact structures
$\sigma_{\beta}\simeq\xi_{\Gamma^{\prime},\Phi(\Gamma_{1},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Phi^{-1}}\sim\triangle^{r}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Phi^{-1}}$
for some $r\in\mathbb{N}$, where $\tilde{\beta}$ and $\delta_{1}$ bound a
trivial bigon. Hence an obvious further isotopy eliminates the trivial bigon
and decreases $\\#(\beta\cap\delta_{1})$ by 2. By applying the above argument
finitely many times, we can reduce to the case where $\beta$ intersects
$\delta_{1}$ in exactly one point, but we have already solved the problem in
this case. We conclude that under the hypothesis at the beginning of this
case, there exists a $\tilde{\beta}$ disjoint with $\delta_{1}$ such that
$\sigma_{\alpha}\ast\sigma_{\beta}\sim\sigma_{\alpha}\ast\sigma_{\tilde{\beta}}\ast\triangle^{l}\ast\xi_{\Phi}$
for some isotopy $\Phi$ and an integer $l\in\mathbb{N}$.
\begin{overpic}[scale={.22}]{cancelintersection2.eps}
\put(1.4,11.5){\tiny{$\Gamma_{1}$}} \put(5.0,13.0){\tiny{$q$}}
\put(13.2,11.7){\tiny{$q_{1}$}} \put(12.8,14.5){\tiny{$q_{2}$}}
\put(25.0,5.0){\tiny{$\beta$}} \put(39.2,7.0){\tiny{$\gamma$}}
\put(100.0,5.0){\tiny{$\beta^{\prime}$}} \put(10.0,-4.5){(a)}
\put(48.0,-4.5){(b)} \put(86.0,-4.5){(c)} \end{overpic}
Figure 22. (a) The convex sphere $(S^{2},\Gamma^{\prime})$ with an admissible
arc $\beta$ intersecting $\delta_{1}$ in at least two points, say, $q_{1}$ and
$q_{2}$. (b) The embedded, oriented loop $\gamma$ approximating the broken
loop $\vec{qq_{1}}\cup\vec{q_{1}q_{2}}\cup\vec{q_{2}q_{1}}\cup\vec{q_{1}q}$.
(c) The pull-up of $\beta$ through the contact structure
$\xi_{\Gamma_{1},D^{2}_{\epsilon},\gamma}$ bounds a trivial bigon with
$\delta_{1}$. \begin{overpic}[scale={.25}]{cancelintersection3.eps}
\put(8.5,20.0){\tiny{$r$}} \put(6.2,17.5){\tiny{$p_{1}$}}
\put(10.3,13.7){\tiny{$r_{1}$}} \put(38.6,18.0){\tiny{$\gamma$}}
\put(15.3,18.0){\tiny{$\beta$}} \put(88.7,15.0){\tiny{$\tilde{\beta}$}}
\put(10.0,-4.5){(a)} \put(47.0,-4.5){(b)} \put(86.0,-4.5){(c)} \end{overpic}
Figure 23. (a) The admissible arc $\beta$, the dividing set $\Gamma^{\prime}$
and $\delta_{1}$ cobound a topological triangle $\triangle rr_{1}p_{1}$, which
may contain other components of the dividing set in the interior. (b) Choose
the disk $D^{2}_{\epsilon}$ to contain all the components of the dividing set
in the topological triangle $\triangle rr_{1}p_{1}$, and an oriented loop
$\gamma$ which intersects $\beta$ in exactly one point. (c) By applying the
isotopy along $\gamma$, the admissible arc $\beta$ becomes $\beta^{\prime}$
which bounds a trivial triangle with the dividing set and $\delta_{1}$.
_Case 2._ Suppose $\beta$ nontrivially intersects the union of the three
components of the dividing set generated by the bypass attachment
$\sigma_{\alpha}$. Without loss of generality, we pick an intersection point
$r$ as depicted in Figure 23(a). Orient $\beta$ so that it starts from $r$.
Let $r_{1}$ be the first intersection point of $\beta$ and $\delta_{1}$. Then
$\beta$, $\delta_{1}$ and $\Gamma^{\prime}$ bound a triangle $\triangle
rr_{1}p_{1}$. By the assumption that there exists no bigon bounded by $\beta$
and $\delta_{1}$, the interior of the triangle $\triangle rr_{1}p_{1}$ does
not intersect with $\beta$. If the interior of the triangle $\triangle
rr_{1}p_{1}$ contains no components of the dividing set, then it is easy to
isotop $\beta$ so that $\\#(\beta\cap\delta_{1})$ decreases by 1. If
otherwise, take a small disk $D^{2}_{\epsilon}\subset\triangle rr_{1}p_{1}$
containing all components of the dividing set $\tilde{\Gamma}$ in $\triangle
rr_{1}p_{1}$, i.e., $\triangle rr_{1}p_{1}\setminus D^{2}_{\epsilon}$ does not
intersect with the dividing set $\Gamma^{\prime}$. Let $\gamma$ be an oriented
loop based at a point in $D^{2}_{\epsilon}$ which does not intersect with the
dividing set, and intersects $\beta$ exactly once. By pulling up the bypass
attachment $\sigma_{\beta}$ through
$\xi_{\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}$, we have (stable)
isotopies of contact structures
$\sigma_{\beta}\simeq\xi_{\Gamma^{\prime},\Phi(\tilde{\Gamma},D^{2}_{\epsilon},\gamma)}\ast\sigma_{\tilde{\beta}}\ast\xi_{\Phi^{-1}}\sim\sigma_{\tilde{\beta}}\ast\triangle^{n}\ast\xi_{\Phi^{-1}}$
so that $\tilde{\beta}$, $\delta_{1}$ and $\Gamma^{\prime}$ bound a trivial
triangle in the sense that the interior of the triangle does not intersect
with the dividing set. Hence we can further isotop $\tilde{\beta}$ to
eliminate the trivial triangle and hence decrease
$\\#(\tilde{\beta}\cap\delta_{1})$ by 1. By applying such isotopies finitely
many times, we get an admissible arc $\tilde{\beta}$ such that
$\\#(\tilde{\beta}\cap\delta_{1})=0$ and satisfy all the conditions of the
proposition. ∎
## 7\. Classification of overtwisted contact structures on $S^{2}\times[0,1]$
We have established enough techniques to classify overtwisted contact
structures on $S^{2}\times[0,1]$.
###### Proposition 7.1.
Let $\xi$ be an overtwisted contact structure on $S^{2}\times[0,1]$ such that
$S^{2}\times\\{0,1\\}$ is convex with
$\Gamma_{S^{2}\times\\{0\\}}=\Gamma_{S^{2}\times\\{1\\}}=S^{1}$. Then
$\xi\sim\triangle^{n}$ for some $n\in\mathbb{N}$, where $\triangle^{n}$
denotes the contact structure on $S^{2}\times[0,1]$ obtained by attaching $n$
bypass triangles to $S^{2}\times\\{0\\}$ with the standard tight neighborhood.
###### Proof.
By Giroux’s criterion of tightness, both $S^{2}\times\\{0\\}$ and
$S^{2}\times\\{1\\}$ have neighborhoods which are tight. Take an increasing
sequence $0=t_{0}<t_{1}<\cdots<t_{n}=1$ such that $\xi$ is isotopic to a
sequence of bypass attachments
$\sigma_{\alpha_{0}}\ast\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{n-1}}$,
where $\alpha_{i}\subset S^{2}\times\\{t_{i}\\}$ are admissible arcs along
which a bypass is attached. Define the complexity of a bypass sequence to be
$c=\max_{0\leq i\leq n}\\#\Gamma_{S^{2}\times\\{t_{i}\\}}$. The idea is to
show that if $c>3$, then we can always decrease $c$ by 2 by isotoping the
bypass sequence and suitably attaching bypass triangles.
To achieve this goal, we divide the admissible arcs on $(S^{2},\Gamma)$ into
four types (I), (II), (III) and (IV), according to the number of components of
$\Gamma$ intersecting the admissible arc as depicted in Figure 24, where we
only draw the dividing set which intersects the admissible arc. Observe that
bypass attachment of type (I) increases $\\#\Gamma$ by 2, bypass attachment of
type (II) and (III) do not change $\\#\Gamma$, and bypass attachment of type
(IV) decreases $\\#\Gamma$ by 2. Hence the complexity of a sequence of bypass
attachments changes only if the types of bypasses in the sequence change. By
repeated application of Lemma 6.3, we may assume that contact structures
induced by isotopies are contained in a neighborhood of $S^{2}\times\\{1\\}$.
By assumption, $S^{2}\times\\{1\\}$ has a tight neighborhood. Hence according
to Remark 5.4, we shall only consider sequences of bypass attachments modulo
contact structures induced by isotopies.
\begin{overpic}[scale={.26}]{Types.eps} \put(15.5,13.0){\tiny{$\alpha$}}
\put(33.0,14.0){\tiny{$\alpha$}} \put(60.0,8.0){\tiny{$\alpha$}}
\put(85.0,8.0){\tiny{$\alpha$}} \put(5.0,-4.0){(I)} \put(28.0,-4.0){(II)}
\put(55.5,-4.0){(III)} \put(86.0,-4.0){(IV)} \end{overpic}
Figure 24. Four types of admissible arcs $\alpha$ on $(S^{2},\Gamma)$.
Claim 1: We can isotop the sequence of bypass attachments such that only
bypasses of type (I) and (IV) appear.
To prove the claim, we first show that a bypass attachment of type (III) can
be eliminated. Take an admissible arc $\alpha$ of type (III). If the bypass
attachment along $\alpha$ is trivial, then by Lemma 3.3, the bypass attachment
$\sigma_{\alpha}$ is induced by an isotopy. Otherwise there exists an
admissible arc $\beta$ disjoint from $\alpha$ as depicted in Figure 25(a)666In
literature, we say $\beta$ is obtained from $\alpha$ by left rotation. such
that if one attaches a bypass along $\alpha$, followed by a bypass attached
along $\beta$, then the later bypass attachment is trivial.
\begin{overpic}[scale={.25}]{ElimTyps.eps} \put(12.0,5.0){\tiny{$\alpha$}}
\put(7.0,11.0){\tiny{$\beta$}} \put(43.0,5.0){\tiny{$\alpha$}}
\put(23.0,9.0){\footnotesize{$\sigma_{\beta}$}}
\put(66.5,5.0){\tiny{$\alpha$}} \put(73.0,13.0){\tiny{$\beta$}}
\put(92.0,5.0){\tiny{$\alpha$}}
\put(78.0,9.0){\footnotesize{$\sigma_{\beta}$}} \put(23.0,-4.0){(a)}
\put(78.0,-4.0){(b)} \end{overpic}
Figure 25.
By the disjointness of admissible arcs $\alpha$ and $\beta$, we get the
following isotopies of contact structures,
$\displaystyle\sigma_{\alpha}$
$\displaystyle\simeq\sigma_{\alpha}\ast\sigma_{\beta}$
$\displaystyle\simeq\sigma_{\beta}\ast\sigma_{\alpha}.$
Observe that $\sigma_{\beta}\ast\sigma_{\alpha}$ is a composition of type (I)
and type (IV) bypass attachments. Hence a finite number of such isotopies will
eliminate all bypass attachments of type (III) in a sequence.
Similarly suppose that $\sigma_{\alpha}$ is the bypass attachment of type (II)
in a sequence and is nontrivial. Then there must exist other components of the
dividing set as shown in Figure 25(b). Choose an admissible arc $\beta$
disjoint from $\alpha$ as depicted in Figure 25(b) such that if one attaches a
bypass along $\alpha$, followed by a bypass attached along $\beta$, then the
later bypass attachment is trivial. By the disjointness of $\alpha$ and
$\beta$ again, we get the following isotopies of contact structures:
$\displaystyle\sigma_{\alpha}$
$\displaystyle\simeq\sigma_{\alpha}\ast\sigma_{\beta}$
$\displaystyle\simeq\sigma_{\beta}\ast\sigma_{\alpha}.$
Observe that $\sigma_{\beta}\ast\sigma_{\alpha}$ is a composition of bypass
attachments both of type (III), hence by a further isotopy will turn
$\sigma_{\alpha}$ into a composition of bypass attachments of type (I) and
(IV). A finite number of such isotopies will eliminate bypasses of type (II).
The claim follows.
From now on, we assume that any bypass attachment in
$\sigma_{\alpha_{0}}\ast\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{n-1}}$
either increases or decreases $\\#\Gamma$ by 2.
Assume that the complexity of the bypass sequence is achieved at level
$S^{2}\times\\{t_{r}\\}$ for some $r\in\\{0,1,\cdots,n\\}$ and is at least 5,
i.e., $\\#\Gamma_{S^{2}\times\\{t_{r}\\}}=c\geq 5$. Then it is easy to see
that $\sigma_{\alpha_{r-1}}$ is type (I) and $\sigma_{\alpha_{r}}$ is type
(IV). By Proposition 6.15, we can always assume that $\alpha_{r}$ is disjoint
from $\alpha_{r-1}$ modulo finitely many bypass triangle attachments. Hence we
can view both $\alpha_{r-1}$ and $\alpha_{r}$ as admissible arcs on
$S^{2}\times\\{t_{r-1}\\}$. To finish the proof of the proposition, it
suffices to prove the following claim.
Claim 2: We can isotop the composition of bypass attachments
$\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}$ such that the local maximum of
$\\#\Gamma$ at $S^{2}\times\\{t_{r}\\}$ decreases by at least 2.
To prove the claim, let $\gamma\subset\Gamma_{S^{2}\times\\{t_{r-1}\\}}$ be
the dividing circle which nontrivially intersects $\alpha_{r-1}$. We do a
case-by-case analysis depending on the number of points $\alpha_{r}$
intersecting with $\gamma$.
Case 1: If $\alpha_{r}$ intersects $\gamma$ in at most one point, then one
easily check that by applying isotopy
$\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\simeq\sigma_{\alpha_{r}}\ast\sigma_{\alpha_{r-1}}$
to the sequence of bypass attachments, $\\#\Gamma_{S^{2}\times\\{t_{r}\\}}$
decreases by 4.
Case 2: If $\alpha_{r}$ intersects $\gamma$ in exactly two points, then once
again we apply the isotopy
$\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\simeq\sigma_{\alpha_{r}}\ast\sigma_{\alpha_{r-1}}$
to the sequence of bypass attachments. Now observe that
$\sigma_{\alpha_{r}}\ast\sigma_{\alpha_{r-1}}$ is a composition of bypass
attachments of type (III). In the proof of the claim above, we see that any
bypass attachment of type (III) is isotopic to a composition of a bypass
attachment of type (IV) followed by a bypass attachment of type (I). Such an
isotopy also decreases the local maximum of $\\#\Gamma$ by 4.
Case 3: If $\alpha_{r}$ also intersects $\gamma$ in three points, we consider
a disk $D$ bounded by $\gamma$ and $\alpha_{r-1}$ as depicted in Figure 26(a).
If $D$ contains no component of the dividing set in the interior, then
$\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}$ is isotopic to a bypass
triangle attachment, more precisely, there exists a trivial bypass along an
admissible arc $\delta$ on $S^{2}\times\\{t_{r}\\}$ such that
$\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\ast\sigma_{\delta}$ is a bypass
triangle attachment along $\alpha_{r-1}$. Suppose $D$ contains at least one
connected component of the dividing set. Let $\beta$ be an admissible arc on
$S^{2}\times\\{t_{r-1}\\}$ disjoint from $\alpha_{r-1}$ and $\alpha_{r}$ such
that it intersects $\gamma$ in two points and the dividing set contained in
$D$ in one point as depicted in Figure 26(b).
\begin{overpic}[scale={.3}]{Case3.eps} \put(29.0,22.0){\tiny{$\alpha_{r-1}$}}
\put(92.5,22.0){\tiny{$\alpha_{r-1}$}} \put(36.0,24.0){\tiny{$\alpha_{r}$}}
\put(99.0,24.0){\tiny{$\alpha_{r}$}} \put(23.0,17.0){\small{$D$}}
\put(0.0,3.0){\tiny{$\gamma$}} \put(63.5,3.0){\tiny{$\gamma$}}
\put(80.5,11.0){\tiny{$\beta$}} \put(17.0,-5.0){(a)} \put(80.0,-5.0){(b)}
\end{overpic}
Figure 26.
We have the following isotopies of contact structures due to Lemma 5.9 and the
disjointness of admissible arcs:
$\displaystyle\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\ast\triangle$
$\displaystyle\simeq\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\ast\triangle_{\beta}$
$\displaystyle=\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\ast\sigma_{\beta}\ast\sigma_{\beta^{\prime}}\ast\sigma_{\beta^{\prime\prime}}$
$\displaystyle\simeq\sigma_{\beta}\ast\sigma_{\alpha_{r-1}}\ast\sigma_{\alpha_{r}}\ast\sigma_{\beta^{\prime}}\ast\sigma_{\beta^{\prime\prime}}$
One can check that the last five bypass attachments above are all of type
(III). Hence we can further isotop as before to eliminate type (III) bypass
attachments to decrease the local maximum of $\\#\Gamma$ by 2.
To summarize, we have proved that any sequence of bypass attachments
$\sigma_{\alpha_{0}}\ast\sigma_{\alpha_{1}}\ast\cdots\ast\sigma_{\alpha_{n-1}}$
on $S^{2}\times[0,1]$ is stably isotopic to another sequence of bypass
attachments whose complexity is at most 3, which is clearly isotopic to a
power of bypass triangle attachments. Thus the proposition is proved. ∎
## 8\. Proof of the main theorem
Now we are ready to finish the proof of Theorem 0.2.
Proof of Theorem 0.2. By Proposition 4.3, we can isotop $\xi$ and
$\xi^{\prime}$ so that they agree in a neighborhood of the 2-skeleton. Without
loss of generality, we can furthermore assume that there exists an embedded
closed ball $B^{3}\subset M$ such that
1. (1)
$\partial B^{3}$ is convex and has a tight neighborhood in $M$ with respect to
both $\xi$ and $\xi^{\prime}$.
2. (2)
$\xi=\xi^{\prime}$ in $M\setminus B^{3}$.
3. (3)
The restriction of $\xi$ and $\xi^{\prime}$ to $M\setminus B^{3}$ and to
$B^{3}$ are all overtwisted.
Take a small ball $B_{\epsilon}^{3}\subset B^{3}$ in a Darboux chart so that
both $\xi|_{B_{\epsilon}^{3}}$ and $\xi^{\prime}|_{B_{\epsilon}^{3}}$ are
tight. We identify $B^{3}\setminus B_{\epsilon}^{3}$ with $S^{2}\times[0,1]$
and represent the contact structures $\xi|_{B^{3}\setminus B_{\epsilon}^{3}}$
and $\xi^{\prime}|_{B^{3}\setminus B_{\epsilon}^{3}}$ by two sequences of
bypass attachments. By Proposition 7.1, both $\xi|_{B^{3}\setminus
B_{\epsilon}^{3}}$ and $\xi^{\prime}|_{B^{3}\setminus B_{\epsilon}^{3}}$ are
stably isotopic to some power of the bypass triangle attachment, in other
words, there are isotopies of contact structures $\xi|_{B^{3}\setminus
B_{\epsilon}^{3}}\ast\triangle^{r}\simeq\triangle^{n+r}$ and
$\xi^{\prime}|_{B^{3}\setminus
B_{\epsilon}^{3}}\ast\triangle^{s}\simeq\triangle^{m+s}$ for some
$n,m,r,s\in\mathbb{N}$. By assumption, the restriction of $\xi$ and
$\xi^{\prime}$ to $M\setminus B^{3}$ are overtwisted, so there exist bypass
triangle attachments along any admissible arc on $\partial B^{3}$ according to
Lemma 3.1. By simultaneously attaching sufficiently many bypass triangles to
$\xi|_{B^{3}\setminus B_{\epsilon}^{3}}$ and $\xi^{\prime}|_{B^{3}\setminus
B_{\epsilon}^{3}}$, we can further assume that $\xi|_{B^{3}\setminus
B_{\epsilon}^{3}}\simeq\triangle^{n}$, $\xi^{\prime}|_{B^{3}\setminus
B_{\epsilon}^{3}}\simeq\triangle^{m}$ and $\xi=\xi^{\prime}$ on $M\setminus
B^{3}$.
Let $d$ be the largest integer such that the Euler class
$e(\xi)=e(\xi^{\prime})\in H^{2}(M;\mathbb{Z})$ divided by $d$ is still an
integral class. Such a $d$ is known as the divisibility of the Euler class.
Combining Proposition 2.11 and Theorem 0.5 in [11], we have $d|(m-n)$. To
complete the proof of the theorem, we need to show that $\xi|_{M\setminus
B^{3}}$ is isotopic to $\xi|_{M\setminus B^{3}}\ast\triangle^{d}$ relative to
the boundary. Since $d=g.c.d.\\{e(\Sigma)|\Sigma\in H_{2}(M)\\}$, it suffices
to prove the following more general fact.
###### Lemma 8.1.
Let $\Sigma$ be a closed surface of genus $g$ and $\eta$ be an $I$-invariant
contact structure on $\Sigma\times[0,1]$. Then $\eta\ast\triangle^{l}$ is
stably isotopic to $\eta$ relative to the boundary, where $l=e(\eta)(\Sigma)$.
###### Proof.
Since we only consider stable isotopies of contact structures, one can
prescribe any dividing set $\Gamma_{\Sigma}$ on $\Sigma$ such that the Euler
class evaluates on $\Sigma$ to $l$. In particular, we consider the dividing
set on $\Sigma$ as depicted in Figure 27, namely, there are $g+1$ circles
$\gamma_{1}\cup\cdots\cup\gamma_{g+1}$ dividing $\Sigma$ into two punctured
disks, in each of which there are $p$ and $q$ isolated circles respectively.
We call the left most circles in the sets of $p$ and $q$ isolated circles
$\Gamma_{0}$ and $\Gamma_{1}$ respectively. We also choose admissible arcs
$\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{p-1}\\}$ and
$\\{\beta_{1},\beta_{2},\cdots,\beta_{q-1}\\}$, and orient $\gamma_{i}$,
$1\leq i\leq g+1$, in a way as depicted in Figure 27.
\begin{overpic}[scale={.35}]{Torsion.eps} \put(8.0,14.0){\tiny{$\gamma_{1}$}}
\put(34.5,14.0){\tiny{$\gamma_{2}$}} \put(88.5,13.5){\tiny{$\gamma_{g+1}$}}
\put(28.0,8.0){\tiny{$\alpha_{1}$}} \put(38.0,6.0){\tiny{$\alpha_{2}$}}
\put(66.0,3.0){\tiny{$\alpha_{p-1}$}} \put(57.0,8.0){$\dots$}
\put(36.5,26.0){\tiny{$\beta_{1}$}} \put(62.5,32.2){\tiny{$\beta_{q-1}$}}
\put(53.0,27.0){$\dots$} \put(60.5,17.5){$\dots$} \put(12.0,8.0){\tiny{$-$}}
\put(12.0,24.0){\tiny{$+$}} \put(22.0,7.3){\tiny{$+$}}
\put(33.3,8.2){\tiny{$+$}} \put(44.7,8.0){\tiny{$+$}}
\put(73.2,8.0){\tiny{$+$}} \put(29.8,26.5){\tiny{$-$}}
\put(42.6,26.3){\tiny{$-$}} \put(68.0,26.3){\tiny{$-$}}
\put(97.0,30.0){$\Sigma$} \put(17.0,10.4){\tiny{$\Gamma_{0}$}}
\put(25.0,30.4){\tiny{$\Gamma_{1}$}} \end{overpic} Figure 27.
An easy calculation shows that $l=2(p-q)$. Choose small disks
$D^{2}_{\epsilon,0}$, $D^{2}_{\epsilon,1}$ in $\Sigma$ such that
$D^{2}_{\epsilon,0}\cap\Gamma_{\Sigma}=\Gamma_{0}$ and
$D^{2}_{\epsilon,1}\cap\Gamma_{\Sigma}=\Gamma_{1}$. Observe that the bypass
triangle attachment along any $\alpha_{i}$ and $\beta_{j}$ consists of three
trivial bypass attachments, hence is isotopic to contact structures induced by
a pure braid of the dividing set. More precisely, let $\gamma^{-}_{i}$,
$i=1,2,\cdots,g+1$, be an oriented loop in the negative region which is
parallel to $\gamma_{i}$. We have the following isotopies of contact
structures
$\triangle_{\alpha_{1}}^{2}\ast\cdots\ast\triangle_{\alpha_{p-1}}^{2}\simeq\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma^{-}_{1}\cup\cdots\cup\gamma^{-}_{g+1})}\simeq\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma^{-}_{1})}\ast\cdots\ast\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma^{-}_{g+1})}$,
where we think of $\gamma^{-}_{1}\cup\cdots\cup\gamma^{-}_{g+1}$ as an
oriented loop homologous to the union of the $\gamma_{i}$’s. Similarly one can
study the bypass triangle attachments along the $\beta_{j}$’s, but with an
opposite orientation. Let $\gamma^{+}_{i}$ be an oriented loop in the positive
region which is parallel to $\gamma_{i}$ for $1\leq i\leq g+1$. We have the
following (stable) isotopies of contact structures
$\triangle_{\beta_{1}}^{-2}\ast\cdots\ast\triangle_{\beta_{q-1}}^{-2}\sim\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma^{+}_{1}\cup\cdots\cup\gamma^{+}_{g+1})}\simeq\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma^{+}_{1})}\ast\cdots\ast\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma^{+}_{g+1})}$.
Here we only have a stable isotopy because of our choice of the orientation of
$\gamma_{i}$. To summarize the computations above, we get the following
(stable) isotopies of contact structures:
$\displaystyle\eta\ast\triangle^{l}$
$\displaystyle\simeq\eta\ast(\triangle_{\alpha_{1}}^{2}\ast\cdots\ast\triangle_{\alpha_{p-1}}^{2})\ast(\triangle_{\beta_{1}}^{-2}\ast\cdots\ast\triangle_{\beta_{q-1}}^{-2})$
$\displaystyle\simeq\eta\ast(\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{1}^{-})}\ast\cdots\ast\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{g+1}^{-})})\ast(\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{1}^{+})}\ast\cdots\ast\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{g+1}^{+})})$
$\displaystyle\simeq\eta\ast(\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{1}^{-})}\ast\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{1}^{+})})\ast\cdots\ast(\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{g+1}^{-})}\ast\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{g+1}^{+})})$
where the last step follows from the fact that isotopies that parallel
transport $D^{2}_{\epsilon,0}$ and $D^{2}_{\epsilon,1}$ are disjoint.
Now it suffices to prove that
$\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{i}^{-})}\ast\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{i}^{+})}$
is stably isotopic to an $I$-invariant contact structure for $1\leq i\leq
g+1$. To see this, take an annular neighborhood $A_{i}$ of $\gamma_{i}$
containing $D^{2}_{\epsilon,0}$ and $D^{2}_{\epsilon,1}$ and an admissible arc
$\delta_{i}$ which intersects $\Gamma_{0}$, $\Gamma_{1}$, and $\gamma_{i}$ as
depicted in Figure 28. We can assume that the isotopies
$\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{i}^{-})$ and
$\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{i}^{+})$ are supported in $A_{i}$.
For simplicity of notation, we denote the composition
$\eta_{\Phi(\Gamma_{0},D^{2}_{\epsilon,0},\gamma_{i}^{-})}\ast\eta_{\Phi(\Gamma_{1},D^{2}_{\epsilon,1},\gamma_{i}^{+})}$
by $\eta_{\gamma_{i}}$.
\begin{overpic}[scale={.28}]{Annulus.eps} \put(10.0,34.0){\tiny{$\Gamma_{0}$}}
\put(31.0,34.0){\tiny{$\Gamma_{1}$}} \put(23.5,65.0){\tiny{$\gamma_{i}$}}
\put(18.43,47.5){\tiny{$\delta_{i}$}} \put(60.0,60.0){\tiny{$+$}}
\put(75.0,75.0){\tiny{$-$}} \put(10.0,44.0){\tiny{$+$}}
\put(31.0,44.0){\tiny{$-$}} \end{overpic} Figure 28. An annulus neighborhood
$A_{i}$ of $\gamma_{i}$ containing $\Gamma_{0}$ and $\Gamma_{1}$.
By pushing down the bypass attachment $\sigma_{\delta_{i}}$ through
$\eta_{\gamma_{i}}$, we have the following isotopies of contact structures:
$\displaystyle\eta_{\gamma_{i}}\ast\triangle_{\delta_{i}}$
$\displaystyle=\eta_{\gamma_{i}}\ast\sigma_{\delta_{i}}\ast\sigma_{\delta^{\prime}_{i}}\ast\sigma_{\delta^{\prime\prime}_{i}}$
$\displaystyle\simeq\sigma_{\tilde{\delta}_{i}}\ast\eta_{\Phi(\gamma_{i})}\ast\sigma_{\delta^{\prime}_{i}}\ast\sigma_{\delta^{\prime\prime}_{i}}$
$\displaystyle\simeq\sigma_{\delta_{i}}\ast\sigma_{\delta^{\prime}_{i}}\ast\sigma_{\delta^{\prime\prime}_{i}}=\triangle_{\delta_{i}}$
where $\tilde{\delta}_{i}$ is the push-down of $\delta_{i}$ which is isotopic
to $\delta_{i}$, and the $\eta_{\Phi(\gamma_{i})}$ is easily seen to be
isotopic to an $I$-invariant contact structure. The argument works for all
$i\in\\{1,2,\cdots,g+1\\}$, hence we establish the stable isotopy as desired.
∎
Acknowledgements. The author is very grateful to Ko Honda for inspiring
conversations throughout this work. The author also thank MSRI for providing
an excellent environment for mathematical research during the academic year
2009-2010.
## References
* [1] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque, 107-108 (1983), 87-161.
* [2] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623-637.
* [3] Y. Eliashberg, Contact 3-manifolds, twenty years since J. Martinet’s work, Ann. Inst. Fourier 42 (1992), 165-192.
* [4] Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Proceeding of Symposium Pure Math., vol.52, Amer. Math. Soc., Providence, RI, (1991), 165-192
* [5] J. Etnyre and K. Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003), 59-74.
* [6] M. Fraser, Classifying Legendrian knots in tight contact 3-manifolds, Ph.D. thesis, 1994.
* [7] H. Geiges, An Introduction to Contact Topology, Cambridge University Press, (2008)
* [8] E. Giroux, Convexité en topologie de contact, Comm. Math. Helv. 66 (1991), 637-677.
* [9] E. Giroux, Sur les transformations de contact au-dessus des surfaces, Essays on geometry and related topics, Vol. 1,2, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, (2001), 329–350.
* [10] K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309–368 (electronic).
* [11] Y. Huang, Bypass attachments and homotopy classes of 2-plane fields in contact topology, preprint 2011. arXiv:1105.2348.
* [12] I. Torisu, On the additivity of the Thurston-Bennequin invariant of Legendrian knots, Pacific J. Math. 210 (2003) 359-365
|
arxiv-papers
| 2011-02-26T09:49:54 |
2024-09-04T02:49:17.303062
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Huang",
"submitter": "Yang Huang",
"url": "https://arxiv.org/abs/1102.5398"
}
|
1102.5421
|
# Networks of gravitational wave detectors and three figures of merit
Bernard F. Schutz1,2 1 Albert Einstein Institute, Potsdam, Germany 2 School
of Physics and Astronomy, University of Cardiff, Wales, UK
bernard.schutz@aei.mpg.de
###### Abstract
This paper develops a general framework for studying the effectiveness of
networks of interferometric gravitational wave detectors and then uses it to
show that enlarging the existing LIGO-VIRGO network with one or more planned
or proposed detectors in Japan (LCGT), Australia, and India brings major
benefits, including much larger detection rate increases than previously
thought. I focus on detecting bursts, i.e. short-duration signals, with
optimal coherent data-analysis methods. I show that the polarization-averaged
sensitivity of any network of identical detectors to any class of sources can
be characterized by two numbers – the visibility distance of the expected
source from a single detector and the minimum signal-to-noise ratio (SNR) for
a confident detection – and one angular function, the antenna pattern of the
network. I show that there is a universal probability distribution function
(pdf) for detected SNR values, which implies that the most likely SNR value of
the first detected event will be 1.26 times the search threshold. For binary
systems, I also derive the universal pdf for detected values of the orbital
inclination, taking into account the Malmquist bias; this implies that the
number of gamma-ray bursts associated with detected binary coalescences should
be 3.4 times larger than expected from just the beaming fraction of the gamma
burst. Using network antenna patterns, I propose three figures of merit that
characterize the relative performance of different networks. These measure (a)
the expected rate of detection by the network and any sub-networks of three or
more separated detectors, taking into account the duty cycle of the
interferometers, (b) the isotropy of the network antenna pattern, and (c) the
accuracy of the network at localizing the positions of events on the sky. I
compare various likely and possible networks, based on these figures of merit.
Adding any new site to the planned LIGO-VIRGO network can dramatically
increase, by factors of 2 to 4, the detected event rate by allowing coherent
data analysis to reduce the spurious instrumental coincident background.
Moving one of the LIGO detectors to Australia additionally improves direction-
finding by a factor of 4 or more. Adding LCGT to the original LIGO-VIRGO
network not only improves direction-finding but will further increase the
detection rate over the extra-site gain by factors of almost 2, partly by
improving the network duty cycle. Including LCGT, LIGO-Australia, and a
detector in India gives a network with position error ellipses a factor of 7
smaller in area and boosts the detected event rate a further 2.4 times above
the extra-site gain over the original LIGO-VIRGO network. Enlarged advanced
networks could look forward to detecting three to four hundred neutron star
binary coalescences per year.
###### pacs:
95.55.Ym,95.45.+i
††: Class. Quantum Grav.
## 1 Introduction: detector networks
### 1.1 Current and future networks of interferometers
The three large gravitational wave detectors of the LIGO project [1], located
at two sites, and the large instrument of the VIRGO project [2], all of which
are expected to reach their Advanced level of sensitivity around 2016,
represent the bare minimum required to realize the potential of gravitational
wave astronomy when detecting signals of short duration. Using gravitational
wave information alone, it is necessary to have at least three separated
detectors for locating such sources on the sky, measuring the intrinsic
amplitude and polarization of the incoming waves [3], and determining
distances to “standard-siren” coalescing compact-object binaries [4]. For
long-duration (continuous-wave) signals, a single detector can use the phase
modulation imprinted by the motion of the Earth to locate sources on the sky.
But if the signal is a “burst”, too short for modulation to be measurable,
then positions must be inferred by time-delay triangulation among at least
three separated detectors. Some of the most important expected signals will be
bursts, such as those from inspiraling and coalescing binaries of neutron
stars and/or black holes.
If one of these delicate interferometers temporarily falls out of observing
mode or experiences a period of unusually high noise, so that one of the three
sites has no functioning detector, or if an incoming gravitational wave
arrives from a location on the sky or with a polarization where one of the
detectors is significantly less sensitive, then an observation by the
remaining detectors will not be able to reconstruct the event completely
unless there is other associated information, for example from a gamma-ray
burst. Although two-detector observations can have enough significance to
identify an event and measure important physical parameters, such as the
stellar masses in a binary system, the aim of building detector networks is to
extract the greatest possible information from the weak and infrequent signals
that we expect to observe with Advanced detectors, and this requires all three
sites operated by LIGO and VIRGO.
Fortunately, this network will be enlarged on a short timescale. Funding has
begun for the LCGT detector in Japan. There are further proposals for
construction in Australia and India. Detectors in Asia or Australia help to
cover sky gaps and operational down-times of the basic three and bring an
added bonus of improved angular resolution, by increasing the length and
number of baselines among detectors of the network. There have been a number
of detailed studies of the observing benefits brought by one or another
detector [5, 6, 7, 8, 9, 10, 11]. These studies usually simulate network
detection by using Monte-Carlo techniques, which provide reliable comparisons
of specific configurations but little insight into what would happen with
other configurations. It would be helpful, therefore, to have general results
applicable to all networks as well as complementary and easily computed ways
of quantifying the extra science brought by one or more further detectors. To
this end I suggest here three relatively simple figures of merit (f.o.m.’s)
that measure the mean performance of different network configurations. They
compare networks’ overall event rates (including allowance for realistic duty
cycles of detectors), the isotropy of their joint antenna patterns, and the
precision with which the networks can measure sky positions of sources. I also
derive two general probability distributions for events detected by any
network: their observed signal-to-noise ratios, and the observed values of the
inclination angle of detected binary systems. The Nissanke et al [11] Monte-
Carlo study of coalescing-binary detection by various networks is particularly
close to the subject of this paper and will provide a useful reference
comparison for various analytic results derived below.
### 1.2 Network coherent analysis
The analysis in this paper assumes that a number of detectors observe
gravitational waves coherently, by combining their data in the most sensitive
way. The earliest detailed study for gravitational waves of what we now call
coherent detection was by Gürsel and Tinto [12]. The papers that placed
coherent network detection on a sound statistical basis were by Flanagan and
Hughes [13] and by Finn [14]. In this paper I shall concentrate on detecting
short-duration signals whose waveform is known in advance, using matched
filtering. Coherent detection can also be used to find signals whose waveform
is not known [15].
Coherent detection is not at present the default method of data analysis. All
the searches carried out so far by the LSC-VIRGO collaboration have involved
coincidence thresholding, which means selecting for further study only
stretches of data that appear to contain signals strong enough to pass a pre-
determined threshold in two or more detectors, where the signals occur within
a maximum time-separation equal to the light-travel time between the detectors
(the coincidence “window”). The experience of current searches has been that
most large events in the individual detector data streams are random
instrumental artifacts (sometimes called “glitches”), and the coincidence test
eliminates almost all of them because the glitches are not correlated in the
data streams of separated detectors. But thresholding is not the optimal
signal detection method against Gaussian noise, and in fact it can be very far
from optimum, as discussed in section 4.2 below. Thresholding is used because,
although most of the noise background in detectors is Gaussian, glitches make
the background far from Gaussian at amplitudes above a few standard
deviations. Interferometer-network searches that use thresholding extend
methods originally developed for networks of bar antennas [16].
However, networks containing three or more detectors – our subject in this
paper – have a degree of redundancy that allows them to veto glitches: once
the time-delays allow identification of the location of the source, the two
polarization waveforms are over-determined by the three or more detector
responses. This means that such networks have linear combinations of detector
outputs that contain no gravitational wave signal, often called null streams
[12, 17, 18, 19]. These can be used to test for and veto glitches, which do
not in general cancel out in the null streams.
Current searches for short-duration signals often follow the
thresholding/coincidence step by doing a coherent analysis of the coincident
events, in order to use the null-stream vetoes and to extract as much
information from them as possible [20, 21]. In fact, the very first analysis
of gravitational wave data from a network of interferometers – the so-called
“Hundred Hour Run” – applied a two-detector null-stream method (after
thresholding) to eliminate glitches and show that the strongest observed
coincident event had a high probability of occurring by chance [22], and
consequently that no gravitational wave event had been observed.
But the glitch vetoes provided by null streams in principle allow three-
detector networks to do fully coherent analysis, without prior thresholding. A
number of studies have therefore explored fully coherent detection or compared
it with coincidence thresholding [6, 10, 18, 19, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33]. It is now clear that coherent detection is already able to
discriminate real gravitational waves from glitches even in a general three-
detector network, and when there are four or more detectors this gets even
better.
Since we will see below that coherent methods are capable of detecting far
more events than coincidence methods, it seems reasonable to assume that fully
coherent detection will become the norm as the detector network grows. This
will not be entirely trivial: one of the principal challenges of introducing
coherent data analysis is that it is very demanding of computing, because one
has to do a signal search for each resolvable location on the sky. But the
payoffs will be worth the effort, especially with the computing power that can
be expected to be available by the time the current network is enlarged. The
purpose of this paper is, therefore, to characterize the performance of
different possible networks when they use coherent detection.
### 1.3 Assumptions and principal results
The detection sensitivity of a detector network is a function of the
sensitivity of the individual detectors and their placement on the earth. An
important part of the sensitivity is the network’s antenna pattern, which
defines up to a radial scaling the region of space around the earth within
which a source should be detected. The overall scale depends on the
sensitivity of the individual detectors and the detection threshold that is
set for discriminating real signals from noise impersonators. It is
conventional in the literature to combine threshold and sensitivity into a
radial measure called the horizon distance, the maximum distance a detector or
network can detect an event, allowing for an optimum alignment. In this paper
I separate threshold from sensitivity by measuring the sensitivity of a
detector or network to a given source in terms of a visibility distance, which
is the distance at which the given source would produce a mean response with a
signal-to-noise ratio 1, averaged over polarizations.
From the properties of the antenna pattern I define the three new f.o.m.’s,
for a rather general source population. The f.o.m.’s are meant to be simple to
compute and to use. They should give a broad-brush characterization of the
effectiveness of networks, but they won’t be precise enough to make fine
discriminations between similar networks. Although the f.o.m.’s can in
principle be computed for any network, I will keep the discussion in this
paper simple by making some assumptions.
* •
Detectors. All the detectors are interferometers with identical sensitivity
and identical duty cycles. The detectors’ noise streams are not correlated
with one another, nor are the times when they drop out of observing mode. The
generalization to detectors with different sensitivity is not difficult.
* •
Networks. The networks are made up of combinations of the Advanced upgrades of
the existing LIGO and VIRGO instruments plus planned and possible instruments
at the locations in Japan, Australia, and India that are given in table 1
below. Only networks containing three or more detectors in different locations
are considered, because, as noted above, fewer detectors do not return sky
position and polarization information from an observation unless there are
associated detections in, say, gamma or optical observatories.
* •
Sources. The gravitational waves all come from an identical population that
are randomly and uniformly distributed in (Euclidean) space and in
polarization. The waves are short bursts, in that the detectors do not move
significantly during the observations, and they are emitted at random times.
The waveforms are identical except that they have different overall
amplitudes, inversely proportional to the distance to the source; they all
have the same polarization evolution (as a function of time) except for a
random rotation in the plane of the sky at the start of the burst. Note that,
according to this definition, binary systems with different inclinations to
the line of sight (different amounts of elliptical polarization) are members
of different populations, but binaries with the same inclination but different
orientations (rotations in the plane of the sky: the angle $\psi$ in figure 1)
are members of the same population. We do not consider stochastic signals or
long continuous-wave signals from GW pulsars.
* •
Analysis. The data are analyzed coherently with a matched filter family
capable of matching the incoming signal perfectly. The data analysis finds the
ideal match by maximizing the log likelihood. Detector noise is purely
Gaussian, at least at the times when events arrive.
Given these assumptions, I summarize here the principal results of this paper:
1. 1.
The sensitivity of a network to a population of identical but randomly
oriented and randomly located sources depends on the signal waveform, the
sensitivity of the detectors (all assumed the same), and the geometry of the
network. The signal and sensitivity contribute only to a scaling factor that
multiplies the antenna pattern, which depends only on the network geometry
(14). Therefore the relative performance of any two networks of similar
detectors observing any given source population is independent of the nature
of the source. This allows us to compare the advantages and disadvantages of
networks without needing to specify much about the signal.
2. 2.
The population of detected events has a universal signal-to-noise
distribution, with a probability density function (p.d.f.) proportional to
$\rho^{-4}$ above the detection threshold, where $\rho$ is the amplitude
signal-to-noise ratio (SNR). The p.d.f. (2.4) depends only on the detection
threshold $\rho_{\rm min}$ set on $\rho$, not on the geometry or sensitivity
of the network.
3. 3.
From this p.d.f it is possible to deduce that the median amplitude SNR of any
detected population will be $2^{1/3}\simeq 1.26$ times the detection threshold
$\rho_{\rm min}$. As we wait for the first detection, this is the most likely
SNR of the first event, provided that coherent data analysis is used for the
search. Similarly, the mean amplitude SNR of the detected population will be
1.5 times the threshold.
4. 4.
Binaries with different inclinations have different maximum detection ranges,
which biases the expected observed distribution of inclinations. I compute the
universal probability distribution for detected inclinations, independent of
the network configuration (28). It peaks around $\pm 30^{o}$.
5. 5.
From this distribution of inclinations one can also deduce another bias,
namely that – provided that mergers involving neutron stars give rise to
narrow-beamed gamma-ray bursts – the number of gamma-ray bursts that will be
detected in association with gravitational wave signals will be 3.4 times
larger than one would expect if there was no correlation between burst
direction and the maximum-power direction of a binary.
6. 6.
The first figure of merit (f.o.m.) is called Triple Detection Rate [3DR]
(section 3.1). It measures the rate at which a network can detect events in
detectors at three or more separated locations. The rate at which events of a
given source population are detected depends of course on the detection volume
accessible to the network, but it also depends on the duty cycle of detectors,
which is the fraction of time they spend in observation mode. The first
introduction of figures of merit into the discussion of networks seems to have
been by Searle, et al [5], who defined a measure of detection rate that
depends effectively only on the detection volume. (See also Searle, et al
[7].) However, especially at the beginning of the operation of Advanced
Detectors, the duty cycle of the detectors will not be 100%. For the full
reconstruction of information about the source, we require at least three
separated detectors to observe the event, so a three-detector subnet of a
larger network can still return detections. Therefore, Triple Detection Rate
as defined here is designed to compute how many events can be detected by sub-
networks of three or more separated detectors, even when some other detectors
in the network may be off the air.
7. 7.
The second f.o.m. is called Sky Coverage [SC] (section 3.2). It measures the
isotropy of the network’s antenna pattern. It is defined as the fraction of
the $4\pi$ sphere that is covered by the network’s antenna pattern at a range
that is $1/\sqrt{2}$ of the maximum. For a given number of detectors of a
standard sensitivity, there is a trade-off between isotropy and overall
detection volume: if the antenna patterns of individual detectors reinforce
each other, then the volume they include will be larger than if they fill in
each other’s directional “holes”. But isotropy might be a desirable thing in
itself. For example, if the source population is anisotropic (perhaps biased
toward the Galactic plane) then an isotropic network might do better than one
with a larger range. Or if the sources are expected to be associated with
objects that can be detected also by a non-gravitational signal, but only if
they are relatively nearby compared to the maximum range of the network (e.g.
supernovae seen with neutrinos), then an isotropic network could do better.
8. 8.
The third f.o.m. is called Directional Precision [DP] (section 3.3). It
measures how well the network localizes events on the sky, its directional
accuracy. Generally speaking, longer baselines improve direction-finding.
Directional Precision uses the measure of solid-angle error introduced by Wen
and Chen [8]. It is proportional to an average over the antenna pattern, not
of the size of the error box, but of its inverse. This prevents the measure
being distorted by small regions where direction-finding is poor; instead it
is weighted more by the regions of the sky where direction-finding is
particularly good.
9. 9.
Enlarging the basic LIGO-VIRGO network with detectors in Japan and/or
Australia also provides a less obvious but perhaps even more important
benefit: it makes coherent data analysis more robust and allows the detection
of events that would not pass the coincidence threshold tests used in the
current LIGO-VIRGO data analysis (section 4.2), where fully coherent analysis
is difficult because of the geometry of the network. This could lead to an
improvement of as much as a factor of 4 in the recovery of signals within a
given detection volume, depending on the effectiveness with which coherent
methods can be introduced into the data analysis of the basic LIGO-VIRGO
network.
By comparing these measures for various possible networks, some simple lessons
emerge. First, if one takes as a baseline the performance of the original
network of Advanced detectors – LIGO Hanford with two full-size
interferometers, LIGO Livingston, and VIRGO – using coherent detection, then
there is a big win in event rate from putting another large detector anywhere
in Asia or Australia. This comes partly from adding more detection volume and
partly from providing greater coverage when individual detectors randomly drop
out of observing mode. A Japanese detector (LCGT) makes the antenna pattern
more isotropic; an extra Australian detector (AIGO) makes its reach go deeper.
If instead of building an extra detector in Australia, one of the LIGO Hanford
instruments is placed in Australia (LIGO Australia), the improvement in
detection rate is not quite as dramatic. The big change then is a significant
improvement in direction-finding. If we take the network that includes LIGO
Australia and LCGT, again there is a very big improvement in the event rate,
and of course it becomes more isotropic as well. This is pretty much a “dream
configuration” in terms of present opportunities. If a project gains traction
in India and a large INDIGO detector is built, then this produces even further
gains that the f.o.m.’s quantify. On top of these improvements due to detector
numbers and geometry, the robustness of coherent detection in an enlarged
network will lead to further striking gains in event rate over the current
coincidence style of analysis.
It is important to remark that these f.o.m.’s should be regarded as rules of
thumb, not as exact measures of the performance of any network. But treated
with a small amount of caution, the measures show how big the science gains
can be from adding further Advanced detectors to the existing three sites.
## 2 Network antenna patterns and the amplitude distribution of detected
events
### 2.1 Antenna pattern and detection volume of a single interferometer
The f.o.m.’s are based on the antenna patterns of the detectors, which
describe their relative sensitivity in different directions. Each detector is
linearly polarized and has a quadrupolar antenna pattern. In the notation of
Sathyaprakash and Schutz [34], we consider a detector in the $x-y$ plane with
arms along the axes, and a gravitational wave coming from a direction given by
the usual spherical coordinates $\theta$ and $\phi$ relative to the detector’s
axes, whose two polarization components $h_{+}$ and $h_{\times}$ are referred
to axes in the plane of the sky that are rotated by an angle $\psi$ relative
to the detector axes (see figure 1, which is taken from figure 3 of
Sathyaprakash and Schutz [34]). Then the strain $\delta L/L$ of the
interferometer is
$\frac{\delta
L(t)}{L}=F_{+}(\theta,\,\phi,\,\psi)h_{+}(t)+F_{\times}(\theta,\,\phi,\,\psi)h_{\times}(t),$
(1)
where the $F_{+}$ and $F_{\times}$ are the antenna pattern functions for the
two polarizations. Using the geometry in figure 1, one can show that
$\displaystyle F_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+\cos^{2}\theta\right)\cos 2\phi\cos
2\psi-\cos\theta\sin 2\phi\sin 2\psi,$ $\displaystyle F_{\times}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(1+\cos^{2}\theta\right)\cos
2\phi\sin 2\psi+\cos\theta\sin 2\phi\cos 2\psi.$ (2)
Figure 1: The relative orientation of the sky and detector frames. From [34].
These are the antenna pattern response functions of the interferometer to the
two polarizations of the wave as defined in the sky plane [35]. Note that the
maximum value of both $F_{+}$ and $F_{\times}$ is 1.
Sometimes the angle $\eta$ between the arms of a detector is not exactly
$\pi/2$, for reasons of local geography or by design. For that reason it is
helpful to orient the detector in the $x$-$y$ coordinate plane by aligning the
bisector of the angle between the arms with the bisector of the angle between
the axes [36]. One also has to multiply the functions $F_{+}$ and $F_{\times}$
in (1) by $\sin\eta$. When we discuss networks we will define the orientation
of the detector to be the geographical direction of the arm bisector.
The expected power signal-to-noise ratio (SNR) of the signal in the detector’s
data stream is, if it can be discovered by ideal matched filtering,
$\rho^{2}=4\int_{0}^{\infty}\frac{|\tilde{\delta L}(f)/L|^{2}}{S_{h}(f)}{\rm
d}f,$ (3)
where $S_{h}(f)$ is the one-sided spectral noise density normalized to the
gravitational wave amplitude, and the time-series strain $\delta L(t)/L$ in
(1) has been Fourier-transformed into $\tilde{\delta L}(f)/L$, which then
depends on the Fourier transforms $\tilde{h}_{+}(f)$ and
$\tilde{h}_{\times}(f)$ of the incoming waves. I will assume from now on that
we are detecting a short burst of gravitational waves, so that the detector
does not change its orientation during the observation. A discussion of
network detection of long-duration signals, such as those from gravitational
wave pulsars, may be found in Cutler and Schutz [37, 38].
We now apply the assumption that the wave has a randomly oriented
polarization. Consider a source which emits wave components $H_{+}(f)$ and
$H_{\times}(f)$, referred to its own frame, defined perhaps by some preferred
axis or plane in the source. Suppose that at the start of the observation this
source frame is different from the detector frame as projected onto the sky by
a rotation angle $\alpha$. During the observation the polarization will rotate
in some way determined by $H_{+}(f)$ and $H_{\times}(f)$. This is of no
interest to us here. The important point is that the ensemble of sources at
the same position in space contains systems with all possible initial angles
$\alpha$. When we average the power SNR in (3) over the ensemble, we will
simply be changing in a uniformly random way the projection of the source’s
intrinsic $+$ and $\times$ components onto the detector’s. The result is that
the mean power SNR over the ensemble (denoted by $\left<\;\right>$) depends
only on the sum of the squares of the sensitivity functions of the detector to
both polarizations:
$\left<\rho^{2}\right>=2\left[F_{+}(\theta,\phi,\psi)^{2}+F_{\times}(\theta,\phi,\psi)^{2}\right]\int_{0}^{\infty}\frac{|H(f)|^{2}}{S_{h}(f)}{\rm
d}f,$ (4)
where $|H(f)|^{2}=|H_{+}|^{2}+|H_{\times}|^{2}$. We call the function
$\displaystyle P(\theta,\phi)$ $\displaystyle=$ $\displaystyle
F_{+}(\theta,\phi,\psi)^{2}+F_{\times}(\theta,\phi,\psi)^{2}$ (5)
$\displaystyle=$
$\displaystyle\frac{1}{4}(1+\cos^{2}\theta)^{2}\cos^{2}2\phi+\cos^{2}\theta\sin^{2}2\phi$
the antenna power pattern of a single interferometer. Note that, from (2.1),
the antenna power pattern is independent of the angle $\psi$ that is the
reference angle for the wave’s polarization, as one would expect after our
ensemble polarization average. It is plotted in the detector coordinate frame
in figure 2. This is often referred to as the “peanut diagram”.
Figure 2: The antenna power pattern (left panel) and its square-root
(amplitude pattern: right panel) of a single interferometer oriented with axes
in the $x$-$y$ plane, averaged over polarizations of the incoming wave. The
amplitude pattern represents the shape of the detection volume of the
instrument, or its maximum detection reach in different directions.
If, for a single detector, there is a detection threshold $\rho_{\rm min}$ on
the amplitude SNR, then a signal from a direction $(\theta,\,\phi)$ can be
expected to be detected if
$2P(\theta,\phi)\int_{0}^{\infty}\frac{|H(f)|^{2}}{S_{h}(f)}{\rm
d}f\geq\rho_{\rm min}^{2}.$ (6)
For the purposes of our discussion, we suppose that the gravitational wave
source has a standard intrinsic amplitude, so that its received amplitude
$H(f)$ is inversely proportional to the distance $r$ to the source. We also
suppose that these sources are randomly distributed in space. Let us normalize
the amplitude by defining (arbitrarily) a standard reference distance $r_{s}$
at which our source would have amplitude $H_{s}(f)$, so that a source at a
distance $r$ has amplitude
$H(f)=\frac{r_{s}}{r}H_{s}(f).$ (7)
This is much the way astronomers distinguish between absolute and apparent
magnitudes, by defining the absolute magnitude to equal the apparent magnitude
of the source if it were at a fixed fiducial distance (10 pc).
Explicitly separating $r$ out in $H(f)$ will be helpful for the volume
integrals below. For example, we can now rewrite (4) as
$\left<\rho^{2}\right>=\frac{2}{r^{2}}P(\theta,\phi)\int_{0}^{\infty}\frac{|r_{s}H_{s}(f)|^{2}}{S_{h}(f)}{\rm
d}f.$ (8)
Note that the product $rH=r_{s}H_{s}$ is independent of the distance to the
source. We use this to define the visibility distance of the source $D_{V}$:
${D_{V}}^{2}=2\int_{0}^{\infty}\frac{|r_{s}H_{s}(f)|^{2}}{S_{h}(f)}{\rm d}f.$
(9)
This is the distance at which the source would have SNR = 1 in a single
detector at its most sensitive location in the sky, namely directly overhead
at $\theta=0$ or $\pi$, after averaging over the sky polarization angle
$\psi$. All the details of filtering and the detector noise curve are hidden
in the single parameter $D_{V}$. This leads to a simple way of writing (8)
$\left<\rho^{2}\right>=P(\theta,\phi)\frac{{D_{V}}^{2}}{r^{2}}.$ (10)
Similarly, I will define the mean horizon distance $R_{0}$ for a single
detector observing this source to be the distance at which the source is on
average just at the detection threshold $\rho_{\rm min}$ when it is overhead,
so that $R_{0}=D_{V}/\rho_{\rm min}$. Then the reach $R$ of the single
detector in any direction ${\theta,\phi}$ is
$R(\theta,\phi)=R_{0}[P(\theta,\phi)]^{1/2}=\frac{D_{V}}{\rho_{\rm
min}}[P(\theta,\phi)]^{1/2}.$ (11)
We call the square-root of the antenna power pattern the antenna amplitude
pattern. The volume bounded by the reach $R(\theta,\phi)$ is called the
detection volume. Its size is determined by the antenna amplitude pattern,
scaled by the mean horizon distance $R_{0}$. The mean horizon distance is
smaller than what is conventionally called the horizon distance, which is the
distance at which an optimally polarized source exactly overhead can just be
detected at threshold.
Note that we are making an approximation here when we define a detection
volume by polarization averaging. Sources at the edge of the volume have only
a 50% chance of being detected, while those that are well inside are detected
with higher probability. Moreover, a number of sources outside this volume
will be detected if they have a favorable polarization. Our approximation is
to replace the real detection probability distribution in space with a fixed
volume that has a hard edge: everything inside is detected, everything outside
is missed. We use this approximation only to study the gross properties of
detection, such as numbers detected, typical position accuracies, and so on,
and only to compare different networks. The comparisons are likely to be
better than the accuracy of the approximation for any single network, since
the errors will systematically affect all networks the same way. The test of
how accurate this approximation is for any specific network is whether it
matches up with Monte-Carlo studies of the real detection problem for that
network.
### 2.2 Antenna pattern of a network of detectors
We now need to generalize these concepts to networks of more than one
detector. I will assume here that the detectors’ noise streams are
uncorrelated. This is a good assumption for all networks except those that
include two detectors at the Hanford LIGO site. Even there, experience has
shown that the correlations can be reduced to a very low level with careful
experimental design. A full treatment of the theory of detection in networks
that have detectors with correlated noise may be found in Finn [14].
When computing the joint antenna pattern of the entire network, the antenna
patterns of the individual detectors must of course be transformed to a common
celestial coordinate system. We take this to be the Earth-based spherical
coordinates, and from now on we denote them by $(\theta,\phi)$. In addition,
there must be a common definition of the incoming wave polarization. I use
here the formulation given in [39], whose expressions were developed for the
problem of long-term observations, where the detector changes orientation with
time. For the present paper we merely need to set $t=0$ in their formulation,
and we shall use conventional spherical sky coordinates rather than
declination and right-ascension.
As shown by Finn [14], the network power SNR is just the sum of the power SNRs
of the individual detectors
$\rho^{2}_{N}=\sum_{k=1}^{N_{D}}\rho^{2}_{k},$ (12)
where $N_{D}$ is the number of detectors and where we define the individual
power SNRs as
$\rho^{2}_{k}=2\int_{0}^{\infty}\frac{|H_{k}(f)|^{2}}{S_{h}(f)}{\rm d}f,$ (13)
where $H_{k}(f)$ is the waveform projected onto the $k$-th detector. Averaging
as before over the random polarization angle, we have
$\left<\rho^{2}_{N}\right>=2\sum_{k}(F_{+,k}^{2}+F_{\times,k}^{2})\int_{0}^{\infty}\frac{|H(f)|^{2}}{S_{h}(f)}{\rm
d}f,$ (14)
where $F_{+,k}$ and $F_{\times,k}$ are the antenna patterns of the individual
detectors. Note that the integral in this equation does not depend on $k$ and
is therefore taken outside the sum. The sum is then the function
$P_{N}(\theta,\phi)=\sum_{k}(F_{+,k}^{2}+F_{\times,k}^{2}),$ (15)
which is called the network antenna power pattern.111If the detectors were not
identical, then one could modify the network antenna pattern simply by
including a single weighting factor consisting of the ratio of $\rho^{2}$ for
each detector to a standard detector $\rho^{2}$ for the particular signal
waveform being considered. The network antenna pattern would then be waveform-
dependent. This is our analytic approximation to the detection sensitivity
found in [7] from Monte-Carlo studies of randomly oriented binary systems. In
terms of $P_{N}$ the network SNR takes the simple and useful form
$\left<\rho^{2}_{N}\right>=P_{N}(\theta,\phi)\frac{D_{V,L}^{2}}{r^{2}},$ (16)
where $D_{V,L}$ is the visibility distance of a single detector, labelled here
as the Livingston detector. Remember, all detectors are assumed identical so
all have the same visibility distance. This assumption is easily dropped if
necessary, but it makes the discussion in the present paper simpler.
It is worth remarking that the polarization-averaged network antenna pattern
does not depend on the local orientation of each detector, since it is the sum
of the individual detector power patterns (15), and for each detector the sum
of the squares of the antenna pattern components is invariant under rotations
of the detector in its plane. It might seem counterintuitive that two co-
located detectors with orthogonal orientation make the same average
contribution to the signal power received by the network as they would if they
were perfectly aligned. When aligned they work well together but miss many
events that one of them would catch when not aligned. When searching for a
stochastic gravitational wave signal, of course, alignment is crucial.
Moreover, even for bursts, the ability to determine polarization and sky
position of a signal will be affected by the relative alignment of the
detectors. I will return to this point later.
The resulting expression for the antenna pattern of an arbitrarily located and
oriented interferometer in our notation is as follows. The source position is
given by the spherical coordinates $(\theta,\phi)$ on the sky, and the frame
for the wave polarization angle $\psi$ is defined to be aligned with this
spherical-coordinate grid. The detector is at latitude $\beta$ and longitude
$\lambda$. It is an interferometer oriented such that the bisector of its arms
points in the direction $\chi$, measured counter-clockwise from East. Its arms
have an opening angle of $\eta$. The celestial coordinates $(\theta,\phi)$ are
aligned with latitude and longitude, so that the equators of both systems
coincide and the celestial point $(\theta=\pi/2,\,\phi=0)$ is in the zenith
direction above the geographic location $(\beta=0,\,\lambda=0)$. The antenna
pattern functions are
$\displaystyle F_{+}$ $\displaystyle=$
$\displaystyle\sin\eta[a\cos(2\psi)+b\sin(2\psi)],$ (17) $\displaystyle
F_{\times}$ $\displaystyle=$
$\displaystyle\sin\eta[b\cos(2\psi)-a\sin(2\psi)],$ (18)
where the functions $a$ and $b$ are given by
$\displaystyle
a=\frac{1}{16}\sin(2\chi)[3-\cos(2\beta)][3-\cos(2\theta)]\cos[2(\phi+\lambda)]+$
$\displaystyle\frac{1}{4}\cos(2\chi)\sin(\beta)[3-\cos(2\theta)]\sin[2(\phi+\lambda)]+$
$\displaystyle\frac{1}{4}\sin(2\chi)\sin(2\beta)\sin(2\theta)\cos(\phi+\lambda)+$
$\displaystyle\frac{1}{2}\cos(2\chi)\cos(\beta)\sin(2\theta)\sin(\phi+\lambda)+\frac{3}{4}\sin(2\chi)\cos^{2}(\beta)\sin^{2}(\theta),$
(19) $\displaystyle
b=\cos(2\chi)\sin(\beta)\cos(\theta)\cos[2(\phi+\lambda)]-\frac{1}{4}\sin(2\chi)[3-\cos(2\beta)]\cos(\theta)\sin[2(\phi+\lambda)]+$
$\displaystyle\cos(2\chi)\cos(\beta)\sin(\theta)\cos(\phi+\lambda)-\frac{1}{2}\sin(2\chi)\sin(2\beta)\sin(\theta)\sin(\phi+\lambda).$
(20)
### 2.3 Detection volume of a network of detectors
The detection volume $V_{N}$ of the network is defined as the region enclosed
by its reach in any direction, which as before is
$R_{N}(\theta,\phi)=R_{0}[P_{N}(\theta,\phi)]^{1/2}=\frac{D_{V,L}}{\rho_{N,{\rm
min}}}P_{N}^{1/2},$ (21)
where $R_{0}$ is defined as before to be the mean horizon distance (maximum
reach) of a single detector for this source at the chosen network detection
threshold SNR $\rho_{N,{\rm min}}$, and where (as before) $D_{V,L}$ is the
single-detector visibility distance (maximum range at SNR = 1). I will assume
that when we compare networks, all of them have the same detection threshold.
We can compute the detection volume explicitly:
$\displaystyle V_{N}$ $\displaystyle=$ $\displaystyle\int{\rm
d}\Omega\int_{0}^{R_{N}(\theta,\phi)}r^{2}dr=\frac{1}{3}\int{\rm d}\Omega
R_{N}^{3}(\theta,\phi)$ (22) $\displaystyle=$
$\displaystyle\frac{1}{3}R_{0}^{3}\int{\rm
d}\Omega[P_{N}(\theta,\phi)]^{3/2}.$
Figure 3: The antenna power patterns of the LIGO and VIRGO detector network
with two detectors at Hanford (HHLV: left panel) and of the network after
including the Japanese detector LCGT (HHJLV: right panel). All detectors are
assumed to be identical. As in Figure 2, the sensitivity is averaged over
polarizations of the incoming wave. Top row: The coordinate system is oriented
with $z$ aligned with geographic North and the $x$-axis at geographic
longitude 0o. In all such plots from now on, the viewer is located at
longitude 40oW and 20oN, above the mid-Atlantic. Note that all antenna
patterns are reflection symmetric through the center of the earth, so that the
hidden side is a mirror image of the side shown in the diagram. Bottom row:
The same data plotted as contour plots. Contours are labeled with values
relative to the maximum. For HHLV on the left, the maximum is 3.03 (square of
mean horizon distance from table 2). For HHJLV on the right, the maximum is
3.31.
Table 1 gives the important parameters of the detector locations that will be
considered in this paper, including the one-letter abbreviation by which the
detectors will be denoted in naming the various networks.
As an illustration, in figure 3 the network antenna power patterns are plotted
for two networks: the planned Advanced network of two LIGO detectors at
Hanford, one at Livingston, and VIRGO; and the same network plus the LCGT
detector in Japan. Notice that the hole in the southwest direction has been
filled in by the Japanese detector.
### 2.4 Universal distribution of detected amplitudes
Because the angular sensitivity of the detectors is totally decoupled from the
dependence of SNR on the distance of the source, which resides in $H(f)$ in
(14), we can work out the expected distribution of SNR for detected events
analytically for any detector network and source population. To do this we
make explicit in (22) the fact that $R_{N}$ is inversely proportional to the
detection threshold $\rho_{N,{\rm min}}$, by using (21):
$V=\frac{D_{V,L}^{3}}{3\rho_{N,{\rm min}}^{3}}\int{\rm
d}\Omega[P_{N}(\theta,\phi)]^{3/2}.$ (23)
The number of detections with SNR larger than any given $\rho_{N}$ is
proportional to the detection volume with $\rho_{N,{\rm min}}$ set equal to
this $\rho_{N}$. This scales as $\rho_{N}^{-3}$. This is a cumulative
distribution: the number of detections with SNR larger than $\rho_{N}$ scales
as $\rho_{N}^{-3}$. It is straightforward from this to show that the universal
probability density function for the distribution of detected SNR values is
$\displaystyle p(\rho_{N}){\rm d}\rho_{N}$ $\displaystyle={}3\rho^{3}_{N,{\rm
min}}\rho_{N}^{-4}{\rm d}\rho_{N},$ $\displaystyle\qquad\rho>\rho_{N,{\rm
min}}$ $\displaystyle={}0,$ $\displaystyle\qquad\rho_{N}<\rho_{N,{\rm min}}.$
From this simple universal distribution one can deduce any of the moments one
wishes. For example, the mean expected amplitude SNR is $1.5\rho_{N,{\rm
min}}$. The mean expected power SNR2 is $3\rho_{N,{\rm min}}^{2}$.
The median of this distribution is of particular interest and can also be
deduced from a simple argument: it is the value of the threshold for which the
detection volume is one-half of the full volume. Since the volume scales as
the inverse cube of the threshold, the median amplitude SNR value will be
$2^{1/3}\rho_{N,{\rm min}}$. The median power SNR2 is $2^{2/3}\rho_{N,{\rm
min}}^{2}$. The importance of the median is that it is the most likely SNR
value of the first signal that will be detected. It has often been remarked
that the rapid increase of volume with distance means that the first source is
likely to be near the detection limit. Here we quantify that statement: the
most likely amplitude SNR of the first detection is $2^{1/3}\simeq 1.26$ times
the threshold of the search. The median source is weaker than either the
amplitude mean or the power mean. That is because the universal distribution
has a peak at the lowest values (at threshold) and has a long tail of strong
but rare events.
Of course, this argument has been made in the context of our antenna pattern
detection criterion, which is an approximation. However, I believe one can
expect that the distribution should be close to the distribution of real
observations, provided the detection criterion depends on coherent addition of
signals against mainly Gaussian noise.
### 2.5 Detection volumes for binary systems
As remarked in the definition of sources in section 1, binary systems with
different inclinations belong to different source populations as far as our
detection volumes are concerned, because the strength of their emitted
radiation depends on inclination, and their own radiation patterns are
anisotropic; in fact, if we were to average the power pattern shown in the
peanut diagram (figure 2) over circles around its long axis we would get a
plot of the radiation power pattern of a binary system. But binaries with
different inclinations are all members of the same physical family, just seen
from different and random directions. Therefore it is interesting here to
consider binary detection as a function of inclination angle $\iota$.
The maximum power is radiated along the rotation axis of the binary, defined
as $\iota=0$, and the minimum power in its orbital plane, $\iota=\pi/2$. For a
general inclination angle it is easy to show from, e.g., Sathyaprakash and
Schutz [34] that the radiated power depends on inclination in the following
way:
$P_{{\rm rad}}(\iota)=F_{\rm rad}(\iota)P_{{\rm rad}}(\iota=0),$ (25)
with
$F_{\rm rad}(\iota)=\frac{1}{8}(1+6\cos^{2}\iota+\cos^{4}\iota).$ (26)
We call this function the binary radiation pattern. As remarked above, this is
the $\phi$-average of the interferometer’s antenna pattern (5). The detection
volume will depend on $\iota$ as
$V_{N}(\iota)=\left[F_{\rm rad}(\iota)\right]^{3/2}V_{N}(\iota=0),$ (27)
This predicts the relative numbers of sources that will be detected, i.e. it
quantifies the bias toward small inclination angles created by the stronger
radiation pattern in those directions. We can derive the probability
distribution function of detected values of $\iota$ by normalizing $F_{\rm
rad}^{3/2}$ over the intrinsic distribution of angles, which has the
probability distribution function $\sin\iota$. The normalizing integral is
$\int_{0}^{\pi}\left[F_{\rm rad}(\iota)\right]^{3/2}\sin\iota\,{\rm
d}\iota=0.58092.$
The probability distribution of detected values of $\iota$ is therefore
$p_{\rm det}(\iota)=0.076076(1+6\cos^{2}\iota+\cos^{4}\iota)^{3/2}\sin\iota.$
(28)
This is plotted in figure 4. Note that this, also, is a universal
distribution, in that it applies to any network doing coherent analysis. As
with the distribution of detected values of SNR, this result is exact only
within the approximation we are making that the polarization-averaged antenna
pattern defines a detection volume with a sharp boundary. This pdf is
completely consistent with the Monte-Carlo result of Nissanke et al [11]
(their figure 3), when allowance is made for the difference between using
$\iota$ as the independent variable (here) and $\cos\iota$ ([11]).
Figure 4: The probability distributions of inclination angle $\iota$ (in
radians) for randomly oriented binaries (the single-peaked curve, which is
just $\sin\iota$) and for detected binaries (the double-peaked curve, from
(28)). The selection bias (essentially the Malmquist bias) toward low
inclinations due to the anisotropic radiation pattern of a binary is clear.
The mean value of $V_{N}(\iota)$ in (27) is $0.29046V_{N}(\iota=0)$. This
means that the expected number of binaries detected, allowing for random
inclination and polarization angles, is about 29% of the number that would be
expected if all the systems were face-on.
Figure 4 also has implications for coincidences between gravitational wave
detections and gamma-ray bursts. If we accept the popular model in which a
coalescence of two neutron stars or a neutron star and a black hole is
accompanied by a gamma-ray burst that is emitted in a narrow cone around the
binary’s rotation axis, then events where the cone points toward us are also
stronger gravitational wave emitters, and so we will see relatively more of
them [40]. The slope of the distribution of detected binaries in figure 4 at
$\iota=0$ is about 1.72, compared with 0.5 for the true distribution, a ratio
of 3.44. Therefore, a coincidence between a gravitational wave event and a
gamma burst with a narrow cone (so that only the linear behavior of the curves
in the figure is relevant) is about 3.4 times more likely than one would
expect by just naively computing the solid angle of the jet. For example, if
jets have a solid angle of $4\pi/100$, then only one out of every one hundred
coalescences would point its jet toward us. But we could expect that one in
every 29 detected coalescences would be accompanied by a gamma-ray burst.
## 3 Figures of merit
### 3.1 Triple Detection Rate: Relative effectiveness of a network
The first of the figures of merit measures the relative effectiveness of a
network at detecting the short bursts of gravitational waves that we assume in
our signal model, using enough detectors to extract the full information
available in the gravitational wave signal. Since all detectors are assumed
identical and the source waveform is the same in each case, only the network
detection volume and the duty cycle need to be used to provide a realistic
measure of the relative rates at which events will be detected by different
networks.
The relative detection volumes of various networks calibrate the volume of
space accessible to the network (often given in current LSC-Virgo papers in
units of MWEG: Milky Way Equivalent Galaxies). But adding extra detectors to a
network does more than increase its detection volume. It also ensures that
there is less time when there are fewer than three detectors in operational
mode. Current interferometers need exquisitely tuned control systems to keep
the interferometry locked on a fringe. During the recent S5 science run [41],
the two big LIGO detectors achieved a duty cycle of about 80%. When the
detectors start up at the advanced level of sensitivity, around 2016, the duty
cycle may well be similar. In principle there is no reason that the duty cycle
could not ultimately be pushed well above 90%, but this will require time and
effort. (The smaller GEO600 detector achieved a 95% duty cycle during S5 and
VIRGO operated at close to that efficiency during its several-month
participation at the end of S5.) If one requires an observation to be
performed by all instruments in a three-detector network with a duty cycle of
80% then they will be observing simultaneously only $0.8^{3}\simeq 51\%$ of
the time. If one adds a fourth detector, the amount of time at least three
detectors will be in observing mode dramatically increases to
$(0.8)^{4}+4(0.2)(0.8)^{3}\simeq 82\%$. Adding a fifth raises this to
$(0.8)^{5}+5(0.2)(0.8)^{4}+10(0.2)^{2}(0.8)^{3}\simeq 94\%$, a further
significant increase. We can expect that these numbers will be realistic
during the first few years of the operation of Advanced detectors, until the
experimental teams can focus their efforts on improving duty cycle instead of
raw sensitivity.
The Triple Detection Rate figure of merit for a given network sums the
detection volumes of all sub-networks containing detectors in three or more
locations, each weighted by the probability that the given sub-network will be
the only one observing at a given time. We do not include the amount of time
that only two detectors are in operation because these cannot fully
reconstruct the event in the absence of other information. Specifically, then,
consider a network of 4 separated detectors, called A, B, C, and D, all of
which are in observing mode for a fraction $f$ of the data-taking time, and
whose down-times are not correlated with one another. We define the Triple
Detection Rate of this network to be the effective available volume, with the
scaling factor $(D_{V,L}/\rho_{N,{\rm min}})^{3}$ removed:
$\displaystyle[3DR]_{\rm ABCD}$ $\displaystyle=$
$\displaystyle\left(\frac{D_{V,L}}{\rho_{N,{\rm
min}}}\right)^{-3}\left[f^{4}V_{\rm ABCD}+(1-f)f^{3}(V_{\rm ABC}+V_{\rm
BCD}+V_{\rm ACD}+V_{\rm ABD})\right],$ (29) $\displaystyle=$
$\displaystyle\frac{f^{4}}{3}\int{\rm
d}\Omega[P_{ABCD}(\theta,\phi)]^{3/2}+\frac{(1-f)f^{3}}{3}\int{\rm
d}\Omega\left\\{[P_{ABC}(\theta,\phi)]^{3/2}+\right.$
$\displaystyle\left.[P_{BCD}(\theta,\phi)]^{3/2}+[P_{ACD}(\theta,\phi)]^{3/2}+[P_{ABD}(\theta,\phi)]^{3/2}\right\\}.$
Triple Detection Rate is thus a measure of the effective three-site detection
volume averaged over a long observing run. The number of events detected by
three or more separated detectors in a network during a given observing period
will be proportional to the network’s value of Triple Detection Rate. The
generalization of (29) to networks with other numbers of detectors is obvious.
The definition of Triple Detection Rate specifies detectors at different sites
because a network of 3 detectors involving two at Hanford cannot resolve sky
positions, and hence cannot infer polarizations, distances, and other
parameters. Therefore, in computing $[3DR]_{\rm HHLV}$, the original four-
detector Advanced network, I do not use (29). Instead of all four three-
detector subnetworks, I include only two, both having the antenna power
pattern HLV, but involving different Hanford detectors. With this assumption
and an 80% duty cycle, we get $[3DR]_{\rm HHLV}=4.86$. This serves as a
reference value for other networks, since it is the basic coverage available
from the presently funded Advanced detectors with a realistic duty cycle for
the initial operation.
By contrast, if one of the Hanford detectors is placed in Australia, we get
the network AHLV, which has $[3DR]_{\rm AHLV}=6.06$ with a duty cycle of 80%.
The rate of events whose locations can be measured goes up by 25% simply by
separating the two Hanford detectors, because doing this creates two more
useful three-detector sub-networks. On the other hand, with a 95% duty cycle,
the difference is not so pronounced: $[3DR]_{\rm HHLV}=7.81$ while $[3DR]_{\rm
AHLV}=8.28$. In this case, most detections occur with all four detectors
working, for which in both configurations there is always a subset of three at
separate locations. We return to compare other interesting specific networks
in section 4 below.
To convert [3DR] back to an effective detection volume in space, multiply by
$(D_{V,L}/\rho_{N,{\rm min}})^{3}$, where $D_{V,L}$ is the visibility distance
of the source for the Livingston detector (the distance at which an optimally
located source has unit SNR), and $\rho_{N,{\rm min}}$ is the network
detection threshold SNR. To convert this effective volume into an expected
detection rate one multiplies by the volume rate of events of this population.
### 3.2 Isotropy
If the antenna patterns of detectors in a network are well-aligned, they
increase the detection volume nonlinearly, since the detection volume of a
small solid angle in any direction depends on the $3/2$ power of the total
antenna power pattern. Where the antenna patterns do not overlap
significantly, they make the network more isotropic. Increasing the detection
volume is obviously an important gain, but there may also be merit in a
network that is more isotropic. Isotropic antenna patterns are better for
coincidence observations with other all-sky survey instruments, particularly
those that are significantly flux-limited with a range shorter than that of
the gravitational wave detectors, as for example neutrino detectors searching
for gravitational collapse events [42, 43]. In such a coincidence observation
the events will be relatively nearby, so the isotropy of the antenna pattern
is more important than its total volume. This illustrates the key point that
the importance attached to different values of the f.o.m.’s depends on one’s
priorities in building a new detector, a point also made in [7].
We define the f.o.m. Sky Coverage to be the fraction of the sky over which the
network’s antenna power pattern is greater than half of its maximum value. By
cutting the sky at this value we are accepting all directions where the reach
of the network is at least $1/\sqrt{2}\simeq 71\%$ of its mean horizon
distance $R_{N}$. The concept of sky coverage was discussed for single
detectors in Sathyaprakash and Schutz [34], but the sky cut was done there at
50% of the mean horizon distance. The place where the cut is made is clearly
arbitrary, but since detection is based on computing SNR2, I use the 50% power
level in defining this f.o.m..
Networks differ greatly in their isotropy. For a single interferometer, [SC]
is just 34%. Aligning antenna patterns keeps them anisotropic, so networks
including the LIGO detectors and an Australian detector tend to have low
values of [SC], while adding in VIRGO or LCGT increases isotropy. Again, this
is illustrated for specific interesting networks in section 4. The AHJLV
network, with detectors in Australia and Japan, reaches 85%, and adding a
detector in India pushes the sky coverage over 90%.
Detector | Label | Longitude | Latitude | Orientation
---|---|---|---|---
LIGO Livingston, LA | L | 90o 46’ 27.3” W | 30o 33’ 46.4” N | 208.0o(WSW)
LIGO Hanford, WA | H | 119o 24’ 27.6” W | 46o 27’ 18.5” N | 279.0o(NW)
VIRGO, Italy | V | 10o 30’ 16” E | 43o 37’ 53” N | 333.5o(NNW)
LCGT, Japan | J | 137o 10’ 48” E | 36o 15’ 00” N | 20.0o(WNW)
AIGO, Australia | A | 115o 42’ 51” E | 31o 21’ 29” S | 45.0o(NE)
INDIGO, India | I | 74o 02’ 59” E | 19o 05’ 47” N | 270.0o(W)
Table 1: Name, abbreviation, geographic location, and orientation of the
various detector positions considered in this paper. The abbreviations will be
used to label functions and diagrams. When there are two instruments at
Hanford we will use HH. The orientation is the geographic compass angle,
measured clockwise from North, of the line bisecting the arms of the detector.
(This decouples the orientation from opening angle for detectors that may not
have perpendicular arms.) For the averages performed in this paper, however,
the orientation will not matter. The data for the LIGO and VIRGO detectors are
for the actual detectors. The data for LCGT are for the planned orientation.
The data for AIGO are from the Australian group (private communication) and
place the detector at Gin-gin. The data for INDIGO are essentially arbitrary;
they correspond to the location of GMRT and an arbitrary orientation. Opening
angles $\eta$ are not listed because all detectors are assumed to have
$\eta=\pi/2$.
### 3.3 Accuracy
The biggest benefit of adding one or more detectors in Asia or Australia is
that they add longer baselines to the existing three detectors, and it is the
baseline that determines the accuracy with which the source can be located on
the sky. Source resolution is achieved by time-delay triangulation, so that
for fixed errors in measuring the time-of-arrival of a signal at different
detectors, longer baselines provide better relative accuracy and smaller sky-
position errors. Position accuracy in turn affects the determination of other
parameters: if the position is wrong then the inferred intrinsic amplitude of
the signal and its polarization will be wrong. This issue has been studied for
specific networks, particularly those containing a detector in Australia,
which offers the longest baselines [8, 44, 45]. These studies sometimes
provide detailed sky maps of error ellipses under various assumptions, and
they show that for any network the angular resolution varies considerably over
the sky. The purpose here is instead to develop a single measure that captures
the general difference in resolution when one compares two different networks.
The f.o.m. called Directional Precision attempts to provide a simple sky-
averaged measure of the relative accuracy with which a given network can
determine positions.
The problem of determining how accurately a network can measure positions has
a long history. Triangulation should produce angular position errors
proportional to the time-of-arrival measurement error divided by the baseline
between two detectors, measured in light-travel time [3]. But since three
detectors need to be involved in order to narrow down the position to a single
location on the sky (or at most two locations), the geometry of the detector
array is key. The first quantitative conjecture on the solid-angle uncertainty
for a network of three gravitational wave detectors appeared in Gürsel and
Tinto [12], who refer to a private communication by K S Thorne. The geometric
characteristic they use is the area $A_{\perp}$ of the triangle of the
detectors projected perpendicular to the direction to the source. The solid
angle error $\delta\Omega$ for a source in a particular direction is,
according to Gürsel and Tinto,
$\delta\Omega=2\frac{(c\delta t_{12})(c\delta t_{13})}{A_{\perp}},$ (30)
where $\delta t_{12}$ and $\delta t_{13}$ are the rms timing errors on two of
the arms of the triangle. This improves when the SNR improves because the
timing errors decrease. No proof of this expression seems to have appeared in
the literature until the recent work of Wen, Fan, and Chen [46, 8], who give a
much more general exact result that reduces to this when the network consists
of three identical detectors. I will base Directional Precision on a
simplification of the Wen-Fan-Chen expressions, which in their full form allow
the exact computation of position errors for networks of any number of non-
identical detectors.
Wen and Chen [8] show that the solid angle uncertainty is given by
$\displaystyle(\delta\Omega)^{-2}=\frac{\sum_{j,k,\ell,m}\xi_{j}\xi_{k}\xi_{\ell}\xi_{m}|(\mathbf{r}_{kj}\times\mathbf{r}_{m\ell})\cdot{\mathbf{n}}|^{2}}{\left[4\sqrt{2}\pi
c^{2}\sum_{j}\xi_{j}\right]^{2}},$ (31)
where the sum is over detectors in the network, $\mathbf{n}$ is the direction
to the source, and $\mathbf{r}_{kj}$ is the vector from detector $k$ to
detector $j$. (It follows that in the sum, $k$ and $j$ are distinct, as are
$m$ and $\ell$.) The symbol $\xi_{j}$ provides the timing accuracy, and for
our case, where we assume we can do perfect matched filtering, it is:
$\xi_{j}=\left<\omega^{2}\right>_{j}\rho^{2}_{j}=(\delta t_{arr,j})^{-2},$
(32)
where $\rho^{2}_{j}$ is the squared SNR in detector $j$, where
$\left<\omega^{2}\right>_{j}$ is the mean squared frequency in the signal,
averaged over the signal waveform in the detector weighted inversely by the
detector noise, and where $\delta t_{arr,j}$ is the r.m.s. time-of-arrival
measurement error in detector $j$ when there are no covariances with other
measurement errors [3]. Notice that (31) depends on the projected areas of all
the various triangles formed by the inter-detector vectors. If there are only
three detectors, there is only one triangle, and this expression essentially
reduces to (30).
The measure (31) is the inverse of one element of the error covariance matrix,
and is therefore an estimate of the inverse of the area of the 1-$\sigma$
error ellipse. It is also related to the Fisher information matrix element for
solid angle. My definition of Directional Precision in (34) below inherits
this: it is to be regarded as an indicator of the 1-$\sigma$ errors in area.
This is an important point to bear in mind when comparing with other authors,
who often quote 90th percentile or 2-$\sigma$ errors.
If we assume that all detectors are identical, then all the
$\left<\omega^{2}\right>_{j}$’s are the same and all the $\xi_{j}$’s are
proportional to the squares of their respective detector antenna pattern,
multiplied by factors that are common to all detectors. Our first
simplification will be to ignore the polarization-dependence of the antenna
patterns for the sources and take
$\xi_{j}=\left<\omega^{2}\right>P_{j}D_{L}^{2}/r^{2}.$
This is not strictly equivalent to taking a polarization average of the solid
angle uncertainty, but when using the expression to compare different networks
on average this should be a small correction. The sum $\sum_{j}\xi_{j}$ is
then proportional to the network power pattern $P_{N}$.
The next simplification is that I will replace each individual detector power
pattern $P_{j}$ by the average of the network power pattern, $P_{N}/N_{D}$.
Again this is in the spirit of finding a simple measure associated with the
network as a whole. It is equivalent to saying that the network power SNR is
equally shared by all detectors.
For the final step we have to decide what it is that we integrate to get a
measure of accuracy. Is it appropriate to find a measure of $|\delta\Omega|$,
$|\delta\Omega|^{2}$, $|\delta\Omega|^{-1}$, $|\delta\Omega|^{-2}$, …? Any of
these might be useful for comparing different networks. I shall opt for
something proportional to (with the previously mentioned simplifications) an
average value of $|\delta\Omega|^{-1}$, mainly for reasons of ease of
computation. This measure is more sensitive to locations where
$|\delta\Omega|$ is small, that is, where the network gives particularly good
directional information. An average of $|\delta\Omega|$ itself would be
dominated by the regions where directions are poor. Given the relationship
between (31) and the Fischer information, the measure used here can also be
thought of as an indicator (no more than that) of the directional information
contained in the measurement: the larger the value of Directional Precision
the more directional information we get.
It follows from these assumptions that
$\left<\left|(\delta\Omega)^{-2}\right|^{1/2}\right>\simeq\frac{R_{\oplus}^{2}\left<\omega^{2}\right>}{4\pi
c^{2}}\rho_{N,{\rm min}}^{2}[DP],$ (33)
where I define, for any network of $N_{D}$ detectors, the Directional
Precision of the network to be
$[DP]=N_{D}^{-2}(V_{N})^{-1}\int{\rm d}\Omega
P_{N}^{3/2}\left[\sum_{k>j,m>\ell}|(\tilde{\mathbf{r}}_{kj}\times\tilde{\mathbf{r}}_{m\ell})\cdot{\mathbf{n}}|^{2}\right]^{1/2}.$
(34)
Here $V_{N}$ the network’s total detection volume, normalized in such a way
that a single interferometer has maximum range 1 ((22) with $R_{0}=1$),
$R_{\oplus}$ is the Earth’s radius, and
$\tilde{\mathbf{r}}_{kj}=\mathbf{r}_{kj}/R_{\oplus}$ is the vector connecting
the locations of detectors $j$ and $k$ on the unit sphere (i.e. in latitude
and longitude).
Larger values of [DP] indicate better direction accuracy. The scale factor in
(33) evaluates straightforwardly to give
$\left<\left|(\delta\Omega)^{-2}\right|^{1/2}\right>\simeq 14\rho^{2}_{N,{\rm
min}}\left(\frac{\left<\omega^{2}\right>}{(2\pi\times 100\;\rm
Hz)^{2}}\right)[DP]\quad\mbox{sr}^{-1}.$ (35)
Note that, in the sum over detectors in (34), the sum is restricted to pairs
where $k$ exceeds $j$ and $m$ exceeds $\ell$. This is justified because, as
noted above, these indices cannot be equal and because including values where
$k<j$ would simply count the same detector pair twice. The coefficient in
front of the sum has been increased by a factor of $\sqrt{2}$ to compensate.
Terms for which $k$ equals $m$ and $j$ equals $\ell$ also vanish because they
involve the cross product of a vector with itself. The sum shown therefore has
$(N_{D}+1)N_{D}(N_{D}-1)(N_{D}-2)/4$ nonvanishing terms. This number of terms,
inside the square root, is roughly compensated by the factor of $N_{D}^{-2}$
outside the integral, which arose from our simplification in which we replaced
each individual detector power pattern $P_{j}$ by the average of the network
power pattern $P_{N}/N_{D}$. The fact that $N_{D}$ roughly cancels out means
that [DP] depends more on the size of the detector triangles than on the
number of detectors in the network: extending the baselines in a network has
more effect on angular accuracy than does adding more detectors with similar
baselines to the existing ones.
It should be noted that [DP] measures the average position accuracy of
detected signals, not the accuracy on a given signal with a fiducial
amplitude. If network A is more sensitive than network B, so that A has a
bigger detection volume, then its position accuracy will be averaged over a
population that includes more distant and weaker sources than those of B. If
we only asked how network A would perform on the detection volume of network
B, its mean direction accuracy would be better than one might guess just by
comparing $[DP]_{A}$ with $[DP]_{B}$. So when using [DP] to compare the
performance of different networks, it is somewhat easier to interpret when it
is used to compare networks with the same number of detectors but different
geometries.
When combined with the values of [DP] we compute in table 2, this is not
inconsistent with the plots of error ellipses in the literature [8, 9, 45,
10]. The dependence on threshold $\rho_{N,{\rm min}}$ is interesting: the
higher the threshold, the stronger the ensemble of detected SNRs, so the
larger the value of [DP], and the better the direction-finding.
Network | Mean Horizon Distance | Detection Volume | Volume Filling Factor | Triple Detection Rate (at 80%) | Triple Detection Rate (at 95%) | Sky Coverage | Directional Precision
---|---|---|---|---|---|---|---
L | 1.00 | 1.23 | 29% | - | - | 33.6% | -
HLV | 1.43 | 5.76 | 47% | 2.95 | 4.94 | 71.8% | 0.68
HHLV | 1.74 | 8.98 | 41% | 4.86 | 7.81 | 47.3% | 0.66
AHLV | 1.69 | 8.93 | 44% | 6.06 | 8.28 | 53.5% | 3.01
HHJLV | 1.82 | 12.1 | 48% | 8.37 | 11.25 | 73.5% | 2.57
HHILV | 1.81 | 12.3 | 50% | 8.49 | 11.42 | 71.8% | 2.18
AHJLV | 1.76 | 12.1 | 53% | 8.71 | 11.25 | 85.0% | 4.24
HHIJLV | 1.85 | 15.8 | 60% | 11.43 | 14.72 | 91.4% | 3.24
AHIJLV | 1.85 | 15.8 | 60% | 11.50 | 14.69 | 94.5% | 4.88
Table 2: Comparison of various networks. A: AIGO or LIGO Australia; H: LIGO
Hanford single detector; HH: LIGO Hanford two detectors; I: INDIGO; J: LCGT;
L: LIGO Livingston; V: VIRGO. Mean Horizon Distance is the maximum detection
distance, scaled to the mean horizon distance (maximum range) of a single
detector observing at the same threshold. Detection Volume is the volume
inside the antenna pattern, on the same scale. Volume Filling Factor is the
ratio between the Detection Volume in column 3 and the volume of a sphere with
radius equal to the Maximum Range in column 2. The remaining columns are the
figures of merit. Triple Detection Rate measures the overall detection rate
and is given for two different values of the duty cycle: 80% to represent a
likely figure at the start of operations, and 95% to represent a reasonable
long-term operation goal. The values of Triple Detection Rate are smaller than
the Detection Volume by factors representing the loss of 3-site observing time
to duty cycle downtime. Sky Coverage measures how isotropic the network
antenna pattern is. Directional Precision reflects angular accuracy: the
typical solid angle uncertainty is inversely proportional to Directional
Precision, so that larger values denote more accurate networks. The first row
of the table is for a single detector, to facilitate comparisons.
## 4 Lessons
### 4.1 Discussion of specific networks
At the present time the only network of Advanced detectors that is fully
approved and funded consists of two LIGO detectors at Hanford and one at
Livingston, plus VIRGO in Italy: HHLV. Working together, these four detectors
have a detection volume of 8.98, more than 7 times that of a single detector
at the same network threshold. But when the duty cycle is 80% the effective
volume [3DR] falls to 4.86. The network covers 47% of the sky at half-power.
Its value of 0.66 for [DP] is the starting point for comparisons of network
accuracy.
In addition to these detectors, funding has started for the LCGT detector in
Japan, so it is reasonable to expect that the Advanced network will include
detectors at Hanford and Livingston in the USA, in Italy, and in Japan. If the
current proposal to move one of the Hanford detectors to Australia becomes
reality, then we should have the network AHJLV. If not, then we are likely to
have HHJLV. In addition, if a proposal to build a detector in India succeeds,
then in the long run we could have AHIJLV or HHIJLV.
To understand the capabilities of these networks it is useful to compare them
with the basic HHLV and with LIGO’s own variant AHLV. These comparisons will
show clearly the considerable benefits brought by the ongoing investment in
Japan and the proposed investment in India.
First we ask, how does AHLV compare to HHLV? The range and volume of AHLV are
very similar to those of HHLV. Its effective detection rate, [3DR], however,
is 25% larger: 6.06 (compared to 4.86) at a duty cycle of 80%, simply because
there are more three-site sub-networks in this array. AHLV is slightly more
isotropic than HHLV, with [SC] equal to 53.5%. This reflects the fact that the
position of the Australian detector at Gingin is very close to being antipodal
to the LIGO detectors. So far these network characteristics are not very
different from HHLV. But the real improvement is in direction finding. The
value of [DP] for AHLV is 3.01, compared with 0.66 for HHLV. This suggests
that the typical error ellipses will be reduced in area by more than 4 if the
detector is moved to Australia. These numbers are consistent with the results
of the much more extensive comparison of these two networks in an unpublished
internal technical report of the LIGO Scientific Collaboration [9] and in a
recent study of coherent detection involving LIGO Australia [10], and they
give a very strong scientific reason for placing the LIGO instrument in
Australia, independently of other detector developments.
Next we examine the improvements brought by the LCGT detector in Japan, with
the simplifying assumption that it will have identical sensitivity to the
other Advanced detectors. If there is no detector in Australia then we will
have the network HHJLV. Its overall detection volume, at 12.1 (figure 5), is
significantly greater that that of HHLV (8.98) and AHLV (8.93), reflecting the
fact that there is one further detector. The improvement in the detection rate
as measured by Triple Detection Rate is even greater: with a Japanese detector
and duty cycles of 80% the rate of detection would be more than 70% higher
than for the basic HHLV, and more than a third higher than AHLV. The network
is also significantly more isotropic as well, with [SC] at 73.5% (figure 6).
Adding the baseline to Japan also greatly improves the direction-finding,
although not by as much as the longer Australian baselines would: for HHJLV
the value of [DP] is 2.57, much better than the 0.66 turned in by HHLV but a
bit below the 3.01 value of AHLV. Nevertheless, the improvement over the basic
HHLV still represents a 4-fold reduction in the typical area of the error
ellipses.
Figure 5: Three network amplitude patterns, which show the true spatial shape
of the detection volumes. As in figure 3, two views are shown, one in
perspective and the other as a contour plot. The networks are: (top row) the
basic network of two instruments at Hanford, one at Livingston, and one at
Pisa; (middle row): the basic network with LCGT in Japan added; (bottom row)
the same after moving one of the Hanford detectors to Australia. Notice that
all these networks have roughly the same maximum range (HHLV: 1.74; HHJLV:
1.82; AHJLV: 1.76), and these are the values to which the contour levels are
scaled. They have different volumes (HHLV: 8.98; HHJLV: 12.1; AHJLV: 12.1)
because of their different isotropy, shown in figure 6. (The numbers are taken
from table 2.)
The Japanese detector may instead operate with a LIGO detector in Australia.
To see the difference with the characteristics we found in the previous
paragraph, we compare AHJLV with HHJLV. In detection volume and event rate the
two networks are essentially indistinguishable (figure 5). Sky coverage goes
up a noticeable amount with the Australian option, from 73.5% to 85% (figure
6). And, as might be expected, the extra baselines to Australia and between
Japan and Australia improve the direction finding. The value of [DP] for AHJLV
is 4.24, compared with 2.57 for HHJLV. So also here the improvement in angular
position information provides a strong reason for putting the LIGO detector in
Australia. Conversely, if one takes the Australian detector as a given and
asks what improvement is brought by the detector in Japan, the comparison is
between AHJLV and AHLV. Here not only is direction-finding significantly
better (4.24 compared to 3.01), but there is a dramatic increase in isotropy
(from 53.5% to 85%) and a factor of 1.4 increase in event rate (from 6.06 to
8.71 at 80% duty cycle).
On the basis of these numbers the network with the Australian option and the
LCGT instrument in Japan looks close to the ideal use of the resources being
invested by the various countries involved. It will have nearly twice as many
detections per year as the basic HHLV would if it could operate in coherent
detection mode (see below), at 80% duty cycle. It will cover nearly twice the
sky area. And its typical direction error ellipses can be a factor of 6
smaller in area. These benefits are brought simply by building one further
detector in Japan and moving a detector from the US to Australia.
Figure 6: Three network isotropy patterns, which show the parts of the unit
sphere where the amplitude sensitivity of the detector is better than
$\sqrt{2}$ of its best sensitivity. The networks are the same as in figure 5.
A nascent project in India might also succeed in building a detector. I have
included it in networks by placing it rather arbitrarily at the site of the
Giant Metrewave Radio Telescope (GMRT) radio telescope. It is interesting to
ask what the properties of networks containing this detector would be. I
include the Japanese detector and consider the two LIGO options: HHIJLV and
AHIJLV. Adding the Indian detector to the existing HHJLV network increases the
event rate by roughly 1/3, regardless of duty cycle. Considering that this is
achieved by adding one detector to a network of 5, which is an investment of
20% on top of the existing expenditure, getting a return of 33% in terms of
science still makes a strong case for this development. The detector in India
also improves isotropy, from 74% to 91%. And the extra baselines improve
position error ellipses, as measured by [DP], by 30%. If the Australian
detector is also built, then we compare AHJLV with AHIJLV. Again the Indian
detector brings an improvement of around 1/3 in event rate and it achieves
nearly complete isotropy, with a value of [SC] of 95%. It brings a 15%
improvement in position error ellipses, as measured by [DP], simply by adding
more baselines to the network.
Several of these networks have recently been studied also by Fairhurst [45],
who concentrated on the localization ability, using a different approach than
that adopted here, and one that is closer to the present methods of data
analysis based on thresholding. His results on comparisons of the localization
abilities of different networks are broadly in agreement with the relative
values of [DP] in table 2, and the typical ellipse areas that the present
treatment gives using (35) are within factors of two of the typical values
obtained by Fairhurst. This gives us confidence that our figures of merit can
be used not only to compare networks but also, to within factors of two, to
characterize the performance of individual networks.
Another useful comparison is between our analytic results and the Monte-Carlo
simulations for coalescing binaries performed by Nissanke, et al [11]. They
take HLV as their baseline network, i.e. assuming only one detector at
Hanford, and they do not allow for duty cycle down-time. They find that AHLV
will detect 1.48 times more events than HLV. In table 2 the appropriate
comparison is between the full detection volumes of HLV and AHLV, whose ratio
is 1.55. We take this to be excellent agreement. Moreover, they measure the
isotropy of various networks by plotting detected event distributions on the
sky (their figure 2). Their conclusions are qualitatively in agreement with
ours in figure 6, and they remark that networks that include LCGT are
noticeably more isotropic, a conclusion also in agreement with our values of
Sky Coverage in table 2.
Note, however, that the true “default” network is HHLV, and in table 2 it is
clear that moving one of the H detectors to Australia hardly changes the total
detection volume. When network duty cycle is taken into account, there is a
net event rate gain (for three-site detections) of up to a factor of $1.24$.
On top of that there is an event rate gain from being able to do coherent data
analysis better, so that the LIGO Australia option not only has better angular
resolution but also a significantly higher detected event rate. This is the
subject of the next section.
### 4.2 Coherent versus coincidence data analysis: implications for event
detection rates
The assumption of this paper is that data analysis is done by fully coherent
combination of the different detectors’ data streams. This is not yet the
practice in the LSC-VIRGO data analysis, mainly because coherent analysis
normally assumes a Gaussian background of instrumental noise, and is therefore
vulnerable to what are often called “glitches”, bursts of noise from
instrumental effects that can masquerade as real signals. Because in present
detectors there is a significant glitch background, data analysis usually
includes a coincidence step, in which events of a sufficient size in single
data streams that occur in coincidence (within a time-window equal to the
light travel times among the various detectors) with events in other detectors
are selected and studied further. This coincidence test eliminates most of the
glitch background.
But a purely coincident analysis also eliminates most of the potentially
detectable signals, i.e. signals that could reliably be detected if the
background noise were ideally Gaussian. The penalty is easy to compute. In a
recent review of the astrophysical evidence for the rates of compact object
binary coalescences, the LSC and VIRGO collaborations predicted a detected
event rate for the HHLV network of Advanced detectors [47]. Their method was
to take the number of events that occur inside the detection volume of a
single detector above the detection threshold $\rho_{\rm min}=8$. They took
the most likely value of the rate of neutron-star coalescences to be
$1\,\rm{Mpc}^{-3}\,{Myr}^{-1}$, or equivalently 100 events per Milky Way
Equivalent Galaxy per million years. With this volume event rate, the most
likely detection rate for these systems came out to be 40 per year. The reason
for counting only events that occur in one detector’s detection volume despite
the fact that the network contains four detectors is to approximate in a rough
(and conservative) way the coincidence criterion.
For the same network, but with coherent data analysis using a network
threshold of the same value ($\rho_{\rm N,\,min}=8$.), the data in table 2
show that the rate would be higher by the ratio of [3DR] which is 4.86
(allowing for an 80% duty cycle), to the volume for a single detector, 1.23.
This ratio is 3.95, which implies that the HHLV network, with perfectly
Gaussian noise, could detect about 160 events per year if it did coherent
analysis. The difference in detection effectiveness between coherent and
coincidence analysis for coalescing binary signals in this basic network is a
factor of 4 in detection rate. This difference is illustrated graphically by
comparing the volumes of space covered by fully coherent analysis and pure
coincidence analysis, in figure 7.
Figure 7: The antenna patterns of the LIGO-VIRGO detectors for (a) coherent
and (b) coincidence analysis methods. The coherent pattern is the HHLV
amplitude pattern. The coincidence pattern is the region in which, for random
polarizations, an event crosses threshold in at least two of the detectors
(but not allowing events that appear only in two Hanford detectors). The
thresholds are assumed to be the same, e.g. if the individual detector
thresholds for the coincidence analysis is 8, then the coherent data analysis
threshold is also set at 8, as discussed in the text.
Naturally this comparison depends on the threshold assumed for the two kinds
of data analysis. The comparison shown in the figure is for equal thresholds:
if, as in [47], the coincidence observation is done with a threshold SNR of 8
in each detector, then we assume that the network coherent threshold is set at
8 as well. This is not unreasonable, since the coherent analysis essentially
fights only against Gaussian noise, where events at $8\sigma$ occur only once
in $10^{5}$ years at an effective sampling rate of 300 Hz. This works well if
coherent methods can eliminate glitches. This may not be fully possible for
HHLV (see below) but it should be possible for the enlarged networks,
including AHLV. Therefore the comparison shown in this figure is relevant for
extrapolations of event rates to the larger networks.
Now, for the existing detectors, instrumentalists are working hard to reduce
the glitch rate, and the LSC-VIRGO analysis teams are bringing in coherent
analysis [28, 19, 29, 30, 33]. Networks containing three or more detectors can
also use their null streams to test for and veto glitches, as described in
section 1.2.
In practice, the analysis teams will begin by mixing coincidence and coherence
methods, by setting a low threshold on coincidences to obtain a population of
possible events, and then using coherent methods (including null streams) to
eliminate the glitch coincidences. Such methods are computationally much less
demanding than purely coherent methods, and they can presumably bridge some of
the gap of the factor of four between pure coincidence and full coherent
methods.
However, the basic HHLV network may not be completely amenable to coherent
analysis, because of the near-perfect alignment of the LIGO Hanford and LIGO
Livingston detectors. While this allows good discrimination against glitches
in one of the LIGO detectors, it reduces the information recoverable from real
events: polarization and sky location can be determined only if VIRGO is
excited comparably strongly to the LIGO detectors, and without a sky location
one cannot define a null stream. This in turn leads to more opportunities for
false alarms, and lowers the significance of real events. It remains to be
seen how much of the full factor of 4 computed above can be recovered by
introducing some degree of coherent analysis into the HHLV network, but
clearly it is a very important step to take. If the nominal detection rate of
40 events per year can be raised even to 80, this will be the most cost-
effective way to improve the baseline network.
It is worth noting that the LSC study of the LIGO Australia option [9] made a
strong recommendation to move to coherent data analysis. The move of one LIGO
detector to Australia breaks the degeneracy of the LIGO instruments,
especially if the new detector is anti-aligned with the existing LIGO
detectors. It should therefore allow fully robust coherent analysis, coming
close to the maximum possible event detection rate of 200 NS-NS events per
year, assuming the most likely rate quoted in [47], and assuming an 80% duty
cycle. The improvement of a factor of up to 5 in the detection rate is
probably the strongest reason for placing a LIGO detector in Australia.
The LCGT detector will add a third null stream to the HHLV or AHLV networks,
and make coherent analysis even more robust. If the most likely coalescence
rates prove to be accurate, and if the network detection threshold is set to
8, the HHJLV network can expect to detect 270 NS-NS coalescences per year, and
the AHJLV network 280. Adding a detector in India raises these numbers to
around 370 events per year. Improving the duty cycle to 95%, which seems
feasible after a few years of operation, increases the five-detector NS-NS
rates to around 360 per year and the six-detector rate nearly to 500 per year.
These rate improvements would qualitatively change the kind of science
obtainable from Advanced detectors.
For coalescences of neutron stars with black holes, the LSC and VIRGO paper
[47] quotes a “best” rate of 10 per year for HHLV with coincidence analysis.
The expected rates for larger networks can therefore be obtained from the NS-
NS rates just quoted by dividing by 4. Similarly, the rates for binary black
hole mergers are expected to be half of the NS-NS rates; black holes have a
much lower number density in the universe, but they can be detected much
further away because of their higher mass. The NS-BH and BH-BH rates are, of
course, much less secure than the NS-NS rates, because there are no observed
binary systems of those types; the rates used in [47] depend exclusively on
population simulations. However, the recent identification of two possible
X-ray binary precursors of BH-BH binaries provides a much-needed observational
normalization of the population. Bulik et al [48] conclude from these systems,
in which a black hole is in a close binary with a Wolf-Rayet star, that the
BH-BH detection rate might in fact be much higher and could even significantly
exceed the NS-NS rate.
One further item is worth noting. Searches for binary signals are optimal if
they incorporate as much prior information as possible, and Bayesian analysis
techniques that do this are becoming standard in the current LSC-VIRGO data
analysis methods. The present study provides three such priors: the network
antenna pattern (a prior on the sky location of the source) and the two
p.d.f.’s: the expected distribution of SNR values (2.4), which is a prior on
the signal amplitude; and (for binaries) the expected distribution of detected
inclination angles ((28) and figure 4), which is a prior that affects the
relative amplitudes and phases of the signal in different detectors. The use
of the antenna pattern as a prior needs to be done with care, because as noted
above there will be a number of sources detected that are outside the “hard”
edge of the detection volume. A polarization-dependent prior is of course even
better than the polarization-averaged antenna pattern computed here.
## 5 Conclusions
In this paper I have developed a framework in which it is possible to compare
networks of gravitational wave interferometers consisting of different numbers
of detectors in different geographical configurations. I have shown that, for
any network, the expected SNR distribution of detected events, once the data
analysis can be done by optimal coherent methods, is a universal $\rho^{-4}$
power law that falls to zero for $\rho$ smaller than the detection threshold.
It follows from this distribution that the most likely SNR of the first
detected signal will be about 1.26 times the threshold of the search. I have
derived the (similarly universal) probability distribution of the inclination
angle of detected binary systems, and I have shown that, if coalescing
binaries are associated with narrowly beamed gamma-ray bursts, then because
the radiated gravitational wave power is correlated with the direction of the
gamma-ray cone, we can expect 3.4 times more detected coincidences than if
they were not correlated. I have suggested three figures of merit that can be
computed for any network and which measure average properties of the network:
its expected event detection rate, its isotropy, and the accuracy of its sky
position measurements. These figures of merit are inevitably crude averages,
and they should not be a substitute for detailed comparisons of networks as
part of the planning for specific new detectors. But they give a clear
indication of the merit of enlarging the network from the originally planned
LIGO and VIRGO detectors to include detectors in Asia and Australia.
It is worth stepping back from the many different options that exist for
enlarging the worldwide interferometer network to consider the net
improvements that are possible if current plans are realized. Consider the
network AHJLV, consisting of LIGO with one detector in Hanford and one in
Livingston, VIRGO in Italy, LCGT in Japan, and LIGO Australia. The numbers in
table 2 show how much more science that network can do than the originally
planned HHLV. Its event rate, with detectors operating on 80% duty cycles,
would be nearly twice as high for all categories of burst sources. It would
cover nearly twice as much of the sky, making it a better bet for coincidence
observations with neutrino detectors. And our measure of the areas of angular
position measurement error ellipses improves by a factor of 6.4, from 0.66 to
4.24, indicating that the typical error ellipse goes down in area by a factor
of more than 6. This will make a huge improvement in follow-up studies with
optical and other telescopes. This network offers much more science than had
been promised in the initial proposals for the existing four large detectors,
at the cost of building only one more detector and moving another to a better
location. The impact of the single extra detector in Japan is so large because
robust gravitational wave astronomy requires a minimum of three detectors in
different locations, so the marginal impact of increases to four and five is
large.
If the project in India gains support and, on a longer timescale, leads to a
sixth Advanced detector, it would create the network AHIJLV, an even bigger
improvement on HHLV. Its event rate would be 2.4 times higher, on a duty cycle
of 80%. It would cover 95% of the sky at half power, and its sky localization
error ellipses would be fbetter than 7 times smaller in area than those of the
presently planned LIGO-VIRGO network.
It is important to realize that both of these enlarged networks have maximum
detection distances that are within 5% of the maximum range of HHLV. Their
large event rate gains come partly from increased isotropy and partly from
having more three-site sub-networks that can detect and localize events even
when one or more detectors has fallen out of observing mode. They survey the
same volume of space more completely than HHLV can. But the big improvements
in sky localization are perhaps the strongest arguments for pursuing these
enlarged networks. The values of Directional Precision we compute here suggest
(using the conversion to steradians given above) that the typical error box in
either network would be smaller than a degree on a side. This not only makes
searching with electromagnetic telescopes for counterparts easier but it
reduces the probability of chance coincidences in a large field of view.
The conclusions in this paper depend strongly on the assumption of coherent
data analysis. If coincidence data analysis is used, where events are selected
for further study only if they cross a particular threshold in each
participating detector, there is no guarantee that the properties described
here will still hold for the different networks. Coherent analysis produces
networks whose antenna patterns are the sum of the power patterns of the
network members. Coincidence analysis produces antenna patterns that are
basically determined by the intersections of the power patterns of network
members. Performing a first cut at the noise by coincidence analysis, even if
it is followed by a coherent follow-up, will not reproduce the assumptions
used here. The reason for coincidence analysis is, of course, to eliminate
rare but strong non-Gaussian noise events, but these can also be identified by
using network null streams, whose number increases with the number of
detectors in the network.
Moving from coincidence to coherent analysis can increase detection rates by
factors of four or more. It is to be expected that network data analysis will
move to fully coherent analysis as the number of detectors increases and as
experimenters manage over time to reduce the frequency and amplitude of non-
Gaussian noise glitches. With such analysis techniques, the full potential of
the enlarged networks, as illustrated by the figures of merit calculated here,
can eventually be realized.
It is a pleasure to acknowledge discussions of the network problem with many
colleagues, including B Krishnan, S Dhurandhar, J Hough, K Kuroda, A
Lazzarini, and J Marx. Special thanks to S Klimenko, S Nissanke, M-A Papa, P
Sutton, B Sathyaprakash, and L Wen for detailed discussions and comments. This
work was stimulated by a kind invitation from K Kuroda to the 58th Fujihara
Seminar in 2009. DFG grant SFB/TR-7 is gratefully acknowledged.
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|
arxiv-papers
| 2011-02-26T14:47:55 |
2024-09-04T02:49:17.315103
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bernard F. Schutz",
"submitter": "Bernard Schutz",
"url": "https://arxiv.org/abs/1102.5421"
}
|
1102.5509
|
TKK Dissertations in Information and Computer Science
Espoo 2010 TKK-ICS-D19
PROBABILISTIC ANALYSIS OF THE HUMAN
TRANSCRIPTOME WITH SIDE INFORMATION
Leo Lahti
Dissertation for the degree of Doctor of Science in Technology to be presented
with due permission of the Faculty of Information and Natural Sciences for
public examination and debate in Auditorium AS1 at the Aalto University School
of Science and Technology (Espoo, Finland) on the 17th of December 2010 at 13
o’clock.
Aalto University School of Science and Technology
Faculty of Information and Natural Sciences
Department of Information and Computer Science
Aalto-yliopiston teknillinen korkeakoulu
Informaatio- ja luonnontieteiden tiedekunta
Tietojenkäsittelytieteen laitos
Distribution:
Aalto University School of Science and Technology
Faculty of Information and Natural Sciences
Department of Information and Computer Science
P.O.Box 15400
FI-00076 Aalto
FINLAND
Tel. +358-9-470 23272
Fax +358-9-470 23277
Email: series@ics.tkk.fi
Copyright ©2010 Leo Lahti
First Edition. Some Rights Reserved.
http://www.iki.fi/Leo.Lahti (leo.lahti@iki.fi)
This thesis is licensed under the terms of Creative Commons Attribution 3.0
Unported license available from http://www.creativecommons.org/. Accordingly,
you are free to copy, distribute, display, perform, remix, tweak, and build
upon this work even for commercial purposes, assuming that you give the
original author credit. See the licensing terms for details. For Appendices
and Figures, consult the separate copyright notices.
ISBN 978-952-60-3367-9 (Print)
ISBN 978-952-60-3368-6 (Online)
ISSN 1797-5050 (Print)
ISSN 1797-5069 (Online)
URL: http://lib.tkk.fi/Diss/2010/isbn9789526033686/
Multiprint Oy
Espoo 2010
ABSTRACT
Lahti, L. (2010): Probabilistic analysis of the human transcriptome with side
information Doctoral thesis, Aalto University School of Science and
Technology, Dissertations in Information and Computer Science, TKK-ICS-D19,
Espoo, Finland.
Keywords: data integration, exploratory data analysis, functional genomics,
probabilistic modeling, transcriptomics
Recent advances in high-throughput measurement technologies and efficient
sharing of biomedical data through community databases have made it possible
to investigate the complete collection of genetic material, the genome, which
encodes the heritable genetic program of an organism. This has opened up new
views to the study of living organisms with a profound impact on biological
research.
Functional genomics is a subdiscipline of molecular biology that investigates
the functional organization of genetic information. This thesis develops
computational strategies to investigate a key functional layer of the genome,
the transcriptome. The time- and context-specific transcriptional activity of
the genes regulates the function of living cells through protein synthesis.
Efficient computational techniques are needed in order to extract useful
information from high-dimensional genomic observations that are associated
with high levels of complex variation. Statistical learning and probabilistic
models provide the theoretical framework for combining statistical evidence
across multiple observations and the wealth of background information in
genomic data repositories.
This thesis addresses three key challenges in transcriptome analysis. First,
new preprocessing techniques that utilize side information in genomic sequence
databases and microarray collections are developed to improve the accuracy of
high-throughput microarray measurements. Second, a novel exploratory approach
is proposed in order to construct a global view of cell-biological network
activation patterns and functional relatedness between tissues across normal
human body. Information in genomic interaction databases is used to derive
constraints that help to focus the modeling in those parts of the data that
are supported by known or potential interactions between the genes, and to
scale up the analysis. The third contribution is to develop novel approaches
to model dependency between co-occurring measurement sources. The methods are
used to study cancer mechanisms and transcriptome evolution; integrative
analysis of the human transcriptome and other layers of genomic information
allows the identification of functional mechanisms and interactions that could
not be detected based on the individual measurement sources. Open source
implementations of the key methodological contributions have been released to
facilitate their further adoption by the research community.
TIIVISTELMÄ
Lahti, L. (2010): Ihmisen geenien ilmentymisen ja taustatiedon tilastollinen
mallitus Väitöskirja, Aalto-yliopiston teknillinen korkeakoulu, Dissertations
in Information and Computer Science, TKK-ICS-D19, Espoo, Suomi.
Avainsanat: aineistojen yhdistely, data-analyysi, toiminnallinen genomiikka,
tilastollinen mallitus, geenien ilmentyminen
Mittausmenetelmien kehitys ja tutkimustiedon laajentunut saatavuus ovat
mahdollistaneet ihmisen perimän eli genomin kokonaisvaltaisen tarkastelun.
Tämä on avannut uusia näkökulmia biologiseen tutkimukseen ja auttanut
ymmärtämään elämän syntyä ja rakennetta uusin tavoin. Toiminnallinen
genomiikka on molekyylibiologian osa-alue, joka tutkii perimän toiminnallisia
ominaisuuksia. Perimän toimintaan liittyvää mittausaineistoa on runsaasti
saatavilla, mutta korkeaulotteisiin mittauksiin liittyy monimutkaisia ja
tuntemattomia taustatekijöitä, joiden huomiointi mallituksessa on
haasteellista. Tehokkaat laskennalliset menetelmät ovat avainasemassa
pyrittäessä jalostamaan uusista havainnoista käyttökelpoista tietoa.
Tässä väitöskirjassa on kehitetty yleiskäyttöisiä laskennallisia menetelmiä,
joilla voidaan tutkia ihmisen geenien ilmentymistä koko perimän tasolla.
Geenien ilmentyminen viittaa lähetti-RNA-molekyylien tuottoon solussa perimän
sisältämän informaation nojalla. Tämä on keskeinen perinnöllisen informaation
säätelytaso, jonka avulla solu säätelee proteiinien tuottoa ja solun toimintaa
ajasta ja tilanteesta riippuen. Tilastollinen oppiminen ja todennäköisyyksin
perustuva probabilistinen mallitus tarjoavat teoreettisen kehyksen, jonka
avulla rinnakkaisiin mittauksiin ja taustatietoihin sisältyvää informaatiota
voidaan käyttää kasvattamaan mallien tilastollista voimaa. Kehitetyt
menetelmät ovat yleiskäyttöisiä laskennallisen tieteen tutkimusvälineitä,
jotka tekevät vähän, mutta selkeästi ilmaistuja mallitusoletuksia ja sietävät
korkeaulotteisiin toiminnallisen genomiikan havaintoaineistoihin sisältyviä
epävarmuuksia.
Väitöskirjassa kehitetyt menetelmät tarjoavat ratkaisuja kolmeen keskeiseen
mallitusongelmaan toiminnallisessa genomiikassa. Luotettavien
esikäsittelymenetelmien kehittäminen on työn ensimmäinen päätulos, jossa
tietokantoihin sisältyvää taustatietoa käytetään perimänlaajuisten
mittausaineistojen epävarmuuksien vähentämiseksi. Toisena päätuloksena
väitöskirjassa kehitetään uusi aliavaruuskasautukseen perustuva menetelmä,
jonka avulla voidaan tutkia ja kuvata solubiologisen vuorovaikutusverkon
käyttäytymistä kokonaisvaltaisesti ihmiskehon eri osissa. Taustatietoa geenien
vuorovaikutuksista käytetään ohjaamaan ja nopeuttamaan mallitusta.
Menetelmällä saadaan uutta tietoa geenien säätelystä ja kudosten
toiminnallisista yhteyksistä. Kolmanneksi väitöskirjatyössä kehitetään uusia
menetelmiä perimänlaajuisten mittausaineistojen yhdistelyyn. Ihmisen geenien
ilmentymisen ja muiden aineistojen riippuvuuksien mallitus mahdollistaa
sellaisten toiminnallisten yhteyksien ja vuorovaikutusten havaitsemisen,
joiden tutkimiseksi yksittäiset havaintoaineistot ovat riittämättömiä.
Aineistojen yhdistelyyn kehitettyjä menetelmiä sovelletaan syöpämekanismien ja
lajien välisten eroavaisuuksien tutkimiseen. Julkaistuilla avoimen lähdekoodin
toteutuksilla on pyritty varmistamaan kehitettyjen menetelmien saatavuus ja
laajempi käyttöönotto laskennallisen biologian tutkimuksessa.
###### Contents
1. Preface
1. LIST OF PUBLICATIONS
2. SUMMARY OF PUBLICATIONS AND THE AUTHOR’S CONTRIBUTION
3. LIST OF ABBREVIATIONS AND SYMBOLS
2. 1 Introduction
1. 1.1 Contributions and organization of the thesis
3. 2 Functional genomics
1. 2.1 Universal genetic code
1. 2.1.1 Protein synthesis
2. 2.1.2 Layers of regulation
2. 2.2 Organization of genetic information
1. 2.2.1 Genome structure
2. 2.2.2 Genome function
3. 2.3 Genomic data resources
1. 2.3.1 Community databases and evolving biological knowledge
2. 2.3.2 Challenges in high-throughput data analysis
4. 2.4 Genomics and health
4. 3 Statistical learning and exploratory data analysis
1. 3.1 Modeling tasks
1. 3.1.1 Central concepts in data analysis
2. 3.1.2 Exploratory data analysis
3. 3.1.3 Statistical learning
2. 3.2 Probabilistic modeling paradigm
1. 3.2.1 Generative modeling
2. 3.2.2 Nonparametric models
3. 3.2.3 Bayesian analysis
3. 3.3 Learning and inference
1. 3.3.1 Model fitting
2. 3.3.2 Generalizability and overlearning
3. 3.3.3 Regularization and model selection
4. 3.3.4 Validation
5. 4 Reducing uncertainty in high-throughput microarray studies
1. 4.1 Sources of uncertainty
2. 4.2 Preprocessing microarray data with side information
3. 4.3 Model-based noise reduction
4. 4.4 Conclusion
6. 5 Global analysis of the human transcriptome
1. 5.1 Standard approaches
2. 5.2 Global modeling of transcriptional activity in interaction networks
3. 5.3 Conclusion
7. 6 Human transcriptome and other layers of genomic information
1. 6.1 Standard approaches for genomic data integration
1. 6.1.1 Combining statistical evidence
2. 6.1.2 Role of side information
3. 6.1.3 Modeling of mutual dependency
2. 6.2 Regularized dependency detection
1. 6.2.1 Cancer gene discovery with dependency detection
3. 6.3 Associative clustering
1. 6.3.1 Exploratory analysis of transcriptional divergence between species
4. 6.4 Conclusion
8. 7 Summary and conclusions
9.
## Preface
This work has been carried out at the Neural Networks Research Centre and
Adaptive Informatics Research Centre of the Laboratory of Computer and
Information Science (Department of Information and Computer Science since
2008), Helsinki University of Technology, i.e., as of 2010 the Aalto
University School of Science and Technology. Part of the work was done at the
Department of Computer Science, University of Helsinki, when I was visiting
there for a year in 2005. I am also pleased to having had the opportunity to
be a part of the Helsinki Institute for Information Technology HIIT. The work
has been supported by the Graduate School of Computer Science and Engineering,
as well as by project funding from the Academy of Finland through the SYSBIO
program and from TEKES through the MultiBio research consortium. The Graduate
School in Computational Biology, Bioinformatics, and Biometry (ComBi) has
supported my participation to scientific conferences and workshops abroad
during the thesis work.
I wish to thank my supervisor, professor Samuel Kaski for giving me the
opportunity to work in a truly interdisciplinary research field with the
freedom and responsibilities of scientific work, and with the necessary amount
of guidance. These have been essential parts of the learning process.
I would also like to express my gratitude to the reviewers of this thesis,
Professor Juho Rousu and Doctor Simon Rogers for their expert feedback.
Research on computational biology has given me the excellent opportunity to
work with and learn from experts in two traditionally distinct disciplines,
computational science and genome biology. I am particularly grateful to
professor Sakari Knuutila for his enthusiasm, curiosity, and personal example
in collaboration and daily research work. Researchers in the Laboratory of
Cytomolecular Genetics at the Haartman Institute have provided a friendly and
inspiring environment for active collaboration during the last years.
My sincere compliments belong to all of my other co-authors, in particular to
Tero Aittokallio, Laura Elo-Uhlgren, Jaakko Hollmén, Juha Knuuttila, Samuel
Myllykangas and Janne Nikkilä. It has been a pleasure to work with you, and
your contributions extend beyond what we wrote together. I would also like to
thank the former and present members of the MI research group for working
beside me through these years, as well as for intriguing discussions about
science and life in general. I would also like to thank the personnel of the
ICS department, in particular professors Erkki Oja and Olli Simula, who have
helped to provide an excellent academic research environment, as well as our
secretaries Tarja Pihamaa and Leila Koivisto, and Markku Ranta and Miki
Sirola, who have given valuable help in so many practical matters during the
years.
Science is a community effort. Open sharing of ideas, knowledge, publication
material, data, software, code, experiences and emotions has had a tremendous
impact to this thesis. I will express my sincere gratitude to the community by
continued participation and contributions.
I would also like to thank my earliest scientific advisors; Reijo, who brought
me writings about the chemistry of life and helped me to grow bacteria and
prepare space dust in the 1980’s, Pekka, who has demonstrated the power of
criticism and emphasized that natural science has to be exact, Tapio, for the
attitude that maths can be just fun, and Risto, for showing how rational
thinking can be applied also in real life. Thanks also go to my science
friends, Manu and Ville; we have shared the passion for natural science, and I
want to thank you for our continuous and inspiring discussions along the way.
I am grateful to my grandfather Osmo, who shared with me the wonder towards
life, science, and humanities, and was willing to discuss it all through days
and nights when I was a child, questioning himself the self-evident truths
again and again, remaining as puzzled as I was. And for Alli and Arja, my
grandmothers, for their understanding, and all support and love.
My Friends. With you I have explored other facets of nature, science, and
life… Thank you for staying with me through all these years and sharing so
many aspects of curiosity, exploration and mutual understanding.
Finally, I am grateful to my parents and sister, Pipsa, Kari, and Tuuli. You
have accepted me and loved me, supported me on the paths that I have chosen to
follow, and understood that freedom can create the strongest ties.
Cambridge, November 23, 2010
Leo Lahti
### LIST OF PUBLICATIONS
This thesis consists of an overview and of the following publications which
are referred to in the text by their Roman numerals.
1. 1.
Laura L. Elo, Leo Lahti, Heli Skottman, Minna Kyläniemi, Riitta Lahesmaa, and
Tero Aittokallio. Integrating probe-level expression changes across
generations of Affymetrix arrays. Nucleic Acids Research, 33(22):e193, 2005.
2. 2.
Leo Lahti, Laura L. Elo, Tero Aittokallio, and Samuel Kaski. Probabilistic
analysis of probe reliability in differential gene expression studies with
short oligonucleotide arrays. IEEE/ACM Transactions on Computational Biology
and Bioinformatics, 8(1):217–225, 2011.
3. 3.
Leo Lahti, Juha E.A. Knuuttila, and Samuel Kaski. Global modeling of
transcriptional responses in interaction networks. Bioinformatics,
26(21):2713–2720, 2010.
4. 4.
Leo Lahti, Samuel Myllykangas, Sakari Knuutila, and Samuel Kaski. Dependency
detection with similarity constraints. In Tülay Adali, Jocelyn Chanussot,
Christian Jutten, and Jan Larsen, editors, Proceedings of the 2009 IEEE
International Workshop on Machine Learning for Signal Processing XIX, pages
89–94. IEEE, Piscataway, NJ, 2009.
5. 5.
Janne Sinkkonen, Janne Nikkilä, Leo Lahti, and Samuel Kaski. Associative
clustering. In Boulicaut, Esposito, Giannotti, and Pedreschi (editors),
Machine Learning: ECML2004 (Proceedings of the ECML’04, 15th European
Conference on Machine Learning), Lecture Notes in Computer Science 3201,
396–406. Springer, Berlin, 2004.
6. 6.
Samuel Kaski, Janne Nikkilä, Janne Sinkkonen, Leo Lahti, Juha E.A. Knuuttila,
and Cristophe Roos. Associative clustering for exploring dependencies between
functional genomics data sets. IEEE/ACM Transactions on Computational Biology
and Bioinformatics: Special Issue on Machine Learning for Bioinformatics –
Part 2, 2(3):203–216, 2005.
### SUMMARY OF PUBLICATIONS AND THE
AUTHOR’S CONTRIBUTION
The publications in this thesis have been a joint effort of all authors; key
contributions by the author of this thesis are summarized below.
Publication 1 introduces a novel analysis strategy to improve the accuracy and
reproducibility of the measurements in genome-wide transcriptional profiling
studies. A central part of the approach is the utilization of side information
in external genome sequence databases. The author participated in the design
of the study, suggested the utilization of external sequence data, implemented
this, as well as participated in preparing the manuscript.
Publication 2 provides a probabilistic framework for probe-level gene
expression analysis. The model combines statistical power across multiple
microarray experiments, and is shown to outperform widely-used preprocessing
methods in differential gene expression analysis. The model provides tools to
assess probe performance, which can potentially help to improve probe and
microarray design. The author had a major role in designing the study. The
author derived the formulation, implemented the model, performed the probe-
level experiments, as well as coordinated the manuscript preparation. The
author prepared an accompanied open source implementation which has been
published in BioConductor, a reviewed open source repository for computational
biology algorithms.
Publication 3 introduces a novel approach for organism-wide modeling of
transcriptional activity in genome-wide interaction networks. The method
provides tools to analyze large collections of genome-wide transcriptional
profiling data. The author had a major role in designing the study. The author
implemented the algorithm, performed the experiments, as well as coordinated
the manuscript preparation. The author participated in and supervised the
preparation of an accompanied open source implementation in BioConductor.
Publication 4 introduces a regularized dependency modeling framework with
particular applications in cancer genomics. The author had a major role in
formulating the biomedical modeling task, and in designing the study. The
theoretical model was jointly developed by the author and S. Kaski. The author
derived and implemented the model, carried out the experiments, and
coordinated the manuscript preparation. The author supervised and participated
in the preparation of an accompanied open source implementation in
BioConductor.
Publication 5 introduces the associative clustering principle, which is a
novel data integration framework for dependency detection with direct
applications in functional genomics. The author participated in implementation
of the method, had the main responsibility in designing and performing the
functional genomics experiments, as well as participated in preparing the
manuscript.
Publication 6 contains the most extensive treatment of the associative
clustering principle. In addition to presenting detailed theoretical
considerations, this work introduces new sensitivity analysis of the results,
and provides a comprehensive validation in bioinformatics case studies. The
author participated in designing the experiments, performed the comparative
functional genomics experiments and technical validation, as well as
participated in preparing the manuscript.
### LIST OF ABBREVIATIONS AND SYMBOLS
In this thesis boldface symbols are used to denote matrices and vectors.
Capital symbols ($\mathbf{X}$) signify matrices and lowercase symbols
($\mathbf{x}$) column vectors. Normal lowercase symbols indicate scalar
variables.
$\mathbb{R}$ Real domain
$\mathbf{X},\mathbf{Y}$ Data matrices ($D\times N$)
$[\mathbf{X};\mathbf{Y}]$ Concatenated data
$\mathbf{x}$, $\mathbf{y}$ Data samples, vectors in $\mathbb{R}^{D}$
$x$, $y$ Scalars in $\mathbb{R}$
$\mathcal{X},\mathcal{Y}$ Random variables
$\mathbf{I}$ Identity matrix
$\boldsymbol{\Sigma},\mathbf{\Psi}$ Covariance matrices
$p(\mathbf{x})$ Probability or probability density of $\mathcal{X}$
$p(\mathbf{X})$ Likelihood
$\mathbb{E}[\cdot]$ Expectation
$\|\cdot\|$ Norm of a matrix or vector
$Tr$ Matrix trace
$I(\mathcal{X};\mathcal{Y})$ Mutual information between random variables
$\mathcal{X}$ and $\mathcal{Y}$
$\text{Beta}(\alpha,\beta)$ Beta distribution with parameters $\alpha$ and
$\beta$
$\text{Dir}(\boldsymbol{\theta})$ Dirichlet distribution with parameter vector
$\boldsymbol{\theta}$
$\text{IG}(\alpha,\beta)$ Inverse Gamma distribution with parameters $\alpha$
and $\beta$
$\text{Mult}(N,\boldsymbol{\theta})$ Multinomial distribution with sample size
$N$ and parameter vector $\boldsymbol{\theta}$
$N(\boldsymbol{\mu},\boldsymbol{\Sigma})$ Normal distribution with mean
$\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$
AC Associative clustering
aCGH Array Comparative genomics hybridization
CCA Canonical correlation analysis
cDNA Complementary DNA
DNA Deoxyribonucleic acid
DP Dirichlet process
EM Expectation – Maximization algorithm
IB Information bottleneck
KL–divergence Kullback-Leibler divergence
MAP Maximum a posteriori
MCMC Markov chain Monte Carlo
ML Maximum likelihood
mRNA Messenger-RNA
tRNA Transfer-RNA
PCA Principal component analysis
RNA Ribonucleic acid
## Chapter 1 Introduction
Revolutions in measurement technologies have led to revolutions in science and
society. Introduction of the microscope in the 17th century opened a new view
to the world of living organisms and enabled the study of life processes at
cellular level. Since then, new techniques have been developed to investigate
ever smaller objects. The discovery of the molecular structure of the DNA in
1953 (Watson and Crick, 1953) led to the establishment of genes as fundamental
units of genetic information that is passed on between generations. The draft
sequence of the human genome, covering three billion DNA base pairs, was
published in 2001 (International human genome sequencing consortium, 2001;
Venter et al., 2001). Modern measurement technologies provide researchers with
large volumes of data concerning the structure, function, and interactions of
genes and their products. Rapid accumulation of genomic data in shared
community databases has accelerated biological research (Cochrane and
Galperin, 2010), but the structural and functional organization of genetic
information is still poorly understood. While functional roles of individual
genes have been characterized, little is known regarding the higher-level
regularities and interactions from which the complexity and diversity of life
emerges. The quest for systems-level understanding of genome function is a
major paradigm in modern biology (Collins et al., 2003).
Computational science has a key role in transforming the genomic data
collections into new biological knowledge (Cohen, 2004). New observations
allow the formulation of new research questions, but also bring new challenges
(Barbour et al., 2005). The sheer size of high-throughput data sets makes them
incomprehensible for human mind, and the complexity of biological phenomena
and high levels of uncontrolled variation set specific challenges for
computational analysis (Tilstone, 2003; Troyanskaya, 2005). Filtering relevant
information from statistically uncertain high-dimensional data is a
challenging task where new computational methods are needed to organize and
summarize the overwhelming volumes of observational data into a comprehensible
form to make new discoveries about the structure of life; computation is a new
microscope for studying massive data sets.
This thesis develops principled exploratory methods to investigate the human
transcriptome. It is a central functional layer of the genome and a
significant source of phenotypic variation. The transcriptome refers to the
complete collection of messenger-RNA transcripts of an organism. The
essentially static genome sequence regulates the time- and context-specific
patterns of transcriptional activity of the genes, and subsequently the
function of living cells through protein synthesis. An average cell contains
over 300,000 mRNA molecules and the expression levels of individual genes span
4-5 orders of magnitude (Carninci, 2009). A wealth of associated genomic
information resources are available in public repositories (Cochrane and
Galperin, 2010). By combining heterogeneous information sources and utilizing
the wealth of background information in public repositories, it is possible to
solve some of the problems that are related to the statistical uncertainties
and small sample size of individual data sets, as well as to form a holistic
picture of the genome (Huttenhower and Hofmann, 2010).
The observational data can provide the starting point to discover novel
research hypotheses of poorly characterized large-scale systems; the analysis
proceeds from general observations of the data toward more detailed
investigations and hypotheses. This differs from traditional hypothesis
testing where the investigation proceeds from hypotheses to measurements that
target particular research questions, in order to support or reject a given
hypothesis. _Exploratory data analysis_ refers to the use of computational
tools to summarize and visualize the data in order to identify potentially
interesting structure, and to facilitate the generation of new research
hypotheses when the search space would be otherwise exhaustively large (Tukey,
1977). When the system is poorly characterized, there is a need for methods
that can adapt to the data and extract features in an automated way. This is
useful since application-oriented models often require careful preprocessing
of the data and a timely model fitting process. They may also require prior
knowledge of the investigated system, which is often not available.
_Statistical learning_ investigates solutions to these problems.
### 1.1 Contributions and organization of the thesis
This thesis introduces computational strategies for genome- and organism-wide
analysis of the human transcriptome. The thesis provides novel tools (i) to
increase the reliability of high-throughput microarray measurements by
combining statistical evidence from genome sequence databases and across
multiple microarray experiments, (ii) to model context-specific
transcriptional activation patterns of genome-scale interaction networks
across normal human body by using background information of genetic
interactions to guide the analysis, and (iii) to integrate measurements of the
human transcriptome to other layers of genomic information with novel
dependency modeling techniques for co-occurring data sources. The three
strategies address widely recognized challenges in functional genomics
(Collins et al., 2003; Troyanskaya, 2005).
Obtaining reliable measurements is the crucial starting point for any data
analysis task. The first contribution of this thesis is to develop
computational strategies that utilize side information in genomic sequence and
microarray data collections in order to reduce noise and improve the quality
of high-throughput observations. Publication 1 introduces a probe-level
strategy for microarray preprocessing, where updated genomic sequence
databases are used in order to remove erroneously targeted probes to reduce
measurement noise. The work is extended in Publication 2, which introduces a
principled probabilistic framework for probe-level analysis. A generative
model for probe-level observations combines evidence across multiple
experiments, and allows the estimation of probe performance directly from
microarray measurements. The model detects a large number of unreliable probes
contaminated by known probe-level error sources, as well as many poorly
performing probes where the source of contamination is unknown and could not
be controlled based on existing probe-level information. The model provides a
principled framework to incorporate prior information of probe performance.
The introduced algorithms outperform widely used alternatives in differential
gene expression studies.
A novel strategy for organism-wide analysis of transcriptional activity in
genome-scale interaction networks in Publication 3 forms the second main
contribution of this thesis. The method searches for local regions in a
network exhibiting coordinated transcriptional response in a subset of
conditions. Constraints derived from genomic interaction databases are used to
focus the modeling on those parts of the data that are supported by known or
potential interactions between the genes. Nonparametric inference is used to
detect a number of physiologically coherent and reproducible transcriptional
responses, as well as context-specific regulation of the genes. The findings
provide a global view on transcriptional activity in cell-biological networks
and functional relatedness between tissues.
The third contribution of the thesis is to integrate measurements of the human
transcriptome to other layers of genomic information. Novel dependency
modeling techniques for co-occurrence data are used to reveal regularities and
interactions, which could not be detected in individual observations. The
regularized dependency modeling framework of Publication 4 is used to detect
associations between chromosomal mutations and transcriptional activity. Prior
biological knowledge is used to constrain the latent variable model and shown
to improve cancer gene detection performance. The associative clustering,
introduced in Publications 5 and 6, provides tools to investigate evolutionary
divergence of transcriptional activity.
Open source implementations of the key methodological contributions of this
thesis have been released in order to guarantee wide access to the developed
algorithmic tools and to comply with the emerging standards of transparency
and reproducibility in computational science, where an increasing proportion
of research details are embedded in code and data accompanying traditional
publications (Boulesteix, 2010; Carey and Stodden, 2010; Ioannidis et al.,
2009) and transparent sharing of these resources can form valuable
contributions to public knowledge (Sommer, 2010; Sonnenburg et al., 2007;
Stodden, 2010).
The thesis is organized as follows: In Chapter 2, there is an overview of
functional genomics, related measurement techniques, and genomic data
resources. General methodological background, in particular of exploratory
data analysis and the probabilistic modeling paradigm, is provided in Chapter
3. The methodological contributions of the thesis are presented in Chapters
4-6. In Chapter 4, strategies to improve the reliability of high-throughput
microarray measurements are presented. In Chapter 5 methods for organism-wide
analysis of the transcriptome are considered. In Chapter 6, two general-
purpose algorithms for dependency modeling are introduced and applied in
investigating functional effects of chromosomal mutations and evolutionary
divergence of transcriptional activity. The conclusions of the thesis are
summarized in Chapter 7.
## Chapter 2 Functional genomics
> _From all we have learnt about the structure of living matter, we must be
> prepared to find it working in a manner that cannot be reduced to the
> ordinary laws of physics - - because the construction is different from
> anything we have yet tested in the physical laboratory._
>
> E. Schrödinger (1956)
Living organisms are controlled not only by natural laws but also by
inheritable genetic programs (Mayr, 2004; Schrödinger, 1944). Such double
causation is a unique feature of life, and in fundamental contrast to purely
physical processes of the inanimate world. Life may have emerged on earth more
than 3.4 billion years ago (Schopf, 2006; Tice and Lowe, 2004). Genetic
information evolves by means of natural selection (Darwin, 1859). Living
organisms maintain homeostasis, adapt to changing environments, respond to
external stimuli, and communicate. Peculiar features of living systems include
metabolism, growth and hierarchical organization, as well as the ability to
replicate and reproduce. All known life forms share fundamental mechanisms at
molecular level, which suggests a common evolutionary origin of the living
organisms.
The complete collection of genetic material, the genome, encodes the heritable
genetic program of an organism. Advances in measurement technology and
computational science have opened up new views to the large-scale organization
of the genome (Carroll, 2003; Lander, 1996). Functional genomics is a
subdiscipline of molecular biology investigating the functional organization
and properties of genetic information. In this thesis, new computational
approaches are developed for investigation of a central functional layer of
the genome of our own species, the human transcriptome. This chapter gives an
overview to the relevant concepts in genome biology in eukaryotic organisms
and associated genomic data resources. For further background in molecular
genome biology, see Alberts et al. (2002); Brown (2006).
### 2.1 Universal genetic code
Cells are fundamental building blocks of living organisms. All known life
forms maintain a carbon-based cellular form that carries the genetic program
(Alberts et al., 2002). Each cell carries a copy of the heritable genetic
code, the genome. The human genome is divided in 23 pairs of chromosomes,
located in the nucleus of the cell, as well as in additional mitochondrial
genome. Chromosomes are macroscopic deoxyribonucleic acid (DNA) molecules in
which the DNA is wrapped around histone molecules and packed into a peculiar
chromatin structure that will ultimately constitute chromosomes. The genetic
code in the DNA consists of four nucleotides: adenosine (A), thymine (T),
guanine (G), and cytosine (C). In ribonucleic acid (RNA), the thymine is
replaced by uracil (U). Ordering of the nucleotides carries genetic
information. Nucleic acid sequences have a peculiar base pairing property,
where only A-T/U and G-C pairs can hybridize with each other. This leads to
the well-known double-stranded structure of the DNA, and forms the basis for
cellular information processing. The _central dogma of molecular biology_
(Crick, 1970) states that DNA encodes the information to construct proteins
through the irreversible process of protein synthesis. This is a central
paradigm in molecular biology, describing the functional organization of life
at the cellular level.
#### 2.1.1 Protein synthesis
Genes are basic units of genetic information. The gene is a sequence of DNA
that contains the information to manufacture a protein or a set of related
proteins. Genetic variation and regulation of gene activity has therefore
major phenotypic consequences. The regulatory region and coding sequence are
two key elements of a gene. The regulatory region regulates gene activity,
while the coding sequence carries the instructions for protein synthesis
(Alberts et al., 2002). Interestingly, the concept of a gene remains
controversial despite comprehensive identification of the protein-coding genes
in the human genome and detailed knowledge of their structure and function
(Pearson, 2006).
Proteins, encoded by the genes, are key functional entities in the cell. They
form cellular structures, and participate in cell signaling and functional
regulation. Protein synthesis refers to the cell-biological process that
converts genetic information into final functional protein products (Figure
2.1A). Key steps in protein synthesis include transcription, pre-mRNA
splicing, and translation. In transcription, the double-stranded DNA is opened
in a proximity of the gene sequence and the process is initiated on the
regulatory region of the gene. The DNA sequence of the gene is then converted
into a complementary pre-mRNA by a polymerase enzyme. The pre-mRNA sequence
contains both protein coding and non-coding segments. These are called exons
and introns, respectively. In pre-mRNA splicing, the introns are removed and
the exons are joined together to form mature messenger-RNA (mRNA). A gene can
encode multiple splice variants, corresponding to different exon definitions
and their combinations; this is called alternative splicing. The mature mRNA
is exported from nucleus to the cell cytoplasm. In translation the mRNA is
converted into a corresponding amino acid sequence in ribosomes based on the
universal genetic code that defines a mapping between nucleic acid triplets,
so-called codons, and amino acids. The code is common for all known life
forms. Each consecutive codon on the mRNA sequence corresponds to an amino
acid, and the corresponding sequence of amino acids constitutes a protein. In
the final stage of protein synthesis, the amino acid sequence folds into a
three-dimensional structure and undergoes post-translational modifications.
The structural characteristics of a protein molecule will ultimately determine
its functional properties (Alberts et al., 2002).
Figure 2.1: A Key steps of protein synthesis. The two key processes in protein
synthesis are called transcription and translation, respectively. In
transcription, the DNA sequence of the gene is transcribed into pre-mRNA based
on the base pairing property of nucleic acid sequences. The pre-mRNA is
modified to produce mature messenger-RNA (mRNA), which is then transported to
cytoplasm. Transfer-RNA (tRNA) carries the mRNA to ribosomes, where it is
translated into an amino acid sequence based on the universal genetic code
where each nucleotide triplet of the mRNA sequence, so-called _codon_ ,
corresponds to a particular amino acid. The amino acid sequence is
subsequently modified to form the final functional protein product. B
Organization of the genetic material in an eukaryotic cell. The nucleotide
base pairs form the double helix structure of DNA. This is wrapped around
histone molecules to form nucleosomes, and the chromatin sequence. The
chromatin is tightly packed to form chromosomes that carry the genetic
material and are located in the cell nucleus. The image has been modified from
http://commons.wikimedia.org/wiki/File:Chromosome_en.svg.
#### 2.1.2 Layers of regulation
Phenotypic changes can rarely be attributed to changes in individual genes;
cell function is ultimately determined by coordinated activation of genes and
other biomolecular entities in response to changes in cell-biological
environment (Hartwell et al., 1999). Gene activity is regulated at all levels
of protein synthesis and cellular processes. A major portion of functional
genome sequence and protein coding-genes themselves participate in the
regulatory system itself (Lauffenburger, 2000).
Epigenetic regulation refers to chemical and structural modifications of
chromosomal DNA, the chromatin, for instance through methylation, acetylation,
and other histone-binding molecules. Such modifications affect the packing of
the DNA molecule around histones in the cell nucleus. The combinatorial
regulation of such modifications regulates access to the gene sequences
(Gibney and Nolan, 2010). Epigenetic changes are believed to be heritable and
they constitute a major source of variation at individual and population level
(Johnson and Tricker, 2010). Transcriptional regulation is the next major
regulatory layer in protein synthesis. So-called transcription factor proteins
can regulate the transcription rate by binding to control elements in gene
regulatory region in a combinatorial fashion. Post-transcriptional
modifications will then regulate pre-mRNA splicing. Up to 95% of human multi-
exon genes are estimated to have alternative splice variants (Pan et al.,
2008). Consequently, a variety of related proteins can be encoded by a single
gene. This contributes to the structural and functional diversity of cell
function (Stetefeld and Ruegg, 2005). Several mechanisms will then affect mRNA
degradation rates. For instance, micro-RNAs that are small, 21-25 basepair
nucleotide sequences can inactivate specific mRNA transcripts through
complementary base pairing, leading to mRNA degradation, or prevention of
translation. Finally, post-translational modifications, protein degradation,
and other mechanisms will affect the three-dimensional structure and life
cycle of a protein. The proteins will participate in further cell-biological
processes. The processes are in continuous interaction and form complex
functional networks, which regulate the life processes of an organism (Alberts
et al., 2002).
### 2.2 Organization of genetic information
The understanding of the structure and functional organization of the genome
is rapidly accumulating with the developing genome-scanning technologies and
computational methods. This section provides an overview to key structural and
functional layers of the human genome.
#### 2.2.1 Genome structure
The genome is a dynamic structure, organized and regulated at multiple levels
of resolution from individual nucleotide base pairs to complete chromosomes
(Figure 2.1B; Brown (2006)). A major portion of heritable variation between
individuals has been attributed to differences in the genomic DNA sequence.
Traditionally, main genetic variation was believed to arise from small point
mutations, so-called single-nucleotide polymorphisms (SNPs), in protein-coding
DNA. Recently, it has been increasingly recognized that structural variation
of the genome makes a remarkable contribution to genetic variation. Structural
variation is observed at all levels of organization from single-nucleotide
polymorphisms to large chromosomal rearrangements, including deletions,
insertions, duplications, copy-number variants, inversions and translocations
of genomic regions (Feuk et al., 2006; Sharp et al., 2006). Such modifications
can directly and indirectly influence transcriptional activity and contribute
to human diversity and health (Collins et al., 2003; Hurles et al., 2008).
The draft DNA sequence of the complete human genome was published in 2001
(International human genome sequencing consortium, 2001; Venter et al., 2001).
The human genome contains three billion base pairs and approximately
20,000-25,000 protein-coding genes (International Human Genome Sequencing
Consortium, 2004). The protein-coding exons comprise less than 1.5% of the
human genome sequence. Approximately 5% of the human genome sequence has been
conserved in evolution for more than 200 million years, including the majority
of protein-coding genes (The ENCODE Project Consortium, 2007; Mouse Genome
Sequencing Consortium, 2002). Half of the genome consists of highly repetitive
sequences. The genome sequence contains structural elements such as
centromeres and telomeres, repetitive and mobile elements, (Prak and Kazazian
Jr., 2000), retroelements (Bannert and Kurth, 2004), and non-coding, non-
repetitive DNA (Collins et al., 2003). The functional role of intergenic DNA,
which forms 75% of the genome, is to a large extent unknown (Venter et al.,
2001). Recent evidence suggests that the three-dimensional organization of the
chromosomes, which is to a large extent regulated by the intergenic DNA is
under active selection, can have a remarkable regulatory role (Lieberman-Aiden
et al., 2009; Parker et al., 2009). Comparison of the human genome with other
organisms, such as the mouse (Mouse Genome Sequencing Consortium, 2002) can
highlight important evolutionary differences between species. For a
comprehensive review of the structural properties of the human genome, see
Brown (2006).
#### 2.2.2 Genome function
In protein synthesis, the gene sequence is transcribed into pre-mRNA, which is
then further modified into mature messenger-RNA and transported to cytoplasm.
An average cell contains over 300,000 mRNA molecules, and the mRNA
concentration, or expression levels of individual genes, vary according to
Zipf’s law, a power-law distribution where most genes are expressed at low
concentrations, perhaps only one or few copies of the mRNA per cell on
average, and a small number of genes are highly expressed, potentially with
thousands of copies per cell (see Carninci, 2009; Furusawa and Kaneko, 2003).
Cell-biological processes are reflected at the transcriptional level.
Transcriptional activity varies by cell type, environmental conditions and
time. Different collections of genes are active in different contexts. Gene
expression, or mRNA expression, refers to the expression level of an mRNA
transcript at particular physiological condition and time point. In addition
to protein-coding mRNA molecules that are the main target of analysis in this
thesis, the cell contains a variety of other functional and non-functional
mRNA transcripts, for instance micro-RNAs, ribosomal RNA and transfer-RNA
molecules (Carninci, 2009; Johnson et al., 2005).
The transcriptome refers to the complete collection of mRNA sequences of an
organism. This is a central functional layer of the genome that regulates
protein production in the cells, with a significant role in creating genetic
variation (Jordan et al., 2005). According to current estimates, up to 90% of
the eukaryotic genome can be transcribed (Consortium, 2005; Gagneur et al.,
2009). The protein-coding mRNA transcripts are translated into proteins at
ribosomes during protein synthesis.
The proteome refers to the collection of protein products of an organism. The
proteome is a main functional layer of the genome. Since the final protein
products carry out a main portion of the actual cell functions, techniques for
monitoring the concentrations of all proteins and their modified forms in a
cell simultaneously would significantly help to improve the understanding of
the cellular systems (Collins et al., 2003). However, sensitive, reliable and
cost-efficient genome-wide screening techniques for measuring protein
expression are currently not available. Therefore genome-wide measurements of
the mRNA expression levels are often used as an indirect estimate of protein
activity.
In addition to the DNA, RNA and proteins, the cell contains a variety of other
small molecules. The extreme functional diversity of living organisms emerges
from the complex network of interactions between the biomolecular entities
(Barabási and Oltvai, 2004; Hartwell et al., 1999). Understanding of these
networks and their functional properties is crucial in understanding cell
function (Collins et al., 2003; Schadt, 2009). However, the systemic
properties of the interactome are poorly characterized and understood due to
the complexity of biological phenomena and incomplete information concerning
the interactions. The cell-biological processes are inherently modular
(Hartwell et al., 1999; Ihmels et al., 2002; Lauffenburger, 2000), and they
exhibit complex pathway cross-talk between the cell-biological processes (Li
et al., 2008). In modular systems, small changes can have significant
regulatory effects (Espinosa-Soto and Wagner, 2010).
### 2.3 Genomic data resources
Systematic observations from the various functional and regulatory layers of
the genome are needed to understand cell-biological systems. Efficient sharing
and integration of genomic information resources through digital media has
enabled large-scale investigations that no single institution could afford.
The public human genome sequencing project (International human genome
sequencing consortium, 2001) is a prime example of such project. Results from
genome-wide transcriptional profiling studies are routinely deposited to
public repositories (Barrett et al., 2009; Parkinson et al., 2009). Sharing of
original data is increasingly accepted as the scientific norm, often following
explicit data release policies. The establishment of large-scale databases and
standards for representing biological information support the efficient use of
these resources (Bammler et al., 2005; Brazma et al., 2006). A continuously
increasing array of genomic information is available in these databases,
concerning aspects of genomic variability across individuals, disease states,
and species (Brent, 2008; Church, 2005; Cochrane and Galperin, 2010; G10KCOS
consortium, 2009; The Cancer Genome Atlas Research Network, 2008).
#### 2.3.1 Community databases and evolving biological knowledge
##### Genomic sequence databases
During the human genome project and preceding sequencing projects DNA sequence
reads were among the first sources of biological data that were collected in
large-scale public repositories, such as GenBank (Benson et al., 2010).
GenBank contains comprehensive sequence information of genomic DNA and RNA for
a number of organisms, as well as a variety of information concerning the
genes, non-coding regions, disease associations, variation and other genomic
features. Online analysis tools, such as the Ensembl Genome browser (Flicek et
al., 2010), facilitate efficient use of these annotation resources. Next-
generation sequencing technologies provide rapidly increasing sequencing
capacity to investigate sequence variation between individuals, populations
and disease states (Ledford, 2010; McPherson, 2009). In particular, the human
and mouse transcriptome sequence collections at the Entrez Nucleotide database
of GenBank are utilized in this thesis, in Publications 1 and 2.
##### Transcriptome databases
Gene expression measurement provides a snapshot of mRNA transcript levels in a
cell population at a specific time and condition, reflecting the activation
patterns of the various cell-biological processes. While gene expression
measurements provide only an indirect view to cellular processes, their wide
availability provides a unique resource for investigating gene co-regulation
on a genome- and organism-wide scale. Versatile collections of microarray data
in public repositories, such as the Gene Expression Omnibus (GEO; Barrett et
al. (2009)) and ArrayExpress (Parkinson et al., 2009) are available for human
and model organisms, and they contain valuable information of cell function
(Consortium, 2005; DeRisi et al., 1997; Russ and Futschik, 2010; Zhang et al.,
2004).
Several techniques are available for quantitative and highly parallel
measurements of mRNA or gene expression, allowing the measurement of the
expression levels of tens of thousands of mRNA transcripts simultaneously
(Bradford et al., 2010). Microarray techniques are routinely used to measure
the expression levels of tens of thousands of mRNA transcripts in a given
sample, and transcriptional profiling is currently a main high-throughput
technique used to investigate gene function at genome- and organism-wide scale
(Gershon, 2005; Yauk et al., 2004). Increasing amounts of transcriptional
profiling data are being produced by sequencing-based methods (Carninci,
2009). A main difference between the microarray- and sequencing-based
techniques is that gene expression arrays have been designed to measure
predefined mRNA transcripts, whereas sequencing-based methods do not require
prior information of the measured sequences, and enable de novo discovery of
expressed transcripts (Bradford et al., 2010; ’t Hoen et al., 2008). Large-
scale microarray repositories provide currently the most mature tools for data
processing and retrieval, and form the main source of transcriptome data in
this thesis.
Microarray technology is based on the base pairing property of nucleic acid
sequences where the DNA or RNA sequences in a sample bind to the complementary
nucleotide sequences on the array. This is called hybridization. The
measurement process begins by the collection of cell samples and isolation of
the sample mRNA. The isolated mRNA is converted to cDNA, labeled with specific
marker molecules, and hybridized on complementary probe sequences on the
array. The array surface may contain hundreds of thousands of spots, each
containing specific probe sequences designed to uniquely match with particular
mRNA sequences. The hybridization level reflects the target mRNA concentration
in the sample, and it is estimated by measuring the intensity of light emitted
by the label molecules with a laser scanner. Short oligonucleotide arrays
(Lockhart et al., 1996) are among the most widely used microarray
technologies, and they are the main source of mRNA expression data in this
thesis. Short oligonucleotide arrays utilize multiple, typically 10-20, probes
for each transcript target that bind to different regions of the same
transcript sequence. Use of several 25-nucleotide probes for each target leads
to more robust estimates of transcript activity. Each probe is expected to
uniquely hybridize with its intended target, and the detected hybridization
level is used as a measure of the activity of the transcript. A short
oligonucleotide array measures absolute expression levels of the mRNA
sequences; relative differences between conditions can be investigated
afterwards by comparing these measurements. A standard whole-genome array
measures typically $\sim$20,000-50,000 unique transcript sequences. A single
microarray experiment can therefore produce hundreds of thousands of raw
observations.
Comparison and integration of individual microarray experiments is often
challenging due to remarkable experimental variation between the experiments.
Common standards have been developed to advance the comparison and integration
(Brazma et al., 2001, 2006). Carefully controlled integrative datasets, so-
called gene expression atlases, contain thousands of genome-wide measurements
of transcriptional activity across diverse conditions in a directly comparable
format. Examples of such data collections include GeneSapiens (Kilpinen et
al., 2008), the human gene expression atlas of the European Bioinformatics
Institute (Lukk et al., 2010), as well as the NCI-60 cell line panel (Scherf
et al., 2000). Integrative analysis of large and versatile transcriptome
collections can provide a holistic view of transcriptional activity of the
various cell-biological processes, and opens up possibilities to discover
previously uncharacterized cellular mechanisms that contribute to human health
and disease.
##### Other types of microarray data
Microarray techniques can also be used to study other functional aspects of
the genome, including epigenetics and micro-RNA regulation, chromosomal
aberrations and polymorphisms, alternative splicing, as well as transcription
factor binding (Butte, 2002; Hoheisel, 2006). For instance, chromosomal
aberrations can be measured with the array comparative genome hybridization
method (aCGH; Pinkel and Albertson 2005), which is based on hybridization of
DNA sequences on the array surface. Copy number changes are a particular type
of chromosomal aberrations, which are a major mechanism for cancer development
and progression. Copy number alterations can cause changes in gene- and micro-
RNA expression, and ultimately cell-biological processes (Beroukhim et al.,
2010). A public repository of copy number measurement data is provided for
instance by the CanGEM database (Scheinin et al., 2008). In Publication 4,
microarray measurements of DNA copy number changes are integrated with
transcriptional profiling data to discover potential cancer genes for further
biomedical analysis.
##### Pathway and interaction databases
Curated information concerning cell-biological processes is valuable in both
experimental design and validation of computational studies (Blake, 2004).
Representation of dynamic biochemical reactions in their full richness is a
challenging task beyond a mere listing of biochemical events; a variety of
proteins and other compounds interact in a hierarchical manner through various
molecular mechanisms (Hartwell et al., 1999; Przytycka et al., 2010).
Standardized database formats such as the BioPAX (BioPAX workgroup, 2005) and
SBML (Strömbäck and Lambrix, 2005) advance the accumulation of highly
structured biological knowledge and automated analysis of such data. A huge
body of information concerning cell-biological processes is available in
public repositories. The most widely used annotation resources include the
Gene Ontology (GO) database (Ashburner et al., 2000) and the KEGG pathway
database (Kanehisa et al., 2010). The GO database provides functional
annotations for genes and can be used for instance to detect enrichment of
certain functional categories among the key findings from computational
analysis, as in Publication 6, where enrichment analysis is used for both
validation and interpretation purposes. Pathways are more structured
representations concerning cellular processes and interactions between
molecular entities. Such prior information can be used to guide computational
modeling, as in Publication 3, where pathway information derived from the KEGG
pathway database is used to guide organism-wide discovery and analysis of
transcriptional response patterns.
##### Evolving biological knowledge
The collective knowledge about genome organization and function is constantly
updated and refined by improved measurement techniques and accumulation of
data (Sebat, 2007). This can alter the analysis and interpretation of results
from large-scale genomic screens. For instance, evolving gene and transcript
definitions are known to significantly affect microarray interpretation. Probe
design on microarray technology relies on sequence annotations that may have
changed significantly after the original array design. Reinterpretation of
microarray data based on updated probe annotations has been shown to improve
the accuracy and comparability of microarray results (Dai et al., 2005; Hwang
et al., 2004; Mecham et al., 2004b). Bioinformatics studies routinely take
into account updates in genome version, genome build, in new analyses. The
constantly refined biological data highlights the need to account for this
uncertainty in computational analyses. In Publications 1 and 2, explicit
computational strategies that are robust against evolving transcript
definitions are developed for microarray data analysis.
#### 2.3.2 Challenges in high-throughput data analysis
High-throughput genetic screens are inherently noisy. Controlling all
potential sources of variation in the measurement process is increasingly
difficult when automated measurement techniques can produce millions of data
points in a single experiment, concerning extremely complex living systems
that are to a large extent poorly understood.
Noise arises from both technical and biological sources (Butte, 2002), and
systematic variation between laboratories, measurement batches and measurement
platforms has to be taken into account when combining the results across
individual studies (Heber and Sick, 2006; MAQC Consortium, 2006). Moreover,
genomic knowledge is constantly evolving, which can potentially change the
interpretation of previous experiments (see e.g. Dai et al., 2005). The
various sources of noise and uncertainty in microarray studies are discussed
in more detail in Chapter 4.
High dimensionality of the data and small sample size form another challenge
for the analysis of high-throughput functional genomics data. Tens of
thousands of transcripts can be measured simultaneously in a single microarray
experiment, which greatly exceeds the number of available samples in most
biomedical studies. Small sample sizes leave considerable uncertainty in the
analyses; few observations contain very limited information concerning the
complex and high-dimensional phenomena and potential interactions between
different parts of the system. Overfitting of the models and the problem of
multiple testing forms considerable challenges in such situations. While
automated analysis methods can generate thousands of hypotheses concerning the
system, prioritizing the findings and characterizing uncertainty in the
predictions become central issues in the analysis. The curse of
dimensionality, coupled with the high levels of noise in functional genomics
studies, is therefore posing particular challenges for computational modeling
(Saeys et al., 2007).
The challenges in controlling the various sources of uncertainty have led to
remarkable problems in reproducing microarray results (Ioannidis et al.,
2009), but maturing technology and the development of common standards and
analytical procedures are constantly improving the reliability of high-
throughput screens (Allison et al., 2006; Reimers, 2010; MAQC Consortium,
2006). The models developed in this thesis combine statistical evidence across
related experiments to improve the reliability of the analysis and to increase
modeling power. Generative probabilistic models provide a rigorous framework
for handling noise and uncertainty in the data and models.
### 2.4 Genomics and health
Genomic variation between individuals has remarkable and to a large extent
unknown contribution to health and disease susceptibility. Large-scale
characterization of the variability between individuals and populations is
expected to elucidate genomic mechanisms associated with disease, as well as
to lead to the discovery of novel medical treatments. High-throughput genomics
can provide new tools to understand disease mechanisms (Braga-Neto and
Marques, 2006; Lage et al., 2008), to ’hack the genome’ (Evanko, 2006) to
treat diseases (Volinia et al., 2010), and to guide personalized therapies
that take into account the individual variability in sensitivity and responses
to treatments (Church, 2005; Downward, 2006; Foekens et al., 2008; Ocana and
Pandiella, 2010; van ’t Veer and Bernards, 2008). Disease signatures are
potentially robust across tissues and experiments (Dudley et al., 2009; Hu et
al., 2006). Genomic screens have revealed new disease subtypes (Bhattacharjee
et al., 2001), and led to the discovery of various diagnostic (Lee et al.,
2008; Su et al., 2009; Tibshirani et al., 2002) and prognostic (Beer et al.,
2002) biomarkers. Diseases cause coordinated changes in gene activity through
biomolecular networks (Cabusora et al., 2005). Integration of chemical,
genomic and pharmacological functional genomics data can also help to predict
new drug targets and responses (Lamb et al., 2006; Yamanishi et al., 2010).
Genomic mutations can also affect genome function and cause diseases (Taylor
et al., 2008). Cancer is an example of a prevalent genomic disease. Boveri
(1914) discovered that cancer cells have chromosomal imbalances, and since
then the understanding of genomic changes associated with cancer has
continuously improved (Stratton et al., 2009; Wunderlich, 2007). For instance,
many human micro-RNA genes are located at cancer-associated genomic regions
and are functionally altered in cancers (see Calin and Croce, 2006). Genomic
changes also affect transcriptional activity of the genes (Myllykangas et al.,
2008). Publication 4 introduces a novel computational approach for screening
cancer-associated DNA mutations with functional implications by genome-wide
integration of chromosomal aberrations and transcriptional activity.
This chapter has provided an overview to central modeling challenges and
research topics in functional genomics. In the following chapters, particular
methodological approaches are introduced to solve research tasks in large-
scale analysis of the human transcriptome. In particular, methods are
introduced to increase the reliability of high-throughput measurements, to
model large-scale collections of transcriptome data and to integrate
transcriptional profiling data to other layers of genomic information. The
next chapter provides general methodological background for these studies.
## Chapter 3 Statistical learning and exploratory data analysis
> _Essentially, all models are wrong, but some are useful._
>
> G.E.P. Box and N.R. Draper (1987)
Models are condensed, simplified representations of observed phenomena. Models
can be used to describe observations and to predict future events. Two key
aspects in modeling are the construction and learning of formal
representations of the observed data. Complex real-world observations contain
large amounts of uncontrolled variation, which is often called noise; all
aspects of the data cannot be described within a single model. Therefore, a
modeling compromise is needed to decide what aspects of data to describe and
what to ignore. The second step in modeling is to fill in, to learn, details
of the formal representation based on the actual empirical observations.
Various learning algorithms are typically available that differ in efficiency
and accuracy. For instance, improvements in computation time can often be
achieved by potential decrease in accuracy. An inference compromise is needed
to decide how to balance between these and other potentially conflicting
objectives of the learning algorithm; the relative importance of each factor
depends on the particular application and available resources, and affects the
choice of the learning procedure. The modeling and inference compromises are
at the heart of data analysis. Ultimately, the value of a model is determined
by its ability to advance the solving of practical problems.
This chapter gives an overview of the key concepts in statistical modeling
central to the topics of this thesis. The objectives of exploratory data
analysis and statistical learning are considered in Section 3.1. The
methodological framework is introduced in Section 3.2, which contains an
overview of central concepts in probabilistic modeling and the Bayesian
analysis paradigm. Key issues in implementing and validating the models are
discussed in Section 3.3.
### 3.1 Modeling tasks
Understanding requires generalization beyond particular observations. While
empirical observations contain information of the underlying process that
generated the data, a major challenge in computational modeling is that
empirical data is always finite and contains only limited information of the
system. Traditional statistical models are based on careful hypothesis
formulation and systematic collection of data to support or reject a given
hypothesis. However, successful hypothesis formulation may require substantial
prior knowledge. When minimal knowledge of the system is available, there is a
need for exploratory methods that can recognize complex patterns and extract
features from empirical data in an automated way (Baldi and Brunak, 1999).
This is a central challenge in computational biology, where the investigated
systems are extremely complex and contain large amounts of poorly
characterized and uncontrolled sources of variation. Moreover, the data of
genomic systems is often very limited and incomplete. General-purpose
algorithms that can learn relevant features from the data with minimal
assumptions are therefore needed, and they provide valuable tools in
functional genomics studies. Classical examples of such exploratory methods
include clustering, classification and visualization techniques. The extracted
features can provide hypotheses for more detailed experimental testing and
reveal new, unexpected findings. In this work, general-purpose exploratory
tools are developed for central modeling tasks in functional genomics.
#### 3.1.1 Central concepts in data analysis
Let us start by defining some of the basic concepts and terminology. _Data
set_ in this thesis refers to a finite collection of observations, or samples.
In experimental studies, as in biology, a sample typically refers to the
particular object of study, for instance a patient or a tissue sample. In
computational studies, sample refers to a numerical observation, or a subset
of observations, represented by a numerical feature vector. Each element of
the feature vector describes a particular feature of the observation. Given
$D$ features and $N$ samples, the data set is presented as a matrix
$\mathbf{X}\in\mathbb{R}^{D\times N}$, where each column vector
$\mathbf{x}\in\mathbb{R}^{D}$ represents a sample and each row corresponds to
a particular feature. The features can represent for instance different
experimental conditions, time points, or particular summaries about the
observations. This is the general structure of the observations investigated
in this work.
The observations are modeled in terms of probability densities; the samples
are modeled as independent instances of a random variable. A central modeling
task is to characterize the underlying probability density of the
observations, $p(\mathbf{x})$. This defines a topology in the sample space and
provides the basis for generalization beyond empirical observations. As
explained in more detail in Section 3.2, the models are formulated in terms of
observations $\mathbf{X}$, model parameters $\boldsymbol{\theta}$, and latent
variables $\mathbf{Z}$ that are not directly observed, but characterize the
underlying process that generated the data.
Ultimately, all models describe relationships between objects. Similarity is
therefore a key concept in data analysis; the basis for characterizing the
relations, for summarizing the observations, and for predicting future events.
Measures of similarity can be defined for different classes of objects such as
feature vectors, data sets, or random variables. Similarity in general is a
vague concept. Euclidean distance, induced by the Euclidean metrics, is a
common (dis-)similarity measure for multivariate observations. Correlation is
a standard choice for univariate random variables. Mutual information is an
information-theoretic measure of statistical dependency between two random
variables, characterizing the decrease in the uncertainty concerning the
realization of one variable, given the other one. The uncertainty of a random
variable $\mathcal{X}$ is measured in terms of entropy111Entropy is defined as
$H(\mathcal{X})=-\int_{\mathbf{x}}p(\mathbf{x})\log p(\mathbf{x})d\mathbf{x}$
for a continuous variable. (Shannon, 1948). The mutual information between two
random variables is then given by
$I(\mathcal{X},\mathcal{Y})=H(\mathcal{X})-H(\mathcal{X}|\mathcal{Y})$ (see
e.g. Gelman et al., 2003). The Kullback-Leibler divergence, or KL–divergence,
is a closely related non-symmetric dissimilarity measure for probability
distributions $p,q$, defined as
$d_{KL}(p,q)=\int_{\mathbf{x}}p(\mathbf{x})\log\frac{p(\mathbf{x})}{q(\mathbf{x})}d\mathbf{x}$
(see e.g. Bishop, 2006). Mutual information between two random variables can
be alternatively formulated as the KL–divergence between their joint density
$p(\mathbf{x},\mathbf{y})$ and the product of their independent marginal
densities, $q(\mathbf{x},\mathbf{y})=p_{x}(\mathbf{x})p_{y}(\mathbf{y})$,
which gives the connection
$I(\mathcal{X},\mathcal{Y})=d_{KL}(p(\mathbf{x},\mathbf{y}),p_{x}(\mathbf{x})p_{y}(\mathbf{y}))$.
Mutual information and KL-divergence are central information-theoretic
measures of dependency employed in the models of this thesis.
It is important to notice that measures of similarity are inherently coupled
to the statistical representation of data and to the goals of the analysis;
different representations can reveal different relationships between
observations. For instance, the Euclidean distance is sensitive to scaling of
the features; representation in natural or logarithmic scale, or with
different units can potentially lead to very different analysis results. Not
all measures are equally sensitive; mutual information can naturally detect
non-linear relationships, and it is invariant to the scale of the variables.
On the other hand, estimating mutual information is computationally demanding.
Feature selection refers to computational techniques for selecting, scaling
and transforming the data into a suitable form for further analysis. Feature
selection has a central role in data analysis, and it is implicitly present in
all analysis tasks in selecting the investigated features for the analysis.
There are no universally optimal stand-alone feature selection techniques,
since the problem is inherently entangled with the analysis task and multiple
equally optimal feature sets may be available for instance in classification
or prediction tasks Guyon and Elisseeff (2003); Saeys et al. (2007).
Successful feature selection can reduce the dimensionality of the data with
minimal loss of relevant information, and focus the analysis on particular
features. This can reduce model complexity, which is expected to yield more
efficient, generalizable and interpretable models. Feature selection is
particularly important in genome-wide profiling studies, where the
dimensionality of the data is large compared to the number of available
samples, and only a small number of features are relevant for the studied
phenomenon. This is also known as the large p, small n problem (West, 2003).
Advanced feature selection techniques can take into account dependencies
between the features, consider weighted combinations of them, and can be
designed to interact with the more general modeling task, as for instance in
the nearest shrunken centroids classifier of Tibshirani et al. (2002). The
constrained subspace clustering model of Publication 3 can be viewed as a
feature selection procedure, where high-dimensional genomic observations are
decomposed into distinct feature subsets, each of which reveals different
relationships of the samples. In Publication 4, identification of maximally
informative features between two data sets forms a central part of a
regularized dependency modeling framework. In Publications 3-4 the procedure
and representations are motivated by biological reasoning and analysis goals.
#### 3.1.2 Exploratory data analysis
Exploratory data analysis refers to the use of computational techniques to
summarize and visualize data in order to facilitate the generation of new
hypotheses for further study when the search space would be otherwise
exhaustively large (Tukey, 1977). The analysis strategy takes the observations
as the starting point for discovering interesting regularities and novel
research hypotheses for poorly characterized large-scale systems without prior
knowledge. The analysis can then proceed from general observations of the data
toward _confirmatory data analysis_ , more detailed investigations and
hypotheses that can be tested in independent data sets with standard
statistical procedures. Exploratory data analysis differs from traditional
hypothesis testing where the hypothesis is given. Light-weight exploratory
tools are particularly useful with large data sets when prior knowledge on the
system is minimal. Standard exploratory approaches in computational biology
include for instance clustering, classification and visualization techniques
(Evanko, 2010; Polanski and Kimmel, 2007).
Cluster analysis refers to a versatile family of methods that partition data
into internally homogeneous groups of similar data points, and often at the
same time minimize the similarity between distinct clusters. Clustering
techniques enable class discovery from the data. This differs from
classification where the target is to assign new observations into known
classes. The partitions provided by clustering can be nested, partially
overlapping or mutually exclusive, and many clustering methods generalize the
partitioning to cover previously unseen data points (Jain and Dubes, 1988).
Clustering can provide compressed representations of the data based on a
shared parametric representation of the observations within each cluster, as
for instance in K-means or Gaussian mixture modeling (see e.g. Bishop, 2006).
Certain clustering approaches, such as the hierarchical clustering (see e.g.
Hastie et al., 2009), apply recursive schemes that partition the data into
internally homogeneous groups without providing a parametric representation of
the clusters. Cluster structure can be also discovered by linear algebraic
operations on the distance matrices, as for instance in spectral clustering.
The different approaches often have close theoretical connections. Clustering
in general is an ill-defined concept that refers to a set of related but
mutually incompatible objectives (Ben-David and Ackerman, 2008; Kleinberg,
2002). Cluster analysis has been tremendously popular in computational
biology, and a comprehensive review of the different applications are beyond
the scope of this thesis. It has been observed, for instance, that genes with
related functions have often similar expression profiles and are clustered
together, suggesting that clustering can be used to formulate hypotheses
concerning the function of previously uncharacterized genes (DeRisi et al.,
1997; Eisen et al., 1998), or to discover novel cancer subtypes with
biomedical implications (Sørlie et al., 2001).
Visualization techniques are another widely used exploratory approach in
computational biology. Visualizations can provide compact and intuitive
summaries of complex, high-dimensional observations on a lower-dimensional
display, for instance by linear projection methods such as principal component
analysis, or by explicitly optimizing a lower-dimensional representation as in
the self-organizing map (Kohonen, 1982). Visualization can provide the first
step in investigating large data sets (Evanko, 2010).
#### 3.1.3 Statistical learning
_Statistical learning_ refers to computational models that can learn to
recognize structure and patterns from empirical data in an automated way.
Unsupervised and supervised models form two main categories of learning
algorithms.
Unsupervised learning approaches seek compact descriptions of the data without
prior knowledge. In probabilistic modeling, unsupervised learning can be
formulated as the task of finding a probability distribution that describes
the observed data and generalizes to new observations. This is also called
density estimation. The parameter values of the model can be used to provide
compact representations of the data. Examples of unsupervised analysis tasks
include methods for clustering, visualization and dimensionality reduction. In
cluster analysis, groups of similar observations are sought from the data.
Dimensionality reduction techniques provide compact lower-dimensional
representations of the original data, which is often useful for subsequent
modeling steps. Not all observations of the data are equally valuable, and
assessing the relevance of the observed regularities is problematic in fully
unsupervised analysis.
In supervised learning the task is to learn a function that maps the inputs
$\mathbf{x}$ to the desired outputs $\mathbf{y}$ based on a set of training
examples in a generalizable fashion, as in regression for continuous outputs,
and classification for discrete output variables. The supervised learning
tasks are inherently asymmetric; the inference proceeds from inputs to
outputs, and prior information of the modeling task is used to supervise the
analysis; the training examples also include a desired output of the model.
The models developed in this thesis can be viewed as unsupervised exploratory
techniques. However, the distinction between supervised and unsupervised
models is not strict, and the models in this thesis borrow ideas from both
categories. The models in Publications 2-3 are unsupervised algorithms that
utilize prior information derived from background databases to guide the
modeling by constraining the solutions. However, since no desired outputs are
available for these models, the modeling tasks differ from supervised
analysis. The dependency modeling algorithms of Publications 4-6 have close
theoretical connections to the supervised learning task. In contrast to
supervised learning, the learning task in these algorithms is symmetric;
modeling of the co-occurring data sets is unsupervised, but coupled. Each data
set affects the modeling of the other data set in a symmetric fashion, and, in
analogy to supervised learning, prediction can then proceed to either
direction. Compared to supervised analysis tasks, the emphasis in the
dependency detection algorithms introduced in this thesis is in the discovery
and characterization of symmetric dependencies, rather than in the
construction of asymmetric predictive models.
### 3.2 Probabilistic modeling paradigm
The main contributions of this thesis follow the generative probabilistic
modeling paradigm. Generative probabilistic models describe the observed data
in terms of probability distributions. This allows the calculation of
expectations, variances and other standard summaries of the model parameters,
and at the same time allows to describe the independence assumptions and
relations between variables, and uncertainty in the modeling process in an
explicit manner. Measurements are regarded as noisy observations of the
general, underlying processes; generative models are used to characterize the
processes that generated the observations.
The first task in modeling is the selection of a _model family_ \- a set of
potential formal representations of the data. As discussed in Section 3.2.2,
the representations can also to some extent be learned from the data. The
second task is to define the _objective function_ , or cost function, which is
used to measure the descriptive power of the models. The third task is to
identify the optimal model within the model family that best describes the
observed data with respect to the objective function. This is called learning
or model fitting. The details of the modeling process are largely determined
by the exact modeling task and particular nature of the observations. The
objectives of the modeling task are encoded in the selected model family, the
objective function and to some extent to the model fitting procedure. The
model family determines the space of possible descriptions for the data and
has therefore a major influence on the final solution. The objective function
can be used to prefer simple models or other aspects in the modeling process.
The model fitting procedure affects the efficiency and accuracy of the
learning process. For further information of these and related concepts, see
Bishop (2006). A general overview of the probabilistic modeling framework is
given in the remainder of this section.
#### 3.2.1 Generative modeling
_Generative probabilistic models_ view the observations as random samples from
an underlying probability distribution. The model defines a probability
distribution $p(\mathbf{x})$ over the feature space. The model can be
parameterized by model parameters $\boldsymbol{\theta}$ that specify a
particular model within the model family. For convenience, we assume that the
model family is given, and leave it out from the notation. In this thesis, the
appropriate model families are selected based on biological hypotheses and
analysis goals. Generative models allow efficient representation of
dependencies between variables, independence assumptions and uncertainty in
the inference (Koller and Friedman, 2009). Let us next consider central
analysis tasks in generative modeling.
##### Finite mixture models
Classical probability distributions provide well-justified and convenient
tools for probabilistic modeling, but in many practical situations the
observed regularities in the data cannot be described with a single standard
distribution. However, a sufficiently rich mixture of standard distributions
can provide arbitrarily accurate approximations of the observed data. In
mixture models, a set of distinct, latent processes, or components, is used to
describe the observations. The task is to identify and characterize the
components and their associations to the individual observations. The standard
formulation assumes independent and identically distributed observations where
each observation has been generated by exactly one component. In a standard
mixture model the overall probability density of the data is modeled as a
weighted sum of component distributions:
$p(\mathbf{x})=\sum_{r=1}^{R}\pi_{r}p_{r}(\mathbf{x}|\boldsymbol{\theta}_{r}),$
(3.1)
where the components are indexed by $r$, and $\int
p(\mathbf{x})d\mathbf{x}=1$. Each mixture component can have a different
distributional form. The mixing proportion, or weight, and model parameters of
each component are denoted by $\pi_{r}$ and $\boldsymbol{\theta}_{r}$,
respectively, with $\sum_{r}\pi_{r}=1$. Many applications utilize convenient
standard distributions, such as Gaussians, or other distributions from the
exponential family. Then the mixture model can be learned for instance with
the EM algorithm described in Section 3.3.1.
In practice, the mixing proportions of the components are often unknown. The
mixing proportions can be estimated from the data by considering them as
standard model parameters to be fitted with a ML estimate. However, the
procedure is potentially prone to overfitting and local optima, i.e., it may
learn to describe the training data well, but fails to generalize to new
observations. An alternative, probabilistic way to determine the weights is to
treat the mixing proportions as latent variables with a prior distribution
$p({\boldsymbol{\pi}})$. A standard choice is a symmetric Dirichlet
prior222Dirichlet distribution is the probability density
$Dir({\boldsymbol{\pi}}|\mathbf{n})\sim\prod_{r}\pi_{r}^{n_{r}-1}$ where the
multivariate random variable ${\boldsymbol{\pi}}$ and the positive parameter
vector $\mathbf{n}$ have their elements indexed by $r$, $0<\pi_{r}<1$, and
$\sum_{r}\pi_{r}=1$. ${\boldsymbol{\pi}}\sim
Dir(\frac{{\boldsymbol{\alpha}}}{R})$. This gives an equal prior weight for
each component and guarantees the standard exchangeability assumption of the
mixture component labels. A label determines cluster identity. Intuitively,
exchangeability corresponds to the assumption that the analysis is invariant
to the ordering of the data samples and mixture components. Compared to
standard mixture models, probabilistic mixture models have increased
computational complexity.
Further prior knowledge can be incorporated in the model by defining prior
distributions for the other parameters of the mixture model. This can also be
used to regularize the learning process to avoid overfitting. A typical prior
distribution for the components of a Gaussian mixture model, parameterized by
$\boldsymbol{\theta}_{r}=\\{\boldsymbol{\mu}_{r},\boldsymbol{\Sigma}_{r}\\}$,
is the normal-inverse-Gamma prior (see e.g. Gelman et al., 2003).
Interpreting the mixture components as clusters provides an alternative,
probabilistic formulation of the clustering task. This has made probabilistic
mixture models a popular choice in the analysis of functional genomics data
sets that typically have high dimensionality but small sample size.
Probabilistic analysis takes the uncertainties into account in a rigorous
manner, which is particularly useful when the sample size is small. The number
of mixture components is often unknown in practical modeling tasks, however,
and has to be inferred based on the data. A straightforward solution can be
obtained by employing a sufficiently large number of components in learning
the mixture model, and selecting the components having non-zero weights as a
post-processing step. An alternative, model-based treatment for learning the
number of mixture components from the data is provided by infinite mixture
models considered in Section 3.2.2.
##### Latent variables and marginalization
The observed variables are often affected by latent variables that describe
relevant structure in the model, but are not directly observed. The latent
variable values can be, to some extent, inferred based on the observed
variables. Combination of latent and observed variables allows the description
of complex probability spaces in terms of simple component distributions and
their relations. Use of simple component distributions can provide an
intuitive and computationally tractable characterization of complex generative
processes underlying the observations.
A generative latent variable model specifies the distributional form and
relationships of the latent and observed variables. As a simple example,
consider the probabilistic interpretation of probabilistic component analysis
(PCA), where the observations $\mathbf{x}$ are modeled with a linear model
$\mathbf{x}=\mathbf{W}\mathbf{z}+\boldsymbol{\varepsilon}$ where a normally
distributed latent variable $\mathbf{z}\sim N(\mathbf{0},\mathbf{I})$ is
transformed with the parameter matrix $\mathbf{W}$ and isotropic Gaussian
noise ($\boldsymbol{\varepsilon}$) is assumed on the observations. More
complex models can be constructed by analogous reasoning. A complete-data
likelihood $p(\mathbf{X},\mathbf{Z}|\boldsymbol{\theta})$ defines a joint
density for the observed and latent variables. Only a subset of variables in
the model is typically of interest for the actual analysis task. For instance,
the latent variables may be central for describing the generative process of
the data, but their actual values may be irrelevant. Such variables are called
nuisance variables. Their integration, or marginalization, provides
probabilistic averaging over the potential realizations. Marginalization over
the latent variables in the complete-data likelihood gives the likelihood
$p(\mathbf{X}|\boldsymbol{\theta})=\int_{\mathbf{Z}}p(\mathbf{X},\mathbf{Z}|\boldsymbol{\theta})d\mathbf{Z}.$
(3.2)
Marginalization over the latent variables collapses the modeling task to
finding optimal values for model parameters $\boldsymbol{\theta}$, in a way
that takes into account the uncertainty in latent variable values. This can
reduce the number of variables in the learning phase, yield more
straightforward and robust inferences, as well as speed up computation.
However, marginalization may lead to analytically intractable integrals. As
certain latent variables may be directly relevant, marginalization depends on
the overall goals of the analysis and may cover only a subset of the latent
variables. In this thesis latent variables are utilized for instance in
Publication 3, which treats the sample-cluster assignments as discrete latent
variables, as well as in Publication 4, where a regularized latent variable
model is introduced to model dependencies between co-occurring observations.
#### 3.2.2 Nonparametric models
Finite mixture models and latent variable models require the specification of
model structure prior to the analysis. This can be problematic since for
instance the number and distributional shape of the generative processes is
unknown in many practical tasks. However, the model structure can also to some
extent be learned from the data. Non-parametric models provide principled
approaches to learn the model structure from the data. In contrast to
parametric models, the number and use of the parameters in nonparametric
models is flexible (see e.g. Hjort et al., 2010; Müller and Quintana, 2004).
The infinite mixture of Gaussians, used as a part of the modeling process in
Publication 3, is an example of a non-parametric model where both the number
of components, as well as mixture proportions of the component distributions
are inferred from the data. Learning of Bayesian network structure is another
example of nonparametric inference, where relations between the model
variables are learned from the data (see e.g. Friedman, 2003). While more
complex models can describe the training data more accurately, an increasing
model complexity needs to be penalized to avoid overfitting and to ensure
generalizability of the model.
Nonparametric models provide flexible and theoretically principled approaches
for data-driven exploratory analysis. However, the flexibility often comes
with an increased computational cost, and the models are potentially more
prone to overfitting than less flexible parametric models. Moreover, complex
models can be difficult to interpret.
Many nonparametric probabilistic models are defined by using the theory of
stochastic processes to impose priors over potential model structures.
Stochastic processes can be used to define priors over function spaces. For
instance, the Dirichlet process (DP) defines a probability density over the
function space of Dirichlet distributions333If $G$ is a distribution drawn
from a Dirichlet process with the probability measure $P$ over the sample
space, $G\sim\mathrm{DP}(P)$, then each finite partition $\\{A_{k}\\}_{k}$ of
the sample space is distributed as $(G(A_{1}),...,G(A_{k}))\sim
Dir(P(A_{1}),...,P(A_{k}))$.. The Chinese Restaurant Process (CRP) provides an
intuitive description of the Dirichlet process in the cluster analysis
context. The CRP defines a prior distribution over the number of clusters and
their size distribution. The CRP is a random process in which $n$ customers
arrive in a restaurant, which has an infinite number of tables. The process
goes as follows: The first customer chooses the first table. Each subsequent
customer $m$ will select a table based on the state $F_{m-1}$ of the
restaurant tables after $m-1$ customers have arrived. The new customer $m$
will select a previously occupied table $i$ with a probability which is
proportional to the number of customers seated at table $i$, i.e.
$p(i|F_{m-i})\propto n_{i}$. Alternatively, the new customer will select an
empty table with a probability which is proportional to a constant $\alpha$.
The model prefers tables with a larger number of customers, and is analogous
to clustering, where the customers and tables correspond to samples and
clusters, respectively. This provides an intuitive prior distribution for
clustering tasks. The prior prefers compact models with relatively few
clusters, but the number of clusters is potentially infinite, and ultimately
determined based on the data.
##### Infinite mixture models
Infinite mixture models are a general class of nonparametric methods where the
number of mixture components are determined in a data-driven manner; the
number of components is potentially infinite (see e.g. Müller and Quintana,
2004; Rasmussen, 2000). An infinite mixture is obtained by letting
$R\rightarrow\infty$ in the finite mixture model of Equation 3.1 and replacing
the Dirichlet distribution prior of the mixing proportions
${\boldsymbol{\pi}}$ by a Dirichlet process. The formal probability
distribution of the Dirichlet process can be intuitively derived with the so-
called stick-breaking presentation. Consider a unit length stick and a stick-
breaking process, where the breakpoint $\beta$ is stochastically determined,
following the beta distribution $\beta\sim Beta(1,\alpha)$, where $\alpha$
tunes the expected breaking point. The process can be viewed as consecutively
breaking off portions of a unit length stick to obtain an infinite sequence of
stick lengths $\pi_{1}=\beta_{1}$;
$\pi_{i}=\beta_{i}\prod_{l=1}^{i-1}(1-\beta_{l})$, with
$\sum_{i=1}^{\infty}\pi_{i}=1$ (Ishwaran and James, 2001). This defines the
probability distribution $\text{Stick}(\alpha)$ over potential partitionings
of the unit stick. A truncated stick-breaking representation considers only
the first $T$ elements. Setting the prior
${\boldsymbol{\pi}}\sim\text{Stick}(\alpha)$, defined by the stick-breaking
representation in Equation 3.1 assigns a prior on the number of mixture
components and their mixing proportions that are ultimately learned from the
observed data. The prior helps to find a compromise between increasing model
complexity and likelihood of the observations.
Traditional approaches used to determine the number mixture components are
based on objective functions that penalize increasing model complexity, for
instance in certain variants of the K-means or in spectral clustering (see
e.g. Hastie et al., 2009). Other model selection criteria include cross-
validation and comparison of the models in terms of their likelihood or
various information-theoretic criteria that seek a compromise between model
complexity and fit (see e.g. Gelman et al., 2003). However, the sample size
may be insufficient for such approaches, and the models may lack a rigorous
framework to account for uncertainties in the observations and model
parameters. Modeling uncertainty in the parameters while learning the model
structure can lead to more robust inference in nonparametric probabilistic
models but also adds inherent computational complexity in the learning
process.
#### 3.2.3 Bayesian analysis
The term ’Bayesian’ refers to interpretation of model parameters as variables.
The uncertainty over the parameter values, arising from limited empirical
evidence, is described in terms of probability distributions. This is in
contrast to the traditional view where parameters have fixed values with no
distribution and the uncertainty is ignored. The Bayesian approach leads to a
learning task where the objective is to estimate the _posterior distribution_
$p(\boldsymbol{\theta}|\mathbf{X})$ of the model parameters
$\boldsymbol{\theta}$, given the observations $\mathbf{X}$. The posterior is
given by the _Bayes’ rule_ (Bayes, 1763):
$p(\boldsymbol{\theta}|\mathbf{X})=\frac{p(\mathbf{X}|\boldsymbol{\theta})p(\boldsymbol{\theta})}{p(\mathbf{X})}.$
(3.3)
The two key elements of the posterior are the likelihood and the prior. The
likelihood $p(\mathbf{X}|\boldsymbol{\theta})$ describes the probability of
the observations, given the parameter values $\boldsymbol{\theta}$. The
parameters can also characterize alternative model structures. The prior
$p(\boldsymbol{\theta})$ encodes prior beliefs about the model and rewards
solutions that match with the prior assumptions or yield simpler models. Such
regularizing properties can be particularly useful when training data is
scarce and there is considerable uncertainty in the parameter estimates. With
strong, informative priors, new observations have little effect on the
posterior. In the limit of large sample size the posterior converges to the
ordinary likelihood $p(\mathbf{X}|\boldsymbol{\theta})$. The Bayesian
inference provides a robust framework for taking the uncertainties into
account when the data is scarce, as it often is in practical modeling tasks.
Moreover, the Bayes’ rule provides a formal framework for sequential update of
beliefs based on accumulating evidence. The prior predictive density
$p(\mathbf{X})=\int p(\mathbf{X},\boldsymbol{\theta})d\boldsymbol{\theta}$ is
a normalizing constant, which is independent of the parameters
$\boldsymbol{\theta}$ and can often be ignored during model fitting.
The involved distributions can have complex non-standard forms and limited
empirical data can only provide partial evidence regarding the different
aspects of the data-generating process. Often only a subset of the parameters
and other variables and their interdependencies can be directly observed. The
Bayesian approach provides a framework for making inferences on the unobserved
quantities through hierarchical models, where the probability distribution of
each variable is characterized by higher-level parameters, so-called
hyperparameters. A similar reasoning can be used to model the uncertainty in
the hyperparameters, until the uncertainties become modeled at an appropriate
detail. Prior information can help to compensate the lack of data on certain
aspects of a model, and explicit models for the noise can characterize
uncertainty in the empirical observations. Distributions can also share
parameters, which provides a basis for pooling evidence from multiple sources,
as for instance in Publication 4. In many applications only a subset of the
parameters in the model are of interest and the modeling process can be
considerably simplified by marginalizing over the less interesting nuisance
variables to obtain an expectation over their potential values.
The Bayesian paradigm provides a principled framework for modeling the
uncertainty at all levels of statistical inference, including the parameters,
the observed and latent variables and the model structure; all information of
the model is incorporated in the posterior distribution, which summarizes
empirical evidence and prior knowledge, and provides a complete description of
the expected outcomes of the data-generating process. When the data does not
contain sufficient information to decide between the alternative model
structures and parameter values, the Bayesian framework provides tools to take
expectations over all potential models, weighted by their relative evidence.
A central challenge in the Bayesian analysis is that the models often include
analytically intractable posterior distributions, and learning of the models
can be computationally demanding. Widely-used approaches for estimating
posterior distributions include Markov Chain Monte Carlo (MCMC) methods and
variational learning. Stochastic MCMC methods provide a widely-used family of
algorithms to estimate intractable distributions by drawing random samples
from these distributions (see e.g. Gelman et al., 2003); a sufficiently large
pool of random samples will converge to the underlying distribution, and
sample statistics can then be used to characterize the distribution. However,
sampling-based methods are computationally intensive and slow. In variational
learning, considered in Section 3.3.1, the intractable distributions are
approximated by more convenient tractable distributions, which yields faster
learning procedure, but potentially less accurate results. While analysis of
the full posterior distribution will provide a complete description of the
uncertainties regarding the parameters, simplified summary statistics, such as
the mean, variance and quantiles of the posterior can provide a sufficient
characterization of the posterior in many practical applications. They can be
obtained for instance by summarizing the output of sampling-based or
variational methods. Moreover, when the uncertainty in the results can be
ignored, point estimates can provide simple, interpretable summaries that are
often useful in further biomedical analysis, as for instance in Publication 2.
Point estimates are single optimal values with no distribution. However, point
estimates are not necessarily sufficient for instance in biomedical
diagnostics and other prediction tasks, where different outcomes are
associated with different costs and it may be crucial to assess the
probabilities of the alternative outcomes. For further discussion on learning
the Bayesian models, see Section 3.3.1.
In this thesis the Bayesian approach provides a formal framework to perform
robust inference based on incomplete functional genomics data sets and to
incorporate prior information of the models in the analysis. The Bayesian
paradigm can alternatively be interpreted as a philosophical position, where
probability is viewed as a subjective concept (Cox, 1946), or considered a
direct consequence of making rational decisions under uncertainty (Bernardo
and Smith, 2000). For further concepts in model selection, comparison and
averaging in the Bayesian analysis, see Gelman et al. (2003). For applications
in computational biology, see Wilkinson (2007).
### 3.3 Learning and inference
The final stage in probabilistic modeling is to learn the optimal statistical
presentation for the data, given the model family and the objective function.
This section highlights central challenges and methodological issues in
statistical learning.
#### 3.3.1 Model fitting
Learning in probabilistic models often focuses on optimizing the model
parameters $\boldsymbol{\theta}$. In addition, posterior distribution of the
latent variables, $p(\mathbf{z}|\mathbf{x},\boldsymbol{\theta})$, can be
calculated. Estimating the latent variable values is called statistical
inference. In the Bayesian analysis, the model parameters can also be treated
as latent variables with a prior probability density, in which case the
distinction between model parameters and latent variables will disappear. A
comprehensive characterization of the variables and their uncertainty would be
achieved by estimating the full posterior distribution. However, this can be
computationally very demanding, in particular when the posterior is not
analytically tractable. The posterior is often approximated with stochastic or
analytical procedures, such as stochastic MCMC sampling methods or variational
approximations, and appropriate summary statistics. In many practical
settings, it is sufficient to summarize the full posterior distribution with a
point estimate. Point estimates do not characterize the uncertainties in the
analysis result, but are often more convenient to interpret than full
posterior distributions.
Various optimization algorithms are available to learn statistical models,
given the learning procedure. The potential challenges in the optimization
include computational complexity and the presence of local optima on complex
probability density topologies, as well as unidentifiability of the models.
Finding a global optimum of a complex model can be computationally exhaustive,
and it can become intractable with increasing sample size. In unidentifiable
models, the data does not contain sufficient information to choose between
alternative models with equal statistical evidence. Ultimately, the
uncertainty in inference arises from limited sample size and the lack of
computational resources.
In the remainder of this section, let us consider more closely the particular
learning procedures central to this thesis: point estimates and variational
approximation, and the standard optimization algorithms used to learn such
representations.
##### Point estimates
Assuming independent and identically distributed observations, the likelihood
of the data, given model parameters, is
$p(\mathbf{X}|\boldsymbol{\theta})=\prod_{i}p(\mathbf{x}_{i}|\boldsymbol{\theta})$.
This provides a probabilistic measure of model fit and the objective function
to maximize. Maximization of the likelihood
$p(\mathbf{X}|\boldsymbol{\theta})$ with respect to $\boldsymbol{\theta}$
yields a maximum likelihood (ML) estimate of the model parameters, and
specifies an optimal model that best describes the data. This is a standard
point estimate used in probabilistic modeling. Practical implementations
typically operate on log-likelihood, the logarithm of the likelihood function.
As a monotone function, this yields the same optima, but has additional
desirable properties: it factorizes the product into a sum and is less prone
to numerical overflows during optimization.
The maximum a posteriori (MAP) estimate additionally takes prior information
of the model parameters into account. While the ML estimate maximizes the
likelihood $p(\mathbf{X}|\boldsymbol{\theta})$ of the observations, the MAP
estimate maximizes the posterior $p(\boldsymbol{\theta}|\mathbf{X})\sim
p(\mathbf{X}|\boldsymbol{\theta})p(\boldsymbol{\theta})$ of the model
parameters. The objective function to maximize is the log-likelihood
$logp(\boldsymbol{\theta}|\mathbf{X})\sim
logp(\mathbf{X}|\boldsymbol{\theta})+logp(\boldsymbol{\theta}).$ (3.4)
The prior is explicit in MAP estimation and the model contains the ML estimate
as a special case; assuming large sample size, or non-informative, uniform
prior $p(\boldsymbol{\theta})\sim 1$, the likelihood of the data
$p(\mathbf{X}|\boldsymbol{\theta})$ will dominate and the MAP estimation
becomes equivalent to optimizing $p(\mathbf{X}|\boldsymbol{\theta})$, yielding
the traditional ML estimate. The ML and MAP estimates are asymptotically
consistent approximations of the posterior distribution, since the posterior
will converge a point distribution with a large sample size. The computation
and interpretation of point estimates is straightforward compared to the use
of posterior distributions in the full Bayesian treatment. The differences
between ML and MAP estimates highlight the role of prior information in the
modeling when training data is limited.
##### Variational inference
In certain modeling tasks the uncertainty in the model parameters needs to be
taken into account. Then point estimates are not sufficient. The uncertainty
is characterized by the posterior distribution
$p(\boldsymbol{\theta}|\mathbf{X})$. However, the posterior distributions are
often intractable and need to be estimated by approximative methods.
Variational approximations provide a fast and principled optimization scheme
(see e.g. Bishop, 2006) that yields only approximative solutions, but can
accelerate posterior inference by orders of magnitude compared to stochastic,
sampling-based MCMC methods that can in principle provide exact solutions,
assuming that infinite computational resources are available. The potential
decrease in accuracy in variational approximations is often acceptable, given
the gains in efficiency. Variational approximation characterizes the
uncertainty in $\boldsymbol{\theta}$ with a tractable distribution
$q(\boldsymbol{\theta})$ that approximates the full, potentially intractable
posterior $p(\boldsymbol{\theta}|\mathbf{X})$,
Variational inference is formulated as an optimization problem where an
intractable posterior distribution
$p(\mathbf{Z},\boldsymbol{\theta}|\mathbf{X})$ is approximated by a more
easily tract-able distribution $q(\mathbf{Z},\boldsymbol{\theta})$ by
minimizing the KL–divergence between the two distributions. This is also shown
to maximize a lower bound of the marginal likelihood $p(\mathbf{X})$, and
subsequently the likelihood of the data, yielding an approximation of the
overall model. The log-likelihood of the data can be decomposed into a sum of
the lower bound ${\cal L}(q)$ of the observed data and the KL–divergence
$d_{KL}(q,p)$ between the approximative and the exact posterior distributions:
$logp(\mathbf{X})={\cal L}(q)+d_{KL}(q,p),$ (3.5)
where
$\displaystyle\begin{array}[]{cll}{\cal
L}(q)&=&\int_{\mathbf{z}}q(\mathbf{Z},\boldsymbol{\theta})log\frac{p(\mathbf{Z},\boldsymbol{\theta},\mathbf{X})}{q(\mathbf{Z},\boldsymbol{\theta})};\\\
d_{KL}(q,p)&=&-\int_{\mathbf{z}}q(\mathbf{Z},\boldsymbol{\theta})log\frac{p(\mathbf{Z},\boldsymbol{\theta}|\mathbf{X})}{q(\mathbf{Z},\boldsymbol{\theta})}.\end{array}$
(3.8)
The KL-divergence is non-negative, and equals to zero if and only if the
approximation and the exact distribution are identical. Therefore ${\cal
L}(q)$ gives a lower bound for the log-likelihood $logp(\mathbf{X})$ in
Equation 3.5. Minimization of $d_{KL}$ with respect to $q$ will provide an
analytically tractable approximation $q(\mathbf{Z},\boldsymbol{\theta})$ of
$p(\mathbf{Z},\boldsymbol{\theta}|\mathbf{X})$. Minimization of $d_{KL}$ will
also maximize the lower bound ${\cal L}(q)$ since the log-likelihood
$logp(\mathbf{X})$ is independent of $q$. The approximation typically assumes
independent parameters and latent variables, yielding a factorized
approximation
$q(\mathbf{Z},\boldsymbol{\theta})=q_{\mathbf{z}}(\mathbf{Z})q_{\boldsymbol{\theta}}(\boldsymbol{\theta})$
based on tractable standard distributions. It is also possible to factorize
$q_{\mathbf{z}}$ and $q_{\boldsymbol{\theta}}$ into further components.
Variational approximations are used for efficient learning of infinite
multivariate Gaussian mixture models in Publication 3.
##### Expectation–Maximization (EM)
The EM algorithm is a general procedure for learning probabilistic latent
variable models (Dempster et al., 1977), and a special case of variational
inference. The algorithm provides an efficient algorithm for finding point
estimates for model parameters in latent variable models. The objective of the
EM algorithm is to maximize the marginal likelihood
$p(\mathbf{X}|\boldsymbol{\theta})=\int_{\mathbf{z}}p(\mathbf{X},\mathbf{Z}|\boldsymbol{\theta})d\mathbf{Z}$
(3.9)
of the observations $\mathbf{X}$ with respect to the model parameters
$\boldsymbol{\theta}$. Marginalization over the probability density of the
latent variables provides an inference procedure that is robust to uncertainty
in the latent variable values. The algorithm iterates between estimating the
posterior of the latent variables, and optimizing the model parameters (see
e.g. Bishop, 2006). Given initial values $\boldsymbol{\theta}_{0}$ of the
model parameters, the expectation step evaluates the posterior density of the
latent variables, $p(\mathbf{z}|\mathbf{x},\boldsymbol{\theta}_{t})$, keeping
$\boldsymbol{\theta}_{t}$ fixed. If the posterior is not analytically
tractable, variational approximation $q(\mathbf{z})$ can be used to obtain a
lower bound for the likelihood in Equation 3.9. The maximization step
optimizes the model parameters $\boldsymbol{\theta}$ with respect to the
following objective function:
$Q(\boldsymbol{\theta},\boldsymbol{\theta}_{t})=\int_{\mathbf{z}}p(\mathbf{Z}|\mathbf{X},\boldsymbol{\theta}_{t})logp(\mathbf{X},\mathbf{Z}|\boldsymbol{\theta})d\mathbf{Z}.$
(3.10)
This is the expectation of the complete-data log-likelihood
$logp(\mathbf{X},\mathbf{Z}|\boldsymbol{\theta})$ over the latent variable
density $p(\mathbf{Z}|\mathbf{X},\boldsymbol{\theta}_{t})$, obtained from the
previous expectation step. The new parameter estimate is then
$\boldsymbol{\theta}_{t+1}=argmax_{\boldsymbol{\theta}}Q(\boldsymbol{\theta},\boldsymbol{\theta}_{t}).$
The expectation and maximization steps determine an iterative learning
procedure where the latent variable density and model parameters are
iteratively updated until convergence. The maximization step will also
increase the target likelihood of Equation 3.9, but potentially with a
remarkably smaller computational cost (Dempster et al., 1977). In contrast to
the marginal likelihood in Equation 3.9, the complete-data likelihood in
Equation 3.10 is logarithmized before integration in the maximization step.
When the joint distribution $p(\mathbf{x},\mathbf{z}|\boldsymbol{\theta})$
belongs to the exponential family, the logarithm will cancel the exponential
in algebraic manipulations. This can considerably simplify the maximization
step. When the likelihoods in the optimization are of suitable form, the
iteration steps can be solved analytically, which can considerably reduce
required evaluations of the objective function. Convergence is guaranteed, if
the optimization can increase the likelihood at each iteration. However, the
identification of a global optimum is not guaranteed in the EM algorithm.
Incorporating prior information of the parameter values through Bayesian
priors can be used to avoid overfitting and focus the modeling on particular
features in the data, as in the regularized dependency modeling framework of
Publication 4, where the EM algorithm is used to learn Gaussian latent
variable models.
##### Standard optimization methods
Optimization methods provide standard tools to implement selected learning
procedures. Optimization algorithms are used to identify parameter values that
minimize or maximize the objective function, either globally, or in local
surroundings of the optimized value. Selection of optimization method depends
on smoothness and continuity properties of the objective function, required
accuracy, and available resources.
Gradient-based approaches optimize the objective function by assuming smooth,
continuous topology over the probability density where setting the derivatives
to zero will yield local optima. If a closed form solution is not available,
it is often possible to estimate gradient directions in a given point.
Optimization can then proceed by updating the parameters towards the desired
direction along the gradient, gradually improving the objective function value
in subsequent gradient ascent steps. So-called quasi-Newton methods use
function values and gradients to characterize the optimized manifold, and to
optimize the parameters along the approximated gradients. An appropriate step
length is identified automatically based on the curvature of the objection
function surface. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) (Broyden, 1970;
Fletcher, 1970; Goldfarb, 1970; Shanno, 1970) method is a quasi-Newton
approach used for standard optimization tasks in this thesis.
#### 3.3.2 Generalizability and overlearning
Probabilistic models are formulated in terms of probability distributions over
the sample space and parameter values. This forms the basis for generalization
to new, unobserved events. A generalizable model can describe essential
characteristics of the underlying process that generated the observations; a
generalizable model is also able to characterize future observations.
Overlearning, or overfitting refers to models that describe the training data
well, but do not generalize to new observations. Such models describe not only
the general processes underlying the observations, but also noise in the
particular observations. Avoiding overfitting is a central aspect in modeling.
Overlearning is particularly likely when training data is scarce. While
overfitting could in principle be avoided by collecting more data, this is
often not feasible since the cost of data collection can be prohibitively
large.
Generalizability can be measured by investigating how accurately the model
describes new observations. A standard approach is to split the data into a
training set, used to learn the model, and a test set, used to measure model
performance on unseen observations that were not used for training. In _cross-
validation_ the test is repeated with several different learning and test sets
to assess the variability in the testing procedure. Cross-validation is used
for instance in Publication 5 of this thesis. _Bootstrap analysis_ (see, for
instance, Efron and Tibshirani, 1994) is another widely used approach to
measure model performance. The observed data is viewed as a finite realization
of an underlying probability density. New samples from the underlying density
are obtained by re-sampling the observed data points with replacement to
simulate variability in the original data; observations from the more dense
regions of the probability space become re-sampled more often than rare
events. Each bootstrap sample resembles the probability density of the
original data. Modeling multiple data sets obtained with the bootstrap helps
to estimate the sensitivity of the model to variations in the data. Bootstrap
is used to assess model performance in Publication 6.
#### 3.3.3 Regularization and model selection
In general, increasing model complexity will yield more flexible models, which
have higher descriptive power but are, on the other hand, more likely to
overfit. Therefore relatively simple models can often outperform more complex
models in terms of generalizability. A compromise between simplicity and
descriptive power can be obtained by imposing additional constraints or soft
penalties in the modeling to prefer compact solutions, but at the same time
retain the descriptive power of the original, flexible model family. This is
called regularization. Regularization is particularly important when the
sample size is small, as demonstrated for instance in Publication 4, where
explicit and theoretically principled regularization is achieved by setting
appropriate priors on the model structure and parameter values. The priors
will then affect the MAP estimate of the model parameters. One commonly used
approach is to prefer sparse solutions that allow only a small number of the
potential parameters to be employed at the same time to model the data (see
e.g. Archambeau and Bach, 2008). A family of probabilistic approaches to
balance between model fit and model complexity is provided by information-
theoretic criteria (see e.g. Gelman et al., 2003). The Bayesian Information
Criterion (BIC) is a widely used information criterion that introduces a
penalty term on the number of model parameters to prefer simpler models. The
log-likelihood ${\cal L}$ of the data, given the model, is balanced by a
measure of model complexity, $qlog(N)$, in the final objective function
$-2{\cal L}+qlog(N)$. Here $q$ denotes the number of model parameters and $N$
is the constant sample size of the investigated data set. The BIC has been
criticized since it does not address changes in prior distributions, and its
derivation is based on asymptotic considerations that hold only approximately
with finite sample size (see e.g. Bishop, 2006). On the other hand, BIC
provides a principled regularization procedure that is easy to implement. In
this thesis, the BIC has been used to regularize the algorithms in Publication
3.
#### 3.3.4 Validation
After learning a probabilistic model, it is necessary to confirm the quality
of the model and verify potential findings in further, independent
experiments. Validation refers to a versatile set of approaches used to
investigate model performance, as well as in model criticism, comparison and
selection. Internal and external approaches provide two complementary
categories for model validation. Internal validation refers to procedures to
assess model performance based on training data alone. For instance, it is
possible to estimate the sensitivity of the model to initialization,
parameterization, and variations in the data, or convergence of the learning
process. Internal analysis can help to estimate the weaknesses and
generalizability of the model, and to compare alternative models. Bootstrap
and cross-validation are widely used approaches for internal validation and
the analysis of model performance (see e.g. Bishop, 2006). Bootstrap can
provide information about the sensitivity of the results to sampling effects
in the data. Cross-validation provides information about the model
generalization performance and robustness by comparing predictions of the
model to real outcomes. External validation approaches investigate model
predictions and fit on new, independent data sets and experiments. Exploratory
analysis of high-throughput data sets often includes massive multiple testing,
and provides potentially thousands of automatically generated hypotheses. Only
a small set of the initial findings can be investigated more closely by human
intervention and costly laboratory experiments. This highlights the need to
prioritize the results and assess the uncertainty in the models.
## Chapter 4 Reducing uncertainty in high-throughput microarray studies
> _As far as the laws of mathematics refer to reality, they are not certain,
> as far as they are certain, they do not refer to reality._
>
> A. Einstein (1956)
Gene expression microarrays are currently the most widely used technology for
genome-wide transcriptional profiling, and they constitute the main source of
data in this thesis. An overview of microarray technology is provided in
Section 2.3.1. Microarray measurements are associated with high levels of
noise from technical and biological sources. Appropriate preprocessing
techniques can help to reduce noise and obtain reliable measurements, which is
the crucial starting point for any data analysis task. This chapter presents
the first main contribution of the thesis, preprocessing techniques that
utilize side information in genomic sequence databases and microarray data
collections in order to improve the accuracy of high-throughput gene
expression data. The chapter is organized as follows: Section 4.1 gives an
overview of the various sources of noise in high-throughput microarray
studies. Section 4.2 introduces a strategy for noise reduction based on side
information in external genomic sequence databases. Section 4.3 extends this
model by describing a model-based approach that additionally combines
statistical evidence across multiple microarray experiments in order to
provide quantitative information of probe performance and utilizes this
information to improve the reliability of high-throughput observations. The
results are summarized in Section 4.4.
### 4.1 Sources of uncertainty
Measurement data obtained with novel high-throughput technologies comes with
high levels of uncontrolled biological and technical variation. This is often
called noise as it obscures the measurements, and adds potential bias and
variance on the signal of interest. Biological noise is associated with
natural biological variation between cell populations, cellular processes and
individuals. Single-nucleotide polymorphisms, alternative splicing and non-
specific hybridization add biological variation in the data (Dai et al., 2005;
Zhang et al., 2005). More technical sources of noise in the measurement
process include RNA extraction and amplification, experiment-specific
variation, as well as platform- and laboratory-specific effects (Choi et al.,
2003; MAQC Consortium, 2006; Tu et al., 2002).
A significant source of noise on gene expression arrays comes from individual
probes that are designed to measure the activity of a given transcript in a
biological sample. Figure 4.1A shows probe-level observations of differential
gene expression for a collection of probes designed to target the same mRNA
transcript. One of the probes is highly contaminated and likely to add
unrelated variation to the analysis. A number of factors affect probe
performance. For instance, it has been reported in Publication 1 and elsewhere
(Hwang et al., 2004; Mecham et al., 2004b) that a large portion of microarray
probes may target unintended mRNA sequences. Moreover, although the probes
have been designed to uniquely hybridize with their intended mRNA target,
remarkable cross-hybridization with the probes by single-nucleotide
polymorphisms (Dai et al., 2005; Sliwerska et al., 2007) and other mRNAs with
closely similar sequences (Zhang et al., 2005) have been reported; high-
affinity probes with high GC-content may have higher likelihood of cross-
hybridization with nonspecific targets (Mei et al., 2003). Alternative
splicing (MAQC Consortium, 2006) and mRNA degradation (Auer et al., 2003) may
cause differences between probes targeting different positions of the gene
sequence. Such effects will contribute to probe-level contamination in a
probe- and condition-specific manner. However, sources of probe-level noise
are still poorly understood (Irizarry et al., 2005; Li et al., 2005) despite
their importance for expression analysis and probe design.
High levels of noise set specific challenges for analysis. Better
understanding of the technical aspects of the measurement process will lead to
improved analytical procedures and ultimately to more accurate biological
results (Reimers, 2010). Publication 2 provides computational tools to
investigate probe performance and the relative contributions of the various
sources of probe-level contamination on short oligonucleotide arrays.
Figure 4.1: A Example of a probe set that contains a probe with high
contamination levels (dashed line) detected by the probabilistic RPA model.
The probe-level observations of differential gene expression for the different
probes that measure the same target transcript are indicated by gray lines.
The black line shows the estimated signal of the target transcript across a
number of conditions. B Increase in the average variance of the probes
associated with the investigated noise sources: mistargeted probes having
errors in the genomic alignment, most 5’/3’ probes of each probe set, GC-rich,
and SNP-associated probes. The variances were estimated by RPA and describe
the noise level of the probes. The results are shown for the individual ALL
and GEA data sets, and for their combined results on both platforms (133A and
95A/Av2). ©IEEE. Reprinted with permission from Publication 2.
### 4.2 Preprocessing microarray data with side information
Preprocessing of the raw data obtained from the original measurements can help
to reduce noise and improve comparability between microarray experiments.
Preprocessing can be defined in terms of statistical transformations on the
raw data, and this is a central part of data analysis in high-throughput
studies. This section outlines the standard preprocessing steps for short
oligonucleotide arrays, the main source of transcriptional profiling data in
this thesis. However, the general concepts also apply to other microarray
platforms (Reimers, 2010).
##### Standard preprocessing steps
A number of preprocessing techniques for short oligonucleotide arrays have
been introduced (Irizarry et al., 2006; Reimers, 2010). The standard
preprocessing steps in microarray analysis include quality control, background
correction, normalization and summarization.
Microarray quality control is used to identify arrays with remarkable
experimental defects, and to remove them from subsequent analysis. The typical
tests consider RNA degradation levels and a number of other summary statistics
to guarantee that the array data is of reasonable quality. The arrays that
pass the microarray quality control are preprocessed further. Each array
typically has spatial biases that vary smoothly across the array, arising from
technical factors in the experiment. Background correction is used to detect
and remove such spatial effects from the array data, and to provide a uniform
background signal, enhancing the comparability of the probe-level observations
between different parts of the array. Moreover, background correction can
estimate the general noise level on the array; this helps to detect probes
whose signal differs significantly from the background noise. Robust multi-
array averaging (RMA) is one of the most widely used approaches for
preprocessing short oligonucleotide array data (Irizarry et al., 2003a). The
background correction in RMA is based on a global model for probe intensities.
The observed intensity, $Y$, is modeled as a sum of an exponential signal
component, $S$ and Gaussian noise $B$. Background corrected data is then
obtained as the expectation $\mathbb{E}_{B}(S|Y)$. While background correction
makes the observations comparable within array, normalization is used to
improve the comparability between arrays. Quantile normalization is a widely
used method that forces all arrays to follow the same empirical intensity
distribution (see e.g. Bolstad et al., 2003). Quantile normalization makes the
measurements across different arrays comparable, assuming that the overall
distribution of mRNA concentration is approximately the same in all cell
populations. This has proven to be a feasible assumption in transcriptional
profiling studies. As always, there are exceptions. For instance, human brain
tissues have systematic differences in gene expression compared to other
organs. On short oligonucleotide arrays, a number of probes target the same
transcript. In the final summarization step, the individual probe-level
observations of each target transcript are summarized into a single summary
estimate of transcript activity. Standard algorithmic implementations are
available for each preprocessing step.
##### Probe-level preprocessing methods
Differences in probe characteristics cause systematic differences in probe
performance. The use of several probes for each target leads to more robust
estimates on transcript activity but it is clear that probe quality may
significantly affect the results of a microarray study (Irizarry et al.,
2003b). Widely used preprocessing algorithms utilize probe-specific parameters
to model probe-specific effects in the probe summarization step. Some of the
first and most well-known probe-level preprocessing algorithms include
dChip/MBEI (Li and Wong, 2001), RMA (Irizarry et al., 2003a), and gMOS (Milo
et al., 2003). Taking probe-level effects into account can considerably
improve the quality of a microarray study (Reimers, 2010). Publications 1 and
2 incorporate side information of the probes to preprocessing, and introduce
improved probe-level analysis methods for differential gene expression
studies.
In order to introduce probe-level preprocessing methods in more detail, let us
consider the probe summarization step of the RMA algorithm (Irizarry et al.,
2003a). RMA has a Gaussian model for probe effects with probe-specific mean
parameters and a shared variance parameter for the probes. The mean parameters
characterize probe-specific binding affinities that cause systematic
differences in the signal levels captured by each probe. Estimating the probe-
specific effects helps to remove this effect in the final probeset-level
summary of the probe-level observations. To briefly outline the algorithm, let
us consider a collection of probes (a probeset) that measure the expression
level of the same target transcript $g$ in condition $i$. The probe-level
observations are modeled as a sum of the true, underlying expression signal
$g_{i}$, which is common to all probes, probe-specific binding affinity
$\mathbf{\mu}_{j}$, and Gaussian noise $\epsilon$. A probe-level observation
for probe $j$ in condition $i$ is then modeled in RMA as
$s_{ij}=g_{i}+\mathbf{\mu}_{j}+\epsilon.$ (4.1)
Measurements from multiple conditions are needed to estimate the probe-
specific effects $\mathbf{\mu}_{j}$. RMA and other models that measure
absolute gene expression have an important drawback: the probe affinity
effects $\\{\mathbf{\mu}_{j}\\}$ are unidentifiable. In order to obtain an
identifiable model, the RMA algorithm includes an additional constraint that
the probe affinity effects are zero on average:
$\Sigma_{j}\mathbf{\mu}_{j}=0$. This yields a well-defined algorithm that has
been shown to produce accurate measurements of gene expression in practical
settings. Further extensions of the RMA algorithm include gcRMA, which has a
more detailed chemical model for the probe effects (Wu and Irizarry, 2004),
refRMA (Katz et al., 2006), which utilizes probe-specific effects derived from
background data collections, and fRMA (McCall et al., 2010), which also models
batch-specific effects in microarray studies. The estimation of unidentifiable
probe affinities is a main challenge for most probe-level preprocessing
models.
RMA and other probe-level models for short oligonucleotide arrays have been
designed to estimate absolute expression levels of the genes. However, gene
expression studies are often ultimately targeted at investigating differential
expression levels, that is, differences in gene expression between
experimental conditions. Measurements of differential expression is obtained
for instance by comparing the expression levels, obtained through the RMA
algorithm or other methods, between different conditions. However, the
summarization of the probe-level values is then performed prior to the actual
comparison. Due to the unidentifiability of the probe affinity parameters in
the RMA and other probe-level models, this is potentially suboptimal.
Publication 1 demonstrates that reversing the order, i.e., calculating
differential gene expression already at the probe level before probeset-level
summarization, leads to improved estimates of differential gene expression.
The explanation is that the procedure circumvents the need to estimate the
unidentifiable probe affinity parameters. This is formally described in
Publication 2, which provides a probabilistic extension of the Probe-level
Expression Change Averaging (PECA) procedure of Publication 1. In PECA, a
standard weighted average statistics summarizes the probe level observations
of differential gene expression. PECA does not model probe-specific effects,
but it is shown to outperform widely used probe-level preprocessing methods,
such as the RMA, in estimating differential expression. Publication 2,
considered in more detail in Section 4.3, provides an extended probabilistic
framework that also models probe-specific effects.
##### Utilizing side information in transcriptome databases
Probe-level preprocessing models and microarray analysis can be further
improved by utilizing external information of the probes (Eisenstein, 2006;
Hwang et al., 2004; Katz et al., 2006). Although any given microarray is
designed on most up-to-date sequence information available, rapidly evolving
genomic sequence data can reveal inaccuracies in probe annotations when the
body of knowledge grows. In recent studies, including Publication 1, a
remarkable number of probes on various oligonucleotide arrays have been
detected not to uniquely match their intended target (Hwang et al., 2004;
Mecham et al., 2004a). A remarkable portion of probes on several popular
microarray platforms in human and mouse did not match with their intended mRNA
target, or were found to target unintended mRNA transcripts in the Entrez
Nucleotide (Wheeler et al., 2005) sequence database in Publication 1 (Table
4.1). The observations are in general concordant with other studies, although
the exact figures vary according to the utilized database and comparison
details (Gautier et al., 2004; Mecham et al., 2004b). In this thesis,
strategies are developed to improve microarray analysis with background
information from genomic sequence databases, and with model-based analysis of
microarray collections.
Probe verification is increasingly used in standard preprocessing, and to
confirm the results of a microarray study. Matching the probe sequences of a
given array to updated genomic sequence databases and constructing an
alternative interpretation of the array data based on the most up-to-date
genomic annotations has been shown to increase the accuracy and cross-platform
consistency of microarray analyses in Publication 1 and elsewhere (Dai et al.,
2005; Gautier et al., 2004).
Publication 1 combines probe verification with a novel probe-level
preprocessing method, PECA, to suggest a novel framework for comparing and
combining results across different microarray platforms. While huge
repositories of microarray data are available, the data for any particular
experimental condition is typically scarce, and coming from a number of
different microarray platforms. Therefore reliable approaches for integrating
microarray data are valuable. Integration of results across platforms has
proven problematic due to various sources of technical variation between array
technologies. Matching of probe sequences between microarray platforms has
been shown to increase the consistency of microarray measurements (Hwang et
al., 2004; Mecham et al., 2004b). However, probe matching between array
platforms guarantees only technical comparability (Irizarry et al., 2005).
Probe verification against external sequence databases is needed to confirm
that the probes are also biologically accurate. This can also improve the
comparability across array platforms, as confirmed by the validation studies
in Publication 1 (Figure 4.2A).
The PECA method of Publication 1 utilizes genomic sequence databases to reduce
probe-level noise by removing erroneous probes based on updated genomic
knowledge. The strategy relies on external information in the databases and
can therefore only remove known sources of probe-level contamination.
Publication 2 introduces a probabilistic framework to measure probe
reliability directly based on microarray data collections. The analysis can
reveal both well-characterized and unknown sources of probe-level
contamination, and leads to improved estimates of gene expression. This model,
coined Robust Probabilistic Averaging (RPA), also provides a theoretically
justified framework for incorporating prior knowledge of the probes into the
analysis.
Array type | Number of probes | Verified probes (%)
---|---|---
HG-U133 Plus2.0 | 604,258 | 58.2
HG-U133A | 247,965 | 82.5
HG-U95Av2 | 199,084 | 82.6
MOE430 2.0 | 496,468 | 68.2
MG-U74Av2 | 197,993 | 73.1
Table 4.1: The proportion of sequence-verified probes on three popular human microarray platforms and two mouse platforms, as observed in Publication 1. Probes that matched to mRNA sequences corresponding to unique genes (defined by a GeneID identifier) in the Entrez database are considered verified. A remarkable portion of the probes on the investigated arrays did not match the Entrez transcript sequences, or had ambiguous targets. A | B
---|---
|
Figure 4.2: A Effect of sequence verification on comparability between
microarray platforms. Correlations between RMA-preprocessed technical
replicates on two array platforms where the same samples have been hybridized
on the two array types. The Pearson correlations were calculated for each pair
of arrays measuring the same biological sample. The gray lines show
correlations obtained with the different probe matching criteria. In the hESC
array comparison, the best match probe sets contained exactly the same probes
on both array generations, which resulted in very high correlations. The
advantages of probe verification and alternative mappings were largest when
arrays with different probe collections were compared in the mCPI, ALL and IM
array comparisons. B Reproducibility of signal estimates in real data sets
between the technical replicates, i.e., the ’best match’ probe sets between
the HG-U95Av2 and HG-U133A platforms. The consistency was measured by the
Pearson correlation between the pairs of arrays, to which the same sample was
hybridized. ©Published by Oxford University Press. Reprinted with permission
from Publication 1.
### 4.3 Model-based noise reduction
Standard approaches for investigating probe performance typically rely on
external information, such as genomic sequence data (see Mecham et al. 2004b;
Zhang et al. 2005 and Publication 1) or physical models (Naef and Magnasco,
2003; Wu et al., 2005). However, such models cannot reveal probes with
uncharacterized sources of contamination, such as cross-hybridization with
alternatively spliced transcripts or closely related mRNA sequences. Vast
collections of microarray data are available in public repositories. These
large-scale data sets contain valuable information of both biological and
technical aspects of gene expression studies. Publication 2 introduces a data-
driven strategy to extract and utilize probe-level information in microarray
data collections.
The model, Robust Probabilistic Averaging (RPA), is a probabilistic
preprocessing procedure that is based on explicit modeling assumptions to
analyze probe reliability and quantify the uncertainty in measurement data
based on gene expression data collections, independently of external
information of the probes. The model can be viewed as a probabilistic
extension of the probe-level preprocessing approach for differential gene
expression studies presented in Publication 1. The explicit Bayesian
formulation quantifies the uncertainty in the model parameters, and allows the
incorporation of prior information concerning probe reliability into the
analysis. RPA provides estimates of probe reliability, and a probeset-level
estimate of differential gene expression directly from expression data and
independently of the noise source. The RPA model is independent of physical
models or external and constantly updated information such as genomic sequence
data, but provides a framework for incorporating such prior information of the
probes in gene expression analysis.
Other probabilistic methods for microarray preprocessing include BGX (Hein et
al., 2005), gMOS (Milo et al., 2003) and its extensions (Liu et al., 2005).
The key difference to the RPA procedure of Publication 2 is that these methods
are designed to provide probeset-level summaries of absolute gene expression
levels, and suffer from the same unidentifiability problem of probe affinity
parameters as the RMA algorithm (Irizarry et al., 2003a). In contrast, RPA
models probe-level estimates of differential gene expression. This removes the
unidentifiability issue, which is advantageous when the objective is to
compare gene expression levels between experimental conditions. Another
important difference is that the other preprocessing methods do not provide
explicit estimates of probe-specific parameters, or tools to investigate probe
performance. Publication 2 assigns an explicit probabilistic measure of
reliability to each probe. This gives tools to analyze probe performance and
to guide probe design.
##### Robust Probabilistic Averaging
Let us now consider in more detail the probabilistic preprocessing framework,
RPA, introduced in Publication 2. Probe performance is ultimately determined
by its ability to accurately measure the expression level of the target
transcript, which is unknown in practical situations. Although the performance
of individual probes varies, the collection of probes designed to measure the
same transcript will provide ground truth for assessing probe performance
(Figure 4.1A). RPA captures the shared signal of the probes within a probeset,
and assumes that the shared signal characterizes the expression of the common
target transcript of the probes. The reliability of individual probes is
estimated with respect to the strongest shared signal of the probes. RPA
assumes normally distributed probe effects, and quantifies probe reliability
based on probe variance around the probeset-level signal across a large number
of arrays. This extends the formulation of the RMA model in Equation 4.1 by
introducing an additional probe-specific Gaussian noise component:
$s_{ij}=g_{i}+\mathbf{\mu}_{j}+\mathbf{\varepsilon}_{ij}.$ (4.2)
In contrast to RMA, the variance is probe-specific in this model, and
distributed as $\mathbf{\varepsilon}_{ij}\sim N(0,\mathbf{\tau}_{j}^{2})$. The
variance parameters $\\{\tau_{j}^{2}\\}$ are of interest in probe reliability
analysis; they reflect the noise level of the probe, in contrast to probe-
level preprocessing methods that focus on estimating the unidentifiable mean
parameter of the Gaussian noise model, corresponding to probe affinity (see
e.g. Irizarry et al., 2003a; Li and Wong, 2001). In Publication 2, probe-level
calculation of differential expression avoids the need to model unidentifiable
probe affinities, the key probe-specific parameter in other probe-level
preprocessing methods. More formally, the unidentifiable probe affinity
parameters $\mu_{.}$ cancel out in RPA when the signal log-ratio between a
user-specified ’reference’ array and the remaining arrays is computed for each
probe: the differential expression signal between arrays $t=\\{1,\dots,T\\}$
and the reference array $c$ for probe $j$ is obtained by
$m_{tj}=s_{tj}-s_{cj}=g_{t}-g_{c}+\varepsilon_{tj}-\varepsilon_{cj}=d_{t}+\varepsilon_{tj}-\varepsilon_{cj}$.
In vector notation, the differential expression profile of probe $j$ across
the $T$ arrays is then written as
$\mathbf{m}_{j}=\mathbf{d}+\boldsymbol{\varepsilon}_{j}$, i.e., a noisy
observation of the true underlying differential expression signal $\mathbf{d}$
and probe-specific noise $\boldsymbol{\varepsilon}_{j}$.
The unidentifiable probe affinity parameters cancel out in the RPA model of
Publication 2. This can partly explain the previous empirical observations
that calculating differential expression already at probe-level improves the
analysis of differential gene expression (Zhang et al., 2002; Elo et al.,
2005). However, the previous models are non-probabilistic preprocessing
methods that do not aim at quantifying the uncertainty in the probes. Use of a
single parameter for probe effects in RPA also gives more straightforward
interpretations of probe reliability.
Posterior estimates of the model parameters are derived to estimate probe
reliability and differential gene expression. The differential expression
vector $\mathbf{d}=\\{d_{t}\\}$ and the probe-specific variances
$\boldsymbol{\tau}^{2}=\\{\mathbf{\tau}_{j}^{2}\\}$ are estimated
simultaneously. The posterior density of the model parameters is obtained from
the likelihood of the data and the prior according to Bayes’ rule (Equation
3.3) as
$p(\mathbf{d},\boldsymbol{\tau}^{2}|\mathbf{m})\sim
p(\mathbf{m}|\mathbf{d},\boldsymbol{\tau}^{2})p(\mathbf{d},\boldsymbol{\tau}^{2}).$
(4.3)
To obtain this posterior, let us consider the likelihood
$p(\mathbf{m}|\mathbf{d},\boldsymbol{\tau}^{2})$ of the data and the prior
$p(\mathbf{d},\boldsymbol{\tau}^{2})$ of the model parameters. The noise on
the selected control array $\varepsilon_{cj}$ is a latent variable, and
marginalized out in the model to obtain the likelihood:
$\begin{split}p(\mathbf{m}|\mathbf{d},\boldsymbol{\tau}^{2})=\prod_{tj}\int
N(m_{tj}|d_{t}-\varepsilon_{cj},\mathbf{\tau}_{j}^{2})N(\varepsilon_{cj}|0,\mathbf{\tau}_{j}^{2})d\varepsilon_{cj}\\\
\sim\prod_{j}(2\pi\mathbf{\tau}_{j}^{2})^{-\frac{T}{2}}exp(-\frac{\sum_{t}(m_{tj}-d_{t})^{2}-\frac{[\sum_{t}(m_{tj}-d_{t})]^{2}}{T+1}}{2\mathbf{\tau}_{j}^{2}}).\end{split}$
(4.4)
Let us assume independent priors,
$p(\mathbf{d},\boldsymbol{\tau}^{2})=p(\mathbf{d})p(\boldsymbol{\tau}^{2})$,
flat non-informative prior $p(\mathbf{d})\sim 1$ and conjugate priors for the
variance parameters in $\boldsymbol{\tau}^{2}$ (inverse Gamma function, see
Gelman et al. 2003). With these standard assumptions, the prior takes the form
$p(\mathbf{d},\boldsymbol{\tau}^{2})\sim\prod_{j}IG(\mathbf{\tau}_{j}^{2};\alpha_{j},\beta_{j}),$
(4.5)
where $\alpha_{j}$ and $\beta_{j}$ are the shape and scale parameters of the
inverse Gamma distribution. Prior information of the probes can be
incorporated in the analysis through these parameters. Probe-level
differential expression is then described by two sets of parameters; the
differential gene expression vector $\mathbf{d}=[d_{1}\dots d_{T}]$, and the
probe-specific variances
$\boldsymbol{\tau}^{2}=[\tau^{2}_{1}\dots\tau^{2}_{J}]$. High variance
$\mathbf{\tau}_{j}^{2}$ indicates that the probe-level observation
$\mathbf{m}_{j}$ is strongly deviated from the estimated true signal
$\mathbf{d}$. Denoting $\hat{\alpha}_{j}=\alpha_{j}+\frac{T}{2}$ and
$\hat{\beta}_{j}=\beta_{j}+\frac{1}{2}\sum_{t}(m_{tj}-d_{t})^{2}-\frac{1}{2}\frac{(\sum_{t}(m_{tj}-d_{t}))^{2}}{T+1}$,
the posterior of the model parameters in Equation 4.3 takes the form
$p(\mathbf{d},\boldsymbol{\tau}^{2}|\mathbf{m})\sim\prod_{j}(\mathbf{\tau}_{j}^{2})^{-(\hat{\alpha}_{j}+1)}exp(-\frac{\hat{\beta}_{j}}{\mathbf{\tau}_{j}^{2}}).$
(4.6)
The formulation allows estimating the uncertainty in the expression estimates
and probe-level parameters. In practice, a MAP point estimate of the
parameters, obtained by maximizing the posterior, is often sufficient. In the
limit of a large sample size ($T\rightarrow\infty$), the model will converge
to estimating ordinary mean and variance parameters. With limited sample sizes
that are typical in microarray studies the prior parameters provide
regularization that makes the probabilistic formulation more robust to
overfitting and local optima, compared to direct estimation of the mean and
variance parameters. Moreover, the probabilistic analysis takes the
uncertainty in the data and model parameters into account in an explicit
manner.
The model also provides a principled framework for incorporating prior
knowledge probe reliability in microarray preprocessing through the probe-
specific hyperparameters $\alpha,\beta$. Estimation and use of probe-specific
effects from external microarray data collections has been previously
suggested in the context of the refRMA method by Katz et al. (2006), where
such side information was shown to improve gene expression estimates. The RPA
method of Publication 2 provides an alternative probabilistic treatment.
##### Model validation
The probabilistic RPA model introduced in Publication 2 was validated by
comparing the preprocessing performance to other preprocessing methods, and
additionally by comparing the estimates of probe-level noise to known sources
of probe-level contamination. The comparison methods include the FARMS
(Hochreiter, 2006), MAS5 (Hubbell et al., 2002), PECA (Publication 1), and RMA
(Irizarry et al., 2003a) preprocessing algorithms. FARMS has a more detailed
model for probe effects than the other methods, and it contains implicitly a
similar probe-specific variance parameter than our RPA model. FARMS is based
on a factor analysis model, and is defined as
$s_{ij}=z_{i}\lambda_{j}+\mathbf{\mu}_{j}+\mathbf{\varepsilon}_{ij}$, where
$z_{i}$ captures the underlying gene expression. In contrast to RMA and RPA
that have a single probe-specific parameter, FARMS has three probe-specific
parameters $\\{\lambda_{j},\mathbf{\mu}_{j},\mathbf{\varepsilon}_{ij}\\}$.
MAS5 is a standard preprocessing algorithm provided by the array manufacturer.
The algorithm performs local background correction, utilizes so-called
mismatch probes to control for non-specific hybridization, and scales the data
from each array to the same average intensity level to improve comparability
across arrays. MAS5 summarizes probe-level observations of absolute gene
expression levels using robust summary statistics, Tukey biweight estimate,
but unlike FARMS, RMA and RPA, MAS5 does not model probe-specific effects.
The preprocessing performance of these methods was investigated in spike-in
experiments where certain target transcripts measured by the array have been
spiked in at known concentrations, as well as on real data sets. The results
from the spike-in experiments were compared in terms of receiver operating
characteristics (ROC). The standard RMA, PECA (Publication 1) and RPA
(Publication 2) had comparable performance in spike-in data, and they
outperformed the MAS5 (Hubbell et al., 2002) and FARMS (Hochreiter, 2006)
preprocessing algorithms in estimating differential gene expression. On real
data sets, PECA and RPA outperformed the other methods, providing higher
reproducibility between technical replicates measured on different microarray
platforms (Figure 4.2B).
In contrast to standard preprocessing algorithms, RPA provides explicit
quantitative estimates of probe performance. The model has been validated on
widely used human whole-genome arrays by comparing the estimates of probe
reliability with known probe-level error sources: errors in probe-genome
alignment, interrogation position of a probe on the target sequence, GC-
content, and the presence of SNPs in the probe target sequences; a good model
for assessing probe reliability should detect probes contaminated by the known
error sources. The results from our analysis can be used to characterize the
relative contribution of different sources of probe-level noise (Figure 4.1B).
In general, the probes with known sources of contamination were more noisy
than the other probes, with 7-39% increase in the average variance, as
detected by RPA. Any single source of error seems to explain only a fraction
of the most highly contaminated probes. A large portion (35-60%) of the
detected least reliable probes were not associated with the investigated known
noise sources. This suggests that previous methods that remove probe-level
noise based on external information, such as genomic alignments will fail to
detect a significant portion of poorly performing probes. The RPA model of
Publication 2 provides rigorous algorithmic tools to investigate the various
probe-level error sources. Better understanding of the factors affecting probe
performance can advance probe design and contribute to reducing probe-related
noise in future generations of gene expression arrays.
### 4.4 Conclusion
The contributions presented in this Chapter provide improved preprocessing
strategies for differential gene expression studies. The introduced techniques
utilize probe-level analysis, as well as side information in sequence and
microarray databases. Probe-level studies have led to the establishment of
probe verification and alternative microarray interpretations as a standard
step in microarray preprocessing and analysis. The alternative interpretations
for microarray data based on updated genomic sequence data (Gautier et al.,
2004; Dai et al., 2005) are now implemented as routine tools in popular
preprocessing algorithms such as the RMA, or the RPA method of Publication 2.
The probe-level analysis strategy has been recently extended to exon array
context, where expression levels of alternative splice variants of the same
genes are compared under particular experimental conditions. The probe-level
approach has shown superior preprocessing performance also with exon arrays
(Laajala et al., 2009). A convenient access to the algorithmic tools developed
in Publications 1 and 2 for microarray preprocessing and probe-level analysis
is provided by the accompanied open source implementation in
BioConductor.111http://www.bioconductor.org/packages/release/bioc/html/RPA.html
## Chapter 5 Global analysis of the human transcriptome
> _When we try to pick out anything by itself, we find that it is bound fast
> by a thousand invisible cords that cannot be broken, to everything in the
> universe._
>
> J. Muir (1869)
Measurements of transcriptional activity provide only a partial view to
physiological processes, but their wide availability provides a unique
resource for investigating gene activity at a genome- and organism-wide scale.
Versatile and carefully controlled gene expression atlases have become
available for normal human tissues, cancer as well as for other diseases (see,
for instance, Kilpinen et al., 2008; Lukk et al., 2010; Roth et al., 2006; Su
et al., 2004). These data sources contain valuable information about shared
and unique mechanisms between disparate conditions, which is not available in
smaller and more specific experiments (Lage et al., 2008; Scherf et al.,
2000). While standard methods for gene expression analysis have focused on
comparisons between particular conditions, versatile transcriptome atlases
allow for global organism-wide characterization of transcriptional activation
patterns (Levine et al., 2006). Novel methodological approaches are needed in
order to realize the full potential of these information sources, as many
traditional methods for expression analysis are not applicable to versatile
large-scale collections. This chapter provides an overview to current
approaches for global transcriptome analysis in Section 5.1 and introduces the
second main contribution of the thesis, a novel exploratory approach that can
be used to investigate context-specific responses in genome-scale interaction
networks across organism-wide collections of measurement data in Section 5.2.
The conclusions are summarized in Section 5.3.
### 5.1 Standard approaches
Global observations of transcriptional activity reflect known and previously
uncharacterized cell-biological processes. Exploratory analysis of the
transcriptome can provide research hypotheses and material for more detailed
investigations. Widely-used standard approaches for global transcriptome
analysis include various clustering, dimensionality reduction and
visualization techniques (see e.g. Huttenhower and Hofmann, 2010; Polanski and
Kimmel, 2007; Quackenbush, 2001). The large data collections open up new
possibilities to investigate functional relatedness between physiological
conditions, disease states, as well as cellular processes, and to discover
previously uncharacterized connections and functional mechanisms (Bergmann et
al., 2004; Kilpinen et al., 2008; Lukk et al., 2010).
Gene expression studies have traditionally focused on the analysis of
relatively small and targeted data sets, such as particular diseases or cell
types. A typical objective is to detect genes, or gene groups, that are
differentially expressed between particular conditions, for instance to
predict disease outcomes, or to identify potentially unknown disease subtypes.
The increasing availability of large and versatile transcriptome collections
that may cover thousands of experimental conditions allows global, data-driven
analysis, and the formulation of novel research questions where the
traditional analysis methods are often insufficient (Huttenhower and Hofmann,
2010).
A variety of approaches have been proposed and investigated in the recent
years in the global transcriptome analysis context. An actively studied
modeling problem in transcriptome analysis is the discovery of transcriptional
modules, i.e., identification of coherent gene groups that show coordinated
transcriptional responses under particular conditions (Segal et al., 2003a,
2004; Stuart et al., 2003). Models have also been proposed to predict gene
regulators (Segal et al., 2003b), and to infer cellular processes and networks
based on transcriptional activation patterns (Friedman, 2004; Segal et al.,
2003c). An increasing number of models are being developed to integrate
transcriptome measurements to other sources of genomic information, such as
regulation and interactions between the genes to detect and characterize
cellular processes and disease mechanisms (Barash and Friedman, 2002; Chari et
al., 2010; Vaske et al., 2010). Findings from transcriptome analysis have
potential biomedical implications, as in Lamb et al. (2006), where chemically
perturbed cancer cell lines were screened to enhance the detection of drug
targets based on shared functional mechanisms between disparate conditions, or
in Sørlie et al. (2001), where cluster analysis of cancer patients based on
genome-wide transcriptional profiling experiments led to the discovery of a
novel breast cancer subtype. In the remainder of this section, the modeling
approaches that are particularly closely related to the contributions of this
thesis are considered in more detail.
##### Investigating known processes
A popular strategy for genome-wide gene expression analysis is to consider
known biological processes and their activation patterns across diverse
collections measurement data from various experimental conditions. Biomedical
databases contain a variety of information concerning genes and their
interactions. For instance, the Gene Ontology database (Ashburner et al.,
2000) provides functional and molecular classifications for the genes in human
and a number of other organisms. Other categories are based on micro-RNA
regulation, chromosomal locations, chemical perturbations and other features
(Subramanian et al., 2005). Joint analysis of functionally related genes can
increase the statistical power of the analysis. So-called gene set-based
approaches are typically designed to test differential expression between two
particular conditions (Goeman and Buhlmann, 2007; Nam and Kim, 2008), but they
can also be used to build global maps of transcriptional activity of the known
processes (Levine et al., 2006). However, gene set-based approaches typically
ignore more detailed information of the interactions between individual genes.
Pathway and interaction databases contain more detailed information concerning
molecular interactions and cell-biological processes (Kanehisa et al., 2008;
Vastrik et al., 2007). Network-based methods utilize relational information of
the genes to guide expression analysis. For instance, Draghici et al. (2007)
demonstrated that taking into account aspects of pathway topology, such as
gene and interaction types, can improve the estimation of pathway activity
between two predefined conditions. Another recent approach which utilizes
pathway topology in inferring pathway activity is PARADIGM (Vaske et al.,
2010), which also integrates other sources of genomic information in pathway
analysis. However, these methods have been designed for the analysis of
particular experimental conditions, rather than comprehensive expression
atlases. MATISSE (Ulitsky and Shamir, 2007) is a network-based approach that
searches for functionally related genes that are connected in the network, and
have correlated expression profiles across many conditions. The potential
shortcoming of this approach is that it assumes global correlation across all
conditions between the interacting genes, while many genes can have multiple,
context-sensitive functional roles. Different conditions induce different
responses in the same genes, and the definition of ’gene set’ is vague
(Montaner et al., 2009; Nacu et al., 2007). Therefore methods have been
suggested to identify ’key condition-responsive genes’ of predefined gene sets
(Lee et al., 2008), or to decompose predefined pathways into smaller and more
specific functional modules (Chang et al., 2009). These approaches rely on
predefined functional classifications for the genes. The data-driven analysis
in Publication 3 provides a complementary approach where the gene sets are
learned directly from the data, guided by prior knowledge of genetic
interactions. This avoids the need to refine suboptimal annotations, and
enables the discovery of new processes. The findings demonstrate that simply
measuring whether a gene set, or a network, is differentially expressed
between particular conditions is often not sufficient for measuring the
activity of cell-biological processes. Since gene function and interactions
are regulated in a context-specific manner, it is important to additionally
characterize how, and in which conditions the expression changes. Global
analysis of transcriptional activation patterns interaction networks,
introduced in Publication 3, can address such questions.
##### Biclustering and subspace clustering
Approaches that are based on previously characterized genes and processes are
biased towards well-characterized phenomena. This limits their value in de
novo discovery of functional patterns. Unsupervised methods provide tools for
such analysis, but often with an increased computational cost and a higher
proportion of false positive findings.
Cluster analysis is widely used for unsupervised analysis of gene expression
data, providing tools for class discovery, gene function prediction and for
visualization purposes. Examples of widely used clustering approaches include
hierarchical clustering and K-means (see e.g. Polanski and Kimmel, 2007).
Clustering of patient samples with similar expression profiles has led to the
discovery of novel cancer subtypes with biomedical implications (Sørlie et
al., 2001); clustering of genes with coordinated activation patterns can be
used, for instance, to predict novel functional associations for poorly
characterized genes (Allocco et al., 2004). The self-organizing map (Kohonen,
1982, 2001) is a related approach that provides efficient tools to visualize
high-dimensional data on lower-dimensional displays, with particular
applications in transcriptional profiling studies (Tamayo et al., 1999;
Törönen et al., 1999). The standard clustering methods are based on comparison
of global expression patterns, and therefore are relatively coarse tools for
analyzing large transcriptome collections. Different genes respond in
different ways, as well as in different conditions. Therefore it is
problematic to find clusters in high-dimensional data spaces, such as in
whole-genome expression profiling studies; different gene groups can reveal
different relationships between the samples. Detection of smaller, coherent
subspaces with a particular structure can be useful in biomedical
applications, where the objective is to identify sets of interesting genes for
further analysis. Both genes and the associated conditions may be unknown, and
the learning task is to detect them from the data. This can help, for
instance, in identifying responses to drug treatments in particular genes
(Ihmels et al., 2002; Tanay et al., 2002), or in identifying functionally
coherent transcriptional modules in gene expression databases (Segal et al.,
2004; Tanay et al., 2005).
Subspace clustering methods (Parsons et al., 2004) provide a family of
algorithms that can be used to identify subsets of dependent features
revealing coherent clustering for the samples; this defines a subspace in the
original feature space. Subspace clustering models are a special case of a
more general family of biclustering algorithms (Madeira and Oliveira, 2004).
Closely related models are also called co-clustering (Cho et al., 2004), two-
way clustering Gad et al. (2000), and plaid models (Lazzeroni and Owen, 2002).
Biclustering methods provide general tools to detect co-regulated gene groups
and associated conditions from the data, to provide compact summaries and to
aid interpretation of transcriptome data collections. Biclustering models
enable the discovery of gene expression signatures (Hu et al., 2006) that have
emerged as a central concept in global expression analysis context. A
signature describes a co-expression state of the genes, associated with
particular conditions. Established signatures have been found to be reliable
indicators of the physiological state of a cell, and commercial signatures
have become available for routine clinical practice (Nuyten and van de Vijver,
2008). However, the established signatures are typically designed to provide
optimal classification performance between two particular conditions. The
problem with the classification-based signatures is that their associations to
the underlying physiological processes are not well understood (Lucas et al.,
2009). In Publication 3 the understanding is enhanced by deriving
transcriptional signatures that are explicitly connected to well-characterized
processes through the network.
##### Role of side information
Standard clustering models ignore prior information of the data, which could
be used to supervise the analysis, to connect the findings to known processes,
as well as to improve scalability. For instance, standard model-based feature
selection, or subspace clustering techniques would consider all potential
connections between the genes or features (Law et al., 2004; Roth and Lange,
2004). Without additional constraints on the solution space they can typically
handle at most tens or hundreds of features, which is often insufficient in
high-throughput genomics applications. Use of side information in clustering
can help to guide unsupervised analysis, for instance based on known or
potential interactions between the genes. This has been shown to improve the
detection of functionally coherent gene groups (Hanisch et al., 2002; Shiga et
al., 2007; Ulitsky and Shamir, 2007; Zhu et al., 2005). However, while these
methods provide tools to cluster the genes, they do not model differences
between conditions. Extensions of biclustering models that can utilize
relational information of the genes include cMonkey (Reiss et al., 2006) and a
modified version of SAMBA biclustering (Tanay et al., 2004). However, cMonkey
and SAMBA are application-oriented tools that rely on additional, organism-
specific information, and their implementation is currently not available for
most organisms, including that of the human. Further application-oriented
models for utilizing side information in the discovery of transcriptional
modules have recently been proposed for instance by Savage et al. (2010) and
Suthram et al. (2010). Publication 3 introduces a complementary method where
the exhaustively large search space is limited with side information
concerning known relations between the genes, derived from genomic interaction
databases. This is a general algorithmic approach whose applicability is not
limited to particular organisms.
##### Other approaches
Prior information on the cellular networks, regulatory mechanisms, and gene
function is often available, and can help to construct more detailed models of
gene function and network analysis, as well as to summarize functional aspects
of genomic data collections (Huttenhower et al., 2009; Segal et al., 2003b;
Troyanskaya, 2005). Versatile transcriptome collections also enable network
reconstruction, i.e., de novo discovery (Lezon et al., 2006; Myers et al.,
2005) and augmentation (Novak and Jain, 2006) of genetic interaction networks.
Other methodological approaches for global transcriptome analysis are provided
by probabilistic latent variable models (Rogers et al., 2005; Segal et al.,
2003a), hierarchical Dirichlet process algorithms (Gerber et al., 2007), as
well as matrix and tensor computations (Alter and Golub, 2005). These methods
provide further model-based tools to identify and characterize transcriptional
programs by decomposing gene expression data sets into smaller, functionally
coherent components.
### 5.2 Global modeling of transcriptional activity in interaction networks
Molecular interaction networks cover thousands of genes, proteins and small
molecules. Coordinated regulation of gene function through molecular
interactions determines cell function, and is reflected in transcriptional
activity of the genes. Since individual processes and their transcriptional
responses are in general unknown (Lee et al., 2008; Montaner et al., 2009),
data-driven detection of condition-specific responses can provide an efficient
proxy for identifying distinct transcriptional states of the network with
potentially distinct functional roles. While a number of methods have been
proposed to compare network activation patterns between particular conditions
(Draghici et al., 2007; Ideker et al., 2002; Cabusora et al., 2005; Noirel et
al., 2008), or to use network information to detect functionally related gene
groups (Segal et al., 2003d; Shiga et al., 2007; Ulitsky and Shamir, 2007),
general-purpose algorithms for a global analysis of context-specific network
activation patterns in a genome- and organism-wide scale have been missing.
Publication 3 introduces and validates two general-purpose algorithms that
provide tools for global modeling of transcriptional responses in interaction
networks. The motivation is similar to biclustering approaches that detect
functionally coherent gene groups that show coordinated response in a subset
of conditions (Madeira and Oliveira, 2004). The network ties the findings more
tightly to cell-biological processes, focusing the analysis and improving
interpretability. In contrast to previous network-based biclustering models
for global transcriptome analysis, such as cMonkey (Reiss et al., 2006) or
SAMBA (Tanay et al., 2004), the algorithms introduced in Publication 3 are
general-purpose tools, and do not depend on organism-specific annotations.
##### A two-step approach
The first approach in Publication 3 is a straightforward extension of network-
based gene clustering methods. In this two-step approach, the functionally
coherent subnetworks, and their condition-specific responses are detected in
separate steps. In the first step, a network-based clustering method is used
to detect functionally coherent subnetworks. In Publication 3, MATISSE, a
state-of-the-art algorithm described in Ulitsky and Shamir (2007), is used to
detect the subnetworks. MATISSE finds connected subgraphs in the network that
have high internal correlations between the genes. In the second step,
condition-specific responses of each identified subnetwork are searched for by
a nonparametric Gaussian mixture model, which allows a data-driven detection
of the responses. However, the two-step approach, coined MATISSE+, can be
suboptimal for detecting subnetworks with particular condition-specific
responses. The main contribution of Publication 3 is to introduce a second
general-purpose algorithm, coined NetResponse, where the detection of
condition-specific responses is used as the explicit key criterion for
subnetwork search.
Figure 5.1: Organism-wide analysis of transcriptional responses in a human
pathway interaction network reveals physiologically coherent activation
patterns and condition-specific regulation. One of the subnetworks and its
condition-specific responses, as detected by the NetResponse algorithm is
shown in the Figure. The expression of each gene is visualized with respect to
its mean level of expression across all samples. ©The Author 2010. Published
by Oxford University Press. Reprinted with permission from Publication 3.
##### The NetResponse algorithm
The network-based search procedure introduced in Publication 3 searches for
local subnetworks, i.e., functionally coherent network modules where the
interacting genes show coordinated responses in a subset of conditions (Figure
5.1). Side information of the gene interactions is used to guide modeling, but
the algorithm is independent of predefined classifications for genes or
measurement conditions. Transcriptional responses of the network are described
in terms of subnetwork activation. Regulation of the subnetwork genes can
involve simultaneous activation and repression of the genes: sufficient
amounts of mRNA for key proteins has to be available while interfering genes
may need to be silenced. The model assumes that a given subnetwork $n$ can
have multiple transcriptional states, associated with different physiological
contexts. A transcriptional state is reflected in a unique expression
signature $\mathbf{s}^{(n)}$, a vector that describes the expression levels of
the subnetwork genes, associated with the particular transcriptional state.
Expression of some genes is regulated at precise levels, whereas other genes
fluctuate more freely. Given the state, expression of the subnetwork genes is
modeled as a noisy observation of the transcriptional state. With a Gaussian
noise model with covariance $\Sigma^{(n)}$, the observation is described by
$\mathbf{x}^{(n)}\sim N(\mathbf{s}^{(n)},\Sigma^{(n)})$. A given subnetwork
can have $R^{(n)}$ latent transcriptional states indexed by $r$. In practice,
the states, including their number $R^{(n)}$, are unknown, and they have to be
estimated from the data. In a specific measurement condition, the subnetwork
$n$ can be in any one of the latent physiological states indexed by $r$.
Associations between the observations and the underlying transcriptional
states are unknown and they are treated as latent variables. Gene expression
in subnetwork $n$ is then modeled with a Gaussian mixture model:
$\mathbf{x}^{(n)}\sim\sum_{r=1}^{R^{(n)}}w_{r}^{(n)}p(\mathbf{x}^{(n)}|\boldsymbol{\theta}_{r}),$
(5.1)
where each component distribution $p$ is assumed to be Gaussian with
parameters
$\boldsymbol{\theta}_{r}=\\{\mathbf{s}_{r}^{(n)},\boldsymbol{\Sigma}_{r}^{(n)}\\}$.
In practice, we assume a diagonal covariance matrix
$\boldsymbol{\Sigma}_{r}^{(n)}$, leaving the dependencies between the genes
unmodeled within each transcriptional state. Use of diagonal covariances is
justified by considerable gains in computational efficiency when the detection
of distinct responses is of primary interest. It is possible, however, that
such simplified model will fail to detect certain subnetworks where the
transcriptional levels of the genes have strong linear dependencies within the
individual transcriptional states; signaling cascades could be expected to
manifest such activation patterns, for instance. More detailed models of
transcriptional activity could help to distinguish the individual states in
particular when the transcriptional states are partially overlapping, but with
increased computational cost. A particular transcriptional response is then
characterized with the triple
$\\{\mathbf{s}_{r}^{(n)},\boldsymbol{\Sigma}_{r}^{(n)},w_{r}^{(n)}\\}$. This
defines the shape, fluctuations and frequency of the associated
transcriptional state of subnetwork $n$. A posterior probability of each
latent state can be calculated for each measurement sample from the Bayes’
rule (Equation 3.3). The posterior probabilities can be interpreted as soft
component memberships for the samples. A hard, deterministic assignment is
obtained by selecting for each sample the component with the highest posterior
probability.
The remaining task is to identify the subnetworks having such distinct
transcriptional states. Detection of the distinct states is now used as a
search criterion for the subnetworks. In order to achieve fast computation, an
agglomerative procedure is used where interacting genes are gradually merged
into larger subnetworks. Initially, each gene is assigned in its own singleton
subnetwork. Agglomeration proceeds by at each step merging the two neighboring
subnetworks where joint modeling of the genes leads to the highest improvement
in the objective function value. Joint modeling of dependent genes reveals
coordinated responses and improves the likelihood of the data in comparison
with independent models, giving the first criterion for merging the
subnetworks. However, increasing subnetwork size tends to increase model
complexity and the possibility of overfitting, since the number of samples
remains constant while the dimensionality (subnetwork size) increases. To
compensate for this effect, the Bayesian information criterion (see Gelman et
al., 2003) is used to penalize increasing model complexity and to determine
optimal subnetwork size. The final cost function for a subnetwork $G$ is
$C(G)=-2{\cal L}+qlog(N)$, where ${\cal L}$ is the (marginal) log-likelihood
of the data, given the mixture model in Equation 5.1, $q$ is the number of
parameters and $N$ denotes sample size. The algorithm then compares
independent and joint models for each subnetwork pair that has a direct link
in the network, and merges at each step the subnetwork pair $G_{i},G_{j}$ that
minimizes the cost
$\Delta\mathcal{C}=-2({\cal L}_{i,j}-({\cal L}_{i}+{\cal
L}_{j}))+(q_{i,j}-(q_{i}+q_{j}))log(N).$ (5.2)
The iteration continues until no improvement is obtained by merging the
subnetworks. The combination of modeling techniques yields a scalable
algorithm for genome- and organism-wide investigations: First, the analysis
focuses on those parts of the data that are supported by known interactions,
which increases modeling power and considerably limits the search space.
Second, the agglomerative scheme finds a fast approximative solution where at
each step the subnetwork pair that leads to the highest improvement in cost
function is merged. Third, an efficient variational approximation is used to
learn the mixture models (Kurihara et al., 2007b). Note that the algorithm
does not necessarily identify a globally optimal solution. However, detection
of physiologically coherent and reproducible responses is often sufficient for
practical applications.
##### Global view on network activation patterns
The NetResponse algorithm introduced in Publication 3 was applied to
investigate transcriptional activation patterns of a pathway interaction
network of 1800 genes based on the KEGG database of metabolic pathways
(Kanehisa et al., 2008) provided by the SPIA package (Tarca et al., 2009)
across 353 gene expression samples from 65 tissues. The two algorithms
proposed in Publication 3, MATISSE+ and NetResponse were shown to outperform
an unsupervised biclustering approach in terms of reproducibility of the
finding. The introduced NetReponse algorithm, where the detection of
transcriptional response patterns is used as a search criterion for subnetwork
identification, was the best-performing method. The algorithm identified 106
subnetworks with 3-20 genes, with distinct transcriptional responses across
the conditions. One of the subnetworks is illustrated in Figure 5.1; the other
findings are provided in the supplementary material of Publication 3. The
detected transcriptional responses were physiologically coherent, suggesting a
potential functional role. The reproducibility of the responses was confirmed
in an independent validation data set, where 80% of the predicted responses
were detected ($p<0.05$). The findings highlight context-specific regulation
of the genes. Some responses are shared by many conditions, while others are
more specific to particular contexts such as the immune system, muscles, or
the brain; related physiological conditions often exhibit similar network
activation patterns. Tissue relatedness can be measured in terms of shared
transcriptional responses of the subnetworks, giving an alternative
formulation of the tissue connectome map suggested by Greco et al. (2008) in
order to highlight functional connectivity between tissues based on the number
of shared differentially expressed genes. In Publication 3, shared network
responses are used instead of shared gene count. The use of co-regulated gene
groups is expected to be more robust to noise than the use of individual
genes. The analysis provides a global view on network activation across the
normal human body, and can be used to formulate novel hypotheses of gene
function in previously unexplored contexts.
### 5.3 Conclusion
Gene function and interactions are often subject to condition-specific
regulation (Liang et al., 2006; Rachlin et al., 2006), but these have been
typically studied only in particular experimental conditions. Organism-wide
analysis can potentially reveal new functional connections and help to
formulate novel hypotheses of gene function in previously unexplored contexts,
and to detect highly specialized functions that are specific to few
conditions. Changes in cell-biological conditions induce changes in the
expression levels of co-regulated genes, in order to produce specific
physiological responses, typically affecting only a small part of the network.
Since individual processes and their transcriptional responses are in general
unknown (Lee et al., 2008; Montaner et al., 2009), data-driven detection of
condition-specific responses can provide an efficient proxy for identifying
distinct transcriptional states of the network, with potentially distinct
functional roles.
Publication 3 provides efficient model-based tools for global, organism-wide
discovery and characterization of context-specific transcriptional activity in
genome-scale interaction networks, independently of predefined classifications
for genes and conditions. The network is used to bring in prior information of
gene function, which would be missing in unsupervised models, and allows data-
driven detection of coordinately regulated gene sets and their context-
specific responses. The algorithm is readily applicable in any organism where
gene expression and pairwise interaction data, including pathways, protein
interactions and regulatory networks, are available. It has therefore a
considerably larger scope than previous network-based models for global
transcriptome analysis, which rely on organism-specific annotations, but lack
implementations for most organisms (Reiss et al., 2006; Tanay et al., 2004).
While biomedical implications of the findings require further investigation,
the results highlight shared and reproducible responses between physiological
conditions, and provide a global view of transcriptional activation patterns
across the normal human body. Other potential applications for the method
include large-scale screening of drug responses and disease subtype discovery.
Implementation of the algorithm is freely available through
BioConductor.111http://bioconductor.org/packages/devel/bioc/html/netresponse.html
## Chapter 6 Human transcriptome and other layers of genomic information
> _The way to deal with the problem of big data is to beat it senseless with
> other big data._
>
> J. Quackenbush (2006)
This chapter presents the third main contribution of the thesis, computational
strategies to integrate measurements of human transcriptome to other layers of
genomic information. Genomic, transcriptomic, proteomic, epigenomic and other
sources of measurement data characterize different aspects of genome
organization (Hawkins et al., 2010; Montaner and Dopazo, 2010; Sara et al.,
2010); any single source provides only a limited view to the cellular system.
Understanding functional organization of the genome and ultimately the cell
function requires integration of data from the various levels of genome
organization and modeling of their dynamical interplay. Such an holistic
approach, which is also called systems biology, is a key to understanding
living organisms, which are “rich in emergent properties because forever new
groups of properties emerge at every level of integration” (Mayr, 2004).
Combining evidence across multiple sources can help to discover functional
mechanisms and interactions, which are not seen in the individual data sets,
and to increase statistical power in noisy and incomplete high-throughput
experiments (Huttenhower and Hofmann, 2010; Reed et al., 2006).
Integration of heterogeneous genomic data comes with a variety of technical
and methodological challenges (Hwang et al., 2005; Troyanskaya, 2005), and the
particular modeling approaches vary according to the analysis task and
particular properties of the investigated measurement sources. Integrative
studies have been limited by poor availability of co-occurring genomic
observations, but suitable data sets are now becoming increasingly available
in both in-house and public biomedical data repositories (The Cancer Genome
Atlas Research Network, 2008). New observations highlight the need for novel,
integrative approaches in functional genomics (Coe et al., 2008). Recent
studies have proposed for instance methods to integrate epigenetic
modifications (Sadikovic et al., 2008), micro-RNA (Qin, 2008), transcription
factor binding (Savage et al., 2010), as well as protein expression (Johnson
et al., 2008). Given the complex stochastic nature of biological systems,
computational efficiency, robustness against uncertainty and interpretability
of the results are key issues. Prior information of biological systems is
often incomplete, and subject to high levels of uncontrolled variation and
complex interdependencies between different parts of the cellular system
(Troyanskaya, 2005). These issues emphasize the need for principled approaches
requiring minimal prior knowledge about the data, as well as minimal model
fitting procedures. Section 6.1 gives an overview of the standard models for
high-throughput data integration methods, which have close connections to the
modeling approaches developed in this work.
### 6.1 Standard approaches for genomic data integration
The integrative approaches can be roughly classified in three categories:
methods that (i) combine statistical evidence across related studies in order
to obtain more accurate inferences of target variables, (ii) utilize side
information in order to guide the analysis of a single, primary data source,
and (iii) detect and characterize dependencies between the measurement sources
in order to discover new functional connections between the different layers
of genomic information. The contributions in Chapters 4 and 5 are associated
with the first two categories; the contributions presented in this chapter,
the regularized dependency detection framework of Publication 4, and
associative clustering of Publications 5 and 6, belong to the third category.
#### 6.1.1 Combining statistical evidence
The first general category of methods for genomic data integration consists of
approaches where evidence across similar studies is combined to increase
statistical power, for instance by comparing and integrating data from
independent microarray experiments targeted at studying the same disease. In
Publications 2 and 3, joint analysis of a large number of commensurable
microarray experiments, where the observed data is directly comparable between
the arrays, helps to increase statistical power and to reveal weak, shared
signals in the data that can not be detected in more restricted experimental
setups and smaller datasets.
However, the related observations are often not directly comparable, and
further methodological tools are needed for integration. Meta-analysis
provides tools for such analysis (Ramasamy et al., 2008). Meta-analysis forms
part of the microarray analysis procedure introduced in Publication 1, where
methods to integrate related microarray measurements across different array
platforms are developed. Meta-analysis emphasizes shared effects between the
studies over statistical significance in individual experiments. In its
standard form, meta-analysis assumes that each individual study measures the
same target variable with varying levels of noise. The analysis starts from
identifying a measure of effect size based on differences, means, or other
summary statistics of the observations such as the Hedges’ g, used in
Publication 1. Weighted averaging of the effect sizes provides the final,
combined result. Weighting accounts for differences in reliability of the
individual studies, for instance by emphasizing studies with large sample
size, or low measurement variance. Averaging is expected to yield more
accurate estimates of the target variable than individual studies. This can be
particularly useful when several studies with small sample sizes are available
for instance from different laboratories, which is a common setting in
microarray analysis context, where the data sets produced by individual
laboratories are routinely deposited to shared community databases.
Ultimately, the quality of meta-analysis results rests on the quality of the
individual studies. Modeling choices, such as the choice of the effect size
measure and included studies will affect the analysis outcome.
Kernel methods (see e.g. Schölkopf and Smola, 2002) provide another widely
used approach for integrating statistical evidence across multiple,
potentially heterogeneous measurement sources. Kernel methods operate on
similarity matrices, and provide a natural framework for combining statistical
evidence to detect similarity and patterns that are supported by multiple
observations. The modeling framework also allows for efficient modeling of
nonlinear feature spaces.
Multi-task learning refers to a class of approaches where multiple, related
modeling tasks are solved simultaneously by combining statistical power across
the related tasks. A typical task is to improve the accuracy of individual
classifiers by taking advantage of the potential dependencies between them
(see e.g. Caruana, 1997).
#### 6.1.2 Role of side information
The second category of approaches for genomic data integration consists of
methods that are asymmetric by nature; integration is used to support the
analysis of one, primary data source. Side information can be used, for
instance, to limit the search space and to focus the analysis to avoid
overfitting, speed up computation, as well as to obtain potentially more
sensitive and accurate findings (see e.g. Eisenstein, 2006). One strategy is
to impose hard constraints on the model, or model family, based on side
information to target specific research questions. In gene expression context,
functional classifications or known interactions between the genes can be used
to constrain the analysis (Goeman and Buhlmann, 2007; Ulitsky and Shamir,
2009). In factor analysis and mixed effect models, clinical annotations of the
samples help to focus the modeling on particular conditions (see e.g. Carvalho
et al., 2008). Hard constraints rely heavily on the accuracy of side
information. Soft, or probabilistic approaches can take the uncertainty in
side information into account, but they are computationally more demanding.
Examples of such methods in the context of transcriptome analysis include for
instance the supervised biclustering models, such as cMonkey and modified
SAMBA, as well as other methods that guide the analysis with additional
information of genes and regulatory mechanisms, such as transcription factor
binding (Reiss et al., 2006; Savage et al., 2010; Tanay et al., 2004).
Publication 3 uses gene interaction network as a hard constraint for modeling
transcriptional co-regulation of the genes, but the condition-specific
responses of the detected gene groups are identified in an unsupervised
manner.
A complementary approach for utilizing side information of the experiments is
provided by multi-way learning. A classical example is the analysis of
variance (ANOVA), where a single data set is modeled by decomposing it into a
set of basic, underlying effects, which characterize the data optimally. The
effects are associated with multiple, potentially overlapping attributes of
the measurement samples, such as disease state, gender and age, which are
known prior to the analysis. Taking such prior knowledge of systematic
variation between the samples into account helps to increase modeling power
and can reveal the attribute-specific effects. An interesting subtask is to
model the interactions between the attributes, so-called interaction effects.
These are manifested only with particular combinations of attributes, and
indicate dependency between the attributes. For instance, simultaneous
cigarette smoking and asbestos exposure will considerably increase the risk of
lung cancer, compared to any of the two risk factors alone (see e.g. Nymark et
al., 2007). Factor analysis is a closely related approach where the
attributes, also called factors, are not given but instead estimated from the
data. Mixed effect models combine the supervised and unsupervised approaches
by incorporating both fixed and random effects in the model, corresponding to
the known and latent attributes, respectively (see e.g. Carvalho et al.,
2008). The standard factorization approaches for individual data sets are
related to the dependency-seeking approaches in Publications 4-6, where co-
occurring data sources are decomposed in an unsupervised manner into
components that are maximally informative of the components in the other data
set.
#### 6.1.3 Modeling of mutual dependency
Symmetric models for dependency detection form the third main category of
methods for genomic data integration, as well as the main topic of this
chapter. Dependency modeling is used to distinguish the shared signal from
dataset-specific variation. The shared effects are informative of the
commonalities and interactions between the observations, and are often the
main focus of interest in integrative analysis. This motivates the development
of methods that can allocate computational resources efficiently to modeling
of the shared features and interactions.
Multi-view learning is a general category of approaches for symmetric
dependency modeling tasks. In multi-view learning, multiple measurement
sources are available, and each source is considered as a different view on
the same objects. The task is to enhance modeling performance by combining the
complementary views. A classical example of such a model is canonical
correlation analysis (Hotelling, 1936). Related approaches that have recently
been applied in functional genomics include for instance probabilistic
variants of meta-analysis (Choi et al., 2007; Conlon et al., 2007),
generalized singular value decomposition (see e.g. Alter et al., 2003; Berger
et al., 2006) and simultaneous non-negative matrix factorization (Badea,
2008).
The dependency modeling approaches in this thesis make an explicit distinction
between statistical representation of data and the modeling task. Let us
denote the representations of two co-occurring multivariate observations,
$\mathbf{x}$ and $\mathbf{y}$, with $f_{x}(\mathbf{x})$ and
$f_{y}(\mathbf{y})$, respectively. The selected representations depend on the
application task. The representation can be for instance used to perform
feature selection as in canonical correlation analysis (CCA) Hotelling (1936),
capture nonlinear features in the data as in kernelized versions of CCA (see
e.g. Yamanishi et al., 2003), or partition the data as in information
bottleneck (Friedman et al., 2001) and associative clustering (Publications
5-6). Statistical independence of the representations implies that their joint
probability density can be decomposed as
$p(f_{x}(\mathbf{x}),f_{y}(\mathbf{y}))=p(f_{x}(\mathbf{x}))p(f_{y}(\mathbf{y}))$.
Deviations from this assumption indicate statistical dependency. The
representations can have a flexible parametric form which can be optimized by
the dependency modeling algorithms to identify dependency structure in the
data.
Recent examples of such dependency-maximizing methods include probabilistic
canonical correlation analysis (Bach and Jordan, 2005), which has close
theoretical connection to the regularized models introduced in Publication 4,
and the associative clustering principle introduced in Publications 5-6.
Canonical correlations and contingency table analysis form the methodological
background for the contributions in Publications 4-6. In the remainder of this
section these two standard approaches for dependency detection are considered
more closely.
##### Classical and probabilistic canonical correlation analysis
Canonical correlation analysis (CCA) is a classical method for detecting
linear dependencies between two multivariate random variables (Hotelling,
1936). While ordinary correlation characterizes the association strength
between two vectors with paired scalar observations, CCA assumes paired
vectorial values, and generalizes correlation to multidimensional sources by
searching for maximally correlating low-dimensional representation of the two
sources, defined by linear projections
$\mathbf{X}\mathbf{v}_{x},\mathbf{Y}\mathbf{v}_{y}$. Multiple projection
components can be obtained iteratively, by finding the most correlating
projection first, and then consecutively the next ones after removing the
dependencies explained by the previous CCA components; the lower-dimensional
representations are defined by projections to linear hyperplanes. The model
can be formulated as a generalized eigenvalue problem that has an analytical
solution with two useful properties: the result is invariant to linear
transformations of the data, and the solution for any fixed number of
components maximizes mutual information between the projections for Gaussian
data (Kullback, 1959; Bach and Jordan, 2002). Extensions of the classical CCA
include generalizations to multiple data sources (Kettenring, 1971; Bach and
Jordan, 2002), regularized solutions with non-negative and sparse projections
(Sigg et al., 2007; Archambeau and Bach, 2008; Witten et al., 2009), and non-
linear extensions, for instance with kernel methods (Bach and Jordan, 2002;
Yamanishi et al., 2003). Direct optimization of correlations in the classical
CCA provides an efficient way to detect dependencies between data sources, but
it lacks an explicit model to deal with the uncertainty in the data and model
parameters.
Recently, the classical CCA was shown to correspond to the ML solution of a
particular generative model where the two data sets are assumed to stem from a
shared Gaussian latent variable $\mathbf{z}$ and normally distributed data-
set-specific noise (Bach and Jordan, 2005). Using linear assumptions, the
model is formally defined as
$\displaystyle\left\\{\begin{array}[]{cl}\mathbf{x}&\sim\mathbf{W}_{x}\mathbf{z}+\boldsymbol{\varepsilon}_{x}\\\
\mathbf{y}&\sim\mathbf{W}_{y}\mathbf{z}+\boldsymbol{\varepsilon}_{y}.\end{array}\right.$
(6.3)
The manifestation of the shared signal in each data set can be different. This
is parameterized by $\mathbf{W}_{x}$ and $\mathbf{W}_{y}$. Assuming a standard
Gaussian model for the shared latent variable,
$\mathbf{z}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ and data set-specific
effects where
$\boldsymbol{\varepsilon}_{x}\sim\mathcal{N}(\mathbf{0},\Psi_{x})$ (and
respectively for $\mathbf{y}$), the correlation-maximizing projections of the
traditional CCA introduced in Section 6.1 can be retrieved from the ML
solution of the model (Archambeau et al., 2006; Bach and Jordan, 2005). The
model decomposes the observed co-occurring data sets into shared and data set-
specific components based on explicit modeling assumptions (Figure 6.1). The
dataset-specific effects can also be described in terms of latent variables as
$\boldsymbol{\varepsilon}_{x}=\mathbf{B}_{x}\mathbf{z}_{x}$ and
$\boldsymbol{\varepsilon}_{y}=\mathbf{B}_{y}\mathbf{z}_{y}$, allowing the
construction of more detailed models for the dataset-specific effects (Klami
and Kaski, 2008). The shared signal $\mathbf{z}$ is treated as a latent
variable and marginalized out in the model, providing the marginal likelihood
for the observations:
$p(\mathbf{X},\mathbf{Y}|\mathbf{W},\Psi)=\int
p(\mathbf{X},\mathbf{Y}|\mathbf{Z},\mathbf{W},\Psi)p(\mathbf{Z})d\mathbf{Z},$
(6.4)
where $\Psi$ denotes the block-diagonal matrix of $\Psi_{x}$, $\Psi_{y}$, and
$\mathbf{W}=[\mathbf{W}_{x};\mathbf{W}_{y}]$. The probabilistic formulation of
CCA has opened up a way to new probabilistic extensions that can treat the
modeling assumptions and uncertainties in the data in a more explicit and
robust manner (Archambeau et al., 2006; Klami and Kaski, 2008; Klami et al.,
2010).
The general formulation provides a flexible modeling framework, where
different modeling assumptions can be used to adapt the models in different
applications. The connection to classical CCA assumes full covariances for the
dataset-specific effects. Simpler models for the dataset-specific effects will
not distinguish between the shared and marginal effects as effectively, but
they have fewer model parameters that can potentially reduce overlearning and
speed up computation. It is also possible to tune the dimensionality of the
shared latent signal. Learning of lower-dimensional models can be faster and
potentially less prone to overfitting. Interpretation of simpler models is
also more straightforward in many applications. The probabilistic formulation
allows rigorous treatment of uncertainties in the data and model parameters
also with small sample sizes that are common in biomedical studies, and allows
the incorporation of prior information through Bayesian priors, as in the
regularized dependency detection framework introduced in Publication 4.
Figure 6.1: A graphical representation of the generative shared latent
variable model in Equation (6.3). The latent source $\mathbf{z}$ is shared by
observations $\mathbf{x}$ and $\mathbf{y}$. The other effects that are
specific to each observation are characterized by $\mathbf{z}_{x}$ and
$\mathbf{z}_{y}$, respectively. Gray shading indicates observed variables.
##### Contingency table analysis
Contingency table analysis is a classical approach used to study associations
between co-occurring categorical observations. The co-occurrences are
represented by cross-tabulating them on a contingency table, the rows and
columns of which correspond to the first and second set of features,
respectively. Various tests are available for measuring dependency between the
rows and columns of the table Yates (1934); Agresti (1992), including the
classical Fisher test (Fisher, 1934), a standard tool for measuring
statistical enrichment of functional categories in gene cluster analysis
(Hosack et al., 2003). While the classical contingency table analysis is used
to measure dependency between co-occurring variables, more recent approaches
use contingency tables to derive objective functions for dependency
exploration tasks. The associative clustering principle introduced in
Publications 5-6 is an example of such approach.
Other approaches that use contingency table dependencies as objective
functions include the information bottleneck (IB) principle (Tishby et al.,
1999) and discriminative clustering (DC) (Sinkkonen et al., 2002; Kaski et
al., 2005). These are asymmetric, dependency-seeking approaches that can be
used to discover cluster structure in a primary data such that it is maximally
informative of another, discrete auxiliary variable. The dependency is
represented on a contingency table, and maximization of contingency table
dependencies provides the objective function for clustering. While the
standard IB operates on discrete data, DC is used to discover cluster
structure in continuous-valued data. The two approaches also employ different
objective functions. In classical IB, a discrete variable $\mathcal{X}$ is
clustered in such a way that the cluster assignments become maximally
informative of another discrete variable $\mathcal{Y}$. The complexity of the
cluster assignments is controlled by minimizing the mutual information between
the cluster indices and the original variables. The task is to find a
partitioning $\tilde{\mathbf{X}}$ that minimizes the cost ${\cal
L}(p(\tilde{\mathbf{X}}|\mathbf{X}))=I(\tilde{\mathbf{X}};\mathbf{X})-\beta
I(\tilde{\mathbf{X}};\mathbf{Y}),$ where $\beta$ controls clustering
resolution. In DC, mutual information is replaced by a Bayes factor between
the two hypotheses of dependent and independent margins. The Bayes factor is
asymptotically consistent with mutual information, but provides an unbiased
estimate for limited sample size (see e.g. Sinkkonen et al., 2005). The
standard information bottleneck and discriminative clustering are asymmetric
methods that treat one of the data sources as the primary target of analysis.
In contrast, the dependency maximization approaches considered in this thesis,
the associative clustering (AC) and regularized versions of canonical
correlation analysis are symmetric and they operate exclusively on continuous-
valued data. CCA is not based on contingency table analysis, but it has close
connections to the Gaussian IB (Chechik et al., 2005) that seeks maximal
dependency between two sets of normally distributed variables. The Gaussian IB
retrieves the same subspace as CCA for one of the data sets. However, in
contrast to the symmetric CCA model, Gaussian IB is a directed method that
finds dependency-maximizing projections for only one of the two data sets. The
second dependency detection approach considered in this thesis, the
associative clustering, is particularly related to the symmetric IB that finds
two sets of clusters, one for each variable, which are optimally compressed
presentations of the original data, and at the same time maximally informative
of each other (Friedman et al., 2001). While the objective function in IB is
derived from mutual information, AC uses the Bayes factor as an objective
function in a similar manner as it is used in the asymmetric discriminative
clustering. Another key difference is that while the symmetric IB operates on
discrete data, AC employs contingency table analysis in order to discover
cluster structure in continuous-valued data spaces.
### 6.2 Regularized dependency detection
Standard unsupervised methods for dependency detection, such as the canonical
correlation analysis or the symmetric information bottleneck, seek maximal
dependency between two data sets with minimal assumptions about the
dependencies. The unconstrained models involve high degrees of freedom when
applied to high-dimensional genomic observations. Such flexibility can easily
lead to overfitting, which is even worse for more flexible nonparametric or
nonlinear, kernel-based dependency discovery methods. Several ways to
regularize the solution have been suggested to overcome associated problems,
for instance by imposing sparsity constraints on the solution space (Bie and
Moor, 2003; Vinod, 1976).
In many applications prior information of the dependencies is available, or
particular types of dependency are relevant for the analysis task. Such prior
information can be used to reduce the degrees of freedom in the model, and to
regularize dependency detection. In the cancer gene discovery application of
Publication 4, DNA mutations are systematically correlated with
transcriptional activity of the genes within the affected region, and
identification of such regions is a biomedically relevant research task. Prior
knowledge of chromosomal distances between the observations can improve the
detection of the relevant spatial dependencies. However, principled approaches
to incorporate such prior information in dependency modeling have been
missing. Publication 4 introduces regularized models for dependency detection
based on classical canonical correlation analysis (Hotelling, 1936) and its
probabilistic formulation (Bach and Jordan, 2005). The models are extended by
incorporating appropriate prior terms, which are then used to reduce the
degrees of freedom based on prior biological knowledge.
##### Correlation-based variant
In order to introduce the regularized dependency detection framework of
Publication 4, let us start by considering regularization of the classical
correlation-based CCA. This searches for arbitrary linear projection vectors
$\mathbf{v}_{x},\mathbf{v}_{y}$ that maximize the correlation between the
projections of the data sets $\mathbf{X},\mathbf{Y}$. Multiple projection
components can be obtained iteratively, by finding the most correlating
projection first, and then consecutively the next ones after removing the
dependencies explained by the previous CCA components. The procedure will
identify maximally dependent linear subspaces of the investigated data sets.
To regularize the solution, Publication 4 couples the projections through a
transformation matrix $\mathbf{T}$ in such a way that
$\mathbf{v}_{y}=\mathbf{T}\mathbf{v}_{x}$. With a completely unconstrained
$\mathbf{T}$ the model reduces to the classical unconstrained CCA; suitable
constraints on can be used to regularize dependency detection.
To enforce regularization one could for instance prefer solutions for
$\mathbf{T}$ that are close to a given transformation matrix,
$\mathbf{T}\sim\mathbf{M}$, or impose more general constraints on the
structure of the transformation matrix that would prefer particular rotational
or other linear relationships. Suitable constraints depend on the particular
applications; the solutions can be made to prefer particular types of
dependency in a soft manner by appropriate penalty terms. In Publication 4 the
completely unconstrained CCA model has been compared with a fully regularized
model with $\mathbf{T}=\mathbf{I}$; this encodes the biological assumption
that probes with small chromosomal distances tend to capture more similar
signal between gene expression and copy number measurements than probes with a
larger chromosomal distance; the projection vectors characterize this
relationship, and are therefore expected to have similar form,
$\mathbf{v}_{x}\sim\mathbf{v}_{y}$. Utilization of other, more general
constraints in related data integration tasks provides a promising topic for
future studies.
The correlation-based treatment provides an intuitive and easily implementable
formulation for regularized dependency detection. However, it lacks an
explicit model for the shared and data-specific effects, and it is likely that
some of the dataset-specific effects are captured by the correlation-
maximizing projections. This is suboptimal for characterizing the shared
effects, and motivates the probabilistic treatment.
##### Probabilistic dependency detection with similarity constraints
The probabilistic approach for regularized dependency detection in Publication
4 is based on an explicit model of the data-generating process formulated in
Equation (6.3). In this model, the transformation matrices $\mathbf{W}_{x}$,
$\mathbf{W}_{y}$ specify how the shared latent variable $\mathbf{Z}$ is
manifested in each data set $\mathbf{X}$, $\mathbf{Y}$, respectively. In the
standard model, the relationship between the transformation matrices is not
constrained, and the algorithm searches for arbitrary linear transformations
that maximize the likelihood of the observations in Equation (6.4). The
probabilistic formulation opens up possibilities to guide dependency search
through Bayesian priors.
In Publication 4, the standard probabilistic CCA model is extended by
incorporating additional prior terms that regularize the relationship by
reparameterizing the transformation matrices as
$\mathbf{W}_{y}=\mathbf{T}\mathbf{W}_{x}$, and setting a prior on
$\mathbf{T}$. The treatment is analogous to the correlation-based variant, but
now the transformation matrices operate on the latent components, rather than
the observations. This allows to distinguish the shared and dataset-specific
effects more explicitly in the model. The task is then to learn the optimal
parameter matrix $\mathbf{W}=[\mathbf{W}_{x};\mathbf{W}_{y}]$, given the
constraint $\mathbf{W}_{y}=\mathbf{T}\mathbf{W}_{x}$. The Bayes’ rule gives
the model likelihood
$p(\mathbf{X},\mathbf{Y},\mathbf{W},\mathbf{\Psi})\sim
p(\mathbf{X},\mathbf{Y}|\mathbf{W},\mathbf{\Psi})p(\mathbf{W},\mathbf{\Psi}).$
(6.5)
The likelihood term $p(\mathbf{X},\mathbf{Y}|\mathbf{W},\mathbf{\Psi})$ can be
calculated based on the model in Equation (6.3). This defines the objective
function for standard probabilistic CCA, which implicitly assumes a flat prior
$p(\mathbf{W},\mathbf{\Psi})\sim 1$ for the model parameters. The formulation
in Equation (6.5) makes the choice of the prior explicit, allowing
modifications on the prior term. To obtain a tractable prior, let us assume
that the prior factorizes as
$p(\mathbf{W},\mathbf{\Psi})=p(\mathbf{W})p(\mathbf{\Psi})$. The first term
can be further decomposed as $p(\mathbf{W})\sim
p(\mathbf{W}_{x})p(\mathbf{T})$, assuming independent priors for
$\mathbf{W}_{x}$ and $\mathbf{T}$. A convenient and tractable prior for
$\mathbf{T}$ is provided by the matrix normal
distribution:111$\mathcal{N}_{m}(\mathbf{T}|\mathbf{M},\mathbf{U},\mathbf{V})\sim
exp\left(-\frac{1}{2}Tr\\{\mathbf{U}^{-1}(\mathbf{T}-\mathbf{M})\mathbf{V}^{-1}(\mathbf{T}-\mathbf{M})^{T}\\}\right)$
where $\mathbf{M}$ is the mean matrix, and $\mathbf{U}$ and $\mathbf{V}$
denote row and column covariances, respectively.
$p(\mathbf{T})=\mathcal{N}_{m}(\mathbf{T}|\mathbf{M},\mathbf{U},\mathbf{V}).$
(6.6)
For computational simplicity, let us assume independent rows and columns with
$\mathbf{U}=\mathbf{V}=\sigma_{T}\mathbf{I}$. The mean matrix $\mathbf{M}$ can
be used to emphasize certain types of dependency between $\mathbf{W}_{x}$ and
$\mathbf{W}_{y}$. Assuming uninformative, flat priors $p(\mathbf{W}_{x})\sim
1$ and $p(\mathbf{\Psi})\sim 1$, as in the standard probabilistic CCA model,
and denoting $\boldsymbol{\Sigma}=\mathbf{W}\mathbf{W}^{T}+\mathbf{\Psi}$, the
negative log-likelihood of the model is
$-logp(\mathbf{X},\mathbf{Y},\mathbf{W},\mathbf{\Psi})\sim
log|\boldsymbol{\Sigma}|+Tr\boldsymbol{\Sigma}^{-1}\tilde{\boldsymbol{\Sigma}}+\frac{\parallel\mathbf{T}-\mathbf{M}\parallel_{F}^{2}}{2\sigma_{T}^{2}}.$
(6.7)
This is the objective function to minimize. Note that this has the same form
as the objective function of the standard probabilistic CCA, except the
additional penalty term
$\frac{\parallel\mathbf{T}-\mathbf{M}\parallel_{F}^{2}}{2\sigma_{T}^{2}}$
arising from the prior $p(\mathbf{T})$. This yields the cost function employed
in Publication 4. In our cancer gene discovery application the choice
$\mathbf{M}=\mathbf{I}$ is used to encode the biological prior constrain
$\mathbf{T}\approx\mathbf{I}$, which states that the observations with a small
chromosomal distance should on average show similar responses in the
integrated data sets, i.e., $\mathbf{W}_{x}\approx\mathbf{W}_{y}$. The
regularization strength can be tuned with $\sigma_{T}^{2}$. A fully
regularized model is obtained with $\sigma_{T}^{2}\rightarrow 0$. When
$\sigma_{T}^{2}\rightarrow\infty$, $\mathbf{W}_{x}$ and $\mathbf{W}_{y}$
become independent a priori, yielding the ordinary probabilistic CCA. The
$\sigma_{T}^{2}$ can be used to regularize the solution between these two
extremes. Note that it is possible to incorporate also other types of prior
information concerning the dependencies into the model through
$p(\mathbf{T})$.
The model parameters $\mathbf{W}$, $\mathbf{\Psi}$ are estimated with the EM
algorithm. The regularized version is not analytically tractable with respect
to $\mathbf{W}$ in the general case, but can be optimized with standard
gradient-based optimization techniques. Special cases of the model have
analytical solutions, which can speed up the model fitting procedure. In
particular, the fully regularized and unconstrained models, obtained with
$\sigma_{T}^{2}=0$ and $\sigma_{T}^{2}=\infty$ respectively, have closed-form
solutions for $\mathbf{W}$. Note that the current formulation assumes that the
regularization parameters $\mathbf{M},\sigma_{T}^{2}$ are defined prior to the
analysis. Alternatively, these parameters could be optimized based on external
criteria, such as cancer gene detection performance in our application, or
learned from the data in a fully Bayesian treatment these parameters would be
treated as latent variables. Incorporation of additional prior information of
the data set-specific effects through priors on $\mathbf{W}_{x}$ and
$\mathbf{\Psi}$ provides promising lines for further work.
#### 6.2.1 Cancer gene discovery with dependency detection
The regularized models provide a principled framework for studying
associations between transcriptional activity and other regulatory layers of
the genome. In Publication 4, the models are used to investigate cancer
mechanisms. DNA copy number changes are a key mechanism for cancer, and
integration of copy number information with mRNA expression measurements can
reveal functional effects of the mutations. While causation may be difficult
to grasp, study of the dependencies can help to identify functionally active
mutations, and to provide candidate biomarkers with potential diagnostic,
prognostic and clinical impact in cancer studies.
The modeling task in the cancer gene discovery application of Publication 4 is
to identify chromosomal regions that show exceptionally high levels of
dependency between gene copy number and transcriptional levels. The model is
used to detect dependency within local chromosomal regions that are then
compared in order to identify the exceptional regions. The dependency is
quantified within a given region by comparing the strength of shared and data
set-specific signal. High scores indicate regions where the shared signal is
particularly high relative to the data-set-specific effects. A sliding-window
approach is used to screen the genome for dependencies. The regions are
defined by the $d$ closest probes around each gene. Then the dimensionality of
the models stays constant, which allows direct comparison of the dependency
measures between the regions without additional adjustment terms that would be
otherwise needed to compensate for differences in model complexity.
Prior information of the dependencies is used to regularize cancer gene
detection. Chromosomal gains and losses are likely to be positively correlated
with the expression levels of the affected genes within the same chromosomal
region or its close proximity; copy number gain is likely to increase the
expression of the associated genes whereas deletion will block gene
expression. The prior information is encoded in the model by setting
$\mathbf{M}=\mathbf{I}$ in the prior term $p(\mathbf{T})$. This accounts for
the expected positive correlations between gene expression and copy number
within the investigated chromosomal region. Regularization based on such prior
information is shown to improve cancer gene detection performance in
Publication 4, where the regularized variants outperformed the unconstrained
models.
A genome-wide screen of 51 gastric cancer patients (Myllykangas et al., 2008)
reveals clear associations between DNA copy number changes and transcriptional
activity. The Figure 6.2 illustrates dependency detection on chromosome arm
17q, where the regularized model reveals high dependency between the two data
sources in a known cancer-associated region. The regularized and unconstrained
models were compared in terms of receiver-operator characteristics calculated
by comparing the ordered gene list from the dependency screen to an expert-
curated list of known genes associated with gastric cancer (Myllykangas et
al., 2008). A large proportion of the most significant findings in the whole-
genome analysis were known cancer genes; the remaining findings with no known
associations to gastric cancer are promising candidates for further study.
Biomedical interpretation of the model parameters is also straightforward. A
ML estimate of the latent variable values $\mathbf{Z}$ characterizes the
strength of the shared signal between DNA mutations and transcriptional
activity for each patient. This allows robust identification of small,
potentially unknown patient subgroups with shared amplification effects. These
would remain potentially undetected when comparing patient groups defined
based on existing clinical annotations. The parameters in $\mathbf{W}$ can
downweigh signal from poorly performing probes in each data set, or probes
that measure genes whose transcriptional levels are not functionally affected
by the copy number change. This provides tools to distinguish between so-
called driver mutations having functional effects from less active passenger
mutations, which is an important task in cancer studies. On the other hand,
the model can combine statistical power across the adjacent measurement
probes, and it captures the strongest shared signal in the two sets of
observations. This is useful since gene expression and copy number data are
typically characterized by high levels of biological and measurement variation
and small sample size.
Figure 6.2: Gene expression, copy number signal, and the dependency score
along the chromosome arm 17q obtained with the regularized latent variable
framework in Equation 6.7. Known cancer-associated genes from an expert-
curated list are marked with black dots.
##### Related approaches
Integration of chromosomal aberrations and transcriptional activity is an
actively studied data integration task in functional genomics. The first
studies with standard statistical tests were carried out by Hyman et al.
(2002) and Phillips et al. (2001) when simultaneous genome-wide observations
of the two data sources had become available. The modeling approaches utilized
in this context can be roughly classified in regression-based, correlation-
based and latent variable approaches. The regression-based models (Adler et
al., 2006; Bicciato et al., 2009; van Wieringen and van de Wiel, 2009)
characterize alterations in gene expression levels based on copy number
observations with multivariate regression or closely related models. The
correlation-based approaches (González et al., 2009; Schäfer et al., 2009;
Soneson et al., 2010) provide symmetric models for dependency detection, based
on correlation and related statistical models. Many of these methods also
regularize the solutions, typically based on sparsity constraints and non-
negativity of the projections (Lê Cao et al., 2009; Waaijenborg et al., 2008;
Witten et al., 2009; Parkhomenko et al., 2009). The correlation-based approach
in Publication 4 introduces a complementary approach for regularization that
constrains the relationship between subspaces where the correlations are
estimated. The latent variable models by Berger et al. (2006); Shen et al.
(2009); Vaske et al. (2010), and Publication 4 are based on explicit modeling
assumptions concerning the data-generating processes. The iCluster algorithm
(Shen et al., 2009) is closely related to the latent variable model considered
in Publication 4. While our model detects continuous dependencies, iCluster
uses a discrete latent variable to partition the samples into distinct
subgroups. The iCluster model is regularized by sparsity constraints on
$\mathbf{W}$, while we tune the relationship between $\mathbf{W}_{x}$ and
$\mathbf{W}_{y}$. Moreover, the model in Publication 4 utilizes full
covariance matrices to model for the dataset-specific effects, whereas
iCluster uses diagonal covariances. The more detailed model for dataset-
specific effects in our model should help to distinguish the shared signal
more accurately. Other latent variable approaches include the iterative method
based on generalized singular-value decomposition (Berger et al., 2006), and
the probabilistic factor graph model PARADIGM (Vaske et al., 2010), which
additionally utilizes pathway topology information in the modeling.
Experimental comparison between the related integrative approaches can be
problematic since they target related, but different research questions where
the biological ground truth is often unknown. For instance, some methods
utilize patient class information in order to detect class-specific
alterations (Schäfer et al., 2009), other methods perform de novo class
discovery (Shen et al., 2009), provide tools for gene prioritization (Salari
et al., 2010), or guide the analysis with additional functional information of
the genes (Vaske et al., 2010). The algorithms introduced in Publication 4 are
particularly useful for gene prioritization and class discovery purposes,
where the target is to identify the most promising cancer gene candidates for
further validation, or to detect potentially novel cancer subtypes. However,
while an increasing number of methods are released as conveniently accessible
algorithmic tools (Salari et al., 2010; Shen et al., 2009; Schäfer et al.,
2009; Witten et al., 2009), implementations of most models are not available
for comparison purposes. Open source implementations of the dependency
detection algorithms developed in this thesis have been released to enhance
transparency and reproducibility of the computational experiments and to
encourage further use of these models (Huovilainen and Lahti, 2010).
### 6.3 Associative clustering
Functions of human genes are often studied indirectly, by studying model
organisms such as the mouse (Davis, 2004; Joyce and Palsson, 2006). Orthologs
are genes in different species that originate from a single gene in the last
common ancestor of these species. Such genes have often retained identical
biological roles in the present-day organisms, and are likely to share the
function (Fitch, 1970). Mutations in the genomic DNA sequence are a key
mechanism in evolution. Consequently, DNA sequence similarity can provide
hypotheses of gene function in poorly annotated species. An exceptional level
of conservation may highlight critical physiological similarities between
species, whereas divergence can indicate significant evolutionary changes
(Jordan et al., 2005). Investigating evolutionary conservation and divergence
will potentially lead to a deeper understanding of what makes each species
unique. Evolutionary changes primarily target the structure and sequence of
genomic DNA. However, not all changes will lead to phenotypic differences. On
the other hand, sequence similarity is not a guarantee of functional
similarity because small changes in DNA can potentially have remarkable
functional implications.
Therefore, in addition to investigating structural conservation of the genes
at the sequence level, another level of investigation is needed to study
functional conservation of the genes and their regulation, which is reflected
at the transcriptome (Jiménez et al., 2002; Jordan et al., 2005).
Transcriptional regulation of the genes is a key regulatory mechanism that can
have remarkable phenotypic consequences in highly modular cell-biological
systems (Hartwell et al., 1999) even when the original function of the
regulated genes would remain intact.
Systematic comparison of transcriptional activity between different species
would provide a straightforward strategy for investigating conservation of
gene regulation (Bergmann et al., 2004; Enard et al., 2002; Zhou and Gibson,
2004). However, direct comparison of individual genes between species may not
be optimal for discovering subtle and complex dependency structures. The
associative clustering principle (AC), introduced in Publications 5-6,
provides a framework for detecting groups of orthologous genes with
exceptional levels of conservation and divergence in transcriptional activity
between two species. While standard dependency detection methods for
continuous data, such as the generalized singular value decomposition (see
e.g. Alter et al., 2003) or canonical correlation analysis (Hotelling, 1936)
detect global linear dependencies between observations, AC searches for
dependent, local groupings to reveal gene groups with exceptional levels of
conservation and divergence in transcriptional activity. The model is free of
particular distributional assumptions about the data, which helps to allocate
modeling resources to detecting dependent subgroups when variation within each
group is less relevant for the analysis. The remainder of this section
provides an overview of the associative clustering principle and its
application to studying evolutionary divergence between species.
Figure 6.3: Principle of associative clustering (AC). AC performs simultaneous
clustering of two data sets, consisting of paired observations, and seeks to
maximize the dependency between the two sets of clusters. The clusters are
defined by cluster centroids in each data space. The clustering results are
represented on a contingency table, where clusters of the two data sets
correspond with the rows and columns of the contingency table, respectively.
These are called the margin clusters of the contingency table. The table cells
are called cross clusters and they contain orthologous genes from the two data
sets. The cluster centroids are optimized to produce a contingency table with
maximal dependency between the margin cluster counts. Cross clusters that show
significant deviation from the null hypothesis of independent margins indicate
dependency. In order to enhance the reliability of the results, the clustering
is repeated with slightly differing bootstrap samples. Then reliable co-
occurrences are identified from a co-occurrence tree with a specified
threshold. Frequently co-occurring orthologues are selected for further
analyzes.
##### The associative clustering principle
The principle of associative clustering (AC) is illustrated in Figure 6.3. AC
performs simultaneous clustering of two data sets to reveal maximally
dependent cluster structure between two sets of observations. The clusters are
defined in each data space by Voronoi parameterization, where the clusters are
defined by cluster centroids to produce connected, internally homogeneous
clusters. Let us denote the two sets of clusters by $\\{V^{(x)}_{i}\\}_{i}$,
$\\{V^{(y)}_{j}\\}_{j}$. A given data point $\mathbf{x}$ is then assigned to
the cluster corresponding to the nearest centroid $\mathbf{m}_{i}$ in the
feature space, with respect to a given distance measure222$\mathbf{x}\in
V_{i}^{(x)}$ if $d(\mathbf{x},\mathbf{m}_{i})\leq
d(\mathbf{x},\mathbf{m}_{k})$ for all $k$. $d$. This divides the space into
non-overlapping Voronoi regions. The regions define a clustering for all
points of the data space. The association between the clusters of the two data
sets can be represented on a contingency table, where the rows and columns
correspond to clusters in the first and second data set, respectively. The
clusters in each data set are called margin clusters. Each pair of co-
occurring observations $(\mathbf{x}_{i},\mathbf{y}_{i})$ maps to one margin
cluster in each data set, and each contingency table cell corresponds to a
pair of margin clusters. These are called cross clusters.
AC searches for a maximally dependent cluster structure by optimizing the
Voronoi centroids in the two data spaces in such a way that the dependency
between the contingency table margins is maximized. Let us denote the number
of samples in cross cluster $i,j$ by $n_{ij}$. The corresponding margin
cluster counts are $n_{i\cdot}=\sum_{j}n_{ij}$ and $n_{\cdot
j}=\sum_{i}n_{ij}$. The observed sample frequencies over the contingency table
margins and cross-clusters are assumed to follow multinomial distribution with
latent parameters $\boldsymbol{\theta}_{i},\boldsymbol{\theta}_{j}$ and
$\boldsymbol{\theta}_{ij}$, respectively. Assuming the model $M_{I}$ of
independent margin clusters, the expected sample frequency in each cross
cluster is given by the outer product of margin cluster frequencies. The model
$M_{d}$ of _dependent margin clusters_ deviates from this assumption. The
Bayes factor (BF) is used to compare the two hypotheses of dependent and
independent margins. This is a rigorously justified approach for model
comparison, which indicates whether the observations provide superior evidence
for either model. Evidence is calculated over all potential values of the
model parameters, marginalized over the latent frequencies. In a standard
setting, the Bayes factor would be used to compare evidence between the
dependent and independent margin cluster models for a given clustering
solution. AC uses the Bayes factor in a non-standard manner; as an objective
function to maximize by optimizing the cluster centroids in each data space;
the centroids define the margin clusters and consequently the margin cluster
dependencies.
The centroids are optimized with a conjugate-gradient algorithm after
smoothing the cluster borders with continuous parameterization. The
hyperparameters $n^{(d)}$, $n^{(x)}$, and $n^{(y)}$ arise from Dirichlet
priors of the two multinomial models $M_{I}$, $M_{D}$ of independent and
dependent margins, respectively. Setting the hyperparameters to unity yields
the classical hypergeometric measure of contingency table dependency (Fisher,
1934; Yates, 1934). With large sample size, the logarithmic Bayes factor
approaches mutual information (Sinkkonen et al., 2005). The Bayes factor is a
desirable choice especially with a limited sample size since a marginalization
over the latent variables makes it robust against uncertainty in the parameter
values, and because finite contingency table counts would give a biased
estimate of mutual information. The number of clusters in each data space is
specified in advance, typically based on the desired level of resolution.
Nonparametric extensions, where the number of margin clusters would be
inferred automatically from the data form one potential topic for further
studies; a closely related approach was recently proposed in Rogers et al.
(2010).
Publication 6 introduces an additional, bootstrap-based procedure to assess
the reliability of the findings (Figure 6.3). The analysis is repeated with
similar, but not identical training data sets obtained by sampling the
original data with replacement. The most frequently detected dependencies are
then investigated more closely. The analysis will emphasize findings that are
not sensitive to small variations in the observed data.
##### Comparison methods
Associative clustering was compared with two alternative methods: standard
K-means on each of the two data sets, and a combination of K-means and
information bottleneck (K-IB). K-means (see e.g. Bishop, 2006) is a classical
clustering algorithm that provides homogeneous, connected clusters based on
Voronoi parameterization. Homogeneity is desirable for interpretation, since
the data points within a given cluster can then be conveniently summarized by
the cluster centroid. On the other hand, K-means considers each data set
independently, which is suboptimal for the dependency modeling task. The two
sets of clusters obtained by K-means, one for each data space, can then be
presented on a contingency table as in associative clustering. The second
comparison method is K-IB introduced in Publication 5. K-IB uses K-means to
partition the two co-occurring, continuous-valued data sets into discrete
atomic regions where each data point is assigned in its own singleton cluster.
This gives two sets of atomic clusters that are mapped on a large contingency
table, filled with frequencies of co-occurring data pairs
$(\mathbf{x}_{k},\mathbf{y}_{k})$. The table is then compressed to the desired
size by aggregating the margin clusters with the symmetric IB algorithm in
order to maximize the dependency between the contingency table margins
(Friedman et al., 2001). Aggregating the atomic clusters provides a flexible
clustering approach, but the resulting clusters are not necessarily
homogeneous and they are therefore difficult to interpret.
AC compared favorably to the other methods. While AC outperformed the standard
K-means in dependency modeling, the cluster homogeneity was not significantly
reduced in AC. The cross clusters from K-IB (Sinkkonen et al., 2003) were more
dependent than in AC. On the other hand, AC produced more easily interpretable
localized clusters, as measured by the sum of intra-cluster variances in
Publication 6. Homogeneity makes it possible to summarize clusters
conveniently, for instance by using the mean expression profiles of the
cluster samples, as in Figure 6.4B. While K-means searches for maximally
homogeneous clusters and K-IB searches for maximally dependent clusters, AC
finds a successful compromise between the goals of dependency and homogeneity.
A B
Figure 6.4: A The contingency table of associative clustering highlights
orthologous gene groups in human (rows) and mouse (columns) with exceptional
levels of conservation (yellow) or divergence (blue) in transcriptional
activity between the two species. B Average expression profiles of a highly
conserved group of testis-specific genes across 21 tissues in man and mouse.
©IEEE. Reprinted with permission from Publication 6.
#### 6.3.1 Exploratory analysis of transcriptional divergence between species
Associative clustering is used in Publications 5 and 6 to investigate
conservation and divergence of transcriptional activity of 2818 orthologous
human-mouse gene pairs across an organism-wide collection of transcriptional
profiling data covering 46 and 45 tissue types in human and mouse,
respectively (Su et al., 2002). AC takes as input two gene expression matrices
with orthologous genes, one for each species, and returns a dependency-
maximizing clustering for the orthologous gene pairs. Interpretation of the
results focuses on unexpectedly large or small cross clusters revealed by the
contingency table analysis of associative clustering. Compared to plain
correlation-based comparisons between the gene expression profiles, AC can
reveal additional cluster structure, where genes with similar expression
profiles are clustered together, and associations between the two species are
investigated at the level of such detected gene groups. The dependency between
each pair of margin clusters can be characterized by comparing the respective
margin cluster centroids that provide a compact summary of the samples within
each cluster.
Biological interpretation of the findings, based on enrichment of Gene
Ontology (GO) categories (Ashburner et al., 2000), revealed genes with
strongly conserved and potentially diverged transcriptional activity. The most
highly enriched categories were associated with ribosomal functions, the high
conservation of which has also been suggested in earlier studies (Jiménez et
al., 2002); ribosomal genes often require coordinated effort of a large group
of genes, and they function in cell maintenance tasks that are critical for
species survival. An exceptional level of conservation was also observed in a
group of testis-specific genes, yielding novel functional hypotheses for
certain poorly annotated genes within the same cross-cluster (Figure 6.4).
Transcriptional divergence, on the other hand, was detected for instance in
genes related to embryonic development.
While general-purpose dependency exploration tools may not be optimal for
studying the specific issue of transcriptional conservation, such tools can
reveal dependency with minimal prior knowledge about the data. This is useful
in functional genomics experiments where little prior knowledge is available.
In Publications 5 and 6, associative clustering has been additionally applied
in investigating dependencies between transcriptional activity and
transcription factor binding, another key regulatory mechanism of the genes.
### 6.4 Conclusion
The models introduced in Publications 4-6 provide general exploratory tools
for the discovery and analysis of statistical dependencies between co-
occurring data sources and tools to guide modeling through Bayesian priors. In
particular, the models consider linear dependencies (Publication 4) and
cluster-based dependency structures (Publications 5-6) between the data
sources. The models are readily applicable to data integration tasks in
functional genomics. In particular, the models have been applied to
investigate dependencies between chromosomal mutations and transcriptional
activity in cancer, and evolutionary divergence of transcriptional activity
between human and mouse. Biomedical studies provide a number of other
potential applications for such general-purpose methods. An increasing number
of co-occurring observations across the various regulatory layers of the
genome are available concerning epigenetic mechanisms, micro-RNAs,
polymorphisms and other genomic features (The Cancer Genome Atlas Research
Network, 2008). Simultaneous observations provide a valuable resource for
investigating the functional properties that emerge from the interactions
between the different layers of genomic information. An open source
implementation in
BioConductor333http://www.bioconductor.org/packages/release/bioc/html/pint.html
provides accessible computational tools for related data integration tasks,
helping to guarantee the utility of the developed models for the computational
biology community.
## Chapter 7 Summary and conclusions
> _Mathematics is biology’s next microscope, only better; biology is
> mathematics’ next physics, only better._
>
> J.E. Cohen (2004)
Following the initial sequencing of the human genome (International human
genome sequencing consortium, 2001; Venter et al., 2001), the understanding of
structural and functional organization of genetic information has extended
rapidly with the accumulation of research data. This has opened up new
challenges and opportunities for making fundamental discoveries about living
organisms and creating a holistic picture about genome organization. The
increasing need to organize the large volumes of genomic data with minimal
human intervention has made computation an increasingly central element in
modern scientific inquiry. It is a paradox of our time that the historical
scale of data in public and proprietary repositories is only revealing how
incomplete our knowledge of the enormous complexity of living systems is. The
particular challenges in data-intensive genomics are associated with the
complex and poorly characterized nature of living systems, as well as with
limited availability of observations. It is possible to solve some of these
challenges by combining statistical power across multiple experiments, and
utilizing the wealth of background information in public repositories.
Exploratory data analysis can help to provide research hypotheses and material
for more detailed investigations based on large-scale genomic observations
when little prior knowledge is available concerning the underlying phenomena;
models that are robust to uncertainty and able to automatically adapt to the
data, can facilitate the discovery of novel biological hypotheses. Statistical
learning and probabilistic models provide a natural theoretical framework for
such analysis.
In this thesis, general-purpose exploratory data analysis methods have been
developed for organism-wide analysis of the human transcriptome, a central
functional layer of the genome. Integrating evidence across multiple sources
of genomic information can help to reveal mechanisms that could not be
investigated based on smaller and more targeted experiments; this is a central
aspect in all contributions. In particular, methods have been developed (i) in
order to improve measurement accuracy of high-throughput observations, (ii) in
order to model transcriptional activation patterns and tissue relatedness in
genome-wide interaction networks at an organism-wide scale, and (iii) in order
to integrate measurements of the human transcriptome with other layers of
genomic information. These results contribute to some of the ’grand
challenges’ in the genomic era by developing strategies to understand cell-
biological systems, genetic contributions to human health and evolutionary
variation (Collins et al., 2003). The computational experiments in this thesis
have been carried out based on publicly available, anonymized data sets that
follow commonly accepted ethical standards in biomedical research. Open access
implementations of the key algorithms have been provided to guarantee wide
access to these tools and to spark new research beyond the original
applications presented in this thesis.
Methodological extensions and application of the developed algorithms to new
data integration tasks in functional genomics and in other fields provide a
promising line for future studies. The methods developed in this thesis are
readily applicable in genome-wide screening studies in cancer and potentially
other diseases. Increasing amounts of co-occurring data concerning various
aspects of the genome have become available, including gene- and micro-RNA
expression, structural variation in the DNA, epigenetic modifications and gene
regulatory networks. It is expected that with small modifications the
introduced methodology can be applied to study further associations between
these and other layers of genome organization, as well as their contributions
to human health. The fundamental research challenges in contemporary genome
biology provide a wide array of applications for statistical learning and
exploratory analysis, and a rich source of ideas for methodological research.
##
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|
arxiv-papers
| 2011-02-27T14:11:30 |
2024-09-04T02:49:17.334076
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leo Lahti",
"submitter": "Leo Lahti",
"url": "https://arxiv.org/abs/1102.5509"
}
|
1102.5621
|
# Anisotropic magnetotransport of superconducting and normal state in an
electron-doped Nd1.85Ce0.15CuO4-δ single crystal
Yue Wang1 yue.wang@pku.edu.cn Hong Gao2 1State Key Laboratory for Mesoscopic
Physics and School of Physics, Peking University, Beijing 100871, People’s
Republic of China 2National Laboratory for Superconductivity, Institute of
Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese
Academy of Sciences, Beijing 100190, People’s Republic of China
###### Abstract
The anisotropic properties of an optimally doped Nd1.85Ce0.15CuO4-δ single
crystal have been studied both below and above the critical temperature
$T_{c}$ via the resistivity measurement in magnetic field $H$ up to 12 T. By
scaling the conductivity fluctuation around the superconducting transition,
the upper critical field $H_{c2}(T)$ has been determined for field parallel to
the $c$-axis or to the basal $ab$-plane. The anisotropy factor
$\gamma=H_{c2}^{\parallel ab}/H_{c2}^{\parallel c}$ is estimated to be about
8. In the normal state ($50\leq T\leq 180$ K), the magnetoresistance (MR)
basically follows an $H^{2}$ dependence and for $H\parallel c$ it is almost 10
times larger than that for $H\parallel ab$. Comparing with hole-doped cuprates
it suggests that the optimally doped Nd1.85Ce0.15CuO4-δ cuprate superconductor
has a moderate anisotropy.
###### pacs:
74.25.F-, 74.25.Op, 74.72.Ek
††preprint: Published version of this paper can be found at Physica C 470, 689
(2010).
## I INTRODUCTION
One of essential features of most high-$T_{c}$ cuprates is their quasi-two-
dimensional crystal structure with CuO2 layer as a key structural unit.
Orenstein00 Under this circumstance, thermodynamic and transport properties
of a given cuprate superconductor usually exhibit the anisotropy when
measuring along the crystallographic $c$-axis or the CuO2 plane ($ab$-plane).
Anisotropy factors in these fundamental physical properties are therefore
crucial to know, not only for characterizing or evaluating the sample but also
as important parameters to be used in theoretical models to describe
high-$T_{c}$ superconductivity and search for its mechanism.
In many cases, however, determination of anisotropy factors is not an easy
task. A well known example is to determine the anisotropy in the upper
critical field $H_{c2}$, i.e., $\gamma=H_{c2}^{\parallel ab}/H_{c2}^{\parallel
c}$, where $H_{c2}^{\parallel ab}$ and $H_{c2}^{\parallel c}$ are the $H_{c2}$
along $ab$-plane and $c$-axis respectively. For most high-$T_{c}$ cuprates the
$H_{c2}$ is exceptionally large and its evaluation is limited by laboratory
accessible magnetic fields $H$ and complicated by some issues such as
superconducting fluctuations, especially for the less explored $H\parallel
ab$-plane ($H\parallel ab$) case. Despite this fact, continuous efforts have
been devoted to extracting this basic parameter, through resistive transport,
magnetization, and other kinds of experiments. Iye88 ; Welp89 ; Farrell89 ;
Panagopoulos03 ; Nagasao08
Comparing with hole-doped counterparts, anisotropic properties of electron-
doped cuprates Ln2-xCexCuO4-δ (Ln = Nd, Pr, …) have been less studied and
controversial results exist. For instance, an early work in Nd2-xCexCuO4-δ
(NCCO) reported a large $\gamma$ of 21, Hidaka89 while subsequent
magnetization measurement in aligned Sm1.85Ce0.15CuO4-δ (SCCO) powders gave a
low $\gamma$ of 3.7 and suggested that previous report might be an
overestimate due to the inaccuracy in determining the $H_{c2}^{\parallel ab}$.
Almasan92 Moreover, it should be noted that in some reports a large $\gamma$
($\sim 30-200$) has been cited for describing and determining the vortex phase
diagram of NCCO. Giller97 ; Nugroho99 In view of these, it is desirable to
redetermine the $\gamma$ in NCCO with the help of high quality crystals and
proper data analysis. In this paper we report the anisotropic magnetotransport
of an optimally doped NCCO single crystal. We succeeded in scaling the
conductivity fluctuation near the superconducting transition in different
magnetic fields for both $H\parallel ab$ and $H\parallel c$-axis ($H\parallel
c$) and this enabled us to obtain $\gamma\simeq 7.5$. Moreover, we also
determined the anisotropy in the normal state transverse magnetoresistance
(MR). Most previous MR measurements have been confined in the $H\parallel c$
configuration. Seng95 ; Gollnik98 Inclusion of the data with $H\parallel ab$
helped us to confirm that the transverse MR in normal state of optimal-doped
NCCO with $H\parallel c$ mainly originates from an orbital effect, namely, the
bending of charge carrier trajectories due to the presence of magnetic field.
## II EXPERIMENT
The optimally doped Nd1.85Ce0.15CuO4-δ single crystal was prepared by
traveling solvent floating-zone technique. Resistivity measurements were
carried out by the standard four-probe method with dc current supplied in the
$ab$-plane. Inset of Fig. 1 shows the temperature dependence of the
resistivity, $\rho(T)$, in zero field. It is found that the crystal shows a
sharp superconducting transition with an onset point at 26.2 K and a
transition width around 0.7 K.
Measurements were performed in an Oxford cryogenic system (Maglab-12) with $H$
up to 12 Tesla. To determine the anisotropy, we have carefully aligned the
crystal to be in $H\parallel c$ or $H\parallel ab$ configuration, by rotating
the sample at fixed $H$ and $T$ in the superconducting state with an angle
resolution of $1^{\circ}$ and tracing the peaks or minima in the angular
dependence of the resistivity. The superconducting transition in magnetic
fields was measured by sweeping temperature at constant $H$, while normal
state MR was done by sweeping magnetic fields at fixed $T$ which was
stabilized within $\sim 10$ mK by a Lakeshore cernox sensor. For both field
directions, the normal state MR was measured with $H$ perpendicular to the
current ($H\perp I$), that is, the transverse MR was obtained.
## III RESULTS and DISCUSSION
### III.1 Superconducting State
Figure 1 shows the superconducting transition curves in different $H$ up to 12
Tesla for $H\parallel ab$ (Fig. 1a) and $H\parallel c$ (Fig. 1b). Upon
applying the field, it is seen that the resistive transition becomes
broadening for $H\parallel ab$, while for $H\parallel c$ a parallel shift of
the transition is more evident. This is not unexpected by assuming a much
larger $H_{c2}$ for $H\parallel ab$. The rounding of the superconducting
transition in magnetic fields, however, implies that it would be difficult to
accurately define the mean-field transition point $T_{c}(H)$ and thus to
extract $H_{c2}(T)$ directly from the experimental curves, as shown in studies
on hole-doped high-$T_{c}$ cuprates. Welp89 In order to reliably determine
$H_{c2}(T)$ of the sample, in the following we performed scaling analysis to
the experimental data based on the Ginzburg$-$Landau (GL) fluctuation theory.
For $H\parallel c$, it is seen that the resistivity shows an upturn at low $T$
in high fields and the $H_{c2}(0)$ could be estimated below 12 T. The low-$T$
upturn in $\rho(T)$ has been widely observed in both hole- and electron-doped
cuprates near optimal-doping, Boebinger96 ; Sekitani01 whose origin however
has remained unclear, with localization or spin effect having been proposed.
Figure 1: (a) Resistivity versus temperature for $H\parallel ab$. The inset
shows the resistivity curve at $\mu_{0}H=0$ T in a wide temperature region.
(b) Resistivity versus temperature for $H\parallel c$.
Scaling analysis of thermodynamic and transport properties around $T_{c}$ has
proved to be an effective way to evaluate $H_{c2}(T)$. Welp91 ; Han92 ; Wen00
; Kacmarcik04 ; Gao06 For superconductors with layered-structure, Ullah and
Dorsey showed that the fluctuation conductivity $\sigma_{fl}$ has a scaling
form
$\sigma_{fl}[\frac{H}{T}]^{1/2}=F_{2D}[\frac{T-T_{c}(H)}{(TH)^{1/2}}]$ (1)
or
$\sigma_{fl}[\frac{H}{T^{2}}]^{1/3}=F_{3D}[\frac{T-T_{c}(H)}{(TH)^{2/3}}]$ (2)
for two-dimensional (2D) and 3D systems, respectively, with $F_{2D}$ and
$F_{3D}$ the unknown scaling functions. Ullah90 ; Ullah91 By using the
appropriate expression to scale the experimental data, we can readily
determine the parameter $T_{c}(H)$ with $T_{c}(0)$ as an additional constraint
and therefore obtain the equivalent $H_{c2}(T)$.
Figure 2: (a) 3D scaling of the fluctuation conductivity, i.e.,
$\sigma_{fl}(H/T^{2})^{1/3}$ versus $[T-T_{c}(H)]/(TH)^{2/3}$ for $H\parallel
ab$ and $2\leq\mu_{0}H\leq 12$ T. The inset shows the same scaling analysis
for $H\parallel c$ and $0.5\leq\mu_{0}H\leq 6$ T. (b) $H_{c2}(T)$ determined
from the fluctuation scaling (solid symbols) for both $H\parallel c$ and
$H\parallel ab$. The dotted lines show the WHH theoretical fitting. The
crossed symbols represent the points in $\rho(T)$ curves at which the
resistivity becomes half the normal state value.
Figure 2(a) shows the scaled curves according to Eq. 2 for both field
directions. The fluctuation conductivity was obtained by subtracting the
normal state conductivity ($\rho_{n}^{-1}$) from the measured conductivity,
$\sigma_{fl}=\rho^{-1}-\rho_{n}^{-1}$, where $\rho_{n}$ at low $T$ was
determined through an extrapolation of the normal state resistivity data
between 40 and 100 K with two-order polynomial fit. For each field, the scaled
data cover the resistive transition region down to temperature at which
$\rho(T)$ becomes half the normal state value. As seen in Fig. 2(a), by
adjusting the $T_{c}(H)$ parameter with the restriction of $T_{c}(0)=26.2$ K,
we have obtained nice scalings of the experimental data for both
$H\parallel~{}ab$ and $H\parallel~{}c$. The resultant $T_{c}(H)$, or
equivalently $H_{c2}(T)$, were plotted as solid symbols in Fig. 2(b). We found
that we could also obtain a 2D scaling of the data with reasonable quality
according to Eq. 1 and roughly the same $T_{c}(H)$, as demonstrated in a
previous study in Sm1.85Ce0.15CuO4-δ for $H\parallel c$. Han92
Figure 2(b) shows that $H_{c2}(T)$ determined from the scaling analysis
exhibits linear temperature dependence in vicinity of $T_{c}$, as in
conventional type-II superconductors. In comparison, we have also determined
the points at which the resistivity becomes 50% of the normal state value and
plotted them as crossed symbols for both field directions in Fig. 2(b). It is
seen that they exhibit positive curvature and are considerably lower than the
determined $H_{c2}(T)$, similar to the observation in hole-doped cuprates.
Welp89 From fitting $H_{c2}(T)$ to the Werthamer$-$Helfand$-$Hohenburg (WHH)
theory Werthamer66 (dotted lines in Fig. 2(b)) we obtain $H_{c2}(0)\simeq 87$
T and 11.6 T, with slope of $H_{c2}(T)$ near $T_{c}$ being $-$4.8 T/K and
$-$0.64 T/K, for $H\parallel~{}ab$ and $H\parallel~{}c$, respectively. This
indicates the in-plane coherence length $\xi_{ab}(0)\simeq 53.3~{}{\AA}$, the
$c$-axis coherence length $\xi_{c}(0)\simeq 7.1~{}{\AA}$ and the anisotropy
factor $\gamma=H_{c2}^{\parallel ab}/H_{c2}^{\parallel c}\simeq 7.5$. We note
that this anisotropy factor is comparable to that determined for hole-doped
YBa2Cu3O7-δ (YBCO) ($\sim 5-8$ in Refs. Welp89, ; Nagasao08, ; Babic99, ) but
smaller than that for La2-xSrxCuO4 (LSCO) ($\sim 20$ in Ref. Panagopoulos03, )
and Bi2Sr2CaCu2O8+δ (Bi2212) ($\sim 60$ in Ref. Farrell89, ) at optimal
doping.
### III.2 Normal State
Figure 3: Normal state MR versus $H^{2}$ for $H\parallel c$ (a) and
$H\parallel ab$ (b).
Now we turn to the normal state above $T_{c}$. Figure 3 shows the MR
$\Delta\rho/\rho_{0}$ ($\Delta\rho=\rho_{H}-\rho_{0}$ with $\rho_{H}$ and
$\rho_{0}$ the resistivity in field $H$ and in zero field respectively) of the
crystal at different temperatures ($T\geq 50$ K), in the plot of
$\Delta\rho/\rho_{0}$ vs. $H^{2}$. For both field directions the positive MR
shows conventional orbital MR behavior in the weak-field regime, Ziman72 that
is, it basically follows an $H^{2}$ dependence and its strength decreases with
increasing temperature. At lower $T$ (50 and 70 K), small deviation from the
$H^{2}$ behavior may come from magnetic-field suppression of superconducting
fluctuations. Ando02 For $H\parallel~{}c$, the MR is rather large with the
order of one percent, similar to previous reports. Gollnik98 This is
contrasted with what we observed in hole-doped YBCO or LSCO single crystals,
where the normal state MR for $H\parallel~{}c$ was about one order of
magnitude smaller at similar temperature and field range. Kimura96 ; Harris94
In hole-doped cuprates the normal state MR in transverse configuration is
usually ascribed to the orbital scattering within a single band picture.
Whereas, in NCCO and other electron-doped cuprates, the large MR, together
with other physical properties such as Hall effect, has been widely
interpreted as an indiction of two-band transport. Seng95 ; Gollnik98
According to classical transport theory, the orbital MR could be enhanced when
different types of charge carriers participating in the electrical conduction.
Ziman72
Figure 4: The ratio of the MR between $H\parallel c$ and $H\parallel ab$ as a
function of $H$. The inset shows the comparison of the MR for both field
directions at 90 K with data for $H\parallel ab$ multiplied by 7.9.
It is seen in Fig. 3 that the MR for $H\parallel~{}ab$ is considerably smaller
than that for $H\parallel~{}c$ at the same temperature and field scale, which
is similar to the observation in hole-doped LSCO single crystals. Kimura96
This anisotropy provides an additional evidence that the measured MR with
$H\parallel c$ is dominated by the orbital contribution. As we know, if the MR
mainly originated from the field coupling to spin degree of charge carriers,
it would be almost isotropic. It is worth mentioning that here we have not
considered the possible effect of field-induced changes in sample’s spin
structure on the MR. In lightly doped Pr1.3-xLa0.7CexCuO4 (PLCCO, $x=0.01$)
and NCCO ($x=0.025$), it was reported that field-induced spin-flop transitions
in the spin structure resulted a much larger, distinct field-dependent MR with
$H\parallel~{}ab$ at low $T$. Lavrov04 ; Li05 Figure 4 shows the anisotropy
of the MR more explicitly, by plotting the ratio
$\zeta=\Delta\rho^{H\parallel~{}c}/\Delta\rho^{H\parallel~{}ab}$ as a function
of $H$. $\zeta$ is around 7 and nearly independent on $H$ and $T$ in present
experiment. Inset of Fig. 4 shows the MR at 90 K as an example, which was
plotted as a function of $H^{2}$. When the MR for $H\parallel~{}ab$ is
multiplied by 7.9, we can see it follows nearly the same line as the MR for
$H\parallel~{}c$.
It may be noted that the normal state MR ratio $\zeta$ is close to the
anisotropy ratio $\gamma$ determined above for the superconducting state. On
the one hand, in our view the closeness of both parameters may be merely
coincidental and have no obvious physical importance. On the other hand,
however, we point out that there should be an internal connection between
$\zeta$ and $\gamma$, since both parameters would relate to the anisotropy of
the effective mass $m^{\ast}$ of charge carriers. In anisotropic GL theory,
$\gamma=H_{c2}^{\parallel ab}/H_{c2}^{\parallel
c}=\sqrt{m_{c}^{\ast}/m_{ab}^{\ast}}$ with $m_{c}^{\ast}$ and $m_{ab}^{\ast}$
the effective mass along $c$-axis and within $ab$-plane respectively. This
implies that the axial effective mass $m_{c}^{\ast}$ is about 50 times heavier
than the in-plane $m_{ab}^{\ast}$ for our NCCO crystal. For the normal state
MR, as indicated in the two-band model, it is governed by the $m^{\ast}$, the
scattering rate $\tau$ and other properties of the carriers. Ziman72 The
anisotropy of the $m^{\ast}$ thus would certainly contribute to the anisotropy
of the MR, namely, the ratio $\zeta$, though a simple relation between them is
difficult to obtain since the MR is determined by aforementioned parameters in
a complicated way, especially with the presence of different types of
carriers. Nevertheless, the present study shows that the optimally doped NCCO
single crystal has a moderate anisotropy in both superconducting and normal
state.
## IV CONCLUSION
In summary, by investigating the upper critical field $H_{c2}$ and the normal
state MR with field $H$ either parallel or perpendicular to the
crystallographic $c$-axis, we have determined the anisotropy properties of an
optimally doped NCCO single crystal. $H_{c2}$ estimated from scaling of the
fluctuation conductivity is about 87 T and 11.6 T for $H\parallel~{}ab$ and
$H\parallel~{}c$ respectively, which yields $\xi_{ab}(0)\simeq 53.3~{}{\AA}$,
$\xi_{c}(0)\simeq 7.1~{}{\AA}$ and the anisotropy factor
$\gamma=H_{c2}^{\parallel ab}/H_{c2}^{\parallel c}\simeq 7.5$. The normal
state MR for $H\parallel~{}ab$ is found to be almost a magnitude smaller than
that for $H\parallel~{}c$. This anisotropy, together with the $H^{2}$
dependence, confirms that the MR with $H\parallel~{}c$ is mainly due to the
orbital scattering. Present findings place optimally doped NCCO cuprate as an
anisotropic 3D superconductor with a moderate anisotropy.
###### Acknowledgements.
We are grateful to Profs. S. L. Li and P. Dai for providing the NCCO single
crystal and for helpful comments. We are also indebted to Prof. H. H. Wen for
experimental support and helpful discussions.
## References
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* (5) C. Panagopoulos, T. Xiang, W. Anukool, J. R. Cooper, Y. S. Wang, and C. W. Chu, Phys. Rev. B 67, 220502 (2003)
* (6) K. Nagasao, T. Masui, and S. Tajima, Physica C 468, 1188 (2008)
* (7) Y. Hidaka and M. Suzuki, Nature 338, 635 (1989)
* (8) C. C. Almasan, S. H. Han, E. A. Early, B. W. Lee, C. L. Seaman, and M. B. Maple, Phys. Rev. B 45, 1056 (1992)
* (9) D. Giller, A. Shaulov, R. Prozorov, Y. Abulafia, Y. Wolfus, L. Burlachkov, Y. Yeshurun, E. Zeldov, V. M. Vinokur, J. L. Peng, and R. L. Greene, Phys. Rev. Lett. 79, 2542 (1997)
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* (11) P. Seng, J. Diehl, S. Klimm, S. Horn, R. Tidecks, K. Samwer, H. Hansel, and R. Gross, Phys. Rev. B 52, 3071 (1995)
* (12) F. Gollnik and M. Naito, Phys. Rev. B 58, 11734 (1998)
* (13) G. S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, and S. Uchida, Phys. Rev. Lett. 77, 5417 (1996)
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* (16) S. H. Han, C. C. Almasan, M. C. de Andrade, Y. Dalichaouch, and M. B. Maple, Phys. Rev. B 46, 14290 (1992)
* (17) H. H. Wen, W. L. Yang, and Z. X. Zhao, Physica C 341
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* (19) H. Gao, C. Ren, L. Shan, Y. Wang, Y. Z. Zhang, S. P. Zhao, X. Yao, and H. H. Wen, Phys. Rev. B 74, 020505(R) (2006)
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|
arxiv-papers
| 2011-02-28T09:06:37 |
2024-09-04T02:49:17.357501
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yue Wang, Hong Gao",
"submitter": "Yue Wang",
"url": "https://arxiv.org/abs/1102.5621"
}
|
1102.5651
|
# Classes of exact static spheroidal Einstein-Maxwell solutions.
S.M.KOZYREV Scientific center gravity wave studies ”Dulkyn”, PB 595, Kazan,
420111, Russian Federation. _Email_ : Sergey@tnpko.ru
###### Abstract
In this paper we study the spheroidal cases of static charged fluid
configurations in general relativity. We consider the effect of the
anisotropic stresses of electromagnetic field on the shape of static charged
self-graviting objects. It is shown that electromagnetic fields can have
significant effect on the structure and properties of self-graviting objects.
## 1 Introduction
Exact solutions of the Einstein-Maxwell field equations are of crucial
importance in relativistic astrophysics. These solutions may be utilised to
model a charged relativistic star as they are matchable to the Reissner-
Nordstrom [1], [2] exterior at the boundary. A wide spread assumption in the
study of stellar structure is that the shape of star can be modeled as a
spherical symmetry object. This approach has been used extensively in the
study of star, star system and galaxies. However, in many systems, deviation
from spherical symmetry may play an important role in determining of them
properties. Physical situation where unspherical shape may be relevant are
very diverse. On the other hand, self-graviting objects resulting from the
coupling electromagnetic field to gravity are a system where anisotropic
pressure occurs naturally.
Anisotropy appears as an extra assumption on the behavior of electromagnetic
field and on the shape of equilibrium configuration. Since we still do not
have a formulation of the possible anisotropic stresses is emerging in these
or other contexts, we take the approach of finding several exact solutions
representing physical situations, modelled by ellipsoid of revolution.
Solutions to the equation in spheroidal coordinates have application to a wide
range of problems in physics [5]. Our goals hear is to find exact spheroidal
solution, offering an analysis of the change in the physical properties of the
stellar and galaxy models due to presence of electromagnetic field.
## 2 Einstein-Maxwell Equations and Static spheroidal configurations.
In this paper we study static, spheroidal solutions of the Einstein-Maxwell
system featuring a spinless charge configurations.The vacuum Einstein-Maxwell
equations, in geometrized units such that $c=8\pi
G=\mu_{0}=\varepsilon_{0}=1$, can be written as [3]
$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=T_{\mu\nu},$ (1) $F^{\mu\nu}_{;\nu}=0,$ (2)
with the electromagnetic energy-mementum tensor given by
$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
F_{\mu\eta}F^{\eta}_{\nu}-\frac{1}{4}g_{\mu\nu}F_{\eta\zeta}F^{\eta\zeta},$
(3)
where
$\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
A_{\nu,\mu}-A_{\mu,\nu},$ (4)
is the electromagnetic field tensor and $A_{\mu}$ is the electromagnetic four
potential.
To start with, note that by using coordinate freedom inherent in general
relativity any static spheroidal geometry can by put into form where are only
two independent metric components typically functions of the coordinates
$\xi$. As we have already mentioned we consider the ansatz static spheroidal
space-time.
The two-dimensional elliptic coordinate system is defined from the set of all
ellipses and all hyperbolas with a common set of two focal points. We denote
the separation of the two focal points by $2c$.
Oblate spheroidal coordinates are derived from elliptic coordinates by
rotating the elliptical coordinate system about the perpendicular bisector of
the focal points. The coordinates are often labelled $\eta$, $\xi$ and
$\theta$ with the transformation to Cartesian coordinates given by
$\displaystyle x=c\eta\xi sin(\theta),$ (5) $\displaystyle
y=c\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)},$ (6) $\displaystyle
z=c\eta\xi sin(\theta)cos(\theta).$ (7)
Similarly, one can obtain the prolate spheroidal coordinates by rotating it
about the parallel bisector.
$\displaystyle x=c\eta\xi,$ (8) $\displaystyle
y=c\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}cos(\theta),$ (9)
$\displaystyle
z=c\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}sin(\theta).$ (10)
Let the spacetime ansatz be described by the spheroidal metric given by
$\displaystyle ds^{2}=-B\left(\xi\right)dt^{2}+A\left(\xi\right)d\Omega^{2}.$
(11)
For this static spheroidal ansatz of space-time we take the electromagnetic
potential as
$\displaystyle A_{\mu}$ $\displaystyle=$ $\displaystyle(\psi,0,0,0),$ (12)
where it is assumed that the electric potential $\psi$ depends on $\xi$ only.
We adopt coordinates that allow us to write spheroidal geometry in prolate
form
$\displaystyle
d\Omega^{2}=c^{2}\frac{\xi^{2}-\eta^{2}}{\xi^{2}-1}d\xi^{2}+c^{2}\frac{\xi^{2}-\eta^{2}}{1-\eta^{2}}d\eta^{2}+c^{2}(\xi^{2}-1)(1-\eta^{2})d\theta^{2},$
(13)
and in oblate form
$\displaystyle
d\Omega^{2}=c^{2}\frac{\xi^{2}-\eta^{2}}{\xi^{2}-1}d\xi^{2}+c^{2}\frac{\xi^{2}-\eta^{2}}{1-\eta^{2}}d\eta^{2}+(c\xi\eta)^{2}d\theta^{2},$
(14)
where $A,B$ are function of $\xi$ only and $\xi\geq 1$, $-1\leq\eta\leq 1$,
$0\leq\theta\leq 2\pi$.
### 2.1 Prolate spheroidal configurations.
After a bit of algebra, the field equations (1) - (2) are explicitly given in
forms of the metric (11) in prolate case.
$\displaystyle\frac{A^{\prime
2}}{4A^{2}}+\frac{A^{\prime}B^{\prime}}{2AB}=-\frac{\varepsilon^{2}}{2(1-\xi^{2})A}=T_{11},$
(15)
$\displaystyle\frac{\xi^{2}-1}{2(\eta^{2}-1)}\left[-\frac{A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}+\frac{A^{\prime
2}}{A^{2}}+\frac{B^{\prime
2}}{2B^{2}}\right]=\frac{\varepsilon^{2}}{2(\xi^{2}-1)(\eta^{2}-1)A}=T_{22},$
(16)
$\displaystyle\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}=0=T_{12},$ (17)
$\displaystyle\frac{(\xi^{2}-1)(\eta^{2}-1)}{2(\xi^{2}-\eta^{2})}\left[-\frac{A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}+\frac{A^{\prime
2}}{A^{2}}+\frac{B^{\prime
2}}{2B^{2}}\right]=-\frac{\varepsilon^{2}(\eta^{2}-1)}{2(\xi^{2}-\eta^{2})A}=T_{33},$
(18)
$\displaystyle\frac{B(\xi^{2}-1)}{c(\xi^{2}-\eta^{2})}\left[-\frac{A^{\prime\prime}}{A^{2}}+\frac{3A^{\prime
2}}{4A^{3}}-\frac{2\xi}{\xi^{2}-1}\frac{A^{\prime}}{A^{2}}\right]=\frac{\varepsilon^{2}B}{2c(\xi^{2}-1)(\xi^{2}-\eta^{2})A^{2}}=T_{00},$
(19)
where prime (’) denoting derivative with respect to the $\xi$ coordinate.
After a simple integration, from (15) - (19) we obtain
$A=\frac{1}{8}\left[a_{0}\pm\varepsilon\ln\left(\frac{\xi+1}{\xi-1}\right)\right]^{2},$
$B=\frac{b_{0}}{A},\\\ $ (20)
where $a_{0},b_{0}$ arbitrary constants.
### 2.2 Oblate spheroidal configurations.
Replacing the line element in the field equations, the oblate set is
$\displaystyle\frac{A^{\prime
2}}{4A^{2}}+\frac{A^{\prime}B^{\prime}}{2AB}=-\frac{\varepsilon^{2}}{2\xi^{2}(\xi^{2}-1)A}=T_{11},$
(21)
$\displaystyle\frac{\xi^{2}-1}{2(\eta^{2}-1)}\left[-\frac{A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}+\frac{A^{\prime
2}}{A^{2}}+\frac{B^{\prime
2}}{2B^{2}}\right]=-\frac{\varepsilon^{2}}{2\xi^{2}(\eta^{2}-1)A}=T_{22},$
(22)
$\displaystyle\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}=0=T_{12},$ (23)
$\displaystyle\frac{\xi^{2}\eta^{2}(\xi^{2}-1)}{2(\xi^{2}-\eta^{2})}\left[\frac{A^{\prime\prime}}{A}+\frac{B^{\prime\prime}}{B}-\frac{A^{\prime
2}}{A^{2}}-\frac{B^{\prime 2}}{2B^{2}}\right]$
$\displaystyle=-\frac{\varepsilon^{2}\eta^{2}}{2(\xi^{2}-\eta^{2})A}=T_{33},$
(24)
$\displaystyle\frac{B(\xi^{2}-1)}{c(\xi^{2}-\eta^{2})}\left[-\frac{A^{\prime\prime}}{A^{2}}+\frac{3A^{\prime
2}}{4A^{3}}-\frac{(2\xi^{2}-1)}{\xi(\xi^{2}-1)}\frac{A^{\prime}}{A^{2}}\right]=\frac{B\varepsilon^{2}}{2c\xi^{2}(\xi^{2}-\eta^{2})A^{2}}=T_{00},$
(25)
Then the solutions of the gravitational field equations take in oblate case
the form
$A=2\left(a_{0}\pm\varepsilon\arctan\sqrt{\frac{\xi^{2}-1}{\xi^{2}+1}}\right)^{2},$
$B=\frac{b_{0}}{A},\\\ $ (26)
where $a_{0},b_{0}$ arbitrary constants.
### 2.3 Analysis
To see that all these metrics is asymptotically flat Minkowski it is enough to
show that the metric components behave in an appropriate way at large
$\xi$-coordinate values, e.g., $g_{\mu\nu}=\eta_{\mu\nu}+O(1/\xi)$ as
$\xi\rightarrow\infty$. By inspection of the coefficients, we verify that this
is so ($a_{0}=1,b_{0}=1$).
In fact, in the present approach, it is easy to show that, in the case of
absent the electromagnetic field $\varepsilon=0$, Einstein’s field equations
yield only the flat space
$A=1,$ $B=1,\\\ $ (27) $\phi=1.$
Therefore, we see that it is possible to explain the shape of spheroidal
configurations by electromagnetic or other fields [4]. This seems to be a
remarkable result, although in a way it should be anticipated since the
directional components of ”equation of state” of electromagnetic field are
anisotropic in the oblate and prolate cases. However, in this case there is a
contribution from the electromagnetic field that makes $T_{\mu\nu}$ nonzero.
On the other hand, it seems natural that we have obtained an ”equation of
state” that describes vacuum, since we do not have matter.
## 3 Discussion
In this article we delineated the qualitative features one would expect from
spheroidal object. It is demonstrated that our model can successfully predict
the spheroidal configuration in terms of a self-gravitating spacetime solution
to the Einstein field equations and reproduce the not spherically-symmetric
shape in terms of the non-trivial energy density and anisotropic pressure of
the electromagnetic field which was absent in the context of empty space.
Hence the approach followed in this paper has proved to be a fruitful avenue
for generating new exact solutions for describing the spacetimes of charged
configurations.
We believe that following this hypotheses the shape of galaxy and rotation
curve may be explained by action of electromagnetic or other fields. The
solution presented here could be a first approximation at the galactic space-
time provided the presence of any physical fields. Therefore, it is necessary
to study how these results modify the standard method of interpretation
rotation data. Further investigation into the nature solutions with view to
separating the real rotational effects from the electromagnetic, scalar or
other fields anisotropy might be rewarding.
## 4 Acknowledgements
I am grateful to S.V. Sushkov and R.A. Daishev for the helpful discussions.
The work was supported in part by the Institute of Applied Problems.
## References
* [1] H .Reissner, ”Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie”. Annalen der Physik 50: 106 120.(1916)
* [2] G. Nordstrom, ”On the Energy of the Gravitational Field in Einstein’s Theory”. Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam 26: 1201 1208.(1918)
* [3] R. C. Tolman Relativity Thermodynamics And Cosmology., Oxford University Press, Oxford, (1934)
* [4] S.M. Kozyrev, New static spheroidal solution in Jordan-Brands-Dicke theory., arXiv:1012.1097 [gr-qc] 2010.
* [5] C. Flammer. Spheroidal Wave Functions. Stanford University Press, 1957.
|
arxiv-papers
| 2011-02-28T12:53:32 |
2024-09-04T02:49:17.363106
|
{
"license": "Public Domain",
"authors": "S.M. Kozyrev",
"submitter": "Sergey Kozyrev",
"url": "https://arxiv.org/abs/1102.5651"
}
|
1102.5711
|
# XMLlab : multimedia publication of simulations applets
using XML and Scilab
Stéphane Mottelet Laboratoire de Mathématiques Appliquées de Compiègne,
Département de Génie Informatique, Université de Technologie de Compiègne, BP
20529, 60205 COMPIEGNE CEDEX, FRANCE André Pauss UMR Génie des Procédés
Industriels, Département de Génie Chimique, Université de Technologie de
Compiègne, BP 20529, 60205 COMPIEGNE CEDEX, FRANCE stephane.mottelet@utc.fr
###### Abstract
We present an XML-based simulation authoring environment. The proposed
description language allows to describe mathematical objects such as systems
of ordinary differential equations, partial differential equations in two
dimensions, or simple curves and surfaces. It also allows to describe the
parameters on which these objects depend. This language is independent of the
target software and allows to ensure the perennity of author’s work, as well
as collaborative work and content reuse. The actual implementation of XMLlab
allows to run the generated simulations within the open source mathematical
software Scilab, either locally when Scilab is installed on the client
machines, or on thin clients running a simple web browser, when XMLlab and
Scilab are installed on a distant server running a standard HTTP server.
###### keywords:
simulation markup language, interoperability , multimedia publication
††thanks: http://xmllab.org
## 1 Introduction
The need to use a simulation tool is in most cases an answer to simple
statements : the user has some equations modeling a physical system. He wants
to solve them, and if possible to be able to easily change some parameters to
see how they influence the results of the simulation and finally save the
parameters and the results (e.g. in a format readable by a spreadsheet
application).
The educational benefit of using simulations, when an adequate tool is used,
is not to be discussed here. But there are very different steps in the
development of a simulation. Once the equations are stated, you firstly have
to make them fit to a particular software, provided this software is adequate
to the disciplinary field of the phenomenon. This first step is not time-
consuming compared to the time which is always spent to develop a graphical
user interface. The author will spend the greater part of his time to polish
the interface, although he could have spent this time to work on another
simulation. Moreover, the more the applet will be polished to fit a particular
case, the less it will be reusable in another close context. The World Wide
Web is a place where a lot of good quality JAVA applets can be found, but
these applets are always difficult to reuse in the context of a particular
course, because modifying them (when the author makes the source code
available) needs abilities in a low level language (JAVA, C++, C), or a high
level script language such as the one used by Matlab or Scilab ([1, 2, 3]).
This kind of work is the concern of craftsmen, and not of an industrial
approach. The author’s work is not reusable in general and its perennity is
not guaranteed, because the work relies on an application using a proprietary
format, and last but not least, the work is exchangeable exclusively with
authors using the same application (and sometimes the same version).
People working on modern documentary applications have already made this
analysis, and this can be seen with the exponential growth of the number of
applications of the XML markup language ([4]). In the field of simulation
applets this work has just begun. One can cite, in the field of biology and
chemistry, the work of a consortium of academic people and authors of
simulation software which has lead to `sbml`, and exchange markup language
modeling biological and chemical systems (with kinetics) using XML ([5, 6]).
Another project is `xmds`, a tool also based on XML allowing to generate
Scilab, Matlab or C++ code (but without any graphical user interface) allowing
to simulate deterministic or stochastic systems ([7]). Compared to our
approach, which will detailed in this paper, the weakness of this tool is a
lack of structuring in the description language (the level of structuring is
not deep enough compared to what XML allows to do).
Concerning other serious XML based simulation modeling projects, we can refer
to [8], where the use of XML markup to describe bond graph models is
considered. This paper has an excellent introduction recalling the essential
features of XML technologies relevant for the description and the processing
of bond graph models, which is also relevant for others high level approaches
for the description of systems, like ours.
In the scope of the XMLlab project, we have chosen to show the benefits of an
approach where the content and the form are well differentiated and are the
concern of different people:
### The content
The content of simulation resides in the equations of the phenomenon, their
description, the associated parameters and their thematic organization. The
description of the content is the concern of the author.
### The form
The form resides in the graphical user interface, the various “widgets” and
menus which reinforce the user-friendliness of the final applet, and the
visualization tools (static curves, animations, sounds). This part of the
applet code is the concern of a high level developer, or the concern of a tool
able to generate this code automatically from the description of the content.
This is the option we have chosen.
The choice of adequate numerical methods is another concern. The author is not
necessarily able to make this choice himself, that’s why this choice has to be
done automatically, knowing which method is the most adequate to solve a given
type of equation.
The purpose of the XMLlab project was not to define a description language by
its own, hence we have chosen a particular “target” application in the early
phases of the project. This application is Scilab, an open source software
developed since 1990 by researchers of INRIA and ENPC. Our goal was to develop
a complete “compilation chain”, allowing to transform the source XML documents
into executable scripts interpreted by the target application. Moreover, the
choice of Scilab is motivated by the fact that this software allows to use the
Tcl/Tk script language ([9, 10]) to generate graphical user interfaces. We
also use the Tcl/Tk message passing system in the XMLlab WebServer, which
allows to dynamically publish the simulations on the web. In fact this paper
extends [11], since the new features of XMLlab generalize its multi-medium
nature.
## 2 The structure of an XMLlab simulation
As specified in the XMLlab DTD, a simulation can be divided in a certain
number of conceptual elements: parameters, mathematical models of objects
(time-dependent or not) and finally a display element to output the results of
the simulation.
### 2.1 Parameters
They are the parameters of the phenomenon and of the mathematical model. The
goal is to allow the user of the simulation to make them vary by means of the
interface which will be generated by the compilation chain. It has to be
possible to simply specify if the value of the parameter is seen in the
interface, but non modifiable by the user. There must also exist hidden
parameters, for internal use only. The parameters are grouped into sections,
e.g. for an ordinary differential equation, the user may want to differentiate
the physical parameters of the phenomenon from the resolution parameters
(final time, number of discretization steps, etc.). This logical structuring
can be then used to graphically structure the interface.
* •
Scalars and matrices : with XMLlab , a simulation the parameters can be
scalars or matrices. The minimum and maximum value of a scalar can be given,
the type of “widget” to use (a slider, or a simple entry field where the user
can modify the value). Each parameter must have symbolic name which can be
reused in the description of the mathematical model, and a default value,
which will be used for the first run of the simulation.
* •
Databases : the user can store many instances of a parameter group. This
allows, e.g. in chemistry, to build a small database of different acids and
alkali, by storing their parameters (acidity constants, charge, etc.). The
database can then be used to generate a menu allowing to choose a given
parameter group in the interface.
### 2.2 Objects with a mathematical model
We now deal with the equations of the phenomenon to be simulated. There are
elements of different levels.
* •
Domains : the most simple describe intervals of $\mathbb{R}$ or domains of
$\mathbb{R}^{2}$. They are simple closed intervals of the type $[a,b]$ (where
the bounds may depend on parameters described in the previous section) or two
dimensional domains. The latter can be rectangles defined by a Cartesian
product of two intervals, or general domains defined by the form of their
boundary by parametric curves (we will discuss curves in the next item). The
user can precise the way these domains have to be discretized, if applicable
(number of discretization points, linearly or logarithmically).
* •
Curves and surfaces : non-parametric curves can be described like this,
$y=f(x),\quad x\in[a,b].$
This definition reuses an interval. This way, it is possible to define many
curves referring to the same interval. Parametric curves can be also defined
like this,
$x=f(t),\;y=g(t),\;t\in[a,b].$
The surfaces can also be of parametric or non-parametric type, namely
$z=f(x,y),\quad(x,y)\in\mathcal{D},$
where $\mathcal{D}$ is a domain of $\mathbb{R}^{2}$, or
$x=f(u,v),\,y=g(u,v),\,z=h(u,v),\;(u,v)\in\mathcal{D}.$
The parameters defined in the previous section can be used at any level, in
the definition of domains or in the equations themselves.
* •
Ordinary differential equations : one can describe systems of ordinary
differential equations, e.g.
$\frac{d}{dt}x(t)=f(x,y,t),\;\frac{d}{dt}y(t)=(x,y,t),\;t\in[a,b],$
with given initial conditions $x(a)=x_{a}$ and $y(a)=y_{a}$. XMLlab allows to
keep the natural description, without having to reformulate each unknown
$x(t)$ or $y(t)$ as the element of a vector $X(t)$ with $x(t)=X_{1}(t)$ and
$y(t)=X_{2}(t)$. The chosen description model consists in:
* –
A time interval (here $[a,b]$)
* –
A list of states. For each state (here $x$ or $y$), its time-derivative and
its initial value are given.
* –
A list of outputs. They are the observations which can be computed by using
the states, e.g. $z(t)=x(t)+y(t)$.
* •
Non linear equations : general systems of non-linear equations can be
described, namely
$f(x,y,z,t,\cdots)=0,\;h(x,y,z,t,\cdots)=0,\;\cdots$
or curves defined by an implicit equation of the form
$f(x,y)=0,x\in[a,b],$
This kind of equation is used in the modeling of acid-alkali titration.
* •
Partial differential equations : XMLlab allows to describe partial
differential equations of diffusion type, namely
$\left\\{\begin{array}[]{c}-\operatorname{div}\left(P\operatorname{grad}u\right)(x)+c(x)u(x)=f(x),\;x\in\Omega,\\\
+\text{\em\ boundary conditions}\end{array}\right.$
The domain $\Omega$ is described from its boundary (parametric curves, defined
earlier). We will give some details on the numerical methods in the next
section. Here again, the parameters can be used at any level, in the
definition of the domain $\Omega$ or in the physical data (diffusion matrix
$P$, source term $f(x)$, proportional coefficient $c(x)$).
### 2.3 Results display
We use a classical hierarchical description, using windows and systems of
axes, where the user just has to precise what he wants to display by making
reference to objects defined in the “Mathematical models” section.
* •
Windows : A window contains systems of axes. The user just has to specify how
they have to be placed if they are more than one (the window is divided within
its height and width).
* •
System of axes : the user has to specify if the system is two or three
dimensional. Each system of axes contains some references to what has to be
represented.
* •
Objects to be represented graphically : reference can be made to a curve, to a
surface or to the state of an equation, by means of its symbolic name. The
chosen structure allows to greatly simplify the number of different elements.
For example, a surface can be referenced in a two or three dimensional system
of axes. In a three dimensional system a perspective projection is used,
although in two dimensions we use a pseudo-color planar representation. In
both cases, the reference to the surface is made identically, only the the
“parent” context is changing.
⬇
1<?xml version=”1.0” ?>
2<!DOCTYPE simulation PUBLIC ”-//UTC//DTD␣XMLlab␣V1.6//EN”
3 ”http://www.xmllab.org/dtd/1.6/fr/simulation.dtd”>
4<simulation>
5 <header>
6 …
7 </header>
8 <notes>
9 …
10 </notes>
11 <parameters>
12 …
13 </parameters>
14 <compute>
15 …
16 </compute>
17 <display>
18 …
19 </display>
20</simulation>
Figure 1: The outline of a simulation showing the high-level elements of a
simulation
$\theta(t)$
Figure 2: The pendulum.
## 3 A typical example of simulation
We give on the figure 1 the skeleton of a simulation document. We will now
explain with details how to build this document to describe a small
simulation.
We consider the pendulum depicted on figure 2. We make the hypothesis that the
line connecting the sphere of mass $M$ to the rotation axis is of negligible
mass compared to $M$. We measure the deviation of the pendulum from the stable
vertical equilibrium position by the angle $\theta(t)$ positively measured as
indicated on figure 2. If one applies the relations of dynamics for bodies
under rotations, we obtain the following ordinary differential equation
$\left\\{\begin{array}[]{rcl}\ddot{\theta}(t)&=&-\displaystyle\frac{g}{L}\sin\theta(t),\quad
t\in[0,T]\\\ \theta(0)&=&\theta_{0},\\\
\dot{\theta}(0)&=&0.\end{array}\right.$
The value of $\theta_{0}$ gives the initial angular deviation of the pendulum,
and we consider that the initial angular velocity is zero. If $\theta_{0}$ is
small, $\theta(t)$ can be approximated by
$\phi(t)=\theta_{0}\cos\left(\sqrt{\frac{g}{L}}t\right)$. We want do describe
a simulation applet allowing to compare $\phi(t)$ and $\theta(t)$ when
$\theta_{0}$ changes. The graphical output should plot $\phi(t)$ and
$\theta(t)$ for $t\in[0,T]$, and the user should have the possibility to
change $\theta_{0}$ in the interval $[-3.14,3.14]$ by moving the point
$(0,\theta_{0})$ interactively on the curve (the point will be represented
with a cross). The XML code fragment containing the description of the
parameters is given in figure 3.
⬇
1<parameters>
2 <section>
3 <title>Parameters of the pendulum</title>
4 <scalar label=”L” unit=”m”>
5 <name>Length of the pendulum</name>
6 <value>1</value>
7 </scalar>
8 <scalar label=”g0” unit=”ms^-2”>
9 <name>Gravity</name>
10 <value>9.81</value>
11 </scalar>
12 <point label=”point0”>
13 <x1 label=”zero”>
14 <value>0</value>
15 </x1>
16 <x2 label=”theta_0”>
17 <value>2</value>
18 </x2>
19 <constraints>
20 <curve ref=”segment”/>
21 </constraints>
22 </point>
23 </section>
24 <section>
25 <title>Resolution parameters</title>
26 <scalar label=”tf” unit=”s” min=”0” max=”10” increment=”1”>
27 <name>Final time</name>
28 <value>2</value>
29 </scalar>
30 </section>
31</parameters>
Figure 3: Code fragment containing the description of the parameters of the
pendulum simulation
Each pair of `section` elements allows to group parameters. The `scalar`
element represents a scalar parameter, containing its full name in the `name`
element and its initial value in the element `value`. The `constraint` element
refers to a curve with label `segment`, which will be described below. For the
final simulation time, corresponding to the parameter with label `tf` (lines
26 to 29 in figure 3), there are bounds (minimum and maximum value) and also a
relevant increment. During the compilation phase, the presence of these three
attributes values will be taken into account and will allow to choose a
particular widget in the graphical user interface.
### 3.1 Mathematical models, compute element
Here are the fragments corresponding to the description of the $[0,T]$
interval :
⬇
1<defdomain1d label=”t” unit=”s”>
2 <name>time</name>
3 <interval>
4 <initialvalue>0</initialvalue>
5 <finalvalue>tf</finalvalue>
6 </interval>
7</defdomain1d>
and the XML code fragment describing the differential equation is given in
figure 4.
⬇
1<ode label=”pendulum”>
2 <refdomain1d ref=”t”/>
3 <states>
4 <state label=”theta” unit=”rad”>
5 <name>Real solution</name>
6 <derivative>theta_dot</derivative>
7 <initialcond>theta_0</initialcond>
8 </state>
9 <state label=”theta_dot” unit=”rad/s”>
10 <name>Derivative of the angle</name>
11 <derivative>-g0/L*sin(theta)</derivative>
12 <initialcond>0</initialcond>
13 </state>
14 </states>
15 <outputs>
16 <output label=”theta_lin”>
17 <name>Harmonic solution</name>
18 <value>theta_0*cos(sqrt(g0/L)*t)</value>
19 </output>
20 </outputs>
21</ode>
Figure 4: Code fragment containing the description of the system of two
differential equations of the pendulum.
The `ode` element (ordinary differential equation) contains an empty element
`refdomain1d` referring to the $[0,T]$ interval (defined earlier by the
`defdomain1d` element, referred by the attribute `ref`), and thus defining the
symbolic name of the integration variable, a `states` element containing the
description of each state ($\theta$ and $\dot{\theta}$) in a `state` element.
Each `state` element contains the name of the state, its time-derivative
`derivative` and its initial value `initialcond`. Until now, the content of
the `derivative` element is not parsed, and will copied verbatim during the
compilation. Since we do not have a dedicated editor, this is easier for the
user to type mathematical formulas like this, but we plan to use Content
MathML (see e.g. [12]) in the future, which will allow an easier a priori
validation of formulas.
The last element `outputs` in `ode` is a list of outputs, which can be
functions of states and/or time. In our example the output explicitly depends
on time but does not depend on the states. Each state or output has a
mandatory attribute `label` which will be referred in the display section.
### 3.2 Graphs, curves and surfaces, graphs element
The code fragment describing the curve mentioned above is given in figure 5.
We only need a simple segment connecting the points $(0,-3.14)$ and
$(0,3.14)$. A point lying on this curve will have its ordinate in the interval
$[-3.14,3.14]$. This curve will not be drawn as it only serves as a constraint
for the value of $\theta_{0}$.
⬇
1<graphs>
2 <polyline label=”segment”>
3 <vertex x1=”0” x2=”-3.14”/>
4 <vertex x1=”0” x2=”3.14”/>
5 </polyline>
6</graphs>
Figure 5: Code fragment describing the segment joining the two points
$(0,-3.14)$ and $(0,3.14)$.
It is not needed to write further XML code to construct the curves of
$\theta(t)$ and $\phi(t)$ versus $t$, as it is possible to refer directly to
the labels `theta` and `theta_lin` in the `<drawcurve2d>` element (see next
section directly below).
### 3.3 Display of results, display element
We want to superimpose two curves in the same axes system, and display a
movable cross at the $(0,\theta_{0})$ coordinate. The XML code corresponding
to this is given in figure 6.
⬇
1<display>
2 <window>
3 <title>Comparison of the two solutions</title>
4 <axis2d>
5 <drawcurve2d ref=”theta”/>
6 <drawcurve2d ref=”thetalin”/>
7 <drawpoints ref=”point0”/>
8 </axis2d>
9 </window>
10<display>
Figure 6: Code fragment describing the display of the two solutions
The `display` element contains only one `window` element, containing itself a
two dimensional system of axes. The two `drawcurve2d` elements within the same
`axis2d` mean that the curves of $\theta$ and its harmonic version will be
superimposed. The `drawpoints` element refers to the `point` element defined
before in the `parameters` element.
### 3.4 Remarks
The different structuration possibilities are constrained by a DTD (Document
Type Definition), allowing an a posteriori validation of a simulation, or can
be used to constrain the edition of a simulation by means of an XML editor.
The figure 7 shows the view that the user can have of its XML file.
Figure 7: The XML file describing the simulation of the pendulum, seen in the
XXE editor, developed by PIXWARE, http://www.xmlmind.com/xmleditor
Within all of the above mentioned elements, some have a particular status. The
`name` and `title` elements can appear several times, with a different `lang`
attribute (`french` or `english` in XMLlab 1.3). The goal is to be able to
generate from the same XML file two different versions of the “compiled”
applet, the language to use being specified as a compilation option.
The `header` element contains some meta-data such as the name of the author
and some keywords. The `notes` element can appear several times with a
different `lang` attribute and allows to write a few paragraphs of text
allowing to describe the simulation and/or to give some help to the user.
## 4 The compilation chain
The compilation chain is entirely based on XML technologies: we use XSL
transformations specified in XSL stylesheets (eXtensible Stylesheet Language).
These transformations are applied to the simulation file by an “XSL
processor”. The XSL technology is well known to allow the display of dynamic
HTML on the World Wide Web, but it is also well fitted to the automatic
generation of scripts. We are here particularly interested in the script
language of Scilab or Matlab, and the script language Tcl/Tk, allowing to
describe graphical user interfaces.
Figure 8: Diagram of the compilation chain. The arrows represent the XSL
transformations, and the italic names the associated stylesheets.
The different phases of the compilation are outlined on the diagram depicted
on figure 8. In the bottom of the diagram, the `pendulum.sce` file is the
Scilab script containing all the computation code, and the display of results.
The `pendulum.tk` file contains the Tcl/Tk code of the interface. The diagram
illustrates the fact that the transformation is not direct and needs an
intermediary step; this particular point needs an explanation.
To allow an easy maintenance of the XSL stylesheets, and especially to allow a
smooth change of target languages (Scilab and Tcl/Tk), we have used “pivot”
XML dialects: the code is generated in a two-step process. We use XSL
transformation to translate XML input into a pseudo-Tcl/Tk and pseudo-Scilab
syntax that are also written using XML dialects. Then we use a second pass to
serialize the XML pseudo-language into the target language. The advantage here
is that the second pass captures all the complexities of formatting clean code
(syntax) while the first pass concentrates on the logical aspects of the
translation (semantics). We use the following dialects :
* •
TKML dialect : as far as the interface Tcl/Tk code is concerned (left side of
the diagram), the intermediary file `pendulum.tkml` is an XML file containing
a logical description of the interface. This file contains the description of
the different widgets (buttons, etc.) and their placement with respect to each
other. Then, this intermediate file is finally translated in Tcl/Tk by means
of a last XSL transformation.We show on figure 9 a small part of the generated
markup. Almost all the parameters will be appear in the graphical user
interface as classical entry widgets (corresponding to the “entry” widget in
Tcl/Tk), where the user has to type the value with the keyboard keys. The
`<scale>` element (lines 7 to 10 in figure 9) describes the widget which will
be used for the final time of the simulation. Since the original markup
describing this parameter in figure 3 gives bounds and an increment, the
logical way of taking into account these constraints is to use a widget with a
moving slider (in the final transformation we use the “scale” widget of
Tcl/Tk). Some aspects of the TKML markup are very similar to the XForms markup
language, which has been chosen by the W3C to develop the next generation of
forms technology for the world wide web, see e.g. [13].
⬇
1 <page name=”id66390” text=”Resolution␣parameters” pady=”4”>
2 <frame packside=”top” anchor=”n” pady=”0” padx=”0” fill=”x” expand=”yes”>
3 <frame packside=”left” padx=”5” expand=”true” fill=”x”>
4 <label anchor=”w” expand=”true”>
5 <text>Final time</text>
6 </label>
7 <scale anchor=”e” variable=”tf” state=”normal” width=”8” from=”1” to=”10”
resolution=”1”>
8 <value>2</value>
9 <command>runScilab</command>
10 </scale>
11 </frame>
12 </frame>
13 </page>
Figure 9: TKML intermediate markup corresponding to the definition of the
Scilab function computing the right-hand side of the system of ordinary
differential equations.
* •
SCIML dialect : for the Scilab code generation, we proceed in the same
manner: we first generate an intermediary file `pendulum.sciml`, written using
a pseudo-Scilab markup, and then transform this file to Scilab code with a
last transformation. The figure 10 shows a fragment of the actual content of
this file.
⬇
1 <function-definition name=”f_pendulum”>
2 <inputs>
3 <parm>_t</parm>
4 <parm>_X</parm>
5 </inputs>
6 <outputs>
7 <parm>lhs</parm>
8 </outputs>
9 <body>
10 <assign>
11 <lhs>t</lhs>
12 <rhs>_t</rhs>
13 </assign>
14 <assign>
15 <lhs>theta</lhs>
16 <rhs>
17 <select matrix=”_X” row=”1:1” col=”1”/>
18 </rhs>
19 </assign>
20 <assign>
21 <lhs>theta_dot</lhs>
22 <rhs>
23 <select matrix=”_X” row=”2:2” col=”1”/>
24 </rhs>
25 </assign>
26 <assign>
27 <lhs>lhs</lhs>
28 <rhs>
29 <list sep=”;”>
30 <parm>(theta_dot)</parm>
31 <parm>(-g0/L*sin(theta))</parm>
32 </list>
33 </rhs>
34 </assign>
35 </body>
36 </function-definition>
Figure 10: SCIML intermediate markup corresponding to the definition of the
Scilab function computing the right-hand side of the system of ordinary
differential equations.
We show on figure 11 a small part of the generated Scilab script
`pendulum.sce` appearing on figure 8, corresponding to the computation of the
right-hand side of the first order differential equation obtained for the
simulation of the pendulum, the call to the ode solver of Scilab and finally
the display of curves (for sake of simplicity we have omitted some parts of
the code).
⬇
1function [lhs]=f_pendulum(_t,_X)
2t=_t;
3theta=_X(1:1,1);
4theta_dot=_X(2:2,1);
5lhs=[(theta_dot);(-g0/L*sin(theta))];
6endfunction
7
8// Time
9t=linspace(0,tf,200)’;
10//␣Script␣code␣for␣the␣pendulum␣ode
11_X0(1:1,1)=theta_0;
12_X0(2:2,1)=0;
13_X=ode(_X0,0,t,f_pendulum);
14theta=_X(1:1,:)’;
15theta_dot=_X(2:2,:)’;
16thetalin=theta_0*cos(sqrt(g0/L)*t);
17
18//␣Display
19plot(t,theta,);
20hold(”on”);
21plot(t,thetalin,;
22hold(”off”);’
Figure 11: Some parts of the generated Scilab script for the pendulum
simulation.
Two ways of distribution can be used: the two Tcl/Tk and Scilab files can be
later used without using the XML source and the compilation chain (thus
protecting the author’s work). However, it would be more profitable to the
community to release the XML source.
The whole compilation chain together with Scilab (except the XML editor), uses
only open-source software packages (`xsltproc` of the Gnome project, Tcl
scripts), and works on any platform (Windows, Mac OS X, Unix).
## 5 The different ways of publishing an XMLlab simulation
In the previous section, we have described the classical way of publishing an
XMLlab simulation, by using a “compilation chain” allowing to transform the
original XML file `pendulum.xml` to a Scilab executable file `pendulum.sce`
and a Tk file `pendulum.tk` which will be run on a local client machine where
Scilab is installed. By using this kind of publication, the interactivity is
maximal; for the pendulum example, the user can move the cross representing
the initial condition and see in real time how the two trajectories diverge
(see figure 13).
### 5.1 Comments on the example
#### 5.1.1 Description of the generated graphical user interface
We make some comments on the figure 13.
* •
User interface : the window of the interface has a central space with a
Notebook-type widget with tabs and a menu bar:
* –
The two tabs named `Parameters of the pendulum` and `Resolution parameters`
correspond to the two parameters groups specified in the XML file. The user
just has to select a given tab to display the corresponding parameter group.
* –
The `Notes` tab gives some some information on the simulation, extracted from
the `header` element: name of the author, date and eventual notes describing
the simulation. The `XMLlab` tab gives some information on the XMLlab project.
* –
The `File` menu contains two interesting items: “Save a session” and “Load a
session”. They allow to save the values of parameters in a text file, and to
load them later. It allows to resume a working session (otherwise the Scilab
script always starts with the initial values of parameters).
* –
The `Languages` menu allows to switch between the languages of the simulation.
In fact, by sake of simplicity we didn’t show the textual elements for each
language in the pendulum example, but giving them in all desired languages
allows to dynamically switch between languages when running a simulation. For
example, when the default language (English here) and French are to be used, a
typical code fragment is the one given in figure 12.
⬇
1 <section>
2 <title>Resolution parameters</title>
3 <title lang=”french”>Paramètres de résolution</title>
4 <scalar label=”T” unit=”s”>
5 <name>Final time</name>
6 <name lang=”french”>Temps final</name>
7 <value>2</value>
8 </scalar>
9 </section>
Figure 12: Code fragment showing textual elements in different languages.
* •
Graphical window : the legend of the two curves is taken from the `name`
elements within the states `theta` and the output `theta_lin`. The abscissa
label is the name of the time variable `t`, and the ordinate label is the unit
(`unit` attribute of element `<state label="theta"`). This window belongs to
Scilab, thus the user has access to the usual menus allowing to save (e.g. in
EPS format) or print the figure.
The user has always the possibility to have access to all variables of the
simulation from the Scilab command line (parameters and results of the
simulation), which remains available during the simulation.
#### 5.1.2 Remark on performances
For this particular example (a system of two scalar ordinary differential
equations), the computation time is negligible compared to the time elapsed by
drawing the curves, and hence the user can see the immediate effect of the
initial angle on the synchronization of the curves. For more intensive
examples (e.g. resolution of a partial differential equation), the response
time can be greater, but the reactivity of the system is always good, even on
a lightweight system (1Ghz Pentium). For these reasons Scilab is really
appreciated, because the built-in functions are well optimized (linear and
nonlinear equations solving, differential equations, sparse matrix algebra,
vectorization of elementary functions for arrays, etc.). Moreover, Scilab
loads in a negligible time (compared to the important startup time of recent
versions of Matlab, because of the use of JAVA for the user interface).
$\begin{array}[]{cc}\includegraphics[width=173.44534pt]{image2}&\leavevmode\nobreak\
\leavevmode\nobreak\ \includegraphics[width=173.44534pt]{image3}\\\
\nobreak\leavevmode\hfil\\\
\begin{minipage}[c]{173.44534pt}\includegraphics[width=173.44534pt]{image5}\end{minipage}&\leavevmode\nobreak\
\leavevmode\nobreak\
\begin{minipage}[c]{173.44534pt}\includegraphics[width=173.44534pt]{image6}\end{minipage}\\\
\end{array}$ Figure 13: Screenshots of the various tabs of the pendulum
simulation running on a local client with Scilab. In the graphical window, the
user can move the cross shaped pointer and see the trajectory changes in real
time.
### 5.2 Offline batch publication of HTML pages
When a large number of simulations have to be deployed in an educational
context, it is possible to automatically generate a tree of HTML files
presenting the simulations e.g. sorted by categories. The HTML files, as well
as some screenshots of the graphical output and of the user interface are
automatically generated in a batch process which only takes a few minutes. The
user can then browse the different pages, take a look at the screenshots, read
the description and finally launch the chosen simulation. The typical HTML
page for a single simulation is given on figure 14. Note: this kind of
publication still needs Scilab on the client machines (see the examples
section of the XMLlab WWW site [14]).
Figure 14: An HTML page describing the pendulum simulation. This kind of page
is dedicated to serve the simulations to clients where Scilab is installed
### 5.3 Online distant publication using the XMLlab WebServer
Still in an educational context, but when it is not possible to have Scilab
installed on all the client machines, it is possible to publish the simulation
towards thin clients running a simple WWW browser. The drawbacks of such an
approach are already known: there is an interactivity loss, but simulations
can be deployed very fast.
We have chosen a rather classical architecture based on a HTTP server and the
Common Gateway Interface. The entry point is a CGI script (written in Tcl)
which processes the user requests (see figure 16). The collection of XML
simulations to be served are stored in the server (running any flavor of Unix)
and the only needed software is:
* •
Scilab with the XMLlab toolbox,
* •
An HTTP server, e.g. the Apache HTTP server,
* •
The VNC virtual X11 server,
Figure 15: Screenshot of the pendulum simulation served by the XMLlab
WebServer on a client machine running the Mozilla web browser on a Linux
machine.
For the pendulum example, the generated HTML page can be seen on figure 15.
The user can interact by changing the parameters values, browse the different
parameter sections and download a PDF version of the graphical output.
We now describe how a typical URL (see on top of figure 16) is processed :
when the first user request is processed, the CGI script associates a session
number to the client and performs the following tasks :
1. 1.
Retrieve the XML file (here `pendulum.xml`) and process it in the XMLlab
compilation chain. This produces two files, a Scilab file `pendulum.sce`
(dedicated to computations and graphical output) and a Tk file `pendulum.tk`
(dedicated to communications with the CGI script and HTML output).
2. 2.
Verify if the X11 VNC server is running (if not, a new server is launched),
and launch a Scilab instance which uses this display for its graphical output.
3. 3.
Run `pendulum.sce` and `pendulum.tk` into Scilab. The obtained result is an
HTML file `pendulum.html` together with image files corresponding with the
first run of the simulation with default values of the parameters.
4. 4.
The HTML output is sent back to the client’s WWW browser.
The red arrows in figure 16 denote tasks which are only done at first user
request. When the user interacts with the simulation, then parameter changes
are sent to the simulation running into Scilab by using the Tk send mechanism.
Figure 16: Internals of the XMLlab WebServer. Red arrows occur only at first
user request.
## 6 Trends and conclusions
The different parameters types and the mathematical objects presented in
section 2 are already present in XMLlab 1.6, but XMLlab is a work in constant
progress, and many extensions and improvements are necessary. However, we
think that the choices we have made are valid, especially concerning the
structuring of the simulations and the architecture of the compilations chain.
The needed development time to add a new type of equation is always limited:
for example, the `stationary-pde` element, allowing to describe an elliptic
partial differential equation (extension of the DTD and associated XSL
stylesheet sections), has been developed in two days (we rely on a Scilab “PDE
toolbox”).
The XMLlab WebServer allowing dynamic publication of the simulation towards
thin clients is the most recent feature we have developed in XMLlab. Since we
have opened the possibility to encapsulate Scilab scripts (making only
computations) the XMLlab WebServer feature is giving to Scilab the equivalent
of what the Matlab WebServer (this product is discontinued) used to provide to
Matlab, but with a completely different approach, since the user doesn’t have
to write any line of HTML. In this case, complex Scilab scripts, eventually
computing some new values of the parameters (see e.g. the linear regression
example in the Mathematics section of XMLlab examples) are embedded in a
`script` element, replacing the `compute` element. Of course all the other
high level elements are present, and allow to do the same multi-medium
publication as for pure XML simulation files (see e.g. the Discrete Cosine
Transform and the Commutation Angles simulations in the examples).
The future developments of XMLlab mainly focus on, discrete time systems
simulation, stochastic systems, generation of Adobe Flash animations, sound
output, and so on. As far as the edition of the XML files is concerned, we
plan to use “Cascading Stylesheets” allowing to edit XML files in a very user-
friendly way in the XXE editor (see [15]). We also plan to migrate the actual
DTD to an XML Schema ([16]), to allow some enhancements in the control of
validity of the different data types contained in elements and attributes. In
the current XMLlab release, this control is made in the XSL stylesheets.
At the time we are writing this paper, XMLlab has already been used for 3
years in chemistry courses at the UTC (150 students by semester), under the
form of demonstrations during the course, and during the labs, together with
experimental acid/alkali titrations. With the help of simulation the students
interpret the experimental curves and are able to answer reasoning questions.
The software is also available in the UTC intranet to allow students to
improve their understanding of phenomenons. A sample survey has been made by
e-mail at the end of each semester and allowed us to conclude that the
software is easy to use, and students do not encounter major technical
problems. The software allows a better understanding of complex phenomenon and
acid/alkali equilibrium, but high level labs assistants are needed (they must
be trained on the software before the labs). XMLlab is also used at the
University Of Picardie Jules Verne since 2007, in the systems biology courses.
XMLlab has also been integrated in the prize-winning SCENARI Platform
editorial chain (SCENARI Platform is an open source application suite for
designing digital editing chains, used for creating professional standard
multimedia documents, see [17, 18]).
Of course, many other examples of simulations have been written to show that
XMLlab can be used in various disciplinary fields : we have examples in the
fields of biology (microbial and enzymatic kinetics), physics (pendulum,
oscillators, two-body system, Poisson equation, etc.), chemistry (Acid/Alkali
equilibriums, chemical kinetics), chemical engineering (ideal reactors), etc.
The complete list of available examples is given in appendix. All these
simulations are available in French, English and Spanish, and the translation
to German is in progress. We hope that a lot of people will contribute and
request some new features, which will help us to make XMLlab fit to new
disciplinary fields.
XMLlab is available at the address `http://xmllab.org`, as a Scilab toolbox
under the GPL license. The distribution is available for all popular platforms
(Windows, Linux, Solaris, Mac OS X) and well integrated, e.g. the applets can
be run directly by double clicking the icon of the file, without having to
launch Scilab before.
The documentation is available in French and English, under the form of a
quick start guide and a reference manual.
XMLlab has been accepted as a contribution by the Scilab Consortium in June
2004, and one of the two authors is now an official contributor member of the
Consortium, also sitting at the steering committee.
## References
* [1] J.-P. Chancelier, F. Delebecque, C. Gomez, R. Goursat, M.and Nikoukah, S. Steer, Introduction à Scilab, Springer, Paris, 2001.
* [2] C. E. Gomez, Engineering and scientific computing with scilab, Birkauser, Boston, 1999.
* [3] P. Motta Pires, D. Rogers, Free/opensource software: an alternative of engineering students, Procceding of the 32nd ASEE/IEEE Frontiers in Education Conference, November 6 - 9.
* [4] T. Bray, J. Paoli, C. e. a. Sperberg-McQueen, Extensible markup language (xml) 1.0, Available via the World Wide Web at http://www.w3.org/TR/2004/REC-xml-20040204.
* [5] M. Hucka, A. Finney, H. Sauro, H. Bolouri, J. Doyle, al., The systems biology markup language (sbml): a medium for representation and exchange of biochemical netwok models, Bioinformatics 19 (2003) 524–531.
* [6] M. Hucka, A. Finney, Systems biology markup language: Level 2 and beyond, Biochem. Soc. Trans. 31 (2003) 1472–1473.
* [7] G. Collecutt, P. Drummond, J. Hope, P. Cochrane, Extensible multi-dimensional simulator (xmds), Available via the World Wide Web at http://www.physics.uq.edu.au/xmds/index.html.
* [8] W. Borutzky, a novel xml format for the exchange and the reuse of bond graph models of engineering systems, Simulation Modelling Practice and Theory 14 (2006) 787–808.
* [9] J. Ousterhout, Scripting: Higher-level programming for the 21st century, IEEE Computer magazine March (1998) 23–30.
* [10] J. Ousterhout, Tcl and the Tk toolkit, Addison-Wesley, 1994.
* [11] S. Mottelet, A. Pauss, Xmllab : un outil générique de simulation basé sur xml et scilab, in: Proceedings of TICE 2004 conference, Université de Technologie de Compiègne, [OAI : oai:edutice.archives-ouvertes.fr:edutice-00000726_v1] - http:`//`edutice.archives-ouvertes.fr/edutice-00000726, 2004, pp. 391–399.
* [12] D. Carlisle, P. Ion, D. Miner, N. Poppelier, Mathematical markup language (mathml) version 2.0 (second edition) recommandation, Available via the World Wide Web at http://www.w3.org/TR/MathML.
* [13] J. Boyer, D. Landwher, R. Merrick, T. Raman, L. Dubinko, L. Klotz, Xforms 1.0 (second edition), Available via the World Wide Web at http://www.w3.org/TR/xforms.
* [14] S. Mottelet, A. Pauss, Xmllab web site, http://xmllab.org.
* [15] B. Bos, H. Wium Lie, C. Lilley, I. E. Jacobs, ascading stylesheets, level 2, css2 specification, Available via the World Wide Web at http://www.w3.org/TR/1998/REC-CSS2-19980512.
* [16] H. Thompson, D. Beech, M. Maloney, N. Mendelsohn, Xml schema part 1: Structures, Available via the World Wide Web at http://www.w3.org/TR/xmlschema-1.
* [17] B. Bachimont, I. Cailleau, M. Crozat, S.and Majada, S. Spinelli, Le procédé scenari : Une chaîne éditoriale pour la production de supports numériques de formation, in: Proceedings of TICE 2002 conference, INSA Lyon, 2002, pp. 183–192.
* [18] S. Crozat, S. Spinelli, Scenari paltform web site, http://scenari-platform.org/projects/scenari/en/pres/co/.
## Acknowledgments
This work has been financed by the groupment “Evaluation of New Technologies
in Education” of the regional Council of Research from Picardie and by the
UNIT consortium (French Numeric University of Engineering and Technology).
IUFM’s teachers and Jules Vernes University of Picardie Professors have
contributed efficiently to the trandisciplinarity of this project.
## Appendix A List of the simulations examples available at the XMLlab WWW
site: xmllab.org
* •
Image Processing
* –
Discrete Cosine Transform.
* •
Engineering
* –
Commutation angles.
* •
Physics
* –
Damped Oscillator. Pendulum. Pendulum (with animation). Earth-Moon system.
Simulation of the Laplace equation.
* •
Maths
* –
Lissajous curve. A simple surface example. Tangent. Osculating polynomials.
Gaussian mix. Lagrange Interpolation. Cycloïd. Linear regression. Inversion of
a matrix. Helix. System of differential equations.
* •
Predation
* –
Kinetics of prey predation by predators. Kinetics of rabbit predation by
foxes.
* •
Chemical kinetics
* –
Nonreversible reaction. Reversible reaction(equilibrium). Simultaneous
reactions (in parallel). Successive reactions with an equilibrium.
* •
Enzymatic kinetics.
* –
Enzymatic kinetics with a competitive inhibition. Enzymatic kinetics with a
incompetitive inhibition. Enzymatic kinetics with a non-competitive
inhibition. Michaelis-Menten’s enzymatic kinetics.
* •
Microbial kinetics
* –
Graef-Andrews’s growth kinetics. Monod’s growth kinetics.
* •
pH titrations
* –
Simulation of acid-alkali titration in water. Simulation of a single acid
titration.
|
arxiv-papers
| 2011-02-28T17:16:06 |
2024-09-04T02:49:17.370900
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "St\\'ephane Mottelet and Andr\\'e Pauss",
"submitter": "St\\'ephane Mottelet",
"url": "https://arxiv.org/abs/1102.5711"
}
|
1103.0090
|
# Wavelet packets and wavelet frame packets on local fields
Biswaranjan Behera (B. Behera) Theoretical Statistics and Mathematics Unit,
Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India
biswa@isical.ac.in and Qaiser Jahan (Q. Jahan) Theoretical Statistics and
Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata,
700108, India qaiser_r@isical.ac.in
###### Abstract.
Using a prime element of a local field $K$ of positive characteristic $p$, the
concepts of multiresolution analysis (MRA) and wavelet can be generalized to
such a field. We prove a version of the splitting lemma for this setup and
using this lemma we have constructed the wavelet packets associated with such
MRAs. We show that these wavelet packets generate an orthonormal basis by
translations only. We also prove an analogue of splitting lemma for frames and
construct the wavelet frame packets in this setting.
###### Key words and phrases:
Wavelet, Multiresolution analysis, Local field, $p$-series field, Wavelet
packet, Wavelet frame packet
###### 2000 Mathematics Subject Classification:
Primary: 42C40; Secondary: 42C15, 43A70, 11S85
Research of the second author is supported by a grant from CSIR, India.
## 1\. Introduction
The concepts of wavelet and multiresolution analysis of $\mathbb{R}^{n}$ has
been extended to many different setups. Dahlke [11] introduced it in locally
compact groups (see also [12, 19, 20, 21]). It was generalized to abstract
Hilbert spaces by Han, Larson, Papadakis, Stavropoulos [13, 26]. Lemarie [22]
extended this concept to stratified Lie groups.
Recently, R. L. Benedetto and J. J. Benedetto ([3, 4]) developed a wavelet
theory for local fields and related groups. Albeverio, Kozyrev, Khrennikov,
Shelkovich, Skopina and their collaborators also discuss about MRA and
wavelets on the $p$-adic field $\mathbb{Q}_{p}$ in a series of papers [1, 16,
17, 18]. Note that $\mathbb{Q}_{p}$ is a local field of characteristic $0$.
Jiang, Li and Jin [15] gave a definition of MRA on a local field of positive
characteristic $p$ and, similar to $\mathbb{R}^{n}$, have constructed the
wavelets from an MRA.
In this article we construct the wavelet packets associated with such an MRA.
We also generalize the wavelet frame packets to this setup. First of all, we
will discuss about wavelet packets very briefly.
Let $\\{V_{j}:j\in\mathbb{Z}\\}$ be an MRA of $L^{2}(\mathbb{R})$ with scaling
function $\varphi$ and wavelet $\psi$. Let $W_{j}$ be the corresponding
wavelet subspaces: $W_{j}=\overline{\rm span}\\{\psi_{jk}:k\in\mathbb{Z}\\}$.
In the construction of a wavelet from an MRA, essentially the space $V_{1}$ is
split into two orthogonal components $V_{0}$ and $W_{0}$. Note that $V_{1}$ is
the closure of the linear span of the functions
$\\{2^{1/2}\varphi(2\cdot-k):k\in\mathbb{Z}\\}$, whereas $V_{0}$ and $W_{0}$
are respectively the closure of the span of
$\\{\varphi(\cdot-k):k\in\mathbb{Z}\\}$ and
$\\{\psi(\cdot-k):k\in\mathbb{Z}\\}$. Since
$\varphi(2\cdot-k)=\varphi\left(2(\cdot-2^{-1}k)\right)$, we see that the
above procedure splits the half integer translates of a function into integer
translates of two functions.
In a similar way, we can split $W_{j}$, which is the span of
$\\{\psi(2^{j}\cdot-k):k\in\mathbb{Z}\\}=\\{\psi\left(2^{j}(\cdot-2^{-j}k)\right):k\in\mathbb{Z}\\}$,
to get two functions whose $2^{-(j-1)}k$ translates will span the same space
$W_{j}$. Repeating the splitting procedure $j$ times, we get $2^{j}$ functions
whose integer translates alone span the space $W_{j}$. If we apply this to
each $W_{j}$, then the resulting basis of $L^{2}(\mathbb{R})$ will consist of
integer translates of a countable number of functions (instead of all
dilations and translations of the wavelet $\psi$). This basis is called the
“wavelet packet basis”. The concept of wavelet packets was introduced by
Coifman, Meyer and Wickerhauser [9, 10]. For a nice exposition of wavelet
packets of $L^{2}(\mathbb{R})$ with dilation 2, we refer to [14].
The concept of wavelet packet was subsequently generalized to $\mathbb{R}^{n}$
by taking tensor products [8]. The non-tensor product versions are due to Shen
[25] for dyadic dilation, and Behera [2] for MRAs with a general dilation
matrix and several scaling functions. Other notable generalizations are the
biorthogonal wavelet packets [7], non-orthogonal version of wavelet packets
[6], the wavelet frame packets [5] on $\mathbb{R}$ for dilation 2, and the
orthogonal, biorthogonal and frame wavelet packets on $\mathbb{R}^{n}$ by Long
and Chen [23] for the dyadic dilation.
We have organized the article as follows. In section 2, we discuss some
preliminary facts about local fields. In section 3, we introduce the concept
of MRA on a local field $K$ of positive characteristic and prove a crucial
lemma called the splitting lemma. We construct the wavelet packets in section
4 and prove that they generate an orthonormal basis for $L^{2}(K)$. In section
5, we prove some basic results needed to prove an analogue of splitting lemma
for wavelet frames on $K$ and finally in section 6 the wavelet frame packets
are constucted.
## 2\. Preliminaries on local fields
Let $K$ be a field and a topological space. Then $K$ is called a _locally
compact field_ or a _local field_ if both $K^{+}$ and $K^{*}$ are locally
compact abelian groups, where $K^{+}$ and $K^{*}$ denote the additive and
multiplicative groups of $K$ respectively.
If $K$ is any field and is endowed with the discrete topology, then $K$ is a
local field. Further, if $K$ is connected, then $K$ is either $\mathbb{R}$ or
$\mathbb{C}$. If $K$ is not connected, then it is totally disconnected. So by
a local field, we mean a field $K$ which is locally compact, nondiscrete and
totally disconnected.
We use the notation of the book by Taibleson [27]. Proofs of all the results
stated in this section can be found in the books [27] or [24].
Let $K$ be a local field. Since $K^{+}$ is a locally compact abelian group, we
choose a Haar measure $dx$ for $K^{+}$. If $\alpha\neq 0,\alpha\in K$, then
$d(\alpha x)$ is also a Haar measure. Let $d(\alpha x)=|\alpha|dx$. We call
$|\alpha|$ the _absolute value_ or _valuation_ of $\alpha$. We also let
$|0|=0$.
The map $x\rightarrow|x|$ has the following properties:
* (a)
$|x|=0$ if and only if $x=0$;
* (b)
$|xy|=|x||y|$ for all $x,y\in K$;
* (c)
$|x+y|\leq\max\\{|x|,|y|\\}$ for all $x,y\in K$.
Property (c) is called the _ultrametric inequality_.
The set $\mathfrak{D}=\\{x\in K:|x|\leq 1\\}$ is called the _ring of integers_
in $K$. It is the unique maximal compact subring of $K$. Define
$\mathfrak{P}=\\{x\in K:|x|<1\\}$. The set $\mathfrak{P}$ is called the _prime
ideal_ in $K$. The prime ideal in $K$ is the unique maximal ideal in
$\mathfrak{D}$. It is principal and prime.
Since $K$ is totally disconnected, the set of values $|x|$ as $x$ varies over
$K$ is a discrete set of the form $\\{s^{k}:k\in\mathbb{Z}\\}\cup\\{0\\}$ for
some $s>0$. Hence, there is an element of $\mathfrak{P}$ of maximal absolute
value. Let $\mathfrak{p}$ be a fixed element of maximum absolute value in
$\mathfrak{P}$. Such an element is called a _prime element_ of $K$. Note that
as an ideal in
$\mathfrak{D},\mathfrak{P}=\left\langle\mathfrak{p}\right\rangle=\mathfrak{p}\mathfrak{D}$.
It can be proved that $\mathfrak{D}$ is compact and open. Hence,
$\mathfrak{P}$ is compact and open. Therefore, the residue space
$\mathfrak{D}/\mathfrak{P}$ is isomorphic to a finite field $GF(q)$, where
$q=p^{c}$ for some prime $p$ and $c\in\mathbb{N}$. For a proof of this fact we
refer to [27].
For a measurable subset $E$ of $K$, let $|E|=\int_{K}\chi_{E}(x)dx$, where
$\chi_{E}$ is the characteristic function of $E$ and $dx$ is the Haar measure
of $K$ normalized so that $|\mathfrak{D}|=1$. Then, it is easy to see that
$|\mathfrak{P}|=q^{-1}$ and $|\mathfrak{p}|=q^{-1}$ (see [27]). It follows
that if $x\neq 0$, and $x\in K$, then $|x|=q^{k}$ for some $k\in\mathbb{Z}$.
Let $\mathfrak{D}^{*}=\mathfrak{D}\setminus\mathfrak{P}=\\{x\in K:|x|=1\\}$.
$\mathfrak{D}^{*}$ is the group of units in $K^{*}$. If $x\neq 0$, we can
write $x=\mathfrak{p}^{k}x^{\prime}$, with $x^{\prime}\in\mathfrak{D}^{*}$.
Recall that $\mathfrak{D}/\mathfrak{P}\cong GF(q)$. Let
$\mathcal{U}=\\{a_{i}\\}_{i=0}^{q-1}$ be any fixed full set of coset
representatives of $\mathfrak{P}$ in $\mathfrak{D}$. Let
$\mathfrak{P}^{k}=\mathfrak{p}^{k}\mathfrak{D}=\\{x\in K:|x|\leq
q^{-k}\\},k\in\mathbb{Z}$. These are called _fractional ideals_. Each
$\mathfrak{P}^{k}$ is compact and open and is a subgroup of $K^{+}$ (see
[24]). Then, if $x\in\mathfrak{P}^{k},k\in\mathbb{Z}$, $x$ can be expressed
uniquely as $x=\sum_{l=k}^{\infty}c_{l}\mathfrak{p}^{l},c_{l}\in\mathcal{U}$.
If $K$ is a local field, then there is a nontrivial, unitary, continuous
character $\chi$ on $K^{+}$. It can be proved that $K^{+}$ is self dual (see
[27]).
Let $\chi$ be a fixed character on $K^{+}$ that is trivial on $\mathfrak{D}$
but is nontrivial on $\mathfrak{P}^{-1}$. We can find such a character by
starting with any nontrivial character and rescaling. We will define such a
character for a local field of positive characteristic. For $y\in K$, we
define $\chi_{y}(x)=\chi(yx)$, $x\in K$.
###### Definition 1.
If $f\in L^{1}(K)$, then the Fourier transform of $f$ is the function
$\hat{f}$ defined by
$\hat{f}(\xi)=\int_{K}f(x)\overline{\chi_{\xi}(x)}~{}dx.$
Note that
$\hat{f}(\xi)=\int_{K}f(x)\overline{\chi(\xi x)}~{}dx=\int_{K}f(x)\chi(-\xi
x)~{}dx.$
Similar to the standard Fourier analysis on the real line, one can prove the
following results.
* (a)
The map $f\rightarrow\hat{f}$ is a bounded linear transformation of $L^{1}(K)$
into $L^{\infty}(K)$, and $\|\hat{f}\|_{\infty}\leq\|f\|_{1}$.
* (b)
If $f\in L^{1}(K)$, then $\hat{f}$ is uniformly continuous.
* (c)
$f\in L^{1}\cap L^{2}(K)$, then $\|\hat{f}\|_{2}=\|f\|_{2}$.
To define the Fourier transform of function in $L^{2}(K)$, we introduce the
functions $\Phi_{k}$. For $k\in\mathbb{Z}$, let $\Phi_{k}$ be the
characteristic function of $\mathfrak{P}^{k}$.
###### Definition 2.
For $f\in L^{2}(K)$, let $f_{k}=f\Phi_{-k}$ and
$\hat{f}(\xi)=\lim\limits_{k\rightarrow\infty}\hat{f}_{k}(\xi)=\lim\limits_{k\rightarrow\infty}\int_{\left|x\right|\leq
q^{k}}f(x)\overline{\chi_{\xi}(x)}~{}d\xi,$
where the limit is taken in $L^{2}(K)$.
We have the following theorem (see Theorem 2.3 in [27]).
###### Theorem 1.
The fourier transform is unitary on $L^{2}(K)$.
Let $\chi_{u}$ be any character on $K^{+}$. Since $\mathfrak{D}$ is a subgroup
of $K^{+}$, the restriction $\chi_{u}|_{\mathfrak{D}}$ is a character on
$\mathfrak{D}$. Also, as character on $\mathfrak{D},\chi_{u}=\chi_{v}$ if and
only if $u-v\in\mathfrak{D}$. That is, $\chi_{u}=\chi_{v}$ if
$u+\mathfrak{D}=v+\mathfrak{D}$ and $\chi_{u}\neq\chi_{v}$ if
$(u+\mathfrak{D})\cap(v+\mathfrak{D})=\phi$. Hence, if
$\\{u(n)\\}_{n=0}^{\infty}$ is a complete list of distinct coset
representative of $\mathfrak{D}$ in $K^{+}$, then
$\\{\chi_{u(n)}\\}_{n=0}^{\infty}$ is a list of distinct characters on
$\mathfrak{D}$. It is proved in [27] that this list is complete. That is, we
have the following proposition.
###### Proposition 1.
Let $\\{u(n)\\}_{n=0}^{\infty}$ be a complete list of (distinct) coset
representatives of $\mathfrak{D}$ in $K^{+}$. Then
$\\{\chi_{u(n)}\\}_{n=0}^{\infty}$ is a complete list of (distinct) characters
on $\mathfrak{D}$. Moreover, it is a complete orthonormal system on
$\mathfrak{D}$.
Given such a list of characters $\\{\chi_{u(n)}\\}_{n=0}^{\infty}$, we define
the Fourier coefficients of $f\in L^{1}(\mathfrak{D})$ as
$\hat{f}(u(n))=\int_{\mathfrak{D}}f(x)\overline{\chi_{u(n)}(x)}dx.$
The series $\sum\limits_{n=0}^{\infty}\hat{f}(u(n))\chi_{u(n)}(x)$ is called
the Fourier series of $f$. From the standard $L^{2}$ theory for compact
abelian groups we conclude that the Fourier series of $f$ converges to $f$ in
$L^{2}(\mathfrak{D})$ and
$\int_{\mathfrak{D}}|f(x)|^{2}dx=\sum\limits_{n=0}^{\infty}|\hat{f}(u(n))|^{2}.$
Also, if $f\in L^{1}(\mathfrak{D})$ and $\hat{f}(u(n))=0$ for all
$n\in\mathbb{N}_{0}$, then $f=0$ a. e.
These results hold irrespective of the ordering of the characters. We now
proceed to impose a natural order on the sequence $\\{u(n)\\}_{n=0}^{\infty}$.
Note that $\Gamma=\mathfrak{D}/\mathfrak{P}$ is isomorphic to the finite field
$GF(q)$ and $GF(q)$ is a $c$-dimensional vector space over the field $GF(p)$.
We choose a set
$\\{1=\epsilon_{0},\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{c-1}\\}\subset\mathfrak{D}^{*}$
such that span$\\{\epsilon_{j}\\}_{j=0}^{c-1}\cong GF(q)$. Let
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. For $n\in\mathbb{N}_{0}$ such that
$0\leq n<q$, we have
$n=a_{0}+a_{1}p+\cdots+a_{c-1}p^{c-1},\quad 0\leq a_{k}<p,k=0,1,\cdots,c-1.$
Define
(1)
$u(n)=(a_{0}+a_{1}\epsilon_{1}+\cdots+a_{c-1}\epsilon_{c-1})\mathfrak{p}^{-1}.$
Now, write
$n=b_{0}+b_{1}q+b_{2}q^{2}+\cdots+b_{s}q^{s},\quad 0\leq
b_{k}<q,k=0,1,2,\cdots,s,$
and define
$u(n)=u(b_{0})+u(b_{1})\mathfrak{p}^{-1}+\cdots+u(b_{s})\mathfrak{p}^{-s}.$
Note that $u(0)=0$ and $\\{u(n)\\}_{n=0}^{q-1}$ is a complete set of coset
representatives of $\mathfrak{D}$ in $\mathfrak{P}^{-1}$ (see [27]). Hence,
$\\{u(n)\mathfrak{p}\\}_{n=0}^{q-1}$ is a complete set of coset
representatives of $\mathfrak{P}$ in $\mathfrak{D}$. Therefore,
$\\{u(n)\mathfrak{p}\\}_{n=0}^{q-1}\cong\mathfrak{D}/\mathfrak{P}\cong
GF(q)\cong{\rm span}\\{\epsilon_{j}\\}_{j=0}^{c-1}.$
In general, it is not true that $u(m+n)=u(m)+u(n)$. But
(2) $u(rq^{k}+s)=u(r)\mathfrak{p}^{-k}+u(s)\quad{\rm if}~{}r\geq 0,k\geq
0~{}{\rm and}~{}0\leq s<q^{k}.$
For brevity, we will write $\chi_{n}=\chi_{u(n)},n\geq 0$. As mentioned
before, $\\{\chi_{n}\\}_{n=0}^{\infty}$ is a complete set of characters on
$\mathfrak{D}$.
Let $\mathcal{U}=\\{a_{i}\\}_{i=0}^{q-1}$ be a fixed set of coset
representatives of $\mathfrak{P}$ in $\mathfrak{D}$. Then every $x\in K$ can
be expressed uniquely as
$x=x_{0}+\sum\limits_{k=1}^{n}b_{k}\mathfrak{p}^{-k},\quad
x_{0}\in\mathfrak{D},b_{k}\in\mathcal{U}.$
Let $K$ be a local field characteristic $p>0$ and
$\epsilon_{0},\epsilon_{1},\dots,\epsilon_{c-1}$ be as above. We define a
character $\chi$ on $K$ as follows:
(3)
$\chi(\epsilon_{\mu}\mathfrak{p}^{-j})=\left\\{\begin{array}[]{lll}\exp(2\pi
i/p),&\mu=0~{}\mbox{and}~{}j=1,\\\ 1,&\mu=1,\cdots,c-1~{}\mbox{or}~{}j\neq
1.\end{array}\right.$
Note that $\chi$ is trivial on $\mathfrak{D}$ but nontrivial on
$\mathfrak{P}^{-1}$.
In order to be able to define the concepts of multiresolution analysis and
wavelets on local fields, we need analogous notions of translation and
dilation. Since
$\bigcup\limits_{j\in\mathbb{Z}}\mathfrak{p}^{-j}\mathfrak{D}=K,$ we can
regard $\mathfrak{p}^{-1}$ as the dilation (note that $|\mathfrak{p}^{-1}|=q$)
and since $\\{u(n):n\in\mathbb{N}_{0}\\}$ is a complete list of distinct coset
representatives of $\mathfrak{D}$ in $K$, the set
$\\{u(n):n\in\mathbb{N}_{0}\\}$ can be treated as the translation set. So we
make the following definition.
###### Definition 3.
A finite set $\\{\psi_{m}:m=1,2,\cdots,M\\}\subset L^{2}(K)$ is called a _set
of basic wavelets_ of $L^{2}(K)$ if the system
$\\{q^{j/2}\psi_{m}(\mathfrak{p}^{-j}\cdot-u(k)):1\leq m\leq
M,j\in\mathbb{Z},k\in\mathbb{N}_{0}\\}$ forms an orthonormal basis for
$L^{2}(K)$.
## 3\. Multiresolution analysis on local fields and the splitting lemma
Similar to $\mathbb{R}^{n}$, wavelets can be constructed from a
multiresolution analysis which we define below (see [15]).
###### Definition 4.
Let $K$ be a local field of characteristic $p>0$, $\mathfrak{p}$ be a prime
element of $K$ and $u(n)\in K$ for $n\in\mathbb{N}_{0}$ be as defined above. A
multiresolution analysis (MRA) of $L^{2}(K)$ is a sequence
$\\{V_{j}\\}_{j\in\mathbb{Z}}$ of closed subspaces of $L^{2}(K)$ satisfying
the following properties:
1. (a)
$V_{j}\subset V_{j+1}$ for all $j\in\mathbb{Z}$;
2. (b)
$\bigcup\limits_{j\in\mathbb{Z}}V_{j}$ is dense in $L^{2}(K)$ and
$\bigcap\limits_{j\in\mathbb{Z}}V_{j}=\\{0\\}$;
3. (c)
$f\in V_{j}$ if and only if $f(\mathfrak{p}^{-1}\cdot)\in V_{j+1}$ for all
$j\in\mathbb{Z}$;
4. (d)
there is a function $\varphi\in V_{0}$, called the _scaling function_ , such
that $\\{\varphi(\cdot-u(k)):k\in\mathbb{N}_{0}\\}$ forms an orthonormal basis
for $V_{0}$.
Given an MRA $\\{V_{j}:j\in\mathbb{Z}\\}$, we define another sequence
$\\{W_{j}:j\in\mathbb{Z}\\}$ of closed subspaces of $L^{2}(K)$ by
$W_{j}=V_{j+1}\ominus V_{j}.$
These subspaces also satisfy
(4) $f\in W_{j}~{}{\rm if~{}and~{}only~{}if}~{}f(\mathfrak{p}^{-1}\cdot)\in
W_{j+1},~{}j\in\mathbb{Z}.$
Moreover, they are mutually orthogonal, and we have the following orthogonal
decompositions:
(5) $\displaystyle L^{2}(K)$ $\displaystyle=$
$\displaystyle\bigoplus\limits_{j\in\mathbb{Z}}W_{j}$ (6) $\displaystyle=$
$\displaystyle V_{0}\oplus\Bigl{(}\bigoplus\limits_{j\geq 0}W_{j}\Bigr{)}.$
Observe that the dilation is induced by $\mathfrak{p}^{-1}$ and
$\left|\mathfrak{p}^{-1}\right|=q$. As in the case of $\mathbb{R}^{n}$, we
expect the existence of $q-1$ number of functions
$\\{\psi_{1},\psi_{2},\cdots,\psi_{q-1}\\}$ to form a set of basic wavelets.
In view of (4) and (5), it is clear that if $\\{\psi_{1},\cdots,\psi_{q-1}\\}$
is a set of function such that $\\{\psi_{m}(\cdot-u(k)):1\leq m\leq
M,k\in\mathbb{N}_{0}\\}$ forms an orthonormal basis for $W_{0}$, then
$\\{q^{j/2}\psi_{m}(\mathfrak{p}^{-j}\cdot-u(k)):1\leq m\leq
M,j\in\mathbb{Z},k\in\mathbb{N}_{0}\\}$ forms an orthonormal basis for
$L^{2}(K)$.
For $f\in L^{2}(K)$, we define
$f_{j,k}=q^{j/2}f(\mathfrak{p}^{-j}x-u(k)),\quad
j\in\mathbb{Z},k\in\mathbb{N}_{0}.$
Then it is easy to see that
$\|f_{j,k}\|_{2}=\|f\|_{2}$
and
$(f_{j,k})^{\wedge}(\xi)=q^{-j/2}\overline{\chi_{k}(\mathfrak{p}^{j}\xi)}\hat{f}(\mathfrak{p}^{j}\xi).$
Since $\varphi\in V_{0}\subset V_{1}$, and
$\\{\varphi_{1,k}:k\in\mathbb{N}_{0}\\}$ is an orthonormal basis in $V_{1}$,
we have
$\varphi(x)=\sum\limits_{k\in\mathbb{N}_{0}}h_{k}q^{1/2}\varphi(\mathfrak{p}^{-1}x-u(k)),$
where $h_{k}=\langle\varphi,\varphi_{1,k}\rangle$ and
$\\{h_{k}:k\in\mathbb{N}_{0}\\}\in\ell^{2}(\mathbb{N}_{0})$. Taking Fourier
transform, we get
(7) $\displaystyle\hat{\varphi}(\xi)$ $\displaystyle=$ $\displaystyle
q^{-1/2}\sum\limits_{k\in\mathbb{N}_{0}}h_{k}\overline{\chi_{k}(\mathfrak{p}\xi)}\hat{\varphi}(\mathfrak{p}\xi)$
$\displaystyle=$ $\displaystyle
m_{0}(\mathfrak{p}\xi)\hat{\varphi}(\mathfrak{p}\xi),$
where
$m_{0}=q^{-1/2}\sum\limits_{k\in\mathbb{N}_{0}}h_{k}\overline{\chi_{k}(\xi)}$.
Let us call a function $f$ on $K$ _integral-periodic_ if
$f(x+u(k))=f(x)~{}\mbox{for all}~{}k\in\mathbb{N}_{0}.$
The following facts were proved in [15].
* (a)
$\chi_{k}(u(l))=\chi(u(k)u(l))=1$ for all $k,l\in\mathbb{N}_{0}$.
* (b)
The function $m_{0}$ is integral-periodic.
* (c)
The system $\\{\varphi(\cdot-u(k)):k\in\mathbb{N}_{0}\\}$ is orthonormal if
and only if
$\sum\limits_{k\in\mathbb{N}_{0}}\left|\widehat{\varphi}(\xi+u(k))\right|^{2}=1$
a.e.
Given an MRA of $L^{2}(K)$, suppose that there exist $q-1$ integral-periodic
functions $m_{l}$, $1\leq l\leq q-1$, such that the matrix
$M(\xi)=\Big{(}m_{l}(\mathfrak{p}\xi+\mathfrak{p}u(k))\Big{)}_{l,k=0}^{q-1}$
is unitary. It was also proved in [15] that
$\\{\psi_{1},\psi_{2},\cdots,\psi_{q-1}\\}$ is a set of basic wavelets of
$L^{2}(K)$ if we define
$\hat{\psi}_{l}(\xi)=m_{l}(\mathfrak{p}\xi)\hat{\varphi}(\mathfrak{p}\xi).$
We now prove a lemma, the splitting lemma, which is essential for the
construction of wavelet packets. With the help of this lemma, we can decompose
a closed subspace of $L^{2}(K)$ into finitely many mutually orthogonal
subspaces in a suitable manner.
###### Lemma 1 (The splitting lemma).
Let $\varphi\in L^{2}(K)$ be such that
$\\{\varphi(\cdot-u(k)):k\in\mathbb{N}_{0}\\}$ is an orthonormal system. Let
$V=\overline{\rm
span}\\{q^{1/2}\varphi(\mathfrak{p}^{-1}\cdot-u(k)):k\in\mathbb{N}_{0}\\}$.
Let $m_{l}=q^{-1/2}\sum_{k=0}^{\infty}h_{k}^{l}\overline{\chi_{k}}(\xi)$,
$0\leq l\leq q-1$, where
$\\{h_{k}^{l}:k\in\mathbb{N}_{0}\\}\in\ell^{2}(\mathbb{N}_{0})$ for $0\leq
l\leq q-1$. Define
$\hat{\psi}_{l}(\xi)=m_{l}(\mathfrak{p}\xi)\hat{\varphi}(\mathfrak{p}\xi)$.
Then $\\{\psi_{l}(\cdot-u(k)):0\leq l\leq q-1,k\in\mathbb{N}_{0}\\}$ is an
orthonormal system in $V$ if and only if the matrix
$M(\xi)=\Bigl{(}m_{l}(\mathfrak{p}\xi+\mathfrak{p}u(k))\Bigr{)}_{l,k=0}^{q-1}$
is unitary for a.e $\xi\in\mathfrak{D}$.
Moreover, $\\{\psi_{l}(\cdot-u(k)):0\leq l\leq q-1,k\in\mathbb{N}_{0}\\}$ is
an orthonormal basis of $V$ whenever it is orthonormal.
###### Proof.
Assume that $M(\xi)$ is unitary for a.e $\xi\in\mathfrak{D}$. Then, for $0\leq
s,t\leq q-1$ and $k,l\in\mathbb{N}_{0}$, we have
$\displaystyle\Bigl{\langle}\psi_{s}\bigl{(}\cdot-u(k)\bigr{)},\psi_{t}\bigl{(}\cdot-u(l)\bigr{)}\Bigr{\rangle}$
$\displaystyle=$
$\displaystyle\Bigl{\langle}\Bigl{(}\psi_{s}\bigl{(}\cdot-u(k)\bigr{)}\Bigr{)}^{\wedge},\Bigl{(}\psi_{t}\bigl{(}\cdot-u(l)\bigr{)}\Bigr{)}^{\wedge}\Bigr{\rangle}$
$\displaystyle=$
$\displaystyle\int_{K}\hat{\psi}_{s}(\xi)\overline{\chi_{k}(\xi)}\overline{\hat{\psi}_{t}(\xi)}\chi_{l}(\xi)~{}d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{D}}\sum\limits_{n\in\mathbb{N}_{0}}\hat{\psi}_{s}\bigl{(}\xi+u(n)\bigr{)}\overline{\hat{\psi}_{t}\bigl{(}\xi+u(n)\bigr{)}}\overline{\chi_{k}(\xi)}\chi_{l}(\xi)~{}d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{D}}\sum\limits_{n\in\mathbb{N}_{0}}m_{s}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(n)\bigr{)}\overline{m_{t}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(n)\bigr{)}}\bigl{|}\hat{\varphi}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(n)\bigr{)}\bigr{|}^{2}\overline{\chi_{k}(\xi)}\chi_{l}(\xi)~{}d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{D}}\sum\limits_{\mu=0}^{q-1}\sum\limits_{n\in\mathbb{N}_{0}}m_{s}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(qn+\mu)\bigr{)}\overline{m_{t}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(qn+\mu)\bigr{)}}$
$\displaystyle\qquad\times\bigl{|}\hat{\varphi}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(qn+\mu)\bigr{)}\bigr{|}^{2}\overline{\chi_{k}(\xi)}\chi_{l}(\xi)~{}d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{D}}\Big{\\{}\sum\limits_{\mu=0}^{q-1}m_{s}(\mathfrak{p}\xi+\mathfrak{p}u(\mu))\overline{m_{t}(\mathfrak{p}\xi+\mathfrak{p}u(\mu))}\Big{\\}}\overline{\chi_{k}(\xi)}\chi_{l}(\xi)~{}d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{D}}\delta_{s,t}\overline{\chi_{k}(\xi)}\chi_{l}(\xi)~{}d\xi$
$\displaystyle=$ $\displaystyle\delta_{s,t}\delta_{k,l}.$
Hence, $\\{\psi_{s}(\cdot-u(k)):0\leq s\leq q-1,k\in\mathbb{N}_{0}\\}$ is an
orthonormal system in $V$. The converse can be proved by reversing the above
steps.
To prove the second part, let $f\in V$ be such that $f$ is orthogonal to
$\psi_{l}(\cdot-u(k))$ for all $l=0,1,\dots,q-1$, $k\in\mathbb{N}_{0}$. We
claim that $f=0$ a. e.
Since $f\in V$, we have
$f(x)=\sum\limits_{m\in\mathbb{N}_{0}}q^{1/2}c_{m}\varphi(\mathfrak{p}^{-1}x-u(m)),$
for some $\\{c_{m}:m\in\mathbb{N}_{0}\\}\in\ell^{2}(\mathbb{N}_{0})$. So there
exists an integral periodic function $m_{f}$ in $L^{2}(\mathfrak{D})$ such
that
$\hat{f}(\xi)=m_{f}(\mathfrak{p}\xi)\hat{\varphi}(\mathfrak{p}\xi).$
Hence, for all $l=0,1,\dots,q-1$, $k\in\mathbb{N}_{0}$, we have (by a similar
calculation)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bigl{\langle}f,\psi_{l}(\cdot-u(k))\bigr{\rangle}$
$\displaystyle=$
$\displaystyle\int_{K}\hat{f}(\xi)\overline{\hat{\psi}_{l}(\xi)}\chi_{k}(\xi)d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{D}}\Bigl{\\{}\sum\limits_{\mu=0}^{q-1}m_{f}(\mathfrak{p}\xi+\mathfrak{p}u(\mu))\overline{m_{l}(\mathfrak{p}\xi+\mathfrak{p}u(\mu))}\Bigr{\\}}\chi_{k}(\xi)d\xi.$
Therefore, for all $l=0,1,\dots,q-1$, we have
$\sum\limits_{\mu=0}^{q-1}m_{f}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(\mu)\bigr{)}\overline{m_{l}\bigl{(}\mathfrak{p}\xi+\mathfrak{p}u(\mu)\bigr{)}}=0.$
Now, for a.e. $\xi$, the vector
$\Bigl{(}m_{f}(\mathfrak{p}\xi+\mathfrak{p}u(\mu))\Bigr{)}_{\mu=0}^{q-1}\in\mathbb{C}^{q}$,
being orthogonal to each row vector of the unitary matrix $M(\xi)$, is the
zero vector. In particular, $m_{f}(\mathfrak{p}\xi)=0$ a.e. This means
$\hat{f}=0$ a. e. and hence $f=0$ a. e. ∎
## 4\. Construction of wavelet packets
Let $\\{V_{j}:j\in\mathbb{Z}\\}$ be an MRA of $L^{2}(K)$ and $\varphi$ be the
corresponding scaling function. Then we have (see (7)),
$\hat{\varphi}(\xi)=m_{0}(\mathfrak{p}\xi)\hat{\varphi}(\mathfrak{p}\xi).$
Applying the splitting lemma to $V=V_{1}$, we get
$\\{\psi_{l}(\cdot-u(k)):0\leq l\leq q-1,k\in\mathbb{N}_{0}\\}$ is an
orthonormal basis for $V_{1}$. Now we will define a sequence
$\\{\omega_{n}:n\geq 0\\}$ of functions. Let
$\omega_{0}=\varphi$
and
$\omega_{n}=\psi_{n}\quad(1\leq n\leq q-1),$
where
(8)
$\hat{\psi_{l}}(\xi)=m_{l}(\mathfrak{p}\xi)\hat{\varphi}(\mathfrak{p}\xi)\quad(1\leq
l\leq q-1).$
Suppose $\omega_{m}$ is defined for $m\geq 0$. For $0\leq r\leq q-1$, define
(9)
$\omega_{r+qm}(x)=q^{1/2}\sum\limits_{k\in\mathbb{N}_{0}}h_{k}^{r}\omega_{m}(\mathfrak{p}^{-1}x-u(k)).$
Note that this defines $\omega_{n}$ for every integer $n\geq 0$. Taking
Fourier Transform, we have
(10)
$(\omega_{r+qm})^{\wedge}(\xi)=m_{r}(\mathfrak{p}\xi)\hat{\omega}_{m}(\mathfrak{p}\xi).$
###### Definition 5.
The functions $\\{\omega_{n}:n\geq 0,\\}$ as defined above will be called the
_wavelet packets_ corresponding to the MRA $\\{V_{j}:j\in\mathbb{Z}\\}$ of
$L^{2}(K)$.
In the following proposition we find an expression for the Fourier transforms
of the wavelet packets in terms of $\hat{\varphi}$.
###### Proposition 2.
For an integer $n\geq 1$, consider the unique expansion of $n$ in the base
$q$:
(11) $n=\mu_{1}+\mu_{2}q+\mu_{3}q^{2}+\cdots+\mu_{j}q^{j-1},$
where $0\leq\mu_{i}\leq q-1$ for all $i=1,2,\dots,j$ and $\mu_{j}\not=0$. Then
(12)
$\hat{\omega}_{n}(\xi)=m_{\mu_{1}}(\mathfrak{p}\xi)m_{\mu_{2}}(\mathfrak{p}^{2}\xi)\cdots
m_{\mu_{j}}(\mathfrak{p}^{j}\xi)\hat{\varphi}(\mathfrak{p}^{j}\xi).$
###### Proof.
We will prove it by induction. If $n$ has an expansion as in (11), then we say
that it has length $j$. Since $\omega_{0}=\varphi$, and
$\omega_{n}=\psi_{n},1\leq n\leq q-1$, it follows from (7) and (8) that the
claim is true for length 1. Assume that it is true for length $j$. Let $m$ be
an integer with an expansion of length $j+1$. Then there exist integers
$\gamma_{1},\gamma_{2},\dots,\gamma_{j+1}$ with
$0\leq\gamma_{1},\gamma_{2},\dots,\gamma_{j+1}\leq q-1$ and
$\gamma_{j+1}\not=0$ such that
$\displaystyle m$ $\displaystyle=$
$\displaystyle\gamma_{1}+\gamma_{2}q+\cdots+\gamma_{j}q^{j-1}+\gamma_{j+1}q^{j}$
$\displaystyle=$ $\displaystyle\gamma_{1}+kq,$
where $k=\gamma_{2}+\gamma_{3}q+\cdots+\gamma_{j+1}q^{j-1}$. Note that $k$ has
length $j$. Hence,
$\displaystyle\hat{\omega}_{m}(\xi)$ $\displaystyle=$
$\displaystyle(\omega_{\gamma_{1}+kq})^{\wedge}(\xi)$ $\displaystyle=$
$\displaystyle
m_{\gamma_{1}}(\mathfrak{p}\xi)\hat{\omega}_{k}(\mathfrak{p}\xi)\qquad\qquad~{}({\rm
by~{}\eqref{e.ftwpkt}})$ $\displaystyle=$ $\displaystyle
m_{\gamma_{1}}(\mathfrak{p}\xi)m_{\gamma_{2}}(\mathfrak{p}^{2}\xi)\cdots
m_{\gamma_{j}+1}(\mathfrak{p}^{j+1}\xi)\hat{\varphi}(\mathfrak{p}^{j+1}\xi).$
Hence the induction is complete. ∎
We will prove the following theorem regarding the wavelet packets.
###### Theorem 2.
Let $\\{\omega_{n}:n\geq 0\\}$ be the basic wavelet packets constructed above.
Then
1. (i)
$\\{\omega_{n}(\cdot-u(k)):q^{j}\leq n\leq q^{j+1}-1,k\in\mathbb{N}_{0}\\}$ is
an orthonormal basis of $W_{j}$, $j\geq 0$.
2. (ii)
$\\{\omega_{n}(\cdot-u(k)):0\leq n\leq q^{j}-1,k\in\mathbb{N}_{0}\\}$ is an
orthonormal basis of $V_{j}$, $j\geq 0$.
3. (iii)
$\\{\omega_{n}(\cdot-u(k)):n\geq 0,k\in\mathbb{N}_{0}\\}$ is an orthonormal
basis of $L^{2}(K)$.
###### Proof.
Since $\\{\omega_{n}:1\leq n\leq q-1\\}$ are wavelets, the case $j=0$ in (i)
is trivial. Assume (i) for $j$. We will prove for $j+1$. By our assumption,
the set of functions $\\{\omega_{n}(\cdot-u(k)):q^{j}\leq n\leq
q^{j+1}-1,k\in\mathbb{N}_{0}\\}$ is an orthonormal basis of $W_{j}$. By (4),
we have
$\\{q^{1/2}\omega_{n}(\mathfrak{p}^{-1}\cdot-u(k)):q^{j}\leq n\leq
q^{j+1}-1,k\in\mathbb{N}_{0}\\}$
is an orthonormal basis for $W_{j+1}$. Let
$E_{n}=\overline{\rm
span}\\{q^{1/2}\omega_{n}(\mathfrak{p}^{-1}\cdot-u(k)):k\in\mathbb{N}_{0}\\}.$
Hence,
(13) $W_{j+1}=\bigoplus\limits_{n=q^{j}}^{q^{j+1}-1}E_{n}.$
Applying the splitting lemma to $E_{n}$, we get the functions
$g^{n}_{l}(x)=\sum\limits_{k=0}^{\infty}h^{l}_{k}q^{1/2}\omega_{n}(\mathfrak{p}^{-1}x-u(k)),\quad(0\leq
l\leq q-1)$
such that $\\{g^{n}_{l}(\cdot-u(k)):0\leq l\leq q-1,k\in\mathbb{N}_{0}\\}$
forms an orthonormal basis for $E_{n}$. Hence, $\\{g^{n}_{l}(\cdot-u(k)):0\leq
l\leq q-1,q^{j}\leq n\leq q^{j+1}-1,k\in\mathbb{N}_{0}\\}$ forms an
orthonormal basis for $W_{j+1}$. But by (9),
$g^{n}_{l}=\omega_{l+qn}.$
This fact, together with (13), shows that
$\displaystyle\\{\omega_{l+qn}(\cdot-u(k)):0\leq l\leq q-1,~{}q^{j}\leq n\leq
q^{j+1}-1,~{}k\in\mathbb{N}_{0}\\}$ $\displaystyle=$
$\displaystyle\\{\omega_{n}(\cdot-u(k)):q^{j+1}\leq n\leq
q^{j+2}-1,~{}k\in\mathbb{N}_{0}\\}$
is an orthonormal basis for $W_{j+1}$. So (i) is proved. Item (ii) follows
from the observation that $V_{j}=V_{0}\oplus W_{0}\oplus\cdots\oplus W_{j-1}$
and (iii) follows from the fact that $\overline{\cup V_{j}}=L^{2}(K)$. ∎
## 5\. Wavelet frame packets
Let $\mathcal{H}$ be a separable Hilbert space. A sequence
$\\{x_{k}:k\in\mathbb{Z}\\}$ of $\mathcal{H}$ is said to be a frame for
$\mathcal{H}$ if there exist constants $C_{1}$ and $C_{2}$, $0<C_{1}\leq
C_{2}<\infty$ such that for all $x\in\mathcal{H}$
(14)
$C_{1}\|x\|^{2}\leq\sum\limits_{k\in\mathbb{Z}}|\left<x,x_{k}\right>|^{2}\leq
C_{2}\|x\|^{2}.$
The largest $C_{1}$ and the smallest $C_{2}$ for which (14) holds are called
the frame bounds.
Suppose that $\Phi=\\{\varphi_{1},\varphi_{2},\dots,\varphi_{N}\\}\subset
L^{2}(K)$ be such that the system $\\{\varphi_{l}(\cdot-u(k)):1\leq l\leq
N,k\in\mathbb{N}_{0}\\}$ is a frame for its closed linear span $S(\Phi)$. Let
$\psi_{1},\psi_{2},\dots,\psi_{N}$ be elements in $S(\Phi)$. A natural
question to ask is: When can we say that $\left\\{\psi_{l}(\cdot-u(k)):1\leq
l\leq N,k\in\mathbb{N}_{0}\right\\}$ is also a frame for $S(\Phi)$?
If $\psi_{j}\in S(\Phi)$, then there exists
$\left\\{p_{jlk}:k\in\mathbb{N}_{0}\right\\}$ in $\ell^{2}(\mathbb{N}_{0})$
such that
$\psi_{j}(x)={\mbox{$\sum\limits_{l=1\,}^{N}$}}\sum\limits_{k\in\mathbb{N}_{0}}p_{jlk}\varphi_{l}(x-u(k)).$
Taking Fourier transform, we get
$\displaystyle\hat{\psi}_{j}(\xi)$ $\displaystyle=$
$\displaystyle{\mbox{$\sum\limits_{l=1\,}^{N}$}}\sum\limits_{k\in\mathbb{N}_{0}}p_{jlk}\overline{\chi_{k}(\xi)}\hat{\varphi}_{l}(\xi)$
$\displaystyle=$
$\displaystyle{\mbox{$\sum\limits_{l=1\,}^{N}$}}P_{jl}(\xi)\hat{\varphi}_{l}(\xi),$
where
$P_{jl}(\xi)=\sum\limits_{k\in\mathbb{N}_{0}}p_{jlk}\overline{\chi_{k}(\xi)}$.
Let $P(\xi)$ be the $N\times N$ matrix:
$P(\xi)=\Bigl{(}P_{jl}(\xi)\Bigr{)}_{1\leq j,l\leq N}.$
Let $S$ and $T$ be two positive definite matrices of order $N$. We say $S\leq
T$ if $T-S$ is positive definite. The following lemma is the generalization of
Lemma 3.1 in [5].
###### Lemma 2.
Let $\varphi_{l},\psi_{l}$ for $1\leq l\leq N$, and $P(\xi)$ be as above.
Suppose that there exist constants $C_{1}$ and $C_{2}$, $0<C_{1}\leq
C_{2}<\infty$ such that
(15) $C_{1}I\leq P^{*}(\xi)P(\xi)\leq C_{2}I\quad
for~{}a.e.~{}\xi\in\mathfrak{D}.$
Then, for all $f\in L^{2}(K)$, we have
$\displaystyle
C_{1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<f,\psi_{l}(\cdot-u(k))\right>$}}\right|^{2}\leq{\mbox{$\sum\limits_{l=1\,}^{N}$}}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<f,\varphi_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq
C_{2}{\mbox{$\sum\limits_{l=1\,}^{N}$}}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<f,\varphi_{l}(\cdot-u(k))\right>$}}\right|^{2}.$
###### Proof.
For $f,g\in L^{2}(K)$, we define
$\left[f,g\right](\xi)=\sum\limits_{l\in\mathbb{N}_{0}}\widehat{f}(\xi+u(l))\overline{\widehat{g}(\xi+u(l))}.$
Then, for $f\in L^{2}(K)$, we have
$\displaystyle\left[f,\psi_{j}\right](\xi)$ $\displaystyle=$
$\displaystyle\sum\limits_{l\in\mathbb{N}_{0}}\hat{f}(\xi+u(l))\overline{\hat{\psi}_{l}(\xi+u(l))}$
$\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{N}\sum\limits_{l\in\mathbb{N}_{0}}\overline{P_{jk}(\xi+u(l))}\hat{f}(\xi+u(l))\overline{\hat{\varphi}_{k}(\xi+u(l))}$
$\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{N}\overline{P}_{jk}(\xi)\left[f,\varphi_{k}\right](\xi),$
since $P_{jk}$ are integral periodic function. Hence
$\sum\limits_{j=1}^{N}\left|\left[f,\psi_{j}\right]\right|^{2}=\sum\limits_{k,k^{\prime}=1}^{N}\sum\limits_{j=1}^{N}\overline{P}_{jk}P_{jk^{\prime}}\left[f,\varphi_{k}\right]\overline{\left[f,\varphi_{k^{\prime}}\right]}=XP^{*}PX^{*},$
where
$X=\bigl{(}\left[f,\varphi_{1}\right],\cdots\left[f,\varphi_{n}\right]\bigr{)}.$
By Plancherel Theorem,
$\sum\limits_{k\in\mathbb{N}_{0}}\sum\limits_{l=1}^{N}\left|\left\langle
f,\varphi_{l}(\cdot-u(k))\right\rangle\right|^{2}=\sum\limits_{l=1}^{N}\int_{\mathfrak{D}}\left|\left[f,\varphi_{l}\right](\xi)\right|^{2}d\xi.$
Hence, inequality (2) is equivalent to
$C_{1}\int_{\mathfrak{D}}XX^{*}\leq\int_{\mathfrak{D}}XP^{*}PX^{*}\leq
C_{2}\int_{\mathfrak{D}}XX^{*},\quad{\rm for~{}all}~{}f\in L^{2}(K).$
This follows from (15). ∎
We now introduce a matrix $E(\xi)$. For $0\leq r,s\leq q-1$ and $1\leq l,j\leq
N$, define for a.e. $\xi$
${\mathcal{E}}^{rs}_{lj}(\xi)=\delta_{lj}q^{-\frac{1}{2}}\overline{\chi\bigl{(}u(r)(\xi+\mathfrak{p}u(s))\bigr{)}}.$
Let
$E^{rs}(\xi)=\Bigl{(}{\mathcal{E}}^{rs}_{lj}(\xi)\Bigr{)}_{1\leq l,j\leq N}$
and
(17) $E(\xi)=\Bigl{(}E^{rs}(\xi)\Bigr{)}_{0\leq r,s\leq q-1}.$
So $E(\xi)$ is a block matrix with $q$ blocks in each row and each column, and
each block is a square matrix of order $N$, so that $E(\xi)$ is a square
matrix of order $qN$.
We have the following lemma which will be useful for the splitting trick for
frames. In the first part of the lemma we use a technique used by Zheng in
[28].
###### Lemma 3.
1. (i)
For $0\leq r,s\leq q-1$,
${\frac{1}{q}}\sum\limits_{t=0}^{q-1}\chi\bigl{(}(u(r)-u(s))\mathfrak{p}u(t)\bigr{)}=\delta_{r,s}.$
2. (ii)
The matrix $E(\xi)$, defined in (17), is unitary for a.e.
$\xi\in\mathfrak{D}$.
###### Proof.
(i) If $r=s$ then $u(r)-u(s)=0$, hence the left hand side equals 1. We assume
$r\neq s$. Let
$r=a_{0}+a_{1}p+\cdots+a_{c-1}p^{c-1}~{}{\rm
and}~{}s=b_{0}+b_{1}p+\cdots+b_{c-1}p^{c-1}$
where $0\leq a_{j},b_{j}\leq p-1$ for $j=0,1,\dots,c-1$. Then (see (1))
$u(r)\mathfrak{p}=a_{0}\epsilon_{0}+a_{1}\epsilon_{1}+\cdots+a_{c-1}\epsilon_{c-1}~{}{\rm
and}~{}u(s)\mathfrak{p}=b_{0}\epsilon_{0}+b_{1}\epsilon_{1}+\cdots+b_{c-1}\epsilon_{c-1}.$
Now, let
$t=d_{0}+d_{1}p+\cdots+d_{c-1}p_{c-1},0\leq d_{j}\leq
p-1~{}\mbox{for}~{}j=0,1,\dots,c-1.$
Observe that as $t$ varies from $0$ to $q-1$, the integers
$d_{0},d_{1},\dots,d_{c-1}$ all vary from $0$ to $p-1$. For each
$j=0,1,\dots,c-1$, we write
$u(r)\mathfrak{p}\epsilon_{j}=\gamma_{r,0}^{j}\epsilon_{0}+\gamma_{r,1}^{j}\epsilon_{1}+\cdots+\gamma_{r,c-1}^{j}\epsilon_{c-1}$
for some unique $\gamma_{r,l}^{j}\in GF(p),0\leq l\leq c-1$. Similarly,
$u(s)\mathfrak{p}\epsilon_{j}=\gamma_{s,0}^{j}\epsilon_{0}+\gamma_{s,1}^{j}\epsilon_{1}+\cdots+\gamma_{s,c-1}^{j}\epsilon_{c-1}$
for some unique $\gamma_{s,l}^{j}\in GF(p),0\leq l\leq c-1$. By the definition
of the character $\chi$ (see (3)), we have
$\chi(u(r)\mathfrak{p}u(t))=\exp\big{(}\tfrac{2\pi
i}{p}(\gamma_{r,0}^{0}d_{0}+\cdots+\gamma_{r,0}^{c-1}d_{c-1})\big{)}$
and
$\chi(u(s)\mathfrak{p}u(t))=\exp\big{(}\tfrac{2\pi
i}{p}(\gamma_{s,0}^{0}d_{0}+\cdots+\gamma_{s,0}^{c-1}d_{c-1})\big{)}.$
Therefore,
$\displaystyle\sum\limits_{t=0}^{q-1}\chi\bigl{(}(u(r)-u(s))\mathfrak{p}u(t)\bigr{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{t=0}^{q-1}\chi\bigl{(}u(r)\mathfrak{p}u(t)\bigr{)}\overline{\chi\big{(}u(s)\mathfrak{p}u(t)\bigr{)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{d_{0}=0}^{p-1}\cdots\sum\limits_{d_{c-1}=0}^{p-1}\exp\Big{(}\tfrac{2\pi
i}{p}(\gamma_{r,0}^{0}d_{0}+\cdots+\gamma_{r,0}^{c-1}d_{c-1})\Big{)}$
$\displaystyle\qquad\qquad\exp\Big{(}\tfrac{-2\pi
i}{p}(\gamma_{s,0}^{0}d_{0}+\cdots+\gamma_{s,0}^{c-1}d_{c-1})\Big{)}$
$\displaystyle=$
$\displaystyle\Bigg{(}\sum\limits_{d_{0}=0}^{p-1}\exp\Bigl{(}\tfrac{2\pi
i}{p}(\gamma_{r,0}^{0}-\gamma_{s,0}^{0})d_{0}\Bigr{)}\Bigg{)}\cdots\Bigg{(}\sum\limits_{d_{c-1}=0}^{p-1}\exp\Bigl{(}\tfrac{2\pi
i}{p}(\gamma_{r,0}^{c-1}-\gamma_{s,0}^{c-1})d_{c-1}\Bigr{)}\Bigg{)}.$
Since $r\neq s$, we claim that $\gamma_{r,0}^{j}\not=\gamma_{s,0}^{j}$ for
some $j$, $0\leq j\leq c-1$. If $\gamma_{r,0}^{j}=\gamma_{s,0}^{j}$ for all
$j$, then, since $u(r)\mathfrak{p}\neq u(s)\mathfrak{p}$, we have
$\displaystyle GF(q)$ $\displaystyle=$ $\displaystyle{\rm
span}\\{(u(r)\mathfrak{p}-u(s)\mathfrak{p})\epsilon_{j}\\}_{j=0}^{c-1}$
$\displaystyle=$ $\displaystyle{\rm
span}\big{\\{}(\gamma_{r,0}^{j}-\gamma_{s,0}^{j})\epsilon_{0},\cdots,(\gamma_{r,c-1}^{j}-\gamma_{s,c-1}^{j})\epsilon_{c-1}\big{\\}}_{j=0}^{c-1}$
$\displaystyle=$ $\displaystyle{\rm
span}\\{\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{c-1}\\}.$
This is a contradiction which proves the claim. Now for any $j$ such that
$\gamma_{r,0}^{j}\not=\gamma_{s,0}^{j}$, we have
$\sum\limits_{d_{j}=0}^{p-1}\exp\Bigl{(}\tfrac{2\pi
i}{p}(\gamma_{r,0}^{j}-\gamma_{s,0}^{j})d_{j}\Bigr{)}=\tfrac{1-\exp\bigl{(}2\pi
i(\gamma_{r,0}^{j}-\gamma_{s,0}^{j})\bigr{)}}{1-\exp\bigl{(}\tfrac{2\pi
i}{p}(\gamma_{r,0}^{j}-\gamma_{s,0}^{j})\bigr{)}}=0,$
since $\gamma_{r,0}^{j}-\gamma_{s,0}^{j}$ is an integer with
$|\gamma_{r,0}^{j}-\gamma_{s,0}^{j}|<p$. This proves (i).
To prove (ii), observe that the $(r,s)$-th block of the matrix
$E(\xi)E^{*}(\xi)$ is
$\sum\limits_{t=0}^{q-1}E^{rt}(\xi)\left(E^{ts}(\xi)\right)^{*}.$
The $(l,j)$-th entry in this block is
$\displaystyle=$
$\displaystyle\sum\limits_{t=0}^{q-1}\sum\limits_{m=0}^{N}{\mathcal{E}}^{rt}_{lm}(\xi)\left({\mathcal{E}}^{ts}_{mj}(\xi)\right)^{*}$
$\displaystyle=$
$\displaystyle\sum\limits_{t=0}^{q-1}\sum\limits_{m=0}^{N}\delta_{lm}q^{-1/2}\overline{\chi\bigl{(}u(r)(\xi+\mathfrak{p}u(t))\bigr{)}}\cdot\delta_{jm}q^{-1/2}\chi\bigl{(}u(s)(\xi+\mathfrak{p}u(t))\bigr{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{m=1}^{N}\delta_{lm}\delta_{jm}q^{-1}\sum\limits_{t=0}^{q-1}\overline{\chi\bigl{(}u(r)(\xi+\mathfrak{p}u(t))\bigr{)}}\chi\bigl{(}u(s)(\xi+\mathfrak{p}u(t))\bigr{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{m=1}^{N}\delta_{lm}\delta_{jm}\chi((u(s)-u(r))\xi)q^{-1}\sum\limits_{t=0}^{q-1}\chi\bigl{(}(u(s)-u(r))\mathfrak{p}u(t))\bigr{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{m=1}^{N}\delta_{lm}\delta_{jm}\delta_{rs},\quad{\rm(by~{}part~{}(i)~{}of~{}the~{}lemma)}$
$\displaystyle=$ $\displaystyle\delta_{lj}\delta_{rs}.$
Hence $E(\xi)E^{*}(\xi)=I$. Similarly, $E(\xi)^{*}E(\xi)=I$. Therefore,
$E(\xi)$ is a unitary matrix. ∎
## 6\. Splitting lemma for wavelet frame packets
Let $\\{\varphi_{j}:1\leq j\leq N\\}$ be functions in $L^{2}(K)$ such that
$\\{\varphi_{j}(\cdot-u(k)):1\leq j\leq N,k\in\mathbb{N}_{0}\\}$ is a frame
for its closed linear span $V$. For $1\leq l\leq N$, $0\leq r\leq q-1$,
suppose that there exist sequences
$\\{h^{r}_{ljk}:k\in\mathbb{Z}\\}\in\ell^{2}(\mathfrak{D})$. Define
$\psi^{r}_{l}(x)=q^{1/2}\sum\limits_{j=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{ljk}\varphi_{j}(\mathfrak{p}^{-1}x-u(k)).$
Taking Fourier transform, we get
$\hat{\psi}_{l}^{r}(\xi)=\sum\limits_{j=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{ljk}q^{-1/2}\overline{\chi_{k}(\mathfrak{p}\xi)}\hat{\varphi}_{j}(\mathfrak{p}\xi)=\sum\limits_{j=1}^{N}h^{r}_{lj}\hat{\varphi}_{j}(\mathfrak{p}\xi),$
where,
$h^{r}_{lj}(\xi)=\sum\limits_{k\in\mathbb{N}_{0}}q^{-1/2}h^{r}_{ljk}\overline{\chi_{k}(\xi)}.$
Let
$H_{r}(\xi)=\bigl{(}h^{r}_{lj}(\xi)\bigr{)}_{1\leq l,j\leq N}$
and
$H(\xi)=\Bigl{(}H_{r}(\xi+\mathfrak{p}u(s))\Bigr{)}_{0\leq r,s\leq q-1}.$
Note that $H(\xi)$ is a square matrix of order $qN$. We can write
$\psi^{r}_{l}$ as
$\displaystyle\psi^{r}_{l}(x)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{ljk}q^{1/2}\varphi_{j}\bigl{(}\mathfrak{p}^{-1}x-u(k)\bigr{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\sum\limits_{s=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{lj,qk+s}q^{1/2}\varphi_{j}\bigl{(}\mathfrak{p}^{-1}x-u(qk+s)\bigr{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\sum\limits_{s=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{lj,qk+s}\varphi_{j}^{(s)}(x-u(k)),$
where
(18) $\varphi_{j}^{(s)}(x)=q^{1/2}\varphi_{j}(\mathfrak{p}^{-1}x-u(s)),\quad
0\leq s\leq q-1.$
Note that $u(qk+s)=\mathfrak{p}^{-1}u(k)+u(s)$ (see eq. (2)). Taking Fourier
transform, we obtain
$\displaystyle(\psi^{r}_{l})^{\wedge}(\xi)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\sum\limits_{s=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{lj,qk+s}\overline{\chi_{k}(\xi)}(\varphi^{(s)}_{j})^{\wedge}(\xi)$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{N}\sum\limits_{s=0}^{q-1}p^{rs}_{lj}(\xi)(\varphi^{(s)}_{j})^{\wedge}(\xi),$
where
$p^{rs}_{lj}(\xi)=\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{lj,qk+s}\overline{\chi_{k}(\xi)}$.
Define the matrices
$P^{rs}(\xi)=\Bigl{(}p^{rs}_{lj}(\xi)\Bigr{)}_{1\leq l,j\leq N}.$
and
$P(\xi)=\Bigl{(}P^{rs}(\xi)\Bigr{)}_{0\leq r,s\leq q-1}.$
###### Proposition 3.
$H(\xi)=P(\mathfrak{p}^{-1}\xi)E(\xi)$, where $E(\xi)$ is the unitary matrix
defined in (17).
###### Proof.
The $(r,s)$-th block of the matrix $P(\mathfrak{p}^{-1}\xi)E(\xi)$ is the
matrix
$\sum\limits_{t=0}^{q-1}P^{rt}(\mathfrak{p}^{-1}\xi)E^{ts}(\xi).$
The $(l,j)$-th entry in this block is equal to
$\displaystyle\sum\limits_{t=0}^{q-1}\sum\limits_{m=1}^{N}p^{rt}_{lm}(\mathfrak{p}^{-1}\xi){\mathcal{E}}^{ts}_{mj}(\xi)$
$\displaystyle=$
$\displaystyle\sum\limits_{t=0}^{q-1}\sum\limits_{m=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{l,m,qk+t}\overline{\chi_{k}(\mathfrak{p}^{-1}\xi)}\delta_{mj}q^{-1/2}\overline{\chi\bigl{(}u(t)(\xi+\mathfrak{p}u(s))\bigr{)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{t=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{l,m,qk+t}\overline{\chi_{k}(\mathfrak{p}^{-1}\xi)}q^{-1/2}\overline{\chi\bigl{(}u(t)(\xi+\mathfrak{p}u(s))\bigr{)}}.$
Now, the $(l,j)$-th entry in the $(r,s)$-th block of $H(\xi)$ is
$\displaystyle h^{r}_{lj}(\xi+pu(s))$ $\displaystyle=$ $\displaystyle
q^{-1/2}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{ljk}\overline{\chi\bigl{(}u(k)(\xi+\mathfrak{p}u(s))\bigr{)}}$
$\displaystyle=$ $\displaystyle
q^{-1/2}\sum\limits_{t=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{l,j,qk+t}\overline{\chi\bigl{(}u(qk+t)(\xi+\mathfrak{p}u(s))\bigr{)}}$
$\displaystyle=$ $\displaystyle
q^{-1/2}\sum\limits_{t=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{l,m,qk+t}\overline{\chi(\mathfrak{p}^{-1}u(k)\xi+u(k)u(s)+u(t)\xi+\mathfrak{p}u(t)u(s))}$
$\displaystyle=$ $\displaystyle
q^{-1/2}\sum\limits_{t=0}^{q-1}\sum\limits_{k\in\mathbb{N}_{0}}h^{r}_{l,m,qk+t}\overline{\chi_{k}(\mathfrak{p}^{-1}\xi)}\overline{\chi\bigl{(}u(t)(\xi+\mathfrak{p}u(s))\bigr{)}}.$
∎
In particular, we have
$H^{*}(\xi)H(\xi)=E^{*}(\xi)P^{*}(\mathfrak{p}^{-1}\xi)P(\mathfrak{p}^{-1}\xi)E(\xi).$
Since $E(\xi)$ is unitary by Lemma 3, $H^{*}(\xi)H(\xi)$ and
$P^{*}(\mathfrak{p}^{-1}\xi)P(\mathfrak{p}^{-1}\xi)$ are similar matrices.
Let $\lambda(\xi)$ and $\Lambda(\xi)$ respectively be the maximal and minimal
eigenvalues of the positive definite matrix $H^{*}(\xi)H(\xi)$, and let
$\lambda=\inf\limits_{\xi}\lambda(\xi)$ and
$\Lambda=\sup\limits_{\xi}\Lambda(\xi)$. Suppose
$0<\lambda\leq\Lambda<\infty$. Then we have
$\lambda I\leq H^{*}(\xi)H(\xi)\leq\Lambda I\quad{\rm
for~{}a.e.}~{}\xi\in\mathfrak{D}.$
This is equivalent to say that
$\lambda I\leq P^{*}(\xi)P(\xi)\leq\Lambda I\quad{\rm
for~{}a.e.}~{}\xi\in\mathfrak{D}.$
Then by Lemma 2, for all $g\in L^{2}(K)$, we have
(19)
$\displaystyle\lambda\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\varphi^{(s)}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{s}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\varphi^{(s)}_{l}(\cdot-u(k))\right>$}}\right|^{2},$
where $\varphi^{(s)}_{l}$ is defined in (18). Since
$\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\varphi_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2}=\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\varphi^{(s)}_{l}(\cdot-u(k))\right>$}}\right|^{2},$
which follows from (18), inequality (19) can be written as
(20)
$\displaystyle\lambda\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\varphi_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{s}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\varphi_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2}.$
This is the _splitting trick_ for frames.
We now apply the splitting trick to the functions $\\{\psi^{s}_{l}:1\leq l\leq
N\\}$ for each $s$, $0\leq s\leq q-1$. We have
(21)
$\displaystyle\lambda\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\psi^{s}_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{r=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{s,r}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\psi^{s}_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2},$
where $\psi^{s,r}_{l},0\leq r\leq q-1$ are defined as:
(22)
$\psi^{s,r}_{l}(x)=\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}h^{s}_{ljk}q^{1/2}\psi^{r}_{j}(\mathfrak{p}^{-1}x-u(k));~{}0\leq
s\leq q-1,1\leq l\leq N.$
Summing (21) over $0\leq s\leq q-1$, we have
$\displaystyle\lambda\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\psi^{s}_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{s=0}^{q-1}\sum\limits_{r=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{s,r}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda\sum\limits_{s=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{1/2}\psi^{s}_{l}(\mathfrak{p}^{-1}\cdot-u(k))\right>$}}\right|^{2}.$
Using (20), we obtain
(23)
$\displaystyle\lambda^{2}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{2/2}\varphi_{l}(\mathfrak{p}^{2}\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{s=0}^{q-1}\sum\limits_{r=0}^{q-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{s,r}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda^{2}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{2/2}\varphi_{l}(\mathfrak{p}^{2}\cdot-u(k))\right>$}}\right|^{2}.$
We now define the wavelet frame packets similar to the orthonormal case. We
start with the functions $\varphi_{1},\varphi_{2},\dots,\varphi_{N}$. Apply
the splitting trick to the space
$\overline{{\rm span}}\\{q^{1/2}\varphi_{l}(\mathfrak{p}^{-1}\cdot-u(k)):1\leq
l\leq N,k\in\mathbb{N}_{0}\\}$
to get the functions $\\{\psi_{l}^{s}:1\leq l\leq N,0\leq s\leq q-1\\}$ (see
(20)). Now for any integer $n\geq 0$, we define $\psi^{n}_{l}$, $1\leq l\leq
N$, recursively as follows. Suppose that $\psi^{r}_{l}$ is already defined for
$r\in\mathbb{N}_{0}$ and $1\leq l\leq N$. Then for $0\leq s\leq q-1$ and
$1\leq l\leq N$, define
$\psi_{l}^{s+qr}=\sum\limits_{j=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}h^{s}_{ljk}q^{1/2}\psi^{r}_{j}(\mathfrak{p}^{-1}\cdot-u(k)).$
Comparing this with equation (22), we see that
$\displaystyle\\{\psi^{s,r}_{l}:0\leq r,s\leq q-1\\}$ $\displaystyle=$
$\displaystyle\\{\psi^{s+qr}_{l}:0\leq r,s\leq q-1\\}$ $\displaystyle=$
$\displaystyle\\{\psi^{n}_{l}:0\leq n\leq q^{2}-1\\}.$
So (23) can be written as
$\displaystyle\lambda^{2}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{2/2}\varphi_{l}(\mathfrak{p}^{-2}\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{n=0}^{q^{2}-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{n}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda^{2}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{2/2}\varphi_{l}(\mathfrak{p}^{-2}\cdot-u(k))\right>$}}\right|^{2}.$
By induction, we get for each $j\geq 1$
(24)
$\displaystyle\lambda^{j}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{j/2}\varphi_{l}(\mathfrak{p}^{-j}\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{n=0}^{q^{j}-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{n}_{l}(\cdot-u(k))\right>$}}\right|^{2}$
$\displaystyle\leq$
$\displaystyle\Lambda^{j}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,q^{j/2}\varphi_{l}(\mathfrak{p}^{-j}\cdot-u(k))\right>$}}\right|^{2}.$
We summarize the above discussion in the following theorem.
###### Theorem 3.
Let $\\{\varphi_{l}:1\leq l\leq N\\}\subset L^{2}(K)$ be such that
$\\{\varphi_{l}(\cdot-u(k)):1\leq l\leq N,k\in\mathbb{N}_{0}\\}$ is a frame
for its closed linear span $V_{0}$, with frame bounds $C_{1}$ and $C_{2}$ .
Let $H(\xi),H_{r}(\xi),\lambda$ and $\Lambda$ be as above. Assume that all
entries of $H(\xi)$ are bounded measurable functions such that
$0<\lambda\leq\Lambda<\infty$. Let $\\{\psi^{n}_{l}:n\geq 0,1\leq l\leq N\\}$
be the wavelet frame packets and let $V_{j}=\\{f\in
L^{2}(K):f(\mathfrak{p}^{j}\cdot)\in V_{0}\\}$. Then for all $j\geq 0$, the
system of functions
$\\{\psi^{n}_{l}(\cdot-u(k)):0\leq n\leq q^{j}-1,1\leq l\leq
N,k\in\mathbb{N}_{0}\\}$
is a frame of $V_{j}$ with frame bounds $\lambda^{j}C_{1}$ and
$\Lambda^{j}C_{2}$.
###### Proof.
Since $\\{\varphi_{l}(\cdot-u(k)):1\leq l\leq N.k\in\mathbb{N}_{0}\\}$ is a
frame of $V_{0}$ with frame bounds $C_{1}$ and $C_{2}$, it is clear that for
all $j$
$\\{q^{j/2}\varphi_{l}(\mathfrak{p}^{-j}\cdot-u(k)):1\leq l\leq
N,k\in\mathbb{N}_{0}\\}$
is a frame of $V_{j}$ with the same bounds. So from (24), we have
$\lambda^{j}C_{1}\|g\|^{2}\leq\sum\limits_{n=0}^{q^{j}-1}\sum\limits_{l=1}^{N}\sum\limits_{k\in\mathbb{N}_{0}}\left|{\mbox{$\left<g,\psi^{n}_{l}(\cdot-u(k))\right>$}}\right|^{2}\leq\Lambda^{j}C_{2}\|g\|^{2}$
for all $g\in V_{j}$. ∎
## References
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* [2] B. Behera, Multiwavelet packets and frame packets of $L^{2}({\mathbb{R}}^{d})$, Proc. Indian Acad. Sci. Math. Sci., 111 (2001) 439–463.
* [3] J. J. Benedetto, R. L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal., 14 (2004) 423–456.
* [4] R. L. Benedetto, Examples of wavelets for local fields, Wavelets, frames and operator theory, 27–47, Contemp. Math., 345, Amer. Math. Soc., 2004.
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* [9] R. Coifman, Y. Meyer, M. V. Wickerhauser, Wavelet analysis and signal procesing, in: M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael (Eds.), Wavelets and Their Applications, Jones and Bartlett, 1992, pp. 153–178.
* [10] R. Coifman, Y. Meyer, M. V. Wickerhauser, Size properties of wavelet packets, in: M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael (Eds.), Wavelets and Their Applications, Jones and Bartlett, 1992, pp. 453–478.
* [11] S. Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups, in: Wavelets, images, and surface fitting, A K Peters, 1994, pp. 141–156.
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* [18] S. Kozyrev, Wavelet theory as $p$-adic spectral analysis (Russian), Izv. Ross. Akad. Nauk Ser. Mat., 66 (2002) 149–158; translation in Izv. Math., 66 (2002) 367–376.
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* [22] P. G. Lemarie, Bases dondelettes sur les groupes de Lie stratifies, Bull. Math. Soc. France, 117 (1989) 211-233.
* [23] R. Long, W. Chen, Wavelet basis packets and wavelet frame packets, J. Fourier Anal. Appl., 3 (1997) 239-256.
* [24] D. Ramakrishnan, R. J. Valenza, Fourier Analysis on Number Fields, Springer-Verlag, 1999.
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* [26] T. Stavropoulos, M. Papadakis, On the multiresolution analyses of abstract Hilbert spaces, Bull. Greek Math. Soc., 40 (1998) 79—92.
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|
arxiv-papers
| 2011-03-01T06:54:02 |
2024-09-04T02:49:17.383363
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Biswaranjan Behera, Qaiser Jahan",
"submitter": "Qaiser Jahan",
"url": "https://arxiv.org/abs/1103.0090"
}
|
1103.0412
|
# Counting large distances in convex polygons:
a computational approach
Filip Morić, David Pritchard EPFL, Lausanne, Switzerland. We gratefully
acknowledge support from the Swiss National Science Foundation (Grant No.
200021-125287/1) and an NSERC Post-Doctoral Fellowship.
###### Abstract
In a convex $n$-gon, let $d_{1}>d_{2}>\dotsb$ denote the set of all distances
between pairs of vertices, and let $m_{i}$ be the number of pairs of vertices
at distance $d_{i}$ from one another. Erdős, Lovász, and Vesztergombi
conjectured that $\sum_{i\leq k}m_{i}\leq kn$. Using a new computational
approach, we prove their conjecture when $k\leq 4$ and $n$ is large; we also
make some progress for arbitrary $k$ by proving that $\sum_{i\leq
k}m_{i}\leq(2k-1)n$. Our main approach revolves around a few known facts about
distances, together with a computer program that searches all distance
configurations of two disjoint convex hull intervals up to some finite size.
We thereby obtain other new bounds such as $m_{3}\leq 3n/2$ for large $n$.
## 1 Introduction
Given a set $S$ of $n$ points in the plane, let $d_{1}>d_{2}>\dotsb$ be the
set of all distances between pairs of points in $S$. It was shown by Hopf and
Pannwitz in 1934 [5] that the distance $d_{1}$ (the diameter of $S$) can occur
at most $n$ times, which is tight (e.g. for a regular polygon of odd order).
In 1987 Vesztergombi [6] showed that the second-largest distance, $d_{2}$, can
occur at most $\frac{3}{2}n$ times; she subsequently [7] considered the
version of the problem when the points are in convex position and showed that
in this case the number of second-largest distances is at most $\frac{4}{3}n$.
She also showed that both results are tight up to additive constants.
Let $m_{i}$ denote the number of times that $d_{i}$ occurs. It is known that
$m_{k}\leq 2kn$ [6], and moreover that $m_{k}\leq kn$ for point sets in convex
position [7], while the following open conjecture would imply $m_{k}\leq 2n$:
###### Conjecture 1.1 (Erdős, Moser [7, 2]).
The number of unit distances generated by $n$ points in convex position cannot
exceed $2n$.
A lower bound of $2n-7$ for this conjecture is known due to Edelsbrunner and
Hajnal [3].
For the rest of the paper we consider only point sets in convex position. One
natural question is to find how large $m_{\leq k}:=\sum_{i\leq k}m_{i}$, i.e.
the number of _top- $k$_ distances, can be in terms of $n$. The conjectured
value is:
###### Conjecture 1.2 (Erdős, Lovász, Vesztergombi [4]).
The number of top-$k$ distances generated by $n$ points in convex position is
at most $kn$, i.e. $m_{\leq k}\leq kn$.
Odd regular polygons prove $m_{\leq k}=kn$ is possible. In [4] the bound
$m_{\leq k}\leq 3kn$ is proven, and $m_{\leq 2}\leq 2n$ was shown in [7],
verifying Conjecture 1.2 for $k=2$.
In this paper we give improved upper bounds on $m_{k}$ and $m_{\leq k}$ for
convex point sets, and more generally bounds for sums of the form $\sum_{t\in
T}m_{t}$. Our first result is the following:
###### Theorem 1.3.
For any $k\geq 1$, the number of top-$k$ distances generated by $n$ points in
convex position is at most $(2k-1)n$, i.e. $m_{\leq k}\leq(2k-1)n$.
Thus we close about half of the gap towards Conjecture 1.2.
Next, by combining several known conditions on distances for convex point
sets, and by using a computer program to carry out an exhaustive search on a
finite abstract version of the problem, we prove the following.
###### Theorem 1.4.
The distances generated by $n$ points in convex position satisfy the following
bounds, for large enough $n$:
* •
$m_{\leq 3}\leq 3n,m_{\leq 4}\leq 4n;$
* •
$m_{3}\leq\frac{3}{2}n,m_{4}\leq\frac{13}{8}n;$
* •
$m_{1}+m_{3}\leq 2n,m_{2}+m_{3}\leq\frac{9}{4}n.$
In particular we verify Conjecture 1.2 for $k\leq 4$ and $n$ large. For
$m_{3}$ and $m_{2}+m_{3}$ the bound is as good as can be obtained by our
abstract version of the problem, as witnessed by periodic patterns achieving
$m_{3}=\frac{3}{2}n$ and $m_{2}+m_{3}=\frac{9}{4}n$, but we do not know if any
convex polygon can realize these distances; we elaborate in Section 6.
The proof of Theorem 1.4 uses a computer program to make certain types of
automatic deductions, as well as the following lemma to eliminate long
distances “near” the boundary:
###### Lemma 1.5.
For any $k\geq 1$ and $\ell\geq 0$, there is a constant $C(k,\ell)$ such that
the following holds: in a convex polygon, if there are $\ell$ or less vertices
between some vertices $a$ and $b$ such that $|ab|\geq d_{k}$, then the number
of top-$k$ distances satisfies $m_{\leq k}\leq n+C(k,\ell)$.
The detailed bound we obtain is of the form $C(k,\ell)=O(k^{2}(k+\ell)^{2})$.
In an earlier version of this paper111http://arxiv.org/abs/1103.0412v1 we
proved results like “$m_{\leq 3}\leq 3n+O(1)$” which are weaker for large $n$
but better for small $n$, using the following alternative lemma:
###### Lemma 1.6.
For any $k\geq 1$ and $\ell\geq 0$, there is a constant $C^{\prime}(k,\ell)$
such that the following holds. In a convex polygon, at most
$C^{\prime}(k,\ell)$ diagonals $ab$ have both (i) $\ell$ or less vertices
between $a$ and $b$ and (ii) $|ab|\geq d_{k}$.
In the latter, $C^{\prime}(k,\ell)=O(k\ell^{2})$. We do not think either lemma
is tight.
In Section 2 we describe _levels_ , a key element in our approach. In Section
3 we collect geometric facts used by the algorithm. We prove Lemma 1.5 in
Section 3.1. The proof of our main result, Theorem 1.4, consists of the
algorithmic approach described in Section 4 together with our computational
results stated in Section 5. We conclude with suggestions for future work.
## 2 Levels
We use the term _diagonal_ to mean any line segment connecting two points of
$S$, including sides of the convex hull of $S$. We will partition the
diagonals into $n$ _levels_ in the following way. Let
$S=\\{a_{1},a_{2},\dots,a_{n}\\}$ be the vertex set of our convex polygon,
ordered clockwise. Then _level_ $i$ is the set of diagonals
$L_{i}:=\\{a_{j}a_{k}\mid j+k\equiv i\bmod{n}\\},$
where the index $i$ can be taken modulo $n$. Equivalently, consider an
auxiliary regular $n$-gon $b_{1}b_{2}\dots b_{n}$, then two diagonals
$a_{i}a_{j}$ and $a_{k}a_{l}$ lie in the same level when the corresponding
segments $b_{i}b_{j}$ and $b_{k}b_{l}$ are parallel. We illustrate this in
Figure 1(a).
Figure 1: (a) Three consecutive levels of diagonals in a convex decagon. (b)
Proof of Fact 3.2.
Levels are used in the following way to prove Theorem 1.3: (i.e., $m_{\leq
k}\leq(2k-1)n$).
###### Proof of Theorem 1.3.
In the next section, we prove Lemma 3.5: in any level, there are at most
$2k-1$ diagonals of length $\geq d_{k}$. Since there are at most $n$ levels,
we are done. ∎
## 3 Geometric Facts
To begin this section, we collect 4 geometric facts from the literature [7, 4,
1], which will be used in our computer program. For completeness, we include
the proofs. The first two facts were used in [7, 4].
###### Fact 3.1.
If $abcd$ is a convex quadrangle, then $|ab|+|cd|<|ac|+|bd|$.
###### Proof.
Let $p$ be the intersection point of the diagonals $ac,bd$. Then by the
triangle inequality,
$|ab|+|cd|<|ap|+|bp|+|cp|+|dp|=|ac|+|bd|\,.$ ∎
###### Fact 3.2.
If $a,b,c,d$ are vertices of a convex polygon in clockwise order, then at
least one of these four cases must occur:
* •
$|ax|>|ad|$ for all vertices $x$ of the polygon between $c$ and $d$, including
$c$;
* •
$|bx|>|bc|$ for all vertices $x$ of the polygon between $c$ and $d$, including
$d$;
* •
$|cx|>|bc|$ for all vertices $x$ of the polygon between $a$ and $b$, including
$a$;
* •
$|dx|>|ad|$ for all vertices $x$ of the polygon between $a$ and $b$, including
$b$.
###### Proof.
Since the sum of the angles of quadrilateral $abcd$ is $2\pi$, at least one
angle is non-acute. Without loss of generality let $\angle
cda\geq\frac{\pi}{2}$. Then for any vertex $x$ of the polygon between $c$ and
$d$ we have that $\angle xda\geq\angle cda\geq\frac{\pi}{2}$, and, thus,
$|ax|>|ad|$ (see Figure 1). ∎
The special case $i=j$ of the following fact appears in [4].
###### Fact 3.3.
If $a,b,c,d$ are vertices of a convex polygon listed in clockwise order, such
that $|bc|\geq d_{i}$ and $|ad|\geq d_{j}$, where $d_{i}$ and $d_{j}$ are the
$i$-th and the $j$-th largest distances among vertices of the polygon, then
either between $a$ and $b$ or between $c$ and $d$ there are no more than
$i+j-3$ other vertices of the polygon.
###### Proof.
Let us denote without loss of generality $a=a_{1},b=a_{x},c=a_{y},d=a_{z}$. We
will show $\min\\{x-1,z-y\\}\leq i+j-2$ which proves the lemma. We use
induction on $i+j$. The base case $i=j=1$ amounts to saying that any two non-
crossing $d_{1}$’s must share a vertex, which follows by Fact 3.1.
For the inductive step, we apply Fact 3.2. Suppose that the 1st of the 4 cases
happens, so $d^{\prime}:=a_{z-1}$ satisfies $|ad^{\prime}|>|ad|$; the other
cases are similar. Consequently, $|ad^{\prime}|\geq d_{j-1}$. By induction,
$\min\\{x-1,(z-1)-y\\}\leq i+(j-1)-3$, from which the desired result follows.
∎
Figure 2: (a) Proof of Fact 3.3, base case $i=2$, $j=1$; (b) Proof of Fact
3.3, inductive step
The following is a strengthening of a result of Altman, obtained by removing
all non-essential conditions from the hypothesis of [1, Lemma 1] but using the
same proof. (He considered only the case where $|a_{1}a_{m}|=d_{1}$.)
###### Fact 3.4.
Let $a_{1}\dots a_{n}$ be a convex polygon. If $1\leq i<j\leq k<\ell<m$ and
$|a_{1}a_{m}|\geq\max\\{|a_{1}a_{k}|,|a_{j}a_{m}|\\}$, then
$|a_{i}a_{\ell}|>\min\\{|a_{i}a_{k}|,|a_{j}a_{\ell}|\\}$.
###### Proof.
Suppose for the sake of contradiction that
$|a_{i}a_{\ell}|\leq\min\\{|a_{i}a_{k}|,|a_{j}a_{\ell}|\\}$. Denote by $x$ and
$y$ the points where $a_{1}a_{j}$ and $a_{m}a_{k}$ intersect $a_{i}a_{\ell}$
(see Figure 3). Repeatedly using the fact that when $s,s^{\prime}$ are two
sides of a triangle, $|s|>|s^{\prime}|$ iff the angle opposite $s$ is larger
than the angle opposite $s^{\prime}$, we have
$\displaystyle\angle a_{j}xa_{\ell}+\angle a_{k}ya_{i}$ $\displaystyle>\angle
a_{j}a_{i}a_{\ell}+\angle a_{k}a_{\ell}a_{i}\geq\angle
a_{i}a_{j}a_{\ell}+\angle a_{\ell}a_{k}a_{i}$ $\displaystyle>\angle
a_{1}a_{j}a_{m}+\angle a_{1}a_{k}a_{m}\geq\angle a_{j}a_{1}a_{m}+\angle
a_{k}a_{m}a_{1}\,.$
However, $\angle a_{j}xa_{\ell}+\angle a_{k}ya_{i}=\angle
a_{j}a_{1}a_{m}+\angle a_{k}a_{m}a_{1}$, which gives a contradiction. ∎
Figure 3: (a) Proof of Fact 3.4. (b) Proof of Lemma 1.5.
### 3.1 Counting Lemmas
First we complete the proof of Theorem 1.3, using Fact 3.3.
###### Lemma 3.5.
In any level there are at most $2k-1$ diagonals of length $\geq d_{k}$.
###### Proof.
Without loss of generality (by relabeling), we consider the level $L_{0}$. The
diagonals of this level are $a_{j}a_{-j}$, with indices modulo $n$, for
$0<j<n/2$. Let $m>0$ (resp. $M$) be the minimal (resp. maximal) $j$ such that
$|a_{j}a_{-j}|\geq d_{k}$. Then by Fact 3.3, we see that $M-m-1\leq k+k-3$. So
the number of top-$k$ diagonals in $L_{0}$ is bounded by
$|\\{m,m+1,\dotsc,M\\}|=M-m+1\leq 2k-1$, which gives the corollary. ∎
Next, we give the proof of Lemma 1.5, which is needed in order to argue that
our computational approach is correct.
###### Proof.
We want to show that if $|ab|\geq d_{k}$, and $a$ and $b$ are separated by at
most $\ell$ vertices, then the number of top-$k$ distances satisfies $m_{\leq
k}\leq n+O(k^{2}(k+\ell)^{2})$. Let $S$ be the interval obtained from this
$[a,b]$ by extending onto $2k$ further points in both directions. By Fact 3.3,
all edges of length $\geq d_{k}$ have at least one endpoint in $S$. Note
$|S|=O(k+\ell).$
We will show an upper bound of $n+O(k^{2}(k+\ell)^{2})$ on the number of edges
$sx$ of length $\geq d_{k}$, with $s\in S,x\in V\backslash S$. This will
complete the proof since the only other top-$k$ distance edges must lie with
both endpoints in $S$, and there are at most $O(k+\ell)^{2}$ such edges.
The key observation is that in the bipartite graph between $S$ and
$V\backslash S$ consisting of these edges, all but a constant number of
vertices in $V\backslash S$ have degree 1. Specifically, if $sx,s^{\prime}x$
are both edges in this graph, then the location of $x$ is uniquely determined
by $s,s^{\prime},|sx|,$ and $|s^{\prime}x|$; it follows that
$\sum_{x}{\tbinom{\deg(x)}{2}}$ is at most $O((k+\ell)^{2}k^{2})$, and
consequently $\sum_{x:\deg(x)>1}\deg(x)=O((k+\ell)^{2}k^{2})$. We are then
done by counting the endpoints of degree-1 vertices, of which there are at
most $n$. ∎
## 4 The Algorithm
The algorithm we use to prove Theorem 1.4 examines distances among finite
configurations of points in the plane. Informally, we examine all possible
configurations of a bounded size, where a configuration includes all
occurrences of top-$k$ distances in a few consecutive levels, and we try to
establish that not too many top-$k$ distances can occur per level, averaged
over a small interval of levels. Thus ultimately, the argument in our proof
decomposes any global point set into local configurations of bounded size.
### 4.1 The Goal
Our computational goal will be to bound the number of long distances which can
occur in a consecutive sequence of several levels. We begin by re-proving (for
large $n$) Vesztergombi’s result on counting the second-largest distances; it
illustrates the type of computational result we need.
###### Proposition 4.1.
We have $m_{2}\leq\frac{4}{3}n$ for large enough $n$.
###### Proof.
We prove the theorem for $n\geq 3\cdot C(16,2)$ with $C$ as in Lemma 1.5. Let
a _special diagonal_ be a diagonal of length $d_{2}$ or longer, whose
endpoints are separated by at most 16 vertices. If there is any special
diagonal, we are done by Lemma 1.5. So we may assume there are no special
diagonals.
Using our computer program, we establish the following lemma.
###### Lemma 4.2.
In every point set $S$ without special diagonals, for every level $i$, at
least one of the following is true:
* •
at most $1=\lfloor 1\cdot\frac{4}{3}\rfloor$ diagonal in level $i$ has length
$d_{2}$;
* •
at most $2=\lfloor 2\cdot\frac{4}{3}\rfloor$ diagonals in levels $i$ and $i+1$
have length $d_{2}$;
* •
at most $4=\lfloor 3\cdot\frac{4}{3}\rfloor$ diagonals in levels
$i,\dotsc,i+2$ have length $d_{2}$;
* •
at most $5=\lfloor 4\cdot\frac{4}{3}\rfloor$ diagonals in levels
$i,\dotsc,i+3$ have length $d_{2}$.
Now let us see how this gives the desired result. Taking $i=1$, the four cases
above establish that for some $1\leq\gamma_{1}\leq 4$, the number of $d_{2}$’s
in levels $1,\dotsc,\gamma_{1}$ is at most $\frac{4}{3}\gamma_{1}$. Applying
the same logic to $i=\gamma_{1}+1$, we get that there is some
$1\leq\gamma_{2}\leq 4$ such that the number of $d_{2}$’s in levels
$\gamma_{1}+1,\dotsc,\gamma_{1}+\gamma_{2}$ is at most
$\frac{4}{3}\gamma_{2}$.
We continue defining further $\gamma_{i}$’s in the same way until
$\sum_{i=1}^{x}\gamma_{i}\equiv\sum_{i=1}^{y}\gamma_{i}\pmod{n}$ for some
$x<y$. Summing a contiguous subset of these bounds, the number of $d_{2}$’s in
levels from $1+\sum_{i=1}^{x}\gamma_{i}$ to $\sum_{i=1}^{y}\gamma_{i}$ is at
most $\frac{4}{3}$ per level on average. But this sum counts each of the $n$
levels an equal number of times, so the number of $d_{2}$’s overall is at most
$\frac{4}{3}n$. ∎
The computer program’s goal is thus to prove a general version of Lemma 4.2:
given a _target ratio_ $\alpha$ and _target distances_ (a subset of
$\\{d_{1},d_{2},\dotsc,d_{k}\\}$), find a constant $m$ so that every level $i$
admits $1\leq m^{\prime}\leq m$ such that $\leq m^{\prime}\cdot\alpha$ target
lengths occur in levels $i,\dotsc,i+m^{\prime}$. The program searches for a
point set with $>\alpha$ target diagonals in level 1, $>2\alpha$ in level 2,
etc. If the search terminates, the above proof shows the number of target
distances is $\leq\alpha n$. The hypothesis that no special diagonals exist is
used only indirectly by the program, explained below.
Our algorithm works with _configurations_ consisting of two disjoint intervals
of points, and an assignment of a distance from
$\\{d_{1},d_{2},\dotsc,d_{k},``<d_{k}"\\}$ to each diagonal spanning the two
intervals. We thereby obtain analogues of Lemma 4.2 by checking all possible
configurations up to some finite size. For this to work, Fact 3.2 is crucial
since it implies that all of the top-$k$ distances in $\ell$ consecutive
levels have all of their endpoints in two intervals of bounded size. We use an
incremental branch-and-bound search: it exhaustively searches all
possibilities, but in an efficient way where large sections of the search
space can be eliminated at once. Each individual step of the algorithm
corresponds to an application of one of the Facts 3.1–3.4. The lack of special
diagonals allows us to focus on _disjoint_ interval pairs. The Java
implementation is available at
http://sourceforge.net/projects/convexdistances/.
### 4.2 Configurations
In more detail, our algorithm maintains a set of _configurations_. Each
configuration has two disjoint intervals of points from $S$; then for each
diagonal generated by one point from each interval, the configuration stores a
set of possible values for the distance between those two points. Arbitrarily
name one interval the _top_ and denote its points as $\\{t_{i}\\}_{i}$, with
$t_{i+1}$ following $t_{i}$ in clockwise order, and name the other interval
the _bottom_ with points $\\{b_{i}\\}_{i}$, and $b_{i-1}$ following $b_{i}$ in
clockwise order. Then we denote the set of possible distances between $t_{i}$
and $b_{j}$ as $D[i,j]$; in each configuration $D[i,j]$ is a subset of
$\\{1,2,\dotsc,k,\infty\\}$ where $x\in D[i,j]$ means that $d_{x}$ is a
possible value for the distance $|t_{i}b_{j}|$, while $\infty\in D[i,j]$ means
that it is possible for $|t_{i}b_{j}|$ to be shorter than $d_{k}$. (So typical
steps in our program use special cases to reason with “$d_{\infty}$” distances
correctly.) Reiterating, a configuration consists of a top interval of
indices, a bottom interval of indices, and for each top-bottom pair a subset
of $\\{1,2,\dotsc,k,\infty\\}$.
We assume that $t_{i}b_{j}$ is in level number $j-i$ (modulo $n$), which is
without loss of generality. To gain some intuition and exhibit the notation,
it is helpful to look at a couple of examples. Our examples will be drawn from
actual point sets and therefore each $D[i,j]$ will be just a singleton, in
contrast to the larger sets $D[i,j]$ typically occurring in the algorithm. The
first example, shown in Figure 4, is a regular polygon of odd order. The
second example, shown in Figure 5, exhibits the extremal construction of
Vesztergombi for second distances [7].
Figure 4: Left: an odd regular polygon, with a top and bottom interval. Right:
the corresponding values of $D$, where entry $x$ in column $i$, row $j$
indicates $D[i,j]=\\{x\\}$. One level is illustrated on the left and circled
on the right.
Figure 5: Left: an illustration of Vesztergombi’s construction with
$m_{2}=\frac{4}{3}n-O(1)$. Some diagonals of length $d_{1}$ and $d_{2}$ are
shown (solid and dotted, respectively). Right: the corresponding
configuration; again, entry $x$ in column $i$, row $j$ indicates
$D[i,j]=\\{x\\}$.
### 4.3 Methodology
Here is an example of a typical step in the algorithm, shown in Figure 6.
Suppose some configuration includes points $t_{1},t_{2},b_{2},b_{1}$, suppose
that $D[1,1]=D[2,2]=\\{2\\}$, $D[1,2]=\\{2,3,\infty\\}$ and that
$D[2,1]=\\{1,2,3,\infty\\}$. Then using Fact 3.1, we know that
$|t_{1}b_{2}|+|t_{2}b_{1}|>|t_{1}b_{1}|+|t_{2}b_{2}|$. As the right-hand side
equals $2d_{2}$ and the maximum possible length of $t_{1}b_{2}$ is $d_{2}$, we
can deduce that $|t_{2}b_{1}|>d_{2}$ and so we may update the configuration
via $D[2,1]:=\\{x\in D[2,1]\mid x<2\\}=\\{1\\}$.
Figure 6: A typical step of the algorithm, using Fact 3.1.
The program uses Facts 3.1–3.4 in ways analogous to the above example.
Whenever one of the facts is applicable, we use it to reduce the size of one
set $D$ in the configuration. We use Fact 3.4 only when $a_{1},a_{i},a_{j}$
lie in the top interval and $a_{k},a_{l},a_{n}$ lie in the bottom or vice-
versa.
Our algorithm also makes use of another easy observation. In any instance $S$,
it cannot be true that both $d_{1}+d_{3}>d_{2}+d_{2}$ and
$d_{1}+d_{3}<d_{2}+d_{2}$. Hence using Fact 3.1, a quadruple
$t,t^{\prime},b^{\prime},b$ (in that cyclic order) with
$|tb|=|t^{\prime}b^{\prime}|=d_{2},|tb^{\prime}|=d_{1},|t^{\prime}b|=d_{3}$
cannot co-exist with another quadruple
$\hat{t},\hat{t}^{\prime},\hat{b}^{\prime},\hat{b}$ with
$|\hat{t}\hat{b}|=d_{1},|\hat{t}^{\prime}\hat{b}^{\prime}|=d_{3},|\hat{t}\hat{b}^{\prime}|=|\hat{t}^{\prime}\hat{b}|=d_{2}$.
More generally, given a configuration we can deduce from any
$i,j,i^{\prime},j^{\prime}$ with each
$D[i,j],D[i,j^{\prime}],D[i^{\prime},j],D[i^{\prime},j^{\prime}]$ singletons
other than $\\{\infty\\}$ that an inequality of the form
$d_{w}+d_{x}>d_{y}+d_{z}$ is true; in testing a configuration for validity our
program will reject any configuration where a contradiction arises from the
set of all such pairwise inequalities. This is done by testing the associated
digraph of $\tbinom{k+1}{2}$ pairs for acyclicity. (We also include arcs of
the form $d_{x}+d_{y}>d_{x}+d_{z}$ whenever $y<z$.)
In some situations none of these facts are applicable; say for example, if
each $D[i,j]$ is equal to $\\{1,2,\infty\\}$, we cannot conclude any further
information. In this case we use an approach which is similar to recursion or
_branch-and-bound_ in this situation, which works as follows. Find some $i,j$
with $|D[i,j]|>1$, let $X$ denote $D[i,j]$. We then replace this configuration
with two new configurations: each of the new ones is almost identical to the
original, except that in one we take $D[i,j]=\min_{x\in X}x$ and in the other
we take $D[i,j]=X\backslash\\{\min_{x\in X}x\\}$. In a little more detail,
while we are examining the levels from 1 to $L$, we only perform branching on
diagonals in levels 1 to $L$, (i.e. only when $1\leq j-i\leq L$) and any other
non-singleton $D[i,j]$ does not entail branching. This was faster in practice
than branching on every $D[i,j]$.
### 4.4 Initializing and Growing Configurations
Recall that our theorems are all of the following form, for a set $T$ of
positive integers and some real $\alpha$:
$\sum_{t\in T}m_{t}\leq\alpha n+O(1).$ ($\spadesuit$)
We call a _target distance_ any distance $d_{t}$ with $t\in T$. We use $k$ to
represent the largest number in $T$.
We begin this detailed section by explaining why it suffices to examine
configurations of bounded size to bound the number of target distances in $L$
consecutive levels. The key tool is Fact 3.3. Namely, suppose $t_{0}b_{1}$ is
any diagonal in level 1 with length $|t_{0}b_{1}|\geq d_{k}$, and consider any
top-$k$ distance diagonal $e$ in levels $1,\dotsc,L$. If $e$ crosses
$t_{0}b_{1}$, then $t_{0}$ (resp. $b_{1}$) is within $L$ steps along the
boundary from an endpoint of $e$ (resp. the other endpoint of $e$). If $e$ and
$t_{0}b_{1}$ don’t cross, one endpoint of $e$ is at most $2k$ steps from
$t_{0}$ or $b_{1}$ by Fact 3.3, and the other endpoint of $e$ is at most
$2k+L$ points away from the other of $t_{0}$ or $b_{1}$. Summarizing, in
either case, $e$ has one endpoint in the interval $I_{t}$ consisting of
vertices at most $2k+L$ steps from $t_{0}$, and $e$’s other endpoint lies in
the interval $I_{b}$ consisting of vertices at most $2k+L$ steps from $b_{1}$;
and this holds for all top-$k$ distance diagonals $e$ in levels $1,\dotsc,L$.
Our program makes valid deductions whenever these intervals are disjoint,
which is false only when $t_{0}$ and $b_{1}$ are within $2(2k+L)$ steps of one
another on the boundary. Set $\ell=2(2k+L)$ and define a _special diagonal_ to
be one with length $\geq d_{k}$ and at most $\ell$ vertices between its
endpoints. Recall that $|t_{0}b_{1}|\geq d_{k}$, so the program’s deductions
are valid unless there was a special diagonal. This explains the choice of
$16=2(2\cdot 2+4)$ in Proposition 4.1 and justifies our general approach.
In the rest of this section we explain some of the implementation details. The
program begins working with a configuration consisting of a single diagonal
$t_{0}b_{1}$ of length $\geq d_{k}$, and we assume without loss of generality
that there are no diagonals $t_{i}b_{i+1}$ such that $i<0$ and
$|t_{i}b_{i+1}|\geq d_{k}$. Thus the top and bottom intervals begins as the
singleton sets $\\{t_{0}\\},\\{b_{1}\\}$.
We will now enlarge these configurations. Reviewing our proof strategy, the
program must enumerate all possible configurations such that level 1 has more
than $\alpha$ diagonals of a target length, _and_ levels 1 and 2 together have
more than $2\alpha$, etc, with the hope being that once the number of levels
is high enough we find that no such configurations exist, since this would
give a result like Lemma 4.2.
Note that, by our choice of $t_{0}$ and $b_{1}$ which normalize our indices,
in any convex point set, all level-1 diagonals of the target distances are of
the form $t_{i}b_{i+1}$ for $i>1$, and by Fact 3.3 they also satisfy $i\leq
2k-2$, so crucially, their possible positions are confined to an interval of
bounded size. We now determine which of these diagonals have target lengths by
_exhaustive guessing_ , a term which simply means trying all possibilities. In
detail, first, exhaustively guess the smallest $i>0$ for which $t_{i}b_{i+1}$
is a target distance, then the second-smallest, etc. When the top and bottom
intervals are enlarged, each new $D[i,j]$ is set to $\\{1,\dotsc,k,\infty\\}$
by default, meaning that no assumptions are made on the distance. When $i$ is
guessed as a minimal new level-1 diagonal for which $t_{i}b_{i+1}$ is a target
distance, rather than the defaults we set $D[i,i+1]=T$ and
$D[i^{\prime},i^{\prime}+1]:=\\{1,\dotsc,k,\infty\\}\backslash T$ for all new
$i^{\prime}<i$.
• Initialize a configuration with intervals $\\{t_{0}\\},\\{b_{1}\\}$ and
$D[0,1]$ set to $T$ (all target distances) • For $L=1,2,\dotsc$ – Extend the
configurations by exhaustively guessing all diagonals of target lengths in
level $L$, extending leftwards first if $L>1$, and then rightwards in all
cases. – Keep only configurations with more than $\alpha L$ target distances
in levels $1,\dotsc,L$. – Stop if no configurations remain. • Upon extending a
configuration, check it: – Use Facts 3.1–3.4 to perform deductions. – Check
that distance pairs are consistent. – If $|D[i,j]|>1$ for some diagonal
$t_{i}b_{j}$ in one of the first $L$ levels, partition it into two
configurations and check both (recursively).
Figure 7: Sketch of the algorithm.
After each new diagonal is added, we re-apply Facts 3.1–3.4 in order to make
additional deductions and eliminate any impossible configuration; and we split
any non-singleton sets $D$ in the first level, as described earlier.
After this exhaustive guessing, we have collected all possible configurations.
We keep only those for which level 1 has more than $\alpha$ diagonals of the
target lengths. If any exist, we grow them in all possible ways to 2-level
configurations, using exhaustive guessing like that explained above, except
that we expand “to the left” before expanding “to the right” (for level 1,
only rightwards expansion was needed due to our choice of $t_{0}$ and
$b_{1}$). Again, we prune those which have no more than $2\alpha$ target
distance in the first two levels.
We repeat the process described in the previous paragraph over and over,
increasing the number of levels by 1 each time. If the program terminates
eventually, it implies a result of the form like Lemma 4.2 and consequently
that ($\spadesuit$ ‣ 4.4) holds for this choice of $T$ and $\alpha$. We give a
high-level review of the algorithm in Figure 7.
## 5 Results: Proof of Theorem 1.4
Each row in Table 1 corresponds to an execution of our program which
terminated. In other words, each execution establishes that an analogue of
Lemma 4.2 holds, and we consequently deduce Theorem 1.4 using reasoning as in
the proof of Proposition 4.1. Each line proves
$\sum_{t\in T}m_{t}\leq\alpha n\textrm{ for $n>C(k,2(2k+L))/(\alpha-1)$},$
($\clubsuit$)
where $k$ is the largest element of $T$, and $C$ is the constant from Lemma
1.5. Note that the first two lines of Table 1 correspond to results that were
already known. The running times are from a computer with a 2 GHz processor.
The program was written in Java, and is available on
SourceForge222http://sourceforge.net/projects/convexdistances/. For
$T=\\{1,2,3,4,5\\}$ or $T=\\{5\\}$ the program ran out of memory before
obtaining any reasonable result.
$T$ | $\alpha$ | $L$ | time (s) | tightness of result
---|---|---|---|---
$\\{1,2\\}$ | $2$ | $2$ | $<1$ | tight (odd regular)
$\\{2\\}$ | $4/3$ | $4$ | $<1$ | tight [7]
$\\{1,2,3\\}$ | $3$ | $3$ | $<1$ | tight (odd regular)
$\\{3\\}$ | $3/2$ | $9$ | $5$ | abstractly tight, Fig. 8
$\\{2,3\\}$ | $9/4$ | $6$ | $1$ | abstractly tight, Fig. 9
$\\{1,3\\}$ | $2$ | $4$ | $<1$ | tight (odd regular)
$\\{1,2,3,4\\}$ | $4$ | $3$ | $68$ | tight (odd regular)
$\\{4\\}$ | $13/8$ | $27$ | $50890$ | unknown
Table 1: The terminating executions of our program, each one proving
($\clubsuit$ ‣ 5) for that $\alpha$ and $T$. _Tight_ means convex point sets
are known with $\sum_{t\in T}m_{t}=\alpha n-O(1)$ and _abstractly tight_ means
some periodic configuration has $\sum_{t\in T}m_{t}=\alpha n$ but we could not
realize it convexly in the plane.
## 6 Abstract Tightness
Our computer program can also generate tight examples. In Figure 8 we show two
periodic configurations with $m_{3}=\frac{3}{2}n$ with periods of 6 and 8
levels, respectively. (No other example has period less than 14.) We were not
able to embed these examples as convex point sets in the plane, and at the
same time we did not disprove that they were embeddable. Based on our
attempts, it seems like there is no simple periodic embedding respecting the
natural symmetries of the distance configurations. A disproof of realizability
could be used in the program to get stronger results. For
$m_{2}+m_{3}=\frac{9}{4}n$ we also have an abstractly tight periodic example
which we could not realize (Fig. 9).
Figure 8: Two unrealized periodic configurations with $m_{3}=\frac{3}{2}n$.
Rows and columns are two intervals of vertices, and entry $i$ (resp. $\infty$)
means distance $d_{i}$ (resp. $<d_{3}$). Figure 9: An unrealized periodic
configurations with $m_{2}+m_{3}=\frac{9}{4}n$.
## 7 Future Directions
Our program is essentially a depth-first search; each configuration examined
by the program has a unique “parent” configuration from which it was grown.
Thus, it would be possible to rewrite the program so as to use a smaller
amount of memory and thereby possibly obtain results with smaller $\alpha$ or
larger $k$; and a distributed implementation should also be straightforward.
It would be good to come up with constructions exhibiting better lower bounds.
For example, no construction is known where $m_{3}/n$ is asymptotically
greater than 4/3.
Our approach constitutes an abstract generalization of the original problem of
bounding sums of the $m_{i}$’s in convex point sets. Vesztergombi [7]
considered an abstraction as well, using only a subset of the facts we applied
here. Can Conjecture 1.1 of Erdős and Moser be violated in either of these
abstractions?
Finally, can the functions $C,C^{\prime}$ in Lemma 1.5 and Lemma 1.6 be
improved?
Acknowldegments. We thank the referees for useful feedback, and K.
Vesztergombi for helpful discussions.
## References
* [1] E. Altman: On a problem of P. Erdős, The American Math. Monthly Vol. 70, No. 2 (Feb., 1963), pp. 148–157.
* [2] P. Brass, W. Moser, J. Pach: Research problems in discrete geometry, Springer Verlag, New York, 2005.
* [3] H. Edelsbrunner, P. Hajnal: A lower bound on the number of unit distances between the vertices of a convex polygon, J. Comb. Theory A 56(2) (1991) 312–316
* [4] P. Erdős, L. Lovász, K. Vesztergombi: On the graph of large distances, Discrete Comput. Geom. 4 (1989) 541–549.
* [5] H. Hopf, E. Pannwitz: _Aufgabe Nr. 167_ , Jahresbericht Deutsch. Math.-Verein. 43(1934) p. 114
* [6] K. Vesztergombi: On large distances in planar sets, Discrete Math. 67 (1987) 191–198
* [7] K. Vesztergombi: On the distribution of distances in finite sets in the plane, Discrete Math. 57 (1985), 129–145
|
arxiv-papers
| 2011-03-02T12:46:34 |
2024-09-04T02:49:17.393656
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Filip Mori\\'c and David Pritchard",
"submitter": "David Pritchard",
"url": "https://arxiv.org/abs/1103.0412"
}
|
1103.0424
|
# Fresnel aperture diffraction: a phase-sensitive probe for superconducting
pairing symmetry
C. S. Liu1,2 and W. C. Wu1 1Department of Physics, National Taiwan Normal
University, Taipei 11677, Taiwan
2Department of Physics, Yanshan University, Qinhuangdao 066004, China
1Department of Physics, National Taiwan Normal University, Taipei 11677,
Taiwan
2Department of Physics, Yanshan University, Qinhuangdao 066004, China
###### Abstract
Fresnel single aperture diffraction (FSAD) is proposed as a phase-sensitive
probe for pairing symmetry and Fermi surface of a superconductor. We consider
electrons injected, through a small aperture, into a thin superconducting (SC)
layer. It is shown that in case of SC gap symmetry
$\Delta(-k_{x},\mathbf{k}_{\parallel})=\Delta(k_{x},\mathbf{k}_{\parallel})$
with $k_{x}$ and $\mathbf{k}_{\parallel}$ respectively the normal and parallel
component of electron Fermi wavevector, quasiparticle FSAD pattern developed
at the image plane is zeroth-order minimum if $k_{x}x=n\pi$ ($n$ is an integer
and $x$ is SC layer thickness). In contrast, if
$\Delta(-k_{x},\mathbf{k}_{\parallel})=-\Delta(k_{x},\mathbf{k}_{\parallel})$,
the corresponding FSAD pattern is zeroth-order maximum. Observable
consequences are discussed for iron-based superconductors of complex multi-
band pairings.
###### pacs:
74.20.Mn, 74.20.Rp, 74.25.Jb, 74.25.Ha
Recently high-$T_{c}$ superconductivity has been observed in several classes
of Fe-pnictide materials kamihara ; Hsu . One key issue for understanding the
superconductivity of pnictides lies on the pairing symmetry of the Cooper
pairs. A conclusive observation of the pairing symmetry remains unsettled to
which both nodal and nodeless order parameters were reported in recent
experiments, however. ARPES measurements indicated clearly a nodeless gap at
all points on Fermi surfaces (FS) Ding_EPL ; zhao and magnetic penetration
depth measurements further suggested the gap being possibly in the $s^{\pm}$
state PhysRevLett.102.127004 ; Martin ; mazin:057003 . The $s^{\pm}$ state is
currently a promising pairing candidate for iron pnictides, which changes sign
between $\alpha$ and $\beta$ bands and can be naturally explained by the spin
fluctuation mechanism mazin:057003 ; wang:047005 ; Tsuei2010 . On the
contrary, the scanning SQUID microscopy measurements seemed to exclude the
spin-triplet pairing states and suggested the order parameter having well-
developed nodes [hicks-2008, ]. In addition, NMR experiments were also in
favor of nodal SC order parameters Matano ; Grafe08 . For phase sensitive
experiments, one point-contact spectroscopy reported was in favor of a nodal
gap Shan , while the other reported was in favor of a nodeless gap chen .
The complex pairing symmetry of these materials suggests that the pairing
mechanism is likely non-universal and may depend strongly on the fine details
of the band structures. With this regard, some possible experiments were
proposed to elucidate these issues zhou09 ; Feng2009 ; Lin2010 . In this
paper, Fresnel single aperture diffraction (FSAD) of electrons is proposed as
a phase-sensitive probe for studying the SC pairing symmetry. Of particular
interest, it is suggested that FSAD could be very useful for studying the
iron-pnictide superconductors of complex multiple FS pairings. It will be
shown that large $Z$ (effective potential barrier) zeroth-order FSAD pattern
is sensitive to both the SC phase and the probing direction and thus can give
an unambiguous signal to distinguish different pairing symmetries among
different FSs.
Fig. 1 sketches the proposed apparatus of a FSAD experiment. A substrate
layer, made of a good conductor, is grown firstly. Next, a sheet of electron-
density sensitive developer for recording the diffraction pattern is
deposited. The developer can be made either by the electron-sensitive material
(like the photographic film) or alternatively by the fluorescent material
(like the TV screen). On top of the recording sheet, a thin layer of SC single
crystal with desired orientation and thickness is assembled. The last step is
to coat an insulating layer on the thin SC layer with a small circular
aperture (by mechanical and/or optical method) for electron beam injection.
While electron beam can be made by natural radioactivity or low-energy
accelerator, it can be alternatively due to a sharp conductive STM tip by an
applied voltage bias. For the latter case, an insulating layer is not needed
because the separation between the tip and the SC thin layer already acts like
an insulating layer between it.
Figure 1: (Color online) Schematic illustration of the Fresnel single aperture
diffraction experiment for a thin superconductor layer (with thickness $x$).
Reflection and transmission processes of a NIS tunnelling junction are shown.
The developer is where the diffraction pattern is recorded.
When electrons tunnel into the superconductor through the circular aperture,
matter waves can interfere constructively or destructively. With enough
electrons passing through, clear diffraction pattern can be recorded in the
developer while extra electrons will flow into the ground (see Fig. 1). To
maintain the coherence for the FSAD signal, the thickness of the thin SC layer
should be made comparable to the SC coherence length. Moreover, quasiparticles
needs to be in the ballistic transport regime or the signal will be less
clear. Scanning tunneling spectroscopy and vortex imaging have revealed that
iron-pnictide superconductors have a short coherence length, $\xi\approx
27.6\pm 2.9$Å Yin:097002 , comparable to that of cuprate superconductors
($\xi\leq 20$Å) Pan . Nevertheless, modern film growing technique has recently
advanced that by improving the quality of the substrate which minimizes the
inverse proximity effect, a nearly perfect ultrathin high-$T_{c}$ SC layer can
be grown as thin as three unit cells only Logvenov . This makes the proposed
FSAD experiment feasible.
Quasiparticle (QP) states of an inhomogeneous superconductor have a coupled
electron-hole character and can be described by the BdG equations Gennes
$\displaystyle E_{\mathbf{k}}u$ $\displaystyle=$ $\displaystyle
h_{0}u+\Delta_{\mathbf{k}}v$ $\displaystyle E_{\mathbf{k}}v$ $\displaystyle=$
$\displaystyle\Delta^{*}_{\mathbf{k}}u-h_{0}v,$ (1)
where $h_{0}\equiv-\hbar^{2}\nabla^{2}/2m-\mu$ with $\mu$ the chemical
potential and $m$ the electron mass. We consider the quantum tunneling in an
NIS junction where the thin SC layer is made normal to the $x$-axis and the
pairing potential is assumed to be $\sim\Delta_{\mathbf{k}}\Theta(x)$ with
$\Theta\left(x\right)$ the Heaviside step function and $\Delta_{\mathbf{k}}$
the SC gap function in the momentum space Hu1526 . For simplicity, proximity
effect of the SC order parameter is ignored at the interface. Under the WKBJ
approximation Bardeen556 , QP wavefunctions for the SC thin layer side ($x>0$)
are
$\left(\begin{array}[]{c}u\\\
v\end{array}\right)=\left(\begin{array}[]{c}{e}^{i\mathbf{k}_{F}\cdot\mathbf{r}}\tilde{u}\\\
{e}^{-i\mathbf{k}_{F}\cdot\mathbf{r}}\tilde{v}\end{array}\right)~{}\mathrm{and}~{}\left(\begin{array}[]{c}\tilde{u}\\\
\tilde{v}\end{array}\right)=e^{-\gamma x}\left(\begin{array}[]{c}\hat{u}\\\
\hat{v}\end{array}\right),$ (2)
where $\mathbf{k}_{F}\equiv(k_{x},\mathbf{k}_{\parallel})$ is the Fermi
wavevector and $\gamma$ is the attenuation constant. With Eq. (2), Eq. (1) can
be reduced to the Andreev equation
$E_{\mathbf{k}}\left(\begin{array}[]{c}\hat{u}\\\
\hat{v}\end{array}\right)=\left(\begin{array}[]{cc}\varepsilon&\Delta_{\mathbf{k}}\\\
\Delta_{\mathbf{k}}&-\varepsilon\end{array}\right)\left(\begin{array}[]{c}\hat{u}\\\
\hat{v}\end{array}\right),$ (3)
where $\varepsilon\equiv i\gamma k_{x}/m$. The wavevector parallel to the
interface, $\mathbf{k}_{\parallel}$, is conserved during the processes
Tanaka3451 .
Solving Eq. (3), one obtains two degenerate eigenstates corresponding
respectively to electron- and hole-like QPs:
$\psi_{\mathbf{k}}^{e}(\mathbf{r})=\left(\begin{array}[]{c}\Delta_{+}\\\
E_{\mathbf{k}}-\varepsilon\end{array}\right){e}^{i\mathbf{k}_{F}\cdot\mathbf{r}};~{}\psi_{\mathbf{k}}^{h}(\mathbf{r})=\left(\begin{array}[]{c}E_{\mathbf{k}}+\varepsilon\\\
\Delta_{-}\end{array}\right){e}^{-i\mathbf{k}_{F}\cdot\mathbf{r}},$ (4)
where $E_{\mathbf{k}}=\sqrt{\Delta_{\mathbf{+}}^{2}+\varepsilon^{2}}$,
$\Delta_{+}\equiv\Delta(k_{x},\mathbf{k}_{\parallel})=\Delta\left(\theta\right)$,
and
$\Delta_{-}\equiv\Delta(-k_{x},\mathbf{k}_{\parallel})=\Delta\left(\pi-\theta\right)$
(scattering angle $\theta$ is defined in Fig. 1). Superposition of the above
two eigenstates will give a resulting wave function for the SC layer
$\psi_{S}(\mathbf{r})=e^{-\gamma
x}\left[c_{1}\psi^{e}_{\mathbf{k}}\left(\mathbf{r}\right)+c_{2}\psi^{h}_{\mathbf{k}}\left(\mathbf{r}\right)\right].$
(5)
The coefficients $c_{1}$ and $c_{2}$ are important which are to be determined
by the boundary conditions. Apart from the factor $e^{-\gamma x}$, Eqs.
(2)–(5) give explicitly
$\begin{array}[]{c}\hat{u}\left(\mathbf{r}\right)=c_{1}\Delta_{+}{e}^{i\mathbf{k}_{F}\cdot\mathbf{r}}+c_{2}\left(E_{\mathbf{k}}+\varepsilon\right){e}^{-i\mathbf{k}_{F}\cdot\mathbf{r}},\\\
\hat{v}\left(\mathbf{r}\right)=c_{1}\left(E_{\mathbf{k}}-\varepsilon\right){e}^{i\mathbf{k}_{F}\cdot\mathbf{r}}+c_{2}\Delta_{-}{e}^{-i\mathbf{k}_{F}\cdot\mathbf{r}}.\end{array}$
(6)
When an electron is injected into the SC layer through the aperture, there are
two types of reflections: normal reflection of electrons (with the coefficient
$r_{N}$) and Andreev reflection of holes (with the coefficient $r_{A}$). In
terms of $r_{N}$ and $r_{A}$, the resulting wave function for the normal side
($x<0$) can be written as
$\psi_{N}\left(\mathbf{r}\right)=\left[\begin{array}[]{c}{e}^{i\mathbf{k}_{F}\cdot\mathbf{r}}+r_{N}{e}^{-i\mathbf{k}_{F}\cdot\mathbf{r}}\\\
r_{A}{e}^{i\mathbf{k}_{F}\cdot\mathbf{r}}\end{array}\right].$ (7)
By applying the following boundary conditions:
$\displaystyle\psi_{N}\left(\mathbf{r}\right)|_{x=0^{-}}=\psi_{S}\left(\mathbf{r}\right)|_{x=0^{+}},$
(8)
$\displaystyle\frac{2mH}{\hbar^{2}}\psi_{S}\left(\mathbf{r}\right)|_{x=0^{+}}=\frac{d\psi_{S}\left(\mathbf{r}\right)}{dz}|_{x=0^{+}}-\frac{d\psi_{N}\left(\mathbf{r}\right)}{dz}|_{x=0^{-}}$
with $H$ the height of the delta-function potential for the barrier,
coefficients in (6) are solved to be $c_{1}=\Delta_{-}(1-iZ)/D$ and
$c_{2}=iZ(E_{\bf k}+\varepsilon)/D$ with
$D=\Delta_{+}\Delta_{-}(1+Z^{2})-Z^{2}(E_{\bf k}-\varepsilon)^{2}$ and
$Z\equiv 2mH/\hbar^{2}k_{F}$ being the effective potential barrier.
In general, the diffraction pattern recorded in the developer will be
proportional to the QP density developed on it. In the current case, the FSAD
intensity is proportional to
$I({\bf
r})=\frac{1}{S}\sum_{i,\mathbf{k}}\left[\left|\hat{u}_{i}\left(\mathbf{r}\right)\right|^{2}f\left(E_{\mathbf{k}}\right)+\left|\hat{v}_{i}\left(\mathbf{r}\right)\right|^{2}f\left(-E_{\mathbf{k}}\right)\right]\Delta
S_{i},$ (9)
where $f(E_{\mathbf{k}})=(e^{\beta E_{\mathbf{k}}}+1)^{-1}$ and considering
the size effect, a spatial average over the aperture (of area $S$) is taken
where $\Delta S_{i}$ denotes a tiny cell within $S$.
_Gap symmetry and barrier $Z$ dependent FSAD_ – For simplicity, we shall
consider the limit such that aperture diameter $d$ is much smaller than the
thickness $x$ of the SC layer. This means that the spatial average in (9) is
not needed. In the limit of $T\rightarrow 0$, interesting results of SC gap
symmetry and barrier $Z$ dependent FSAD will be obtained. Knowing the
coefficients $c_{1}$ and $c_{2}$, QP wavefunctions developed at ${\bf
r}=(x,0,0)$ on the developer are obtained to be
$\displaystyle\hat{u}(x)$ $\displaystyle=$ $\displaystyle\hat{v}(x)$ (10)
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}e^{ik_{x}x}-2iZ\sin(k_{x}x),&~{}\mathrm{if}~{}\Delta_{-}=\Delta_{+}\\\
e^{ik_{x}x}-2iZ\cos(k_{x}x),&~{}\mathrm{if}~{}\Delta_{-}=-\Delta_{+}~{}.\end{array}\right.$
(13)
There are many observable consequences out of the above symmetry-dependent
results. In the following, we illuminate how one can probe the pairing
symmetry and Fermi surface of a superconductor based on Eq. (10).
When barrier is low, $Z\ll 1$,
$\hat{u}(x)=\hat{v}(x)=\exp\left(ik_{x}x\right)$ for both even
($\Delta_{-}=\Delta_{+}$) and odd ($\Delta_{-}=-\Delta_{+}$) symmetries. In
this limit, FSAD pattern recorded in the developer makes no difference between
the two symmetries. In this case, $I=1$ and a zeroth-order maximum FSAD
pattern (Airy disk) will occur. Nevertheless, the $Z\ll 1$ FSAD pattern can be
used to measure the FS of the SC sample. Using the well-known formula
$\sin\theta=1.22\lambda/d$ ($d$ is the aperture diameter) that locates first
minimum of the Airy pattern, one can measure $\theta$ which determines the de
Broglie wavelength of electrons, $\lambda$, and hence unambiguously identify
the Fermi vector along the particular direction via the relation,
$k_{x}=2\pi/\lambda$. It should be emphasized that the above result remains
valid even when $Z$ is not too small ($Z\lesssim 1$).
Figure 2: (Color online) Illustration of SC gap symmetry dependent FSAD in the
large-$Z$ limit.
Of most interest is when the barrier is high, $Z\gg 1$, to which _only the
first Fresnel zone appears when $d$ is small enough_. In this limit,
$\hat{u}(x)=\hat{v}(x)\sim\sin(k_{x}x)$ for even symmetry and
$\sim\cos(k_{x}x)$ for odd symmetry. Consequently
$I\sim\left\\{\begin{array}[]{ll}\sin^{2}(k_{x}x),&~{}~{}~{}~{}\mathrm{if}~{}\Delta_{-}=\Delta_{+}\\\
\cos^{2}(k_{x}x),&~{}~{}~{}~{}\mathrm{if}~{}\Delta_{-}=-\Delta_{+}\end{array}\right.$
(14)
and the zeroth-order FSAD pattern developed behaves drastically different
between the two symmetries. Experimentally to create a high barrier $Z$ a thin
insulating layer can be coated on the SC layer in assembling the FSAD
apparatus. Alternatively, $Z$ can be tuned by adding a bias voltage in the
substrate layer relative to the ground, in addition to the bias voltage
between the tip and the substrate layer. Moreover, for the large-$Z$ limit, it
is well-known that for even-parity pairing, maximum conductance occurs when
incident electron energy equals the gap amplitude, $E\approx\Delta$. While for
odd-parity pairing, owing to the zero-bias bound (midgap) state, maximum
conductance occurs at $E\approx 0$ Tanaka3451 . Thus for the present FSAD
experiment, one can try to tune the incident electron energy to gain maximum-
intensity signal.
Knowing the Fermi vector $k_{x}$ (at particular direction), one may grow the
SC film for the FSAD experiment with the desired thickness $x$ which satisfies
$k_{x}x=n\pi$ ($n$ is an integer) and is comparable to the coherence length
$\xi$. Consequently for even symmetry, $I\sim\sin^{2}\pi$ and FSAD will show a
zeroth-order _minimum_ at the developer. In contrast, for odd symmetry,
$I\sim\cos^{2}\pi$ and the FSAD will show a zeroth-order _maximum_.
Fig. 2 illustrates the large-$Z$ gap-symmetry dependent FSAD pattern for
different symmetries. While iron-pinicides seem to be spin-singlet
superconductors with possibly $s$\- and/or $d$-wave pairing symmetries
[hicks-2008, ], for completeness and for references to a spin-triplet
superconductor of interest, we have also considered the cases of $p$-wave
symmetry in Fig. 2 (and also in Table 1). As is shown, for all cases we
consider that the direction of injected electron, $\mathbf{k}$, is pointing
near the $k_{x}$ axis (with an angle $\theta$). For $s$-wave gap,
$\Delta_{-}$= $\Delta_{+}$ and the corresponding FSAD will be a zeroth-order
minimum, which is apparently independent of the direction of electron
injected. However, for nodal superconductors such as $p$\- and $d$-wave cases,
the results are two folds. If $\mathbf{k}$ is pointing such that $\Delta_{-}$=
$\Delta_{+}$ (for example the $p_{y}$ and $d_{x^{2}-y^{2}}$ symmetries in Fig.
2), the corresponding FSAD will show a zeroth-order minimum. In contrast, if
$\mathbf{k}$ is pointing such that $\Delta_{-}$= $-\Delta_{+}$ (for example
the $p_{x}$ and $d_{xy}$ symmetries in Fig. 2), the corresponding FSAD will
show a zeroth-order maximum. It is worth noting that for all $p$\- and
$d$-wave nodal cases, if $\mathbf{k}$ is pointing right at the nodes,
$\Delta_{+}=\Delta_{-}=0$, the corresponding FSAD pattern will show a zeroth-
order maximum, analogous to the case of a normal metal.
Figure 3: (Color online) Schematic of the Fermi surfaces of Fe-pnictide
superconductors in folded Brillouin zone. Important incident electron
directions for FSAD experiment are shown.
Moreover, for both $d_{x^{2}-y^{2}}$ and $d_{xy}$ symmetries, zeroth-order
FSAD pattern could change from maximum (minimum) to minimum (maximum) if the
SC layer is grown with $\pi/4$ rotated about the $c$-axis (assuming that SC
gap mainly develops in the $ab$ plane). Similarly, for both $p_{x}$ and
$p_{y}$ symmetries, zeroth-order FSAD pattern could change from maximum
(minimum) to minimum (maximum) if the SC layer is grown with $\pi/2$ rotated
about the $c$-axis. This gives another machinery for FSAD to distinguish
between $s$-, $p$-, and $d$-wave pairing symmetries.
We now discuss possible schemes of FSAD patterns for multiband iron-pnictide
superconductors. The so-called $\alpha$ sheets are concentric and nearly
circular hole pockets around the $\Gamma$ point. While the $\beta$ sheets are
nearly circular electron pockets around the M points singh:237003 ; cao:220506
. These FS sheets are sketched in Fig. 3. If the pairing originates from the
same mechanism, most likely $\alpha_{1}$ and $\alpha_{2}$ bands will have the
same pairing symmetry. Similarly $\beta_{1}$ and $\beta_{2}$ bands will also
likely have the same pairing symmetry. However, pairing symmetries may differ
between $\alpha$ and $\beta$ bands. Among other experiments, one can actually
perform FSAD experiment to test the pairing symmetry of each _individual_ band
by carefully tuning the energy of incident electrons for maximum intensity
with desired orientation and layer thickness.
Table 1: Possible FSAD patterns for various pairing symmetries and incident directions shown in Fig. 3. symmetry | $\Gamma$X | $\Gamma$X′ | $\Gamma$M | $\Gamma$M′ | $\Gamma$X⊥
---|---|---|---|---|---
$s$ | min | min | min | min | min
$p_{x}$ | max | max | max | max | max
$p_{y}$ | max | min | min | min | min
$d_{x^{2}-y^{2}}$ | min | min | max | min | min
$d_{xy}$ | max | max | max | max | max
More explicitly, one can first grow a set of thin layers with different
crystal directions, and then perform small-$Z$ FSAD experiments to measure and
compile the FSs. With the knowing FSs, one can grow another set of thin layers
with desired thickness $x$ and crystal directions. For instance, if one thin
layer has $x$ simultaneously satisfying $k_{1}x=n_{1}\pi$ and
$k_{2}x=n_{2}\pi$ with $n_{1},n_{2}$ both integers and $k_{1}$ and $k_{2}$ the
corresponding Fermi vectors of $\alpha_{1}$ and $\alpha_{2}$ (or $\beta_{1}$
and $\beta_{2}$) bands, one can then perform large-$Z$ FSAD experiment on this
thin layer to sort out the pairing symmetry on $\alpha$ and/or $\beta$ bands.
Taking LaO1-xFxFeAs as an example, if $k_{x}\simeq 0.22\pi/a$ for $\beta$-band
with lattice constant $a\simeq 0.4$nm raghu_prb_08 ; Takahashi_nature_08 , the
thickness of the SC film can be better taken to be $x=n\pi/k_{x}\simeq
n(4.55a)\simeq n(1.84\mathrm{nm})$. For $n=2$, $x\simeq
3.68\mathrm{nm}=36.8$Å.
In Fig. 3, important directions of incident electrons of FSAD experiment are
indicated for iron-pnictides. Possible FSAD patterns for various incident
directions and pairing symmetries are listed in Table 1. Note that it is also
important to do the FSAD experiment for the $\Gamma$X′ and $\Gamma$M′
directions which are slightly deviated from the $\Gamma$X and $\Gamma$M
directions. In view of Table 1, if the zeroth-order FSAD pattern changes from
maximum for $\Gamma$X to minimum for $\Gamma$X′ direction, pairing symmetry is
likely to be $p_{y}$-wave. Similarly, it is likely to be
$d_{x^{2}-y^{2}}$-wave if it changes from maximum for $\Gamma$M to minimum for
$\Gamma$M′ direction.
The proposed FSAD experiment is sensitive to the pairing gap symmetry on one
particular FS. Due to the nature of a zero momentum transfer probe, it cannot
link the pairing gap symmetries on two distant FSs. For iron-pnictides, one
can use the experiment to check whether it’s $s$-wave on both $\alpha$ and
$\beta$ bands which is consistent with the $s_{\pm}$ state, or $s$-wave on one
band and $d$-wave on the other band. However, it is not able to tell if there
is a sign change between the two bands. To verify the sign change, other
experiments which can link the pairing symmetries on two distant FSs are in
demand.
In summary, we propose that Fresnel single aperture diffraction (FSAD) could
be a useful phase-sensitive probe for the pairing symmetry of a
superconductor. It is demonstrated that FSAD pattern is intimately related to
the SC pairing symmetry and the direction of incident electrons. Possible
designs of FSAD experiment are suggested and discussed for iron-pnictide
superconductors of complex multiple Fermi surface pairings. It is noted that
the same scheme discussed in the present paper can also be applied to other
phase-sensitive experiments, such as Young’s interference and Fresnel lens.
###### Acknowledgements.
This work was supported by National Science Council of Taiwan (Grant No.
99-2112-M-003-006), Hebei Provincial Natural Science Foundation of China
(Grant No. A2010001116), and National Natural Science Foundation of China
(Grant No. 10974169). We also acknowledge the support from the National Center
for Theoretical Sciences, Taiwan.
## References
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|
arxiv-papers
| 2011-03-02T13:12:11 |
2024-09-04T02:49:17.399084
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. S. Liu, W. C. Wu",
"submitter": "Cheng Shi Liu",
"url": "https://arxiv.org/abs/1103.0424"
}
|
1103.0435
|
# Two are better than one: Fundamental parameters of frame coherence
Waheed U. Bajwa Robert Calderbank Dustin G. Mixon Department of Electrical
and Computer Engineering, Rutgers, The State University of New Jersey,
Piscataway, New Jersey 08854, USA Department of Electrical and Computer
Engineering, Duke University, Durham, North Carolina 27708, USA Program in
Applied and Computational Mathematics, Princeton University, Princeton, New
Jersey 08544, USA
###### Abstract
This paper investigates two parameters that measure the coherence of a frame:
worst-case and average coherence. We first use worst-case and average
coherence to derive near-optimal probabilistic guarantees on both sparse
signal detection and reconstruction in the presence of noise. Next, we provide
a catalog of nearly tight frames with small worst-case and average coherence.
Later, we find a new lower bound on worst-case coherence; we compare it to the
Welch bound and use it to interpret recently reported signal reconstruction
results. Finally, we give an algorithm that transforms frames in a way that
decreases average coherence without changing the spectral norm or worst-case
coherence.
###### keywords:
frames , worst-case coherence , average coherence , Welch bound , sparse
signal processing
††journal: Applied and Computational Harmonic Analysis
## 1 Introduction
Many classical applications, such as radar and error-correcting codes, make
use of over-complete spanning systems [46]. Oftentimes, we may view an over-
complete spanning system as a _frame_. Take $F=\\{f_{i}\\}_{i\in\mathcal{I}}$
to be a collection of vectors in some separable Hilbert space $\mathcal{H}$.
Then $F$ is a frame if there exist _frame bounds_ $A$ and $B$ with $0<A\leq
B<\infty$ such that $A\|x\|^{2}\leq\sum_{i\in\mathcal{I}}|\langle
x,f_{i}\rangle|^{2}\leq B\|x\|^{2}$ for every $x\in\mathcal{H}$. When $A=B$,
$F$ is called a _tight frame_. For finite-dimensional unit norm frames, where
$\mathcal{I}=\\{1,\ldots,N\\}$, the _worst-case coherence_ is a useful
parameter:
$\mu_{F}:=\max_{\begin{subarray}{c}i,j\in\\{1,\ldots,N\\}\\\ i\neq
j\end{subarray}}|\langle f_{i},f_{j}\rangle|.$ (1)
Note that orthonormal bases are tight frames with $A=B=1$ and have zero worst-
case coherence. In both ways, frames form a natural generalization of
orthonormal bases.
In this paper, we only consider finite-dimensional frames. Those not familiar
with frame theory can simply view a finite-dimensional frame as an $M\times N$
matrix of rank $M$ whose columns are the frame elements. With this view, the
tightness condition is equivalent to having the spectral norm be as small as
possible; for an $M\times N$ unit norm frame $F$, this equivalently means
$\|F\|_{2}^{2}=\frac{N}{M}$.
Throughout the literature, applications require finite-dimensional frames that
are nearly tight and have small worst-case coherence [11, 21, 31, 37, 46, 47,
50, 56]. Among these, a foremost application is sparse signal processing,
where frames of small spectral norm and/or small worst-case coherence are
commonly used to analyze sparse signals [11, 21, 47, 50, 56]. In general,
sparse signal processing deals with measurements of the form
$y=Fx+e,$
where $F$ is $M\times N$ with $M\ll N$, $x$ has at most $K$ nonzero entries,
and $e$ is some sort of noise. When given measurements $y$ of $x$, one might
be asked to reconstruct the original sparse vector $x$, or to find the
locations of its nonzero entries, or to simply determine whether $x$ is
nonzero—each of these is a sparse signal processing problem. In some
applications, the signal $x$ is sparse in the identity basis, in which case
$F$ represents the measurement process. In other applications, $x$ is sparse
in an orthonormal basis or an overcomplete dictionary $G$ [10]. In this case,
$F$ is a composition of $A$, the frame resulting from the measurement process,
and $G$, the sparsifying dictionary, i.e., $F=AG$. We do not make a
distinction between the two formulations in this paper, but our results are
most readily interpretable in a physical setting for the former case.
Recently, [5] introduced another notion of frame coherence called _average
coherence_ :
$\nu_{F}:=\tfrac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}\langle f_{i},f_{j}\rangle\bigg{|}.$ (2)
Note that, in addition to having zero worst-case coherence, orthonormal bases
also have zero average coherence. Intuitively, worst-case coherence is a
measure of dissimilarity between frame elements, whereas average coherence
measures how well the frame elements are distributed in the unit hypersphere.
In sparse signal processing, there are a number of performance guarantees that
depend only on worst-case coherence [20, 23, 25, 47]. These guarantees at best
allow for sparsity levels on the order of $\sqrt{M}$. Compressed sensing has
brought guarantees that depend on the Restricted Isometry Property, which is
much more difficult to check, but the guarantees allow for sparsity levels on
the order of $\smash{\frac{M}{\log N}}$ [6, 13, 14]. Recently, [5] used worst-
case and average coherence to produce _probabilistic_ guarantees that also
allow for sparsity levels on the order of $\smash{\frac{M}{\log N}}$; these
guarantees require that worst-case and average coherence together satisfy the
following property:
###### Definition 1.
We say an $M\times N$ unit norm frame $F$ satisfies the _Strong Coherence
Property_ if
$\mbox{(SCP-1)}~{}~{}~{}\mu_{F}\leq\tfrac{1}{164\log
N}\qquad\mbox{and}\qquad\mbox{(SCP-2)}~{}~{}~{}\nu_{F}\leq\tfrac{\mu_{F}}{\sqrt{M}},$
where $\mu_{F}$ and $\nu_{F}$ are given by (1) and (2), respectively.
The reader should know that the constant $164$ is not particularly essential
to the above definition; it is used in [5] to simplify some analysis and make
certain performance guarantees explicit, but the constant is by no means
optimal. This in mind, the requirement (SCP-1) can be interpreted more
generally as $\mu_{F}=O(\tfrac{1}{\log N})$. In the next section, we will use
the Strong Coherence Property to continue the work of [5]. Where [5] provided
guarantees for noiseless reconstruction, we will produce near-optimal
guarantees for signal detection and reconstruction from _noisy_ measurements
of sparse signals. These guarantees are related to those in [11, 21, 49, 50],
and we will also elaborate on this relationship.
The results given in [5] and Section 2, as well as the applications discussed
in [11, 21, 31, 37, 46, 47, 50, 56] demonstrate a pressing need for nearly
tight frames with small worst-case and average coherence, especially in the
area of sparse signal processing. This paper offers three additional
contributions in this regard. In Section 3, we provide a sizable catalog of
frames that exhibit small spectral norm, worst-case coherence, and average
coherence. With all three frame parameters provably small, these frames are
guaranteed to perform well in relevant applications. Next, performance in many
applications is dictated by worst-case coherence [11, 21, 31, 37, 46, 47, 50,
56]. It is therefore particularly important to understand which worst-case
coherence values are achievable. To this end, the Welch bound [46] is commonly
used in the literature. However, the Welch bound is only tight when the number
of frame elements $N$ is less than the square of the spatial dimension $M$
[46]. Another lower bound, given in [38, 54], beats the Welch bound when there
are more frame elements, but it is known to be loose for real frames [18].
Given this context, Section 4 gives a new lower bound on the worst-case
coherence of real frames. Our bound beats both the Welch bound and the bound
in [38, 54] when the number of frame elements far exceeds the spatial
dimension. Finally, since average coherence is so new, there is currently no
intuition as to when (SCP-2) is satisfied. In Section 5, we use ideas akin to
the switching equivalence of graphs to transform a frame that satisfies
(SCP-1) into another frame with the same spectral norm and worst-case
coherence that additionally satisfies (SCP-2).
Throughout the paper, we make use of certain notations that we address here.
Recall, with big-O notation, that $f(n)=O(g(n))$ if there exists positive $C$
and $n_{0}$ such that for all $n>n_{0}$, $f(n)\leq Cg(n)$. Also,
$f(n)=\Omega(g(n))$ if $g(n)=O(f(n))$, and $f(n)=\Theta(g(n))$ if
$f(n)=O(g(n))$ and $g(n)=O(f(n))$. Additionally, we use $F_{\mathcal{K}}$ to
denote the matrix whose columns are taken from the matrix $F$ according to the
index set $\mathcal{K}$. Similarly, we use $x_{\mathcal{K}}$ to denote the
column vector whose entries are taken from the column vector $x$ according to
the index set $\mathcal{K}$. The column vector of the $T$ largest entries in
column vector $x$ is denoted by $x_{T}$. We also use $\|x\|$ to denote the
$\ell^{2}$ norm of a vector $x$, while $\|F\|_{2}$ is the spectral norm of a
matrix $F$. Lastly, we use a star ($*$) to denote the matrix adjoint, a dagger
($\dagger$) to denote the matrix pseudoinverse, and $\mathrm{I}_{K}$ to denote
the $K\times K$ identity matrix.
## 2 Worst-case and average coherence: Applications to sparse signal
processing
Frames with small spectral norm, worst-case coherence, and/or average
coherence have found use in recent years with applications involving sparse
signals. Donoho et al. used the worst-case coherence in [21] to provide
uniform bounds on the signal and support recovery performance of combinatorial
and convex optimization methods and greedy algorithms. Later, Tropp [50] and
Candès and Plan [11] used both the spectral norm and worst-case coherence to
provide tighter bounds on the signal and support recovery performance of
convex optimization methods for most support sets under the additional
assumption that the sparse signals have independent nonzero entries with zero
median. Recently, Bajwa et al. [5] made use of the spectral norm and both
coherence parameters to report tighter bounds on the noisy model selection and
noiseless signal recovery performance of an incredibly fast greedy algorithm
called _one-step thresholding (OST)_ for most support sets and _arbitrary_
nonzero entries. In this section, we discuss further implications of the
spectral norm and worst-case and average coherence of frames in applications
involving sparse signals.
### 2.1 The Weak Restricted Isometry Property
A common task in signal processing applications is to test whether a
collection of measurements corresponds to mere noise [33]. For applications
involving sparse signals, one can test measurements $y\in\mathbb{C}^{M}$
against the null hypothsis $H_{0}:y=e$ and alternative hypothesis
$H_{1}:y=Fx+e$, where the entries of the noise vector $e\in\mathbb{C}^{M}$ are
independent, identical zero-mean complex-Gaussian random variables and the
signal $x\in\mathbb{C}^{N}$ is $K$-sparse. The performance of such signal
detection problems is directly proportional to the energy in $Fx$ [19, 27,
33]. In particular, existing literature on the detection of sparse signals
[19, 27] leverages the fact that $\|Fx\|^{2}\approx\|x\|^{2}$ when $F$
satisfies the Restricted Isometry Property (RIP) of order $K$. In contrast, we
now show that the Strong Coherence Property also guarantees
$\|Fx\|^{2}\approx\|x\|^{2}$ for most $K$-sparse vectors. We start with a
definition:
###### Definition 2.
We say an $M\times N$ frame $F$ satisfies the _$(K,\delta,p)$ -Weak Restricted
Isometry Property (Weak RIP)_ if for every $K$-sparse vector
$y\in\mathbb{C}^{N}$, a random permutation $x$ of $y$’s entries satisfies
$(1-\delta)\|x\|^{2}\leq\|Fx\|^{2}\leq(1+\delta)\|x\|^{2}$ (3)
with probability exceeding $1-p$.
At first glance, it may seem odd that we introduce a random permutation when
we might as well define Weak RIP in terms of a $K$-sparse vector whose support
is drawn randomly from all $\smash{\binom{N}{K}}$ possible choices. In fact,
both versions would be equivalent in distribution, but we stress that in the
present definition, the values of the nonzero entries of $x$ are _not_ random;
rather, the only randomness we have is in the locations of the nonzero
entries. We wish to distinguish our results from those in [11], which
explicitly require randomness in the values of the nonzero entries. We also
note the distinction between RIP and Weak RIP—Weak RIP requires that $F$
preserves the energy of _most_ sparse vectors. Moreover, the manner in which
we quantify “most” is important. For each sparse vector, $F$ preserves the
energy of most permutations of that vector, but for different sparse vectors,
$F$ might not preserve the energy of permutations with the same support. That
is, unlike RIP, Weak RIP is _not_ a statement about the singular values of
submatrices of $F$. Certainly, matrices for which most submatrices are well-
conditioned, such as those discussed in [49, 50], will satisfy Weak RIP, but
Weak RIP does not require this. That said, the following theorem shows, in
part, the significance of the Strong Coherence Property.
###### Theorem 3.
Any $M\times N$ unit norm frame $F$ that satisfies the Strong Coherence
Property also satisfies the $(K,\delta,\frac{4K}{N^{2}})$-Weak Restricted
Isometry Property provided $N\geq 128$ and
$\smash{2K\log{N}\leq\min\\{\frac{\delta^{2}}{100\mu_{F}^{2}},M\\}}$.
###### Proof.
Let $x$ be as in Definition 2. Note that (3) is equivalent to
$\smash{\big{|}\|Fx\|^{2}-\|x\|^{2}\big{|}\leq\delta\|x\|^{2}}$. Defining
$\mathcal{K}:=\\{n:|x_{n}|>0\\}$, then the Cauchy-Schwarz inequality gives
$\big{|}\|Fx\|^{2}-\|x\|^{2}\big{|}=|x_{\mathcal{K}}^{*}(F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K})x_{\mathcal{K}}|\leq\|x_{\mathcal{K}}\|~{}\|(F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K})x_{\mathcal{K}}\|\leq\sqrt{K}~{}\|x_{\mathcal{K}}\|~{}\|(F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K})x_{\mathcal{K}}\|_{\infty},$
(4)
where the last inequality uses the fact that
$\|\cdot\|\leq\sqrt{K}~{}\|\cdot\|_{\infty}$ in $\mathbb{C}^{K}$. We now
consider [5, Lemma 3], which states that for any $\epsilon\in[0,1)$ and $a\geq
1$,
$\|(F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K})x_{\mathcal{K}}\|_{\infty}\leq\epsilon\|x_{\mathcal{K}}\|$
with probability exceeding
$\smash{1-4K\mathrm{e}^{-(\epsilon-\sqrt{K}\nu_{F})^{2}/16(2+a^{-1})^{2}\mu_{F}^{2}}}$
provided $\smash{K\leq\min\\{\epsilon^{2}\nu_{F}^{-2},(1+a)^{-1}N\\}}$. We
claim that (4) together with [5, Lemma 3] guarantee
$\smash{\big{|}\|Fx\|^{2}-\|x\|^{2}\big{|}\leq\delta\|x\|^{2}}$ with
probability exceeding $\smash{1-\frac{4K}{N^{2}}}$. In order to establish this
claim, we fix $\epsilon=10\mu\sqrt{2\log{N}}$ and $a=2\log{128}-1$. It is then
easy to see that (SCP-1) gives $\epsilon<1$, and also that (SCP-2) and
$2K\log{N}\leq M$ give $K\leq\epsilon^{2}\nu_{F}^{-2}/9$. Therefore, since the
assumption that $N\geq 128$ together with $2K\log{N}\leq M$ implies
$K\leq(1+a)^{-1}N$, we obtain
$\smash{\mathrm{e}^{-(\epsilon-\sqrt{K}\nu_{F})^{2}/16(2+a^{-1})^{2}\mu_{F}^{2}}\leq\frac{1}{N^{2}}}$.
The result now follows from the observation that
$\smash{2K\log{N}\leq\frac{\delta^{2}}{100\mu_{F}^{2}}}$ implies
$\sqrt{K}\epsilon\leq\delta$. ∎
This theorem shows that having small worst-case and average coherence is
enough to guarantee Weak RIP. This contrasts with related results by Tropp
[49, 50] that require $F$ to be nearly tight. In fact, the proof of Theorem 3
does not even use the full power of the Strong Coherence Property; instead of
(SCP-1), it suffices to have $\smash{\mu_{F}\leq 1/(15\\!\sqrt{\log N})}$,
part of what [5] calls the Coherence Property. Also, if $F$ has worst-case
coherence $\smash{\mu_{F}=O(1/\\!\sqrt{M})}$ and average coherence
$\nu_{F}=O(1/M)$, then even if $F$ has large spectral norm, Theorem 3 states
that $F$ preserves the energy of most $K$-sparse vectors with $K=O(M/\log N)$,
i.e., the sparsity regime which is linear in the number of measurements.
### 2.2 Reconstruction of sparse signals from noisy measurements
Another common task in signal processing applications is to reconstruct a
$K$-sparse signal $x\in\mathbb{C}^{N}$ from a small collection of linear
measurements $y\in\mathbb{C}^{M}$. Recently, Tropp [50] used both the worst-
case coherence and spectral norm of frames to find bounds on the
reconstruction performance of _basis pursuit (BP)_ [17] for most support sets
under the assumption that the nonzero entries of $x$ are independent with zero
median. In contrast, [5] used the spectral norm and worst-case and average
coherence of frames to find bounds on the reconstruction performance of OST
for most support sets and _arbitrary_ nonzero entries. However, both [5] and
[50] limit themselves to recovering $x$ in the absence of noise, corresponding
to $y=Fx$, a rather ideal scenario.
Our goal in this section is to provide guarantees for the reconstruction of
sparse signals from noisy measurements $y=Fx+e$, where the entries of the
noise vector $e\in\mathbb{C}^{M}$ are independent, identical complex-Gaussian
random variables with mean zero and variance $\sigma^{2}$. In particular, and
in contrast with [21], our guarantees will hold for arbitrary unit norm frames
$F$ without requiring the signal’s sparsity level to satisfy
$K=O(\mu_{F}^{-1})$. The reconstruction algorithm that we analyze here is the
OST algorithm of [5], which is described in Algorithm 1. The following theorem
extends the analysis of [5] and shows that the OST algorithm leads to near-
optimal reconstruction error for certain important classes of sparse signals.
Before proceeding further, we first define some notation. We use
$\textsf{{snr}}:=\|x\|^{2}/\mathbb{E}[\|e\|^{2}]$ to denote the _signal-to-
noise ratio_ associated with the signal reconstruction problem. Also, we use
$\smash{\mathcal{T}_{\sigma}(t):=\\{n:|x_{n}|>\frac{2\sqrt{2}}{1-t}\sqrt{2\sigma^{2}\log{N}}\\}}$
for any $t\in(0,1)$ to denote the locations of all the entries of $x$ that,
roughly speaking, lie above the _noise floor_ $\sigma$. Finally, we use
$\smash{\mathcal{T}_{\mu}(t):=\\{n:|x_{n}|>\frac{20}{t}\mu_{F}\|x\|\sqrt{2\log{N}}\\}}$
to denote the locations of entries of $x$ that, roughly speaking, lie above
the _self-interference floor_ $\mu_{F}\|x\|$.
Algorithm 1 One-Step Thresholding (OST) for sparse signal reconstruction [5]
Input: An $M\times N$ unit norm frame $F$, a vector $y=Fx+e$, and a threshold
$\lambda>0$
Output: An estimate $\hat{x}\in\mathbb{C}^{N}$ of the true sparse signal $x$
$\hat{x}\leftarrow 0$ {Initialize}
$z\leftarrow F^{*}y$ {Form signal proxy}
$\hat{\mathcal{K}}\leftarrow\\{n:|z_{n}|>\lambda\\}$ {Select indices via OST}
$\hat{x}_{\hat{\mathcal{K}}}\leftarrow(F_{\hat{\mathcal{K}}})^{\dagger}y$
{Reconstruct signal via least-squares}
###### Theorem 4 (Reconstruction of sparse signals).
Take an $M\times N$ unit norm frame $F$ which satisfies the Strong Coherence
Property, pick $t\in(0,1)$, and choose
$\smash{\lambda=\sqrt{2\sigma^{2}\log{N}}~{}\max\\{\frac{10}{t}\mu_{F}\sqrt{M~{}\textsf{{snr}}},\frac{\sqrt{2}}{1-t}\\}}$.
Further, suppose $x\in\mathbb{C}^{N}$ has support $\mathcal{K}$ drawn
uniformly at random from all possible $K$-subsets of $\\{1,\ldots,N\\}$. Then
provided
$K\leq\tfrac{N}{c_{1}^{2}\|F\|_{2}^{2}\log{N}},$ (5)
Algorithm 1 produces $\hat{\mathcal{K}}$ such that
$\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}\subseteq\mathcal{K}$
and $\hat{x}$ such that
$\|x-\hat{x}\|\leq
c_{2}\sqrt{\sigma^{2}|\hat{\mathcal{K}}|\log{N}}+c_{3}\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|$
(6)
with probability exceeding $1-10N^{-1}$. Finally, defining
$T:=|\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)|$, we further have
$\|x-\hat{x}\|\leq c_{2}\sqrt{\sigma^{2}K\log{N}}+c_{3}\|x-x_{T}\|$ (7)
in the same probability event. Here, $c_{1}=37\mathrm{e}$,
$c_{2}=\frac{2}{1-\mathrm{e}^{-1/2}}$, and
$c_{3}=1+\frac{\mathrm{e}^{-1/2}}{1-\mathrm{e}^{-1/2}}$ are numerical
constants.
###### Proof.
To begin, note that since $\smash{\|F\|_{2}^{2}\geq\frac{N}{M}}$, we have from
(5) that $K\leq M/(2\log{N})$. It is then easy to conclude from [5, Theorem 5]
that $\smash{\hat{\mathcal{K}}}$ satisfies
$\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}\subseteq\mathcal{K}$
with probability exceeding $1-6N^{-1}$. Therefore, conditioned on the event
$\smash{\mathcal{E}_{1}:=\\{\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}\subseteq\mathcal{K}\\}}$,
we can make use of the triangle inequality to write
$\|x-\hat{x}\|\leq\|x_{\hat{\mathcal{K}}}-\hat{x}_{\hat{\mathcal{K}}}\|+\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|.$
(8)
Next, we may use (5) and the fact that $F$ satisfies the Strong Coherence
Property to conclude from [49] (see, e.g., [5, Proposition 3]) that
$\|F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K}\|_{2}<\mathrm{e}^{-1/2}$
with probability exceeding $1-2N^{-1}$. Hence, conditioning on
$\mathcal{E}_{1}$ and
$\smash{\mathcal{E}_{2}:=\\{\|F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K}\|_{2}<\mathrm{e}^{-1/2}\\}}$,
we have that
$\smash{(F_{\hat{\mathcal{K}}})^{\dagger}=(F_{\hat{\mathcal{K}}}^{*}F_{\hat{\mathcal{K}}})^{-1}F_{\hat{\mathcal{K}}}^{*}}$
since $F_{\hat{\mathcal{K}}}$ is a submatrix of a full column rank matrix
$F_{\mathcal{K}}$. Therefore, given $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$,
we may write
$\hat{x}_{\hat{\mathcal{K}}}=(F_{\hat{\mathcal{K}}})^{\dagger}(Fx+e)=x_{\hat{\mathcal{K}}}+(F_{\hat{\mathcal{K}}})^{\dagger}F_{\mathcal{K}\setminus\hat{\mathcal{K}}}x_{\mathcal{K}\setminus\hat{\mathcal{K}}}+(F_{\hat{\mathcal{K}}})^{\dagger}e,$
(9)
and so substituting (9) into (8) and applying the triangle inequality gives
$\displaystyle\|x-\hat{x}\|$
$\displaystyle\leq\|(F_{\hat{\mathcal{K}}})^{\dagger}F_{\mathcal{K}\setminus\hat{\mathcal{K}}}x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|+\|(F_{\hat{\mathcal{K}}})^{\dagger}e\|+\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|$
$\displaystyle\leq\Big{(}1+\|(F_{\hat{\mathcal{K}}}^{*}F_{\hat{\mathcal{K}}})^{-1}\|_{2}\|F_{\hat{\mathcal{K}}}^{*}F_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|_{2}\Big{)}\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|+\|(F_{\hat{\mathcal{K}}}^{*}F_{\hat{\mathcal{K}}})^{-1}\|_{2}\|F_{\hat{\mathcal{K}}}^{*}e\|.$
(10)
Since, given $\mathcal{E}_{1}$, we have that
$\smash{F_{\hat{\mathcal{K}}}^{*}F_{\hat{\mathcal{K}}}-\mathrm{I}_{K}}$ and
$\smash{F_{\hat{\mathcal{K}}}^{*}F_{\mathcal{K}\setminus\hat{\mathcal{K}}}}$
are submatrices of
$\smash{F_{\mathcal{K}}^{*}F_{\mathcal{K}}-\mathrm{I}_{K}}$, and since the
spectral norm of a matrix provides an upper bound for the spectral norms of
its submatrices, we have the following given $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$:
$\smash{\|F_{\hat{\mathcal{K}}}^{*}F_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|_{2}\leq\mathrm{e}^{-1/2}}$
and
$\smash{\|(F_{\hat{\mathcal{K}}}^{*}F_{\hat{\mathcal{K}}})^{-1}\|_{2}\leq\tfrac{1}{1-\mathrm{e}^{-1/2}}}$.
We can now substitute these bounds into (10) and make use of the fact that
$\smash{\|F_{\hat{\mathcal{K}}}^{*}e\|\leq|\hat{\mathcal{K}}|^{1/2}\|F_{\hat{\mathcal{K}}}^{*}e\|_{\infty}}$
to conclude that
$\|x-\hat{x}\|\leq\tfrac{|\hat{\mathcal{K}}|^{1/2}}{1-\mathrm{e}^{-1/2}}\|F_{\hat{\mathcal{K}}}^{*}e\|_{\infty}+\Big{(}1+\tfrac{\mathrm{e}^{-1/2}}{1-\mathrm{e}^{-1/2}}\Big{)}\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|,$
given $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. At this point, define the event
$\smash{\mathcal{E}_{3}=\\{\|F_{\hat{\mathcal{K}}}^{*}e\|_{\infty}\leq
2\sqrt{\sigma^{2}\log{N}}\\}}$ and note from [5, Lemma 6] that
$\smash{\Pr(\mathcal{E}_{3}^{\mathrm{c}})\leq
2(\sqrt{2\pi\log{N}}~{}N)^{-1}}$. A union bound therefore gives (6) with
probability exceeding $1-10N^{-1}$. For (7), note that
$\hat{\mathcal{K}}\subseteq\mathcal{K}$ implies $|\hat{\mathcal{K}}|\leq K$,
and so
$\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t)\subseteq\hat{\mathcal{K}}$
implies that
$\|x_{\mathcal{K}\setminus\hat{\mathcal{K}}}\|\leq\|x_{\mathcal{K}\setminus(\mathcal{T}_{\sigma}(t)\cap\mathcal{T}_{\mu}(t))}\|=\|x-x_{T}\|$.
∎
A few remarks are in order now for Theorem 4. First, if $F$ satisfies the
Strong Coherence Property _and_ $F$ is nearly tight, then OST handles sparsity
that is almost linear in $M$: $K=O(M/\log{N})$ from (5). Second, we do not
impose any control over the size of $T$, but rather we state the result in
generality in terms of $T$; its size is determined by the signal class $x$
belongs to, the worst-case coherence of the frame $F$ we use to measure $x$,
and the magnitude of the noise that perturbs $Fx$. Third, the $\ell_{2}$ error
associated with the OST algorithm is the near-optimal (modulo the $\log$
factor) error of $\smash{\sqrt{\sigma^{2}K\log{N}}}$ _plus_ the best $T$-term
approximation error caused by the inability of the OST algorithm to recover
signal entries that are smaller than $\smash{O(\mu_{F}\|x\|\sqrt{2\log{N}})}$.
In particular, if the $K$-sparse signal $x$, the worst-case coherence
$\mu_{F}$, and the noise $e$ together satisfy
$\|x-x_{T}\|=O(\smash{\sqrt{\sigma^{2}K\log{N}}})$, then the OST algorithm
succeeds with a near-optimal $\ell_{2}$ error of
$\smash{\|x-\hat{x}\|=O(\sqrt{\sigma^{2}K\log{N}})}$. To see why this error is
near-optimal, note that a $K$-dimension vector of random entries with mean
zero and variance $\sigma^{2}$ has expected squared norm $\sigma^{2}K$; in our
case, we pay an additional log factor to find the locations of the $K$ nonzero
entries among the entire $N$-dimensional signal. It is important to recognize
that the optimality condition
$\|x-x_{T}\|=O(\smash{\sqrt{\sigma^{2}K\log{N}}})$ depends on the signal
class, the noise variance, and the worst-case coherence of the frame; in
particular, the condition is satisfied whenever
$\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=O(\smash{\sqrt{\sigma^{2}K\log{N}}})$,
since
$\|x-x_{T}\|\leq\|x_{\mathcal{K}\setminus\mathcal{T}_{\sigma}(t)}\|+\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=O\Big{(}\sqrt{\sigma^{2}K\log{N}}\Big{)}+\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|.$
The following lemma provides classes of sparse signals that satisfy
$\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=O(\smash{\sqrt{\sigma^{2}K\log{N}}})$
given sufficiently small noise variance and worst-case coherence, and
consequently the OST algorithm is near-optimal for the reconstruction of such
signal classes.
###### Lemma 5.
Take an $M\times N$ unit norm frame $F$ with worst-case coherence
$\smash{\mu_{F}\leq\frac{c_{0}}{\sqrt{M}}}$ for some $c_{0}>0$, and suppose
that $\smash{K\leq\frac{N}{c_{1}^{2}\|F\|_{2}^{2}\log N}}$ for some $c_{1}>0$.
Fix a constant $\beta\in(0,1]$, and suppose the magnitudes of $\beta K$
nonzero entries of $x$ are some $\alpha=\Omega(\sqrt{\sigma^{2}\log{N}})$,
while the magnitudes of the remaining $(1-\beta)K$ nonzero entries are not
necessarily same, but are smaller than $\alpha$ and scale as
$\smash{O(\sqrt{\sigma^{2}\log{N}})}$. Then
$\smash{\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|=O(\sqrt{\sigma^{2}K\log{N}})}$,
provided $\smash{c_{0}\leq\frac{tc_{1}}{20\sqrt{2}}}$.
###### Proof.
Let $\mathcal{K}$ be the support of $x$, and define
$\mathcal{I}:=\\{n:|x_{n}|=\alpha\\}$. We wish to show that
$\mathcal{I}\subseteq\mathcal{T}_{\mu}(t)$, since this implies
$\smash{\|x_{\mathcal{K}\setminus\mathcal{T}_{\mu}(t)}\|\leq\|x_{\mathcal{K}\setminus\mathcal{I}}\|=O(\sqrt{\sigma^{2}K\log{N}})}$.
In order to prove $\mathcal{I}\subseteq\mathcal{T}_{\mu}(t)$, notice that
$\|x\|^{2}=\|x_{\mathcal{I}}\|^{2}+\|x_{\mathcal{K}\setminus\mathcal{I}}\|^{2}<\beta
K\alpha^{2}+(1-\beta)K\alpha^{2}=K\alpha^{2},$
and so combining this with the fact that $\|F\|_{2}^{2}\geq\frac{N}{M}$ gives
$\mu_{F}\|x\|\sqrt{\log{N}}<\tfrac{c_{0}}{\sqrt{M}}\sqrt{K}\alpha\sqrt{\log{N}}\leq\tfrac{c_{0}}{\sqrt{M}}\sqrt{\tfrac{N}{c_{1}^{2}\|F\|_{2}^{2}\log
N}}~{}\alpha\sqrt{\log{N}}\leq\tfrac{c_{0}}{c_{1}}\alpha.$
Therefore, provided $\smash{c_{0}\leq\frac{tc_{1}}{20\sqrt{2}}}$, we have that
$\mathcal{I}\subseteq\mathcal{T}_{\mu}(t)$. ∎
In words, Lemma 5 implies that OST is near-optimal for those $K$-sparse
signals whose entries above the noise floor have roughly the same magnitude.
This subsumes a very important class of signals that appears in applications
such as multi-label prediction [32], in which all the nonzero entries take
values $\pm\alpha$. To the best of our knowledge, Theorem 4 is the first
result in the sparse signal processing literature that does not require RIP
and still provides near-optimal reconstruction guarantees for such signals
from noisy measurements, while using either random or deterministic frames,
even when $K=O(M/\log{N})$.
We note that our techniques can be extended to reconstruct noisy signals, that
is, we may consider measurements of the form $y=F(x+n)+e$, where
$n\in\mathbb{C}^{N}$ is also a noise vector of independent, identical zero-
mean complex-Gaussian random variables. In particular, if the frame $F$ is
tight, then our measurements will not color the noise, and so noise in the
signal may be viewed as noise in the measurements: $y=Fx+(Fn+e)$; if the frame
is not tight, then the noise will become correlated in the measurements, and
performance would be depend nontrivially on the frame’s Gram matrix. Also, the
authors have had some success with generalizing Theorem 4 to approximately
sparse signals; the analysis follows similiar lines, but is rather cumbersome,
and it appears as though the end result is only strong enough in the case of
very nearly sparse signals. As such, we omit this result.
## 3 Frame constructions
In this section, we consider a range of nearly tight frames with small worst-
case and average coherence. We investigate various ways of selecting frames at
random from different libraries, and we show that for each of these frames,
the spectral norm, worst-case coherence, and average coherence are all small
with high probability. Later, we will consider deterministic constructions
that use Gabor and chirp systems, spherical designs, equiangular tight frames,
and error-correcting codes. For the reader’s convenience, all of these
constructions are summarized in Table 1. Before we go any further, recall the
following lower bound on worst-case coherence:
###### Theorem 6 (Welch bound [46]).
Every $M\times N$ unit norm frame $F$ has worst-case coherence
$\mu_{F}\geq\sqrt{\tfrac{N-M}{M(N-1)}}$.
We will use the Welch bound in the proof of the following lemma, which gives
three different sufficient conditions for a frame to satisfy (SCP-2). These
conditions will prove quite useful in this section and throughout the paper.
###### Lemma 7.
For any $M\times N$ unit norm frame $F$, each of the following conditions
implies $\nu_{F}\leq\frac{\mu_{F}}{\sqrt{M}}$:
1. (i)
$\langle f_{k},\sum_{n=1}^{N}f_{n}\rangle=\frac{N}{M}$ for every
$k=1,\ldots,N$,
2. (ii)
$N\geq 2M$ and $\sum_{n=1}^{N}f_{n}=0$,
3. (iii)
$N\geq M^{2}+3M+3$ and $\|\sum_{n=1}^{N}f_{n}\|^{2}\leq N$.
###### Proof.
For condition (i), we have
$\nu_{F}=\tfrac{1}{N-1}\max_{i}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\langle
f_{i},f_{j}\rangle\bigg{|}=\tfrac{1}{N-1}\max_{i}\bigg{|}\bigg{\langle}f_{i},\sum_{j=1}^{N}f_{j}\bigg{\rangle}-1\bigg{|}=\tfrac{1}{N-1}\big{(}\tfrac{N}{M}-1\big{)}.$
The Welch bound therefore gives
$\nu_{F}=\tfrac{1}{N-1}\big{(}\tfrac{N}{M}-1\big{)}=\tfrac{N-M}{M(N-1)}\leq\mu_{F}\sqrt{\tfrac{N-M}{M(N-1)}}\leq\tfrac{\mu_{F}}{\sqrt{M}}$.
For condition (ii), we have
$\nu_{F}=\tfrac{1}{N-1}\max_{i}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\langle
f_{i},f_{j}\rangle\bigg{|}=\tfrac{1}{N-1}\max_{i}\bigg{|}\bigg{\langle}f_{i},\sum_{j=1}^{N}f_{j}\bigg{\rangle}-1\bigg{|}=\tfrac{1}{N-1}.$
Considering the Welch bound, it suffices to show
$\frac{1}{N-1}\leq\frac{1}{\sqrt{M}}\sqrt{\frac{N-M}{M(N-1)}}$. Rearranging
equivalently gives
$N^{2}-(M+1)N-M(M-1)\geq 0.$ (11)
When $N=2M$, the left-hand side of (11) becomes $(M-1)^{2}$, which is
trivially nonnegative. Otherwise, we have
$N\geq 2M+1\geq
M+1+\sqrt{M(M-1)}\geq\tfrac{M+1}{2}+\sqrt{\big{(}\tfrac{M+1}{2}\big{)}^{2}+M(M-1)}.$
In this case, by the quadratic formula and the fact that the left-hand side of
(11) is concave up in $N$, we have that (11) is indeed satisfied. For
condition (iii), we use the triangle and Cauchy-Schwarz inequalities to get
$\nu_{F}=\tfrac{1}{N-1}\max_{i}\bigg{|}\bigg{\langle}f_{i},\sum_{j=1}^{N}f_{j}\bigg{\rangle}-1\bigg{|}\leq\tfrac{1}{N-1}\bigg{(}\max_{i}\bigg{|}\bigg{\langle}f_{i},\sum_{j=1}^{N}f_{j}\bigg{\rangle}\bigg{|}+1\bigg{)}\leq\tfrac{\sqrt{N}+1}{N-1}.$
Considering the Welch bound, it suffices to show
$\smash{\frac{\sqrt{N}+1}{N-1}\leq\frac{1}{\sqrt{M}}\sqrt{\frac{N-M}{M(N-1)}}}$.
Taking $x:=\sqrt{N}$ and rearranging gives a polynomial:
$x^{4}-(M^{2}+M+1)x^{2}-2M^{2}x-M(M-1)\geq 0$. By convexity and monotonicity
of the polynomial in $[M+\frac{3}{2},\infty)$, it can be shown that the
largest real root of this polynomial is always smaller than
$\smash{M+\frac{3}{2}}$. Also, considering it is concave up in $x$, it
suffices that $\smash{\sqrt{N}=x\geq M+\frac{3}{2}}$, which we have since
$N\geq M^{2}+3M+3\geq(M+\frac{3}{2})^{2}$. ∎
### 3.1 Normalized Gaussian frames
Construct a matrix with independent, Gaussian-distributed entries that have
zero mean and unit variance. By normalizing the columns, we get a matrix
called a _normalized Gaussian frame_. This is perhaps the most widely studied
type of frame in the signal processing and statistics literature. To be clear,
the term “normalized” is intended to distinguish the results presented here
from results reported in earlier works, such as [5, 6, 13, 52], which only
ensure that Gaussian frame elements have unit norm in expectation. In other
words, normalized Gaussian frame elements are independently and uniformly
distributed on the unit hypersphere in $\mathbb{R}^{M}$. That said, the
following theorem characterizes the spectral norm and the worst-case and
average coherence of normalized Gaussian frames.
###### Theorem 8 (Geometry of normalized Gaussian frames).
Build a real $M\times N$ frame $G$ by drawing entries independently at random
from a Gaussian distribution of zero mean and unit variance. Next, construct a
normalized Gaussian frame $F$ by taking
$\smash{f_{n}:=\frac{g_{n}}{\|g_{n}\|}}$ for every $n=1,\ldots,N$. Provided
$\smash{60\log{N}\leq M\leq\frac{N-1}{4\log{N}}}$, then the following
inequalities simultaneously hold with probability exceeding $1-11N^{-1}$:
1. (i)
$\mu_{F}\leq\frac{\sqrt{15\log{N}}}{\sqrt{M}-\sqrt{12\log{N}}}$,
2. (ii)
$\nu_{F}\leq\frac{\sqrt{15\log{N}}}{M-\sqrt{12M\log{N}}}$,
3. (iii)
$\|F\|_{2}\leq\frac{\sqrt{M}+\sqrt{N}+\sqrt{2\log{N}}}{\sqrt{M-\sqrt{8M\log{N}}}}$.
###### Proof.
Theorem 8(i) can be shown to hold with probability exceeding $1-2N^{-1}$ by
using a bound on the norm of a Gaussian random vector in [34, Lemma 1] and a
bound on the magnitude of the inner product of two independent Gaussian random
vectors in [26, Lemma 6]. Specifically, pick any two distinct indices
$i,j\in\\{1,\dots,N\\}$, and define probability events
$\mathcal{E}_{1}:=\\{|\langle g_{i},g_{j}\rangle|\leq\delta_{1}\\}$,
$\mathcal{E}_{2}:=\\{\|g_{i}\|^{2}\geq M(1-\delta_{2})\\}$, and
$\mathcal{E}_{3}:=\\{\|g_{j}\|^{2}\geq M(1-\delta_{2})\\}$ for
$\smash{\delta_{1}=\sqrt{15M\log{N}}}$ and
$\smash{\delta_{2}=\sqrt{(12\log{N})/M}}$. Then it follows from the union
bound that
$\Pr\bigg{(}|\langle
f_{i},f_{j}\rangle|>\tfrac{\delta_{1}}{M(1-\delta_{2})}\bigg{)}=\Pr\bigg{(}\tfrac{|\langle
g_{i},g_{j}\rangle|}{\|g_{i}\|~{}\|g_{j}\|}>\tfrac{\delta_{1}}{M(1-\delta_{2})}\bigg{)}\leq\Pr(\mathcal{E}_{1}^{\mathrm{c}})+\Pr(\mathcal{E}_{2}^{\mathrm{c}})+\Pr(\mathcal{E}_{3}^{\mathrm{c}}).$
One can verify that
$\Pr(\mathcal{E}_{2}^{\mathrm{c}})=\Pr(\mathcal{E}_{3}^{\mathrm{c}})\leq
N^{-3}$ because of [34, Lemma 1], and we further have
$\Pr(\mathcal{E}_{1}^{\mathrm{c}})\leq 2N^{-3}$ because of [26, Lemma 6] and
the fact that $M\geq 60\log{N}$. Thus, for any fixed $i$ and $j$,
$\smash{|\langle
f_{i},f_{j}\rangle|\leq\\!\sqrt{15\log{N}}/(\\!\sqrt{M}-\\!\sqrt{12\log{N}})}$
with probability exceeding $1-4N^{-3}$. It therefore follows by taking a union
bound over all $\smash{\binom{N}{2}}$ choices for $i$ and $j$ that Theorem
8(i) holds with probability exceeding $1-2N^{-1}$.
Theorem 8(ii) can be shown to hold with probability exceeding $1-6N^{-1}$ by
appealing to the preceding analysis and Hoeffding’s inequality for a sum of
independent, bounded random variables [30]. Specifically, fix any index
$i\in\\{1,\dots,N\\}$, and define random variables
$Z^{i}_{j}:=\frac{1}{N-1}\langle f_{i},f_{j}\rangle$. Next, define the
probability event
$\mathcal{E}_{4}:=\bigcap_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\bigg{\\{}|Z^{i}_{j}|\leq\tfrac{1}{N-1}~{}\tfrac{\sqrt{15\log{N}}}{\sqrt{M}-\sqrt{12\log{N}}}\bigg{\\}}.$
Using the analysis for the worst-case coherence of $F$ and taking a union
bound over the $N-1$ possible $j$’s gives
$\Pr(\mathcal{E}_{4}^{\mathrm{c}})\leq 4N^{-2}$. Furthermore, taking
$\delta_{3}:=\sqrt{15\log{N}}/(M-\sqrt{12M\log{N}})$, then elementary
probability analysis gives
$\Pr\bigg{(}\Big{|}\sum_{j\not=i}Z^{i}_{j}\Big{|}>\delta_{3}\bigg{)}\leq\Pr\Bigg{(}\Big{|}\sum_{j\not=i}Z^{i}_{j}\Big{|}>\delta_{3}~{}\Bigg{|}~{}\mathcal{E}_{4}\Bigg{)}+\Pr(\mathcal{E}_{4}^{\mathrm{c}})\leq\int_{S^{M-1}}\\!\\!\\!\Pr\Bigg{(}\Big{|}\sum_{j\not=i}Z^{i}_{j}\Big{|}>\delta_{3}~{}\Bigg{|}~{}\mathcal{E}_{4},f_{i}=x\Bigg{)}~{}p_{f_{i}}(x)~{}\mathrm{dH}^{M-1}(x)+4N^{-2},$
(12)
where $S^{M-1}$ denotes the unit hypersphere in $\mathbb{R}^{M}$,
$\mathrm{H}^{M-1}$ denotes the $(M-1)$-dimensional Hausdorff measure on
$S^{M-1}$, and $p_{f_{i}}(x)$ denotes the probability density function for the
random vector $f_{i}$. The first thing to note here is that the random
variables $\\{Z^{i}_{j}:j\not=i\\}$ are bounded and jointly independent when
conditioned on $\mathcal{E}_{4}$ and $f_{i}$. This assertion mainly follows
from Bayes’ rule and the fact that $\\{f_{j}:j\not=i\\}$ are jointly
independent when conditioned on $f_{i}$. The second thing to note is that
$\smash{\mathbb{E}[Z^{i}_{j}~{}|~{}\mathcal{E}_{4},f_{i}]=0}$ for every $j\neq
i$. This comes from the fact that the random vectors
$\smash{\\{f_{n}\\}_{n=1}^{N}}$ are independent and have a uniform
distribution over $\smash{S^{M-1}}$, which in turn guarantees that the random
variables $\\{Z^{i}_{j}:j\not=i\\}$ have a symmetric distribution around zero
when conditioned on $\mathcal{E}_{4}$ and $f_{i}$. We can therefore make use
of Hoeffding’s inequality [30] to bound the probability expression inside the
integral in (12) as
$\Pr\Bigg{(}\Big{|}\sum_{j\not=i}Z^{i}_{j}\Big{|}>\delta_{3}~{}\Bigg{|}~{}\mathcal{E}_{4},f_{i}=x\Bigg{)}\leq
2\mathrm{e}^{-(N-1)/2M},$ (13)
which is bounded above by $2N^{-2}$ provided
$\smash{M\leq\frac{N-1}{4\log{N}}}$. We can now substitute (13) into (12) and
take the union bound over the $N$ possible choices for $i$ to conclude that
Theorem 8(ii) holds with probability exceeding $1-6N^{-1}$.
Lastly, Theorem 8(iii) can be shown to hold with probability exceeding
$1-3N^{-1}$ by using a bound on the spectral norm of standard Gaussian random
matrices reported in [41] along with [34, Lemma 1]. Specifically, define an
$N\times N$ diagonal matrix
$D:=\mathrm{diag}(\|g_{1}\|^{-1},\dots,\|g_{N}\|^{-1})$, and note that the
entries of $G:=FD^{-1}$ are independently and normally distributed with zero
mean and unit variance. We therefore have from (2.3) in [41] that
$\Pr\Big{(}\|G\|_{2}>\sqrt{M}+\sqrt{N}+\sqrt{2\log{N}}\Big{)}\leq 2N^{-1}.$
(14)
In addition, we can appeal to the preceding analysis for the probability bound
on Theorem 8(i) and conclude using [34, Lemma 1] and a union bound over the
$N$ possible choices for $i$ that
$\Pr\Big{(}\|D\|_{2}>\Big{(}M-\sqrt{8M\log{N}}\Big{)}^{-1/2}\Big{)}\leq
N^{-1}.$ (15)
Finally, since $\|F\|_{2}\leq\|G\|_{2}\|D\|_{2}$, we can take a union bound
over (14) and (15) to argue that Theorem 8(iii) holds with probability
exceeding $1-3N^{-1}$.
The complete result now follows by taking a union bound over the failure
probabilities for the conditions (i)-(iii) in Theorem 8. ∎
###### Example 9.
To illustrate the bounds in Theorem 8, we ran simulations in MATLAB. Picking
$N=50000$, we observed $30$ realizations of normalized Gaussian frames for
each $M=700,900,1100$. The distributions of $\mu_{F}$, $\nu_{F}$, and
$\|F\|_{2}$ were rather tight, so we only report the ranges of values
attained, along with the bounds given in Theorem 8:
$\begin{array}[]{rrcll}M=700:&\qquad\mu_{F}&\in&[0.1849,0.2072]&\qquad\leq
0.8458\\\ &\qquad\nu_{F}&\in&[0.5643,0.6613]\times 10^{-3}&\qquad\leq
0.0320\\\ &\qquad\|F\|_{2}&\in&[8.0521,8.0835]&\qquad\leq 11.9565\\\ \\\
M=900:&\qquad\mu_{F}&\in&[0.1946,0.2206]&\qquad\leq 0.6848\\\
&\qquad\nu_{F}&\in&[0.5800,0.7501]\times 10^{-3}&\qquad\leq 0.0229\\\
&\qquad\|F\|_{2}&\in&[8.4352,8.4617]&\qquad\leq 10.3645\\\ \\\
M=1100:&\qquad\mu_{F}&\in&[0.1807,0.1988]&\qquad\leq 0.5852\\\
&\qquad\nu_{F}&\in&[0.5260,0.6713]\times 10^{-3}&\qquad\leq 0.0177\\\
&\qquad\|F\|_{2}&\in&[7.7262,7.7492]&\qquad\leq 9.2927\end{array}$
These simulations seem to indicate that our bounds on $\mu_{F}$ and
$\|F\|_{2}$ reflect real-world behavior, at least within an order of
magnitude, whereas the bound on $\nu_{F}$ is rather loose.
### 3.2 Random harmonic frames
Random harmonic frames, constructed by randomly selecting rows of a discrete
Fourier transform (DFT) matrix and normalizing the resulting columns, have
received considerable attention lately in the compressed sensing literature
[12, 14, 42]. However, to the best of our knowledge, there is no result in the
literature that shows that random harmonic frames have small worst-case
coherence. To fill this gap, the following theorem characterizes the spectral
norm and the worst-case and average coherence of random harmonic frames.
###### Theorem 10 (Geometry of random harmonic frames).
Let $U$ be an $N\times N$ non-normalized discrete Fourier transform matrix,
explicitly, $U_{k\ell}:=\mathrm{e}^{2\pi\mathrm{i}k\ell/N}$ for each
$k,\ell=0,\ldots,N-1$. Next, let $\\{B_{i}\\}_{i=0}^{N-1}$ be a collection of
independent Bernoulli random variables with mean $\smash{\frac{M}{N}}$, and
take $\mathcal{M}:=\\{i:B_{i}=1\\}$. Finally, construct an
$|\mathcal{M}|\times N$ harmonic frame $F$ by collecting rows of $U$ which
correspond to indices in $\mathcal{M}$ and normalize the columns. Then $F$ is
a unit norm tight frame: $\smash{\|F\|_{2}^{2}=\frac{N}{|\mathcal{M}|}}$.
Furthermore, provided $\smash{16\log{N}\leq M\leq\frac{N}{3}}$, the following
inequalities simultaneously hold with probability exceeding
$1-4N^{-1}-N^{-2}$:
1. (i)
$\frac{1}{2}M\leq|\mathcal{M}|\leq\frac{3}{2}M$,
2. (ii)
$\nu_{F}\leq\frac{\mu_{F}}{\sqrt{|\mathcal{M}|}}$,
3. (iii)
$\mu_{F}\leq\sqrt{\frac{118(N-M)\log{N}}{MN}}$.
###### Proof.
The claim that $F$ is tight follows trivially from the fact that the rows of
$U$ are orthogonal and that the rows of $F$ correspond to a subset of the rows
of $U$. Next, we define the probability events
$\smash{\mathcal{E}_{1}:=\\{|\mathcal{M}|\leq\tfrac{3}{2}M\\}}$ and
$\smash{\mathcal{E}_{2}:=\\{|\mathcal{M}|\geq\tfrac{1}{2}M\\}}$, and claim
that
$\smash{\Pr(\mathcal{E}_{1}^{\mathrm{c}}\cup\mathcal{E}_{2}^{\mathrm{c}})\leq
N^{-1}+N^{-2}}$. The proof of this claim follows from a Bernstein-like large
deviation inequality. Specifically, note that
$\smash{|\mathcal{M}|=\sum_{i=0}^{N-1}B_{i}}$ with
$\mathbb{E}[|\mathcal{M}|]=M$, and so we have from [3, Theorem A.1.12, Theorem
A.1.13] and [42, pp. 4] that for any $\delta_{1}\in[0,1)$,
$\Pr\Big{(}|\mathcal{M}|>(1+\delta_{1})M\Big{)}\leq\mathrm{e}^{-M\delta_{1}^{2}(1-\delta_{1})/2}\qquad\mbox{and}\qquad\Pr\Big{(}|\mathcal{M}|<(1-\delta_{1})M\Big{)}\leq\mathrm{e}^{-M\delta_{1}^{2}/2}.$
(16)
Taking $\delta_{1}:=\tfrac{1}{2}$, then a union bound gives
$\Pr(\mathcal{E}_{1}^{\mathrm{c}}\cup\mathcal{E}_{2}^{\mathrm{c}})\leq
N^{-1}+N^{-2}$ provided $M\geq 16\log{N}$. Conditioning on
$\mathcal{E}_{1}\cap\mathcal{E}_{2}$, we have that Theorem 10(i) holds
trivially, while Theorem 10(ii) follows from Lemma 7. Specifically, we have
that $\frac{N}{3}\geq M$ guarantees $N\geq 2|\mathcal{M}|$ because of the
conditioning on $\mathcal{E}_{1}\cap\mathcal{E}_{2}$, which in turn implies
that $F$ satisfies either condition (i) or (ii) of Lemma 7, depending on
whether $0\in\mathcal{M}$. This therefore establishes that Theorem 10(i)-(ii)
simultaneously hold with probability exceeding $1-N^{-1}-N^{-2}$.
The only remaining claim is that
$\mu_{X}\leq\delta_{2}:=\sqrt{(118(N-M)\log{N})/MN}$ with high probability. To
this end, define $p:=\frac{M}{N}$, and pick any two distinct indices
$i,j\in\\{0,\dots,N-1\\}$. Note that
$\langle
f_{i},f_{j}\rangle=\tfrac{1}{|\mathcal{M}|}\sum_{k=0}^{N-1}B_{k}U_{ki}\overline{U_{kj}}=\tfrac{1}{|\mathcal{M}|}\sum_{k=0}^{N-1}(B_{k}-p)U_{ki}\overline{U_{kj}},$
(17)
where the last equality follows from the fact that $U$ has orthogonal columns.
Next, we write
$\smash{U_{ki}\overline{U_{kj}}=\cos(\theta_{k})+\mathrm{i}\sin(\theta_{k})}$
for some $\theta_{k}\in[0,2\pi)$. Then applying the union bound to (17) and to
the real and imaginary parts of $\smash{U_{ki}\overline{U_{kj}}}$ gives
$\displaystyle\Pr\Big{(}|\langle f_{i},f_{j}\rangle|>\delta_{2}\Big{)}$
$\displaystyle\leq\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)U_{ki}\overline{U_{kj}}\Big{|}>\tfrac{M\delta_{2}}{2\sqrt{2}}\bigg{)}+\Pr\Big{(}|\mathcal{M}|<\tfrac{M}{2\sqrt{2}}\Big{)}$
$\displaystyle\leq\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)\cos(\theta_{k})\Big{|}>\tfrac{M\delta_{2}}{4}\bigg{)}+\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)\sin(\theta_{k})\Big{|}>\tfrac{M\delta_{2}}{4}\bigg{)}+N^{-3},$
(18)
where the last term follows from (16) and the fact that $M\geq 16\log{N}$.
Define random variables $Z_{k}:=(B_{k}-p)\cos(\theta_{k})$. Note that the
$Z_{k}$’s have zero mean and are jointly independent. Also, the $Z_{k}$’s are
bounded by $1-p$ almost surely since
$|(B_{k}-p)\cos(\theta_{k})|\leq\max\\{p,1-p\\}$ and $N\geq 2M$. Moreover, the
variance of each $Z_{k}$ is bounded: $\mathrm{var}(Z_{\ell})\leq p(1-p)$.
Therefore, we may use the Bernstein inequality for a sum of independent,
bounded random variables [8] to bound the probability that
$|\sum_{k=0}^{N-1}Z_{k}|$ deviates from $\delta_{3}:=\frac{M\delta_{2}}{4}$:
$\Pr\bigg{(}\Big{|}\sum_{k=0}^{N-1}(B_{k}-p)\cos(\theta_{k})\Big{|}>\delta_{3}\bigg{)}\leq
2\mathrm{e}^{-\delta_{3}^{2}/(2Np(1-p)+2(1-p)\delta_{3}/3)}\leq 2N^{-3}.$
Similarly, the probability that
$|\sum_{k=0}^{N-1}(B_{k}-p)\sin(\theta_{k})|>\delta_{3}$ is also bounded above
by $2N^{-3}$. Substituting these probability bounds into (18) gives $|\langle
f_{i},f_{j}\rangle|>\delta_{2}$ with probability at most $5N^{-3}$ provided
$M\geq 16\log{N}$. Finally, we take a union bound over the
$\smash{\binom{N}{2}}$ possible choices for $i$ and $j$ to get that Theorem
10(iii) holds with probability exceeding $1-3N^{-1}$.
The result now follows by taking a final union bound over
$\mathcal{E}_{1}^{\mathrm{c}}\cup\mathcal{E}_{2}^{\mathrm{c}}$ and
$\\{\mu_{X}>\delta_{2}\\}$. ∎
As stated earlier, random harmonic frames are not new to sparse signal
processing. Interestingly, for the application of compressed sensing, [13, 42]
provides performance guarantees for both random harmonic and Gaussian frames,
but requires more rows in a random harmonic frame to accommodate the same
level of sparsity. This suggests that random harmonic frames may be inferior
to Gaussian frames as compressed sensing matrices, but practice suggests
otherwise [22]. In a sense, Theorem 10 helps to resolve this gap in
understanding; there exist compressed sensing algorithms whose performance is
dictated by worst-case coherence [5, 21, 47, 50], and Theorem 10 states that
random harmonic frames have near-optimal worst-case coherence, being on the
order of the Welch bound with an additional $\sqrt{\log N}$ factor.
###### Example 11.
To illustrate the bounds in Theorem 10, we ran simulations in MATLAB. Picking
$N=5000$, we observed $30$ realizations of random harmonic frames for each
$M=1000,1250,1500$. The distributions of $|\mathcal{M}|$, $\nu_{F}$, and
$\mu_{F}$ were rather tight, so we only report the ranges of values attained,
along with the bounds given in Theorem 10. Notice that Theorem 10 gives a
bound on $\nu_{F}$ in terms of both $|\mathcal{M}|$ and $\mu_{F}$. To simplify
matters, we show that
$\smash{\nu_{F}\leq\frac{\min\mu_{F}}{\sqrt{\max|\mathcal{M}|}}\leq\frac{\mu_{F}}{\sqrt{|\mathcal{M}|}}}$,
where the minimum and maximum are taken over all realizations in the sample:
$\begin{array}[]{rrcll}M=1000:&\qquad|\mathcal{M}|&\in&[961,1052]&\qquad\subseteq[500,1500]\\\
&\qquad\nu_{F}&\in&[0.2000,0.8082]\times 10^{-3}&\qquad\leq
0.0023\approx\tfrac{0.0746}{\sqrt{1052}}\\\
&\qquad\mu_{F}&\in&[0.0746,0.0890]&\qquad\leq 0.8967\\\ \\\
M=1250:&\qquad|\mathcal{M}|&\in&[1207,1305]&\qquad\subseteq[625,1875]\\\
&\qquad\nu_{F}&\in&[0.2000,0.6273]\times 10^{-3}&\qquad\leq
0.0018\approx\tfrac{0.0623}{\sqrt{1305}}\\\
&\qquad\mu_{F}&\in&[0.0623,0.0774]&\qquad\leq 0.7766\\\ \\\
M=1500:&\qquad|\mathcal{M}|&\in&[1454,1590]&\qquad\subseteq[750,2250]\\\
&\qquad\nu_{F}&\in&[0.2000,0.4841]\times 10^{-3}&\qquad\leq
0.0015\approx\tfrac{0.0571}{\sqrt{1590}}\\\
&\qquad\mu_{F}&\in&[0.0571,0.0743]&\qquad\leq 0.6849\end{array}$
The reader may have noticed how consistently the average coherence value of
$\nu_{F}\approx 0.2000\times 10^{-3}$ was realized. This occurs precisely when
the zeroth row of the DFT is not selected, as the frame elements sum to zero
in this case:
$\nu_{F}:=\tfrac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}\langle
f_{i},f_{j}\rangle\bigg{|}=\tfrac{1}{N-1}\max_{i\in\\{1,\ldots,N\\}}\bigg{|}\bigg{\langle}f_{i},\sum_{j=1}^{N}f_{j}\bigg{\rangle}-\|f_{i}\|^{2}\bigg{|}=\tfrac{1}{N-1}.$
These simulations seem to indicate that our bounds on $|\mathcal{M}|$,
$\nu_{F}$, and $\mu_{F}$ leave room for improvement. The only bound that lies
within an order of magnitude of real-world behavior is our bound on
$|\mathcal{M}|$.
### 3.3 Gabor and chirp frames
Gabor frames constitute an important class of frames, as they appear in a
variety of applications such as radar [29], speech processing [53], and
quantum information theory [43]. Given a nonzero seed function
$f:\mathbb{Z}_{M}\rightarrow\mathbb{C}$, we produce all time- and frequency-
shifted versions: $f_{xy}(t):=f(t-x)\mathrm{e}^{2\pi\mathrm{i}yt/M}$,
$t\in\mathbb{Z}_{M}$. Viewing these shifted functions as vectors in
$\mathbb{C}^{M}$ gives an $M\times M^{2}$ Gabor frame. The following theorem
characterizes the spectral norm and the worst-case and average coherence of
Gabor frames generated from either a deterministic Alltop vector [1] or a
random Steinhaus vector.
###### Theorem 12 (Geometry of Gabor frames).
Take an Alltop function defined by
$\smash{f(t):=\frac{1}{\sqrt{M}}\mathrm{e}^{2\pi\mathrm{i}t^{3}/M}}$,
$t\in\mathbb{Z}_{M}$. Also, take a random Steinhaus function defined by
$\smash{g(t):=\frac{1}{\sqrt{M}}\mathrm{e}^{2\pi\mathrm{i}\theta_{t}}}$,
$t\in\mathbb{Z}_{M}$, where the $\theta_{t}$’s are independent random
variables distributed uniformly on the unit interval. Then the $M\times M^{2}$
Gabor frames $F$ and $G$ generated by $f$ and $g$, respectively, are unit norm
and tight, that is, $\|F\|_{2}=\|G\|_{2}=\sqrt{M}$, and both frames have
average coherence $\smash{\leq\frac{1}{M+1}}$. Furthermore, if $M\geq 5$ is
prime, then $\smash{\mu_{F}=\frac{1}{\sqrt{M}}}$, while if $M\geq 13$, then
$\mu_{G}\leq\sqrt{(13\log{M})/M}$ with probability exceeding $1-4M^{-1}$.
###### Proof.
The tightness claim follows from [35], in which it was shown that Gabor frames
generated by nonzero seed vectors are tight. The bound on average coherence is
a consequence of [5, Theorem 7] concerning arbitrary Gabor frames. The claim
concerning $\mu_{F}$ follows directly from [46], while the claim concerning
$\mu_{G}$ is a simple consequence of [40, Theorem 5.1]. ∎
Instead of taking all translates and modulates of a seed function, [16]
constructs _chirp frames_ by taking all powers and modulates of a chirp
function. Picking $M$ to be prime, we start with a chirp function
$h_{M}:\mathbb{Z}_{M}\rightarrow\mathbb{C}$ defined by
$\smash{h_{M}(t):=\mathrm{e}^{\pi\mathrm{i}t(t-M)/M}}$, $t\in\mathbb{Z}_{M}$.
The $M^{2}$ frame elements are then defined entrywise by
$\smash{h_{ab}(t):=\frac{1}{\sqrt{M}}h_{M}(t)^{a}\mathrm{e}^{2\pi\mathrm{i}bt/M}}$,
$t\in\mathbb{Z}_{M}$. Certainly, chirp frames are, at the very least, similar
in spirit to Gabor frames. As a matter of fact, the chirp frame is in some
sense equivalent to the Gabor frame generated by the Alltop function: it is
easy to verify that
$h_{(-6x,y-3x^{2})}(t)=\mathrm{e}^{2\pi\mathrm{i}(t^{3}+x^{3})/M}f_{xy}(t)$,
and when $M\geq 5$, the map $(x,y)\mapsto(-6x,y-3x^{2})$ is a permutation over
$\mathbb{Z}_{M}^{2}$. Using terminology from Definition 28, we say the chirp
frame is _wiggling equivalent_ to a unitary rotation of permuted Alltop Gabor
frame elements. As such, by Lemma 29, the chirp frame has the same spectral
norm and worst-case coherence as the Alltop Gabor frame, but the average
coherence may be different. In this case, the average coherence still
satisfies (SCP-2). Indeed, adding the frame elements gives
$\sum_{a=0}^{M-1}\sum_{b=0}^{M-1}h_{ab}(t)=\tfrac{1}{\sqrt{M}}\sum_{a=0}^{M-1}h_{M}(t)^{a}\sum_{b=0}^{M-1}\mathrm{e}^{2\pi\mathrm{i}bt/M}=\tfrac{1}{\sqrt{M}}\sum_{a=0}^{M-1}h_{M}(t)^{a}M\delta_{0}(t)=\sqrt{M}\bigg{(}\sum_{a=0}^{M-1}h_{M}(0)^{a}\bigg{)}~{}\delta_{0}(t)=M^{3/2}\delta_{0}(t),$
and so $\langle
h_{a^{\prime}b^{\prime}},\sum_{a=0}^{M-1}\sum_{b=0}^{M-1}h_{ab}\rangle=\langle
h_{a^{\prime}b^{\prime}},M^{3/2}\delta_{0}\rangle=M^{3/2}h_{a^{\prime}b^{\prime}}(0)=M=\frac{M^{2}}{M}$.
Therefore, Lemma 7(i) gives the result:
###### Theorem 13 (Geometry of chirp frames).
Pick $M$ prime, and let $H$ be the $M\times M^{2}$ frame of all powers and
modulates of the chirp function $f_{M}$. Then $H$ is a unit norm tight frame
with $\|H\|_{2}=\sqrt{M}$, and has worst case coherence
$\smash{\mu_{H}=\frac{1}{\sqrt{M}}}$ and average coherence
$\smash{\nu_{H}\leq\frac{\mu_{H}}{\sqrt{M}}}$.
###### Example 14.
To illustrate the bounds in Theorems 12 and 13, we consider the examples of an
Alltop Gabor frame and a chirp frame, each with $M=5$. In this case, the Gabor
frame has $\smash{\nu_{F}\approx 0.1348\leq 0.1667\approx\frac{1}{M+1}}$,
while the chirp frame has
$\smash{\nu_{H}=\frac{1}{6}\leq\frac{1}{5}=\frac{\mu_{H}}{\sqrt{M}}}$. Note
the Gabor and chirp frames have different average coherences despite being
equivalent in some sense. For the random Steinhaus Gabor frame, we ran
simulations in MATLAB and observed $30$ realizations for each $M=60,70,80$.
The distributions of $\nu_{G}$ and $\mu_{G}$ were rather tight, so we only
report the ranges of values attained, along with the bounds given in Theorem
12:
$\begin{array}[]{rrcll}M=60:&\qquad\nu_{G}&\in&[0.3916,0.5958]\times
10^{-2}&\qquad\leq 0.0164\\\ &\qquad\mu_{G}&\in&[0.3242,0.4216]&\qquad\leq
0.9419\\\ \\\ M=70:&\qquad\nu_{G}&\in&[0.3151,0.4532]\times 10^{-2}&\qquad\leq
0.0141\\\ &\qquad\mu_{G}&\in&[0.2989,0.3814]&\qquad\leq 0.8883\\\ \\\
M=80:&\qquad\nu_{G}&\in&[0.2413,0.3758]\times 10^{-2}&\qquad\leq 0.0124\\\
&\qquad\mu_{G}&\in&[0.2711,0.3796]&\qquad\leq 0.8439\end{array}$
These simulations seem to indicate that bound on $\nu_{G}$ is conservative by
an order of magnitude.
### 3.4 Spherical 2-designs
Lemma 7(ii) leads one to consider frames of vectors that sum to zero. In [31],
it is proved that real unit norm tight frames with this property make up
another well-studied class of vector packings: spherical 2-designs. To be
clear, a collection of unit-norm vectors $F\subseteq\mathbb{R}^{M}$ is called
a spherical $t$-design if, for every polynomial $g(x_{1},\ldots,x_{M})$ of
degree at most $t$, we have
$\tfrac{1}{\mathrm{H}^{M-1}(S^{M-1})}\int_{S^{M-1}}g(x)~{}\mathrm{d}\mathrm{H}^{M-1}(x)=\tfrac{1}{|F|}\sum_{f\in
F}g(f),$
where $S^{M-1}$ is the unit hypersphere in $\mathbb{R}^{M}$ and
$\mathrm{H}^{M-1}$ denotes the $(M-1)$-dimensional Hausdorff measure on
$S^{M-1}$. In words, vectors that form a spherical $t$-design serve as good
representatives when calculating the average value of a degree-$t$ polynomial
over the unit hypersphere. Today, such designs find application in quantum
state estimation [28].
Since real unit norm tight frames always exist for $N\geq M+1$, one might
suspect that spherical 2-designs are equally common, but this intuition is
faulty—the sum-to-zero condition introduces certain issues. For example, there
is no spherical 2-design when $M$ is odd and $N=M+2$. In [36], spherical
2-designs are explicitly characterized by construction. The following theorem
gives a construction based on harmonic frames:
###### Theorem 15 (Geometry of spherical 2-designs).
Pick $M$ even and $N\geq 2M$. Take an $\frac{M}{2}\times N$ harmonic frame $G$
by collecting rows from a discrete Fourier transform matrix according to a set
of nonzero indices $\mathcal{M}$ and normalize the columns. Let $m(n)$ denote
$n$th largest index in $\mathcal{M}$, and define a real $M\times N$ frame $F$
by
$F_{k\ell}:=\left\\{\begin{array}[]{ll}\sqrt{\frac{2}{M}}\cos(\frac{2\pi
m((k+1)/2)\ell}{N}),&k\mbox{ odd}\\\ \sqrt{\frac{2}{M}}\sin(\frac{2\pi
m(k/2)\ell}{N}),&k\mbox{ even}\end{array}\right.,\qquad
k=1,\ldots,M,~{}\ell=0,\ldots,N-1.$
Then $F$ is unit norm and tight, i.e., $\|F\|_{2}^{2}=\frac{N}{M}$, with
worst-case coherence $\mu_{F}\leq\mu_{G}$ and average coherence
$\nu_{F}\leq\frac{\mu_{F}}{\sqrt{M}}$.
###### Proof.
It is easy to verify that $F$ is a unit norm tight frame using the geometric
sum formula. Also, since the frame elements sum to zero and $N\geq 2M$, the
claim regarding average coherence follows from Lemma 7(ii). It remains to
prove $\mu_{F}\leq\mu_{G}$. For each distinct pair of indices
$i,j\in\\{1,\ldots,N\\}$, we have
$\langle
f_{i},f_{j}\rangle=\tfrac{2}{M}\sum_{m\in\mathcal{M}}\Big{(}\cos(\tfrac{2\pi
mi}{N})\cos(\tfrac{2\pi mj}{N})+\sin(\tfrac{2\pi mi}{N})\sin(\tfrac{2\pi
mj}{N})\Big{)}=\tfrac{2}{M}\sum_{m\in\mathcal{M}}\cos(\tfrac{2\pi
m(i-j)}{N})=\mathrm{Re}\langle g_{i},g_{j}\rangle,$
and so $|\langle f_{i},f_{j}\rangle|=|\mathrm{Re}\langle
g_{i},g_{j}\rangle|\leq|\langle g_{i},g_{j}\rangle|$. This gives the result. ∎
###### Example 16.
To illustrate the bounds in Theorem 15, we consider the spherical 2-design
constructed from a $9\times 37$ harmonic equiangular tight frame [54].
Specifically, we take a $37\times 37$ DFT matrix, choose nonzero row indices
$\mathcal{M}=\\{1,7,9,10,12,16,26,33,34\\},$
and normalize the columns to get a harmonic frame $G$ whose worst-case
coherence achieves the Welch bound:
$\smash{\mu_{G}=\sqrt{\frac{37-9}{9(37-1)}}\approx 0.2940}$. Following Theorem
15, we produce a spherical 2-design $F$ with $\mu_{F}\approx
0.1967\leq\mu_{G}$ and $\smash{\nu_{F}\approx 0.0278\leq
0.0464\approx\frac{\mu_{F}}{\sqrt{M}}}$.
### 3.5 Steiner equiangular tight frames
We now consider a construction that dates back to Seidel with [44], and was
recently developed further in [24]. Here, a special type of block design is
used to build an equiangular tight frame (ETF), that is, a tight frame in
which the modulus of every inner product between frame elements achieves the
Welch bound. Let’s start with a definition:
###### Definition 17.
A $(t,k,v)$-_Steiner system_ is a $v$-element set $V$ with a collection of
$k$-element subsets of $V$, called _blocks_ , with the property that any
$t$-element subset of $V$ is contained in exactly one block. The
$\\{0,1\\}$-_incidence matrix_ $A$ of a Steiner system has entries $A_{ij}$,
where $A_{ij}=1$ if the $i$th block contains the $j$th element, and otherwise
$A_{ij}=0$.
One example of a Steiner system is a set with all possible two-element blocks.
This forms a $(2,2,v)$-Steiner system because every pair of elements is
contained in exactly one block. The following theorem details how [24]
constructs ETFs using Steiner systems.
###### Theorem 18 (Constructing Steiner equiangular tight frames [24]).
Every $(2,k,v)$-Steiner system can be used to build a
$\smash{\frac{v(v-1)}{k(k-1)}\times v(1+\frac{v-1}{k-1})}$ equiangular tight
frame $F$ according the following procedure:
1. (i)
Let $A$ be the $\frac{v(v-1)}{k(k-1)}\times v$ incidence matrix of a
$(2,k,v)$-Steiner system.
2. (ii)
Let $H$ be the $(1+\frac{v-1}{k-1})\times(1+\frac{v-1}{k-1})$ discrete Fourier
transform matrix.
3. (iii)
For each $j=1,\ldots,v$, let $F_{j}$ be a
$\frac{v(v-1)}{k(k-1)}\times(1+\frac{v-1}{k-1})$ matrix obtained from the
$j$th column of $A$ by replacing each of the one-valued entries with a
distinct row of $H$, and every zero-valued entry with a row of zeros.
4. (iv)
Concatenate and rescale the $F_{j}$’s to form
$F=(\frac{k-1}{v-1})^{\frac{1}{2}}[F_{1}\cdots F_{v}]$.
As an example, we build an ETF from a (2,2,3)-Steiner system. In this case,
the incidence matrix is
$A=\left[\begin{array}[]{ccc}+&+&\\\ +&&+\\\ &+&+\end{array}\right].$
For this matrix, each row represents a block. Since each block contains two
elements, each row of the matrix has two ones. Also, any two elements
determines a unique common row, and so any two columns have a single one in
common. To form the corresponding $3\times 9$ ETF $F$, we use the $3\times 3$
DFT matrix. Letting $\omega=\mathrm{e}^{2\pi\mathrm{i}/3}$, we have
$H=\left[\begin{array}[]{lll}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\end{array}\right].$
Finally, we replace the two ones in each column of $A$ with the second and
third rows of $H$. Normalizing the columns gives $3\times 9$ ETF:
$F=\tfrac{1}{\sqrt{2}}\left[\begin{array}[]{lllllllll}1&\omega&\omega^{2}&1&\omega&\omega^{2}&&&\\\
1&\omega^{2}&\omega&&&&1&\omega&\omega^{2}\\\
&&&1&\omega^{2}&\omega&1&\omega^{2}&\omega\end{array}\right].$ (19)
Several infinite families of $(2,k,v)$-Steiner systems are already known, and
Theorem 18 says that each one can be used to build an ETF. See [24] for a
complete discussion of this construction and how it relates to each known
family of Steiner systems. Interestingly, every Steiner ETF satisfies $N\geq
2M$. If, in step (iii) of Theorem 18, we choose the distinct rows to be the
$\frac{v-1}{k-1}$ rows of the DFT $H$ that are not all-ones, then the sum of
columns of each $F_{j}$ is zero, meaning the sum of columns of $F$ is also
zero. This was done in the example above, and the columns sum to zero,
accordingly. Therefore, by Lemma 7(ii), Steiner ETFs satisfy (SCP-2). This
gives the following theorem:
###### Theorem 19 (Geometry of Steiner equiangular tight frames).
Build an $M\times N$ matrix $F$ according to Theorem 18, and in step (iii),
choose rows from the discrete Fourier transform matrix $H$ that are not all-
ones. Then $F$ is an equiangular tight frame, meaning
$\|F\|_{2}^{2}=\frac{N}{M}$ and $\mu_{F}^{2}=\frac{N-M}{M(N-1)}$, and has
average coherence $\nu_{F}\leq\frac{\mu_{F}}{\sqrt{M}}$.
###### Example 20.
To illustrate the bound in Theorem 19, we note that the example given in (19)
has
$\smash{\nu_{F}=\frac{1}{8}\leq\frac{1}{2\sqrt{3}}=\frac{\mu_{F}}{\sqrt{M}}}$.
### 3.6 Code-based frames
Many structures in coding theory are also useful in frame theory. In this
section, we build frames from a code that originally emerged with Berlekamp in
[9], and found recent reincarnation with [55]. We build a $2^{m}\times
2^{(t+1)m}$ frame, indexing rows by elements of $\mathbb{F}_{2^{m}}$ and
indexing columns by $(t+1)$-tuples of elements from $\mathbb{F}_{2^{m}}$. For
$x\in\mathbb{F}_{2^{m}}$ and $\alpha\in\mathbb{F}_{2^{m}}^{t+1}$, the
corresponding entry of the matrix $F$ is given by
$F_{x\alpha}=\tfrac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}},$
(20)
where $\mathrm{Tr}:\mathbb{F}_{2^{m}}\rightarrow\mathbb{F}_{2}$ denotes the
trace map, defined by $\mathrm{Tr}(z)=\sum_{i=0}^{m-1}z^{2^{i}}$. The
following theorem gives the spectral norm and the worst-case and average
coherence of this frame.
Name | $\mathbb{R}/\mathbb{C}$ | Size | $\mu_{F}$ | $\nu_{F}$ | Restrictions | Probability
---|---|---|---|---|---|---
Normalized Gaussian | $\mathbb{R}$ | $M\times N$ | $\leq\frac{\sqrt{15\log{N}}}{\sqrt{M}-\sqrt{12\log{N}}}$ | $\leq\frac{\sqrt{15\log{N}}}{M-\sqrt{12M\log{N}}}$ | $60\log N\leq M\leq\frac{N-1}{4\log N}$ | $\geq 1-\frac{11}{N}$
Random harmonic | $\mathbb{C}$ | $|\mathcal{M}|\times N$, $\frac{1}{2}M\leq|\mathcal{M}|\leq\frac{3}{2}M$ | $\leq\sqrt{\frac{118(N-M)\log{N}}{MN}}$ | $\leq\frac{\mu_{F}}{\sqrt{|\mathcal{M}|}}$ | $16\log{N}\leq M\leq\frac{N}{3}$ | $\geq 1-\frac{4}{N}-\frac{1}{N^{2}}$
Alltop Gabor | $\mathbb{C}$ | $M\times M^{2}$ | $=\frac{1}{\sqrt{M}}$ | $\leq\frac{1}{M+1}$ | $M\geq 5$ prime | Deterministic
Steinhaus Gabor | $\mathbb{C}$ | $M\times M^{2}$ | $\leq\sqrt{\frac{13\log M}{M}}$ | $\leq\frac{1}{M+1}$ | $M\geq 13$ | $\geq 1-\frac{4}{M}$
Chirp | $\mathbb{C}$ | $M\times M^{2}$ | $=\frac{1}{\sqrt{M}}$ | $\leq\frac{\mu_{F}}{\sqrt{M}}$ | $M$ prime | Deterministic
$\overset{\mbox{Spherical 2-design}}{\mbox{from harmonic }G}$ | $\mathbb{R}$ | $M\times N$ | $\leq\mu_{G}$ | $\leq\frac{\mu_{F}}{\sqrt{M}}$ | $M$ even, $N\geq 2M$ | Deterministic
Steiner | $\mathbb{C}$ | $M\times N$, $M=\frac{v(v-1)}{k(k-1)}$, $N=v(1+\frac{v-1}{k-1})$ | $=\sqrt{\frac{N-M}{M(N-1)}}$ | $\leq\frac{\mu_{F}}{\sqrt{M}}$ | $\exists(2,k,v)$-Steiner system | Deterministic
Code-based | $\mathbb{R}$ | $2^{m}\times 2^{(t+1)m}$ | $\leq\frac{1}{\sqrt{2^{m-2t-1}}}$ | $\leq\frac{\mu_{F}}{\sqrt{2^{m}}}$ | None | Deterministic
Table 1: Eight constructions detailed in this paper. All of these are unit
norm tight frames except for the normalized Gaussian frame, which has squared
spectral norm
$\|F\|_{2}^{2}\leq(\\!\sqrt{M}+\\!\sqrt{N}+\\!\sqrt{2\log{N}})^{2}/(M-\\!\sqrt{8M\log{N}})$
in the same probability event as is measured above.
###### Theorem 21 (Geometry of code-based frames).
The $2^{m}\times 2^{(t+1)m}$ frame defined by (20) is unit norm and tight,
i.e., $\|F\|_{2}^{2}=2^{tm}$, with worst-case coherence
$\mu_{F}\leq\frac{1}{\sqrt{2^{m-2t-1}}}$ and average coherence
$\smash{\nu_{F}\leq\frac{\mu_{F}}{\sqrt{2^{m}}}}$.
###### Proof.
For the tightness claim, we use the linearity of the trace map to write the
inner product of rows $x$ and $y$:
$\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}\\!\\!\tfrac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}\tfrac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}y+\sum_{i=1}^{t}\alpha_{i}y^{2^{i}+1}\big{]}}=\tfrac{1}{2^{m}}\bigg{(}\\!\sum_{\alpha_{0}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}[\alpha_{0}(x+y)]}\bigg{)}\\!\\!\sum_{\alpha_{1}\in\mathbb{F}_{2^{m}}}\\!\\!\cdots\\!\\!\sum_{\alpha_{t}\in\mathbb{F}_{2^{m}}}\\!\\!(-1)^{\mathrm{Tr}\big{[}\sum_{i=1}^{t}\alpha_{i}(x^{2^{i}+1}+y^{2^{i}+1})\big{]}}.$
This expression is $2^{tm}$ when $x=y$. Otherwise, note that
$\alpha_{0}\mapsto(-1)^{\mathrm{Tr}[\alpha_{0}(x+y)]}\in\\{\pm 1\\}$ defines a
homomorphism on $\mathbb{F}_{2^{m}}$. Since $(x+y)^{-1}\mapsto-1$, the inverse
images of $\pm 1$ under this homomorphism must form two cosets of equal size,
and so
$\sum_{\alpha_{0}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}[\alpha_{0}(x+y)]}=0$,
meaning distinct rows in $F$ are orthogonal. Thus, $F$ is a unit norm tight
frame.
For the worst-case coherence claim, we first note that the linearity of the
trace map gives
$(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}(-1)^{\mathrm{Tr}\big{[}\alpha^{\prime}_{0}x+\sum_{i=1}^{t}\alpha^{\prime}_{i}x^{2^{i}+1}\big{]}}=(-1)^{\mathrm{Tr}\big{[}(\alpha_{0}+\alpha^{\prime}_{0})x+\sum_{i=1}^{t}(\alpha_{i}+\alpha^{\prime}_{i})x^{2^{i}+1}\big{]}},$
i.e., every inner product between columns of $F$ is a sum over another column.
Thus, there exists $\alpha\in\mathbb{F}_{2^{m}}^{t+1}$ such that
$2^{2m}\mu_{F}^{2}=\bigg{(}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}\bigg{)}^{2}=2^{m}+\sum_{x\in\mathbb{F}_{2^{m}}}\sum_{\begin{subarray}{c}y\in\mathbb{F}_{2^{m}}\\\
y\neq
x\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}(x+y)+\sum_{i=1}^{t}\alpha_{i}\big{(}(x+y)^{2^{i}+1}+\sum_{j=0}^{i-1}(xy)^{2^{j}}(x+y)^{2^{i}-2^{j+1}+1}\big{)}\big{]}},$
where the last equality is by the identity
$(x+y)^{2^{i}+1}=x^{2^{i}+1}+y^{2^{i}+1}+\sum_{j=0}^{i-1}(xy)^{2^{j}}(x+y)^{2^{i}-2^{j+1}+1}$,
whose proof is a simple exercise of induction. From here, we perform a change
of variables: $u:=x+y$ and $v:=xy$. Notice that $(u,v)$ corresponds to $(x,y)$
for some $x\neq y$ whenever $(z+x)(z+y)=z^{2}+uz+v$ has two solutions, that
is, whenever $\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$. Since $(u,v)$
corresponds to both $(x,y)$ and $(y,x)$, we must correct for under-counting:
$\displaystyle 2^{2m}\mu_{F}^{2}$
$\displaystyle=2^{m}+2\sum_{\begin{subarray}{c}u\in\mathbb{F}_{2^{m}}\\\ u\neq
0\end{subarray}}\sum_{\begin{subarray}{c}v\in\mathbb{F}_{2^{m}}\\\
\mathrm{Tr}(v/u^{2})=0\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}u+\sum_{i=1}^{t}\alpha_{i}\big{(}u^{2^{i}+1}+\sum_{j=0}^{i-1}v^{2^{j}}u^{2^{i}-2^{j+1}+1}\big{)}\big{]}}$
$\displaystyle=2^{m}+2\sum_{\begin{subarray}{c}u\in\mathbb{F}_{2^{m}}\\\ u\neq
0\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}u+\sum_{i=1}^{t}\alpha_{i}u^{2^{i}+1}\big{]}}\sum_{\begin{subarray}{c}v\in\mathbb{F}_{2^{m}}\\\
\mathrm{Tr}(v/u^{2})=0\end{subarray}}(-1)^{\mathrm{Tr}\big{[}\big{(}\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{-j}}u^{2^{i-j}-2+2^{-j}}\big{)}v\big{]}}$
$\displaystyle\leq 2^{m}+2\sum_{\begin{subarray}{c}u\in\mathbb{F}_{2^{m}}\\\
u\neq
0\end{subarray}}~{}\bigg{|}\\!\\!\\!\sum_{\begin{subarray}{c}v\in\mathbb{F}_{2^{m}}\\\
\mathrm{Tr}(v/u^{2})=0\end{subarray}}\\!\\!\\!(-1)^{\mathrm{Tr}[p(u)v]}~{}\bigg{|},$
(21)
where the second equality is by repeated application of
$\mathrm{Tr}(z)=\mathrm{Tr}(z^{2})$, and
$\smash{p(u):=\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{-j}}u^{2^{i-j}-2+2^{-j}}}$.
To bound $\mu_{F}$, we will count the $u$’s that produce nonzero summands in
(21).
For each $u\neq 0,$ we have a homomorphism
$\smash{\chi_{u}:\\{v\in\mathbb{F}_{2^{m}}:\mathrm{Tr}(\frac{v}{u^{2}})=0\\}\rightarrow\\{\pm
1\\}}$ defined by $\chi_{u}(v):=(-1)^{\mathrm{Tr}[p(u)v]}$. Pick $u\neq 0$ for
which there exists a $v$ such that both
$\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$ and $\mathrm{Tr}[p(u)v]=1$. Then
$\chi_{u}(v)=-1$, and so the kernel of $\chi_{u}$ is the same size as the
coset
$\smash{\\{v\in\mathbb{F}_{2^{m}}:\mathrm{Tr}(\frac{v}{u^{2}})=0,\chi_{u}(v)=-1\\}}$,
meaning the summand associated with $u$ in (21) is zero. Hence, the nonzero
summands in (21) require $\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$ and
$\mathrm{Tr}[p(u)v]=0$. This is certainly possible whenever $p(u)=0$.
Exponentiation gives
$p(u)^{2^{t-1}}=\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{t-j-1}}u^{2^{t+i-j-1}-2^{t}+2^{t-j-1}},$
which has degree $2^{2t-1}-2^{t-1}$. Thus, $p(u)=0$ has at most
$2^{2t-1}-2^{t-1}$ solutions, and each such $u$ produces a summand in (21) of
size $2^{m-1}$. Next, we consider the $u$’s for which
$\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$, $\mathrm{Tr}[p(u)v]=0$, and
$p(u)\neq 0$. In this case, the hyperplanes defined by
$\smash{\mathrm{Tr}(\frac{v}{u^{2}})=0}$ and $\mathrm{Tr}[p(u)v]=0$ are
parallel, and so $\smash{p(u)=\frac{1}{u^{2}}}$. Here,
$1=(u^{2}p(u))^{2^{t-1}}=\sum_{i=1}^{t}\sum_{j=0}^{i-1}\alpha_{i}^{2^{t-j-1}}u^{2^{t+i-j-1}+2^{t-j-1}},$
which has degree $2^{2t-1}+2^{t-1}$. Thus, $\smash{p(u)=\frac{1}{u^{2}}}$ has
at most $2^{2t-1}+2^{t-1}$ solutions, and each such $u$ produces a summand in
(21) of size $2^{m-1}$. We can now continue the bound from (21):
$2^{2m}\mu_{F}^{2}\leq 2^{m}+2(2^{2t-1}-2^{t-1}+2^{2t-1}+2^{t-1})2^{m-1}\leq
2^{m+2t+1}$. From here, isolating $\mu_{F}$ gives the claim.
Lastly, for the average coherence, pick some $x\in\mathbb{F}_{2^{m}}$. Then
summing the entries in the $x$th row gives
$\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}\tfrac{1}{\sqrt{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\alpha_{0}x+\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}=\tfrac{1}{\sqrt{2^{m}}}\bigg{(}\sum_{\alpha_{0}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}(\alpha_{0}x)}\bigg{)}\sum_{\alpha_{1}\in\mathbb{F}_{2^{m}}}\cdots\sum_{\alpha_{t}\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}\big{[}\sum_{i=1}^{t}\alpha_{i}x^{2^{i}+1}\big{]}}=\left\\{\begin{array}[]{lc}2^{(t+1/2)m},&x=0\\\
0,&x\neq 0\end{array}\right..$
That is, the frame elements sum to a multiple of an identity basis element:
$\smash{\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}f_{\alpha}=2^{(t+1/2)m}\delta_{0}}$.
Since every entry in row $x=0$ is $\smash{\frac{1}{\sqrt{2^{m}}}}$, we have
$\smash{\langle
f_{\alpha^{\prime}},\sum_{\alpha\in\mathbb{F}_{2^{m}}^{t+1}}f_{\alpha}\rangle=\frac{2^{(t+1)m}}{2^{m}}}$
for every $\alpha^{\prime}\in\mathbb{F}_{2^{m}}^{t+1}$, and so by Lemma 7(i),
we are done. ∎
###### Example 22.
To illustrate the bounds in Theorem 21, we consider the example where $m=4$
and $t=1$. This is a $16\times 256$ code-based frame $F$ with
$\smash{\mu_{F}=\frac{1}{2}\leq\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2^{m-2t-1}}}}$
and
$\smash{\nu_{F}=\frac{1}{17}\leq\frac{1}{8}=\frac{\mu_{F}}{\sqrt{2^{m}}}}$.
## 4 Fundamental limits on worst-case coherence
In many applications of frames, performance is dictated by worst-case
coherence [5, 11, 21, 31, 37, 46, 47, 50, 56]. It is therefore particularly
important to understand which worst-case coherence values are achievable. To
this end, the Welch bound is commonly used in the literature. When worst-case
coherence achieves the Welch bound, the frame is equiangular and tight [46];
one of the biggest open problems in frame theory concerns equiangular tight
frames [43]. However, equiangular tight frames cannot have more vectors than
the square of the spatial dimension [46], meaning the Welch bound is not tight
whenever $N>M^{2}$. When the number of vectors $N$ is exceedingly large, the
following theorem gives a better bound:
###### Theorem 23 ([2, 39]).
Every sufficiently large $M\times N$ unit norm frame $F$ with $N\geq 2M$ and
worst-case coherence $\mu_{F}<\frac{1}{2}$ satisfies
$\mu_{F}^{2}\log\big{(}\tfrac{1}{\mu_{F}}\big{)}\geq\tfrac{C\log N}{M}$ (22)
for some constant $C>0$.
For a fixed worst-case coherence $\mu_{F}<\frac{1}{2}$, this bound indicates
that the number of vectors $N$ cannot exceed some exponential in the spatial
dimension $M$, that is, $N\leq a^{M}$ for some $a>0$. However, since the
constant $C$ is not established in this theorem, it is unclear which base $a$
is appropriate for each $\mu_{F}$. The following theorem is a little more
explicit in this regard:
###### Theorem 24 ([38, 54]).
Every $M\times N$ unit norm frame $F$ has worst-case coherence $\mu_{F}\geq
1-2N^{-1/(M-1)}$. Furthermore, taking $N=\Theta(a^{M})$, this lower bound goes
to $1-\frac{2}{a}$ as $M\rightarrow\infty$.
For many applications, it does not make sense to use a complex frame, but the
bound in Theorem 24 is known to be loose for real frames [18]. We therefore
improve Theorems 23 and 24 for the case of real unit norm frames:
###### Theorem 25.
Every real $M\times N$ unit norm frame $F$ has worst-case coherence
$\mu_{F}\geq\cos\bigg{[}\pi\Big{(}\tfrac{M-1}{N\pi^{1/2}}~{}\tfrac{\Gamma(\frac{M-1}{2})}{\Gamma(\frac{M}{2})}\Big{)}^{\frac{1}{M-1}}\bigg{]}.$
(23)
Furthermore, taking $N=\Theta(a^{M})$, this lower bound goes to
$\cos(\frac{\pi}{a})$ as $M\rightarrow\infty$.
Before proving this theorem, we first consider the special case where the
spatial dimension is $M=3$:
###### Lemma 26.
Given $N$ points on the unit sphere $S^{2}\subseteq\mathbb{R}^{3}$, the
smallest angle between points is $\leq 2\cos^{-1}\big{(}1-\frac{2}{N}\big{)}$.
###### Proof.
We first claim there exists a closed spherical cap in $S^{2}$ with area
$\smash{\frac{4\pi}{N}}$ that contains two of the $N$ points. Suppose
otherwise, and take $\gamma$ to be the angular radius of a spherical cap with
area $\smash{\frac{4\pi}{N}}$. That is, $\gamma$ is the angle between the
center of the cap and every point on the boundary. Since the cap is closed, we
must have that the smallest angle $\alpha$ between any two of our $N$ points
satisfies $\alpha>2\gamma$. Let $C(p,\theta)$ denote the closed spherical cap
centered at $p\in S^{2}$ of angular radius $\theta$, and let $P$ denote our
set of $N$ points. Then we know for $p\in P$, the $C(p,\gamma)$’s are
disjoint, $\frac{\alpha}{2}>\gamma$, and $\bigcup_{p\in
P}C(p,\tfrac{\alpha}{2})\subseteq S^{2}$, and so taking 2-dimensional
Hausdorff measures on the sphere gives
$\mathrm{H}^{2}(S^{2})=4\pi=\mathrm{H}^{2}\bigg{(}\bigcup_{p\in
P}C(p,\gamma)\bigg{)}<\mathrm{H}^{2}\bigg{(}\bigcup_{p\in
P}C(p,\tfrac{\alpha}{2})\bigg{)}\leq\mathrm{H}^{2}(S^{2}),$
a contradiction.
Since two of the points reside in a spherical cap of area
$\smash{\frac{4\pi}{N}}$, we know $\alpha$ is no more than twice the radius of
this cap. We use spherical coordinates to relate the cap’s area to the radius:
$\smash{\mathrm{H}^{2}(C(\cdot,\gamma))=2\pi\int_{0}^{\gamma}\sin\phi~{}\mathrm{d}\phi=2\pi(1-\cos\gamma)}$.
Therefore, when $\smash{\mathrm{H}^{2}(C(\cdot,\gamma))=\frac{4\pi}{N}}$, we
have $\gamma=\cos^{-1}(1-\frac{2}{N})$, and so $\alpha\leq 2\gamma$ gives the
result. ∎
###### Theorem 27.
Every real $3\times N$ unit norm frame $F$ has worst-case coherence
$\mu_{F}\geq 1-\frac{4}{N}+\frac{2}{N^{2}}$.
###### Proof.
Packing $N$ unit vectors in $\mathbb{R}^{3}$ corresponds to packing $2N$
antipodal points in $S^{2}$, and so Lemma 26 gives $\alpha\leq
2\cos^{-1}(1-\frac{1}{N})$. Applying the double angle formula to
$\mu_{F}=\cos\alpha\geq\cos[2\cos^{-1}(1-\frac{1}{N})]$ gives the result. ∎
$N$$\mu_{F}$Numerically optimalWelch boundTheorem 24Theorem 25Theorem 27
Figure 1: Different bounds on worst-case coherence for $M=3$, $N=3,\ldots,55$.
Stars give numerically determined optimal worst-case coherence of $N$ real
unit vectors, found in [18]. Dotted curve gives Welch bound, dash-dotted curve
gives bound from Theorem 24, dashed curve gives bound from Theorem 25, and
solid curve gives bound from Theorem 27.
Now that we understand the special case where $M=3$, we tackle the general
case:
###### Proof of Theorem 25.
As in the proof of Theorem 27, we relate packing $N$ unit vectors to packing
$2N$ points in the hypersphere $S^{M-1}\subseteq\mathbb{R}^{M}$. The argument
in the proof of Lemma 26 generalizes so that two of the $2N$ points must
reside in some closed hyperspherical cap of hypersurface area
$\frac{1}{2N}\mathrm{H}^{M-1}(S^{M-1})$. Therefore, the smallest angle
$\alpha$ between these points is no more than twice the radius of this cap.
Let $C(\gamma)$ denote a hyperspherical cap of angular radius $\gamma$. Then
we use hyperspherical coordinates to get
$\displaystyle\mathrm{H}^{M-1}(C(\gamma))$
$\displaystyle=\int_{\phi_{1}=0}^{\gamma}\int_{\phi_{2}=0}^{\pi}\cdots\int_{\phi_{M-2}=0}^{\pi}\int_{\phi_{M-1}=0}^{2\pi}\sin^{M-2}(\phi_{1})\cdots\sin^{1}(\phi_{M-2})~{}\mathrm{d}\phi_{M-1}\cdots\mathrm{d}\phi_{1}$
$\displaystyle=2\pi\bigg{(}\prod_{j=1}^{M-3}\pi^{1/2}\tfrac{\Gamma(\frac{j+1}{2})}{\Gamma(\frac{j}{2}+1)}\bigg{)}\int_{0}^{\gamma}\sin^{M-2}\phi~{}\mathrm{d}\phi$
$\displaystyle=\tfrac{2\pi^{(M-1)/2}}{\Gamma(\frac{M-1}{2})}\int_{0}^{\gamma}\sin^{M-2}\phi~{}\mathrm{d}\phi.$
(24)
We wish to solve for $\gamma$, but analytically inverting
$\int_{0}^{\gamma}\sin^{M-2}\phi~{}\mathrm{d}\phi$ is difficult. Instead, we
use $\sin\phi\geq\frac{2\phi}{\pi}$ for $\phi\in[0,\frac{\pi}{2}]$. Note that
we do not lose generality by forcing $\gamma\leq\frac{\pi}{2}$, since this is
guaranteed with $N\geq 2$. Continuing (24) gives
$\mathrm{H}^{M-1}(C(\gamma))\geq\tfrac{2\pi^{(M-1)/2}}{\Gamma(\frac{M-1}{2})}\int_{0}^{\gamma}\big{(}\tfrac{2\phi}{\pi}\big{)}^{M-2}\mathrm{d}\phi=\tfrac{(2\gamma)^{M-1}}{(M-1)\pi^{(M-3)/2}\Gamma(\frac{M-1}{2})}.$
(25)
Using the formula for a hypersphere’s hypersurface area, we can express the
left-hand side of (25):
$\tfrac{(2\gamma)^{M-1}}{(M-1)\pi^{(M-3)/2}\Gamma(\frac{M-1}{2})}\leq\mathrm{H}^{M-1}(C(\gamma))=\tfrac{1}{2N}\mathrm{H}^{M-1}(S^{M-1})=\tfrac{\pi^{M/2}}{N\Gamma(\frac{d}{2})}.$
Isolating $2\gamma$ above and using $\alpha\leq 2\gamma$ and $\mu=\cos\alpha$
gives (23). The second part of the result comes from a simple application of
Stirling’s approximation. ∎
In [18], numerical results are given for $M=3$, and we compare these results
to Theorems 24 and 25 in Figure 1. Considering this figure, we note that the
bound in Theorem 24 is inferior to the maximum of the Welch bound and the
bound in Theorem 25, at least when $M=3$. This illustrates the degree to which
Theorem 25 improves the bound in Theorem 24 for real frames. In fact, since
$\cos(\frac{\pi}{a})\geq 1-\frac{2}{a}$ for all $a\geq 2$, the bound for real
frames in Theorem 25 is asymptotically better than the bound for complex
frames in Theorem 24. Moreover, for $M=2$, Theorem 25 says
$\mu\geq\cos(\frac{\pi}{N})$, and [7] proved this bound to be tight for every
$N\geq 2$. Lastly, Figure 1 illustrates that Theorem 27 improves the bound in
Theorem 25 for the case $M=3$.
In many applications, large dictionaries are built to obtain sparse
reconstruction, but the known guarantees on sparse reconstruction place
certain requirements on worst-case coherence. Asymptotically, the bounds in
Theorems 24 and 25 indicate that certain exponentially large dictionaries will
not satisfy these requirements. For example, if $N=\Theta(3^{M})$, then
$\mu_{F}=\Omega(\frac{1}{3})$ by Theorem 24, and if the frame is real, we have
$\mu_{F}=\Omega(\frac{1}{2})$ by Theorem 25. Such a dictionary will only work
for sparse reconstruction if the sparsity level $K$ is sufficiently small;
deterministic guarantees require $K<\mu_{F}^{-1}$ [21, 48], while
probabilistic guarantees require $K<\mu_{F}^{-2}$ [5, 49], and so in this
example, the dictionary can, at best, only accommodate sparsity levels that
are smaller than 10. Unfortunately, in real-world applications, we can expect
the sparsity level to scale with the signal dimension. This in mind, Theorems
24 and 25 tell us that dictionaries can only be used for sparse reconstruction
if $N=O((2+\epsilon)^{M})$ for some sufficiently small $\epsilon>0$. To
summarize, the Welch bound is known to be tight only if $N\leq M^{2}$, and
Theorems 24 and 25 give bounds which are asympotically better than the Welch
bound whenever $N=\Omega(2^{M})$. When $N$ is between $M^{2}$ and $2^{M}$, the
best bound to date is the (loose) Welch bound, and so more work needs to be
done to bound worst-case coherence in this parameter region.
## 5 Reducing average coherence
In [5], average coherence is used to derive a number of guarantees on sparse
signal processing. Since average coherence is so new to the frame theory
literature, this section will investigate how average coherence relates to
worst-case coherence and the spectral norm. We start with a definition:
###### Definition 28 (Wiggling and flipping equivalent frames).
We say the frames $F$ and $G$ are _wiggling equivalent_ if there exists a
diagonal matrix $D$ of unimodular entries such that $G=FD$. Furthermore, they
are _flipping equivalent_ if $D$ is real, having only $\pm 1$’s on the
diagonal.
The terms “wiggling” and “flipping” are inspired by the fact that individual
frame elements of such equivalent frames are related by simple unitary
operations. Note that every frame with $N$ nonzero frame elements belongs to a
flipping equivalence class of size $2^{N}$, while being wiggling equivalent to
uncountably many frames. The importance of this type of frame equivalence is,
in part, due to the following lemma, which characterizes the shared geometry
of wiggling equivalent frames:
###### Lemma 29 (Geometry of wiggling equivalent frames).
Wiggling equivalence preserves the norms of frame elements, the worst-case
coherence, and the spectral norm.
###### Proof.
Take two frames $F$ and $G$ such that $G=FD$. The first claim is immediate.
Next, the Gram matrices are related by $G^{*}G=D^{*}F^{*}FD$. Since
corresponding off-diagonal entries are equal in modulus, we know the worst-
case coherences are equal. Finally,
$\|G\|_{2}^{2}=\|GG^{*}\|_{2}^{2}=\|FDD^{*}F^{*}\|_{2}=\|FF^{*}\|_{2}=\|F\|_{2}^{2}$,
and so we are done. ∎
Wiggling and flipping equivalence are not entirely new to frame theory. For a
real equiangular tight frame $F$, the Gram matrix $F^{*}F$ is completely
determined by the sign pattern of the off-diagonal entries, which can in turn
be interpreted as the Seidel adjacency matrix of a graph $G_{F}$. As such,
flipping a frame element $f\in F$ has the effect of negating the corresponding
row and column in the Gram matrix, which further corresponds to _switching_
the adjacency rule for that vertex $v_{f}\in V(G_{F})$ in the graph—vertices
are adjacent to $v_{f}$ after switching precisely when they were not adjacent
before switching. Graphs are called _switching equivalent_ if there is a
sequence of switching operations that produces one graph from the other; this
equivalence was introduced in [51] and was later extensively studied by Seidel
in [44, 45]. Since flipping equivalent real equiangular tight frames
correspond to switching equivalent graphs, the terms have become
interchangeable. For example, [15] uses switching (i.e., wiggling and
flipping) equivalence to make progress on an important problem in frame theory
called the _Paulsen problem_ , which asks how close a nearly unit norm, nearly
tight frame must be to a unit norm tight frame.
Now that we understand wiggling and flipping equivalence, we are ready for the
main idea behind this section. Suppose we are given a unit norm frame with
acceptable spectral norm and worst-case coherence, but we also want the
average coherence to satisfy (SCP-2). Then by Lemma 29, all of the wiggling
equivalent frames will also have acceptable spectral norm and worst-case
coherence, and so it is reasonable to check these frames for good average
coherence. In fact, the following theorem guarantees that at least one of the
flipping equivalent frames will have good average coherence, with only modest
requirements on the original frame’s redundancy.
###### Theorem 30 (Constructing frames with low average coherence).
Let $F$ be an $M\times N$ unit norm frame with $\smash{M<\frac{N-1}{4\log
4N}}$. Then there exists a frame $G$ that is flipping equivalent to $F$ and
satisfies $\smash{\nu_{G}\leq\frac{\mu_{G}}{\sqrt{M}}}$.
###### Proof.
Take $\\{R_{n}\\}_{n=1}^{N}$ to be a Rademacher sequence that independently
takes values $\pm 1$, each with probability $\frac{1}{2}$. We use this
sequence to randomly flip $F$; define
$Z:=F~{}\mathrm{diag}\\{R_{n}\\}_{n=1}^{N}$. Note that if
$\smash{\Pr(\nu_{Z}\leq\frac{\mu_{F}}{\sqrt{M}})>0}$, we are done. Fix some
$i\in\\{1,\ldots,N\\}$. Then
$\Pr\Bigg{(}\tfrac{1}{N-1}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\langle
z_{i},z_{j}\rangle\bigg{|}>\tfrac{\mu_{F}}{\sqrt{M}}\Bigg{)}=\Pr\Bigg{(}\bigg{|}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}R_{j}\langle
f_{i},f_{j}\rangle\bigg{|}>\tfrac{(N-1)\mu_{F}}{\sqrt{M}}\Bigg{)}.$ (26)
We can view $\sum_{j\neq i}R_{j}\langle f_{i},f_{j}\rangle$ as a sum of $N-1$
independent zero-mean complex random variables that are bounded by $\mu_{F}$.
We can therefore use a complex version of Hoeffding’s inequality [30] (see,
e.g., [4, Lemma 3.8]) to bound the probability expression in (26) as $\leq
4\mathrm{e}^{-(N-1)/4M}$. From here, a union bound over all $N$ choices for
$i$ gives $\Pr(\nu_{Z}\leq\frac{\mu_{F}}{\sqrt{M}})\geq
1-4N\mathrm{e}^{-(N-1)/4M}$, and so $M<\frac{N-1}{4\log 4N}$ implies
$\Pr(\nu_{Z}\leq\frac{\mu_{F}}{\sqrt{M}})>0$, as desired. ∎
While Theorem 30 guarantees the existence of a flipping equivalent frame with
good average coherence, the result does not describe how to find it.
Certainly, one could check all $2^{N}$ frames in the flipping equivalence
class, but such a procedure is computationally slow. As an alternative, we
propose a linear-time flipping algorithm (Algorithm 2). The following theorem
guarantees that linear-time flipping will produce a frame with good average
coherence, but it requires the original frame’s redundancy to be higher than
what suffices in Theorem 30.
Algorithm 2 Linear-time flipping
Input: An $M\times N$ unit norm frame $F$
Output: An $M\times N$ unit norm frame $G$ that is flipping equivalent to $F$
$g_{1}\leftarrow f_{1}$ {Keep first frame element}
for $n=2$ to $N$ do
if $\|\sum_{i=1}^{n-1}g_{i}+f_{n}\|\leq\|\sum_{i=1}^{n-1}g_{i}-f_{n}\|$ then
$g_{n}\leftarrow f_{n}$ {Keep frame element to make sum length shorter}
else
$g_{n}\leftarrow-f_{n}$ {Flip frame element to make sum length shorter}
end if
end for
###### Theorem 31.
Suppose $N\geq M^{2}+3M+3$. Then Algorithm 2 outputs an $M\times N$ frame $G$
that is flipping equivalent to $F$ and satisfies
$\nu_{G}\leq\frac{\mu_{G}}{\sqrt{M}}$.
###### Proof.
Considering Lemma 7(iii), it suffices to have $\|\sum_{n=1}^{N}g_{n}\|^{2}\leq
N$. We will use induction to show $\|\sum_{n=1}^{k}g_{n}\|^{2}\leq k$ for
$k=1,\ldots,N$. Clearly, $\|\sum_{n=1}^{1}g_{n}\|^{2}=\|f_{n}\|^{2}=1\leq 1$.
Now assume $\|\sum_{n=1}^{k}g_{n}\|^{2}\leq k$. Then by our choice for
$g_{k+1}$ in Algorithm 2, we know that
$\|\sum_{n=1}^{k}g_{n}+g_{k+1}\|^{2}\leq\|\sum_{n=1}^{k}g_{n}-g_{k+1}\|^{2}$.
Expanding both sides of this inequality gives
$\bigg{\|}\sum_{n=1}^{k}g_{n}\bigg{\|}^{2}+2\mathrm{Re}\bigg{\langle}\sum_{n=1}^{k}g_{n},g_{k+1}\bigg{\rangle}+\|g_{k+1}\|^{2}\leq\bigg{\|}\sum_{n=1}^{k}g_{n}\bigg{\|}^{2}-2\mathrm{Re}\bigg{\langle}\sum_{n=1}^{k}g_{n},g_{k+1}\bigg{\rangle}+\|g_{k+1}\|^{2},$
and so $\mathrm{Re}\langle\sum_{n=1}^{k}g_{n},g_{k+1}\rangle\leq 0$.
Therefore,
$\bigg{\|}\sum_{n=1}^{k+1}g_{n}\bigg{\|}^{2}=\bigg{\|}\sum_{n=1}^{k}g_{n}\bigg{\|}^{2}+2\mathrm{Re}\bigg{\langle}\sum_{n=1}^{k}g_{n},g_{k+1}\bigg{\rangle}+\|g_{k+1}\|^{2}\leq\bigg{\|}\sum_{n=1}^{k}g_{n}\bigg{\|}^{2}+\|g_{k+1}\|^{2}\leq
k+1,$
where the last inequality uses the inductive hypothesis. ∎
###### Example 32.
As an example of how linear-time flipping reduces average coherence, consider
the following matrix:
$F:=\frac{1}{\sqrt{5}}\left[\begin{array}[]{cccccccccc}+&+&+&+&-&+&+&+&+&-\\\
+&-&+&+&+&-&-&-&+&-\\\ +&+&+&+&+&+&+&+&-&+\\\ -&-&-&+&-&+&+&-&-&-\\\
-&+&+&-&-&+&-&-&-&-\end{array}\right].$
Here, $\smash{\nu_{F}\approx 0.3778>0.2683\approx\frac{\mu_{F}}{\sqrt{M}}}$.
Even though $N<M^{2}+3M+3$, we run linear-time flipping to get the flipping
pattern $D:=\mathrm{diag}(+-+--++-++)$. Then $FD$ has average coherence
$\smash{\nu_{FD}\approx
0.1556<\frac{\mu_{F}}{\sqrt{M}}=\frac{\mu_{FD}}{\sqrt{M}}}$. This example
illustrates that the condition $N\geq M^{2}+3M+3$ in Theorem 31 is sufficient
but not necessary.
## Acknowledgments
The authors thank the anonymous referees for their helpful suggestions,
Matthew Fickus for his insightful comments on chirp frames, and Samuel Feng
and Michael A. Schwemmer for their help with using the computer clusters in
Princeton’s mathematics department. This work was supported by the Office of
Naval Research under grant N00014-08-1-1110, by the Air Force Office of
Scientific Research under grants FA9550-09-1-0551 and FA 9550-09-1-0643, and
by NSF under grant DMS-0914892. Mixon was supported by the A.B. Krongard
Fellowship. The views expressed in this article are those of the authors and
do not reflect the official policy or position of the United States Air Force,
Department of Defense, or the U.S. Government.
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|
arxiv-papers
| 2011-03-02T14:18:12 |
2024-09-04T02:49:17.405803
|
{
"license": "Public Domain",
"authors": "Waheed U. Bajwa, Robert Calderbank, Dustin G. Mixon",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1103.0435"
}
|
1103.0617
|
# On the absolute matrix summability factors
H. S. ÖZARSLAN and T. ARI
Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
E-mail:seyhan@erciyes.edu.tr and tkandefer@erciyes.edu.tr
###### Abstract
In this paper, we have obtained a necessary and sufficient condition on
$(\lambda_{n})$ for the series $\sum\lambda_{n}a_{n}$ to be
$\left|A\right|_{k}$ summable, $k\geq 1$, whenever $\sum a_{n}$ is
$\left|A\right|$ summable. As a consequence we extend some known results of
Sarıgöl [2].
1\. Introduction
Let $\sum a_{n}$ be a given infinite series with the partial sums
$\left(s_{n}\right)$, and let $A=(a_{nv})$ be a normal matrix, i.e., a lower
triangular matrix of nonzero diagonal entries. Then $A$ defines the sequence-
to-sequence transformation, mapping the sequence $s=(s_{n})$ to
$As=\left(A_{n}(s)\right)$, where
$\displaystyle A_{n}(s)=\sum_{v=0}^{n}a_{nv}s_{v},\quad n=0,1,...$ (1)
The series $\sum a_{n}$ is said to be summable $\left|A\right|_{k}\,,k\geq 1$,
if (see [3])
$\displaystyle\sum_{n=1}^{\infty}n^{k-1}\
\left|\bar{\Delta}A_{n}(s)\right|^{k}<\infty,$ (2)
where
$\displaystyle\bar{\Delta}A_{n}(s)=A_{n}(s)-A_{n-1}(s)$
and it is said to be $\left|R,p_{n}\right|_{k}$ summable (see [5])if (2) holds
when $A$ is a Riesz matrix.
Key Words: Absolute summability, absolute matrix summability, infinite series.
2010 AMS Subject Classification: 40D25, 40F05, 40G99.
By a Riesz matrix we mean one such that
$\displaystyle a_{nv}=\frac{p_{v}}{P_{n}},\quad for\quad 0\leq v\leq n,\quad
and\quad a_{nv}=0\quad for\quad v>n,$
where $(p_{n})$ is a sequence of positive real numbers such that
$\displaystyle
P_{n}=\sum_{v=0}^{n}p_{v}\rightarrow\infty,\quad(n\rightarrow\infty),\quad\left(P_{-i}=p_{-i}=0,\quad
i\geq 1\right).$
Sarıgöl [2] has proved the following theorem for $\left|R,p_{n}\right|_{k}$
summability method.
Theorem A. Suppose that $(p_{n})$ and $(q_{n})$ are positive sequences with
$P_{n}\rightarrow\infty$ and $Q_{n}\rightarrow\infty$ as $n\rightarrow\infty$.
Then $\sum a_{n}\lambda_{n}$ is summable $\left|R,q_{n}\right|_{k}$, $k\geq
1$, whenever $\sum a_{n}$ is summable $\left|R,p_{n}\right|$, if and only if
$\displaystyle\textbf{(a)}\ \
\lambda_{n}=O\left\\{n^{\frac{1}{k}-1}\frac{q_{n}P_{n}}{p_{n}Q_{n}}\right\\},$
$\displaystyle\textbf{(b)}\ \
W_{n}\triangle\left(Q_{n-1}\lambda_{n}\right)=O\left(\frac{p_{n}}{P_{n}}\right),$
(3) $\displaystyle\textbf{(c)}\ \ Q_{n}\lambda_{n+1}W_{n}=O(1),$
where, provided that
$\displaystyle
W_{n}=\left\\{\sum_{v=n+1}^{\infty}v^{k-1}\left(\frac{q_{v}}{Q_{v}Q_{v-1}}\right)^{k}\right\\}^{\frac{1}{k}}<\infty.$
Lemma. ([4]) $A=(a_{nv})\in(l_{1},l_{k})$ if and only if
$\displaystyle\sup_{v}\sum_{n=1}^{\infty}|a_{nv}|^{k}<\infty$ (4)
for the cases $1\leq k<\infty$, where $(l_{1},l_{k})$ denotes the set of all
matrices $A$ which map $l_{1}$ into $l_{k}=\\{x=(x_{n})\ :\
\sum|x_{n}|^{k}<\infty\\}$.
2\. The main result. The aim of this paper is to generalize Theorem $A$ for
absolute matrix summability. Before stating the main theorem we must first
introduce some further notations.
Given a normal matrix $A=(a_{nv})$, we associate two lover semimatrices
$\bar{A}=(\bar{a}_{nv})$ and $\hat{A}=(\hat{a}_{nv})$ as follows:
$\displaystyle\bar{a}_{nv}=\sum_{i=v}^{n}a_{ni},\quad n,v=0,1,...$ (5)
and
$\displaystyle\hat{a}_{00}=\bar{a}_{00}=a_{00},\quad\hat{a}_{nv}=\bar{a}_{nv}-\bar{a}_{n-1,v}\quad
n=1,2,...$ (6)
It may be noted that $\bar{A}$ and $\hat{A}$ are the well-known matrices of
series-to-sequence and series-to-series transformations, respectively. Then,
we have
$\displaystyle A_{n}(s)$ $\displaystyle=$
$\displaystyle\sum_{v=0}^{n}a_{nv}s_{v}=\sum_{v=0}^{n}a_{nv}\sum_{i=0}^{v}a_{i}$
(7) $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}a_{i}\sum_{v=i}^{n}a_{nv}=\sum_{i=0}^{n}\bar{a}_{ni}a_{i}$
and
$\displaystyle\bar{\Delta}A_{n}(s)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\bar{a}_{ni}a_{i}-\sum_{i=0}^{n-1}\bar{a}_{n-1,i}a_{i}$
(8) $\displaystyle=$
$\displaystyle\bar{a}_{nn}a_{n}+\sum_{i=0}^{n-1}(\bar{a}_{ni}-\bar{a}_{n-1,i})a_{i}$
$\displaystyle=$
$\displaystyle\hat{a}_{nn}a_{n}+\sum_{i=0}^{n-1}\hat{a}_{ni}a_{i}=\sum_{i=0}^{n}\hat{a}_{ni}a_{i}.$
If $A$ is a normal matrix, then $A^{\prime}=(a^{\prime}_{nv})$ will denote the
inverse of $A$. Clearly, if $A$ is normal then $\hat{A}=(\hat{a}_{nv})$ is
normal and it has two-sided inverse
$\hat{A}^{\prime}=(\hat{a}^{\prime}_{nv})$, which is also normal (see [1]).
Now we shall prove the following theorem.
Theorem. Let $k\geq 1$, $A=(a_{nv})$ and $B=(b_{nv})$ be two positive normal
matrices. In order that $\sum a_{n}\lambda_{n}$ is summable
$\left|B\right|_{k}$, whenever $\sum a_{n}$ is summable $\left|A\right|$ it is
necessary that
$\displaystyle|\lambda_{n}|=O\left\\{n^{\frac{1}{k}-1}\frac{a_{nn}}{b_{nn}}\right\\},$
(9)
$\displaystyle\sum_{n=v+1}^{\infty}n^{k-1}|\Delta_{v}(\hat{b}_{nv}\lambda_{v})|^{k}=O(a_{vv})^{k},$
(10)
$\displaystyle\sum_{n=v+1}^{\infty}n^{k-1}|\hat{b}_{n,v+1}\lambda_{v+1}|^{k}=O(1),$
(11) $\displaystyle a_{n-1,v}\geq a_{nv},\quad for\quad n\geq v+1,$ (12)
$\displaystyle\bar{a}_{n0}=1,\quad n=0,1,2,...\ .$ (13)
Then (9)-(11) and
$\displaystyle\bar{b}_{n0}=1,\quad n=0,1,2,...,$ (14) $\displaystyle
a_{nn}-a_{n+1,n}=O(a_{nn}\ a_{n+1,n+1}),$ (15)
$\displaystyle\sum_{v=r+2}^{n}\left|\hat{b}_{nv}\right|\left|\hat{a}^{\prime}_{vr}\lambda_{v}\right|=O(\frac{b_{nn}}{a_{nn}}|\lambda_{n}|)$
(16)
are also sufficient.
It should be noted that if we take $a_{nv}=\frac{p_{v}}{P_{n}}$ and
$b_{nv}=\frac{q_{v}}{Q_{n}}$, then we get Theorem A.
Proof of the theorem.
Necessity. Let $(x_{n})$ and $(y_{n})$ denote $A$-transform and $B$-transform
of the series $\sum a_{n}$ and $\sum a_{n}\lambda_{n}$, respectively. Then, by
(7) and (8), we have
$\displaystyle\overline{\Delta}x_{n}=\sum_{v=0}^{n}\hat{a}_{nv}a_{v}\ and\
\overline{\Delta}y_{n}=\sum_{v=0}^{n}\hat{b}_{nv}a_{v}\lambda_{v}.$ (17)
For $k\geq 1$, we define
$\displaystyle A=\left\\{(a_{i}):\sum a_{i}\ is\ summable\ |A|\right\\},$
$\displaystyle B=\left\\{(a_{i}\lambda_{i}):\sum a_{i}\lambda_{i}\ is\
summable\ |B|_{k}\right\\}.$
Then it is routine to verify that these are BK-spaces, if normed by
$\displaystyle\left\|X\right\|=\left\\{\sum_{n=0}^{\infty}\mid{\overline{\Delta}x_{n}}\mid\right\\}$
(18)
and
$\displaystyle\left\|Y\right\|=\left\\{\sum_{n=0}^{\infty}n^{k-1}\mid{\overline{\Delta}y_{n}}\mid^{k}\right\\}^{\frac{1}{k}}$
(19)
respectively.
Since $\sum a_{n}$ is summable $|A|$ implies $\sum a_{n}\lambda_{n}$ is
summable $|B|_{k}$, by the hypothesis of the theorem,
$\displaystyle\left\|X\right\|<\infty\Rightarrow\left\|Y\right\|<\infty.$
Now consider the inclusion map c: A$\rightarrow$B defined by c(x)=x. This is
continous, which is immediate as A and B are BK-spaces. Thus there exists a
constant M such that
$\displaystyle\left\|Y\right\|\leq M\,\left\|X\right\|.$ (20)
By applying (17) to $a_{v}=e_{v}-e_{v+1}$ ( $e_{v}$ is the v-th coordinate
vector), we have
$\overline{\Delta}x_{n}=\left\\{\begin{array}[]{cl}0&,\mbox{ if $n<v$}\\\
\hat{a}_{nv}&,\mbox{ if $n=v$}\\\ \Delta_{v}\hat{a}_{nv}&,\mbox{ if
$n>v$}\end{array}\right.$
and
$\overline{\Delta}y_{n}=\left\\{\begin{array}[]{cl}0&,\mbox{ if $n<v$}\\\
\hat{b}_{nv}\lambda_{v}&,\mbox{ if $n=v$}\\\
\Delta_{v}(\hat{b}_{nv}\lambda_{v})&,\mbox{ if $n>v$}.\end{array}\right.$
So (18) and (19) give us
$\displaystyle\left\|X\right\|=\left\\{a_{vv}+\sum_{n=v+1}^{\infty}\mid{\Delta_{v}\hat{a}_{nv}}\mid\right\\}$
and
$\displaystyle\left\|Y\right\|=\left\\{v^{k-1}b_{vv}\mid{\lambda_{v}}\mid^{k}+\sum_{n=v+1}^{\infty}n^{k-1}\mid{\Delta_{v}\left(\hat{b}_{nv}\lambda_{v}\right)}\mid^{k}\right\\}^{\frac{1}{k}}.$
Hence it follows from (20) that
$\displaystyle
v^{k-1}b_{vv}\mid{\lambda_{v}}\mid^{k}+\sum_{n=v+1}^{\infty}n^{k-1}\mid{\Delta_{v}\hat{b}_{nv}\lambda_{v}}\mid^{k}$
$\displaystyle\leq$ $\displaystyle
M^{k}a_{vv}^{k}+M^{k}\sum_{n=v+1}^{\infty}\mid{\Delta_{v}\hat{a}_{nv}}\mid^{k}.$
Using (12), we can find
$\displaystyle
v^{k-1}b_{vv}\mid{\lambda_{v}}\mid^{k}+\sum_{n=v+1}^{\infty}n^{k-1}\mid{\Delta_{v}(\hat{b}_{nv}\lambda_{v})}\mid^{k}=O\left\\{a_{vv}^{k}\right\\}.$
The above inequality will be true iff each term on the left hand side is
$O\left\\{a_{vv}^{k}\right\\}$. Taking the first term,
$\displaystyle
v^{k-1}b_{vv}\mid{\lambda_{v}}\mid^{k}=O\left\\{a_{vv}^{k}\right\\}$
then
$\displaystyle\mid{\lambda_{v}}\mid=O\left\\{v^{\frac{1}{k}-1}\frac{a_{vv}}{b_{vv}}\right\\}$
which verifies that (9) is necessary.
Using the second term we have,
$\displaystyle\sum_{n=v+1}^{\infty}n^{k-1}\mid{\Delta_{v}(\hat{b}_{nv}\lambda_{v})}\mid^{k}=O\left\\{\mid{a_{vv}}\mid^{k}\right\\}$
which is condition (10).
Now if we apply (17) to $a_{v}=e_{v+1}$, we have,
$\overline{\Delta}x_{n}=\left\\{\begin{array}[]{cl}0&,\mbox{ if $n\leq v$}\\\
\hat{a}_{n,v+1}&,\mbox{ if $n>v$}\end{array}\right.$
and
$\overline{\Delta}y_{n}=\left\\{\begin{array}[]{cl}0&,\mbox{ if $n\leq v$}\\\
\hat{b}_{n,v+1}\lambda_{v+1}&,\mbox{ if $n>v$}\end{array}\right.$
respectively.
Hence
$\displaystyle\left\|X\right\|=\left\\{\sum_{n=v+1}^{\infty}\mid{\hat{a}_{n,v+1}}\mid\right\\},$
$\displaystyle\left\|Y\right\|=\left\\{\sum_{n=v+1}^{\infty}n^{k-1}\mid{\hat{b}_{n,v+1}\lambda_{v+1}}\mid^{k}\right\\}^{\frac{1}{k}}.$
Hence it follows from (20) that
$\displaystyle\sum_{n=v+1}^{\infty}n^{k-1}\mid{\hat{b}_{n,v+1}\lambda_{v+1}}\mid^{k}\leq
M^{k}\left\\{\sum_{n=v+1}^{\infty}\mid{\hat{a}_{n,v+1}}\mid\right\\}^{k}.$
Using (13) we can find
$\displaystyle\sum_{n=v+1}^{\infty}n^{k-1}\mid{\hat{b}_{n,v+1}\lambda_{v+1}}\mid^{k}=O(1)$
which is condition (11).
Sufficiency. We use the notations of necessity. Then
$\displaystyle\overline{\Delta}x_{n}=\sum_{v=0}^{n}\hat{a}_{nv}a_{v}$ (21)
which implies
$\displaystyle a_{v}=\sum_{r=0}^{v}\hat{a}^{\prime}_{vr}\
\overline{\Delta}x_{r}.$ (22)
In this case
$\displaystyle\bar{\Delta}y_{n}=\sum_{v=0}^{n}\hat{b}_{nv}a_{v}\lambda_{v}=\sum_{v=0}^{n}\hat{b}_{nv}\lambda_{v}\
\sum_{r=0}^{v}\hat{a}^{\prime}_{vr}\bar{\Delta}x_{r}.$
On the other hand, since
$\displaystyle\hat{b}_{n0}=\bar{b}_{n0}-\bar{b}_{n-1,0}$
by (14), we have
$\displaystyle\bar{\Delta}y_{n}$ $\displaystyle=$
$\displaystyle\sum_{v=1}^{n}\hat{b}_{nv}\lambda_{v}\\{\sum_{r=0}^{v}\hat{a}^{\prime}_{vr}\
\bar{\Delta}x_{r}\\}$ (23) $\displaystyle=$
$\displaystyle\sum_{v=1}^{n}\hat{b}_{nv}\lambda_{v}\\{\hat{a}^{\prime}_{vv}\
\bar{\Delta}x_{v}+\hat{a}^{\prime}_{v,v-1}\
\bar{\Delta}x_{v-1}+\sum_{r=0}^{v-2}\hat{a}^{\prime}_{vr}\
\bar{\Delta}x_{r}\\}$ $\displaystyle=$
$\displaystyle\sum_{v=1}^{n}\hat{b}_{nv}\lambda_{v}\ \hat{a}^{\prime}_{vv}\
\bar{\Delta}x_{v}+\sum_{v=1}^{n}\hat{b}_{nv}\lambda_{v}\
\hat{a}^{\prime}_{v,v-1}\
\bar{\Delta}x_{v-1}+\sum_{v=1}^{n}\hat{b}_{nv}\lambda_{v}\sum_{r=0}^{v-2}\hat{a}^{\prime}_{vr}\
\bar{\Delta}x_{r}$ $\displaystyle=$ $\displaystyle\hat{b}_{nn}\lambda_{n}\
\hat{a}^{\prime}_{nn}\
\bar{\Delta}x_{n}+\sum_{v=1}^{n-1}(\hat{b}_{nv}\lambda_{v}\
\hat{a}^{\prime}_{vv}+\ \hat{b}_{n,v+1}\lambda_{v+1}\
\hat{a}^{\prime}_{v+1,v})\ \bar{\Delta}x_{v}$
$\displaystyle+\sum_{r=0}^{n-2}\bar{\Delta}x_{r}\sum_{v=r+2}^{n}\hat{b}_{nv}\lambda_{v}\
\hat{a}^{\prime}_{vr}.$
By considering the equality
$\displaystyle\sum_{k=v}^{n}\hat{a}^{\prime}_{nk}\hat{a}_{kv}=\delta_{nv}$
where $\delta_{nv}$ is the Kronocker delta, we have that
$\displaystyle\hat{b}_{nv}\lambda_{v}\
\hat{a}^{\prime}_{vv}+\hat{b}_{n,v+1}\lambda_{v+1}\ \hat{a}^{\prime}_{v+1,v}$
$\displaystyle=$
$\displaystyle\frac{\hat{b}_{nv}\lambda_{v}}{\hat{a}_{vv}}+\hat{b}_{n,v+1}\lambda_{v+1}\
(-\frac{\hat{a}_{v+1,v}}{\hat{a}_{vv}\ \hat{a}_{v+1,v+1}})$ $\displaystyle=$
$\displaystyle\frac{\hat{b}_{nv}\lambda_{v}}{a_{vv}}-\frac{\hat{b}_{n,v+1}\lambda_{v+1}\
(\bar{a}_{v+1,v}-\bar{a}_{v,v})}{a_{vv}\ a_{v+1,v+1}}$ $\displaystyle=$
$\displaystyle\frac{\hat{b}_{nv}\lambda_{v}}{a_{vv}}-\frac{\hat{b}_{n,v+1}\lambda_{v+1}\
(a_{v+1,v+1}+a_{v+1,v}-a_{vv})}{a_{vv}\ a_{v+1,v+1}}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{v}\left(\hat{b}_{nv}\lambda_{v}\right)}{a_{vv}}+\hat{b}_{n,v+1}\lambda_{v+1}\
\frac{a_{vv}-a_{v+1,v}}{a_{vv}\ a_{v+1,v+1}}$
and so
$\displaystyle\bar{\Delta}y_{n}$ $\displaystyle=$
$\displaystyle\frac{b_{nn}\lambda_{n}}{a_{nn}}\
\bar{\Delta}x_{n}+\sum_{v=1}^{n-1}\
\frac{\Delta_{v}\left(\hat{b}_{nv}\lambda_{v}\right)}{a_{vv}}\
\bar{\Delta}x_{v}+\sum_{v=1}^{n-1}\hat{b}_{n,v+1}\lambda_{v+1}\
\frac{a_{vv}-a_{v+1,v}}{a_{vv}\ a_{v+1,v+1}}\ \bar{\Delta}x_{v}$
$\displaystyle+$
$\displaystyle\sum_{r=0}^{n-2}\bar{\Delta}x_{r}\sum_{v=r+2}^{n}\hat{b}_{nv}\lambda_{v}\
\hat{a}^{\prime}_{vr}.$
Let
$\displaystyle T_{n}(1)=\frac{b_{nn}\lambda_{n}}{a_{nn}}\
\bar{\Delta}x_{n}+\sum_{v=1}^{n-1}\
\frac{\Delta_{v}\left(\hat{b}_{nv}\lambda_{v}\right)}{a_{vv}}\
\bar{\Delta}x_{v}+\sum_{v=1}^{n-1}\hat{b}_{n,v+1}\lambda_{v+1}\
\frac{a_{vv}-a_{v+1,v}}{a_{vv}\ a_{v+1,v+1}}\ \bar{\Delta}x_{v},$
$\displaystyle
T_{n}(2)=\sum_{r=0}^{n-2}\bar{\Delta}x_{r}\sum_{v=r+2}^{n}\hat{b}_{nv}\lambda_{v}\
\hat{a}^{\prime}_{vr}.$
Since
$\displaystyle\left|T_{n}(1)+T_{n}(2)\right|^{k}\leq
2^{k}\left(\left|T_{n}(1)\right|^{k}+\left|T_{n}(2)\right|^{k}\right)$
to complete the proof of theorem, it is sufficient to show that
$\displaystyle\sum_{n=1}^{\infty}n^{k-1}\left|T_{n}(i)\right|^{k}<\infty\quad
for\quad i=1,2.$
Then
$\displaystyle\overline{T_{n}(1)}$ $\displaystyle=$ $\displaystyle
n^{1-\frac{1}{k}}\ T_{n}(1)$ $\displaystyle=$ $\displaystyle
n^{1-\frac{1}{k}}\frac{b_{nn}\lambda_{n}}{a_{nn}}\
\bar{\Delta}x_{n}+n^{1-\frac{1}{k}}\sum_{v=1}^{n-1}\
\frac{\Delta_{v}\left(\hat{b}_{nv}\lambda_{v}\right)}{a_{vv}}\
\bar{\Delta}x_{v}+n^{1-\frac{1}{k}}\sum_{v=1}^{n-1}\hat{b}_{n,v+1}\lambda_{v+1}\
\frac{a_{vv}-a_{v+1,v}}{a_{vv}\ a_{v+1,v+1}}\ \bar{\Delta}x_{v}$
$\displaystyle=$ $\displaystyle\sum_{v=1}^{\infty}c_{nv}\bar{\Delta}x_{v}$
where
$c_{nv}=\left\\{\begin{array}[]{cl}n^{1-\frac{1}{k}}\left(\frac{\Delta_{v}\left({b}_{nv}\lambda_{v}\right)}{a_{vv}}+\hat{b}_{n,v+1}\lambda_{v+1}\
\frac{a_{vv}-a_{v+1,v}}{a_{vv}\ a_{v+1,v+1}}\right)&,\mbox{ if $1\leq v\leq
n-1$}\\\ n^{1-\frac{1}{k}}\frac{{b}_{nn}\lambda_{n}}{a_{nn}}&,\mbox{ if
$v=n$}\\\ 0&,\mbox{ if $v>n.$}\end{array}\right.$
Now
$\displaystyle\sum|\overline{T_{n}(1)}|^{k}<\infty\ \ \textmd{whenever}\ \
\sum|\bar{\Delta}x_{n}|<\infty$
is equivalently
$\displaystyle\sup_{v}\sum_{n=1}^{\infty}|c_{nv}|^{k}<\infty$ (24)
by Lemma. But (24) is equivalent to
$\displaystyle\sum_{n=v}^{\infty}|c_{nv}|^{k}$ $\displaystyle=$ $\displaystyle
O(1)\left\\{n^{1-\frac{1}{k}}|\frac{{b}_{nn}\lambda_{n}}{a_{nn}}|^{k}+\sum_{n=v+1}^{\infty}n^{1-\frac{1}{k}}\left|\frac{\Delta_{v}\left(\hat{b}_{nv}\lambda_{v}\right)}{a_{vv}}+\hat{b}_{n,v+1}\lambda_{v+1}\
\frac{a_{vv}-a_{v+1,v}}{a_{vv}\ a_{v+1,v+1}}\right|^{k}\right\\}$ (25)
$\displaystyle=$ $\displaystyle O(1)\ \ as\ \ v\rightarrow\infty.$
Finally
$\displaystyle\sum_{n=2}^{\infty}n^{k-1}\left|T_{n}(2)\right|^{k}$
$\displaystyle=$
$\displaystyle\sum_{n=2}^{\infty}n^{k-1}\left|\sum_{r=0}^{n-2}\bar{\Delta}x_{r}\sum_{v=r+2}^{n}\hat{b}_{nv}\
\hat{a}^{\prime}_{vr}\lambda_{v}\right|^{k}$ $\displaystyle=$ $\displaystyle
O(1)\sum_{n=2}^{\infty}n^{k-1}\left|\sum_{r=0}^{n-2}\bar{\Delta}x_{r}\frac{b_{nn}\lambda_{n}}{a_{nn}}\right|^{k}.$
Then as in $T_{n}(1)$, we have that
$\displaystyle\overline{T_{n}(2)}$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{n-2}n^{1-\frac{1}{k}}\bar{\Delta}x_{r}\frac{b_{nn}|\lambda_{n}|}{a_{nn}}$
$\displaystyle=$ $\displaystyle\sum_{r=1}^{\infty}d_{nr}\bar{\Delta}x_{r}$
where
$d_{nr}=\left\\{\begin{array}[]{cl}n^{1-\frac{1}{k}}\frac{b_{nn}\lambda_{n}}{a_{nn}}&,\mbox{
if $0\leq r\leq n-2$}\\\ 0&,\mbox{ if $r>n-2.$}\end{array}\right.$
Now
$\displaystyle\sum|\overline{T_{n}(2)}|^{k}<\infty\ \ whenever\ \
\sum|\bar{\Delta}x_{n}|<\infty$
is equivalently
$\displaystyle\sup_{r}\sum_{n=1}^{\infty}|d_{nr}|^{k}<\infty$ (26)
by Lemma. But (26) is equivalent to
$\displaystyle\sum_{n=r}^{\infty}|d_{nr}|^{k}=O(1)\sum_{n=r+2}^{\infty}\left|n^{1-\frac{1}{k}}\frac{b_{nn}\lambda_{n}}{a_{nn}}\right|^{k}=O(1).$
(27)
Therefore, we have
$\displaystyle\sum_{n=1}^{\infty}n^{k-1}\left|T_{n}(i)\right|^{k}<\infty\quad
for\quad i=1,2.$
This completes the proof of theorem.
## References
* [1] R. G. Cooke, Infinite matrices and sequence spaces, Macmillan, (1950).
* [2] M. A. Sarıgöl, On the absolute riesz summability factors of infinite series, Indian J. Pure Appl. Math., 23 (12) (1992), 881-886.
* [3] N.Tanovi$\breve{c}$-Miller, On strong summability, Glasnik Matematicki, 34 (1979), 87-97.
* [4] I. J. Maddox, Elements of functional analysis, Cambridge University Press, (1970).
* [5] C. Orhan, On Equivalence of Summability Methods, Math Slovaca, 40 (1990), 171-175.
|
arxiv-papers
| 2011-03-03T08:34:03 |
2024-09-04T02:49:17.416540
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. S. Ozarslan and T. Ari",
"submitter": "Tuba Ari",
"url": "https://arxiv.org/abs/1103.0617"
}
|
1103.0659
|
# Limits on possible new spin-spin interactions between neutrons from
measurements of the Longitudinal Spin Relaxation Rate of Polarized 3He Gas
Changbo Fu Center for Exploration of Energy and Matter, Indiana University,
Bloomington, IN 47408 W. M. Snow wsnow@indiana.edu Center for Exploration of
Energy and Matter, Indiana University, Bloomington, IN 47408
(August 27, 2024)
###### Abstract
New particles with masses in the sub-eV range have been predicted by various
theories beyond the Standard Model. Some can induce new spin-spin interactions
between fermions. Existing constraints on such interactions between nucleons
with mesoscopic ranges (millimeters to nanometers) are quite poor. Polarized
3He gas is an especially clean system to use to constrain these possible new
spin-spin interactions because the spin-independent atomic potential between
helium atoms is well-characterized experimentally. The small effects from
binary atomic collisions in a polarized gas from magnetic dipole-dipole and
other possible weak spin-spin interactions which lead to spin relaxation can
be calculated perturbatively. We compare existing measurements of the
longitudinal spin relaxation rate $\Gamma_{1}$ of polarized 3He gas with
theoretical calculations and set a $1\sigma$ upper bound on the pseudoscalar
coupling strength $g_{p}$ for possible new neutron-neutron dipole-dipole
interactions of $g_{p}^{(n)}g_{p}^{(n)}/4\pi\leq 1.7\times 10^{-3}$ for
distances larger than 100 nm. We also set new direct limits on possible
gravitational torsion interactions between neutrons.
###### pacs:
13.75.Cs, 13.88.+e, 14.20.Dh, 14.70.Pw, 14.80.Va
## I Introduction
New spin-dependent interactions of nature with very weak couplings to matter
and ranges of “mesoscopic” scale (millimeters to nanometers) are poorly
constrained by experiment and are attracting more theoretical attentionMoody84
; Raf90 ; Ros00 ; Jae10 . Particles which might transmit such interactions are
starting to be referred to as WISPs (weakly-interacting sub-eV particles)
Jae10 . Symmetries broken at a high energy scale generically lead to weakly-
coupled light particles with long-range interactions through Goldstone’s
theorem. Theoretical attempts to explain dark matter and dark energy can also
produce new weakly-coupled long-range interactions. In both cases there are
many examples in which the new interactions are spin-dependent. The fact that
the dark energy density of order (1 meV)4 corresponds to a
$\leavevmode\nobreak\ 100$ $\mu$m length scale by dimensional analysis also
encourages searches for new phenomena around this scale Ade03 ; Ade09 . Taken
together, these developments suggest that it is reasonable to conduct
experimental searches for possible new spin-dependent interactions which act
over mesoscopic distance scales.
Many searches for new spin-dependent interactions have been motivated by the
idea of axions Pec77 ; Wei78 ; Wilczek78 ; Ros00 ; Raf90 ; You96 , which can
induce a $P$-odd and $T$-odd interaction between polarized and unpolarized
particles proportional to ${\vec{s}}\cdot{\vec{r}}$, where ${\vec{r}}$ is the
distance between the particles and ${\vec{s}}$ is the spin of the polarized
particle. Several other ideas can generate exotic spin-dependent interactions
Hehl76 ; Shapiro02 ; Hammond02 ; Kostelec04 ; Arkani04 ; Arkani05 ; Georgi07 ;
Kostelec08 . However the idea to search for new spin-dependent interactions
can be considered within a more general theoretical context. Dobrescu and
Mocioiu Dob06 recently performed a general classification of interactions
between nonrelativistic spin $1/2$ fermions assuming only rotational
invariance. This analysis emphasized the rich variety of possibilities for new
spin-dependent interactions. Of the 16 different terms in the elastic
scattering amplitude uncovered in this analysis, 15 involve either one or both
of the spins of the fermions.
In this paper we will consider constraints on possible new spin-dependent,
velocity-independent forces between nucleons. For one boson exchange between
two nonrelativistic spin $1/2$ fermions there are 9 types of potentials
involving both spins. Six depend on the relative velocities of the particles
and the remaining three ($V_{2},V_{3}$, and $V_{11}$ in the notation of
Dobrescu and Mocioiu) are velocity-independent:
$\displaystyle V_{2}=\frac{\hbar c}{4\pi
r}\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}\,e^{-r/\lambda},$ (1) $\displaystyle
V_{3}$ $\displaystyle=$ $\displaystyle\frac{\hbar^{3}}{4\pi
m^{2}r^{3}c}\\{(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{\sigma}}_{2})\left(1+\frac{r}{\lambda}\right)$
(2)
$\displaystyle-3(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{r}})(\hat{\mathbf{\sigma}}_{2}\cdot\hat{\mathbf{r}})\left(1+\frac{r}{\lambda}+\frac{r^{2}}{3\lambda^{2}}\right)\\}e^{-r/\lambda},$
and
$\displaystyle V_{11}=\frac{\hbar^{2}}{4\pi
mr^{2}}(\hat{\sigma}_{1}\times\hat{\sigma}_{2})\cdot\hat{r}\left(1+\frac{r}{\lambda}\right)e^{-r/\lambda},$
(3)
where $m$ is the mass, ${\vec{s}_{i}}=\hbar\hat{\sigma}_{i}/2$ is the spin of
the polarized particle, $\hbar$ is Planck constant, $\lambda$ is the
interaction range, and $\hat{\mathbf{r}}={\mathbf{r}}/r$ is the unit vector
between the particles.
The existing constraints on new spin-spin interactions between nucleons at
distance scales below 1 cm are generally rather poor. It is not hard to
understand why: for shorter-range interactions both the number of particles
that can be brought within the range of the interaction becomes smaller and
smaller, and the required precision with which one can understand the large
backgrounds from the electromagnetic fields that accompany any macroscopically
polarized medium becomes more and more difficult to achieve. Several
measurements Wine91 ; Gle08 ; Vas09 constrain $V_{2}$ and $V_{3}$ at
relatively large distances. The best constraints at atomic distance scales
come from the work of Ramsey Ram79 who used spectroscopy in molecular
hydrogen to constrain $V_{2}$ and $V_{3}$ interactions between protons.
Recently Kimball and coauthors Kim10 have used measurements Soboll72 ;
Borel03 and calculations Wal89 ; Tscherbul09 of cross sections for spin
exchange collisions between 3He and Na atoms to constrain $V_{2}$, $V_{3}$,
and $V_{8}$ between neutrons and protons. $V_{8}$ is a spin-dependent and
velocity-dependent potential of the form
$\displaystyle V_{8}=\frac{\hbar c}{4\pi
r}(\vec{\sigma}_{1}\cdot\vec{v})(\vec{\sigma}_{2}\cdot\vec{v})\,e^{-r/\lambda}$
(4)
where $\vec{v}$ is the relative velocity of the particles (such a potential
can also influence atomic spin exchange collisions). In atomic spin exchange
collisions the spin-dependent part of the interaction is a small perturbation
on the dominant spin-independent atom-atom potential, and theoretical
calculations of the spin-exchange cross section can be performed with high
accuracy given sufficiently precise data on atomic potentials. The theoretical
calculations are simpler for systems involving light atoms, and experimental
data on spin exchange cross sections exist under conditions dominated by fast
binary atom-atom collisions which minimize possible contributions from three-
body collisions and the spin-rotation interaction Wal89 ; Walker97 . In
combination with existing constraints from spectroscopy in molecular hydrogen
Ram79 mentioned above, these authors were also able to set indirect
constraints for new interactions between neutrons.
Measurements on ensembles of polarized 3He gas atoms have been used in several
recent studies which constrain monopole-dipole interactionsSer09 ; Ig09 ;
Pok10 ; Fu10 ; Pet10 ; Fu11 , which involve the spin of one of the two
particles. In this paper we show that polarized 3He can also be used to
improve existing constraints on possible new nucleon spin-dependent
interactions involving the spins of both particles. The 3He nucleus is
isolated enough from external influences by the inert closed electron shell
that other weak interactions involving the spin of the nucleus can manifest
themselves. The interactions between the 3He atoms in a gas at room
temperature are dominated by binary atomic collisions whose dynamics have been
accurately calculated using the well-measured He-He atomic potential, and the
extra effects from weak spin-dependent interactions can be treated to high
accuracy as weak perturbations. Unlike the spin exchange collisions between
noble gas atoms and alkali metal atoms, there is no contribution from
polarized electrons. Experimental measurement coupled with theoretical
analysis shows that the polarization of the 3He nucleus is dominated as one
would expect by the polarization of the neutron Friar90 , and therefore any
limit derived from this system can be attributed directly to a limit on
neutron-neutron interactions.
The spin exchange cross section between 3He atoms can be calculated with
relatively high accuracy using the well-measured atomic potentials since (as
for Na-3He) the spin-dependent part of 3He-3He scattering is also a small
perturbation on the dominant spin-independent part. The spin relaxation rate
$\Gamma_{1}^{(1)}$ is simply related to the 3He-3He spin-exchange cross
section $\sigma_{1,E}^{(1)}$Chapman75
$\displaystyle\Gamma_{1}^{(1)}=n\left(\frac{2}{\pi\mu(k_{B}T)^{3}}\right)^{1/2}\int_{0}^{\infty}e^{-E/k_{B}T}\sigma_{1,E}^{(1)}\,E{\rm
d}E,$ (5)
where $k_{B}$ is the Boltzmann constant, $E$ is the energy of the particles,
$\mu$ is the reduced mass, $T$ is the temperature, and $n$ is the gas density.
Furthermore, there is extensive data on the longitudinal spin relaxation rate
$\Gamma_{1}$ of ensembles of polarized 3He gas atoms under conditions in which
this rate is dominated by binary 3He-3He spin exchange collisions. By using
special glass cells to suppress the loss of polarization from interaction with
the container walls, the measured spin relaxation rate of polarized 3He gas in
certain cells is so slow (relaxation times on the order of several hundred
hours) that the measured rate closely approaches the rate calculated from
magnetic dipole-dipole interactions. Since the events which lead to the
$\Gamma_{1}$ relaxation rate come from a large number of binary atom-atom
collisions between many pairs of polarized atoms, $\Gamma_{1}$ measurements
have the potential to be more sensitive to new interactions than measurements
of spin exchange cross sections, which involve single binary collisions
between atoms.
In this work, we compare the measured longitudinal spin relaxation rates
$\Gamma_{1}$ of polarized 3He gas with theoretical calculations of
$\Gamma_{1}$ from magnetic dipole-dipole interactions to set a limit for
possible new spin-spin couplings between neutrons. The rest of this paper is
organized as follows. In Sec. II we discuss the physical mechanisms which can
lead to $\Gamma_{1}$ spin relaxation in an ensemble of polarized gas atoms and
argue that the dominant contributions for the data considered in this paper
come from spin exchange collisions and interactions of the polarized atoms
with the cell walls. We also outline the calculation of the contribution to
$\Gamma_{1}$ from the magnetic dipole-dipole interaction. The spin dependent
potential $V_{3}$ described above is directly proportional to the magnetic
dipole-dipole interaction in the $\lambda\to\infty$ limit. We observe that
there is a distance scale beyond which the difference between the radial
dependence of the matrix elements involved in the calculation of the spin
exchange cross section for the two interactions is negligible. In Sec. III we
present experimental data on $\Gamma_{1}$ spin relaxation rates and use this
data to set limits on the pseudoscalar coupling $g_{p}$ which generate the
$V_{3}$ potential and on possible contributions from gravitational torsion
between neutrons. Sec. IV has our conclusions and suggestions for further
work.
## II $\Gamma_{1}$ Spin Relaxation Mechanisms in Polarized 3He Gas Cells
Interactions which can cause longitudinal spin relaxation in an ensemble of
polarized 3He atoms include: (1) a possible electric dipole moment, (2) the
spin-rotation interaction in 3He-3He collisions, (3) wall
relaxation($\Gamma_{1}^{(wall)}$), (4) magnetic field
gradients($\Gamma_{1}^{(\partial B)}$), (5) magnetic dipole-dipole
interactions($\Gamma_{1}^{(1)}$), and (6) a new dipole-dipole
interaction($\Gamma_{1}^{(2)}$). The experimental upper bounds on atomic
electric dipole moments in general and on 3He in particular make mechanism (1)
utterly negligible Purcell60 and we shall not consider it further.
The spin-rotation interaction (mechanism 2 above) is proportional to
${\vec{S}}\cdot{\vec{N}}$ with ${\vec{N}}$ the orbital angular momentum coming
from the motional magnetic fields seen by the polarized nucleus during atomic
collisions Walker97 . The interatomic interaction distorts the charge
distribution of the atom and creates a fluctuating field. At room temperature
the average kinetic energy of the colliding 3He atoms is small compared to the
atomic binding energy and therefore these distortions are relatively small. In
addition, the 3He-3He interaction is weak enough that there are no molecular
bound states which can allow the atoms to experience several revolutions and
amplify the spin-rotation interaction, as happens for other atomic species.
This effect has been investigated Chapman75 for 3He and is negligible (<1%)
compared with magnetic dipole-dipole interaction at room temperature.
The longitudinal relaxation rate $\Gamma_{1}^{(wall)}$ due to atomic
collisions with the cell wall is significant. The detailed physics involved in
the wall relaxation remains poorly understood. Nevertheless careful
preparation of the surfaces of certain types of aluminosilicate glasses can
reduce the relaxation rate from this process to be small compared to dipole-
dipole relaxation, as will be shown in section III.
Spin relaxation $\Gamma_{1}^{(\partial B)}$ from the motion of the polarized
nuclei in magnetic field gradients can be calculated to beCat88
$\Gamma_{1}^{(\partial B)}=D\frac{|\triangledown B_{\bot}|^{2}}{B_{||}^{2}},$
(6)
where $D$ is the diffusion constant of the polarized gas and $B_{||}\
(B_{\bot})$ are the magnetic fields parallel (perpendicular) to the spin’s
direction. For a polarized 3He cell of pressure 1 bar at room temperature in
an external field gradient of ${\delta B_{\bot}/B_{||}}=10^{-4}$ cm-1, which
is typically achieved in the Helmholtz coil arrangements used in the
measurements described in section III, $\Gamma_{1}^{(\partial B)}=2\times
10^{-8}$ s-1McIver09 , which is small compared to the dipole-dipole relaxation
in the cells discussed in section III. Internal field gradients induced by the
magnetization of the polarized gas are proportional to the gas density and
polarization and also depend on the geometry of the gas container. A field
generated by polarized gas in a spherical cell is uniform and thus has no
effect on spin relaxation. For the gas cells discussed in this paper the
relaxation from self-generated internal field gradients is very small compared
to relaxation from external field gradients.
The spin relaxation $\Gamma_{1}^{(1)}$ due to magnetic dipole-dipole
interactions in binary 3He-3He collisions dominates the bulk relaxation in the
gas. It can be calculated by first solving for the 3He-3He scattering
amplitude using the measured spin-independent atom-atom potential $V^{(0)}$
and then adding the hyperfine interaction as a perturbation. The spin-
dependent magnetic dipole-dipole interaction potential $V^{(1)}$ has the form
$\displaystyle
V^{(1)}(r)=\frac{f^{(1)}}{r^{3}}\left[(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{\sigma}}_{2})-3(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{r}})(\hat{\mathbf{\sigma}}_{2}\cdot\hat{\mathbf{r}})\right],$
(7)
where $f^{(1)}=\alpha\frac{\hbar^{3}g^{2}}{16m_{e}^{2}c}$, $g=-0.002317$ is
g-factor of 3He, $\alpha$ is the fine structure constant, and $m_{e}$ is the
mass of electron. The expression for $\Gamma_{1}$ from binary collisions of
polarized atoms in a gas in thermal equilibrium at temperature $T$ is shown in
Eq. 5, where the spin exchange cross section can be written as Mullin90 ;
Shizgal73 ; New93 ,
$\displaystyle\sigma_{1,E}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{48\pi
m^{2}}{5\hbar^{4}}\left(f^{(1)}\right)^{2}$ (8)
$\displaystyle\times\sum_{ll^{\prime}(odd)}(2l+1)(2l^{\prime}+1)C^{2}(ll^{\prime}2;00)\left<\frac{1}{(kr)^{3}}\right>^{2}_{ll^{\prime}}.$
where $C(ll^{\prime}2;00)$ are Clebsch-Cordon coefficients, $k$ is the
wavenumber, and the matrix elements
$\left<\frac{1}{(kr)^{3}}\right>_{ll^{\prime}}$ corresponding to $l\rightarrow
l^{\prime}$ transitions can be obtained by solving the Schrodinger equation
for the two-body scattering statesNew93 .
For atom-atom collisions at room temperature only partial waves with small $l$
make significant contributions to the spin exchange cross section. Equating
the centrifugal barrier at a distance corresponding to the 3He atomic diameter
of $0.06$ nm with the kinetic energy, $l(l+1)\hbar^{2}/2\mu r^{2}=3k_{B}T/2$
(where $\mu$ is the reduced mass of 3He) to find the orbital angular momentum
associated with the closest approach of the atoms yields $l\simeq 3$. At room
temperature partial waves with $l>3$ do not penetrate the centrifugal barrier
and therefore see mainly the long-range Van der Waals interaction.
The potential energy from the possible new dipole-dipole interaction which we
propose to constrain has the formMoody84 ,
$\displaystyle V^{(2)}(r)$ $\displaystyle=$ $\displaystyle
f^{(2)}\frac{e^{-r/\lambda}}{r^{3}}\\{(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{\sigma}}_{2})\left(1+\frac{r}{\lambda}\right)$
(9)
$\displaystyle-3(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{r}})(\hat{\mathbf{\sigma}}_{2}\cdot\hat{\mathbf{r}})\left(1+\frac{r}{\lambda}+\frac{r^{2}}{3\lambda^{2}}\right)\\},$
where $f^{(2)}=g_{p}^{2}\hbar^{3}/(16\pi m_{n}^{2}c)$, $m_{n}$ is the mass,
$c$ is the speed of light, $\lambda$ is the interaction range, and $g_{p}$ is
the coupling constant. Because $V^{(2)}<V^{(1)}\ll V^{(0)}$, $V^{(2)}$ can
also be treated as a perturbation and one can follow the same procedure as in
Eq. (8) to obtain the matrix elements of $\left<V^{(2)}\right>_{ll^{\prime}}$.
Note that the dipole-dipole potential under exchange of a finite-mass particle
in Eq. (9) reduces to the same form as the usual electromagnetic dipole-dipole
potential in Eq. (7) as the particle becomes massless ($\lambda\to\infty$). In
this limit the analysis required to set a bound on $V_{3}$ is greatly
simplified. For the data considered in this paper this limiting case is
reached already for interaction ranges $\lambda>100$ nm. The interatomic
potential $V^{(0)}(r)$ falls quickly outside the atomic diameter, and
collisions with impact parameters of this size between 3He atoms at room
temperature correspond to $l\approx 3$. Partial waves with $l>3$ feel only the
weak long-range part of the atom-atom potential, which can be calculated in
perturbation theory and makes a small contribution to the matrix element. The
lower partial waves encounter the hard core repulsion. For interaction ranges
$\lambda>100$ nm, however, the Yukawa term in the potential is slowly varying
and the radial dependence of $V^{(2)}$ and $V^{(1)}$ are therefore the same to
high accuracy for small $r$. One can show numerically that beyond $r=10$ nm
the lower partial waves approach their asymptotic forms and make negligible
contributions to the matrix element. Therefore one can choose an upper cutoff
of $r=10$ nm in the radial integral for the calculation of matrix element
$\left<V^{(2)}\right>_{ll^{\prime}}$ with negligible uncertainty. In this case
$\lambda\gg r$ in the matrix elements and Eq. (9) can be simplified to
$\displaystyle V^{(2)}(r)$ $\displaystyle\simeq$
$\displaystyle\frac{f^{(2)}}{r^{3}}\left[(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{\sigma}}_{2})-3(\hat{\mathbf{\sigma}}_{1}\cdot\hat{\mathbf{r}})(\hat{\mathbf{\sigma}}_{2}\cdot\hat{\mathbf{r}})\right].$
(10)
In this limit $V^{(2)}$ and $V^{(1)}$ have the same form, so the contribution
of the potentials $V^{(1)}$ and $V^{(2)}$ to the longitudinal spin relaxation
rate is
$\displaystyle\Gamma_{1}^{(1,2)}=n\left(\frac{2}{\pi\mu(\kappa
T)^{3}}\right)^{1/2}\int_{0}^{\infty}e^{-E/\kappa
T}\sigma_{1,E}^{(1,2)}\,EdE,$ (11)
where $\sigma_{1,E}^{(1,2)}$ is
$\displaystyle\sigma_{1,E}^{(1,2)}$ $\displaystyle=$ $\displaystyle\frac{48\pi
m^{2}}{5\hbar^{4}}\left(f^{(1)}+f^{(2)}\right)^{2}$ (12)
$\displaystyle\times\sum_{ll^{\prime}(odd)}(2l+1)(2l^{\prime}+1)C^{2}(ll^{\prime}2;00)\left<\frac{1}{(kr)^{3}}\right>^{2}_{ll^{\prime}}.$
Using Eq. (5) and Eq. (11), we have,
$g_{p}^{2}/4\pi=\frac{\alpha\,g^{2}(m_{n}/m_{e})^{2}}{4}\left[\left(\frac{\Gamma_{1}^{(1,2)}}{\Gamma_{1}^{(1)}}\right)^{1/2}-1\right].$
(13)
The experimentally measured longitudinal relaxation rate $\Gamma_{1}^{(exp)}$
must satisfy $\Gamma_{1}^{(exp)}\geq\Gamma_{1}^{(1,2)}$. Therefore, using Eq.
(13) we can derive a lower limit for the product of the couplings $g_{p}^{2}$:
$\displaystyle g_{p}^{2}/4\pi$ $\displaystyle\leq$
$\displaystyle\frac{\alpha\,g^{2}(m_{n}/m_{e})^{2}}{4}\left[\left(\frac{\Gamma_{1}^{(exp)}}{\Gamma_{1}^{(1)}}\right)^{1/2}-1\right]$
(14) $\displaystyle=$ $\displaystyle 0.033R,$
where
$R\equiv\left[\left(\Gamma_{1}^{(exp)}/\Gamma_{1}^{(1)}\right)^{1/2}-1\right]$
is an upper bound on the strength of the new dipole-dipole interaction
relative to the magnetic dipole-dipole interaction: $V^{(2)}/V^{(1)}\leq R$.
The uncertainty on the constraint on $g_{p}^{2}/4\pi$ is determined by the
experimental uncertainty of $\Gamma_{1}^{(exp)}$ and the theoretical
uncertainty in the calculated value for $\Gamma_{1}^{(1)}$.
## III Measurements of $\Gamma_{1}^{(exp)}$ Relaxation Rates of Polarized 3He
Gas
The technology of laser optical pumping to produce macroscopic quantities of
gas with high polarization has undergone extensive development for scientific
applications in neutron scattering, medical imaging, and nuclear and particle
physics Babcock09 ; Chen07 ; Babcock06 ; Holmes08 ; Slifer08 . There exist two
widely-used methods to polarize 3He gas, metastability-exchange optical
pumping (MEOP) Colegrove63 and spin-exchange optical pumping (SEOP) Walker97
. We use SEOP data in this paper. In addition to the 3He, SEOP cells also
contain a small amount (normally <0.1 g) of Rb and/or K for optical pumping
and a small amount of N2 gas to nonradiatively relax the optically-pumped
alkali atoms to prevent radiation trapping. It has been found experimentally
that certain aluminosilicate glasses can have wall relaxation rates which are
small compared to the dipole-dipole relaxation rate, and it is
$\Gamma_{1}^{(exp)}$ measurements in these cells that we use to set our
limits.
To our knowledge the most accurate comparisons between theory and experiment
for the $\Gamma_{1}$ spin relaxation rates of polarized 3He gas in a SEOP cell
were performed by Newbury et al. New93 ; Newthesis and Rich et al. Rich02 .
For 3He at room temperature $\Gamma_{1}^{(1)}$ is New93
$\Gamma_{1}^{(1)}=3.73\times 10^{-7}\cdot[n]\cdot{\rm s^{-1}},$ (15)
where the 3He density $[n]$ is in units of amagats. Some of the theoretical
uncertainties in this result come from the uncertainty in the measurements of
the interatomic potential (which differ slightlyAziz87 ) and produce
corresponding uncertainties in the calculated spin exchange rate. The
relaxation rates calculated with these different interatomic potentials
$V^{(0)}$ differ by 1-2%. We assign a 2% relative standard uncertainty in
$\Gamma_{1}^{(1)}$ from experimental knowledge in interatomic potentials and a
1% relative standard uncertainty from other sources related to the numerical
calculation. In addition there are uncertainties in the theoretical prediction
associated with uncertainties in the knowledge of the temperature and density
of the 3He in the cells. In Ref. New93 , the densities of the cells are
8.37(19) amagat for cell 808, and 4.67(11) amagat for cell 842, where the
numbers in parentheses denote the standard uncertainties in the last digit(s).
$\Gamma_{1}^{(1)}$ is proportional to the density, and near room temperature
$\Gamma_{1}\propto T^{1/2}$Mullin90 . We assume a temperature and
corresponding standard uncertainty of $(297\pm 5)\ ^{\circ}$C, which yields a
$\Gamma_{1}^{(1)}$ relative standard uncertainty of 0.8%. The relative
standard uncertainty in the theoretical prediction for $\Gamma_{1}$ for the
Newbury cells was therefore
${\Delta\Gamma_{1}^{(1)}\over\Gamma_{1}^{(1)}}=3\%$. The measured relaxation
rates are $3.19(4)\times 10^{-6}$/s for cell 808 and $1.84(2)\times 10^{-6}$/s
for cell 842. Using these numbers and equation 14, we set a limit of
$g_{p}^{(n)}g_{p}^{(n)}/4\pi<1.7\times 10^{-3}$ at the $1\sigma$ confidence
level.
A similar upper bound is also obtained in an independent measurement at a
lower gas density. In Ref. Rich02 , the cell Wilma with gas density of 0.73(3)
amagat (the original paper mistakenly stated 0.78 amagat for the density) was
measured to have spin relaxation rate $3.31(6)\times 10^{-7}$/s. Using these
numbers in equation 14 would set a slightly less stringent limit of
$g_{p}^{(n)}g_{p}^{(n)}/4\pi<5\times 10^{-3}$ at the $1\sigma$ confidence
level. We also note that $\Gamma_{1}^{(exp)}$ measurements for several other
3He SEOP cells have been conducted over the last decade Smith98 ; JCNS2010
with densities ranging from $0.7$ amagat to $2$ amagat and with different
types of glasses as the cell materials. Although many of these cells possess
wall relaxation rates comparable to those discussed above, none of the
$\Gamma_{1}^{(exp)}$ measurements in these cells is less than the calculated
$\Gamma_{1}^{(1)}$.
Figure 1: Comparison of $1\sigma$ upper bounds on the coupling constant
combination $g_{p}^{2}/4\pi$ for possible new pseudoscalar dipole-dipole
interactions from the following sources: 1). using $\Gamma_{1}$ measurements
in a 3He-129Xe maserGle08 , 2). this work, using measurements in 3He SEOP
cells, 3). using interaction between 3He and K species in two separate SEOP
cellsVas09 , and 4). using hydrogen molecular spectroscopyRam79 .
In Figure 1 we show the limits on $g_{p}^{2}/4\pi$ for neutrons extracted via
Eq. (14) using the comparison between the $\Gamma_{1}^{(exp)}$ measurements
with theory. Limits from measurements in a 3He-129Xe maserGle08 and from
measurements involving separate SEOP cells of 3He and KVas09 are the most
stringent for neutron-neutron interactions at distances greater than 1 cm.
Hydrogen molecular spectroscopyRam79 provides a stringent direct constraint
on proton-proton interactions. The indirect limits on neutron-neutron
interactions set by Kimball et al. on $g_{p}^{2}/4\pi$ for the $V_{3}$
interaction are above the top of the vertical axis of the plot. The limits
from 3He-3He, which as mentioned in the introduction are cleanly interpretable
as direct limits on neutron-neutron interactions, are the best direct limits
to our knowledge for distance scales from 100 nm to a few mm, corresponding to
exchange particles with masses from 1 eV to 0.1 meV.
We can also use this data to constrain short-range gravitational torsion
between neutrons. Torsion, an additional warping of spacetime with spin as its
source, is required for the conservation of angular momentum in general
relativity when intrinsic spin is included Hammond10 . Recently the
experimental constraints on long-range gravitational torsion have been
tightened considerably with the realization Kostelec04 that a background
torsion field violates effective local Lorentz invariance. Kostelecký and
coauthors Kostelec08 were able to constrain $19$ of the $24$ components of
the torsion tensor for the first time, and these methods have since been
adopted Heckel08 to further improve constraints on $4$ of these $19$
components. As for short-range spin-dependent interactions, experimental
constraints on short-range torsion Neville80 ; Neville82 ; Carroll94 ;
Hammond95 are poor. The spin-spin interaction generated by a short-range
torsion field is of the same form ($V_{3}$ in the Dobrescu notation)
considered above Hammond95 ; Ade09 scaled by a parameter $\beta$ where
$\displaystyle\beta^{2}=(g_{p}^{2}/4\pi\hbar c)({{2\hbar c}\over
9Gm_{n}^{2}})$ (16)
Figure 2 shows the constraints on short-range torsion from the work of Kimball
et al Kim10 , Ramsey Ram79 , and our work, which limits $\beta^{2}<2\times
10^{37}$ for neutron-neutron interactions. Over the distance range between 100
nm and 1 cm these works set to our knowledge the best experimental limits on
the torsion parameter for neutron-proton, neutron-neutron, and proton-proton
interactions, respectively. At distance scales $>50$ cm a much more stringent
limit of $\beta^{2}<2\times 10^{28}$ comes from the work of Romalis and co-
workers Vas09 .
Figure 2: Comparison of $1\sigma$ upper bounds on gravitation torsion
couplings between nucleons from the following sources: 1). from the analysis
of 3He-Na spin exchange cross section measurementsKim10 , 2). this work, 3).
using hydrogen molecular spectroscopyRam79 .
## IV Conclusions
By comparing theory and experiment for the longitudinal spin relaxation rate
$\Gamma_{1}$ of polarized 3He gas, we have derived a $1\sigma$ upper bound of
$g_{p}^{(n)}g_{p}^{(n)}/4\pi<1.7\times 10^{-3}$ on the product of the
couplings for a possible new pseudoscalar spin-spin interaction between
neutrons with a dipole-dipole form $V_{3}$ at distance scales larger than 100
nm. This constraint limits such interactions to a size no more than 4% of the
magnetic dipole-dipole interactions between the polarized 3He atoms. Although
these limits are certainly much less stringent than those for spin-spin
interactions of macroscopic range, they are to our knowledge the best direct
limits from laboratory experiments on spin-spin interactions of this form
between neutrons. Attempts to improve the constraints using this approach
would be limited both by the spin exchange cross section uncertainties and our
ignorance of the physics of 3He wall relaxation rates, and we do not expect
that it will be possible to significantly improve the accuracy of either the
theoretical calculations or the understanding of the physics of the wall
relaxation in the near future. However it should be possible to use this same
data to set the best constraints on the $V_{2}$ and $V_{11}$ spin-dependent
potentials and the velocity and spin-dependent potential $V_{8}$ for distance
scales larger than 100 nm. We plan to present these limits in a forthcoming
publication.
## V Acknowledgements
We thank Tom Gentile, Wangchun Chen, Xilin Zhang for extensive discussions.
This work was supported in part by NSF PHY-0116146. The work of WMS was
supported in part by the Indiana University Center for Spacetime Symmetries.
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|
arxiv-papers
| 2011-03-03T11:47:17 |
2024-09-04T02:49:17.421565
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Changbo Fu and W. M. Snow",
"submitter": "Changbo Fu",
"url": "https://arxiv.org/abs/1103.0659"
}
|
1103.0713
|
11institutetext: University of Liverpool, and the Cockcroft Institute, UK
# Maxwell’s equations for magnets
Andrzej Wolski
###### Abstract
Magnetostatic fields in accelerators are conventionally described in terms of
multipoles. We show that in two dimensions, multipole fields do provide
solutions of Maxwell’s equations, and we consider the distributions of
electric currents and geometries of ferromagnetic materials required (in
idealized situations) to generate specified multipole fields. Then, we
consider how to determine the multipole components in a given field. Finally,
we show how the two-dimensional multipole description may be extended to three
dimensions; this allows fringe fields, or the main fields in such devices as
undulators and wigglers, to be expressed in terms of a set of modes, where
each mode provides a solution to Maxwell’s equations.
## 0.1 Maxwell’s equations
Maxwell’s equations may be written in differential form as follows:
$\displaystyle\textrm{div}\,\vec{D}$ $\displaystyle=$ $\displaystyle\rho,$ (1)
$\displaystyle\textrm{div}\,\vec{B}$ $\displaystyle=$ $\displaystyle 0,$ (2)
$\displaystyle\textrm{curl}\,\vec{H}$ $\displaystyle=$
$\displaystyle\vec{J}+\frac{\partial\vec{D}}{\partial t},$ (3)
$\displaystyle\textrm{curl}\,\vec{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial\vec{B}}{\partial t}.$ (4)
The fields $\vec{B}$ (magnetic flux density) and $\vec{E}$ (electric field
strength) determine the force on a particle of charge $q$ travelling with
velocity $\vec{v}$ (the Lorentz force equation):
$\vec{F}=q\left(\vec{E}+\vec{v}\times\vec{B}\right).$
The electric displacement $\vec{D}$ and magnetic intensity $\vec{H}$ are
related to the electric field and magnetic flux density by:
$\displaystyle\vec{D}$ $\displaystyle=$ $\displaystyle\varepsilon\vec{E},$
$\displaystyle\vec{B}$ $\displaystyle=$ $\displaystyle\mu\vec{H}.$
The electric permittivity $\varepsilon$ and magnetic permeability $\mu$ depend
on the medium within which the fields exist. The values of these quantities in
vacuum are fundamental physical constants. In SI units:
$\displaystyle\mu_{0}$ $\displaystyle=$ $\displaystyle 4\pi\times
10^{-7}\,\textrm{Hm}^{-1},$ $\displaystyle\varepsilon_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\mu_{0}c^{2}},$
where $c$ is the speed of light in vacuum. The permittivity and permeability
of a material characterize the response of that material to electric and
magnetic fields. In simplified models, they are often regarded as constants
for a given material; however, in reality the permittivity and permeability
can have a complicated dependence on the fields that are present. Note that
the _relative permittivity_ $\varepsilon_{r}$ and the _relative permeability_
$\mu_{r}$ are frequently used. These are dimensionless quantities, defined by:
$\varepsilon_{r}=\frac{\varepsilon}{\varepsilon_{0}},\quad\mu_{r}=\frac{\mu}{\mu_{0}}.$
(5)
That is, the relative permittivity is the permittivity of a material relative
to the permittivity of free space, and similarly for the relative
permeability.
The quantities $\rho$ and $\vec{J}$ are respectively the electric charge
density (charge per unit volume) and electric current density
($\vec{J}\cdot\vec{n}$ is the charge crossing unit area perpendicular to unit
vector $\vec{n}$ per unit time). Equations (2) and (4) are independent of
$\rho$ and $\vec{J}$, and are generally referred to as the “homogeneous”
equations; the other two equations, (1) and (3) are dependent on $\rho$ and
$\vec{J}$, and are generally referred to as the “inhomogeneous” equations. The
charge density and current density may be regarded as _sources_ of
electromagnetic fields. When the charge density and current density are
specified (as functions of space, and, generally, time), one can integrate
Maxwell’s equations (1)–(3) to find possible electric and magnetic fields in
the system. Usually, however, the solution one finds by integration is not
unique: for example, the field within an accelerator dipole magnet may be
modified by propagating an electromagnetic wave through the magnet. However,
by imposing certain constraints (for example, that the fields within a magnet
are independent of time) it is possible to obtain a unique solution for the
fields in a given system of electric charges and currents.
Most realistic situations are sufficiently complicated that solutions to
Maxwell’s equations cannot be obtained analytically. A variety of computer
codes exist to provide solutions numerically, once the charges, currents, and
properties of the materials present are all specified, see for example
References [1, 2, 3]. Solving for the fields in realistic (three-dimensional)
systems often requires a reasonable amount of computing power; some
sophisticated techniques have been developed for solving Maxwell’s equations
numerically with good efficiency [4]. We do not consider such techniques here,
but focus instead on the analytical solutions that may be obtained in
idealized situations. Although the solutions in such cases may not be
sufficiently accurate to complete the design of a real accelerator magnet, the
analytical solutions do provide a useful basis for describing the fields in
real magnets, and provide also some important connections with the beam
dynamics in an accelerator.
An important feature of Maxwell’s equations is that, for systems containing
materials with constant permittivity and permeability (i.e. permittivity and
permeability that are independent of the fields present), the equations are
_linear_ in the fields and sources. That is, each term in the equations
involves a field or a source to (at most) the first power, and products of
fields or sources do not appear. As a consequence, the _principle of
superposition_ applies: if $\vec{B}_{1}$ and $\vec{B}_{2}$ are solutions of
Maxwell’s equations with the current densities $\vec{J}_{1}$ and
$\vec{J}_{2}$, then the field $\vec{B}_{T}=\vec{B}_{1}+\vec{B}_{2}$ will be a
solution of Maxwell’s equations, with the source given by the total current
density $\vec{J}_{T}=\vec{J}_{1}+\vec{J}_{2}$. This means that it is possible
to represent complicated fields as superpositions of simpler fields. An
important and widely used analysis technique for accelerator magnets is to
decompose the field (determined from either a magnetic model, or from
measurements of the field in an actual magnet) into a set of multipoles. While
it is often the ideal to produce a field consisting of a single multipole
component, this is never perfectly achieved in practice: the multipole
decomposition indicates the extent to which components other than the
“desired” multipole are present. Multipole decompositions also produce useful
information for modelling the beam dynamics. Although the principle of
superposition strictly only applies in systems where the permittivity and
permeability are independent of the fields, it is always possible to perform a
multipole decomposition of the fields in free space (e.g. in the interior of a
vacuum chamber), since in that region the permittivity and permeability are
constants. However, it should be remembered that for nonlinear materials
(where the permeability, for example, depends on the magnetic field strength),
the field inside the material comprising the magnet will not necessarily be
that expected if one were simply to add together the fields corresponding to
the multipole components.
Solutions to Maxwell’s equations lead to a rich diversity of phenomena,
including the fields around charges and currents in certain simple
configurations, and the generation, transmission and absorption of
electromagnetic radiation. Many existing texts cover these phenomena in
detail; see, for example, the authoritative text by Jackson [5]. Therefore, we
consider only briefly the electric field around a point charge and the
magnetic field around a long straight wire carrying a uniform current: our
main purpose here is to remind the reader of two important integral theorems
(Gauss’ theorem, and Stokes’ theorem), of which we shall make use later. In
the following sections, we will discuss analytical solutions to Maxwell’s
equations for situations relevant to some of the types of magnets commonly
used in accelerators. These include multipoles (dipoles, quadrupoles,
sextupoles, and so on), solenoids, and insertion devices (undulators and
wigglers). We consider only static fields. We begin with two-dimensional
fields, that is fields that are independent of one coordinate (generally, the
coordinate representing the direction of motion of the beam). We will show
that multipole fields are indeed solutions of Maxwell’s equations, and we will
derive the current distributions needed to generate “pure” multipole fields.
We then discuss multipole decompositions, and compare techniques for
determining the multipole components present in a given field from numerical
field data (from a model, or from measurements). Finally, we consider how the
two-dimensional multipole decomposition may be extended to three-dimensional
fields, to include (for example) insertion devices, and fringe fields in
multipole magnets.
## 0.2 Integral theorems and the physical interpretation of Maxwell’s
equations
### 0.2.1 Gauss’ theorem and Coulomb’s law
Guass’ theorem states that for any smooth vector field $\vec{a}$:
$\int_{V}\textrm{div}\,\vec{a}\,dV=\int_{\partial V}\vec{a}\cdot d\vec{S},$
where $V$ is a volume bounded by the closed surface $\partial V$. Note that
the area element $d\vec{S}$ is oriented to point _out_ of $V$.
Gauss’ theorem is helpful for obtaining physical interpretations of two of
Maxwell’s equations, (1) and (2). First, applying Gauss’ theorem to (1) gives:
$\int_{V}\textrm{div}\,\vec{D}\,dV=\int_{\partial V}\vec{D}\cdot d\vec{S}=q,$
(6)
where $q=\int_{V}\rho\,dV$ is the total charge enclosed by $\partial V$.
Suppose that we have a single isolated point charge in an homogeneous,
isotropic medium with constant permittivity $\varepsilon$. In this case, it is
interesting to take $\partial V$ to be a sphere of radius $r$. By symmetry,
the magnitude of the electric field must be the same at all points on
$\partial V$, and must be normal to the surface at each point. Then, we can
perform the surface integral in (6):
$\int_{\partial V}\vec{D}\cdot d\vec{S}=4\pi r^{2}D.$
This is illustrated in Fig. 1: the outer circle represents a cross-section of
a sphere ($\partial V$) enclosing volume $V$, with the charge $q$ at its
centre. The black arrows in Fig. 1 represent the electric field lines, which
are everywhere perpendicular to the surface $\partial V$. Since
$\vec{D}=\varepsilon\vec{E}$, we find Coulomb’s law for the magnitude of the
electric field around a point charge:
$E=\frac{q}{4\pi\varepsilon r^{2}}.$
Figure 1: Electric field lines from a point charge $q$. The field lines are
everywhere perpendicular to a spherical surface centered on the charge.
Applied to Maxwell’s equation (2), Gauss’ theorem leads to:
$\int_{V}\textrm{div}\,\vec{B}\,dV=\int_{\partial V}\vec{B}\cdot d\vec{S}=0.$
In other words, the magnetic flux integrated over any closed surface must
equal zero – at least, until we discover magnetic monopoles. Lines of magnetic
flux occur in closed loops; whereas lines of electric field can start (and
end) on electric charges.
### 0.2.2 Stokes’ theorem and Ampère’s law
Stokes’ theorem states that for any smooth vector field $\vec{a}$:
$\int_{S}\textrm{curl}\,\vec{a}\cdot d\vec{S}=\int_{\partial S}\vec{a}\cdot
d\vec{l},$ (7)
where the loop $\partial S$ bounds the surface $S$. Applied to Maxwell’s
equation (3), Stokes’ theorem leads to:
$\int_{\partial S}\vec{H}\cdot d\vec{l}=\int_{S}\vec{J}\cdot d\vec{S},$ (8)
which is Ampère’s law. From Ampère’s law, we can derive an expression for the
strength of the magnetic field around a long, straight wire carrying current
$I$. The magnetic field must have rotational symmetry around the wire. There
are two possibilities: a radial field, or a field consisting of closed
concentric loops centred on the wire (or some superposition of these fields).
A radial field would violate Maxwell’s equation (2). Therefore, the field must
consist of closed concentric loops; and by considering a circular loop of
radius $r$, we can perform the integral in Eq. (8):
$2\pi rH=I,$
where $I$ is the total current carried by the wire. In this case, the line
integral is performed around a loop $\partial S$ centered on the wire, and in
a plane perpendicular to the wire: essentially, this corresponds to one of the
magnetic field lines, see Fig. 2. The total current passing through the
surface $S$ bounded by the loop $\partial S$ is simply the total current $I$.
Figure 2: Magnetic field lines around a long straight wire carrying a current
$I$.
In an homogeneous, isotropic medium with constant permeability $\mu$,
$\vec{B}=\mu_{0}\vec{H}$, and we obtain the expression for the magnetic flux
density at distance $r$ from the wire:
$B=\frac{I}{2\pi\mu r}.$ (9)
This result will be useful when we come to consider how to generate specified
multipole fields from current distributions.
Finally, applying Stokes’ theorem to the homogeneous Maxwell’s equation (4),
we find:
$\int_{\partial S}\vec{E}\cdot d\vec{l}=-\frac{\partial}{\partial
t}\int_{S}\vec{B}\cdot d\vec{S}.$ (10)
Defining the electromotive force $\mathscr{E}$ as the integral of the electric
field around a closed loop, and the magnetic flux $\Phi$ as the integral of
the magnetic flux density over the surface bounded by the loop, Eq. (10)
gives:
$\mathscr{E}=-\frac{\partial\Phi}{\partial t},$ (11)
which is Faraday’s law of electromagnetic induction. Faraday’s law is
significant for magnets with time-dependent fields, such as pulsed magnets
(used for injection and extraction), and magnets that are “ramped” (for
example, when changing the beam energy in a storage ring). The change in
magnetic field will induce a voltage across the coil of the magnet, that must
be taken into account when designing the power supply. Also, the induced
voltages can induce eddy currents in the core of the magnet, or in the coils
themselves, leading to heating. This is an issue for superconducting magnets,
which must be ramped slowly to avoid quenching [6].
### 0.2.3 Boundary conditions
Gauss’ theorem and Stokes’ theorem can be applied to Maxwell’s equations to
derive constraints on the behaviour of electromagnetic fields at boundaries
between different materials. Here, we shall focus on the boundary conditions
on the magnetic field: these conditions will be useful when we consider
multipole fields in iron-dominated magnets.
Figure 3: (a) Left: “Pill box” surface for derivation of the boundary
conditions on the normal component of the magnetic flux density at the
interface between two media. (b) Right: Geometry for derivation of the
boundary conditions on the tangential component of the magnetic intensity at
the interface between two media.
Consider first a short cylinder or “pill box” that crosses the boundary
between two media, with the flat ends of the cylinder parallel to the
boundary, see Fig. 3 (a). Applying Gauss’ theorem to Maxwell’s equation (2)
gives:
$\int_{V}\textrm{div}\,\vec{B}\,dV=\int_{\partial V}\vec{B}\cdot d\vec{S}=0,$
where the boundary $\partial V$ encloses the volume $V$ within the cylinder.
If we take the limit where the length of the cylinder ($2h$) approaches zero,
then the only contributions to the surface integral come from the flat ends;
if these have infinitesimal area $dS$, then since the orientations of these
surfaces are in opposite directions on opposite sides of the boundary, and
parallel to the normal component of the magnetic field, we find:
$-B_{1\perp}\,dS+B_{2\perp}\,dS=0,$
where $B_{1\perp}$ and $B_{2\perp}$ are the normal components of the magnetic
flux density on either side of the boundary. Hence:
$B_{1\perp}=B_{2\perp}.$ (12)
In other words, the normal component of the magnetic flux density is
continuous across a boundary.
A second boundary condition, this time on the component of the magnetic field
parallel to a boundary, can be obtained by applying Stokes’ theorem to
Maxwell’s equation (3). In particular, we consider a surface $S$ bounded by a
loop $\partial S$ that crosses the boundary of the material, see Fig. 3 (b).
If we integrate both sides of Eq. (3) over that surface, and apply Stokes’
theorem (7), we find:
$\int_{S}\textrm{curl}\,\vec{H}\cdot d\vec{S}=\int_{\partial S}\vec{H}\cdot
d\vec{l}=I+\frac{\partial}{\partial t}\int_{S}\vec{D}\cdot d\vec{S},$
where $I$ is the total current flowing through the surface $S$. Now, let the
surface $S$ take the form of a thin strip, with the short ends perpendicular
to the boundary, and the long ends parallel to the boundary. In the limit that
the length of the short ends goes to zero, the area of $S$ goes to zero: both
the current flowing through the surface $S$, and the electric displacement
integrated over $S$ become zero. However, there are still contributions to the
integral of $\vec{H}$ around $\partial S$ from the long sides of the strip.
Thus, we find that:
$H_{1\parallel}=H_{2\parallel},$ (13)
where $H_{1\parallel}$ is the component of the magnetic intensity parallel to
the boundary at a point on one side of the boundary, and $H_{2\parallel}$ is
the component of the magnetic intensity parallel to the boundary at a nearby
point on the other side of the boundary. In other words, the _tangential_
component of the magnetic intensity $\vec{H}$ is continuous across a boundary.
We can derive a stronger contraint on the magnetic field at a boundary in the
case that the material on one side of the boundary has infinite permeability
(which can provide a reasonable model for some ferromagnetic materials). Since
$\vec{B}=\mu\vec{H}$, it follows from (13) that:
$\frac{B_{1\parallel}}{\mu_{1}}=\frac{B_{2\parallel}}{\mu_{2}},$
and in the limit $\mu_{2}\to\infty$, while $\mu_{1}$ remains finite, we must
have:
$B_{1\parallel}=0.$ (14)
In other words, the magnetic flux density at the surface of a material of
infinite permeability must be perpendicular to that surface. Of course, the
permeability of a material characterizes its response to an applied external
magnetic field: in the case that the permeability is infinite, a material
placed in an external magnetic field acquires a magnetization that exactly
cancels any component of the external field at the surface of the material.
## 0.3 Two-dimensional multipole fields
Consider a region of space free of charges and currents; for example, the
interior of an accelerator vacuum chamber (at least, in an ideal case, and
when the beam is not present). If we further exclude propagating
electromagnetic waves, then any magnetic field generated by steady currents
outside the vacuum chamber must satisfy:
$\displaystyle\textrm{div}\,\vec{B}$ $\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle\textrm{curl}\,\vec{B}$ $\displaystyle=$ $\displaystyle 0.$ (16)
Eq. (15) is just Maxwell’s equation (2), and Eq. (16) follows from Maxwell’s
equation (3) given that $\vec{J}=0$, $\vec{B}=\mu_{0}\vec{H}$, and derivatives
with respect to time vanish.
We shall show that a magnetic field $\vec{B}=(B_{x},B_{y},B_{z})$ with $B_{z}$
constant, and $B_{x}$, $B_{y}$ given by:
$B_{y}+iB_{x}=C_{n}(x+iy)^{n-1}$ (17)
where $i=\sqrt{-1}$ and $C_{n}$ is a (complex) constant, satisfies Eqs. (15)
and (16). Note that the field components $B_{x}$ and $B_{y}$ are real, and are
obtained from the imaginary and real parts of the right hand side of Eq. (17).
To show that the above field satisfies Eqs. (15) and (16), we apply the
differential operator
$\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}$ (18)
to each side of Eq. (17). Applied to the left hand side, we find:
$\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial
y}\right)\left(B_{y}+iB_{x}\right)=\left(\frac{\partial B_{y}}{\partial
x}-\frac{\partial B_{x}}{\partial y}\right)+i\left(\frac{\partial
B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}\right).$ (19)
Applied to the right hand side of Eq. (17), the differential operator (18)
gives:
$\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial
y}\right)C_{n}(x+iy)^{n-1}=C_{n}(n-1)(x+iy)^{n-2}+i^{2}C_{n}(n-1)(x+iy)^{n-2}=0.$
(20)
Combining Eqs. (17), (19) and (20), we find:
$\displaystyle\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial
y}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle\frac{\partial
B_{y}}{\partial x}-\frac{\partial B_{x}}{\partial y}$ $\displaystyle=$
$\displaystyle 0.$
Finally, we note that $B_{z}$ is constant, so any derivatives of $B_{z}$
vanish; furthermore, $B_{x}$ and $B_{y}$ are independent of $z$, so any
derivatives of these coordinates with respect to $z$ vanish. Thus, we conclude
that for the field (17):
$\displaystyle\textrm{div}\,\vec{B}$ $\displaystyle=$ $\displaystyle 0,$ (21)
$\displaystyle\textrm{curl}\,\vec{B}$ $\displaystyle=$ $\displaystyle 0,$ (22)
and that this field is therefore a solution to Maxwell’s equations within the
vacuum chamber. Of course, this analysis tells us only that the field is a
_possible_ physical field: it does not tell us how to generate such a field.
The problem of generating a field of the form Eq. (17) we shall consider in
Section 0.4.
Fields of the form (17) are known as _multipole fields_. The index $n$ (an
integer) indicates the _order_ of the multipole: $n=1$ is a dipole field,
$n=2$ is a quadrupole field, $n=3$ is a sextupole field, and so on. A solenoid
field has $C_{n}=0$ for all $n$, and $B_{z}$ non-zero; usually, a solenoid
field is not considered a multipole field, and we assume (unless stated
otherwise) that $B_{z}=0$ in a multipole magnet. Note that we can apply the
principle of superposition to deduce that a more general magnetic field can be
constructed by adding together a set of multipole fields:
$B_{y}+iB_{x}=\sum_{n=1}^{\infty}C_{n}(x+iy)^{n-1}.$ (23)
A “pure” multipole field of order $n$ has $C_{n}\neq 0$ for only that one
value of $n$.
The coefficients $C_{n}$ in Eq. 23 characterise the strength and orientation
of each multipole component in a two-dimensional magnetic field. It is
sometimes more convenient to express the field using polar coordinates, rather
than Cartesian coordinates. Writing $x=r\cos\theta$ and $y=r\sin\theta$, we
see that Eq. (23) becomes:
$B_{y}+iB_{x}=\sum_{n=1}^{\infty}C_{n}r^{n-1}e^{i(n-1)\theta}.$
By writing the multipole expansion in this form, we see immediately that the
strength of the field in a pure multipole of order $n$ varies as $r^{n-1}$
with distance from the magnetic axis. We can go a stage further, and express
the field in terms of polar components:
$B_{y}+iB_{x}=B_{r}\sin\theta+B_{\theta}\cos\theta+iB_{r}\cos\theta-
iB_{\theta}\sin\theta=\left(B_{\theta}+iB_{r}\right)e^{-i\theta},$
thus:
$B_{\theta}+iB_{r}=\sum_{n=1}^{\infty}C_{n}r^{n-1}e^{in\theta}.$ (24)
By writing the field in this form, we see that for a pure multipole of order
$n$, rotation of the magnet through $\pi/n$ around the $z$ axis simply changes
the sign of the field. We also see that if we write:
$C_{n}=\left|C_{n}\right|\,e^{in\phi_{n}}$
then the value of $\phi_{n}$ (the phase of $C_{n}$) determines the orientation
of the field. Conventionally, a pure multipole with $\phi_{n}=0$ is known as a
“normal” multipole, while a pure multipole with $\phi_{n}=\pi/2$ is known as a
“skew” multipole.
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Figure 4: “Pure” multipole fields. Top: dipole. Middle: quadrupole. Bottom:
sextupole. Fields on the left are normal ($a_{n}$ positive); those on the
right are skew ($b_{n}$ positive). The positive $y$ axis is vertically up; the
positive $x$ axis is horizontal and to the right.
The units of $C_{n}$ depend on the order of the multipole. In SI units, for a
dipole, the units of $C_{1}$ are tesla (T); for a quadrupole, the units of
$C_{2}$ are Tm-1; for a sextupole, the units of $C_{3}$ are Tm-2, and so on.
It is sometimes preferred to specify multipole components in dimensionless
units. In that case, we introduce a reference field, $B_{\textrm{ref}}$, and a
reference radius, $R_{\textrm{ref}}$. The multipole expansion is then written:
$B_{y}+iB_{x}=B_{\textrm{ref}}\sum_{n=1}^{\infty}(a_{n}+ib_{n})\left(\frac{x+iy}{R_{\textrm{ref}}}\right)^{n-1}.$
(25)
This is a standard notation for multipole fields, see for example [7]. In
polar coordinates:
$B_{y}+iB_{x}=B_{\textrm{ref}}\sum_{n=1}^{\infty}(a_{n}+ib_{n})\left(\frac{r}{R_{\textrm{ref}}}\right)^{n-1}e^{i(n-1)\theta}.$
(26)
The reference field and reference radius can be chosen arbitrarily, but must
be specified if the coefficients $a_{n}$ and $b_{n}$ are to be interpreted
fully.
Note that for a pure multipole field of order $n$, the coefficients $a_{n}$
and $b_{n}$ are related to the derivates of the field components with respect
to the $x$ and $y$ coordinates. Thus, for a normal multipole:
$\frac{\partial^{n-1}B_{y}}{\partial
x^{n-1}}=(n-1)!\frac{B_{\textrm{ref}}}{R_{\textrm{ref}}^{n-1}}a_{n},$
and for a skew multipole:
$\frac{\partial^{n-1}B_{x}}{\partial
x^{n-1}}=(n-1)!\frac{B_{\textrm{ref}}}{R_{\textrm{ref}}^{n-1}}b_{n}.$
A normal dipole has a uniform vertical field; a normal quadrupole has a
vertical field for $y=0$, that increases linearly with $x$; a normal sextupole
has a vertical field for $y=0$ that increases as the square of $x$; and so on.
## 0.4 Generating multipole fields
Given a system of electric charges and currents, we can integrate Maxwell’s
equations to find the electric and magnetic fields generated by those charges
and currents. In general, the integration must be done numerically; but for
simple systems it is possible to find analytical solutions. We considered two
such cases in Section 0.2: the electric field around an isolated point charge,
and the magnetic field around a long straight wire carrying a constant
current.
It turns out that we can combine the magnetic fields from long, straight,
parallel wires to generate pure multipole fields. It is also possible to
generate pure multipole fields using high-permeability materials with the
appropriate geometry. We consider both methods in this section. For the
moment, we deal with “idealised” geometries without practical constraints. We
discuss the impact of some of the practical limitations in later sections.
### 0.4.1 Current distribution for a multipole field
Our goal is to determine a current distribution that will generate a pure
multipole field of specified order. As a first step, we derive the multipole
components in the field around a long straight wire carrying a uniform
current. We already know, from Ampère’s law (9) that the field at distance $r$
from a long straight wire carrying current $I$ in free space has magnitude
given by:
$B=\frac{I}{2\pi\mu_{0}r},$
and that the direction of the field describes a circle centred on the wire. To
derive the multipole components in the field, we first derive an expression
for the field components at an arbitrary point $(x,y)$ from a wire carrying
current $I$, passing through a point $(x_{0},y_{0})$ and parallel to the $z$
axis.
Since we are working in two dimensions, we can represent the components of a
vector by the real and imaginary parts of a complex number. Thus, the vector
from $(x_{0},y_{0})$ to a point $(x,y)$ is given by
$re^{i\theta}-r_{0}e^{i\theta_{0}}$, and the magnitude of the field at $(x,y)$
is:
$B=\frac{I}{2\pi\mu_{0}}\,\frac{1}{\left|re^{i\theta}-r_{0}e^{i\theta_{0}}\right|}.$
The geometry is shown in Fig. 5. The direction of the field is perpendicular
to the line from $(x_{0},y_{0})$ to $(x,y)$. Since a rotation through 90∘ can
be represented by a multiplication by $i$, we can write:
$B_{x}+iB_{y}=\frac{I}{2\pi\mu_{0}}\,\frac{i\left(re^{i\theta}-r_{0}e^{i\theta_{0}}\right)}{\left|re^{i\theta}-r_{0}e^{i\theta_{0}}\right|^{2}},$
and hence:
$B_{y}+iB_{x}=\frac{I}{2\pi\mu_{0}}\,\frac{\left(re^{-i\theta}-r_{0}e^{-i\theta_{0}}\right)}{\left|re^{i\theta}-r_{0}e^{i\theta_{0}}\right|^{2}}=\frac{I}{2\pi\mu_{0}}\,\frac{1}{re^{i\theta}-r_{0}e^{i\theta_{0}}}.$
Figure 5: Geometry for calculation of multipole components in the field around
a long, straight wire carrying a uniform current. The wire passes through
$r_{0}e^{i\theta_{0}}$, and is parallel to the $z$ axis (the direction of the
current is pointing out of the page).
Now, we write the magnetic field as:
$B_{y}+iB_{x}=-\frac{I}{2\pi\mu_{0}r_{0}}\,\frac{e^{-i\theta_{0}}}{1-\frac{r}{r_{0}}e^{i(\theta-\theta_{0})}}.$
and use the Taylor series expansion for $(1-\zeta)^{-1}$, where $\zeta$ is a
complex number with $|\zeta|<1$:
$\frac{1}{1-\zeta}=\sum_{n=0}^{\infty}\zeta^{n},$
to write:
$B_{y}+iB_{x}=-\frac{I}{2\pi\mu_{0}r_{0}}\,e^{-i\theta_{0}}\,\sum_{n=1}^{\infty}\left(\frac{r}{r_{0}}\right)^{n-1}e^{i(n-1)(\theta-\theta_{0})}.$
(27)
Eq. (27) is valid for $r<r_{0}$. Comparing with the standard multipole
expansion, Eq. (26), we see that if we choose for the reference field
$B_{\textrm{ref}}$ and the reference radius $R_{\textrm{ref}}$:
$\displaystyle B_{\textrm{ref}}$ $\displaystyle=$
$\displaystyle\frac{I}{2\pi\mu_{0}r_{0}},$ $\displaystyle R_{\textrm{ref}}$
$\displaystyle=$ $\displaystyle r_{0},$
then the coefficients for the multipole components in the field are given by:
$b_{n}+ia_{n}=-e^{-in\theta_{0}}.$
The field around a long straight wire can be represented as an infinite sum
over all multipoles.
Now we consider a current flowing on the surface of a cylinder of radius
$r_{0}$. Suppose that the current flowing in a section of the cylinder at
angle $\theta_{0}$ and subtending angle $d\theta_{0}$ at the origin is
$I\\!(\theta_{0})\,d\theta_{0}$. By the principle of superposition, we can
obtain the total field by summing the contributions from the currents at all
values of $\theta_{0}$:
$B_{y}+iB_{x}=-\frac{1}{2\pi\mu_{0}r_{0}}\,\sum_{n=1}^{\infty}\left(\frac{r}{r_{0}}\right)^{n-1}e^{i(n-1)\theta}\int_{0}^{2\pi}e^{-in\theta_{0}}\,I\\!(\theta_{0})\,d\theta_{0}.$
(28)
We see that the multipole components are related to the Fourier components in
the current distribution over the cylinder of radius $r_{0}$. In particular,
if we consider a current distribution with just a single Fourier component:
$I\\!(\theta_{0})=I_{0}\cos\left(n_{0}\theta_{0}-\phi\right),$ (29)
the integral in the right hand side of Eq. (28) vanishes except for $n=n_{0}$,
and we find:
$B_{y}+iB_{x}=-\frac{I_{0}}{2\pi\mu_{0}r_{0}}\,\left(\frac{r}{r_{0}}\right)^{n_{0}-1}e^{i(n_{0}-1)\theta}\pi
e^{-i\phi}.$
The current distribution (29) generates a pure multipole field of order
$n_{0}$. If we choose, as before:
$\displaystyle B_{\textrm{ref}}$ $\displaystyle=$
$\displaystyle\frac{I_{0}}{2\pi\mu_{0}r_{0}},$ $\displaystyle
R_{\textrm{ref}}$ $\displaystyle=$ $\displaystyle r_{0},$
then the multipole coefficients are:
$b_{n}+ia_{n}=-\pi e^{-i\phi}.$
The parameter $\phi$ gives the “angle” of the current distribution. For
$\phi=0$ or $\phi=\pi$, the current generates a normal multipole; for
$\phi=\pm\pi/2$, the current generates a skew multipole.
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Figure 6: Current distributions for generating pure multipole fields. Top:
dipole. Middle: quadrupole. Bottom: sextupole. Fields on the left are normal
($a_{n}$ positive); those on the right are skew ($b_{n}$ positive). The
positive $y$ axis is vertically up; the positive $x$ axis is horizontal and to
the right. The deviation of the red line from the circular boundary shows the
local current density. Current is flowing in the positive $z$ direction (out
of the page) for increased radius, and in the negative $z$ direction for
reduced radius.
The fact that a sinusoidal current distribution on a cylinder can generate a
pure multipole field is not simply of academic interest. By winding wires in
an appropriate pattern on a cylinder, it is possible to approximate a
sinusoidal current distribution closely enough to produce a multipole field of
acceptable quality for many applications. Usually, several layers of windings
are used with a different pattern of wires in each layer, to improve the
approximation to a sinusoidal current distribution. Superconducting wires can
be used to achieve strong fields: an example of superconducting quadrupoles in
the LHC is shown in Fig. 7.
Figure 7: Superconducting quadrupoles in the LHC.
### 0.4.2 Geometry of an iron-dominated multipole magnet
Normal-conducting magnets usually use iron cores to increase the flux density
achieved by a given current. In such a magnet, the shape of the magnetic field
depends mainly on the geometry of the iron. In this section, we shall derive
the geometry required to generate a pure multipole of given order. To simplify
the problem, we will make some approximations: in particular, we shall assume
that the iron core has uniform cross-section and infinite extent along $z$;
that there are no limits to the iron in $x$ or $y$; and that the iron has
infinite permeability. The field in a more realistic magnet will generally
need to be calculated numerically; however, the characteristics derived from
our idealized model are often a good starting point for the design of an iron-
dominated multipole magnet.
We base our analysis on the magnetic scalar potential, $\varphi$, which is
related to the magnetic field $\vec{B}$ by:
$\vec{B}=-\textrm{grad}\,\varphi.$ (30)
Note that the curl of the field in this case is zero, for any function
$\varphi$: this is a consequence of the mathematical properties of the grad
and curl operators. Therefore, it follows from Maxwell’s equation (3) that a
magnetic field can only be derived from a scalar potential if: (i) there is no
current density at the location where the field is to be calculated; (ii)
there is no time-dependent electric displacement at the location where the
field is to be calculated. Where there exists an electric current or a time-
dependent electric field, it is more appropriate to use a vector potential (in
which case, the magnetic flux density is found from the curl of the vector
potential). However, for multipole fields, we have already shown that both the
divergence and the curl of the field vanish, Eqs. (21) and (22). Since the
curl of the grad of any function is identically zero, Eq. (22) is
automatically satisfied for any field $\vec{B}$ derived using (30). From Eq.
(21), we find:
$\nabla^{2}\varphi=0,$ (31)
where $\nabla^{2}$ is the laplacian operator. Eq. (31) is Poisson’s equation:
the scalar potential in a particular case is found by solving this equation
with given boundary conditions.
To determine the geometry of iron required to generate a pure multipole field,
we shall start by writing down the scalar potential for a pure multipole
field. Since the magnetic flux density $\vec{B}$ is obtained from the gradient
of the scalar potential, the flux density at any point must be perpendicular
to a surface of constant scalar potential. However, we already know, from Eq.
(14), that the magnetic flux density at the surface of a material with
infinite permeability must be perpendicular to that surface. Hence, to
generate a pure multipole field in a magnet containing material of infinite
permeability, we just need to shape the material so that its surface follows a
surface of constant magnetic scalar potential for the required field.
We therefore look for a potential $\varphi$ that satisfies:
$-\left(\frac{\partial}{\partial y}+i\frac{\partial}{\partial
x}\right)\varphi=B_{y}+iB_{x}=C_{n}(x+iy)^{n-1}.$
As we shall now show, an appropriate solution is:
$\varphi=-\left|C_{n}\right|\frac{r^{n}}{n}\sin(n\theta-\phi_{n})$ (32)
where:
$x+iy=re^{i\theta},$
and so:
$\displaystyle x$ $\displaystyle=$ $\displaystyle r\cos\theta,$ $\displaystyle
y$ $\displaystyle=$ $\displaystyle r\sin\theta.$
That Eq. (32) is indeed the potential for a pure multipole of order $n$ can be
shown as follows. In polar coordinates, the gradient can be written:
$\textrm{grad}\,\varphi=\hat{r}\frac{\partial\varphi}{\partial
r}+\frac{\hat{\theta}}{r}\frac{\partial\varphi}{\partial\theta},$ (33)
where $\hat{r}$ and $\hat{\theta}$ are unit vectors in the directions of
increasing $r$ and $\theta$, respectively. Using:
$\displaystyle\hat{r}$ $\displaystyle=$
$\displaystyle\hat{x}\cos\theta+\hat{y}\sin\theta,$
$\displaystyle\hat{\theta}$ $\displaystyle=$
$\displaystyle-\hat{x}\sin\theta+\hat{y}\cos\theta,$
it follows from Eq. (33) that:
$\displaystyle-\textrm{grad}\,\varphi$ $\displaystyle=$
$\displaystyle(\hat{x}\cos\theta+\hat{y}\sin\theta)\left|C_{n}\right|r^{n-1}\sin(n\theta-\phi_{n})-(\hat{x}\sin\theta-\hat{y}\cos\theta)\left|C_{n}\right|r^{n-1}\cos(n\theta-\phi_{n}),$
$\displaystyle=$
$\displaystyle\hat{x}\sin\left((n-1)\theta-\phi_{n}\right)\left|C_{n}\right|r^{n-1}+\hat{y}\cos\left((n-1)\theta-\phi_{n}\right)\left|C_{n}\right|r^{n-1}.$
Thus, the field derived from the potential (32) can be written:
$B_{y}+iB_{x}=\left|C_{n}\right|e^{-i\phi_{n}}r^{n-1}e^{i(n-1)\theta}.$
Therefore, if:
$C_{n}=\left|C_{n}\right|e^{-i\phi_{n}},$
then:
$B_{y}+iB_{x}=C_{n}r^{n-1}e^{i(n-1)\theta}=C_{n}(x+iy)^{n-1},$
and we see that the potential (32) does indeed generate a pure multipole field
of order $n$.
From the above argument, we can immediately conclude that to generate a pure
multipole field, we can shape a high permeability material such that the
surface of the material follows the curve given (in parametric form, with
parameter $\theta$) by:
$r^{n}\sin(n\theta-\phi_{n})=r_{0}^{n},$ (34)
where $r_{0}$ is a constant giving the minimum distance between the surface of
the material and the origin. The cross-sections of iron-dominated multipole
magnets of orders 1, 2 and 3 are shown in Fig. 8. Note that $r\to\infty$ for
$n\theta-\phi_{n}\to\textrm{integer}\times\pi$. Treating each region between
infinite values of $r$ as a separate pole, we see that a pure multipole of
order $n$ has 2$n$ poles. We also see that the potential changes sign when
moving from one pole to either adjacent pole: that is, poles alternate between
“north” and “south”. The field must be generated by currents flowing along
wires between the poles, parallel to the $z$ axis: to avoid direct
contribution from the field around the wires, these wires should be located a
large (in fact, infinite) distance from the origin.
|
---|---
|
|
Figure 8: Pole shapes for generating pure multipole fields. Top: dipole.
Middle: quadrupole. Bottom: sextupole. Fields on the left are normal ($a_{n}$
positive); those on the right are skew ($b_{n}$ positive). The positive $y$
axis is vertically up; the positive $x$ axis is horizontal and to the right.
The poles, shown as black (north) or grey (south), are constructed from
material with infinite permeability.
Note that it is possible to determine the shape of the pole face for a magnet
containing any specified set of multipoles, by summing the potentials for the
different multipole components, and then solving for $r$ as a function of
$\theta$, for a fixed value of the scalar potential. Magnets designed to have
more than one multipole component are often known as “combined function”
magnets. Perhaps the most common type of combined function magnet is a dipole
with a quadrupole component: such magnets can be used to steer and focus a
beam simultaneously. The shape of the pole faces and the field lines in a
dipole with (strong) quadrupole component is shown in Fig. 9.
Figure 9: Pole shapes for dipole magnet with additional quadrupole component.
In practice, some variation from the “ideal” geometry is needed, to account
for the fact that the material used in the magnet has finite permeability, and
finite extent transversely and longitudinally. The wires carrying the current
that generates the magnetic flux are arranged in coils around each pole; as we
shall see, the strength of the field is determined by the number of ampere-
turns in each coil. An iron-dominated electromagnetic quadrupole is shown in
Fig. 10.
Figure 10: Iron-dominated quadrupole magnet for the EMMA Fixed-Field
Alternating Gradient accelerator at Daresbury Laboratory. Left: magnet cross-
section [8]. Right: magnet prototype [9].
To complete our discussion of methods to generate multipole fields, we derive
an expression for the field strength in an iron dominated magnet with a given
number of ampere-turns in the coil around each pole. To do this, we consider a
line integral as shown in Fig. 11. In the figure, we show a quadrupole;
however the generalisation to other orders of multipole is straightforward.
Note that, in principle, the coils carrying the electric current, and the line
segment $C_{3}$, are an infinite distance from the origin (the centre of the
magnet).
Figure 11: Contour for line integral used to calculate the field strength in
an iron-dominated quadrupole.
Using Maxwell’s equation (3), with constant (zero) electric displacement, and
integrating over the surface $S$ bounded by the curve $C_{1}+C_{2}+C_{3}$
gives:
$\int_{S}\textrm{curl}\,\vec{H}\cdot d\vec{S}=\int_{S}\vec{J}\cdot
d\vec{S}=-NI.$
Note that the surface is oriented so that the normal is parallel to the
positive $z$ axis; and the coil around each pole consists of $N$ turns of wire
carrying current $I$. Applying Stokes’ theorem (7) gives:
$\int_{C_{1}}\vec{H}\cdot d\vec{l}+\int_{C_{2}}\vec{H}\cdot
d\vec{l}+\int_{C_{3}}\vec{H}\cdot d\vec{l}=-NI.$
We know, from Eq. (12), that the normal component of the magnetic flux density
$\vec{B}$ is continuous across a boundary. Then, since $\vec{B}=\mu\vec{H}$,
it follows that for a finite field between the poles, and for $\mu\to\infty$,
the magnetic intensity $\vec{H}$ vanishes within the poles. Also, the field is
perpendicular to the line segment $C_{2}$. Thus, the only part of the integral
that makes a non-zero contribution, is the integral along $C_{1}$ from the
face of the pole to the origin. Hence:
$\int_{0}^{r_{0}}\frac{B_{r}(r)}{\mu_{0}}dr=NI.$ (35)
The contour $C_{1}$ is chosen so that along this contour, the field has only a
radial component, parallel to the contour. From Eq. (24), we see that for a
multipole of order $n$, along this contour we have:
$B_{r}=\left|C_{n}\right|r^{n-1}.$
Thus, we find by performing the integral in Eq. (35):
$\left|C_{n}\right|=\mu_{0}NI\frac{n}{r_{0}^{n}}.$
For a normal multipole, the field is given by:
$B_{y}+iB_{x}=\frac{\mu_{0}nNI}{r_{0}}\left(\frac{x+iy}{r_{0}}\right)^{n-1}.$
For example, in a normal quadrupole ($n=2$), the field gradient is given by:
$\frac{\partial B_{y}}{\partial x}=\frac{2\mu_{0}NI}{r_{0}^{2}}.$ (36)
## 0.5 Multipole decomposition
In the previous section, we derived the current density distributions and
material geometries needed to generate a pure multipole field of a given
order. However, the distributions and geometries required are not perfectly
achievable in practice: the currents and materials have infinite longitudinal
extent; and we require either a current that exists purely on the surface of a
cylinder, or infinite permeability materials with infinite transverse extent.
Real multipole magnets, therefore, will not consist of a single multipole
component, but a superposition of (in general) an infinite number of multipole
fields. The exact shape of the field can have a significant impact on the beam
dynamics in an accelerator. In many simulation codes for accelerator beam
dynamics, the magnets are specified by the multipole coefficients: this is
because simple techniques exist for approximating the effect, for example, of
sextupole, octopule, and other higher-order components in the field of a
quadrupole magnet. The question then arises how to determine the multipole
components in a given magnetic field.
At this point, we can make a distinction between the _design_ field of a
magnet, and the field that exists within a fabricated magnet. The design field
is one that is still in some sense “ideal”; though the design field for a
quadrupole magnet (for example) will contain other multipole components,
because the design has to respect practical constraints, i.e. the magnet will
have finite longitudinal and transverse extent, any currents will flow in
wires of non-zero dimension, and any materials present will have finite (and
often non-linear) permeability. Usually, one attempts to optimize the design
to minimize the strengths of the multipole components apart from the one
required: the residual strengths are generally known as _systematic_ multipole
errors. These errors will be present in any fabricated magnet, although,
because of construction tolerances, the errors will vary between any two
magnets of the same nominal design. The differences between the multipole
components in the design field and the components in a particular magnet are
known as _random_ multipole errors.
The effects of systematic and random multipole errors on an accelerator, and
hence the specification of upper limits on these quantities, can usually only
be properly understood by running beam dynamics simulations. Therefore,
accelerator magnet (and lattice) design often proceeds iteratively. Some
initial estimate of the limits on the errors is often needed to guide the
magnet design; but then any design that is developed must be studied by
further beam dynamics simulations to determine whether improvements are
needed.
It is therefore important to be able to determine the multipole components in
a magnetic field from numerical field data: these data may come from either a
magnetic model (i.e. from the design of a magnet), or from measurements on a
real device. There are different procedures that can be used to achieve the
“decomposition” of a field into its multipole components. In this section, we
shall consider methods based on Cartesian and polar representations of two-
dimensional fields (i.e. fields that are independent of the longitudinal
coordinate). In Section 0.6 we shall consider decompositions of three-
dimensional fields (i.e. fields that have explicit dependence on longitudinal
as well as transverse coordinates). However, we first consider an important
concept in the discussion of multipole field errors, namely how the symmetry
of a multipole magnet leads to “allowed” and “forbidden” higher-order
multipoles.
### 0.5.1 Multipole symmetry, “allowed” and “forbidden” higher-order
multipoles
A pure multipole field of order $n$ can be written:
$B_{y}+iB_{x}=\left|C_{n}\right|e^{-i\phi_{n}}r^{n-1}e^{i(n-1)\theta}.$ (37)
The parameter $\phi_{n}$ characterises the angular orientation of the magnet
around the $z$ axis. In particular, from Eq. (34), we see that a change in
$\phi_{n}$ by $n\alpha$ is equivalent to a rotation of the coordinates (a
change in $\theta$) by $-\alpha$. Thus, a rotation of a magnet around the $z$
axis by angle $\alpha$ may be represented by a change in $\phi_{n}$ by
$n\alpha$. In particular, if the magnet is rotated by $\pi/n$, then from Eq.
(37), we see that the field at any point simply changes sign:
$\textrm{if }\phi_{n}\mapsto\phi_{n}+\pi,\textrm{ then
}\vec{B}\mapsto-\vec{B}.$ (38)
This property of the magnetic field is imposed by the symmetry of the magnet.
In a real magnet, it will not be satisfied exactly, because random variations
in the geometry will break the symmetry. However, it is possible to maintain
the symmetry exactly in the design of the magnet; this means that although
higher order multipoles will in general be present, only those multipoles
satisfying the symmetry constraint (38) can be present. These are the
“allowed” multipoles. Other multipoles, which must be completely absent, are
the “forbidden” multipoles.
We can derive a simple expression for the allowed multipoles in a magnet
designed with symmetry for a multipole of order $n$. Consider an additional
multipole (a “systematic error”) in this field, of order $m$. By the principle
of superposition, the total field can be written as:
$B_{y}+iB_{x}=\left|C_{n}\right|e^{-i\phi_{n}}r^{n-1}e^{i(n-1)\theta}+\left|C_{m}\right|e^{-i\phi_{m}}r^{m-1}e^{i(m-1)\theta}.$
The geometry is such that under a rotation about the $z$ axis through $\pi/n$,
the magnet looks the same, except that all currents have reversed direction:
therefore the field simply changes sign. Under this rotation
$\phi_{n}\mapsto\phi_{n}+\pi$; however, $\phi_{m}\mapsto\phi_{m}+m\pi/n$. This
means that we must have:
$e^{-i\frac{m}{n}\pi}=-1.$
Therefore, $m/n$ must be an odd integer. Assuming that $m\neq n$ (i.e. the
multipole error is of a different order than the “main” multipole field),
then:
$\frac{m}{n}=3,5,7,\dots$ (39)
Thus, for a dipole, the allowed higher order multipoles are sextupole,
decapole, etc.; for a quadrupole, the allowed higher order multipoles are
dodecapole, 20-pole, etc. The fact that the allowed higher order multipoles
have an order given by an odd integer multiplied by the order of the main
multipole is a consequence of the fact that magnetic poles always occur in
north-south pairs. This is illustrated for a quadrupole in Fig. 12; here, we
see that to maintain the correct rotational symmetry (with the field changing
sign under a rotation through $\pi/2$) the first higher-order multipole must
be constructed by “splitting” each main pole into three, then into five, and
so on.
Figure 12: Normal quadrupole field (left) and dodecapole field (right). The
dodecapole is the first higher-order multipole with the same rotational
symmetry as the quadrupole (under a rotation by $\pi/2$, north and south poles
interchange).
The field in a real magnet will contain all higher order multipoles, not just
the ones allowed by symmetry. However, it is often the case that the allowed
multipoles dominate over the forbidden multipoles.
### 0.5.2 Fitting multipoles: Cartesian basis
Suppose we have obtained a set of numerical field data, either from a magnetic
model, or from measurements on a real magnet. To determine the effect of the
field on the beam dynamics in an accelerator, it is helpful to know the
multipole components in the field. One way to compute the multipole components
is to fit a polynomial to the field data. For example, if we consider a normal
multipole (coefficients $C_{n}$ are all real), the vertical field along the
$x$ axis (i.e. for $y=0$) is given by:
$B_{y}=\sum_{n=1}^{\infty}C_{n}x^{n-1}.$ (40)
The number of data points determines the highest order multipole that can be
fitted. Fitting may be achieved using, for example, a routine that minimises
the squares of the residuals between the data and the fitted function.
However, although this procedure can, in principle, produce good results, it
is not very robust. In particular, the presence of multipoles of higher order
than those included in the fit can affect the values determined for those
multipoles that are included in the fit. We can illustrate this as follows.
Let us construct a quadrupole field ($n=$2), and add to it higher order
multipoles of order 3, 4, 5 and 6. The values of the coefficients $a_{n}$
(actual values, and fitted values in two different cases) are given in Table
1. The field $B_{y}/B_{\textrm{ref}}$ is plotted as a function of
$x/R_{\textrm{ref}}$ in Fig. 13: the field data are shown as points, while the
fit, including multipoles up to order 6, is shown as a line. Also shown is the
deviation $\Delta B_{y}/B_{\textrm{ref}}$ from an ideal quadrupole field, i.e.
$\Delta B_{y}$ is the contribution of the higher order multipoles. We see that
if we base the fit on all the multipoles that are present (i.e. up to order
6), then we obtain accurate values for all multipole coefficients.
Table 1: Actual and fitted multipole values for a quadrupole field with artificially constructed multipole errors. $n$ | actual coefficient $a_{n}$ | fitted coefficient $a_{n}$
---|---|---
| | $(n\leq 6)$ | $(n\leq 5)$
2 | 1.000 | 1.000 | 0.9972
3 | 0.010 | 0.010 | 0.0100
4 | 0.001 | 0.001 | 0.0131
5 | 0.010 | 0.010 | 0.0100
6 | 0.010 | 0.010 | —
Figure 13: Measured (points) and fitted (line) field in a quadrupole with
higher-order multipole errors of order 3, 4, 5 and 6. Multipoles up to order 6
are fitted. Left: total field. Right: deviation from quadrupole field.
However, in general, multipoles of all orders are present, while our fit is
based on a finite number of multipoles. If we try to fit the data in our
illustrative case using multipoles up to order 5 only (i.e. omitting the order
6 multipole that is present), then we see that there is an impact on the
accuracy with which we determine the lower-order multipoles. This can be seen
in the final column of Table 1: there is even an error in the value that we
determine for the quadrupole strength. When we plot the fit against the field
data, we see that there is some small residual deviation between the data and
the fit: this is to be expected, since the function we are using to obtain the
fit does not match exactly the function used to generate the data. Although
not visible in the total field, plotted in Fig. 14, the difference between the
fit and the data is apparent in the plot of the deviation from the quadrupole
field.
Figure 14: Measured (points) and fitted (line) field in a quadrupole with
higher-order multipole errors of order 3, 4, 5 and 6. Multipoles up to order 5
are fitted. Left: total field. Right: deviation from quadrupole field.
Our concern is that the presence of higher-order multipoles has affected the
accuracy with which we determine the lower-order multipoles, even down to the
quadrupole field strength. This can have significant implications for beam
dynamics: the effect of a linear focusing error in a beam line (from some
variation in the quadrupole strength) can be very different from the effects
of higher-order multipole errors. For example, if one is measuring the
betatron tunes or the beta functions in a storage ring, these can be very
sensitive to linear focusing errors, and relatively insensitive to higher
order multipoles. Determining the multipole coefficients using a polynomial
fit can lead to inaccurate predictions of the linear behaviour of the beam
line, depending on the higher-order multipoles present in the magnets.
The problem is that we have based our fit on monomials, i.e. powers of $x$.
Our fit is a sum of these monomials, with coefficients determined from the
data. However, it is possible to obtain a fit to data generated using one
monomial, with a different monomial. For example, if one constructs data which
is purely linear in $x$, then one can obtain a fit using a monomial $x^{3}$
(even though the fit will not be as good as one obtained using a monomial
$x$). Mathematically, the basis functions we are using (monomials in this
case) are not orthogonal: the coefficients we determine depend on the which
set of basis functions we choose to use. A more robust technique would use
basis functions that are orthogonal, i.e. the coefficients we determine will
be the same, no matter which set of functions we choose. Fortunately, there
exists an appropriate set of functions that provides an orthogonal basis for
multipole fields. We discuss this basis in the following section, 0.5.3. The
advantage of orthogonal basis functions is that the coefficients we determine
for different terms in the fit are _independent_ of which terms we include in
the fit; for example, the quadrupole strength that we find in a particular
magnet will be the same, irrespective of which higher-order terms we include
in the fit, and which higher-order terms are actually present.
### 0.5.3 Fitting multipoles: polar basis
From Eq. (24) we know that the field in a multipole magnet can be written in
polar coordinates as:
$B_{\theta}+iB_{r}=\sum_{n=1}^{\infty}C_{n}r^{n-1}e^{in\theta}.$
We see that if we make a set of measurements of $B_{r}$ and $B_{\theta}$ at
different values of $\theta$ and fixed radial distance $r$, then we can obtain
the coefficients $C_{n}$ by a discrete Fourier transform.
Suppose we make $M$ measurements of the field, at $\theta=\theta_{m}$, where:
$\theta_{m}=2\pi\frac{m}{M},\qquad m=0,1,2\dots M-1.$ (41)
We write the measurement at $\theta=\theta_{m}$ as $B_{m}$; note that $B_{m}$
is a complex number, whose real and imaginary parts are given by the azimuthal
and radial components of the field at $\theta=\theta_{m}$.
Now we construct, for a chosen integer $n^{\prime}$:
$\sum_{m=0}^{M-1}B_{m}e^{-2\pi
in^{\prime}\frac{m}{M}}=\sum_{m=0}^{M-1}\sum_{n=1}^{\infty}C_{n}r_{0}^{n-1}e^{2\pi
i(n-n^{\prime})\frac{m}{M}},$
where $r_{0}$ is the radial distance at which the field measurements are made.
The summation over $m$ on the right hand side vanishes, unless $n=n^{\prime}$.
Thus, we can write:
$\sum_{m=0}^{M-1}B_{m}e^{-2\pi
in^{\prime}\frac{m}{M}}=MC_{n^{\prime}}r_{0}^{n^{\prime}-1}.$
If we relabel $n^{\prime}$ as $n$, then we see that the multipole coefficients
$C_{n}$ are given by:
$C_{n}=\frac{1}{Mr_{0}^{n-1}}\sum_{m=0}^{M-1}B_{m}e^{-2\pi in\frac{m}{M}}.$
(42)
The advantage of this technique over that in section 0.5.2 is that the basis
functions used to construct the fit are of the form $e^{in\theta}$, for
integer $n$. These functions are orthogonal: mathematically, this means that:
$\int_{0}^{2\pi}e^{in\theta}e^{-in^{\prime}\theta}\,d\theta=2\pi\delta_{nn^{\prime}},$
where the Kronecker delta function $\delta_{nn^{\prime}}=1$ if $n=n^{\prime}$,
and $\delta_{nn^{\prime}}=0$ if $n\neq n^{\prime}$. The important consequence
for us, is that the value we determine for any given multipole using Eq. (42)
is independent of the presence of any other multipoles, of higher or lower
order.
A further advantage of using the polar basis instead of the Cartesian basis,
comes from the dependence of the field on the radial distance. Suppose that
the field data are measured (or obtained from a model) with accuracy $\Delta
B_{m}$. Then the accuracy in the multipole coefficients will be:
$\Delta C_{n}\approx\frac{\Delta B_{m}}{r_{0}^{n-1}}.$
We obtain better accuracy in the multipole coefficients if we choose the
radius $r_{0}$ on which the measurements are made, to be as large as possible.
Furthermore, the accuracy in the fitted field will be:
$\Delta B\approx\Delta C_{n}\left(\frac{r}{r_{0}}\right)^{n-1}.$
We obtain _improved_ accuracy in the field for $r<r_{0}$; but the accuracy
reduces quickly (particularly for higher-order multipoles) for $r>r_{0}$. It
is important to choose the radial distance $r_{0}$ large enough to enclose all
particles likely to pass through the magnet, otherwise results from tracking
may not be accurate.
### 0.5.4 Multipole decomposition: some comments
In this section, we have considered two techniques for deriving the multipole
components of two-dimensional magnetostatic fields. We have seen that while
the multipole components can be obtained, in principle, from a simple least-
squares fit of a polynomial to the field components along one or other of the
coordinate axes, there are advantages to basing the fit on field data obtained
on a circle enclosing the origin, with as large a radius as possible. In the
next section, we shall see how the idea of a multipole expansion can be
generalised to three dimensions, and how a multipole decomposition can be
performed in that case. However, it is worth pausing to consider in a little
more detail some of the reasons for wishing to represent a field as a set of
(multipole) modes.
It is of course possible to represent a magnetic field using a set of
numerical field data, giving the three field components on points forming a
“mesh” covering the region of interest. In some ways, this is a very
convenient representation, since it is the one usually provided directly by a
magnetic modelling code: further processing is usually required to arrive at
other representations. However, while a numerical field map in two dimensions
is often a practical representation, in three dimensions the amount of data in
even a relatively simple magnet can become extremely large, especially if a
high resolution is required for the mesh. A multipole representation, on the
other hand, provides the description of a magnetic field as a relatively small
set of coefficients, from which the field components at any point can be
reconstructed, using the basis functions. In other words, a multipole
representation is more “portable” than a numerical field map.
Secondly, a representation based on a multipole expansion lends itself to
further manipulation in ways that a numerical field map does not. For example,
any noise in the data (from measurement or computational errors) can be
“smoothed” by suppression of higher-order modes. Conversely, random errors can
be introduced into data based on a model with perfect symmetry by introducing
multipole coefficients corresponding to “forbidden” harmonics. There will of
course be issues surrounding the suppression or enhancement of errors by
adjusting the multipole coefficients; however, one benefit of this approach is
that for _any_ set of multipole coefficients, the field is at least a physical
field, in the sense of satisfying Maxwell’s equations. The same will, for
example, not usually be true if a general smoothing algorithm is applied to a
numerical field map.
Finally, one of the main motivations for performing a multipole decomposition
of a field is to provide data in a format appropriate for many beam dynamics
codes. Accuracy is one criterion often important for beam dynamics codes:
efficiency is another. Characterisation of a storage ring frequently requires
tracking of thousands of particles over hundreds or thousands of turns,
through a beam line that can easily consist of hundreds of magnetic elements.
Numerical integration of the equations of motion for a particle in a numerical
field map is generally too slow to be a practical method. There are many
techniques that can be used to improve the efficiency of particle tracking in
accelerators: one of the most common is the “thin lens” method. The dynamical
effects of dipoles and quadrupoles usually need to be represented with high
accuracy. Fortunately, for these magnets, it is possible to write down
accurate solutions to the equations of motion in closed form, allowing
tracking through a magnet of given length to be performed in a single step.
The same is not true for sextupoles, or higher-order multipoles; however, it
is usually sufficiently accurate to represent such magnets by a model in which
the length of the magnet approaches zero, but where the integrated strength
(the multipole coefficient multiplied by the length) remains constant. For
such a “thin lens” it is possible to write down exact solutions to the
equations of motion, allowing tracking again to be performed in a single step.
A quadrupole with higher-order multipole errors can be represented as a “long”
perfect quadrupole field, with a set of “thin” multipoles at one end, or at
the centre. However, construction of such a model for a tracking code requires
a multipole decomposition of the field obtained from a magnet modelling code.
We should emphasise that in our discussion of multipole decomposition, here
and in sections 0.5.2 and 0.5.3, we have made no clear distinction between
field data obtained from a computational model, or from measurement of a real
magnet. Of course, it is much easier to obtain the data required from a
computational model: it is then quite straightforward to perform the required
decomposition to determine the values of the various multipole coefficients.
Unfortunately, the data do not include manufacturing errors, which can be very
important. Measurements provide more realistic data: however, many other
issues need to be addressed, including accuracy of field measurements,
alignment of the measurement instruments with respect to the magnet, etc. Such
issues are beyond the scope of our discussion.
## 0.6 Three-dimensional fields
In the previous sections, we have restricted ourselves to the case where the
magnetic field is independent of the longitudinal coordinate. The multipole
modes that we can use for such fields actually provide a good description for
many accelerator multipole magnets, even though such magnets of course have
finite length. The ends or “fringe fields” of dipoles, quadrupoles and so on,
where the field strengths often vary rapidly with longitudinal position,
cannot be accurately represented by two-dimensional fields; however, in many
accelerators, only the fringe fields of dipoles have a significant impact on
the dynamics.
However, there are cases where a full three-dimensional description of a
magnetic field is desirable, or even necessary. For example, the fields of
insertion devices (wigglers and undulators) are often represented as a
sequence of short dipoles of alternating polarity; however, where the period
becomes small compared with the aperture, the three-dimensional nature of the
field can start to have effects that cannot be ignored. There can even be
cases where “conventional” multipoles designed for special situations (for
example, where very wide aperture is required, and where the length of the
magnet needs to be short, because of space constraints) can have fringe fields
that affect the dynamics to a significant extent.
It is therefore of somewhat more than purely academic interest to consider how
two-dimensional multipole representations may be generalised to three
dimensions. As usual, there are many different ways to approach the problem:
the method that is used will often depend on the problem to be solved. In the
following sections, we describe two rather general methods that may be of use
in many situations arising in accelerators. First, we consider a field
expansion based on Cartesian modes. While this provides some nice
illustrations, the Cartesian expansion does have some disadvantages. To
address these disadvantages, we describe how a field expansion based on polar
coordinates can be performed.
### 0.6.1 Cartesian modes
Consider the field given by:
$\displaystyle B_{x}$ $\displaystyle=$ $\displaystyle-
B_{0}\frac{k_{x}}{k_{y}}\sin k_{x}x\sinh k_{y}y\sin k_{z}z,$ (43)
$\displaystyle B_{y}$ $\displaystyle=$ $\displaystyle B_{0}\cos k_{x}x\cosh
k_{y}y\sin k_{z}z,$ (44) $\displaystyle B_{z}$ $\displaystyle=$ $\displaystyle
B_{0}\frac{k_{z}}{k_{y}}\cos k_{x}x\sinh k_{y}y\cos k_{z}z.$ (45)
As may easily be verified, this field satisfies:
$\textrm{curl}\,\vec{B}=0.$
Furthermore, the equation:
$\textrm{div}\,\vec{B}=0$
is satisfied if:
$k_{y}^{2}=k_{x}^{2}+k_{z}^{2}.$ (46)
We conclude that, as long as the constraint (46) is satisfied, that the fields
(43)–(45) provide solutions to Maxwell’s equations in regions with constant
permeability, and static (or zero) electric fields. Of course, it is possible
to find similar sets of equations but with different “phase” along each of the
coordinate axes, and with the hyperbolic trigonometric function appearing for
the dependence on $x$ or $z$, rather than $y$. By superposing fields, with
appropriate variations on the form given by Eqs. (43)–(45), it is possible to
construct quite general three-dimensional magnetic fields. For example, a
slightly more general field than that given by Eqs. (43)–(45) can be obtained
simply by superposing fields with different mode numbers and amplitudes:
$\displaystyle B_{x}$ $\displaystyle=$
$\displaystyle-\int\\!\\!\\!\int\tilde{B}(k_{x},k_{z})\frac{k_{x}}{k_{y}}\sin
k_{x}x\sinh k_{y}y\sin k_{z}z\,dk_{x}\,dk_{z},$ (47) $\displaystyle B_{y}$
$\displaystyle=$ $\displaystyle\int\\!\\!\\!\int\tilde{B}(k_{x},k_{z})\cos
k_{x}x\cosh k_{y}y\sin k_{z}z\,dk_{x}\,dk_{z},$ (48) $\displaystyle B_{z}$
$\displaystyle=$
$\displaystyle\int\\!\\!\\!\int\tilde{B}(k_{x},k_{z})\frac{k_{z}}{k_{y}}\cos
k_{x}x\sinh k_{y}y\cos k_{z}z\,dk_{x}\,dk_{z}.$ (49)
In this form, we see already how to perform a mode decomposition, i.e. how we
can determine the coefficients $\tilde{B}(k_{x},k_{z})$ as functions of the
“mode numbers” $k_{x}$ and $k_{z}$. If we consider in particular the vertical
field component on the plane $y=y_{0}$, then we have from (48):
$\frac{B_{y}}{\cosh k_{y}y_{0}}=\int\\!\\!\\!\int\tilde{B}(k_{x},k_{z})\cos
k_{x}x\sin k_{z}z\,dk_{x}\,dk_{z}.$
Hence, $\tilde{B}(k_{x},k_{z})$ may be obtained from an inverse Fourier
transform of $B_{y}(x,z)/\\!\cosh k_{y}y_{0}$. Given field data on a grid over
$x$ and $z$, then we can perform numerically an inverse discrete Fourier
transform, to obtain a set of coefficients $\tilde{B}(k_{x},k_{z})$. Note that
once we have obtained these coefficients, then we can reconstruct all field
components at all points in space. This is an important consequence of the
strong constraints on the fields provided by Maxwell’s equations: in general,
for a static field, if we know how one field component varies over a two-
dimensional plane, then we can deduce how all the field components vary over
all space (on and off the plane).
Let us consider an example. To keep things simple, we shall again work with
the case where the field is independent of one coordinate: now, however, we
shall assume that the fields are independent of the horizontal transverse,
rather than the longitudinal coordinate. This may be a suitable model for a
planar wiggler or undulator with very wide poles. The model may of course be
extended to include dependence of the fields on the horizontal transverse
coordinate: although our immediate example strictly deals with a two-
dimensional field, the extension to three dimensions is quite straightforward.
Suppose that the mode amplitude function $\tilde{B}(k_{x},k_{z})$ has the
form:
$\tilde{B}(k_{x},k_{z})=\delta(k_{x})\tilde{B}(k_{z}),$ (50)
where $\delta(k_{x})$ is the Dirac delta function. The delta function has the
property that, for any function $f(k_{x})$:
$\int_{-\infty}^{\infty}\delta(k_{x})f(k_{x})\,dk_{x}=f(0).$
Using (50) in Eqs. (47) to (49) gives:
$\displaystyle B_{x}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B_{y}$ $\displaystyle=$ $\displaystyle\int\tilde{B}(k_{z})\cosh k_{z}y\sin
k_{z}z\,dk_{z},$ $\displaystyle B_{z}$ $\displaystyle=$
$\displaystyle\int\tilde{B}(k_{z})\sinh k_{z}y\cos k_{z}z\,dk_{z}.$
There is no horizontal transverse field component, and the vertical and
longitudinal field components have no dependence on $x$: we have a two-
dimensional field. In a plane defined by a particular value for the vertical
coordinate, $y=y_{0}$, the vertical field component is given by:
$B_{y}(z)=\cosh k_{z}y_{0}\int\tilde{B}(k_{z})\sin k_{z}z\,dk_{z}.$
The mode amplitude function can be obtained from a Fourier transform of the
vertical component of the magnetic field on the plane $y=y_{0}$. Usually, we
will have a finite set of field data, obtained from a magnet modelling code,
or from measurements on a real device.
Suppose that we have a data set of $2M+1$ vertical field measurements, taken
at locations:
$y=y_{0},\qquad z=\frac{m}{M}\hat{z},$ (51)
where $m$ is an integer in the range $-M\leq m\leq M$. The field at any point
is given by:
$B_{y}(y,z)=\sum_{m=-M}^{M}\tilde{B}_{m}\cosh mk_{z}y\sin mk_{z}z,$
where:
$k_{z}=\frac{2\pi}{2\hat{z}}.$
Note that in this case, the field is antisymmetric about $z=0$, i.e.
$B_{y}(y,-z)=-B_{y}(y,z).$
The mode amplitudes $\tilde{B}_{m}$ are obtained by:
$\tilde{B}_{m}=\frac{1}{\cosh
mk_{z}y_{0}}\frac{1}{2M}\sum_{m^{\prime}=-M}^{M}B_{y}(y_{0},z)\sin
m^{\prime}k_{z}z.$
Note that, because of the antisymmetry of the field:
$\tilde{B}_{-m}=-\tilde{B}_{m}.$
As a specific numerical example, let us construct an “artificial” data set
along a line $y=y_{0}=0.25$, and with $\hat{z}=3$. The data are constructed
using a function that gives a sinusoidal variation in the field along $z$ up
to $|z|<1.25$; then a continuous and smooth (continuous first derivative)
fall-off to zero field for $|z|>1.5$. For this numerical example, we do not
worry unduly about units: the reader may assume lengths in cm, fields in kG,
or any other preferred units. Initially, we take $M=40$, i.e. we assume we
have 81 measurements of the field (or, we have computed the field from a model
at 81 equally-spaced points along $z$; strictly speaking, because we are
dealing with the case that the field is antisymmetric in $z$, we need only
half this number of field measurements or computations). The field “data”, the
fitted field (reconstructed using the mode amplitudes) in $y=0.25$, and the
mode amplitudes, are shown in Fig. 15.
Figure 15: Left: Field data (points) and fit (line) in a magnet with
dependence of the field on longitudinal coordinate $z$. Right: Mode
amplitudes. The field data consist of 81 measurements (or computations) at
equally-spaced points from $z=-3$ to $z=+3$, and $y=0.25$.
It is interesting to compare with the situation where we have only 31 field
measurements or computations, i.e. $M=15$. Using the same function that we
used to construct the data set with 81 data points, we produce the fit and the
mode amplitudes shown in Fig. 16. Comparing Figs. 15 and 16, we see that in
both cases, the fitted field does pass exactly through all the data points.
This is a necessary consequence of the fit, which is based on a discrete
Fourier transform of the data points. However, using only 31 data points,
there is a significant oscillation of the fitted field between the data points
in the region $1.5<|z|<3$, where the field is actually zero (by construction).
This is a consequence of the fact that we have “truncated” some modes with
non-negligible amplitude. The mode amplitudes in both cases are the same for
mode numbers $-15\leq m\leq m$; but with 81 data points we can determine
amplitudes for a larger number of modes, which gives us a more accurate
interpolation between the data points.
Figure 16: Left: Field data (points) and fit (line) in a magnet with
dependence of the field on longitudinal coordinate $z$. Right: Mode
amplitudes. The field data consist of 31 measurements (or computations) at
equally-spaced points from $z=-3$ to $z=+3$, and $y=0.25$.
Having obtained fits to the field in the plane $y=0.25$, we can reconstruct
the field at any point, on or off the plane. It is often of interest to look
at the mid-plane; usually, this is defined by $y=0$. In this plane, we do not
have any field data. However, we can compare the field produced by the fits
with 81 data points and with 31 data points: these fields are shown in Fig.
17. We see that the fit based on 31 data points produces an essentially
identical field on $y=0$ as the fit based on 81 data points. (The data points
in each case are taken on the plane $y=0.25$). This is a consequence of the
“suppression” of higher-order modes, that arises from the hyperbolic
dependence of the field on the $y$ coordinate.
Figure 17: Field on the plane $y=0$ determined from fits to the data shown in
Figs. 15 and 16. Left: fit determined from data set with 81 data points.
Right: fit determined from data set with 31 data points.
To emphasise the significance of the hyperbolic dependence of the field on the
vertical coordinate, we can look at the variation of $B_{y}$ with $y$, for a
given value of $z$. We choose $z=0.25$, which corresponds to a peak in the
vertical field component as a function of $z$. The variation of $B_{y}$ with
$y$ for the two cases (fit based on 81 data points, and fit based on 31 data
points) is shown in Fig. 18.
Figure 18: Vertical field component as a function of $y$, for $z=0.25$. The
field is determined from fits to the data shown in Figs. 15 and 16. Left: fit
determined from data set with 81 data points. Right: fit determined from data
set with 31 data points.
Up to $y=0.25$, the two fits give essentially the same field. However, if we
try to extrapolate beyond this plane (the plane on which the fit was
performed), we see dramatically different behaviour. Fig. 19 compares the
vertical field component obtained from the two fits (81 data points and 31
data points), again at $z=0.25$, but now with a range of $y$ from 0 to 0.5. In
one case (81 data points), the field increases to a maximum before dropping
rapidly. In the other case (31 data points), the field increases monotonically
over the range. The reason for the different behaviour is the additional modes
in the fit to the set of 81 data points. These higher order modes make only a
small additional contribution to the field for $|y|<0.25$; but for values of
the vertical coordinate beyond this value, because of the hyperbolic
dependence of $y$, the contribution of these modes becomes increasingly
significant, and eventually, dominant.
Figure 19: Vertical field component as a function of $y$, for $z=0.25$. The
field is determined from fits to the data shown in Figs. 15 and 16. Left: fit
determined from data set with 81 data points. Right: fit determined from data
set with 31 data points.
The behaviour of the field fits for $|y|>0.25$ is a clear illustration of why
it is dangerous to extrapolate the fit beyond the region enclosed by the plane
of the fit. In this case, because of the symmetry in the vertical direction,
the region enclosed is between the planes $y=-0.25$ and $y=+0.25$. The “safe”
region is also bounded in $z$, by $z=-0.3$ and $z=+0.3$; because we use
discrete mode numbers in $z$, the fitted fields will in fact be periodic in
$z$, and will repeat with period $z=0.6$. In general, there will be similar
periodicity in $x$; however, in this particular example, we analysed a field
that was independent of $x$, so the “safe” region of the fit is unbounded in
$x$.
### 0.6.2 Cylindrical modes
The Caretsian modes discussed in Section 0.6.1 are often useful for describing
fields in insertion devices, particularly those that have weak variation of
the field with $x$, and periodic behaviour in $z$ (over some range): because
the modes “reflect” the geometry, it is often possible to achieve good fits to
a given field using a small number of modes. To maximise the region over which
the fit is reliable, one needs to choose a plane with a value of $y$ as large
as possible, with $x$ and $z$ extending out as far as possible on this plane.
For a planar undulator or wiggler, it is often possible to choose a plane
close to the pole tips in which $x$ in particular extends over the entire
vacuum chamber.
However, for other geometries, the Cartesian modes may not provide a
convenient basis. For example, if the magnet has a circular aperture, then the
plane that provides the largest range in $x$ is the mid-plane, $y=0$, and as
$y$ increases, the available range in $x$ decreases. To base the fit on the
Cartesian basis requires some compromise between the range of reliability in
the horizontal transverse and vertical directions.
Fortunately, it is possible to choose an alternative basis for magnets with
circular aperture, in which the field fit can be based on the surface of a
cylinder inscribed through the magnet. In that case, the radius of the
cylinder can be close to the aperture limit, maximising the range of
reliability of the fit. The appropriate modes in this case are most easily
expressed in cylindrical polar coordinates.
A field with zero divergence and curl (and hence satisfying Maxwell’s
equations for static fields in regions with uniform permeability) is given by:
$\displaystyle B_{r}$ $\displaystyle=$ $\displaystyle\int
dk_{z}\sum_{n}\tilde{B}_{n}(k_{z})\,I^{\prime}_{n}(k_{z}r)\sin n\theta\cos
k_{z}z,$ (52) $\displaystyle B_{\theta}$ $\displaystyle=$ $\displaystyle\int
dk_{z}\sum_{n}\tilde{B}_{n}(k_{z})\,\frac{n}{k_{z}r}I_{n}(k_{z}r)\cos
n\theta\cos k_{z}z,$ (53) $\displaystyle B_{z}$ $\displaystyle=$
$\displaystyle-\int dk_{z}\sum_{n}\tilde{B}_{n}(k_{z})\,I_{n}(k_{z}r)\sin
n\theta\sin k_{z}z.$ (54)
Here, $I_{n}(k_{z}r)$ is the modified Bessel function of the first kind, of
order $n$. Modified Bessel functions of the first kind for order $n=0$ to
$n=3$ are plotted in Fig. 20. For small values of the argument $\xi$, the
modified Bessel function of order $n$ has the series expansion:
$I_{n}(\xi)=\frac{\xi^{n}}{2^{n}\Gamma\\!(1+n)}+O(n+1).$ (55)
For larger values of the argument, the modified Bessel functions $I_{n}(\xi)$
increase exponentially. This is significant: it means that if we fit a field
to data on the surface of a cylinder of given radius, then residuals of the
fit will decrease exponentially within the cylinder towards $r=0$, and
increase exponentially outside the cylinder with increasing $r$. The “safe”
region of the fit will be within the cylinder.
Figure 20: Modified Bessel functions of the first kind, of order $n=0$ to
$n=3$.
Note that Eqs. (52)–(54) may be generalised to include different “phases” in
the azimuthal angle $\theta$ and the longitudinal coordinate $z$.
An attractive feature of the polar basis is that it is possible to draw a
direct connection between the three-dimensional modes in this basis and the
multipole components in a two-dimensional field. Consider a mode amplitude
$\tilde{B}_{n}(k_{z})$ given (for some particular value of $n$) by:
$\tilde{B}_{n}(k_{z})=2^{n}\Gamma\\!(1+n)C_{n}\frac{\delta(k_{z})}{nk_{z}^{n-1}},$
(56)
where $\delta()$ is the Dirac delta function, and $C_{n}$ is a constant.
Substituting these mode amplitudes into Eqs. (52)–(54), using the expansion
(55), and performing the integral over $k_{z}$ gives:
$\displaystyle B_{r}$ $\displaystyle=$ $\displaystyle\sum_{n}C_{n}r^{n-1}\sin
n\theta,$ $\displaystyle B_{\theta}$ $\displaystyle=$
$\displaystyle\sum_{n}C_{n}r^{n-1}\cos n\theta,$ $\displaystyle B_{z}$
$\displaystyle=$ $\displaystyle 0.$
Comparing with Eq. (24), we see that this is a multipole field of order $n$.
Thus, a two-dimensional multipole field is a special case of a three-
dimensional field (52)–(54), with mode coefficient given by Eq. (56).
In general, the mode coefficients $\tilde{B}_{n}(k_{z})$ may be obtained by a
Fourier transform of the field on the surface of a cylinder of given radius.
For example, it follows from Eq. (52) that:
$\frac{B_{r}}{I^{\prime}_{n}(k_{z}r)}=\int
dk_{z}\sum_{n}\tilde{B}_{n}(k_{z})\,\sin n\theta\cos k_{z}z.$
An elegant feature of the polar basis, as compared to the Cartesian basis
discussed in Section 0.6.1, is that the modes reflect the real periodicity of
the field in the angle coordinate $\theta$. In the Cartesian basis, the modes
were periodic in $x$, although the field, in general, would not have any
periodicity in $x$.
Since the mode coefficients $\tilde{B}_{n}(k_{z})$ are related to the
multipole coefficients in a two-dimensional field, we can use these
coefficients to extend the idea of a multipole to a three-dimensional field.
Strictly speaking, the mode coefficients $\tilde{B}_{n}(k_{z})$ are related to
the field by a two-dimensional Fourier transform; however, we can perform a
one-dimensional inverse Fourier transform (in the $z$ variable) to obtain a
set of functions which represent, in some sense, the “multipole components” of
a three-dimensional field as a function of $z$. Here, we use the term
“multipole components” rather loosely, since a multipole field is strictly
defined only in the two-dimensional case (i.e. for a field that is independent
of the longitudinal coordinate). A quantity that is perhaps easier to
interpret is the contribution to the field at any point made by the mode
coefficients $\tilde{B}_{n}(k_{z})$ with a given $n$. For $n=1$, the field
components $B_{r}$ and $B_{\theta}$ at any point in $z$ will behave as for a
dipole field; for $n=2$, $B_{r}$ and $B_{\theta}$ will behave as for a
quadrupole field, and so on.
As an illustrative example, we consider the field in a specific device: the
wiggler in a damping ring for TESLA (a proposed linear collider) [10]. This
wiggler has a peak field of 1.6 T and period 400 mm; the total length of
wiggler in each of the TESLA damping rings would be over 400 m. The field in
the wiggler has been studied extensively, because of concerns that dynamical
effects associated with the nonlinear components in the field would limit the
acceptance of the damping ring [11]. A model was constructed for one quarter
period of the magnet, which allowed the field at any point within the body of
the magnet to be computed. Effects associated with the ends of the wiggler
were neglected, but could in principle be included in the study. By performing
a mode decomposition using the techniques described above, it was possible to
construct an accurate dynamical model allowing fast tracking to characterise
the acceptance of the damping ring. The methods used for the dynamical
analysis are beyond the scope of the present discussion; however, we present
the results of the analysis relating directly to the field, to illustrate the
methods described in this section.
A model of the wiggler was used to compute the magnetic field on a mesh of
points bounded by a cylinder of radius 9 mm, within one quarter period of the
wiggler. Although all field components were computed on the mesh, which
covered the interior of the cylinder as well as the surface, only the radial
field component on the surface of the cylinder was used to calculate the mode
amplitudes. The fit can be validated by comparing the field “predicted” by the
fit with the field data (from the computational model) not used directly in
the fitting procedure. A fit achieved using 7 azimuthal and 100 longitudinal
modes is shown in Fig. 21. Each plot shows the variation of the vertical field
as a function of one Cartesian coordinate, with the other two coordinates
fixed at zero. In the vertical direction, the range shown is from the mid-
plane of the wiggler to close to the pole tip. Note that the variation in the
field in the transverse ($x$ and $y$) directions is very small, less than 0.1%
of the maximum field. It appears from Fig. 21, that there is very good
agreement between the fit (line) and the field data (circles) within the
cylinder on the surface of which the fit was performed.
Figure 21: Fit to the field of one quarter of one period of the TESLA damping
ring wiggler.
The quality of the fit can be further illustrated by plotting the residuals,
i.e. the difference between the fitted field and the field data. The residuals
for the vertical field component on two horizontal planes, $y=$ 0 mm and $y=$
6 mm are shown in Fig. 22. Note that to produced “smooth” surface plots, we
interpolate between the mesh points used in the computational model. On the
mid-plane of the wiggler, the residuals are less than 1 gauss (recall that the
peak field is 1.6 T); the region shown in the left hand plot in Fig. 22 lies
entirely within the surface of the cylinder used in the fit. On the plane $y=$
6 mm, the residuals are somewhat larger, and show an exponential increase for
large values of the horizontal transverse coordinate: but note that for values
of $x$ larger than about 6.7 mm, the points in the plot are _outside_ the
surface of the cylinder used for the fit. In the longitudinal direction, the
residuals appear to be dominated by very high frequency modes: this suggests
that it may be possible to reduce the residuals still further by increasing
the number of longitudinal modes used in the fit. However, this fit was
considered to be of sufficient quality to allow an accurate determination of
the effect of the wiggler on the beam dynamics to be made.
Figure 22: Residuals of the fit to the field of one quarter of one period of
the TESLA damping ring wiggler.
The tools used for study of the beam dynamics were based on the mode
coefficients determined by the fitting procedure. Once a fit has been obtained
and shown to be of good quality, then, strictly speaking, further analysis of
the field is not required. However, it is interesting to compute, from the
mode amplitudes, the contribution to the field in the wiggler from different
“multipole” components, as a function of longitudinal position. As described
above, the contribution of a multipole of order $n$ is obtained by a one-
dimensional (in the longitudinal dimension) inverse Fourier transform of the
mode amplitudes $\tilde{B}_{n}(k_{z})$. To obtain non-zero values for the
contributions from multipoles higher than order $n=$ 1 (dipole), we need to
choose non-zero values for either the $x$ or $y$ coordinates at which we
compute the field. We choose (arbitrarily) $x=$ 8 mm, and $y=$ 0 mm. The
contributions to the vertical field component from multipoles of order 1
through 7 are shown in Fig. 23. Note that multipoles of even order are
forbidden by the symmetry of the wiggler (see Section 0.5.1). We see from Fig.
23 that the dominant contribution by far is, as expected, the dipole
component. The sextupole component is not insignificant; the contributions of
higher order multipoles are extremely small, and the high-frequency
“oscillation” as a function of longitudinal position is probably unphysical,
and the result of noise in the fitting.
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Figure 23: Multipole contributions to the field in the TESLA damping wiggler
as a function of longitudinal position, at $x=$8 mm and $y=$0 mm, for orders 1
(dipole), 3 (sextupole), 5 (decapole) and 7.
It is worth making a few final remarks about mode decompositions for three-
dimensional fields. First, as already mentioned, in many cases a full three-
dimensional mode decomposition will not be necessary. While this does provide
a detailed description of the field in a form suitable for beam dynamics
studies, three-dimensional decompositions do rely on a large number of
accurate and detailed field measurements. While such “measurements” may be
conveniently obtained from a model, it may be difficult or impractical to make
such measurements on a real magnet. Fortunately, in many cases, a two-
dimensional field description in terms of multipoles is sufficient. Generally,
a three-dimensional analysis only need be undertaken where there are grounds
to believe that the three-dimensional nature of the field is likely to have a
significant impact on the beam dynamics.
Second, we have already emphasised that to obtain an accurate description of
the field within some region in terms of a mode decomposition, the mode
amplitudes should be determined by a fit on a surface enclosing the region of
interest. Outside the region bounded by the surface of the fit, the fitted
field can be expected to diverge exponentially from the real field. However,
in choosing the surface for the fit, the geometry of the magnet will impose
some constraints. A magnet with a wide rectangular aperture may lend itself to
a description using a Cartesian basis (fitting on the surface of a rectangular
box); a circular aperture, however, is more likely to require use of a polar
basis (fitting on the surface of a cylinder with circular cross-section). Both
cases have been described above. It may be appropriate in other cases to
perform a fit on the surface of a cylinder with elliptical cross-section. The
basis functions in this case involve Mathieu functions. For further details,
the reader is referred to work by Dragt [12] and by Dragt and Mitchell [13].
## .7 The vector potential
Our analysis of iron-dominated multipole magnets in Section 0.4.2 was based on
the magnetic scalar potential, $\varphi$. The magnetic flux density can be
derived from a scalar potential:
$\vec{B}=-\textrm{grad}\,\varphi$
in the case that the flux density has vanishing divergence and curl:
$\textrm{div}\,\vec{B}=\textrm{curl}\,\vec{B}=0.$
More generally (in particular, where the flux density has non-vanishing curl)
one derives the magnetic flux density from a vector potential $\vec{A}$,
using:
$\vec{B}=\textrm{curl}\,\vec{A}.$ (57)
Although we have not required the vector potential in our discussion of
Maxwell’s equations for accelerator magnets, it is sometimes used in analysis
of beam dynamics. In particular, descriptions of the dynamics based on
Hamiltonian mechanics generally use the vector potential rather than the
magnetic flux density or the magnetic scalar potential. We therefore include
here a brief discussion of the vector potential, paying attention to aspects
relevant to the descriptions we have developed for two-dimensional and three-
dimensional magnet fields.
First, we note that the divergence of any curl is identically zero:
$\textrm{div}\,\textrm{curl}\,\vec{V}\equiv 0,$
for any differentiable vector field $\vec{V}$. Thus, if we write
$\vec{B}=\textrm{curl}\,\vec{A}$, then Maxwell’s equation (2):
$\textrm{div}\,\vec{B}=0,$
is automatically satisfied. Maxwell’s equation (3) in uniform media (constant
permeability), with zero current and static electric fields gives:
$\textrm{curl}\,\vec{B}=\mu\vec{J},$ (58)
where $\vec{J}$ is the current density. This leads to the equation for the
vector potential:
$\textrm{curl}\,\textrm{curl}\,\vec{A}\equiv\textrm{grad}\,(\textrm{div}\,\vec{A})-\nabla^{2}\vec{A}=\mu\vec{J}.$
(59)
Eq. (59) is a second-order differential equation for the vector potential in a
medium with permeability $\mu$, where the current density is $\vec{J}$. This
appears harder to solve than the first-order differential equation for the
magnetic flux density, Eq. (58). However, Eq. (59) may be simplified
significantly, if we apply an appropriate _gauge condition_. To understand
what this means, recall that the magnetic flux density is given by the curl of
the vector potential, and that the curl of the gradient of any scalar field is
identically zero. Thus, we can add the gradient of a scalar field to a vector
potential, and obtain a new vector potential that gives the same flux density
as the old one. That is, if:
$\vec{B}=\textrm{curl}\,\vec{A},$
and:
$\vec{A}^{\prime}=\vec{A}+\textrm{grad}\,\psi,$ (60)
for an arbitrary differentiable scalar field $\psi$, then:
$\textrm{curl}\,\vec{A}^{\prime}=\textrm{curl}\,\vec{A}+\textrm{curl}\,\textrm{grad}\,\psi=\textrm{curl}\,\vec{A}=\vec{B}.$
In other words, the vector potential $\vec{A}^{\prime}$ leads to exactly the
same flux density as the vector potential $\vec{A}$. Since the dynamics of a
given system are determined by the fields rather than the potentials, either
$\vec{A}^{\prime}$ or $\vec{A}$ is a valid choice for the description of the
system. Eq. (60) is known as a _gauge transformation_. The consequence of
having the freedom to make a gauge transformation means that the vector
potential for any given system is not uniquely defined: given some particular
vector potential, it is always possible to make a gauge tranformation without
any change in the physical observables of a system. The analogue in the case
of electric fields, of course, is that the “zero” of the electric scalar
potential can be chosen arbitrarily: only _changes_ in potential (i.e. energy)
are observable, so given some particular scalar potential field, it is
possible to add a constant (that is, a quantity independent of position) and
obtain a new scalar potential that gives the same physical observables as the
original scalar potential.
For magnetostatic fields, we can use a gauge transformation to simplify Eq.
(59). Suppose we have obtained a vector potential $\vec{A}$ for some
particular physical system. Define a scalar field $\psi$, which satisfies:
$\nabla^{2}\psi=-\textrm{div}\,\vec{A}.$ (61)
Then define:
$\vec{A}^{\prime}=\vec{A}+\textrm{grad}\,\psi.$
Since $\vec{A}^{\prime}$ and $\vec{A}$ are related by a gauge transformation,
they lead to the same magnetic flux density, and the same physical observables
for the system. However, the divergence of $\vec{A}^{\prime}$ vanishes:
$\textrm{div}\,\vec{A}^{\prime}=\textrm{div}\,\vec{A}+\textrm{div}\,\textrm{grad}\,\psi=-\nabla^{2}\psi+\nabla^{2}\psi=0,$
where we have used Eq. (61). Thus, given any vector potential, we can make a
gauge transformation to find a new vector potential that gives the same
magnetic flux density, but has vanishing divergence. The _gauge condition_ :
$\textrm{div}\,\vec{A}=0,$ (62)
is known as the _Coulomb gauge_. It is possible to work with other gauge
conditions (for example, for time-dependent electromagnetic fields the Lorenz
gauge condition is often more appropriate); however, for our present purposes,
the Coulomb gauge leads to a simplification of Eq. (59), which now becomes:
$\nabla^{2}\vec{A}=-\mu\vec{J}.$ (63)
Eq. (63) is Poisson’s equation for a vector field. Note that despite being a
second-order differential equation, it is in a sense simpler than Maxwell’s
equation (3), since we have “decoupled” the components of the vectors; that
is, we have a set of three uncoupled second-order differential equations,
where each equation relates a component of the vector potential to the
corresponding component of the current density. Eq. (63) has the solution:
$\vec{A}(\vec{r})=-\frac{\mu}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{|\vec{r}-\vec{r}^{\,\prime}|}\,d^{3}r^{\prime}.$
In this form, we see that the potential at a point in space is inversely
proportional to the distance from the source.
Now, consider the potential given by:
$A_{x}=0,\quad A_{y}=0,\quad A_{z}=-\textrm{Re}\,\frac{C_{n}(x+iy)^{n}}{n}.$
(64)
Taking derivatives, we find that:
$\displaystyle\frac{\partial A_{z}}{\partial x}$ $\displaystyle=$
$\displaystyle-\textrm{Re}\,C_{n}(x+iy)^{n-1},$ $\displaystyle\frac{\partial
A_{z}}{\partial y}$ $\displaystyle=$
$\displaystyle\textrm{Im}\,C_{n}(x+iy)^{n-1}.$
Then, since $A_{x}$ and $A_{y}$ are zero, we have:
$\vec{B}=\textrm{curl}\,\vec{A}=\left(\frac{\partial A_{z}}{\partial
y},-\frac{\partial A_{z}}{\partial x},0\right).\\\ $
Hence:
$B_{y}+iB_{x}=C_{n}(x+iy)^{n-1},$ (65)
which is just the multipole field. Thus, Eq. (64) is a potential that gives a
multipole field. Note also that, since $A_{z}$ is independent of $z$, this
potential satisfies the Coulomb gauge condition (62):
$\textrm{div}\,\vec{A}=\frac{\partial A_{x}}{\partial x}+\frac{\partial
A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z}=0.$
An advantage of working with the vector potential in the Coulomb gauge is
that, for multipole fields, the transverse components of the vector potential
are both zero. This simplifies, to some extent, the Hamiltonian equations of
motion for a particle moving through a multipole field. However, note that the
longitudinal component $B_{z}$ of the magnetic flux density is zero in this
case. To generate a solenoidal field, with $B_{z}$ equal to a non-zero
constant, we need to introduce non-zero components for $A_{x}$, or $A_{y}$, or
both. For example, a solenoid field with flux density $B_{\textrm{sol}}$ may
be derived from the vector potential:
$A_{x}=-\frac{1}{2}B_{\textrm{sol}}y,\quad
A_{y}=\frac{1}{2}B_{\textrm{sol}}x.$
Let us return for a moment to the case of multipole fields. If we work in a
gauge in which the transverse components of the vector potential are both
zero, then the field components are given by:
$B_{y}=-\frac{\partial A_{z}}{\partial x},\quad B_{x}=\frac{\partial
A_{z}}{\partial y}.$
From these expressions, we see that if we take any two points with the same
$y$ coordinate, then the difference in the vector potential between these two
points is given by the “flux” passing through a line between these points:
$\Delta A_{z}=-\int B_{y}\,dx.$
Similarly for any two points with the same $x$ coordinate:
$\Delta A_{z}=\int B_{x}\,dy.$
In general, for a field that is independent of $z$, and working in a gauge
where $A_{x}=A_{y}=0$, we can write:
$\Delta A_{z}=\frac{\Delta\Phi}{\Delta z},$ (66)
where $\Delta A_{z}$ is the change in the vector potential between two points
$P_{1}$ and $P_{2}$ in a given plane $z=z_{0}$; and $\Delta\Phi$ is the
magnetic flux through a rectangular “loop” with vertices $P_{1}$, $P_{2}$,
$P_{3}$ and $P_{4}$: see Fig. 24. $P_{3}$ and $P_{4}$ are points obtained by
transporting $P_{1}$ and $P_{2}$ a distance $\Delta z$ parallel to the $z$
axis. Eq. (66) can also be obtained by applying Stokes’ theorem to the loop
$P_{1}P_{2}P_{3}P_{4}$, with the relationship (57) between $\vec{B}$ and
$\vec{A}$:
$\int\vec{A}\cdot d\vec{l}=\int\textrm{curl}\,\vec{A}\cdot
d\vec{S}=\int\vec{B}\cdot d\vec{S},$
hence:
$A_{z}(P_{2})\Delta z-A_{z}(P_{1})\Delta z=\Delta\Phi.$
Figure 24: Interpretation of the vector potential in a two-dimensional
magnetic field (i.e. a field that is independent of $z$). The change in the
vector potential between $P_{1}$ and $P_{2}$ is equal to the flux of the
magnetic field through the loop $P_{1}P_{2}P_{3}P_{4}$, divided by $\Delta z$.
Finally, we give the vector potentials corresponding to three-dimensional
fields. In the Cartesian basis, with the field given by Eqs. (43)–(45), a
possible vector potential (in the Coulomb gauge) is:
$\displaystyle A_{x}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
A_{y}$ $\displaystyle=$ $\displaystyle B_{0}\frac{k_{z}}{k_{x}k_{y}}\sin
k_{x}x\sinh k_{y}y\cos k_{z}z,$ $\displaystyle A_{z}$ $\displaystyle=$
$\displaystyle-B_{0}\frac{1}{k_{x}}\sin k_{x}x\cosh k_{y}y\sin k_{z}z.$
In the polar basis, with the field given by Eqs. (52)–(54), a possible vector
potential is:
$\displaystyle A_{r}$ $\displaystyle=$
$\displaystyle-\frac{r}{m}I_{m}\\!(k_{z}r)\cos n\theta\sin k_{z}z,$
$\displaystyle A_{\theta}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
A_{z}$ $\displaystyle=$
$\displaystyle-\frac{r}{2m}I_{m}^{\prime}\\!(k_{z}r)\cos n\theta\sin k_{z}z.$
However, note that this potential does not satisfy the Coulomb gauge
condition.
## References
* [1] Vector Fields Software, http://www.cobham.com.
* [2] Computer Simulation Technology, http://www.cst.com.
* [3] ESRF Insertion Devices Group, http://www.esrf.eu/Accelerators/Groups/InsertionDevices/Software/Radia.
* [4] S. Russenschuck, “Foundation of numerical field computation,” Proceedings of the CERN Accelerator School Course on Magnets, Bruges, Belgium (2009).
* [5] J. D. Jackson, “Classical electrodynamics,” John Wiley and Sons, 3rd Edition (1998).
* [6] L. Bottura, “Superconducting magnets,” Proceedings of the CERN Accelerator School Course on Magnets, Bruges, Belgium (2009).
* [7] A. Chao and M. Tigner (editors), “Handbook of accelerator physics and engineering,” World Scientific Publishing (1999).
* [8] B. J. A. Shepherd, N. Marks, “Quadrupole magnets for the 20 MeV FFAG, EMMA,” Proceedings of PAC07, Albuquerque, New Mexica, USA (2007).
* [9] N. Marks et al., “Development and adjustment of the EMMA quadrupoles,” Proceedings of EPAC08, Genoa, Italy (2008).
* [10] TESLA Techncial Design Report (2001). http://tesla.desy.de/new_pages/TDR_CD/start.html.
* [11] A. Wolski, J. Gao and S. Guiducci, (editors) “Configuration studies and recommendations for the ILC damping rings,” LBNL–59449 (2006).
* [12] A. J. Dragt, “Lie methods for nonlinear dynamics with applications to accelerator physics,” available at the URL http://www.physics.umd.edu/dsat/dsatliemethods.html.
* [13] C. E. Mitchell and A. J. Dragt, “Computation of transfer maps from magnetic field data in wigglers and undulators,” ICFA Beam Dynamics Newsletter, 42, p. 65 (April 2007).
|
arxiv-papers
| 2011-03-03T15:23:34 |
2024-09-04T02:49:17.428251
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Andrzej Wolski",
"submitter": "Andrzej Wolski",
"url": "https://arxiv.org/abs/1103.0713"
}
|
1103.0897
|
# Multiple Kernel Learning: A Unifying Probabilistic Viewpoint
Hannes Nickisch hannes@nickisch.org
Max Planck Institute for Intelligent Systems, Spemannstraße 38, 72076
Tübingen, Germany
Matthias Seeger matthias.seeger@epfl.ch
Ecole Polytechnique Fédérale de Lausanne, INJ 339, Station 14, 1015 Lausanne,
Switzerland
###### Abstract
We present a probabilistic viewpoint to multiple kernel learning unifying
well-known regularised risk approaches and recent advances in approximate
Bayesian inference relaxations. The framework proposes a general objective
function suitable for regression, robust regression and classification that is
lower bound of the marginal likelihood and contains many regularised risk
approaches as special cases. Furthermore, we derive an efficient and provably
convergent optimisation algorithm.
Keywords: Multiple kernel learning, approximate Bayesian inference, double
loop algorithms, Gaussian processes
## 1 Introduction
Nonparametric kernel methods, cornerstones of machine learning today, can be
seen from different angles: as regularised risk minimisation in function
spaces (Schölkopf and Smola, 2002), or as probabilistic Gaussian process
methods (Rasmussen and Williams, 2006). In these techniques, the kernel (or
equivalently covariance) function encodes interpolation characteristics from
observed to unseen points, and two basic statistical problems have to be
mastered. First, a latent function must be predicted which fits data well, yet
is as smooth as possible given the fixed kernel. Second, the kernel function
parameters have to be learned as well, to best support predictions which are
of primary interest. While the first problem is simpler and has been addressed
much more frequently so far, the central role of learning the covariance
function is well acknowledged, and a substantial number of methods for
“learning the kernel”, “multiple kernel learning”, or “evidence maximisation”
are available now. However, much of this work has firmly been associated with
one of the “camps” (referred to as _regularised risk_ and _probabilistic_ in
the sequel) with surprisingly little crosstalk or acknowledgments of prior
work across this boundary. In this paper, we clarify the relationship between
major regularised risk and probabilistic kernel learning techniques precisely,
pointing out advantages and pitfalls of either, as well as algorithmic
similarities leading to novel powerful algorithms.
We develop a common analytical and algorithmical framework encompassing
approaches from both camps and provide clear insights into the optimisation
structure. Even though, most of the optimisation is non convex, we show how to
operate a provably convergent “almost Newton” method nevertheless. Each step
is not much more expensive than a gradient based approach. Also, we do not
require any foreign optimisation code to be available. Our framework unifies
kernel learning for regression, robust regression and classification.
The paper is structured as follows: In section 2, we introduce the regularised
risk and the probabilistic view of kernel learning. In increasing order of
generality, we explain multiple kernel learning (MKL, section 2.1), maximum a
posteriori estimation (MAP, section 2.2) and marginal likelihood maximisation
(MLM, section 2.3). A taxonomy of the mutual relations between the approaches
and important special cases is given in section 2.4. Section 3 introduces a
general optimisation scheme and section 4 draws a conclusion.
## 2 Kernel Methods and Kernel Learning
Kernel-based algorithms come in many shapes, however, the primary goal is –
based on training data $\\{(\mathbf{x}_{i},y_{i})\,|\,i=1..n\\}$,
$\mathbf{x}_{i}\in{\cal X}$, $y_{i}\in\mathcal{Y}$ and a parametrised kernel
function $k_{\bm{\theta}}(\mathbf{x},\mathbf{x}^{\prime})$ – to predict the
output $y_{*}$ for unseen inputs $\mathbf{x}_{*}$. Often, linear
parametrisations
$k_{\bm{\theta}}(\mathbf{x},\mathbf{x}^{\prime})=\sum_{m=1}^{M}\theta_{m}k_{m}(\mathbf{x},\mathbf{x}^{\prime})$
are used, where the $k_{m}$ are fixed positive definite functions, and
$\bm{\theta}\succeq\mathbf{0}$. Learning the kernel means finding
$\bm{\theta}$ to best support this goal. In general, kernel methods employ a
postulated latent function $u:{\cal X}\to\mathbb{R}$ whose smoothness is
controlled via the function space squared norm
$\|u(\cdot)\|_{k_{\bm{\theta}}}^{2}$. Most often, smoothness is traded against
data fit, either enforced by a _loss function_ $\ell(y_{i},u(\mathbf{x}_{i}))$
or modeled by a _likelihood_ $\mathbb{P}(y_{i}|u_{i})$. Let us define kernel
matrices
$\mathbf{K}_{\bm{\theta}}:=[k_{\bm{\theta}}(\mathbf{x}_{i},\mathbf{x}_{j})]_{ij}$,
and $\mathbf{K}_{m}:=[k_{m}(\mathbf{x}_{i},\mathbf{x}_{j})]_{ij}$ in
$\mathbb{R}^{n\times n}$ and the vectors
$\mathbf{y}:=[y_{i}]_{i}\in\mathcal{Y}^{n}$,
$\mathbf{u}:=[u(\mathbf{x}_{i})]_{i}\in\mathbb{R}^{n}$ collecting outputs and
latent function values, respectively.
The _regularised risk_ route to kernel prediction, which is followed by any
support vector machine (SVM) or ridge regression technique, yields
$\|u(\cdot)\|_{k_{\bm{\theta}}}^{2}+\frac{C}{n}\sum_{i=1}^{n}\ell(y_{i},u_{i})$
as criterion, enforcing smoothness of $u(\cdot)$ as well as good data fit
through th _e_ loss function $\frac{C}{n}\ell(y_{i},u(\mathbf{x}_{i}))$. By
the representer theorem, the minimiser can be written as
$u(\cdot)=\sum_{i}\alpha_{i}k_{\bm{\theta}}(\cdot,\mathbf{x}_{i})$, so that
$\|u(\cdot)\|_{k_{\bm{\theta}}}^{2}=\bm{\alpha}^{\top}\mathbf{K}_{\bm{\theta}}\bm{\alpha}$
(Schölkopf and Smola, 2002). As
$\mathbf{u}=\mathbf{K}_{\bm{\theta}}\bm{\alpha}$, the regularised risk problem
is given by
$\min_{\mathbf{u}}\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\frac{C}{n}\sum_{i=1}^{n}\ell(y_{i},u_{i}).$
(1)
A _probabilistic_ viewpoint of the same setting is based on the notion of a
Gaussian process (GP) (Rasmussen and Williams, 2006): a Gaussian random
function $u(\cdot)$ with mean function
$\mathbb{E}[u(\mathbf{x})]=m(\mathbf{x})\equiv 0$ and covariance function
$\mathbb{V}[u(\mathbf{x}),u(\mathbf{x}^{\prime})]=\mathbb{E}[u(\mathbf{x})u(\mathbf{x}^{\prime})]=k_{\bm{\theta}}(\mathbf{x},\mathbf{x}^{\prime})$.
In practice, we only use finite-dimensional snapshots of the process
$u(\cdot)$: for example,
$\mathbb{P}(\mathbf{u};\bm{\theta})=\mathcal{N}(\mathbf{u}|\mathbf{0},\mathbf{K}_{\bm{\theta}})$,
a zero-mean joint Gaussian with covariance matrix $\mathbf{K}_{\bm{\theta}}$.
We adopt this GP as prior distribution over $u(\cdot)$, estimating the latent
function as maximum of the posterior process
$\mathbb{P}(u(\cdot)|\mathbf{y};\bm{\theta})\propto\mathbb{P}(\mathbf{y}|\mathbf{u})\mathbb{P}(u(\cdot);\bm{\theta})$.
Since the likelihood depends on $u(\cdot)$ only through the finite subset
$\\{u(\mathbf{x}_{i})\\}$, the posterior process has a finite-dimensional
representation:
$\mathbb{P}(u(\cdot)|\mathbf{y},\mathbf{u})=\mathbb{P}(u(\cdot)|\mathbf{u})$,
so that
$\mathbb{P}(u(\cdot)|\mathbf{y};\bm{\theta})=\int\mathbb{P}(u(\cdot)|\mathbf{u})\mathbb{P}(\mathbf{u}|\mathbf{y};\bm{\theta})\text{d}\mathbf{u}$
is specified by the joint distribution
$\mathbb{P}(\mathbf{u}|\mathbf{y};\bm{\theta})$, a probabilistic equivalent of
the representer theorem. Kernel prediction amounts to _maximum a posteriori_
(MAP) estimation
$\max\nolimits_{\mathbf{u}}\mathbb{P}(\mathbf{u}|\mathbf{y};\bm{\theta})\equiv\max\nolimits_{\mathbf{u}}\mathbb{P}(\mathbf{u};\bm{\theta})\mathbb{P}(\mathbf{y}|\mathbf{u})\equiv\min\nolimits_{\mathbf{u}}\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}-2\ln\mathbb{P}(\mathbf{y}|\mathbf{u})+\ln|\mathbf{K}_{\bm{\theta}}|,$
(2)
ignoring an additive constant. Minimising equations (1) and (2) for any fixed
kernel matrix $\mathbf{K}$ gives the same minimiser $\hat{\mathbf{u}}$ and
prediction
$u(\mathbf{x}_{*})=\hat{\mathbf{u}}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}[k_{\bm{\theta}}(\mathbf{x}_{i},\mathbf{x}_{*})]_{i}$.
The correspondence between likelihood and loss function bridges probabilistic
and regularised risk techniques. More specifically, any likelihood
$\mathbb{P}(\mathbf{y}|\mathbf{u})$ induces a loss function
$\ell(\mathbf{y},\mathbf{u})$ via
$-2\ln\mathbb{P}(\mathbf{y}|\mathbf{u})=-2\sum_{i}\ln\mathbb{P}(y_{i}|u_{i})\rightsquigarrow\frac{C}{n}\sum_{i=1}^{n}\ell(y_{i},u_{i})=\ell(\mathbf{y},\mathbf{u}),$
however some loss functions cannot be interpreted as a negative log likelihood
as shown in table (2) and as discussed for the SVM by Sollich (2000). If, the
likelihood is a _log-concave_ function of $\mathbf{u}$, it corresponds to a
convex loss function (Boyd and Vandenberghe, 2002, Sect. 3.5.1). Common loss
functions and likelihoods for classification $\mathcal{Y}=\\{\pm 1\\}$ and
regression $\mathcal{Y}=\mathbb{R}$ are listed in table (2).
$\mathcal{Y}$ | Loss function | $\ell(y_{i},u_{i})$ | $\mathbb{P}(y_{i}|u_{i})$ | Likelihood
---|---|---|---|---
$\\{\pm 1\\}$ | SVM Hinge loss | $\max(0,1-y_{i}u_{i})$ | $\nexists$
$\\{\pm 1\\}$ | Log loss | $\ln(\exp(-y_{i}u_{i})+1)$ | $1/(\exp(-\tau y_{i}u_{i})+1)$ | Logistic
$\mathbb{R}$ | SVM $\epsilon$-insensitive loss | $\max(0,|y_{i}-u_{i}|/\epsilon-1)$ | $\nexists$
$\mathbb{R}$ | Quadratic loss | $(y_{i}-u_{i})^{2}$ | $\mathcal{N}(y_{i}|u_{i},\sigma^{2})$ | Gaussian
$\mathbb{R}$ | Linear loss | $|y_{i}-u_{i}|$ | $\mathcal{L}(y_{i}|u_{i},\tau)$ | Laplace
Table 1: Relations between loss functions and likelihoods
In the following, we discuss several approaches to learn the kernel parameters
$\bm{\theta}$ and show how all of them can be understood as instances of or
approximations to Bayesian evidence maximisation. Although the exposition MKL
section 2.1 and MAP section 2.2 use a linear parametrisation
$\bm{\theta}\mapsto\mathbf{K}_{\bm{\theta}}=\sum_{m=1}^{M}\theta_{m}\mathbf{K}_{m}$,
much of the results in MLM 2.3 and all the aforementioned discussion are still
applicable to non-linear parametrisations.
### 2.1 Multiple Kernel Learning
A widely adopted regularised risk principle, known as _multiple kernel
learning_ (MKL) (Christianini et al., 2001; Lanckriet et al., 2004; Bach et
al., 2004), is to minimise equation (1) w.r.t. the kernel parameters
$\bm{\theta}$ as well. One obvious caveat is that for any fixed $\mathbf{u}$,
equation (1) becomes ever smaller as $\theta_{m}\to\infty$: it cannot per se
play a meaningful statistical role. In order to prevent this, researchers
constrain the domain of $\bm{\theta}\in\bm{\Theta}$ and obtain
$\min_{\bm{\theta}\in\bm{\Theta}}\min_{\mathbf{u}}\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\ell(\mathbf{y},\mathbf{u}),$
where $\bm{\Theta}=\\{\bm{\theta}\succeq\mathbf{0},\>\|\bm{\theta}\|_{2}\leq
1\\}$ or
$\bm{\Theta}=\\{\bm{\theta}\succeq\mathbf{0},\>\mathbf{1}^{\top}\bm{\theta}\leq
1\\}$ (Varma and Ray, 2007). Notably, these constraints are imposed
independently of the statistical problem, the model and of the parametrization
$\bm{\theta}\mapsto\mathbf{K}_{\bm{\theta}}$. The Lagrangian form of the MKL
problem with parameter $\lambda$ and a general $p$-norm unit ball constraint
where $p\geq 1$ (Kloft et al., 2009) is given by
$\min_{\bm{\theta}\succeq\mathbf{0}}\phi_{\text{MKL}}(\bm{\theta}),\;\text{
where
}\phi_{\text{MKL}}(\bm{\theta}):=\min_{\mathbf{u}}\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\ell(\mathbf{y},\mathbf{u})+\underbrace{\lambda\cdot\mathbf{1}^{\top}\bm{\theta}^{p}}_{\rho(\bm{\theta})},\;\lambda>0.$
(3)
Since, the _regulariser_ $\rho(\bm{\theta})$ for the kernel parameter
$\bm{\theta}$ is convex, the map
$(\mathbf{u},\mathbf{K})\mapsto\mathbf{u}^{\top}\mathbf{K}^{-1}\mathbf{u}$ is
jointly convex for $\mathbf{K}\succeq\mathbf{0}$ (Boyd and Vandenberghe, 2002)
and the parametrisation $\bm{\theta}\mapsto\mathbf{K}_{\bm{\theta}}$ is
linear, MKL is a jointly convex problem for $\bm{\theta}\succeq\mathbf{0}$
whenever the loss function $\ell(\mathbf{y},\mathbf{u})$ is convex.
Furthermore, there are efficient algorithms to solve equation (3) for large
models (Sonnenburg et al., 2006).
### 2.2 Joint MAP Estimation
Adopting a probabilistic MAP viewpoint, we can minimise equation (2) w.r.t.
$\mathbf{u}$ and $\bm{\theta}\succeq\mathbf{0}$:
$\min_{\bm{\theta}\succeq\mathbf{0}}\phi_{\text{MAP}}(\bm{\theta}),\;\text{
where
}\phi_{\text{MAP}}(\bm{\theta}):=\min_{\mathbf{u}}\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}-2\ln\mathbb{P}(\mathbf{y}|\mathbf{u})+\ln|\mathbf{K}_{\bm{\theta}}|.$
(4)
While equation (3) and equation (4) share the “inner solution”
$\hat{\mathbf{u}}$ for fixed $\mathbf{K}_{\bm{\theta}}$ – in case the loss
$\ell(\mathbf{y},\mathbf{u})$ corresponds to a likelihood
$\mathbb{P}(\mathbf{y}|\mathbf{u})$ – they are different when it comes to
optimising $\bm{\theta}$. The _joint MAP_ problem is not in general jointly
convex in $(\bm{\theta},\mathbf{u})$, since
$\bm{\theta}\mapsto\ln|\mathbf{K}_{\bm{\theta}}|$ is concave, see figure 2.
However, it is always a well-posed statistical procedure, since
$\ln|\mathbf{K}_{\bm{\theta}}|\to\infty$ as $\theta_{m}\to\infty$ for all $m$.
Figure 1: Convex upper bounds on (the concave non-decreasing)
$\ln|\mathbf{K}_{\bm{\theta}}|$
By Fenchel duality, we can represent any concave non-decreasing function and
hence the log determinant function by
$\ln|\mathbf{K}_{\bm{\theta}}|=\min_{\bm{\lambda}\succeq\mathbf{0}}\bm{\lambda}^{\top}|\bm{\theta}|^{p}-g^{*}(\bm{\lambda})$.
As a consequence, we obtain a piecewise polynomial upper bound for any
particular value $\bm{\lambda}$.
We show in the following, how the regularisers
$\rho(\bm{\theta})=\lambda\left\|\bm{\theta}\right\|_{p}^{p}$ of equation (3)
can be related to the probabilistic term
$f(\bm{\theta})=\ln|\mathbf{K}_{\bm{\theta}}|$. In fact, the same reasoning
can be applied to any concave non-decreasing function.
Since the function $\bm{\theta}\mapsto
f(\bm{\theta})=\ln|\mathbf{K}_{\bm{\theta}}|$, $\bm{\theta}\succeq\mathbf{0}$
is jointly concave, we can represent it by
$f(\bm{\theta})=\min_{\bm{\lambda}\succeq\mathbf{0}}\bm{\lambda}^{\top}\bm{\theta}-f^{*}(\bm{\lambda})$
where $f^{*}(\bm{\lambda})$ denotes Fenchel dual of $f(\bm{\theta})$.
Furthermore, the mapping
$\bm{\vartheta}\mapsto\ln|\sum_{m=1}^{M}\sqrt[p]{\vartheta_{m}}\mathbf{K}_{m}|=f(\sqrt[p]{\bm{\vartheta}})=g(\bm{\vartheta})$,
$\bm{\vartheta}\succeq\mathbf{0}$ is jointly concave due to the composition
rule (Boyd and Vandenberghe, 2002, §3.2.4), because
$\bm{\vartheta}\mapsto\sqrt[p]{\bm{\vartheta}}$ is jointly concave and
$\bm{\theta}\mapsto f(\bm{\theta})$ is non-decreasing in all components
$\theta_{m}$ as all matrices $\mathbf{K}_{m}$ are positive (semi-)definite
which guarantees that the eigenvalues of $\mathbf{K}_{\bm{\theta}}$ increase
as $\theta_{m}$ increases. Thus we can – similarly to Zhang (2010) – represent
$\ln|\mathbf{K}_{\bm{\theta}}|$ as
$\ln|\mathbf{K}_{\bm{\theta}}|=f(\bm{\theta})=g(\bm{\vartheta})=\min_{\bm{\lambda}\succeq\mathbf{0}}\bm{\lambda}^{\top}\bm{\vartheta}-g^{*}(\bm{\lambda})=\min_{\bm{\lambda}\succeq\mathbf{0}}\bm{\lambda}^{\top}|\bm{\theta}|^{p}-g^{*}(\bm{\lambda}).$
Choosing a particular value $\bm{\lambda}=\lambda\cdot\mathbf{1}$, we obtain
the bound
$\ln|\mathbf{K}_{\bm{\theta}}|\leq\lambda\cdot\left\|\bm{\theta}\right\|_{p}^{p}-g^{*}(\lambda\cdot\mathbf{1})$.
Figure 1 illustrates the bounds for $p=1$ and $p=2$. The bottom line is that
one can interpret the regularisers
$\rho(\bm{\theta})=\lambda\left\|\bm{\theta}\right\|_{p}^{p}$ in equation (3)
as corresponding to parametrised upper bounds to the
$\ln|\mathbf{K}_{\bm{\theta}}|$ part in equation (4), hence
$\phi_{\text{MKL}}(\bm{\theta})=\psi_{\text{MAP}}(\bm{\theta},\bm{\lambda}=\lambda\cdot\mathbf{1})$,
where
$\phi_{\text{MAP}}(\bm{\theta})=\min_{\bm{\lambda}\succeq\mathbf{0}}\psi_{\text{MAP}}(\bm{\theta},\bm{\lambda})$.
Far from an ad hoc choice to keep $\bm{\theta}$ small, the
$\ln|\mathbf{K}_{\bm{\theta}}|$ term embodies the Occam’s razor concept behind
MAP estimation: overly large $\bm{\theta}$ are ruled out, since their
explanation of the data $\mathbf{y}$ is extremely unlikely under the prior
$\mathbb{P}(\mathbf{u};\bm{\theta})$. The Occam’s razor effect depends
crucially on the proper normalization of the prior (MacKay, 1992). For
example, the weighting parameter $C$ of $k$ ($k=C\tilde{k}$) can be learned by
joint MAP: if $C=e^{c}$, then equation (4) is convex in $c$ for any fixed
$\mathbf{u}$. If kernel-regularised estimation equation (1) is interpreted as
MAP estimation under a GP prior equation (2), the correct extension to kernel
learning is joint MAP: the MKL criterion equation (3) lacks prior
normalization, which renders MAP w.r.t. $\bm{\theta}$ meaningful in the first
place. From a non-probabilistic viewpoint, the $\ln|\mathbf{K}_{\bm{\theta}}|$
term comes with a model and data dependent structure at least as complex as
the rest of equation (3).
While the MKL objective, equation (3), enjoys the benefit of being convex in
the (linear) kernel parameters $\bm{\theta}$, this does not hold true for
joint MAP estimation, equation (4), in general. We illustrate the differences
in figure 2. The function $\psi_{\text{MAP}}(\bm{\theta},\mathbf{u})$ is a
building block of the MAP objective
$\phi_{\text{MAP}}(\bm{\theta})=\min_{\mathbf{u}}[\psi_{\text{MAP}}(\bm{\theta},\mathbf{u})-2\ln\mathbb{P}(\mathbf{y}|\mathbf{u})]$,
where
$\psi_{\text{MAP}}(\bm{\theta},\mathbf{u})=\underbrace{\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}}_{\psi_{\cup}(\bm{\theta},\mathbf{u})}+\underbrace{\ln|\mathbf{K}_{\bm{\theta}}|}_{\psi_{\cap}(\bm{\theta})}\leq\psi_{\text{MKL}}(\bm{\theta},\mathbf{u})-g^{*}(\lambda\cdot\mathbf{1}),\>\psi_{\text{MKL}}(\bm{\theta},\mathbf{u})=\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\lambda\left\|\bm{\theta}\right\|_{p}^{p}.$
More concretely, $\psi_{\text{MAP}}(\bm{\theta},\mathbf{u})$ is a sum of a
nonnegative, jointly convex function $\psi_{\cup}(\bm{\theta},\mathbf{u})$
that is strictly decreasing in every component $\theta_{m}$ and a concave
function $\psi_{\cap}(\bm{\theta})$ that is strictly increasing in every
component $\theta_{m}$. Both functions $\psi_{\cup}(\bm{\theta},\mathbf{u})$
and $\psi_{\cap}(\bm{\theta})$ alone do not have a stationary point due to
their monotonicity in $\theta_{m}$. However, their sum can have (even
multiple) stationary points as shown in figure 2 on the left left. We can
show, that the map
$\mathbf{K}\mapsto\mathbf{u}^{\top}\mathbf{K}^{-1}\mathbf{u}+\ln|\mathbf{K}|$
is _invex_ i.e. every stationary point $\hat{\mathbf{K}}$ is a global minimum.
Using the convexity of
$\mathbf{A}\mapsto\mathbf{u}^{\top}\mathbf{A}\mathbf{u}-\ln|\mathbf{A}|$ (Boyd
and Vandenberghe, 2002) and the fact that the derivative of
$\mathbf{A}\mapsto\mathbf{A}^{-1}$ for $\mathbf{A}\in\mathbb{R}^{n\times n}$,
$\mathbf{A}\succ\mathbf{0}$ has full rank $n^{2}$, we see by Mishra and Giorgi
(2008, theorem 2.1) that
$\mathbf{K}\mapsto\mathbf{u}^{\top}\mathbf{K}^{-1}\mathbf{u}+\ln|\mathbf{K}|$
is indeed invex.
Often, the MKL objective for the case $p=1$ is motivated by the fact that the
optimal solution $\bm{\theta}^{\star}$ is _sparse_ (e.g. Sonnenburg et al.,
2006), meaning that many components $\theta_{m}$ are zero. Figure 2
illustrates that $\phi_{\text{MAP}}(\bm{\theta})$ also yields sparse
solutions; in fact it enforces even more sparsity. In MKL,
$\phi_{\text{MAP}}(\bm{\theta})$ is simply relaxed to a convex objective
$\phi_{\text{MKL}}(\bm{\theta})$ at the expense of having only a single less
sparse solution.
Figure 2: Convex and non-convex building blocks of the MKL and MAP objective
function
#### Intuition for the Gaussian Case
We can gain further intuition about the criteria $\phi_{\text{MKL}}$ and
$\phi_{\text{MAP}}$ by asking which _matrices_ $\mathbf{K}$ minimise them. For
simplicity, assume that
$\mathbb{P}(\mathbf{y}|\mathbf{u})=\mathcal{N}(\mathbf{y}|\mathbf{u},\sigma^{2}\mathbf{I})$
and $n/C=\sigma^{2}$, hence
$\ell(\mathbf{y},\mathbf{u})=\frac{1}{\sigma^{2}}\left\|\mathbf{y}-\mathbf{u}\right\|_{2}^{2}$.
The inner minimiser $\hat{\mathbf{u}}$ for both $\phi_{\text{MKL}}$ and
$\phi_{\text{MAP}}$ is given by
$\mathbf{K}_{\bm{\theta}}^{-1}\hat{\mathbf{u}}=(\mathbf{K}_{\bm{\theta}}+\sigma^{2}\mathbf{I})^{-1}\mathbf{y}$.
With $\sigma^{2}\to 0$, we find for joint MAP that
$\frac{\partial}{\partial\mathbf{K}}\phi_{\text{MAP}}=\mathbf{0}$ results in
$\hat{\mathbf{K}}=\mathbf{y}\mathbf{y}^{\top}$. While this “nonparametric”
estimate requires smoothing to be useful in practice, closeness to
$\mathbf{y}\mathbf{y}^{\top}$ is fundamental to covariance estimation and can
be found in regularised risk kernel learning work (Christianini et al., 2001).
On the other hand, for $\text{tr}(\mathbf{K}_{m})=1$ and hence
$\rho(\bm{\theta})=\lambda\text{tr}(\mathbf{K}_{\bm{\theta}})=\lambda\left\|\bm{\theta}\right\|_{1}$,
$\frac{\partial}{\partial\mathbf{K}}\phi_{\text{MKL}}=\mathbf{0}$ leads to
$\hat{\mathbf{K}}^{2}=\lambda^{-1}\mathbf{y}\mathbf{y}^{\top}$: an odd way of
estimating covariance, not supported by any statistical literature we are
aware of.
### 2.3 Marginal Likelihood Maximisation
While the joint MAP criterion uses a properly normalised prior distribution,
it is still not probabilistically consistent. Kernel learning amounts to
finding a value $\hat{\bm{\theta}}$ of high data likelihood, no matter what
the latent function $u(\cdot)$ is. The correct likelihood to be maximised is
_marginal_ :
$\mathbb{P}(\mathbf{y}|\bm{\theta})=\int\mathbb{P}(\mathbf{y}|\mathbf{u})\mathbb{P}(\mathbf{u}|\bm{\theta})\text{d}\mathbf{u}$
(“max-sum”), while joint MAP employs the plug-in surrogate
$\max_{\mathbf{u}}\mathbb{P}(\mathbf{y}|\mathbf{u})\mathbb{P}(\mathbf{u}|\bm{\theta})$
(“max-max”). _Marginal likelihood maximisation_ (MLM) is also known as
Bayesian estimation, and it underlies the EM algorithm or maximum likelihood
learning of conditional random fields just as well: complexity is controlled
(and overfitting avoided) by averaging over unobserved variables $\mathbf{u}$
(MacKay, 1992), rather than plugging in some point estimate $\hat{\mathbf{u}}$
$\phi_{\text{MLM}}(\bm{\theta}):=-2\ln\int\mathcal{N}(\mathbf{u}|\mathbf{0},\mathbf{K}_{\bm{\theta}})\mathbb{P}(\mathbf{y}|\mathbf{u})\text{d}\mathbf{u}.$
(5)
#### The Gaussian Case
Before developing a general MLM approximation, we note an important
analytically solvable exception: for Gaussian likelihood
$\mathbb{P}(\mathbf{y}|\mathbf{u})=\mathcal{N}(\mathbf{y}|\mathbf{u},\sigma^{2}\mathbf{I})$,
$\mathbb{P}(\mathbf{y}|\bm{\theta})=\mathcal{N}(\mathbf{y}|\mathbf{0},\mathbf{K}_{\bm{\theta}}+\sigma^{2}\mathbf{I})$,
and MLM becomes
$\phi_{\text{GAU}}(\bm{\theta}):=\mathbf{y}^{\top}(\mathbf{K}_{\bm{\theta}}+\sigma^{2}\mathbf{I})^{-1}\mathbf{y}+\ln|\mathbf{K}_{\bm{\theta}}+\sigma^{2}\mathbf{I}|.$
(6)
Even if the primary purpose is classification, the Gaussian likelihood is used
for its analytical simplicity (Kapoor et al., 2009). Only for the Gaussian
case, joint MAP and MLM have an analytically closed form. From the product
formula of Gaussians (Brookes, 2005, §5.1)
$\mathbb{Q}(\mathbf{u}):=\mathcal{N}(\mathbf{u}|\mathbf{0},\mathbf{K}_{\bm{\theta}})\mathcal{N}(\mathbf{y}|\mathbf{u},\bm{\Gamma})=\mathcal{N}(\mathbf{y}|\mathbf{0},\mathbf{K}_{\bm{\theta}}+\bm{\Gamma})\mathcal{N}(\mathbf{u}|\mathbf{m},\mathbf{V}),$
where $\mathbf{V}=(\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1})^{-1}$ and
$\mathbf{m}=\mathbf{V}\bm{\Gamma}^{-1}\mathbf{y}$ we can deduce that
$-2\ln\int\mathbb{Q}(\mathbf{u})\text{d}\mathbf{u}=\ln|\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1}|+\min_{\mathbf{u}}[-2\ln\mathbb{Q}(\mathbf{u})]-n\ln|2\pi|.$
(7)
Using $\sigma^{2}\mathbf{I}=\bm{\Gamma}$ and
$\min_{\mathbf{u}}[-2\ln\mathbb{Q}(\mathbf{u})]=-2\ln\mathbb{Q}(\mathbf{m})$,
we see that by
$\phi_{\text{MAP/GAU}}(\bm{\theta}):\stackrel{{\scriptstyle\text{c}}}{{=}}\phi_{\text{GAU}}(\bm{\theta})-\ln|\mathbf{K}_{\bm{\theta}}^{-1}+\sigma^{-2}\mathbf{I}|\stackrel{{\scriptstyle\text{c}}}{{=}}\mathbf{y}^{\top}(\mathbf{K}_{\bm{\theta}}+\sigma^{2}\mathbf{I})^{-1}\mathbf{y}+\ln|\mathbf{K}_{\bm{\theta}}|$
(8)
MLM and MAP are very similar for the Gaussian case.
The “ridge regression” approximation is also used together with $p$-norm
constraints instead of the $\ln|\mathbf{K}_{\bm{\theta}}|$ term (Cortes et
al., 2009)
$\phi_{\text{RR}}(\bm{\theta}):=\mathbf{y}^{\top}(\mathbf{K}_{\bm{\theta}}+\sigma^{2}\mathbf{I})^{-1}\mathbf{y}+\lambda\left\|\bm{\theta}\right\|_{p}^{p}.$
(9)
Unfortunately, most GP methods to date work with a Gaussian likelihood for
simplicity, a restriction which often proves short-sighted. Gaussian-linear
models come with unrealistic properties, and benefits of MLM over joint MAP
cannot be realised.
Kernel parameter learning has been an integral part of probabilistic GP
methods from the very beginning. Williams and Rasmussen (1996) proposed MLM
for Gaussian noise equation 6, fifteen years ago. They treated sums of
exponential and linear kernels as well as learning lengthscales (ARD),
predating recent proposals such as “products of kernels” (Varma and Babu,
2009).
#### The General Case
In general, joint MAP always has the analytical form equation 4, while
$\mathbb{P}(\mathbf{y}|\bm{\text{$\theta$}})$ can only be approximated. For
non-Gaussian $\mathbb{P}(\mathbf{y}|\mathbf{u})$, numerous approximate
inference methods have been proposed, specifically motivated by learning
kernel parameters via MLM. The simplest such method is Laplace’s
approximation, applied to GP binary and multi-way classification by Williams
and Barber (1998): starting with convex joint MAP,
$\ln\mathbb{P}(\mathbf{y},\mathbf{u})$ is expanded to second order around the
posterior mode $\hat{\mathbf{u}}$. More recent approximations Girolami and
Rogers (2005); Girolami and Zhong (2006) can be much more accurate, yet come
with non-convex problems and less robust algorithms (Nickisch and Rasmussen,
2008). In this paper, we concentrate on the variational lower bound relaxation
(VB) by Jaakkola and Jordan (2000), which is convex for log-concave
likelihoods $\mathbb{P}(\mathbf{y}|\mathbf{u})$ (Nickisch and Seeger, 2009),
providing a novel simple and efficient algorithm. While our VB approximation
to MLM is more expensive to run than joint MAP for non-Gaussian likelihood
(even using Laplace’s approximation), the implementation complexity of our VB
algorithm is comparable to what is required in the Gaussian noise case
equation 6.
More, specifically, we exploit that super-Gaussian of likelihoods
$\mathbb{P}(y_{i}|u_{i})$ can be lower bounded by scaled Gaussians
$\mathcal{N}_{\gamma_{i}}$ of any width $\gamma_{i}$:
$\mathbb{P}(y_{i}|u_{i})=\max_{\gamma_{i}>0}\mathcal{N}_{\gamma_{i}}=\max_{\gamma_{i}>0}\exp\left(\beta_{i}u_{i}-\frac{u_{i}^{2}}{2\gamma_{i}}-\frac{1}{2}h_{i}(\gamma_{i})\right),$
where $\beta_{i}\propto y_{i}$ are constants, and $h_{i}(\cdot)$ is convex
(Nickisch and Seeger, 2009) whenever the likelihood $\mathbb{P}(y_{i}|u_{i})$
is log-concave. If the posterior distribution is
$\mathbb{P}(\mathbf{u}|\mathbf{y})=Z^{-1}\mathbb{P}(\mathbf{y}|\mathbf{u})\mathbb{P}(\mathbf{u})$,
then $\ln Z\geq Ce^{-\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})/2}$ by plugging
in these bounds, where $C$ is a constant and
$\phi_{\text{VB}}(\bm{\theta}):=\min_{\bm{\gamma}\succ\mathbf{0}}\psi_{\text{VB}}(\bm{\theta},\bm{\gamma}),\quad\psi_{\text{VB}}(\bm{\theta},\bm{\gamma}):=h(\bm{\gamma})-2\ln\int\mathcal{N}(\mathbf{u}|\mathbf{0},\mathbf{K}_{\bm{\theta}})e^{\mathbf{u}^{\top}(\bm{\beta}-\frac{1}{2}\bm{\Gamma}{}^{-1}\mathbf{u})}\text{d}\mathbf{u},$
(10)
$h(\bm{\gamma}):=\sum_{i}h_{i}(\gamma_{i})$,
$\bm{\Gamma}:=\text{dg}(\bm{\gamma})$. The variational
relaxation111Generalisations to other super-Gaussian potentials (log-concave
or not) or models including linear couplings and mixed potentials are given by
Nickisch and Seeger (2009). amounts to maximising the lower bound, which means
that $\mathbb{P}(\mathbf{u}|\mathbf{y})$ is fitted by the _Gaussian_
approximation $\mathbb{Q}(\mathbf{u}|\mathbf{y};\bm{\gamma})$ with covariance
matrix $\mathbf{V}=(\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}{}^{-1})^{-1}$
(Nickisch and Seeger, 2009). Alternatively, we can interpret
$\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})$ as an upper bound on the Kullback-
Leibler divergence
$\text{KL}(\mathbb{Q}(\mathbf{u}|\mathbf{y};\bm{\gamma})||\mathbb{P}(\mathbf{u}|\mathbf{y}))$
(Nickisch, 2010, §2.5.9), a measure for the dissimilarity between the exact
posterior $\mathbb{P}(\mathbf{u}|\mathbf{y})$ and the parametrised Gaussian
approximation $\mathbb{Q}(\mathbf{u}|\mathbf{y};\bm{\gamma})$.
Finally, note that by equation (7),
$\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})$ can also be written as
$\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})=\ln|\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1}|+h(\bm{\gamma})+\min_{\mathbf{u}}R(\mathbf{u},\bm{\theta},\bm{\gamma})+\ln|\mathbf{K}_{\bm{\theta}}|,$
(11)
where
$R(\mathbf{u},\bm{\theta},\bm{\gamma})=\mathbf{u}^{\top}(\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1})\mathbf{u}-2\bm{\beta}^{\top}\mathbf{u}$.
Using the concavity of
$\bm{\gamma}^{-1}\mapsto\ln|\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1}|$
and Fenchel duality
$\ln|\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1}|=\min_{\mathbf{z}\succ\mathbf{0}}\mathbf{z}^{\top}\bm{\gamma}^{-1}-g_{\bm{\theta}}^{*}(\mathbf{z})=\hat{\mathbf{z}}_{\bm{\theta}}^{\top}\bm{\gamma}^{-1}-g_{\bm{\theta}}^{*}(\hat{\mathbf{z}}_{\bm{\theta}})$,
with the optimal value $\hat{\mathbf{z}}_{\bm{\theta}}=\text{dg}(\mathbf{V})$,
we can reformulate $\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})$ as
$\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})=\min_{\mathbf{z}\succ\mathbf{0}}[\mathbf{z}^{\top}\bm{\gamma}^{-1}-g_{\bm{\theta}}^{*}(\mathbf{z})]+h(\bm{\gamma})+\min_{\mathbf{u}}R(\mathbf{u},\bm{\theta},\bm{\gamma})+\ln|\mathbf{K}_{\bm{\theta}}|,$
which allows to perform the minimisation w.r.t. $\bm{\gamma}$ in closed form
(Nickisch, 2010, §3.5.6):
$\phi_{\text{VB}}(\bm{\theta})=\min_{\mathbf{z}\succ\mathbf{0}}\psi_{\text{VB}}(\bm{\theta},\mathbf{z}),\quad\psi_{\text{VB}}(\bm{\theta},\mathbf{z})=\min_{\mathbf{u}}\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\tilde{\ell}_{\mathbf{z}}(\mathbf{y},\mathbf{u})-g_{\bm{\theta}}^{*}(\mathbf{z})+\ln|\mathbf{K}_{\bm{\theta}}|,$
(12)
where
$\tilde{\ell}_{\mathbf{z}}(\mathbf{y},\mathbf{u}):=2\bm{\beta}^{\top}(\mathbf{v}-\mathbf{u})-2\ln\mathbb{P}(\mathbf{y}|\mathbf{v})$
and finally
$\mathbf{v}=\text{sign}(\mathbf{u})\odot\sqrt{\mathbf{u}^{2}+\mathbf{z}}$.
Note that for $\mathbf{z}=\mathbf{0}$, we exactly recover joint MAP
estimation, equation (4), as $\mathbf{z}=\mathbf{0}$ implies
$\mathbf{u}=\mathbf{v}$ and
$\tilde{\ell}_{\mathbf{z}}(\mathbf{y},\mathbf{u})=\ell(\mathbf{y},\mathbf{u})$.
For fixed $\bm{\theta}$, the optimal value
$\hat{\mathbf{z}}_{\bm{\theta}}=\text{dg}(\mathbf{V})$ corresponds to the
marginal variances of the Gaussian approximation
$\mathbb{Q}(\mathbf{u}|\mathbf{y};\bm{\gamma})$: Variational inference
corresponds to variance-smoothed joint MAP estimation (Nickisch, 2010) with a
loss function $\tilde{\ell}(\mathbf{y},\mathbf{u},\bm{\theta})$ that
explicitly depends on the kernel parameters $\bm{\theta}$. We have two
equivalent representations of the loss
$\tilde{\ell}(\mathbf{y},\mathbf{u},\bm{\theta})$ that directly follow from
equations (11) and (12):
$\displaystyle\tilde{\ell}(\mathbf{y},\mathbf{u},\bm{\theta})$
$\displaystyle=$
$\displaystyle\min_{\bm{\gamma}\succ\mathbf{0}}[\ln|\mathbf{K}_{\bm{\theta}}^{-1}+\bm{\Gamma}^{-1}|+h(\bm{\gamma})+\mathbf{u}^{\top}\bm{\Gamma}^{-1}\mathbf{u}-2\bm{\beta}^{\top}\mathbf{u}],\>\text{and}$
$\displaystyle\tilde{\ell}(\mathbf{y},\mathbf{u},\bm{\theta})$
$\displaystyle=$
$\displaystyle\min_{\mathbf{z}\succ\mathbf{0}}[2\bm{\beta}^{\top}(\mathbf{v}-\mathbf{u})-2\ln\mathbb{P}(\mathbf{y}|\mathbf{v})-g_{\bm{\theta}}^{*}(\mathbf{z})],\;\mathbf{v}=\text{sign}(\mathbf{u})\odot\sqrt{\mathbf{u}^{2}+\mathbf{z}}.$
Our VB problem is
$\min_{\bm{\theta}\succeq\mathbf{0},\bm{\gamma}\succ\mathbf{0}}\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})$
or equivalently
$\min_{\bm{\theta}\succeq\mathbf{0},\mathbf{z}\succ\mathbf{0}}\psi_{\text{VB}}(\bm{\theta},\mathbf{z})$.
The inner variables here are $\bm{\gamma}$ and $\mathbf{z}$, in addition to
$\mathbf{u}$ in joint MAP. There are further similarities: since
$\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})=-2\ln\int
e^{-R(\mathbf{u},\bm{\gamma},\bm{\theta})}\text{d}\mathbf{u}+h(\bm{\gamma})+\ln|2\pi\mathbf{K}_{\bm{\theta}}|$,
$(\bm{\gamma},\bm{\theta})\mapsto\psi_{\text{VB}}-\ln|\mathbf{K}_{\bm{\theta}}|$
is jointly convex for $\bm{\gamma}\succ\mathbf{0}$,
$\bm{\theta}\succeq\mathbf{0}$, by the joint convexity of
$(\mathbf{u},\bm{\gamma},\bm{\theta})\mapsto R$ and Prékopa’s theorem (Boyd
and Vandenberghe, 2002, §3.5.2). Joint MAP and VB share the same convexity
structure. In contrast, approximating $\mathbb{P}(\mathbf{y}|\bm{\theta})$ by
other techniques like Expectation Propagation (Minka, 2001) or general
Variational Bayes (Opper and Archambeau, 2009) does not even constitute convex
problems for fixed $\bm{\theta}$.
### 2.4 Summary and Taxonomy
Name | Objective function
---|---
Marginal Likelihood Maximisation | $\phi_{\text{MLM}}(\bm{\theta})=-2\ln\left[\int\mathcal{N}(\mathbf{u}|\mathbf{0},\mathbf{K}_{\bm{\theta}})\mathbb{P}(\mathbf{y}|\mathbf{u})\text{d}\mathbf{u}\right]$
Variational Bounds | $\phi_{\text{VB}}(\bm{\theta})=\min_{\bm{\gamma}\succ\mathbf{0}}\psi_{\text{VB}}(\bm{\theta},\bm{\gamma})\geq\phi_{\text{MLM}}(\bm{\theta})$ by $\mathbb{P}(y_{i}|u_{i})\geq\mathcal{N}_{\gamma_{i}}$
Maximum A Posteriori | $\phi_{\text{MAP}}(\bm{\theta})=-2\ln\left[\max_{\mathbf{u}}\mathcal{N}(\mathbf{u}|\mathbf{0},\mathbf{K}_{\bm{\theta}})\mathbb{P}(\mathbf{y}|\mathbf{u})\right]=\psi_{\text{VB}}(\bm{\theta},\mathbf{z}=\mathbf{0})$
Multiple Kernel Learning | $\phi_{\text{MKL}}(\bm{\theta})=\phi_{\text{MAP}}(\bm{\theta})+\lambda\left\|\bm{\theta}\right\|_{p}^{p}-\ln|\mathbf{K}_{\bm{\theta}}|=\psi_{\text{MAP}}(\bm{\theta},\bm{\lambda}=\lambda\cdot\mathbf{1})$
| General $\mathbb{P}(y_{i}|u_{i})$ | | Gaussian $\mathbb{P}(y_{i}|u_{i})$ |
---|---|---|---|---
| $\phi_{\text{MLM}}(\bm{\theta})$, eq. (5) | $\longrightarrow$ | $\phi_{\text{GAU}}(\bm{\theta})$, eq. (6) |
Super-Gaussian Bounding | $\downarrow$ | | $\downarrow$ | Bound is tight
| $\phi_{\text{VB}}(\bm{\theta})$, eq. (10) | $\longrightarrow$ | $\phi_{\text{GAU}}(\bm{\theta})$, eq. (6) |
Maximum instead of integral | $\downarrow$ | | $\downarrow$ | Mode $\equiv$ mean
| $\phi_{\text{MAP}}(\bm{\theta})$, eq. (4) | $\longrightarrow$ | $\phi_{\text{MAP/GAU}}(\bm{\theta})$, eq. (8) |
Bound $\ln|\mathbf{K}_{\bm{\theta}}|\leq\lambda\left\|\bm{\theta}\right\|_{p}^{p}-g^{*}(\lambda\mathbf{1})$ | $\downarrow$ | | $\downarrow$ |
| $\phi_{\text{MKL}}(\bm{\theta})$, eq. (3) | $\longrightarrow$ | $\phi_{\text{RR}}(\bm{\theta})$, eq. (9) |
Table 2: Taxonomy of kernel learning objective functions
The upper table visualises the relationship between several kernel learning
objective functions for arbitrary likelihood/loss functions: Marginal
likelihood maximisation (MLM) can be bounded by variational bounds (VB) and
maximum a posteriori estimation (MAP) is a special case
$\mathbf{z}=\mathbf{0}$ thereof. Finally multiple kernel learning (MKL) can be
understood as an upper bound to the MAP estimation objective
$\bm{\lambda}=\lambda\cdot\mathbf{1}$. The lower table complements the upper
table by also covering the analytically important Gaussian case.
In the last paragraphs, we have detailed how a variety of kernel learning
approaches can be obtained from Bayesian marginal likelihood maximisation in a
sequence of nested upper bounding steps. Table 2.4 nicely illustrates how many
kernel learning objectives are related to each other – either by upper bounds
or by Gaussianity assumptions. We can clearly see, that
$\phi_{\text{VB}}(\bm{\theta})$ – as an upper bound to the negative log
marginal likelihood – can be seen as the mother function. For a special case,
$\mathbf{z}=\mathbf{0}$, we obtain joint maximum a posteriori estimation,
where the loss functions does not depend on the kernel parameters. Going
further, a particular instance $\bm{\lambda}=\lambda\cdot\mathbf{1}$ yields
the widely use multiple kernel learning objective that becomes convex in the
kernel parameters $\bm{\theta}$. In the following, we will concentrate on the
optimisation and computational similarities between the approaches.
## 3 Algorithms
In this section, we derive a simple, provably convergent and efficient
algorithm for MKL, joint MAP and VB. We use the Lagrangian form of equation
(3) and $\ell(\mathbf{y},\mathbf{u}):=-2\ln\mathbb{P}(\mathbf{y}|\mathbf{u})$:
$\displaystyle\psi_{\text{MKL}}(\bm{\theta},\mathbf{u})$ $\displaystyle=$
$\displaystyle\mathbf{u}^{\top}\mathbf{K}^{-1}\mathbf{u}+\qquad\;\ell(\mathbf{y},\mathbf{u})\qquad\qquad\qquad\qquad\qquad\;\,+\lambda\cdot\mathbf{1}^{\top}\bm{\theta},\>\lambda>0,$
$\displaystyle\psi_{\text{MAP}}(\bm{\theta},\mathbf{u})$ $\displaystyle=$
$\displaystyle\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\qquad\;\ell(\mathbf{y},\mathbf{u})\qquad\qquad\qquad\qquad\qquad\;\,+\ln|\mathbf{K}_{\bm{\theta}}|,\quad\text{and}$
$\displaystyle\psi_{\text{VB}}(\bm{\theta},\mathbf{u})$ $\displaystyle=$
$\displaystyle\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\min_{\mathbf{z}\succ\mathbf{0}}\left[\ell(\mathbf{y},\mathbf{v})+2\bm{\beta}^{\top}(\mathbf{v}-\mathbf{u})-g_{\bm{\theta}}^{*}(\mathbf{z})\right]+\ln|\mathbf{K}_{\bm{\theta}}|,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{where}\;\mathbf{v}=\text{sign}(\mathbf{u})\odot\sqrt{\mathbf{u}^{2}+\mathbf{z}}.$
Many previous algorithms use alternating minimization, which is easy to
implement but tends to converge slowly. Both $\phi_{\text{VB}}$ and
$\phi_{\text{MAP}}$ are jointly convex up to the concave
$\bm{\theta}\mapsto\ln|\mathbf{K}_{\bm{\theta}}|$ part. Since
$\ln|\mathbf{K}_{\bm{\theta}}|=\min_{\bm{\lambda}\succ\mathbf{0}}\bm{\lambda}^{\top}\bm{\theta}-f^{*}(\bm{\lambda})$
(Legendre duality, Boyd and Vandenberghe, 2002), joint MAP becomes
$\min_{\bm{\lambda}\succ\mathbf{0},\bm{\theta}\succeq\mathbf{0},\mathbf{u}}\phi_{\bm{\lambda}}(\bm{\theta},\mathbf{u})$
with
$\phi_{\bm{\lambda}}:=\mathbf{u}^{\top}\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{u}+\ell(\mathbf{y},\mathbf{u})+\bm{\lambda}^{\top}\bm{\theta}-f^{*}(\bm{\lambda})$
which is jointly convex in $(\bm{\theta},\mathbf{u})$. Algorithm 1 iterates
between refits of $\bm{\lambda}$ and joint Newton updates of
$(\bm{\theta},\mathbf{u})$.
0: Criterion
$\psi_{\\#}(\bm{\theta},\mathbf{u})=\tilde{\psi}_{\\#}(\bm{\theta},\mathbf{u})+\ln|\mathbf{K}_{\bm{\theta}}|$
to minimise for
$(\mathbf{u},\bm{\theta})\in\mathbb{R}^{n}\times\mathbb{R}_{+}^{M}$.
repeat
Newton $\min_{\mathbf{u}}\psi_{\\#}$ for fixed $\bm{\theta}$ (optional; few
steps).
Refit upper bound:
$\bm{\lambda}\leftarrow\nabla_{\bm{\theta}}\ln|\mathbf{K}_{\bm{\theta}}|=[\text{tr}(\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{K}_{1}),..,\text{tr}(\mathbf{K}_{\bm{\theta}}^{-1}\mathbf{K}_{M})]^{\top}$.
Compute joint Newton search direction $\mathbf{d}$ for
$\psi_{\bm{\lambda}}:=\tilde{\psi}_{\\#}+\bm{\lambda}^{\top}\bm{\theta}$:
$\nabla_{[\bm{\theta};\mathbf{u}]}^{2}\psi_{\bm{\lambda}}\mathbf{d}=-\nabla_{[\bm{\theta};\mathbf{u}]}\psi_{\bm{\lambda}}$.
Linesearch: Minimise $\psi_{\\#}(\alpha)$ i.e.
$\psi_{\\#}(\bm{\theta},\mathbf{u})$ along
$[\bm{\theta};\mathbf{u}]+\alpha\mathbf{d}$, $\alpha>0$.
until Outer loop converged
Algorithm 1 Double loop algorithm for joint MAP, MKL and VB.
The Newton direction costs $O(n^{3}+M\,n^{2})$, with $n$ the number of data
points and $M$ the number of base kernels. All algorithms discussed in this
paper require $O(n^{3})$ time, apart from the requirement to store the base
matrices $\mathbf{K}_{m}$. The convergence proof hinges on the fact that
$\phi$ and $\phi_{\bm{\lambda}}$ are tangentially equal (Nickisch and Seeger,
2009). Equivalently, the algorithm can be understood as Newton’s method, yet
dropping the part of the Hessian corresponding to the $\ln|\mathbf{K}|$ term
(note that
$\nabla_{(\mathbf{u},\bm{\theta})}\phi_{\bm{\lambda}}=\nabla_{(\mathbf{u},\bm{\theta})}\phi$
for the Newton direction computation). Exact Newton for MKL.
In practice, we use
$\mathbf{K}_{\bm{\theta}}=\sum_{m}\theta_{m}\mathbf{K}_{m}+\varepsilon\mathbf{I},\>\varepsilon=10^{-8}$
to avoid numerical problems when computing $\bm{\lambda}$ and
$\ln|\mathbf{K}_{\bm{\theta}}|$. We also have to enforce
$\bm{\theta}\succeq\mathbf{0}$ in algorithm 1, which is done by the barrier
method (Boyd and Vandenberghe, 2002). We minimise
$t\phi+\mathbf{1}^{\top}(\ln\bm{\theta})$ instead of $\phi$, increasing $t>0$
every few outer loop iterations.
A variant algorithm 1 can be used to solve VB in a different parametrisation
($\bm{\gamma}\succ\mathbf{0}$ replaces $\mathbf{u}$), which has the same
convexity structure as joint MAP. Transforming equation (10) similarly to
equation (6), we obtain
$\phi_{\text{VB}}(\bm{\theta})=\min_{\bm{\gamma}\succ\mathbf{0}}\ln|\mathbf{C}|-\ln|\bm{\Gamma}|+\bm{\beta}^{\top}\bm{\Gamma}\mathbf{C}^{-1}\bm{\Gamma}\bm{\beta}-\bm{\beta}^{\top}\bm{\Gamma}\bm{\beta}+h(\bm{\gamma})$
(13)
with $\mathbf{C}:=\mathbf{K}_{\bm{\theta}}+\bm{\Gamma}$, computed using the
Cholesky factorisation $\mathbf{C}=\mathbf{L}\mathbf{L}{}^{\top}$. They cost
$O(M\,n^{3})$ to compute, which is more expensive than for joint MAP or MKL.
Note that the cost $O(M\,n^{3})$ is not specific to our particular relaxation
or algorithm e.g. the Laplace MLM approximation (Williams and Barber, 1998),
solved using gradients w.r.t. $\bm{\theta}$ only, comes with the same
complexity.
## 4 Conclusion
We presented a unifying probabilistic viewpoint to multiple kernel learning
that derives regularised risk approaches as special cases of approximate
Bayesian inference. We provided an efficient and provably convergent
optimisation algorithm suitable for regression, robust regression and
classification.
Our taxonomy of multiple kernel learning approaches connected many previously
only loosely related ideas and provided insights into the common structure of
the respective optimisation problems. Finally, we proposed an algorithm to
solve the latter efficiently.
## References
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* Rasmussen and Williams (2006) Carl Edward Rasmussen and Christopher K. I. Williams. _Gaussian Processes for Machine Learning_. MIT Press, 2006.
* Schölkopf and Smola (2002) Bernhard Schölkopf and Alex Smola. _Learning with Kernels_. MIT Press, 1st edition, 2002.
* Sollich (2000) Peter Sollich. Probabilistic methods for support vector machines. In _NIPS_ , 2000.
* Sonnenburg et al. (2006) Sören Sonnenburg, Gunnar Rätsch, Christin Schäfer, and Bernhard Schölkopf. Large scale multiple kernel learning. _JMLR_ , 7:1531–1565, 2006.
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|
arxiv-papers
| 2011-03-04T13:32:28 |
2024-09-04T02:49:17.440954
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hannes Nickisch and Matthias Seeger",
"submitter": "Hannes Nickisch",
"url": "https://arxiv.org/abs/1103.0897"
}
|
1103.0927
|
# Conditional control of quantum beats in a cavity QED system
D G Norris A D Cimmarusti and L A Orozco Joint Quantum Institute, Department
of Physics, University of Maryland and National Institute of Standards and
Technology, College Park, MD 20742-4111, U.S.A. lorozco@umd.edu
###### Abstract
We probe a ground-state superposition that produces a quantum beat in the
intensity correlation of a two-mode cavity QED system. We mix drive with
scattered light from an atomic beam traversing the cavity, and effectively
measure the interference between the drive and the light from the atom. When a
photon escapes the cavity, and upon detection, it triggers our feedback which
modulates the drive at the same beat frequency but opposite phase for a given
time window. This results in a partial interruption of the beat oscillation in
the correlation function, that then returns to oscillate.
## 1 Introduction
Quantum feedback [1] and quantum control [2] are important disciplines with
relationships to quantum information science. The question of how to control a
quantum system without disturbing it remains open in general [3], but the
search for efficient protocols continues and the experimental realization is
now using weak quantum measurements. (See for example the recent paper by
Gillett et al. [4]).
The detection of a photon escaping a quantum system at a random time heralds
the preparation of a conditional quantum state. Manipulation of these states
is essential in the field of quantum feedback. The preferential probe of this
conditional measurement in quantum optics is the intensity correlation
function which has been used since the pioneer work of Kimble et al. on
resonance fluorescence [5].
This paper presents the preliminary implementation of indirectly coupled
quantum feedback in our cavity quantum electrodynamical (QED) system. It acts
on the ground state coherences we recently observed [6]. However, it builds up
on extensive literature that has looked into the evolution and control of
quantum states such as Refs. [7, 8]. This work closely follows our previous
studies [9, 10, 11], except our conditional state manipulation is long-lived
($\sim 5$ $\mu$s) and it consists of a ground state superposition detected
through a homodyne measurement done in photon counting.
Wiseman [12] established the connection between homodyne measurements and weak
measurements in cavity QED. Weak measurements reduce the problems of back
action in quantum feedback [4, 13]. Our previous work with conditional
homodyne detection [14, 15, 16] used a strong local oscillator. Recent
measurements perform homodyne detection of resonance fluorescence with a weak
local oscillator [17]. Our work is moving on that direction and we expect to
improve our ability to control the quantum states with new forms of feedback.
## 2 Description of the quantum beats
Quantum beats are amongst the first phenomena to be fully accounted for by
quantum mechanics [18]. They consist of oscillations in the radiation
intensity of a group of excited atoms due to interfering emission pathways.
Usually the atomic systems that exhibit quantum beats have the “Type I” (V
system) energy level structure: two excited states and a ground state. The
atoms are initially prepared in a superposition of the excited states, by for
example a broadband excitation pulse. They then decay to the ground state, and
beating occurs between the two decay paths at the difference between the
frequencies of emission (the excited state splitting). For “Type II”
($\Lambda$ system) atoms the situation is reversed. We now have only one
excited state and two ground states. References [18, 19, 20] show that no beat
is possible for “Type-II” atoms in which the two orthogonal ground states are
non-degenerate. However, while this is true for the mean intensity, beats may
still lie in the fluctuations [21]. Our recent work [6] shows the ground-state
quantum beats observed in the conditional evolution of a cavity quantum
electrodynamical (QED) system. They show dynamics lost on the average but
retained in the variance.
Figure 1: (color online) Simplified atomic energy structure with Zeeman
levels. The drive induces the $\pi$ transitions. (a) The atom decays back to
the ground state, but is now in a superposition. (b) The drive reexcites the
atom and it decays back to the initial state. (Figure based on Ref.[22].
We use an optical cavity QED system in the intermediate coupling regime (i.e.
the dipole coupling constant is of the same order as the cavity and
spontaneous emission decay rates) for the study of the beats. Consider a
single atom with hyperfine and Zeeman structure in the ground and excited
states. The atom interacts with two orthogonally polarized cavity modes,
vertical (V) and horizontal (H). We work in the weak, continuous drive regime
for the V mode. The excitation is such that we can keep up to two photons in
the undriven H mode for intensity correlation measurements. The transitions
take place between $F\rightarrow F^{\prime}$ ($F\neq F^{\prime}$). A weak
magnetic field defines the quantization axis in the direction of V. This mode
drives $\pi$ transitions. The ground ($\delta$) and excited
($\delta^{\prime}$) state Zeeman frequency shifts may be different, and we
limit the discussion to the six central Zeeman sublevels of the manifold as
indicated in Fig. 1. The cavity decay rate is such that photon leakage can
occur before reabsorption, so we neglect the latter [6, 22].
$\begin{split}\lvert\psi_{i}\rangle&=\lvert
b_{0},0\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(a)\\\
\lvert\psi_{i}^{c}(t)\rangle&=\alpha e^{i\delta t}\lvert b_{-1},0\rangle+\beta
e^{-i\delta t}\lvert b_{+1},0\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(b)\\\
\lvert\psi_{f}^{c}(t)\rangle&=\alpha e^{i\delta t}\lvert b_{0},1\rangle+\beta
e^{-i\delta t}\lvert
b_{0},1\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(c)\end{split}$ (1)
Figure 1 shows a simplified model for the physical origin of the quantum
beats. The system starts in state $\lvert\psi_{i}\rangle$ (see Eq. 1a and Fig.
1a) with the atom at the center of the Zeeman manifold and no photons in the
$H$ mode. There is a continuous excitation with $\pi$ light that at some point
excites the atom to $\lvert e_{0}\rangle$. Once there, the atom will
spontaneously decay to the ground state, emitting $\pi$, $\sigma^{+}$ or
$\sigma^{-}$ photons. In the case of $\pi$ polarized light, the radiation adds
to V, but both $\sigma$ polarizations have components in the H mode. Since the
helicity and frequency of a $\sigma$ photon cannot be determined in the H,V
basis, the detection of a photon escaping the H mode heralds that the atom is
in a superposition state of the $m=\pm 1$ ground states:
$\lvert\psi_{i}^{c}(t)\rangle$ (see Eq. 1b), that we label as the initial
$(i)$ conditional $(c)$ state. This functions as the first step in our quantum
eraser realization [23].
The coefficients $\alpha$ and $\beta$ of the superposition depend on Clebsch-
Gordan coefficients, the Zeeman ground and excited state shift difference, and
the excited state linewidth [6]. The atom, now in the ground state, but with
angular momentum perpendicular to the magnetic field, undergoes Larmor
precession with a time-dependent phase $\phi(t)=\delta t$ (see Eq. (1b), but
the continuous drive V can reexcite it (see Fig. 1b). The atom then can
spontaneously decay back to $\lvert b_{0}\rangle$, emitting a second photon
into the H mode( Eq. 1c) and leaving the system in the final ($f$) conditional
($c$) state $\lvert\psi_{f}^{c}(t)\rangle$. This second photon erases the path
information present in the intermediate state and represents the second step
in our quantum eraser realization.
The probability for the second emission depends on the phase $\phi(t)$
acquired since the detection of the first $H$ photon, so the quantum beats
only manifest themselves in the second-order intensity correlation function,
$g^{(2)}(\tau)$; they are not visible in the mean transmitted intensity. Eq.
(2) shows the calculation of the conditional intensity $\langle
I_{1}(t)\rangle_{c}$ of a second photon starting with the conditional final
state $\lvert\psi_{f}^{c}(t)\rangle$. This gives the unnormalized second-order
correlation function [24].
$\begin{split}\langle I_{1}(t)\rangle_{c}&=\langle\psi_{f}^{c}(t)\lvert
a^{\dagger}a\rvert\psi_{f}^{c}(t)\rangle\\\ &=(\alpha^{*}\beta e^{-2i\delta
t}+\alpha\beta^{*}e^{2i\delta
t}+\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}\\\
&=2\lvert\alpha\rvert\lvert\beta\rvert\cos(2\delta
t+\phi_{1})+\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}\end{split}$ (2)
Here $\phi_{1}$ is a possible complex phase difference between $\alpha$ and
$\beta$. The normalized second-order correlation function recovers the quantum
beats as an oscillation at frequency $2\delta$, twice the Larmor precession
frequency.
The real experiment is more complex, as we use an atomic beam rather than a
single atom. There can be many atoms in the cavity mode at any given time,
with a random distribution in the Gaussian transverse profile and standing
wave. In addition, small amounts of light from the driven mode may be coupled
into the orthogonal mode through cavity birefringence or optical elements.
Ref. [6] shows that even with these complications, the beats do survive, but
can come from three different physical mechanisms.
We use the work of Carmichael et al. [25] who give the analytical form (Eq. 3)
of the measured average second-order correlation function
$\overline{g_{s}^{(2)}}(\tau)$ for the problem of resonance fluorescence,
taking into account atomic number fluctuations in a beam. Although our system
is not strictly this, since atoms in a cavity mode are not fully independent,
their treatment is approximately valid under the assumption of no reabsorption
of an emitted $H$ photon in our cavity:
$\overline{g_{s}^{(2)}}(\tau)=1+\frac{1}{\left(1+\Upsilon/\bar{N}\right)^{2}}\left(\frac{g_{A}^{(2)}(\tau)}{\bar{N}}+\left|g_{A}^{(1)}(\tau)\right|^{2}f(A)+\frac{2\Upsilon}{\bar{N}}\text{Re}\left(g_{A}^{(1)}(\tau)\right)f_{D}(A)\right)$
(3)
Here $\Upsilon$ is the background-to-signal ratio for a single atom (the
background can consist of a small amount of mixed drive from the V mode),
$\bar{N}$ is the mean number of atoms in the mode, and $g_{A}^{(1)}(\tau)$ and
$g_{A}^{(2)}(\tau)$ are the normalized single-atom first- and second-order
correlation functions. The functions $f(A)$ and $f_{D}(A)$ quantify the
spatial coherence within a detection area $A$ for terms containing products of
fields from different sources. Since we collect light from a single cavity
spatial and polarization mode, H, each has a value of unity.
The first source of beats, described previously in detail in Eqs 1 and 2, is
the $g_{A}^{(2)}(\tau)$ term in Eq. (3). The second source of beats is the
$\left|g_{A}^{(1)}(\tau)\right|^{2}$ term, a two-atom contribution arising
from interference in the time-ordering of emissions from indistinguishable
atoms which we associate to a conditional intensity $\langle
I_{2}(t)\rangle_{c}$. (This is the same term that gives photon bunching in
thermal light [26], as observed by Hanbury-Brown and Twiss [27].) The last
contribution comes from the $\text{Re}\left(g_{A}^{(1)}(\tau)\right)$ term.
This is an interference between the background $\Upsilon$ and the light
emitted by a single atom. We refer to it as a homodyne beat, and it occurs at
the single Larmor frequency. We recover it in the correlation function when
the background is large enough so that this term dominates.
Equation 4 shows a simplified way to obtain this third term (again we consider
only a single atom in the cavity, so there will be no two-atom contributions.)
We calculate the conditional intensity $\langle I^{\prime}(t)\rangle_{c}$ (Eq.
4a,b) for a second photon starting from the atomic superposition of
$\lvert\psi_{f}^{c}(t)\rangle$ that has already deposited a photon in the $H$
cavity mode and we mix a certain amount of the drive $\eta$ into the H mode
such that we now have the conditional state $\lvert\psi^{\prime
c}(t)\rangle=\lvert\psi_{f}^{c}(t)\rangle+\eta\lvert b_{0},1\rangle$. The
result shows that there are two parts: We recover the single atom contribution
$\langle I_{1}(t)\rangle_{c}$, but there are also other terms that we collect
and label $\langle I_{3}(t)\rangle_{c}$:
$\begin{split}\langle I^{\prime}(t)\rangle_{c}&=\langle\psi^{\prime
c}(t)\lvert a^{\dagger}a\rvert\psi^{\prime
c}(t)\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(a)\\\
&=\langle I_{1}(t)\rangle_{c}+2\lvert\eta\rvert\lvert\alpha\rvert\cos(\delta
t+\phi_{3})+2\lvert\eta\rvert\lvert\beta\rvert\cos(\delta
t+\phi^{\prime}_{3})+\lvert\eta\rvert^{2},~{}~{}~{}~{}~{}~{}~{}(b)\\\ \langle
I_{3}(t)\rangle_{c}&=2\lvert\eta\rvert\lvert\alpha\rvert\cos(\delta
t+\phi_{3})+2\lvert\eta\rvert\lvert\beta\rvert\cos(\delta
t+\phi^{\prime}_{3})+\lvert\eta\rvert^{2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(c)\end{split}$
(4)
Here $\phi_{3}$ and $\phi^{\prime}_{3}$ are respectively the complex phase
differences between $\alpha$ and $\eta$ and $\beta$ and $\eta$. All these
parameters have stable values because they are in a cavity. This homodyne term
is different from the others in that it can take negative values and thus
cause $\overline{g_{s}^{(2)}}(\tau)$ to dip below unity. (See in particular
Fig. 4e in Ref. [6] and the relevant discussion in the text.)
The crux of our feedback protocol lies in the manipulation of this homodyne
beat. We mix light from the V mode with light from the H mode on the cavity
output. The increasing fraction of V mode light causes the homodyne beat term
to dominate in $\overline{g_{s}^{(2)}}(\tau)$. While the feedback should in
principle work equally well when the other terms dominate, the larger
visibility of the homodyne term makes it easier to work with experimentally.
## 3 Experimental setup
Figure 2 shows the main features of the experiment. A 780 nm linearly
polarized laser beam drives the TEM00 $V$ mode of a Fabry-Perot optical cavity
in resonance with the $D_{2}$ line of 85Rb. A cold beam of atoms goes through
the cavity at near perpendicular incidence with respect to the mode. As the
lifetime of the excited state of Rb is only 26 ns, the atoms undergo multiple
excitations during the transit time of $\sim 5$ $\mu$s. The mirrors sit 2.2 mm
apart, forming a 11,000 finesse cavity with decay rate $\kappa/2\pi=3.2\times
10^{6}~{}$s-1 comparable to the atomic decay rate $\gamma/2\pi=6\times
10^{6}~{}$s-1 and dipole coupling constant $g/2\pi=1.5$ MHz, for the
transition $5S_{1/2}~{}(F,m)=(3,0)\rightarrow
5P_{3/2}~{}(F^{\prime},m^{\prime})=(4,0)$. For a more detailed description of
the apparatus, see Ref. [28].
Figure 2: (color online) Schematic of the apparatus. HWP: Half-Wave Plate,
APD: Avalanche Photo-Diode, PBS: Polarizing Beam Splitter, BS: Beam Splitter,
AOM: Acousto-Optic Modulator
Atoms enter the cavity optically pumped to $5S_{1/2}~{}F=3,m=0$ which
corresponds to our $\lvert b_{0}\rangle$. A Glan-Thompson polarizer and zero-
order half-wave plate (HWP) placed before the cavity linearly polarize the
drive with a very good extinction ratio that can reach better than $5\times
10^{-5}$. After the cavity another HWP aligns the output polarization to a
Wollaston polarizing beam splitter (PBS) to separate the H and V modes. The H
light passes through a regular beam splitter (BS) which divides the light
between two avalanche photodiodes (APD). Both detector outputs then go to a
correlator card (Becker and Hickl DPC-230) which records a continuous stream
of detection times with a resolution of 164 ps.
The rest of the components in Fig. 2 enable the feedback protocol. The
“electronics box” represents the following: The pulse from the ‘start’ APD
(designated arbitrarily) is split into two and passed through a Lecroy 688AL
level adaptor to produce a clean TTL pulse. This triggers an HP 33120 signal
generator whose output controls the amplitude modulation port of an Isomet
D323B radio frequency driver box. The driver connects to an 80 MHz Crystal
Technology 3080-122 acousto-optical modulator (AOM), whose first-order
diffracted beam drives the cavity. In this way, the intensity of the drive can
be modulated conditionally, based on the trigger from the ‘start’ APD.
## 4 Preliminary results
Figure 3: (color online) $g^{(2)}(\tau)$ exhibiting a quantum beat oscillation
with $f=860$ KHz, corresponding to a magnetic field strength of 1.8 G
We measure the intensity correlation function ${g^{(2)}}(\tau)$ from our
cavity in a regime where the homodyne quantum beat term dominates, which we
achieve by changing the angle on the HWP after the cavity by approximately 2
degrees away from maximum drive extinction, which increases the value of
$\Upsilon$. The effective number of maximally coupled atoms in the mode is
approximately 2. Fig. 3 shows our normalized second-order correlation function
due primarily to the beating against the drive; this is apparent because it
dips below one.
The basic idea for control is simple. We rely on conditional measurements to
set the initial phase of the quantum beat. Since the intensity of the detected
light is proportional to the drive intensity (from both the atomic spontaneous
emission and the driven mode response), we can modulate the drive at the same
frequency as the conditional output signal but with opposite phase. This way
the beat will cancel as long as the modulation amplitude is chosen correctly.
Figure 4: (color online) Calculated $g^{(2)}(\tau)$ signal from the feedback
model with parameters extracted from the experiment. (a) The red squares are
model without feedback. The blue trace is model calculation exhibiting the
effects of our feedback. The brown trace at the bottom identifies the time
window where we apply the feedback. (b) Shows in green the difference between
the red squares (no feedback) and the blue line (with feedback).
We are able to model the signal (after 0.5 $\mu$s) with a simple function that
contains an oscillation ($\cos{\Omega t}$) at frequency $\Omega/2\pi$=860 kHz,
Gaussian damping ($\exp{-(t^{2}/\sigma^{2})}$) with $\sigma=1.8~{}\mu$s, and
amplitude and time offsets; the intent is to capture the basic physics, not to
fit the exact form. The oscillation corresponds to the Larmor frequency and
the characteristic time of the Gaussian reflects the transit time of the atoms
through the Gaussian transverse profile of the mode. The sharp peak at the
origin is a multiatom contribution (see Fig. 4d in Ref. [6]) that we are not
taking into account. We obtain the numbers for the model by looking at the
fast Fourier transform (FFT) of the data in Fig. 3 as well as at the long term
($\approx$ 8 $\mu$s) value of the background. The width of the resonance in
the FFT fits well to a Gaussian, but there is an asymmetry on the
characteristic width; we average the two numbers and use that for the model.
There are other frequencies visible on the FFT, coming from the standing wave
modulation of the dipole coupling constant and from the harmonics of the
Larmor frequency; we ignore these in the model.
Figure 4a illustrates the usual signal (red squares) and the signal with the
feedback protocol (continuous blue line) based in the model of the signal that
we just presented. It is clear that there is a modification of the response
during the time that the pulse is applied, but the cancellation is not
perfect. The difference Fig. 4b between the trace with feedback and that
without recovers the applied modulation to the input drive.
A photon “click” in the ‘start’ detector triggers the signal generator, which
outputs a sinusoidal voltage pulse whose amplitude-to-offset ratio is 8.5%, in
a voltage region where the AOM and driver amplitude response is linear. The
delay in the application of the modulation to the drive has an intrinsic
($\sim 1.5$ $\mu$s) contribution from the signal generator, and a variable
part which we use to adjust the phase to match that of the quantum beats.
Figure 5: (color online) Experimental measurements of (a) $g^{(2)}(\tau)$. The
red squares is the negative-$\tau$ portion reflected back across the vertical
axis of the data. The blue trace is the positive-$\tau$ portion, exhibiting
the effects of our feedback. (b) shows in green the difference between the red
squares (no feedback) and blue traces (with feedback).
The feedback pulse lasts for one period of the quantum beat oscillation ($\sim
1.2$ $\mu$s), after which the beat returns with the same phase as before. We
obtain a partial attenuation of the oscillations (See blue line in Fig. 5a),
owing primarily to the mismatch between the shape of the applied pulse and the
measured $g^{(2)}(\tau)$. Performance can be improved with use of a
programmable pulse generator that matches more carefully the shape of the
decaying exponential. In addition, trigger events missed due to signal
generator dead time decrease the effects of the feedback.
## 5 Future work
We wish to fully and deterministically manipulate the quantum beats exhibited
by $g^{(2)}(\tau)$, in the sense of controlling the amplitude, phase, or
frequency of the atoms in the ground state. One possibility is to use laser
pulses from the cavity side to conditionally transport half of the ground-
state superposition to a different state. This excitation would interrupt the
coherent evolution, with the possibility of bringing it back with
deterministic phase by use of coherent Raman transitions. Reference [22] shows
applications in quantum error correction.
Preliminary attempts with a laser perpendicular to the cavity mode have
produced large mechanical effects in the atoms, pushing them out of the
cavity. Modifications to the apparatus should allow retroflection of the beam
to cancel such effects, as well as the application of arbitrary polarization
states.
## 6 Conclusions
Our cavity QED system exhibits a long-lived homodyne quantum beat which has
great potential for studies in the feedback and control of ground-state
coherences. A simple feedback mechanism that modulates the drive shows
moderate control of the conditional quantum beats in the intensity of the
photon correlations. More elaborate techniques will further probe the nature
of quantum feedback in our system. This work was supported by the National
Science Foundation (NSF). We thank Pablo Barberis-Blostein and Howard
Carmichael for their stimulating discussions and continued theoretical
support.
## References
## References
* [1] Wiseman H M and Milburn G J 2009 Quantum Measurement and Control (Cambridge: Cambridge University Press)
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|
arxiv-papers
| 2011-03-04T15:46:06 |
2024-09-04T02:49:17.448346
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D G Norris, A D Cimmarusti, L A Orozco",
"submitter": "Andres Cimmarusti",
"url": "https://arxiv.org/abs/1103.0927"
}
|
1103.1038
|
# Maximum Principle for Quasi-linear Backward Stochastic Partial Differential
Equations111Supported by NSFC Grant #10325101, by Basic Research Program of
China (973 Program) Grant # 2007CB814904, by the Science Foundation of the
Ministry of Education of China Grant #200900071110001, and by WCU (World Class
University) Program through the Korea Science and Engineering Foundation
funded by the Ministry of Education, Science and Technology
(R31-2009-000-20007).
Jinniao Qiu 222Department of Finance and Control Sciences, School of
Mathematical Sciences, Fudan University, Shanghai 200433, China. E-mail:
071018032@fudan.edu.cn (Jinniao Qiu), sjtang@fudan.edu.cn (Shanjian Tang). and
Shanjian Tang22footnotemark: 2 333Graduate Department of Financial
Engineering, Ajou University, San 5, Woncheon-dong, Yeongtong-gu, Suwon,
443-749, Korea.
###### Abstract
In this paper we are concerned with the maximum principle for quasi-linear
backward stochastic partial differential equations (BSPDEs for short) of
parabolic type. We first prove the existence and uniqueness of the weak
solution to quasi-linear BSPDE with the null Dirichlet condition on the
lateral boundary. Then using the De Giorgi iteration scheme, we establish the
maximum estimates and the global maximum principle for quasi-linear BSPDEs. To
study the local regularity of weak solutions, we also prove a local maximum
principle for the backward stochastic parabolic De Giorgi class.
AMS Subject Classification: 60H15; 35R60
Keywords: Stochastic partial differential equation, Backward stochastic
partial differential equation, De Giorgi iteration, Backward stochastic
parabolic De Gigorgi class
## 1 Introduction
In this paper we investigate the following quasi-linear BSPDE:
$\left\\{\begin{array}[]{l}\begin{split}-du(t,x)=\,&\displaystyle\biggl{[}\partial_{x_{j}}\Bigl{(}a^{ij}(t,x)\partial_{x_{i}}u(t,x)+\sigma^{jr}(t,x)v^{r}(t,x)\Bigr{)}+b^{j}(t,x)\partial_{x_{j}}u(t,x)\\\
&\displaystyle+c(t,x)u(t,x)+\varsigma^{r}(t,x)v^{r}(t,x)+g(t,x,u(t,x),\nabla
u(t,x),v(t,x))\\\ &\displaystyle+\partial_{x_{j}}f^{j}(t,x,u(t,x),\nabla
u(t,x),v(t,x))\biggr{]}\,dt\\\
&\displaystyle-v^{r}(t,x)\,dW_{t}^{r},\quad(t,x)\in
Q:=[0,T]\times\mathcal{O};\\\ u(T,x)=\,&G(x),\quad
x\in\mathcal{O}.\end{split}\end{array}\right.$ (1.1)
Here and in the following we use Einstein’s summation convention,
$T\in(0,\infty)$ is a fixed deterministic terminal time,
$\mathcal{O}\subset\mathbb{R}^{n}$ is a bounded domain with
$\partial\mathcal{O}\in C^{1}$,
$\nabla=(\partial_{x_{1}},\cdots,\partial_{x_{n}})$ is the gradient operator
and $(W_{t})_{t\in[0,T]}$ is an $m$-dimensional standard Brownian motion in
the filtered probability space $(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\geq
0},P)$. A solution of BSPDE (1.1) is a pair of random fields $(u,v)$ defined
on $\Omega\times[0,T]\times\mathcal{O}$ such that (1.1) holds in a weak sense
(see Definition 2.2).
The study of backward stochastic partial differential equations (BSPDEs) can
be dated back about thirty years ago (see Bensoussan [2] and Pardoux [19]).
Such BSPDE arises in many applications of probability theory and stochastic
processes, for instance in the nonlinear filtering and stochastic control
theory for processes with incomplete information, as an adjoint equation of
the Duncan-Mortensen-Zakai filtration equation (for instance, see [2, 14, 15,
23, 26, 27]). In the dynamic programming theory, some nonlinear BSPDEs as the
so-called backward stochastic Hamilton-Jacobi-Bellman equations, are also
introduced in the study of non-Markovian control problems (see Peng [20] and
Englezos and Karatzas [12]).
The maximum principle is a powerful tool to study the regularity of solutions,
and constitutes a beautiful chapter of the classical theory of deterministic
second-order elliptic and parabolic partial differential equations. Using the
technique of Moser’s iteration, Aronson and Serrin proved the maximum
principle and local bound of weak solutions for deterministic quasi-linear
parabolic equations (see [1, Theorems 1 and 2]), which are stated in the
backward form as the following two theorems.
###### Theorem 1.1.
Let $u$ be a weak solution of a quasi-linear parabolic equation
$\begin{array}[]{l}\begin{split}-\partial_{t}u=\partial_{x_{i}}\mathscr{A}_{i}(t,x,u,\nabla
u)+\mathscr{B}(t,x,u,\nabla u)\end{split}\end{array}$ (1.2)
in the bounded cylinder $Q=(0,T)\times\mathcal{O}\subset\mathbb{R}^{1+n}$ such
that $u\leq M$ on the parabolic boundary
$\left((0,T]\times\mathcal{O}\right)\cup\left(\\{T\\}\times\mathcal{O}\right)$.
Then almost everywhere in $Q$
$u\leq M+C\Xi(\mathscr{A},\mathscr{B})$
where the constant $C$ depends only on $T,|\mathcal{O}|$ and the structure
terms of the equation, while $\Xi(\mathscr{A},\mathscr{B})$ is expressed in
terms of some quantities related to the coefficients $\mathscr{A}$ and
$\mathscr{B}$.
###### Theorem 1.2.
Let $u$ be a weak solution of (1.2) in $Q$. Suppose that the set $Q_{3\rho}$
is contained in $Q$. Then almost everywhere in $Q_{\rho}$ we have
$|u(t,x)|\leq
C\left(\rho^{-(n+2)/2}\|u\|_{W^{2}(Q(3\rho))}+\rho^{\theta}\Xi_{1}(\mathscr{A},\mathscr{B})\right)$
where the constant $C$ depends only on $\rho$ and the structure terms of
(1.2), $Q_{\rho}:=(\bar{t},\bar{t}+\rho^{2})\times B_{\rho}(\bar{x})$,
$\theta\in(0,1)$ is one of the structure terms of (1.2) and
$\Xi_{1}(\mathscr{A},\mathscr{B})$ is expressed in terms of some quantities
related to the coefficients $\mathscr{A}$ and $\mathscr{B}$. In particular,
weak solutions of (1.2) must be locally essentially bounded.
In contrast with the deterministic one, the stochastic maximum principle has
received rather few discussions. We note that Denis and Matoussi [6], and
Denis, Matoussi, and Stoica [7] gave a stochastic version of Aronson and
Serrin’s above results, and obtained via Moser’s iteration scheme a stochastic
maximum principle, which claims an $L^{p}$ estimate for the time and space
maximal norm of weak solutions to forward quasi-linear stochastic partial
differential equations (SPDEs). Any stochastic maximum principle seems to be
lacking for backward ones in the literature, which then becomes quite
interesting to know.
In this paper, we concern the maximum principle of a weak solution to BSPDE
(1.1). Using the De Giorgi iteration scheme, we establish the global maximum
principle and the local boundedness theorem for quasi-linear BSPDEs (1.1),
which include the above two theorems as particular cases. As highlighted by
the classical theory of deterministic parabolic PDEs, our stochastic maximum
principle for BSPDEs is expected to be used in the study of Hölder continuity
of the solutions of BSPDEs and further in the study of more general quasi-
linear BSPDEs.
It is worth noting that our estimates for weak solutions are uniform with
respect to $w\in\Omega$. In contrast to Denis, Matoussi, and Stoica’s $L^{p}$
estimate ($p\in(2,\infty)$) for the time and space maximal norm of weak
solutions of (forward) quasi-linear SPDEs, we prove an $L^{\infty}$ estimate
for that of quasi-linear BSPDE (1.1). This distinction comes from the
essential difference between SPDEs and BSPDEs: the diffusion $v$ in BSPDE
(1.1) is endogenous, while the diffusion in the SPDEs is exogenous, which
makes impossible any $L^{\infty}$ estimate for a forward SPDE due to the
active white noise. On the other hand, indeed, the technique of Moser’s
iteration can also be used to study the behavior of weak solutions of BSPDE
(1.1) and to obtain the global and local maximum principles. However, as the
De Giorgi iteration scheme works for the degenerate parabolic case, we prefer
De Giorgi’s method in this paper and leave the application of Moser’s method
as an exercise to the interested reader.
Many works have been devoted to the linear and semi-linear BSPDEs either in
the whole space or in a domain (see, for instance, [8, 9, 10, 14, 24, 26,
27]). A theory of solvability of quausi-linear BSPDEs is recently established
in an abstract framework in Qiu and Tang [22]. However, it is prevailing in
these works to assume that the coefficients $b,c$ and $\varsigma$ are
essentially bounded. To inherit in our stochastic maximum principle the
general structure of admitting the unbounded coefficients $b$ and $c$ in the
deterministic maximum principle, we prove by approximation in Section 4 the
existence and uniqueness result (Theorem 4.1) for the weak solution to the
quasi-linear BSPDE (1.1) with the null Dirichlet condition on the lateral
boundary, under a new rather general framework. This result is invoked to
prove Proposition 4.3 as the Itô’s formula for the composition of solutions of
BSDEs into a class of time-space smooth functions, which is the starting point
of the De Giorgi scheme in the proof of subsequent stochastic maximum
principles.
This paper is organized as follows. In Section 2, we set notations, hypotheses
and the notion of the weak solution to BSPDE (1.1). In Section 3, we prepare
several auxiliary results, including a generalized Itô formula, which will be
used to establish Proposition 4.3 below as a key step in the study of our
stochastic maximum principle. In Section 4 we prove the existence and
uniqueness of the weak solution to BSPDE (1.1). Finally, in Section 5, we
establish the maximum principles for quasi-linear BSPDEs. In the first
subsection, we use the De Giorgi iteration scheme to obtain the global maximum
principles for BSPDEs (1.1) and in the second subsection, we prove the local
maximum principle for our backward stochastic parabolic De Giorgi class.
## 2 Preliminaries
Let $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ be a
complete filtered probability space on which is defined an $m$-dimensional
standard Brownian motion $W=\\{W_{t}:t\in[0,\infty)\\}$ such that
$\\{\mathscr{F}_{t}\\}_{t\geq 0}$ is the natural filtration generated by $W$
and augmented by all the $\mathbb{P}$-null sets in $\mathscr{F}$. We denote by
$\mathscr{P}$ the $\sigma$-algebra of the predictable sets on
$\Omega\times[0,T]$ associated with $\\{\mathscr{F}_{t}\\}_{t\geq 0}$.
Denote by $\mathbb{Z}$ the set of all the integers and by $\mathbb{N}$ the set
of all the positive integers. Denote by $|\cdot|$ and
$\langle\cdot,\cdot\rangle$ the norm and scalar product in a finite-dimension
Hilbert space. Like in $\mathbb{R},\mathbb{R}^{k},\mathbb{R}^{k\times l}$ with
$k,l\in\mathbb{N}$, we have defined
$|x|:=\left(\sum_{i=1}^{k}x^{2}_{i}\right)^{\frac{1}{2}}\quad\textrm{and}\quad|y|:=\left(\sum_{i=1}^{k}\sum_{j=1}^{l}y^{2}_{ij}\right)^{\frac{1}{2}}\quad\textrm{for}~{}(x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{k\times
l}.$
For the sake of convenience, we denote
$\partial_{s}:=\frac{\partial}{\partial{s}}\ \,{\rm and}\
\,\partial_{st}:=\frac{\partial^{2}}{\partial s\partial t}.$
Let $V$ be a Banach space equipped with norm $\|\cdot\|_{V}$. For real
$p\in(0,\infty)$, $\mathcal{S}^{p}(V)$ is the set of all the $V$-valued,
adapted and c$\grave{\textrm{a}}$dl$\grave{\textrm{a}}$g processes
$(X_{t})_{t\in[0,T]}$ such that
$\|X\|_{\mathcal{S}^{p}(V)}:=\left(E[\sup_{t\in[0,T]}\|X_{t}\|_{V}^{p}]\right)^{1\wedge\frac{1}{p}}<\infty.$
It is worth noting that ($\mathcal{S}^{p}(V)$,
$\|\cdot\|_{\mathcal{S}^{p}(V)}$) is a Banach space for $p\in[1,\infty)$ and
for $p\in(0,1)$, $dis(X,X^{\prime}):=\|X-X^{\prime}\|_{\mathcal{S}^{p}(V)}$ is
a metric of $\mathcal{S}^{p}(V)$ under which $\mathcal{S}^{p}(V)$ is complete.
Define the parabolic distance in $\mathbb{R}^{1+n}$ as follows:
$\delta(X,Y):=\max\\{|t-s|^{1/2},|x-y|\\},$
for $X:=(t,x)$ and $Y:=(s,y)\in\mathbb{R}^{1+n}$. Denote by $Q_{r}(X)$ the
ball of radius $r>0$ and center $X:=(t,x)\in\mathbb{R}^{1+n}$ with
$x\in\mathbb{R}^{n}$:
$\begin{split}Q_{r}(X):=&\,\\{Y\in\mathbb{R}^{1+d}:\delta(X,Y)<r\\}=(t-r^{2},t+r^{2})\times
B_{r}(x),\\\ B_{r}(x):=&\,\\{y\in\mathbb{R}^{n}:|y-x|<r\\},\end{split}$
and by $|Q_{r}(X)|$ the volume.
Denote by $\partial\Pi$ the boundary of domain $\Pi\subset\mathbb{R}^{n}$.
Throughout this paper, we assume $\partial\mathcal{O}\in C^{1}$. The set
$S_{T}:=[0,T]\times\partial\mathcal{O}$ is called the lateral boundary of $Q$
and the set $\partial_{\rm p}Q:=S_{T}\cup(\\{T\\}\times\mathcal{O})$ is called
the parabolic boundary of $Q$.
For domain $\Pi\subset\mathbb{R}^{n}$, we denote by $C_{c}^{\infty}(\Pi)$ the
totality of infinitely differentiable functions of compact supports in $\Pi$,
and the spaces like $L^{\infty}(\Pi),L^{p}(\Pi)$ and $W^{k,p}(\Pi)$ are
defined as usual for integer $k$ and real number $p\in[1,\infty)$. We denote
by $\ll\cdot,~{}\cdot\gg_{\Pi}$ the inner product of $L^{2}(\Pi)$ and the
subscript $\Pi$ will be omitted for $\Pi=\mathcal{O}$. Set
$\Pi_{t}:=[t,T]\times\Pi$ for $t\in[0,T)$. For each integer $k$ and real
number $p\in[1,\infty)$, we denote by $W^{k,p}_{\mathscr{F}}(\Pi_{t})$ the
totality of the $W^{k,p}(\Pi)$-valued predictable processes $u$ on $[t,T]$
such that
$\|u\|_{W^{k,p}_{\mathscr{F}}(\Pi_{t})}:=\left(E\left[\int_{t}^{T}\|u(s,\cdot)\|_{W^{k,p}(\Pi)}^{p}ds\right]\right)^{1/p}<\infty.$
Then
$(W^{k,p}_{\mathscr{F}}(\Pi_{t}),~{}\|\cdot\|_{W^{k,p}_{\mathscr{F}}(\Pi_{t})})$
is a Banach space.
###### Definition 2.1.
For $(p,t,k)\in[1,\infty)\times[0,T)\times\mathbb{Z}$, define
$\mathcal{M}^{k,p}(\Pi_{t})$ as the totality of $u\in
W^{k,p}_{\mathscr{F}}(\Pi_{t})$ such that
$\|u\|_{k,p;\Pi_{t}}:=\left(\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{s\in[t,T]}E\left[\int_{s}^{T}\|u(\omega,\tau,\cdot)\|^{p}_{W^{k,p}(\Pi)}d\tau\big{|}\mathscr{F}_{s}\right]\right)^{1/p}<\infty.$
For $u\in W^{k,p}_{\mathscr{F}}(\Pi_{t})$, we deduce from [3, Theorem 6.3]
that the process
$\left\\{1_{[t,T]}(s)E\left[\int_{s}^{T}\|u(\omega,\tau,\cdot)\|^{p}_{W^{k,p}(\Pi)}d\tau\big{|}\mathscr{F}_{s}\right],\,\,s\in[0,T]\right\\}\quad\in
S^{\beta}(\mathbb{R})\textrm{ for any }\beta\in(0,1).$
This shows that the norm $\|\cdot\|_{k,p;\Pi_{t}}$ in the preceding definition
makes a sense. Moreover, ($\mathcal{M}^{k,p}(\Pi_{t})$,
$\|\cdot\|_{k,p;\Pi_{t}}$) is a Banach space.
To simplify notations, $k=0$ appearing in either superscript or subscript of
spaces or norms will be omitted and therefore the notations
$W^{0,p}_{\mathscr{F}}(\Pi_{t}),~{}\|\cdot\|_{W^{0,p}_{\mathscr{F}}(\Pi_{t})},~{}\mathcal{M}^{0,p}(\Pi_{t})$
and $\|\cdot\|_{0,p;\Pi_{t}}$ will be abbreviated as
$W^{p}_{\mathscr{F}}(\Pi_{t}),~{}\|\cdot\|_{W^{p}_{\mathscr{F}}(\Pi_{t})},~{}\mathcal{M}^{p}(\Pi_{t})$
and $\|\cdot\|_{p;\Pi_{t}}$. Note that $W^{0,p}(\Pi)\equiv L^{p}(\Pi)$.
Moreover, we introduce the following spaces of random fields.
$\mathcal{L}^{\infty}(\Pi_{t})$ is the totality of $u\in
W^{p}_{\mathscr{F}}(\Pi_{t})$ such that
$\|u\|_{\infty;\Pi_{t}}:=\operatorname*{ess\,sup}_{(\omega,s,x)\in\Omega\times[t,T]\times\Pi}|u(\omega,s,x)|<\infty.$
$\mathcal{L}^{\infty,p}(\Pi_{t})$ is the totality of $u\in
W^{p}_{\mathscr{F}}(\Pi_{t})$ such that
$\|u\|_{\infty,p;\Pi_{t}}:=\operatorname*{ess\,sup}_{(\omega,s)\in\Omega\times[t,T]}\|u(\omega,s,\cdot)\|_{L^{p}(\Pi)}<\infty.$
$\mathcal{V}_{2}(\Pi_{t})$ is the totality of $u\in
W^{1,2}_{\mathscr{F}}(\Pi_{t})$ such that
$\|u\|_{\mathcal{V}_{2}(\Pi_{t})}:=\left(\|u\|^{2}_{\infty,2;\Pi_{t}}+\|\nabla
u\|^{2}_{2;\Pi_{t}}\right)^{1/2}<\infty.$ (2.1)
$\mathcal{V}_{2,0}(\Pi_{t})$, equipped with the norm (2.1), is the totality of
$u\in\mathcal{V}_{2}(\Pi_{t})$ such that
$\lim_{r\rightarrow 0}\|u(s+r,\cdot)-u(s,\cdot)\|_{L^{2}(\Pi)}=0,\textrm{ for
all }s,s+r\in[t,T]$
holds almost surely. We denote by $\dot{\mathcal{V}}_{2}(Q)$
($\dot{\mathcal{V}}_{2,0}(\Pi_{t})$, $\dot{W}^{1,p}_{\mathscr{F}}(\Pi_{t})$
and $\dot{\mathcal{M}}^{1,p}(\Pi_{t})$, respectively) all the random fields
$u\in\mathcal{V}_{2}(Q)$ ($\mathcal{V}_{2,0}(\Pi_{t})$,
$W^{1,p}_{\mathscr{F}}(\Pi_{t})$ and $\mathcal{M}^{1,p}(\Pi_{t})$,
respectively), satisfying
$u(\omega,s,\cdot)|_{\partial\Pi}=0,\quad
a.e.~{}(\omega,s)\in\Omega\times[t,T].$
By convention, we treat elements of spaces defined above like $W^{k,p}(\Pi)$
and $\mathcal{M}^{k,p}(\Pi_{t})$ as functions rather than distributions or
classes of equivalent functions, and if we know that a function of this class
has a modification with better properties, then we always consider this
modification. For example, if $u\in W^{1,p}(\Pi)$ with $p>n$, then $u$ has a
modification lying in $C^{\alpha}(\Pi)$ for $\alpha\in(0,\frac{p-n}{p})$, and
we always adopt the modification $u\in W^{1,p}(\Pi)\cap C^{\alpha}(\Pi)$. By
saying a finite dimensional vector-valued function
$v:=(v_{i})_{i\in\mathcal{I}}$ belongs to a space like $W^{k,p}(\Pi)$, we mean
that each component $v_{i}$ belongs to the space and the norm is defined by
$\|v\|_{W^{k,p}(\Pi)}=\left(\sum_{i\in\mathcal{I}}\|v_{i}\|^{p}_{W^{k,p}(\Pi)}\right)^{1/p}.$
Consider quasi-linear BSPDE (1.1). We define the following assumptions.
$({\mathcal{A}}1)$ The pair of random functions
$f(\cdot,\cdot,\cdot,\vartheta,y,z):~{}\Omega\times[0,T]\times\mathcal{O}\rightarrow\mathbb{R}^{n}\textrm{
and
}g(\cdot,\cdot,\cdot,\vartheta,y,z):~{}\Omega\times[0,T]\times\mathcal{O}\rightarrow\mathbb{R}$
are $\mathscr{P}\otimes\mathcal{B}(\mathcal{O})$-measurable for any
$(\vartheta,y,z)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{m}$. There
exist positive constants $L,\kappa$ and $\beta$ such that for all
$(\vartheta_{1},y_{1},z_{1}),(\vartheta_{2},y_{2},z_{2})\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n\times
m}$ and $(\omega,t,x)\in\Omega\times[0,T]\times\mathcal{O}$
$\begin{split}|f(\omega,t,x,\vartheta_{1},y_{1},z_{1})-f(\omega,t,x,\vartheta_{2},y_{2},z_{2})|\leq&L|\vartheta_{1}-\vartheta_{2}|+\frac{\kappa}{2}|y_{1}-y_{2}|+\beta^{1/2}|z_{1}-z_{2}|,\\\
|g(\omega,t,x,\vartheta_{1},y_{1},z_{1})-g(\omega,t,x,\vartheta_{2},y_{2},z_{2})|\leq&L(|\vartheta_{1}-\vartheta_{2}|+|y_{1}-y_{2}|+|z_{1}-z_{2}|).\end{split}$
$({\mathcal{A}}2)$ The pair functions $a$ and $\sigma$ are
$\mathscr{P}\otimes\mathcal{B}(\mathcal{O})$-measurable. There exist positive
constants $\varrho>1,\lambda$ and $\Lambda$ such that the following hold for
all $\xi\in\mathbb{R}^{n}$ and
$(\omega,t,x)\in\Omega\times[0,T]\times\mathcal{O}$
$\begin{split}&\lambda|\xi|^{2}\leq(2a^{ij}(\omega,t,x)-\varrho\sigma^{ir}\sigma^{jr}(\omega,t,x))\xi^{i}\xi^{j}\leq\Lambda|\xi|^{2};\\\
&|a(\omega,t,x)|+|\sigma(\omega,t,x)|\leq\Lambda;\\\ &\hbox{ \rm and
}\lambda-\kappa-\varrho^{\prime}\beta>0\textrm{ \rm with
}\varrho^{\prime}:=\frac{\varrho}{\varrho-1}.\end{split}$
$({\mathcal{A}}3)$ $G\in
L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))$. There exist two real
numbers $p>n+2$ and $q>(n+2)/2$ such that
$f_{0}:=f(\cdot,\cdot,\cdot,0,0,0)\in\mathcal{M}^{p}(Q),\
g_{0}:=g(\cdot,\cdot,\cdot,0,0,0)\in\mathcal{M}^{\frac{p(n+2)}{p+n+2}}(Q),$
and
$\left(b^{i}\right)^{2},\left(\varsigma^{r}\right)^{2},c\in\mathcal{M}^{q}(Q)$,
$i=1,\cdots,n$; $r=1,\cdots,m$. Define
$\Lambda_{0}:=B_{q}(b,c,\varsigma):=\||b|^{2}\|_{q;Q}+\|c\|_{q;Q}+\||\varsigma|^{2}\|_{q;Q}.$
(2.2)
$({\mathcal{A}}3)_{0}$ $\quad G\in
L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O})),\,f_{0}\in\mathcal{M}^{2}(Q),\,g_{0}\in\mathcal{M}^{2}(Q)$
and $b,\,\varsigma,\,c\in\mathcal{L}^{\infty}(Q)$.
$({\mathcal{A}}4)$ There exists a nonnegative constant $L_{0}$ such that
$c\leq L_{0}$.
For $p\in[2,\infty)$, define the functional $A_{p}$:
$A_{p}(u,v):=\|u\|_{p;Q}+\|v\|_{{\frac{p(n+2)}{p+n+2}};Q},\quad(u,v)\in\mathcal{M}^{p}(Q)\times\mathcal{M}^{\frac{p(n+2)}{p+n+2}}(Q),$
and the functional $H_{p}$:
$H_{p}(u,v):=\|u\|_{p;Q}+\|v\|_{p;Q},\quad(u,v)\in\mathcal{M}^{p}(Q)\times\mathcal{M}^{p}(Q).$
###### Definition 2.2.
A pair of processes $(u,v)\in W_{\mathscr{F}}^{1,2}(Q)\times
W_{\mathscr{F}}^{2}(Q)$ is called a weak solution to BSPDE (1.1) if it holds
in the weak sense, i.e. for any $\varphi\in C_{c}^{\infty}(\mathcal{O})$ there
holds almost surely
$\begin{split}&\ll\varphi,\,u(t)\gg\\\
=&\ll\varphi,\,G\gg-\int_{t}^{T}\ll\varphi,\,v^{r}(s)\gg
dW_{s}^{r}+\int_{t}^{T}\ll\varphi,\,g(s,\cdot,u(s),\nabla u(s),v(s))\gg ds\\\
&-\int_{t}^{T}\ll\partial_{x_{j}}\varphi,\quad
a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)+f^{j}(s,\cdot,u(s),\nabla
u(s),v(s))\gg ds\\\
&+\int_{t}^{T}\ll\varphi,\,b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)\gg
ds,\quad\forall\,t\in[0,T].\\\ \end{split}$ (2.3)
Denote by $\mathscr{U}\times\mathscr{V}(G,f,g)$ the set of all the weak
solutions $(u,v)\in\mathcal{V}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$ of BSPDE
(1.1).
###### Remark 2.1.
Let $(u,v)\in W_{\mathscr{F}}^{1,2}(Q)\times W_{\mathscr{F}}^{2}(Q)$ be a weak
solution to BSPDE (1.1). For each $\zeta(t,x)=\psi(t)\varphi(x)$ with
$\varphi\in C_{c}^{\infty}(\mathcal{O})$ and $\psi\in
C_{c}^{\infty}(\mathbb{R})$, in view of (2.3), we have almost surely
$\begin{split}&\ll\zeta(s^{\prime\prime}),\,u(s^{\prime\prime})\gg-\ll\zeta(s^{\prime}),\,u(s^{\prime})\gg\\\
=&\ll\zeta(s^{\prime\prime})-\zeta(s^{\prime}),\,u(s^{\prime\prime})\gg+\ll\zeta(s^{\prime}),\,u(s^{\prime\prime})-u(s^{\prime})\gg\\\
=&[\psi(s^{\prime\prime})-\psi(s^{\prime})]\ll\varphi,\,u(s^{\prime\prime})\gg+\psi(s^{\prime})\left(\ll\varphi,\,u(s^{\prime\prime})\gg-\ll\varphi,\,u(s^{\prime})\gg\right)\\\
=&[\psi(s^{\prime\prime})-\psi(s^{\prime})]\ll\varphi,\,u(s^{\prime\prime})\gg\\\
&-\psi(s^{\prime})\bigg{(}\int_{s^{\prime}}^{s^{\prime\prime}}\ll\varphi,\,g(s,\cdot,u(s),\nabla
u(s),v(s))\gg ds-\int_{s^{\prime}}^{s^{\prime\prime}}\ll\varphi,\,v^{r}(s)\gg
dW_{s}^{r}\\\
&-\int_{s^{\prime}}^{s^{\prime\prime}}\ll\partial_{x_{j}}\varphi,\,a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)+f^{j}(s,\cdot,u(s),\nabla
u(s),v(s))\gg ds\\\
&+\int_{s^{\prime}}^{s^{\prime\prime}}\ll\varphi,\,b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)\gg
ds\bigg{)}\end{split}$
for $s^{\prime\prime}=t_{i+1}$ and $s^{\prime}=t_{i}$, where
$t=t_{0}<t_{1}<t_{2}<\cdots<t_{N}=T,\,\,2\\!<N\in\mathbb{N}$ and
$t_{i+1}-t_{i}=T/N$, $i=1,2,\cdots,N$. Summing up both sides of these
equations and passing to the limit, we have almost surely
$\begin{split}&\ll\zeta(t),\,u(t)\gg\\\
=&\ll\zeta(T),\,G\gg-\int_{t}^{T}\ll\partial_{s}\zeta(s),\,u(s)\gg
ds-\int_{t}^{T}\ll\zeta(s),\,v^{r}(s)\gg dW_{s}^{r}\\\
&-\int_{t}^{T}\ll\partial_{x_{j}}\zeta(s),\quad
a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)+f^{j}(s,\cdot,u(s),\nabla
u(s),v(s))\gg ds\\\
&+\int_{t}^{T}\ll\zeta(s),\,b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)\gg
ds\\\ &+\int_{t}^{T}\ll\zeta(s),\,g(s,\cdot,u(s),\nabla u(s),v(s))\gg
ds,\quad\forall\ t\in[0,T].\end{split}$ (2.4)
Since the linear space
$\left\\{\sum_{i=1}^{N}\psi_{i}(t)\varphi_{i}(x),(t,x)\in\mathbb{R}\times\mathcal{O}:N\in\mathbb{N},\,(\varphi_{i},\psi_{i})\in
C_{c}^{\infty}(\mathcal{O})\times
C_{c}^{\infty}(\mathbb{R}),\,i=1,2,\cdots,N\right\\}$
is dense in $C_{c}^{\infty}(\mathbb{R})\otimes C_{c}^{\infty}(\mathcal{O})$,
(2.4) holds for any test function $\zeta\in C_{c}^{\infty}(\mathbb{R})\otimes
C_{c}^{\infty}(\mathcal{O})$.
Under assumptions $({\mathcal{A}}1),({\mathcal{A}}2)$ and
$({\mathcal{A}}3)_{0}$, we deduce from [22, Theorem 2.1] that there exists a
unique weak solution $(u,v)\in(\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap
S^{2}(L^{2}(\mathcal{O})))\times W^{2}_{\mathscr{F}}(Q)$, which admits
$L^{2}(\mathcal{O})$-valued continuous trajectories for $u$, and which is also
said to satisfy the null Dirichlet condition on the lateral boundary since $u$
vanishes in a generalized sense on the boundary $\partial\mathcal{O}$. Denote
by $\dot{\mathscr{U}}\times\dot{\mathscr{V}}(G,f,g)$ all the random fields
lying in $\mathscr{U}\times\mathscr{V}(G,f,g)$ which satisfy the null
Dirichlet boundary condition.
## 3 Auxiliary results
In what follows, $C>0$ is a constant which may vary from line to line and
$C(a_{1},a_{2},\cdots)$ is a constant to depend on the parameters
$a_{1},a_{2},\cdots$.
First, we give the following embedding lemma.
###### Lemma 3.1.
For $u\in\dot{\mathcal{V}}_{2}(\Pi_{t})$ with $t\in[0,T)$, we have
$u\in\mathcal{M}^{\frac{2(n+2)}{n}}(\Pi_{t})$ and
$\begin{split}\|u\|_{\frac{2(n+2)}{n};\Pi_{t}}\leq\,&\ C(n)~{}\|\nabla
u\|_{2;\Pi_{t}}^{n/(n+2)}\operatorname*{ess\,sup}_{(\omega,s)\in\Omega\times[t,T]}\|u(\omega,s,\cdot)\|_{L^{2}(\Pi)}^{2/(n+2)}\leq\
C(n)~{}\|u\|_{\mathcal{V}_{2}(\Pi_{t})}.\end{split}$
###### Proof.
By the well known Gagliard-Nirenberg inequality (c.f. [13], [16] or [18]), we
have
$\|u(\omega,s,\cdot)\|_{L^{q}(\Pi)}^{q}\leq\ C~{}\|\nabla
u(\omega,s,\cdot)\|_{L^{2}(\Pi)}^{\alpha
q}\|u(\omega,s,\cdot)\|_{L^{2}(\Pi)}^{q(1-\alpha)},\quad a.e.\
(\omega,s)\in\Omega\times[t,T],$
where $\alpha=n/(n+2)$ and $q=2(n+2)/n$. Integrating on $[\tau,T]$ for
$\tau\in[t,T)$ and taking conditional expectation, we obtain almost surely
$\begin{split}E\left[\int_{\Pi_{\tau}}|u(s,x)|^{q}dxds\Big{|}\mathscr{F}_{\tau}\right]\leq&\
C~{}\|\nabla
u\|_{2;\Pi_{t}}^{2}\operatorname*{ess\,sup}_{(\omega,s)\in\Omega\times[t,T]}\|u(\omega,s,\cdot)\|_{L^{2}(\Pi)}^{(1-\alpha)q}\leq\
C~{}\|u\|^{q}_{\mathcal{V}_{2}(\Pi_{t})}.\end{split}$
Therefore, $u\in\mathcal{M}^{\frac{2(n+2)}{n}}(\Pi_{t})$ and
$\begin{split}\|u\|_{\frac{2(n+2)}{n};\Pi_{t}}\leq&\ C~{}\|\nabla
u\|_{2;\Pi_{t}}^{n/(n+2)}\operatorname*{ess\,sup}_{(\omega,s)\in\Omega\times[t,T]}\|u(\omega,s,\cdot)\|_{L^{2}(\Pi)}^{2/(n+2)}\leq\
C~{}\|u\|_{\mathcal{V}_{2}(\Pi_{t})}\end{split}$
with $C$ only depending on $n$. ∎
###### Lemma 3.2.
For any $r\in\mathbb{R}$ and $u\in\mathcal{V}_{2,0}(\Pi_{t})$ with $t\in[0,T)$
we have
$(u-r)^{+}:=(u-r)\vee 0\in\mathcal{V}_{2,0}(\Pi_{t}).$
Moreover, if $\\{u_{k},k\in\mathbb{N}\\}$ is a Cauchy sequence in
$\mathcal{V}_{2,0}(\Pi_{t})$ with limit $u\in\mathcal{V}_{2,0}(\Pi_{t})$, then
$\lim_{k\to\infty}\|(u_{k}-r)^{+}-(u-r)^{+}\|_{\mathcal{V}_{2}(\Pi_{t})}=0.$
###### Proof.
It can be checked that $(u-r)^{+}\in\mathcal{V}_{2}(\Pi_{t})$. Since
$|(u-r)^{+}-(v-r)^{+}|\leq|u-v|,$
Then we have
$\|(u-r)^{+}(s+h)-(u-r)^{+}(s)\|_{L^{2}(\Pi)}\leq\|u(s+h)-u(s)\|_{L^{2}(\Pi)},\quad\forall
s,\,s+h\in[t,T].$
Hence, the continuity of $u$ implies that of $(u-r)^{+}$. The other assertions
follow in a similar way. We complete our proof. ∎
In contrast to the deterministic case, the integrand of Itô’s stochastic
integral is required to be adapted, and the technique of Steklov time average
(see [17, page 100]) finds difficulty in our stochastic situation. We directly
establish some Itô formula to get around the difficulty.
###### Lemma 3.3.
Let
$\phi:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\longrightarrow\mathbb{R}$
be a continuous function which is twice continuously differentiable such that
$\phi^{\prime}(t,x,0)=0$ for any $(t,x)\in\mathbb{R}\times\mathbb{R}^{n}$ and
there exists a constant $M\in(0,\infty)$ such that
$\sup_{(t,x)\in\mathbb{R}^{n+1},s\in\mathbb{R}\setminus\\{0\\}}\left\\{\left|\phi^{\prime\prime}(t,x,s)\right|+\frac{1}{|s|}\sum_{i=1}^{n}\left|\partial_{x_{i}}\phi^{\prime}(t,x,s)\right|+\frac{1}{s^{2}}\left|\partial_{t}\phi(t,x,s)-\partial_{t}\phi(t,x,0)\right|\right\\}<M,$
where $\phi^{\prime}(t,x,s):=\partial_{s}\phi(t,x,s)$ and
$\phi^{\prime\prime}(t,x,s):=\partial_{ss}\phi(t,x,s)$. Assume that the
equation
$\begin{split}u(t,x)=\,&u(T,x)+\int_{t}^{T}\left(h^{0}(s,x)+\partial_{x_{i}}h^{i}(s,x)\right)\,ds-\int_{t}^{T}z^{r}(s,x)\,dW_{s}^{r},\quad
t\in[0,T]\end{split}$ (3.1)
holds in the weak sense of Definition 2.2, where $u(T)\in
L^{2}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}));\ h^{i}\in
W^{2}_{\mathscr{F}}(Q),i=0,1,\cdots,n;$ and $z\in W^{2}_{\mathscr{F}}(Q)$. If
$u\in\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap S^{2}(L^{2}(\mathcal{O}))$, we have
almost surely
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,u(t,x))\,dx\\\
=&\int_{\mathcal{O}}\phi(T,x,u(T,x))\,dx-\int_{t}^{T}\int_{\mathcal{O}}\partial_{s}\phi(s,x,u(s,x))\,dxds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\,z^{r}(s)\gg
dW_{s}^{r}+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\,h^{0}(s)\gg ds\\\
&-\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u(s))\partial_{x_{i}}u(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u(s)),\,h^{i}(s)\gg
ds\\\
&-\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u(s)),\,|z(s)|^{2}\gg
ds,~{}\forall t\in[0,T].\end{split}$ (3.2)
###### Remark 3.1.
Lemma 3.3 extends Itô formulas of [7] and [21] to our more general case where
the test function $\phi$ is allowed to depend on both time and space
variables. The extension is motivated by the subsequent study of the local
maximum principle where Itô formula for truncated solutions of BSDEs is
required.
###### Proof of Lemma 3.3.
All the integrals in (3.2) are well defined. In particular, the stochastic
integral
$I(t):=\int_{0}^{t}\ll\phi^{\prime}(s,\cdot,u(s)),\,z^{r}(s)\gg
dW_{s}^{r},\quad t\in[0,T]$
is a martingale since
$\begin{split}E\left[\sup_{t\in[0,T]}\left|I(t)\right|\right]\leq&\,CE\left[\left(\int_{0}^{T}\Bigm{|}\ll|\phi^{\prime}(s,\cdot,u(s))|,\,|z(s)|\gg\Bigm{|}^{2}ds\right)^{1/2}\right]\\\
\leq&\,CM\|u\|_{S^{2}(L^{2}(\mathcal{O}))}\|z\|_{W^{2}_{\mathscr{F}}(Q)}.\end{split}$
We extend the random fields $u,h^{0},h^{1},\cdots,h^{n}$ and $z$ from their
domain $\Omega\times[0,T]\times\mathcal{O}$ to
$\Omega\times[0,T]\times\mathbb{R}^{n}$ by setting them all to be zero outside
$\mathcal{O}$, and we still use themselves to denote their respective
extensions. Since $u$ satisfies the null Dirichlet condition on the lateral
boundary and $\partial\mathcal{O}\in C^{1}$, we have $u\in
W^{1,2}_{\mathscr{F}}([0,T]\times\mathbb{R}^{n})$. It is obvious that all the
extensions $h^{0},h^{1},\cdots,h^{n}$ and $z$ lie in
$W^{2}_{\mathscr{F}}([0,T]\times\mathbb{R}^{n})$.
Step 1. Consider $h^{i}\in\dot{W}^{1,2}_{\mathscr{F}}(\mathcal{O})$,
$i=1,2,\cdots,n$. Choose a sufficiently large positive integer $N_{0}$ so that
$\\{x\in\mathcal{O}:dis(x,\partial\mathcal{O})>1/N_{0}\\}$ is a nonempty sub-
domain of $\mathcal{O}$. For integer $N>N_{0}$, define
$\mathcal{O}^{N}:=\\{x\in\mathcal{O}:dis(x,\partial\mathcal{O})>1/N\\}.$
Let $\rho\in C_{c}^{\infty}(\mathbb{R}^{n})$ be a nonnegative function such
that
$\textrm{supp}({\rho})\subset B_{1}(0)\hbox{ \rm and
}\int_{\mathbb{R}^{n}}\rho(x)\,dx=1.$
Define for each positive integer $k$,
$\rho_{k}(x):=(2Nk)^{n}\rho(2Nkx),\quad
u_{k}(s,x):=u(s)\ast\rho_{k}(x):=\int_{\mathbb{R}^{n}}\rho_{k}(x-y)u(s,y)\,dy.$
In a similar way, we write
$z_{k}(s,x):=z(s)\ast\rho_{k}(x)\hbox{ \rm and
}h^{i}_{k}(s,x):=h^{i}(s)\ast\rho_{k}(x),\,\,i=1,2,\cdots,n.$
Then for each $x\in\mathcal{O}^{N}$, we have almost surely
$u_{k}(t,x)=u_{k}(T,x)+\int_{t}^{T}\left(\partial_{x_{i}}h_{k}^{i}(s,x)+h_{k}^{0}(s,x)\right)\,ds-\int_{t}^{T}z_{k}^{r}(s,x)\,dW_{s}^{r},~{}\forall
t\in[0,T].$
By using Itô formula for each $x\in\mathcal{O}^{N}$ and then integrating over
$\mathcal{O}^{N}$ with respect to $x$ , we obtain
$\begin{split}&\int_{\mathcal{O}^{N}}\phi(t,x,u_{k}(t,x))\,dx\\\
=&\int_{\mathcal{O}^{N}}\\!\phi(T,x,u_{k}(T,x))\,dx-\int_{t}^{T}\\!\\!\\!\int_{\mathcal{O}^{N}}\partial_{s}\phi(s,x,u_{k}(s,x))\,dxds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\
\,h^{0}(s)\gg_{\mathcal{O}^{N}}ds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\
\,\partial_{x_{i}}h_{k}^{i}(s)\gg_{\mathcal{O}^{N}}ds\\\
&-\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u_{k}(s)),\
\,|z_{k}(s)|^{2}\gg_{\mathcal{O}^{N}}ds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\
\,z_{k}^{r}(s)\gg_{\mathcal{O}^{N}}dW_{s}^{r}.\end{split}$ (3.3)
For the sake of convenience, we define
$\begin{split}\delta\phi_{k}(t,x):=&\phi(t,x,u(t,x))-\phi(t,x,u_{k}(t,x))\\\
\delta u_{k}(t,x):=&u(t,x)-u_{k}(t,x).\end{split}$
and in a similar way, we define
$\delta\phi_{k}^{\prime},\delta\phi_{k}^{\prime\prime},\delta h_{k}^{i}$ and
$\delta z_{k}^{r}$ $i=0,1,\cdots,n;r=1,\cdots,m$.
Since for almost all $(\omega,s)\in\Omega\times[0,T]$
$\begin{split}&\|u_{k}(\omega,s)\|_{W^{1,2}(\mathbb{R}^{n})}\leq\|u(\omega,s)\|_{W^{1,2}(\mathbb{R}^{n})},~{}\lim_{k\rightarrow\infty}\|\delta
u_{k}(\omega,s)\|_{W^{1,2}(\mathbb{R}^{n})}\rightarrow 0;\\\
&\|h^{0}_{k}(\omega,s)\|_{L^{2}(\mathbb{R}^{n})}\leq\|h^{0}(\omega,s)\|_{L^{2}(\mathbb{R}^{n})},~{}\lim_{k\rightarrow\infty}\|\delta
h^{0}_{k}(\omega,s)\|_{L^{2}(\mathbb{R}^{n})}\rightarrow 0;\\\
&\|h^{i}_{k}(\omega,s)\|_{W^{1,2}(\mathbb{R}^{n})}\leq\|h^{i}(\omega,s)\|_{W^{1,2}(\mathbb{R}^{n})},~{}\lim_{k\rightarrow\infty}\|\delta
h^{i}_{k}(\omega,s)\|_{W^{1,2}(\mathbb{R}^{n})}\rightarrow 0,i=1,2,\cdots;\\\
&\|z_{k}(\omega,s)\|_{L^{2}(\mathbb{R}^{n})}\leq\|z(\omega,s)\|_{L^{2}(\mathbb{R}^{n})},~{}\lim_{k\rightarrow\infty}\|\delta
z_{k}(\omega,s)\|_{L^{2}(\mathbb{R}^{n})}\rightarrow 0,\end{split}$
by Lebesgue’s dominated convergence theorem, we have as $k\rightarrow\infty$
$\begin{split}&\sum_{i=1}^{n}\|\delta
h^{i}_{k}(s)\|^{2}_{W_{\mathscr{F}}^{1,2}([0,T]\times\mathbb{R}^{n})}+\|\delta
h^{0}_{k}(s)\|^{2}_{W^{2}_{\mathscr{F}}([0,T]\times\mathbb{R}^{n})}+\|\delta
z_{k}(s)\|_{W_{\mathscr{F}}^{2}([0,T]\times\mathbb{R}^{n})}^{2}\\\ &+\|\delta
u_{k}(s)\|_{W_{\mathscr{F}}^{1,2}([0,T]\times\mathbb{R}^{n})}^{2}\rightarrow
0,\end{split}$
$\begin{split}&E\left[\int_{0}^{T}\\!\\!\\!\int_{\mathcal{O}}\left|\delta\phi_{k}(t,x)\right|\,dxdt\right]\leq
E\left[\int_{0}^{T}M\ll|u_{k}(t)|+|u(t)|,\,|\delta u_{k}(t)|\gg
dt\right]\rightarrow 0,\end{split}$
$\begin{split}&E\left[\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{n}}\big{|}\phi^{\prime}(s,x,u_{k}(s,x))\partial_{x_{i}}h_{k}^{i}(s,x)-\phi^{\prime}(s,x,u(s,x))\partial_{x_{i}}h^{i}(s,x)\big{|}\,dxds\right]\\\
\leq&\
E\bigg{[}\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{n}}\Big{(}M\big{|}\delta
u_{k}(s,x)\partial_{x_{i}}h^{i}_{k}(s,x)\big{|}+M|u(s,x)|\left|\partial_{x_{i}}(\delta
h^{i}_{k})(s,x)\right|\Big{)}\,dxds\bigg{]}\rightarrow 0,\\\ &\
i=1,\cdots,n\end{split}$
and
$\begin{split}&E\left[\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{n}}|\phi^{\prime}(s,x,u_{k}(s,x))h_{k}^{0}(s,x)-\phi^{\prime}(s,x,u(s,x))h^{0}(s,x)|\,dxds\right]\\\
\leq&E\left[\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{n}}\big{(}M|\delta
u_{k}(s,x)h^{0}_{k}(s,x)|+M|u(s,x)||\delta
h^{0}_{k}(s,x)|\big{)}\,dxds\right]\rightarrow 0.\end{split}$
Since the convergence
$\lim_{k\rightarrow\infty}\|\delta
u_{k}\|_{W^{1,2}_{\mathscr{F}}([0,T]\times\mathbb{R}^{n})}=0$
implies that $u_{k}(\omega,t,x)$ converges to $u(\omega,t,x)$ in measure
$dP\otimes dt\otimes dx$, from the dominated convergence theorem we conclude
that
$\lim_{k\rightarrow\infty}E\left[\int_{0}^{T}\\!\\!\\!\int_{\mathcal{O}}\,|{\partial_{s}}\phi(s,x,u_{k}(s,x))\,-\partial_{s}\phi(s,x,u(s,x))|\,\,dxds\right]=0.$
In a similar way, we obtain
$\begin{split}E\left[\int_{0}^{T}\\!\\!\\!\int_{\mathcal{O}}\Big{|}\phi^{\prime\prime}(s,x,u(s,x))|z(s,x)|^{2}-\phi^{\prime\prime}(s,x,u_{k}(s,x))|z_{k}(s,x)|^{2}\Big{|}\,dxds\right]\rightarrow
0\end{split}$
and
$\begin{split}&E\left[\sup_{t\in[0,T]}\left|\sum_{r=1}^{m}\int_{t}^{T}\\!\\!\\!\int_{\mathbb{R}^{n}}\left(\phi^{\prime}(s,x,u_{k}(s,x))z^{r}_{k}(s,x)-\phi^{\prime}(s,x,u(s,x))z^{r}(s,x)\right)\,dxdW^{r}_{s}\right|\right]\\\
\leq&\ CE\left[\left(\int_{0}^{T}\left|\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\
z_{k}(s)\gg_{\mathbb{R}^{n}}-\ll\phi^{\prime}(s,\cdot,u(s)),\
z(s)\gg_{\mathbb{R}^{n}}\right|^{2}\,ds\right)^{1/2}\right]\\\ \leq&\
CE\biggl{[}\Bigl{(}\int_{0}^{T}\big{(}\|\delta
u_{k}(s)\|_{L^{2}(\mathbb{R}^{n})}^{2}\|z(s)\|^{2}_{L^{2}(\mathbb{R}^{n})}+\|\phi^{\prime}(s,u_{k}(s))\|^{2}_{L^{2}(\mathbb{R}^{n})}\|\delta
z_{k}(s)\|^{2}_{L^{2}(\mathbb{R}^{n})}\big{)}\,ds\Bigr{)}^{1/2}\biggr{]}\\\
&\longrightarrow 0\textrm{ as }k\rightarrow\infty.\end{split}$
Hence taking limits in $L^{1}(\Omega\times[0,T],\mathscr{P})$ as
$k\rightarrow\infty$ on both sides of (3.3) and noting the path-wise
continuity of $u$, we have almost surely
$\begin{split}&\int_{\mathcal{O}^{N}}\phi(t,x,u(t,x))\,dx\\\
=&\int_{\mathcal{O}^{N}}\phi(T,x,u(T,x))\,dx-\int_{t}^{T}\\!\\!\\!\int_{\mathcal{O}^{N}}\partial_{s}\phi(s,x,u(s,x))\,dxds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\
\,h^{0}(s)\gg_{\mathcal{O}^{N}}ds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\
\,\partial_{x_{i}}h^{i}(s)\gg_{\mathcal{O}^{N}}ds\\\
&-\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u(s)),\
\,|z(s)|^{2}\gg_{\mathcal{O}^{N}}ds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\
\,z^{r}(s)\gg_{\mathcal{O}^{N}}dW_{s}^{r},\quad\forall\ t\in[0,T].\end{split}$
(3.4)
Passing to the limit in $L^{1}(\Omega\times[0,T],\mathscr{P})$ by letting
$N\rightarrow\infty$ on both sides of (3.4), in view of the path-wise
continuity of $u$ and the integration-by-parts formula, we conclude (3.2).
Step 2. For the general $h^{i}\in W_{\mathscr{F}}^{2}(Q)$, we choose sequences
$\\{h^{i}_{k}\\}$, $\\{z^{r}_{k}\\}$ and $\\{u_{k}\\}$ from
$S^{2}(\mathbb{R})\otimes C_{c}^{\infty}(\mathcal{O})$ such that
$\begin{split}\lim_{k\rightarrow\infty}\bigg{\\{}&\sum_{i=0}^{n}\|\delta
h_{k}^{i}\|_{W^{2}_{\mathscr{F}}(Q)}+\|\delta
z_{k}\|_{W_{\mathscr{F}}^{2}(Q)}+\|\delta
u_{k}\|_{W^{1,2}_{\mathscr{F}}(Q)}+\|\delta
u_{k}(0)\|_{L^{2}(\mathcal{O})}\bigg{\\}}=0.\end{split}$
Consider
$\begin{split}\bar{u}(t,x)=\,&u(0,x)+\int_{0}^{t}\left(\Delta\bar{u}(s,x)+\partial_{x_{i}}\tilde{h}^{i}(s,x)-h^{0}(s,x)\right)\,ds\\\
&+\int_{0}^{t}z^{r}(s,x)\,dW^{r}_{s},\quad t\in[0,T]\end{split}$ (3.5)
with
$\tilde{h}^{i}(s,x):=-\partial_{x_{i}}u(s,x)-h^{i}(s,x).$
From Remark 2.1 and [5, Theorem 2.1], there are unique weak solutions
$u\in\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap S^{2}(L^{2}(\mathcal{O}))$ to SPDE
(3.5) in the sense of [5, Definition 1] or equivalently [7, Definition 4]),
and $u^{k}\in\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap S^{2}(L^{2}(\mathcal{O}))$ to
SPDE (3.5) with $u(0,x)$, $z(s,x)$ and $\tilde{h}^{i}(s,x)$ being replaced by
$u_{k}(0,x)$, $z_{k}(s,x)$ and
$\tilde{h}_{k}^{i}(s,x):=-\partial_{x_{i}}u_{k}(s,x)-h_{k}^{i}(s,x),\quad
k=1,2,\cdots.$
Then we deduce from [5, Propositions 6 and 7, and Theorem 9] that $u^{k}\in
W^{2,2}_{\mathscr{F}}(Q)\cap\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap
S^{2}(L^{2}(\mathcal{O}))$ and
$\begin{split}&\lim_{k\rightarrow\infty}\\{\|u^{k}-u\|_{W_{\mathscr{F}}^{1,2}(Q)}+\|u^{k}-u\|_{S^{2}(L^{2}(\mathcal{O}))}\\}\\\
\leq&\ C\lim_{k\rightarrow\infty}\\{\|\delta
u_{k}\|_{W_{\mathscr{F}}^{2}(Q)}+\|\delta
z_{k}\|_{W_{\mathscr{F}}^{2}(Q)}+\|\delta
u_{k}(0)\|_{L^{2}(\mathcal{O})}+\sum_{i=0}^{n}\|\delta
h_{k}^{i}\|_{W_{\mathscr{F}}^{2}(Q)}\\}\\\ =&\ 0\end{split}$ (3.6)
with the constant $C$ being independent of $k$. For each $k$, by Step 1 we
have
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,u^{k}(t,x))\,dx\\\
=&\int_{\mathcal{O}}\phi(T,x,u^{k}(T,x))\,dx-\int_{t}^{T}\\!\\!\\!\int_{\mathcal{O}}\partial_{s}\phi(s,x,u^{k}(s,x))\,dxds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u^{k}(s)),\ \,h_{k}^{0}(s)\gg ds\\\
&+\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{k}(s))\partial_{x_{i}}u^{k}(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u^{k}(s)),\
\,\partial_{x_{i}}u^{k}(s)\gg ds\\\
&+\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{k}(s))\partial_{x_{i}}u^{k}(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u^{k}(s)),\
\,\tilde{h}_{k}^{i}(s)\gg ds\\\
&-\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{k}(s)),\,|z_{k}(s)|^{2}\gg
ds-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u^{k}(s)),\,z_{k}^{r}(s)\gg
dW_{s}^{r},\end{split}$
for all $t\in[0,T]$, $P$-a.s.. By taking limits as $k\rightarrow\infty$, we
complete our proof. ∎
###### Remark 3.2.
Let
$\psi:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\longrightarrow\mathbb{R}$
be a continuous function satisfying the assumptions on $\phi$ in Lemma 3.3
except that for each $(t,y)$, $\psi^{\prime\prime}(t,y,s)$ may be not
continuous with respect to $s$. Then if there exists a sequence
$\\{\phi^{k},k\in\mathbb{R}\\}$ of functions satisfying the assumptions on
$\phi$ in Lemma 3.3, such that
$\lim_{k\rightarrow\infty}\phi^{k}(t,y,s)=\psi(t,y,s)\hbox{ \rm for each
}(t,y,s)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R},$
the assertion in Lemma 3.3 still holds for $\psi$.
Rewritting (3.1) into
$\begin{split}u(t,x)=u(0,x)+\int_{0}^{t}\left(\Delta
u(s,x)+\partial_{x_{i}}\tilde{h}^{i}(s,x)-h^{0}(s,x)\right)\,ds+\int_{0}^{t}z^{r}(s,x)\,dW_{s}^{r}\end{split}$
with
$\tilde{h}^{i}(s,x):=-\partial_{x_{i}}u(s,x)-h^{i}(s,x),$
we obtain
###### Lemma 3.4.
Let all the assumptions on $\phi$ of Lemma 3.3 be satisfied and (3.1) hold in
the weak sense of Definition 2.2 with $u(T)\in
L^{2}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))$, $z\in
W^{2}_{\mathscr{F}}(Q)$, $h^{i}\in W^{2}_{\mathscr{F}}(Q),i=1,\cdots,n$ and
$h^{0}\in W^{1}_{\mathscr{F}}(Q)$. We assume further that
$\phi^{\prime}(s,x,r)\leq M$ for any
$(s,x,r)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$. If
$u\in\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap S^{2}(L^{2}(\mathcal{O}))$, then (3.2)
holds almost surely for all $t\in[0,T]$.
The proof is very similar to that of [6, Proposition 2] and is omitted here.
The only difference is that to prove Lemma 3.4 we use Lemma 3.3 instead of [7,
Lemma 7].
Through a standard procedure we obtain by Lemma 3.3 the following
###### Lemma 3.5.
Let all the assumptions on $\phi$ of Lemma 3.3 be satisfied. If the function
$u$ in (3.1) belongs to $W^{1,2}_{\mathscr{F}}(Q)\cap S^{2}(L^{2}(Q))$ with
$u^{+}\in\dot{W}^{1,2}_{\mathscr{F}}(Q)$, we have almost surely
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,u^{+}(t,x))\,dx+\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{+}(s)),\,|z^{u}(s)|^{2}\gg\,ds\\\
=&\int_{\mathcal{O}}\phi(T,x,u^{+}(T,x))\,dx-\int_{t}^{T}\\!\\!\\!\int_{\mathcal{O}}\partial_{s}\phi(s,x,u^{+}(s,x))\,dxds\\\
&+\int_{t}^{T}\ll\phi(s,\cdot,u^{+}(s)),\,h^{0,u}(s)\gg\,ds\\\
&-\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{+}(s))\partial_{x_{i}}u^{+}(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u^{+}(s)),\quad
h^{i,u}(s)\gg\,ds\\\ &-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u^{+}(s)),\,\
z^{r,u}(s)\gg\,dW_{s}^{r},\,\quad t\in[0,T]\end{split}$ (3.7)
with
$h^{i,u}:=1_{\\{u>0\\}}h^{i},\quad i=0,1,\cdots,n$
and
$z^{r,u}=1_{\\{u>0\\}}z^{r},\quad r=1,\cdots,m;\quad
z^{u}:=(z^{1,u},\cdots,z^{m,u}).$
###### Remark 3.3.
Note that the assumption $u^{+}\in\dot{W}_{\mathscr{F}}^{1,2}(Q)$ does not
imply that $u$ vanishes in a generalized sense on the boundary
$\partial\mathcal{O}$ and therefore Lemma 3.3 can not be applied directly to
get the corresponding equation (3.1) for $u^{+}$.
###### Sketch of the proof.
Step 1. For $k\in\mathbb{N}$, define
$\psi(s)=\psi_{k}(s):=\left\\{\begin{array}[]{l}\begin{split}0,\quad&s\in(-\infty,\frac{1}{k});\\\
\frac{k}{2}(s-\frac{1}{k})^{2},\quad&s\in[\frac{1}{k},\frac{2}{k}];\\\
s-\frac{3}{2k},\quad&s\in(\frac{2}{k},+\infty).\end{split}\end{array}\right.$
(3.8)
Then the assumptions on $u^{+}$ imply that
$\psi(u)\in\dot{W}_{\mathscr{F}}^{1,2}(Q)$.
Take $\varphi\in C_{c}^{\infty}(\mathcal{O})$ and set $\mathscr{V}:=\varphi
u$. Then $\mathscr{V}\in\dot{W}_{\mathscr{F}}^{1,2}(Q)$. Since (3.1) holds in
the weak sense of Definition 2.2, we have almost surely for any $\xi\in
C_{c}^{\infty}(\mathcal{O})$
$\begin{split}&\ll\xi,\ \varphi u(t)\gg\\\ =\ &\ll\xi,\ \varphi
u(T)\gg+\int_{t}^{T}\ll\xi,\ \varphi h^{0}(s)-\partial_{x_{i}}\varphi
h^{i}(s)\gg ds\\\ &-\int_{t}^{T}\ll\partial_{x_{i}}\xi,\ \varphi h^{i}(s)\gg
ds-\int_{t}^{T}\ll\xi,\ \varphi z^{r}(s)\gg dW_{s}^{r},\ \ \forall
t\in[0,T].\end{split}$
Hence, there holds
$\begin{split}\mathscr{V}(t,x)=&\mathscr{V}(T,x)+\int_{t}^{T}\left[\varphi(x)h^{0}(s,x)-\partial_{x_{i}}\varphi(x)h^{i}(s,x)+\partial_{x_{i}}\left(\varphi(x)h^{i}(s,x)\right)\right]ds\\\
&-\int_{t}^{T}\varphi(x)z^{r}(s,x)dW^{r}_{s},\quad t\in[0,T]\end{split}$
in the weak sense of Definition 2.2.
For $\tilde{\varphi}\in C_{c}^{\infty}(O)$, by Lemma 3.3 and Remark 3.2 we
have almost surely
$\begin{split}&\ll\psi(\mathscr{V}(t)),\ \
\tilde{\varphi}\gg+\frac{1}{2}\int_{t}^{T}\ll\psi^{\prime\prime}(\mathscr{V}(s))\tilde{\varphi},\quad|\varphi
z(s)|^{2}\gg\,ds\\\ =&\ll\psi(\mathscr{V}(T)),\ \
\tilde{\varphi}\gg+\int_{t}^{T}\ll\psi^{\prime}(\mathscr{V}(s))\tilde{\varphi},\quad\varphi
h^{0}(s)\gg\,ds\\\
&-\int_{t}^{T}\ll\partial_{x_{i}}(\tilde{\varphi}\psi^{\prime}(\mathscr{V}(s))\varphi),\
\,h^{i}(s)\gg ds\\\
&-\int_{t}^{T}\ll\psi^{\prime}(\mathscr{V}(s))\tilde{\varphi},\ \,\varphi
z^{r}(s)\gg\,dW^{r}_{s},\quad\forall t\in[0,T].\end{split}$ (3.9)
Choosing $\varphi$ such that $\varphi\equiv 1$ in an open subset
$\mathcal{O}^{\prime}\Subset\mathcal{O}$ (i.e.,
$\overline{\mathcal{O}^{\prime}}\subset\mathcal{O}$ ) and
$supp(\tilde{\varphi})\subset\mathcal{O}^{\prime}$, we have almost surely
$\begin{split}&\ll\tilde{\varphi},\
\psi(u(t))\gg+\frac{1}{2}\int_{t}^{T}\ll\tilde{\varphi},\quad\psi^{\prime\prime}(u(s))|z(s)|^{2}\gg\,ds\\\
=&\ll\tilde{\varphi},\,\psi(u(T))\gg+\int_{t}^{T}\ll\tilde{\varphi},\ \
\psi^{\prime}(u(s))h^{0}(s)\gg\,ds\\\
&-\int_{t}^{T}\ll\partial_{x_{i}}(\tilde{\varphi}\psi^{\prime}(u(s))),\quad
h^{i}(s)\gg\,ds-\int_{t}^{T}\ll\tilde{\varphi},\ \
\psi^{\prime}(u(s))z^{r}(s)\gg\,dW^{r}_{s}\\\
=&\ll\tilde{\varphi},\,\psi(u(T))\gg+\int_{t}^{T}\ll\tilde{\varphi},\ \
\psi^{\prime}(u(s))h^{0}(s)\gg\,ds\\\ &-\int_{t}^{T}\ll\tilde{\varphi},\ \
\psi^{\prime}(u(s))z^{r}(s)\gg\,dW^{r}_{s}-\int_{t}^{T}\ll\partial_{x_{i}}\tilde{\varphi},\quad\psi^{\prime}(u(s))h^{i}(s)\gg\,ds\\\
&-\int_{t}^{T}\ll\tilde{\varphi},\ \
\psi^{\prime\prime}(u(s))\partial_{x_{i}}u(s)h^{i}(s)\gg\,ds,\quad\forall
t\in[0,T].\end{split}$
Since $\tilde{\varphi}$ is arbitrary, we have
$\begin{split}\psi(u(t,x))=&\psi(u(T,x))+\int_{t}^{T}\\!\\!\psi^{\prime}(u(s,x))h^{0}(s,x)\,ds-\frac{1}{2}\int_{t}^{T}\\!\\!\psi^{\prime\prime}(u(s,x))|z(s,x)|^{2}\,ds\\\
&-\int_{t}^{T}\psi^{\prime}(u(s,x))z^{r}(s,x)\,dW^{r}_{s}-\int_{t}^{T}\psi^{\prime\prime}(u(s,x))\partial_{x_{i}}u(s,x)h^{i}(s,x)\,ds\\\
&+\int_{t}^{T}\partial_{x_{i}}(\psi^{\prime}(u(s,x))h^{i}(s,x))\,ds\end{split}$
(3.10)
holds in the weak sense of Definition 2.2.
Step 2. It is sufficient to prove this lemma for test functions $\phi$ of
bounded first and second derivatives. Since (3.10) holds for $\psi=\psi_{k}$,
$k=1,2,\cdots$, in view of Lemma 3.4 we obtain
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,\psi_{k}(u(t,x)))\,dx+\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,\psi_{k}(u(s))),\
\ |\psi_{k}^{\prime}(u(s))z^{u}(s)|^{2}\gg\,ds\\\
=&\int_{\mathcal{O}}\phi(T,x,\psi_{k}(u(T,x)))\,dx+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,\psi_{k}(u(s))),\
\ \psi_{k}^{\prime}(u(s))h^{0,u}(s)\gg\,ds\\\
&-\int_{t}^{T}\\!\\!\\!\int_{\mathcal{O}}\partial_{s}\phi(s,x,\psi_{k}(u(s,x)))\,dxds-\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,\psi_{k}(u(s))),\
\ \psi_{k}^{\prime\prime}(u(s))|z(s)|^{2}\gg\,ds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,\psi_{k}(u(s))),\quad\psi_{k}^{\prime\prime}(u(s))\partial_{x_{i}}u(s)h^{i}(s)\gg\,ds\\\
&-\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,\psi_{k}(u(s)))\psi_{k}^{\prime}(u(s))\partial_{x_{i}}u(s)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\partial_{x_{i}}\phi^{\prime}(s,\cdot,\psi_{k}(u(s))),\quad\psi_{k}^{\prime}(u(s))h^{i,u}(s)\gg\,ds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,\psi_{k}(u(s))),\quad\psi_{k}^{\prime}(u(s))z^{r,u}(s)\gg\,dW_{s}^{r}\end{split}$
holds almost surely for all $t\in[0,T]$. From properties of $\phi$, we have
$\phi^{\prime}(t,x,r)\leq M|r|$ for any
$(t,x,r)\in[0,T]\times\mathcal{O}\times\mathbb{R}$. It follows that for any
$(s,x)\in[0,T]\times\mathcal{O}$,
$\begin{split}\left|\phi^{\prime}(s,x,\psi_{k}(u(s,x)))\psi_{k}^{\prime\prime}(u(s,x))\right|\leq&\
M\left|\psi_{k}(u(s,x))\right|\left|\psi_{k}^{\prime\prime}(u(s))\right|\\\
=&\
\frac{Mk}{2}\left|u(s,x)-\frac{1}{k}\right|^{2}k1_{[\frac{1}{k},\frac{2}{k}]}(u(s,x))\\\
\leq&\ M1_{[\frac{1}{k},\frac{2}{k}]}(u(s,x)).\end{split}$ (3.11)
On the other hand, we check that
$\lim_{k\rightarrow\infty}\|\psi_{k}(u)-u^{+}\|_{W^{1,2}_{\mathscr{F}}(Q)}=0$.
Therefore, by the dominated convergence theorem and taking limits in
$L^{1}([0,T]\times\Omega,\mathscr{P},\mathbb{R})$ on both sides of the above
equation, we prove our assertion. ∎
## 4 Solvability of Equation (1.1)
Before the solvability of equation (1.1), we give a useful lemma which is
borrowed from [11, Corollary B1] and called the stochastic Gronwall-Bellman
inequality.
###### Lemma 4.1.
Let $(\Omega,\mathcal{F},\mathbb{F},P)$ be a filtered probability space whose
filtration $\mathbb{F}=\\{\mathcal{F}_{t}:t\in[0,T]\\}$ satisfies the usual
conditions. Suppose $\\{Y_{s}\\}$ and $\\{X_{s}\\}$ are optional integrable
processes and $\alpha$ is a nonnegative constant. If for all $t$,
$s\rightarrow E[Y_{s}|\mathcal{F}_{t}]$ is continuous almost surely and
$Y_{t}\leq(\geq)E[\int_{t}^{T}(X_{s}+\alpha Y_{s})ds|\mathcal{F}_{t}]+Y_{T}$,
then for all $t$,
$Y_{t}\leq(\geq)e^{\alpha(T-t)}E[Y_{T}|\mathcal{F}_{t}]+E\left[\int_{t}^{T}e^{\alpha(s-t)}X_{s}ds|\mathcal{F}_{t}\right]\quad
a.s..$
###### Theorem 4.2.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}3)$ be satisfied and
$\left\\{h^{i},i=0,1,\cdots,n\right\\}\subset\mathcal{M}^{2}(Q)$. Then
$\dot{\mathscr{U}}\times\dot{\mathscr{V}}(G,f+h,g+h^{0})$ (with
$h=(h^{1},\cdots,h^{n})$) admits one and only one element $(u,v)$ which
satisfies the following estimate
$\begin{split}\|u\|_{\mathcal{V}_{2}(Q)}+\|v\|_{\mathcal{M}^{2}(Q)}\leq
C\left\\{\|G\|_{L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))}+A_{p}(f_{0},g_{0})+H_{2}(h,h^{0})\right\\},\end{split}$
(4.1)
where $C$ is a constant depending on
$n,p,q,\kappa,\lambda,\beta,\varrho,\Lambda_{0},T,|\mathcal{O}|$ and $L$.
###### Proof. .
Step 1. Let $({\mathcal{A}}3)_{0}$ be satisfied. From [22, Theorem 2.1], there
is a unique weak solution $(u,v)$ in the space
$(\dot{W}^{1,2}_{\mathscr{F}}(Q)\cap S^{2}(L^{2}(\mathcal{O})))\times
W^{2}_{\mathscr{F}}(Q)$.
$Claim~{}(*):~{}(u,v)\in\dot{\mathscr{U}}\times\dot{\mathscr{V}}(G,f+h,g+h^{0})$.
We shall prove $Claim~{}(*)$ in Step 2. By Lemma 3.3, we have almost surely
$\begin{split}&\|u(t)\|_{L^{2}(\mathcal{O})}^{2}+\int_{t}^{T}\|v(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds\\\
=&\|G\|^{2}_{L^{2}(\mathcal{O})}+2\int_{t}^{T}\ll u(s),\
\,b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)+h^{0}(s)\gg ds\\\
&-2\int_{t}^{T}\ll\partial_{x_{j}}u(s),\ \
a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+f^{j}(s,\cdot,u(s),\nabla
u(s),v(s))+h^{j}(s)\gg\,ds\\\ &-2\int_{t}^{T}\ll u(s),\
\,v^{r}(s)\gg\,dW^{r}_{s}+2\int_{t}^{T}\ll u(s),\ \,g(s,\cdot,u(s),\nabla
u(s),v(s))\gg\,ds\end{split}$ (4.2)
for all $t\in[0,T]$. Therefore, we obtain that almost surely
$\begin{split}&E\left[\|u(t)\|_{L^{2}(\mathcal{O})}^{2}+\int_{t}^{T}\|v(s)\|_{L^{2}(\mathcal{O})}^{2}ds|\mathscr{F}_{t}\right]\\\
=&\
E\left[\|G\|^{2}_{L^{2}(\mathcal{O})}|\mathscr{F}_{t}\right]+2E\left[\int_{t}^{T}\ll
u(s),g(s,\cdot,u(s),\nabla u(s),v(s))\gg ds\big{|}\mathscr{F}_{t}\right]\\\
&+2E\left[\int_{t}^{T}\ll u(s),\
\,b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)+h^{0}(s)\gg
ds\big{|}\mathscr{F}_{t}\right]\\\
&-2E\Big{[}\int_{t}^{T}\ll\partial_{x_{j}}u(s),\
\,a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)\\\
&~{}~{}~{}~{}~{}~{}~{}+f^{j}(s,\cdot,u(s),\nabla u(s),v(s))+h^{j}(s)\gg
ds\big{|}\mathscr{F}_{t}\Big{]},\,\ \forall t\in[0,T].\end{split}$ (4.3)
Using the Lipschitz condition and Hölder inequality, we get the following
estimates
$\begin{split}&2E\left[\int_{t}^{T}\left(\ll u(s),\
h^{0}(s)\gg-\ll\partial_{x_{j}}u(s),\
h^{j}(s)\gg\right)ds\big{|}\mathscr{F}_{t}\right]\\\ \leq&\
E\left[\int_{t}^{T}\\!\left(\|u(s)\|^{2}_{L^{2}(\mathcal{O})}+\|h^{0}(s)\|_{L^{2}(\mathcal{O})}^{2}+\varepsilon^{-1}\|h(s)\|^{2}_{L^{2}(\mathcal{O})}+\varepsilon\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}\right)ds\big{|}\mathscr{F}_{t}\right],\end{split}$
(4.4)
$\begin{split}&\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll
u(s),\,g(s,\cdot,u(s),\nabla u(s),v(s))\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\ \leq&\
\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll
u(s),\,g_{0}(s)+L(|u(s)|+|\nabla u(s)|+|v(s)|)\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\ \leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+C(\varepsilon,\varepsilon_{1},L)\|u\|_{2;\mathcal{O}_{t}}^{2}\\\
&+\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll|u(s)|,\,|g_{0}(s)|\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\ \leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+C(\varepsilon,\varepsilon_{1},L)\|u\|_{2;\mathcal{O}_{t}}^{2}+2|\mathcal{O}_{t}|^{\frac{1}{2}-\frac{1}{p}}\|g_{0}\|_{\frac{p(n+2)}{n+2+p};\mathcal{O}_{t}}\|u\|_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\\\
\leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+C(\varepsilon,\varepsilon_{1},L)\|u\|_{2;\mathcal{O}_{t}}^{2}+c(n)|\mathcal{O}_{t}|^{\frac{1}{2}-\frac{1}{p}}\|g_{0}\|_{\frac{p(n+2)}{n+2+p};\mathcal{O}_{t}}\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}\\\
\leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+C(\varepsilon,\varepsilon_{1},L)\|u\|_{2;\mathcal{O}_{t}}^{2}+\delta\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}\\\
&+C(\delta,n,p,|Q|)\|g_{0}\|_{\frac{p(n+2)}{n+2+p};\mathcal{O}_{t}}^{2}\end{split}$
(4.5)
and
$\begin{split}&\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll
u(s),\ b^{i}(s)\partial_{x_{i}}u(s)+c(s)\,u(s)+\varsigma^{r}(s)v^{r}(s)\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\ \leq&\
(\varepsilon^{-1}+\varepsilon^{-1}_{1})\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}E\left[\int_{\tau}^{T}\ll|b(s)|^{2}+|c(s)|+|\varsigma(s)|^{2},\
u^{2}(s)\gg ds\big{|}\mathscr{F}_{\tau}\right]\\\ &+\varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}\\\
\leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+(\varepsilon^{-1}+\varepsilon^{-1}_{1})B_{q}(b,c,\varsigma)\|u\|^{2}_{\frac{2q}{q-1};\mathcal{O}_{t}}\\\
\leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+(\varepsilon^{-1}+\varepsilon^{-1}_{1})B_{q}(b,c,\varsigma)\|u\|^{2\alpha}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\|u\|^{2(1-\alpha)}_{2;\mathcal{O}_{t}}\\\
\leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}\\\ &\
+(\varepsilon^{-1}+\varepsilon^{-1}_{1})B_{q}(b,c,\varsigma)\left(C(n)\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}\right)^{2\alpha}\|u\|^{2(1-\alpha)}_{2;\mathcal{O}_{t}}\textrm{
(by Lemma \ref{lem emmbedding for space V})}\\\ \leq&\ \varepsilon\|\nabla
u\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v\|_{2;\mathcal{O}_{t}}^{2}+\delta\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}\\\
&+C(\delta,n,q)\left|(\varepsilon^{-1}+\varepsilon_{1}^{-1})B_{q}(b,c,\varsigma)\right|^{\frac{1}{1-\alpha}}\|u\|_{2;\mathcal{O}_{t}}^{2}\end{split}$
(4.6)
with $\alpha:=\frac{n+2}{2q}\in(0,1)$ and the three positive small parameters
$\varepsilon$, $\varepsilon_{1}$ and $\delta$ waiting to be determined later.
Also, there exists a constant
$\theta>\varrho^{\prime}=\frac{\varrho}{\varrho-1}$ such that
$\lambda-\kappa-\beta\theta>0$ and
$\begin{split}-&E\left[\int_{t}^{T}2\ll\partial_{x_{j}}u(s),a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)+f^{j}(s,u(s),\nabla
u(s),v(s))\gg ds\big{|}\mathscr{F}_{t}\right]\\\
\leq&-E\left[\int_{t}^{T}\ll\partial_{x_{j}}u(s),\,(2a^{ij}(s)-\varrho\sigma^{jr}(s)\sigma^{ir}(s))\partial_{x_{i}}u(s)\gg
ds\big{|}\mathscr{F}_{t}\right]\\\
&+\frac{1}{\varrho}E\left[\int_{t}^{T}\|v(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\right]\\\
&+2E\left[\int_{t}^{T}\ll|\nabla u(s)|,\,L|u(s)|+\frac{\kappa}{2}|\nabla
u(s)|+\beta^{\frac{1}{2}}|v(s)|+|f_{0}(s)|\gg
ds\big{|}\mathscr{F}_{t}\right]\\\
\leq&-(\lambda-\kappa-\beta\theta-\varepsilon)E\left[\int_{t}^{T}\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds\big{|}\mathscr{F}_{t}\right]+C(\varepsilon)\|f_{0}\|^{2}_{2;\mathcal{O}_{t}}\\\
&+\left(\frac{1}{\varrho}+\frac{1}{\theta}\right)E\left[\int_{t}^{T}\|v(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\right]+C(\varepsilon,L)E\left[\int_{t}^{T}\|u(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\right]\\\
\leq&-(\lambda-\kappa-\beta\theta-\varepsilon)E\left[\int_{t}^{T}\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds\big{|}\mathscr{F}_{t}\right]\\\
&+C(\varepsilon,|Q|,p,L)\left\\{E\left[\int_{t}^{T}\|u(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\right]+\|f_{0}\|_{p;\mathcal{O}_{t}}^{2}\right\\}\\\
&+\left(\frac{1}{\varrho}+\frac{1}{\theta}\right)E\left[\int_{t}^{T}\|v(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\right],\
\,\forall t\in[0,T]\ \,a.s..\end{split}$ (4.7)
Choosing $\varepsilon$ and $\varepsilon_{1}$ to be small enough, we get
$\begin{split}&\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}+\|v\|_{2;\mathcal{O}_{t}}^{2}\\\
\leq&\
3~{}\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}\left\\{\|u(\tau)\|_{L^{2}(\mathcal{O})}^{2}+E\left[\int_{\tau}^{T}(\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}+\|v(s)\|_{L^{2}(\mathcal{O})}^{2})\,ds\big{|}\mathscr{F}_{\tau}\right]\right\\}\\\
\leq&\
C_{1}\biggl{\\{}\|G\|_{L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))}^{2}+\|f_{0}\|_{p;\mathcal{O}_{t}}^{2}+\left|H_{2}(h,h^{0})\right|^{2}\\\
&+\delta\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}+C(\delta,n,q,\Lambda_{0})\int_{t}^{T}\|u(s)\|_{\mathcal{V}_{2}(\mathcal{O}_{s})}^{2}\,ds+C(\delta,n,p,|Q|)\|g_{0}\|_{\frac{p(n+2)}{n+2+p};\mathcal{O}_{t}}^{2}\biggr{\\}}\end{split}$
with the constant $C_{1}$ being independent of $\delta$. Then by choosing
$\delta$ to be so small that $C_{1}\delta<1/2$, we obtain
$\begin{split}&\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}+\|v\|_{2;\mathcal{O}_{t}}^{2}\\\
\leq&\
C\left\\{\|G\|_{L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))}^{2}+\int_{t}^{T}\|u(s)\|_{\mathcal{V}_{2}(\mathcal{O}_{s})}^{2}\,ds+\left|A_{p}(f_{0},g_{0})\right|^{2}+\left|H_{2}(h,h^{0})\right|^{2}\right\\}.\end{split}$
(4.8)
Thus, it follows from Gronwall inequality that
$\|u\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}+\|v\|_{2;\mathcal{O}_{t}}^{2}\leq
C\left\\{\|G\|_{L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))}^{2}+\left|A_{p}(f_{0},g_{0})\right|^{2}+\left|H_{2}(h,h^{0})\right|^{2}\right\\}$
(4.9)
with the constant $C$ depending on
$T,L,\Lambda_{0},\lambda,\beta,\kappa,\varrho,n,p,q$ and $|Q|$.
Step 2. We prove $Claim~{}(*)$. It is sufficient to prove
$(u,v)\in\dot{\mathcal{V}}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$. Making estimates
like (4.4) and (4.7), we obtain
$\begin{split}&\|u(t)\|_{L^{2}(\mathcal{O})}^{2}+E\left[\int_{t}^{T}\|v(s)\|_{L^{2}(\mathcal{O})}^{2}ds|\mathscr{F}_{t}\right]\\\
=&\
E\left[\|G\|^{2}_{L^{2}(\mathcal{O})}|\mathscr{F}_{t}\right]+2E\left[\int_{t}^{T}\ll
u(s),g(s,\cdot,u(s),\nabla u(s),v(s))\gg ds\big{|}\mathscr{F}_{t}\right]\\\
&+2E\left[\int_{t}^{T}\ll u(s),\
\,b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)+h^{0}(s)\gg
ds\big{|}\mathscr{F}_{t}\right]\\\
&-2E\Big{[}\int_{t}^{T}\ll\partial_{x_{j}}u(s),\
\,a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}(s)\\\
&~{}~{}~{}~{}~{}~{}~{}+f^{j}(s,\cdot,u(s),\nabla u(s),v(s))+h^{j}(s)\gg
ds\big{|}\mathscr{F}_{t}\Big{]}\\\ \leq&\
-(\lambda-\kappa-\beta\theta-\varepsilon)E\left[\int_{t}^{T}\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds\big{|}\mathscr{F}_{t}\right]\\\
&+\left(\frac{1}{\varrho}+\frac{1}{\theta}+\varepsilon\right)E\left[\int_{t}^{T}\|v(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\right]\\\
&+E\left[\|G\|^{2}_{L^{2}(\mathcal{O})}|\mathscr{F}_{t}\right]+C(\varepsilon)\left(\left|H_{2}(f_{0},g_{0})\right|^{2}+\left|H_{2}(h,h^{0})\right|^{2}\right)\\\
&+C(\varepsilon,\lambda,\beta,\kappa,\varrho,L,\||b|\|_{\mathcal{L}^{\infty}(Q)},\|c\|_{\mathcal{L}^{\infty}(Q)},\||\varsigma|\|_{\mathcal{L}^{\infty}(Q)})E\left[\int_{t}^{T}\|u(s)\|_{L^{2}(\mathcal{O})}^{2}ds\big{|}\mathscr{F}_{t}\right]\end{split}$
(4.10)
with the positive constant $\varepsilon$ waiting to be determined later.
Letting $\varepsilon$ be small enough, we have almost surely
$\begin{split}&\|u(t)\|_{L^{2}(\mathcal{O})}^{2}+E\left[\int_{t}^{T}\left(\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}+\|v(s)\|_{L^{2}(\mathcal{O})}^{2}\right)ds|\mathscr{F}_{t}\right]\\\
\leq&\
C\left\\{\|G\|^{2}_{L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))}+\left|H_{2}(f_{0},g_{0})\right|^{2}+\left|H_{2}(h,h^{0})\right|^{2}+E\left[\int_{t}^{T}\|u(s)\|_{L^{2}(\mathcal{O})}^{2}ds\big{|}\mathscr{F}_{t}\right]\right\\}\end{split}$
for all $t\in[0,T]$. Then, by Lemma 4.1 we obtain
$\begin{split}&\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{t\in[0,T]}\left\\{\|u(t)\|_{L^{2}(\mathcal{O})}^{2}+E\left[\int_{t}^{T}\left(\|\nabla
u(s)\|_{L^{2}(\mathcal{O})}^{2}+\|v(s)\|_{L^{2}(\mathcal{O})}^{2}\right)ds|\mathscr{F}_{t}\right]\right\\}\\\
\leq&\
C\left\\{\|G\|^{2}_{L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))}+\left|H_{2}(f_{0},g_{0})\right|^{2}+\left|H_{2}(h,h^{0})\right|^{2}\right\\}\end{split}$
with the constant $C$ depending on
$\lambda,\beta,\kappa,\varrho,L,T,\||b|\|_{\mathcal{L}^{\infty}(Q)},\|c\|_{\mathcal{L}^{\infty}(Q)},\||\varsigma|\|_{\mathcal{L}^{\infty}(Q)}$.
Hence, $(u,v)\in\dot{\mathcal{V}}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$. We
complete the proof of $Claim~{}(*)$.
Step 3. Now we consider the general case of assumption $({\mathcal{A}}3)$. The
existence of the solution can be shown by approximation. As $p>n+2$ and
$\mathcal{M}^{p}(Q)\subset\mathcal{M}^{2}(Q)$, $f_{0}\in\mathcal{M}^{2}(Q)$.
We approximate the functions $b$, $c$, $\varsigma$ and $g$ by
$b_{k}:=b1_{\\{|b|\leq k\\}},\ c_{k}:=c1_{\\{|c|\leq k\\}},\
\varsigma_{k}:=\varsigma 1_{\\{|\varsigma|\leq k\\}}\ {\rm and}\
g^{k}:=g-g_{0}+g^{k}_{0},$ (4.11)
with $g^{k}_{0}=g_{0}1_{\\{|g_{0}|\leq k\\}}$. Then we have
$\lim_{k\rightarrow\infty}B_{q}(b-b_{k},c-c_{k},\varsigma-\varsigma_{k})+A_{p}(0,g_{0}-g_{0}^{k})=0.$
Let $(u_{k},v_{k})\in\dot{\mathcal{V}}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$ be
the unique weak solution to (1.1) with $(b,c,\varsigma,f,g)$ being replaced by
$(b_{k},c_{k},\varsigma_{k},f+h,g^{k}+h^{0})$. Then by estimate (4.9), there
exists a positive constant $C_{0}$ such that
$\sup_{k\in\mathbb{N}}\left\\{\|u_{k}\|_{\mathcal{V}_{2}(Q)}^{2}+\|v_{k}\|_{2;Q}^{2}\right\\}<C_{0}.$
For $k,l\in\mathbb{N}$, the pair of random fields
$(u_{kl},v_{kl}):=(u_{k}-u_{l},v_{k}-v_{l})\in\dot{\mathcal{V}}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$
is the weak solution to the following BSPDE:
$(k,l)~{}~{}\left\\{\begin{array}[]{l}\begin{split}-du_{kl}(t,x)=&\displaystyle\biggl{[}\partial_{x_{j}}\Bigl{(}a^{ij}(t,x)\partial_{x_{i}}u_{kl}(t,x)+\sigma^{jr}(t,x)v_{kl}^{r}(t,x)\Bigr{)}+b_{k}^{j}(t,x)\partial_{x_{j}}u_{kl}(t,x)\\\
&\displaystyle+c_{k}(t,x)u_{kl}(t,x)+\varsigma^{r}_{k}(t,x)v^{r}_{kl}(t,x)\\\
&\displaystyle+b_{kl}^{j}(t,x)\partial_{x_{j}}u_{l}(t,x)+c_{kl}(t,x)u_{l}(t,x)+\varsigma_{kl}^{r}(t,x)v^{r}_{l}(t,x)\\\
&\displaystyle+\bar{g}_{kl}(t,x,u_{kl}(t,x),\nabla u_{kl}(t,x),v_{kl}(t,x))\\\
&\displaystyle+\partial_{x_{j}}\bar{f}_{kl}^{j}(t,x,u_{kl}(t,x),\nabla
u_{kl}(t,x),v_{kl}(t,x))\biggr{]}\,dt\\\ &\displaystyle-
v_{kl}^{r}(t,x)\,dW_{t}^{r},\quad(t,x)\in Q:=[0,T]\times\mathcal{O};\\\
u_{kl}(T,x)=&0,\quad x\in\mathcal{O}\end{split}\end{array}\right.$
with
$\begin{split}\bar{f}_{kl}(t,x,R,Y,Z):=&f(t,x,R+u_{l}(t,x),Y+\nabla
u_{l}(t,x),Z+v_{l}(t,x))\\\ &-f(t,x,u_{l}(t,x),\nabla
u_{l}(t,x),v_{l}(t,x)),\\\
\bar{g}_{kl}(t,x,R,Y,Z):=&g^{k}(t,x,R+u_{l}(t,x),Y+\nabla
u_{l}(t,x),Z+v_{l}(t,x))\\\ &-g^{l}(t,x,u_{l}(t,x),\nabla
u_{l}(t,x),v_{l}(t,x)),\\\ (b_{kl},\ c_{kl},\
\varsigma_{kl})(t,x):=&(b_{k}-b_{l},\ c_{k}-c_{l},\
\varsigma_{k}-\varsigma_{l})(t,x).\end{split}$
Since
$\begin{split}&\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll
u_{kl}(s),\
b_{kl}^{i}\partial_{x_{i}}u_{l}(s)+c_{kl}\,u_{l}(s)+\varsigma^{r}_{kl}v_{l}^{r}(s)\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\ \leq&\
2{\bar{\varepsilon}}^{-1}\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}E\left[\int_{\tau}^{T}\\!\\!\ll\left|b_{kl}(s)\right|^{2}+|c_{kl}(s)|+\left|\varsigma_{kl}(s)\right|^{2},\
u_{kl}^{2}(s)\gg ds\big{|}\mathscr{F}_{\tau}\right]\\\
&+\bar{\varepsilon}\left(\|\nabla
u_{l}\|_{2;\mathcal{O}_{t}}^{2}+\|v_{l}\|_{2;\mathcal{O}_{t}}^{2}\right)\\\
\leq&\
\bar{\varepsilon}\left(\|u_{l}\|_{\mathcal{V}_{2}(Q)}^{2}+\|v_{l}\|_{2;Q}^{2}\right)+2\bar{\varepsilon}^{-1}B_{q}(b_{kl},c_{kl},\varsigma_{kl})\|u_{kl}\|^{2}_{\frac{2q}{q-1};\mathcal{O}_{t}}\\\
\leq&\
\bar{\varepsilon}C_{0}+\delta\|u_{kl}\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}+C(\delta,n,q)\left|\bar{\varepsilon}^{-1}B_{q}(b_{kl},c_{kl},\varsigma_{kl})\right|^{\frac{2q}{2q-n-2}}\|u_{kl}\|_{2;\mathcal{O}_{t}}^{2}\textrm{
(by Lemma \ref{lem emmbedding for space V})},\end{split}$
in a similar way to the derivation of (4.8), we obtain
$\begin{split}&\|u_{kl}\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}^{2}+\|v_{kl}\|_{2;\mathcal{O}_{t}}^{2}\\\
\leq&\
C\bigg{\\{}\bar{\varepsilon}+\left|A_{p}(0,g_{0}^{k}-g_{0}^{l})\right|^{2}+\left(1+\left|\bar{\varepsilon}^{-1}B_{q}(b_{kl},c_{kl},\varsigma_{kl})\right|^{\frac{2q}{2q-n-2}}\right)\int_{t}^{T}\|u_{kl}(s)\|_{\mathcal{V}_{2}(\mathcal{O}_{s})}^{2}\,ds\bigg{\\}}\end{split}$
which, by Gronwall inequality, implies
$\begin{split}&\|u_{kl}\|_{\mathcal{V}_{2}(Q)}^{2}+\|v_{kl}\|_{2;Q}^{2}\\\
\leq\
&C\left(\bar{\varepsilon}+\left|A_{p}(0,g_{0}^{k}-g_{0}^{l})\right|^{2}\right)\exp{\left[T\left(1+\left|\bar{\varepsilon}^{-1}B_{q}(b_{kl},c_{kl},\varsigma_{kl})\right|^{\frac{2q}{2q-n-2}}\right)\right]}\end{split}$
(4.12)
with the constant $C$ being independent of $k$, $l$ and $\bar{\varepsilon}$.
By choosing $\bar{\varepsilon}$ to be small and then $k$ and $l$ to be
sufficiently large, we conclude that $(u_{k},v_{k})$ is a Cauchy sequence in
$\dot{\mathcal{V}}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$. Passing to the limit, we
check that the limit
$(u,v)\in\dot{\mathscr{U}}\times\dot{\mathscr{V}}(G,f+h,g+h^{0})$. In view of
estimate (4.9) we prove estimate (4.1).
Step 4. It remains to prove the uniqueness. Assume that
$(u^{\prime},v^{\prime})$ and $(u,v)$ are two weak solutions in
$\dot{\mathcal{V}}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$. Then their difference
$(\bar{u},\bar{v}):=(u-u^{\prime},v-v^{\prime})\in\dot{\mathscr{U}}\times\dot{\mathscr{V}}(0,\bar{f},\bar{g})$
with
$\begin{split}\bar{f}(t,x,R,Y,Z):=&f(t,x,R+u^{\prime}(t,x),Y+\nabla
u^{\prime}(t,x),Z+v^{\prime}(t,x))\\\ &-f(t,x,u^{\prime}(t,x),\nabla
u^{\prime}(t,x),v^{\prime}(t,x)),\\\
\bar{g}(t,x,R,Y,Z):=&g(t,x,R+u^{\prime}(t,x),Y+\nabla
u^{\prime}(t,x),Z+v^{\prime}(t,x))\\\ &-g(t,x,u^{\prime}(t,x),\nabla
u^{\prime}(t,x),v^{\prime}(t,x)).\end{split}$
Since $\bar{f}_{0}=0,\ \bar{g}_{0}=0$ and $\bar{u}(T)=0$, we deduce from (4.9)
that $\bar{u}=0$ and $\bar{v}=0$. The proof is complete. ∎
###### Remark 4.1.
On the basis of the monotone operator theory, Qiu and Tang in [22] established
a theory of solvability for quasi-linear BSPDEs in an abstract framework.
However even for the linear case $(f,g)\equiv(f_{0},g_{0})$, our BSPDE (1.1)
under assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}3)$ falls beyond the
framework of Qiu and Tang [22] since our $b,c,$ and $\varsigma$ may be
unbounded.
###### Corollary 4.3.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}3)$ be true,
$\left\\{h^{i},i=0,1,\cdots,n\right\\}\subset\mathcal{M}^{2}(Q)$ and
$(u,v)\in\dot{\mathscr{U}}\times\dot{\mathscr{V}}(G,f+h,g+h^{0})$ with
$h=(h^{1},\cdots,h^{n})$. Let
$\phi:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\longrightarrow\mathbb{R}$
satisfy the assumptions of Lemma 3.3. Then we have almost surely
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,u(t,x))\,dx+\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u(s)),\
|v(s)|^{2}\gg ds\\\
=&\int_{\mathcal{O}}\phi(T,x,G(x))\,dx-\int_{t}^{T}\int_{\mathcal{O}}\partial_{s}\phi(s,x,u(s,x))\,dxds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\
b^{i}\partial_{x_{i}}u(s)+c\,u(s)+\varsigma^{r}v^{r}(s)+h^{0}(s)\gg ds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\ g(s,\cdot,u(s),\nabla
u(s),v(s))\gg ds\\\
&-\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u(s))\partial_{x_{i}}u(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u(s)),\quad
a^{ji}\partial_{x_{j}}u(s)+\sigma^{ri}v^{r}(s)\\\
&\quad\quad+f^{i}(s,\cdot,u(s),\nabla u(s),v(s))+h^{i}(s)\gg ds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u(s)),\ v^{r}(s)\gg dW_{s}^{r},\,\
\forall t\in[0,T].\end{split}$
The proof of the corollary is rather standard and is sketched below.
###### Remark 4.2.
In a similar way to Remark 3.2, our corollary also holds for $\psi$ in Remark
3.2.
###### Sketch of the proof.
First, one can check that all the terms involved in our assertion is well
defined. Similar to the proof of Theorem 4.2, we still approximate
$(b,c,\varsigma,g)$ by $(b_{k},c_{k},\varsigma_{k},g^{k})$ which is defined in
(4.11). By Theorem 4.2, there is a unique weak solution $(u_{k},v_{k})$ to
(1.1) with $(b,c,\varsigma,f,g)$ being replaced by
$(b_{k},c_{k},\varsigma_{k},f+h,g^{k}+h^{0})$. Then by Lemma 3.3, we have for
each $k\in\mathbb{N}$,
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,u_{k}(t,x))\,dx+\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u_{k}(s)),\,|v_{k}(s)|^{2}\gg
ds\\\
=&\int_{\mathcal{O}}\phi(T,x,G(x))\,dx-\int_{t}^{T}\\!\int_{\mathcal{O}}\partial_{s}\phi(s,x,u_{k}(s,x))\,dxds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\
b_{k}^{i}\partial_{x_{i}}u_{k}(s)+c_{k}\,u_{k}(s)+\varsigma_{k}^{r}v_{k}^{r}(s)+h^{0}(s)\gg
ds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\,g^{k}(s,\cdot,u_{k}(s),\nabla
u_{k}(s),v_{k}(s))\gg ds\\\
&-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u_{k}(s)),\,v_{k}^{r}(s)\gg
dW_{s}^{r}\\\
&-\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u_{k}(s))\partial_{x_{i}}u_{k}(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u_{k}(s)),\quad
a^{ji}\partial_{x_{j}}u_{k}(s)+\sigma^{ri}v_{k}^{r}(s)\\\
&\quad\quad+f^{i}(s,\cdot,u_{k}(s),\nabla u_{k}(s),v_{k}(s))+h^{i}(s)\gg
ds\end{split}$ (4.13)
almost surely for all $t\in[0,T]$. On the other hand, from the proof of
Theorem 4.2 it follows that
$\lim_{k\rightarrow\infty}\left\\{\|u-u_{k}\|_{\mathcal{V}_{2}(Q)}+\|v-v_{k}\|_{2;Q}\right\\}=0.$
Hence passing to the limit in $L^{1}(\Omega,\mathscr{F})$ and taking into
account the path-wise continuity of $u$, we prove our assertion. ∎
We have
###### Proposition 4.4.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}3)$ be satisfied,
$\left\\{h^{i},i=0,1,\cdots,n\right\\}\subset\mathcal{M}^{2}(Q)$ and
$(u,v)\in\mathscr{U}\times\mathscr{V}(G,f+h,g+h^{0})$ with
$h=(h^{1},\cdots,h^{n})$ and $u^{+}\in\dot{\mathcal{V}}_{2,0}(Q)$. Let
$\phi:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\longrightarrow\mathbb{R}$
satisfy the assumptions of Lemma 3.3. Then, with probability 1, the following
relation
$\begin{split}&\int_{\mathcal{O}}\phi(t,x,u^{+}(t,x))\,dx+\frac{1}{2}\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{+}(s)),\,|v^{u}(s)|^{2}\gg
ds\\\
=&\int_{\mathcal{O}}\phi(T,x,G^{+}(x))\,dx-\int_{t}^{T}\int_{\mathcal{O}}\partial_{s}\phi(s,x,u^{+}(s,x))\,dxds\\\
&-\int_{t}^{T}\ll\phi^{\prime\prime}(s,\cdot,u^{+}(s))\partial_{x_{i}}u^{+}(s)+\partial_{x_{i}}\phi^{\prime}(s,\cdot,u^{+}(s)),\quad
a^{ji}(s)\partial_{x_{j}}u^{+}(s)\\\
&\quad\quad+\sigma^{ri}(s)v^{r,u}(s)+f^{i,u}(s)\gg\,ds\\\
&+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u^{+}(s)),\
b^{i}(s)\partial_{x_{i}}u^{+}(s)+c(s)\,u^{+}(s)+\varsigma^{r}(s)v^{r,u}(s)\gg
ds\\\ &+\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u^{+}(s)),\,g^{u}(s)\gg
ds-\int_{t}^{T}\ll\phi^{\prime}(s,\cdot,u^{+}(s)),\,v^{r,u}\gg
dW_{s}^{r}\end{split}$
holds almost surely for all $t\in[0,T]$ where
$\begin{split}&g^{u}(s,x):=1_{\\{(s,x):u(s,x)>0\\}}(s,x)\left(h^{0}(s,x)+g(s,x,u(s,x),\nabla
u(s,x),v(s,x))\right);\\\
&f^{i,u}(s,x):=1_{\\{(s,x):u(s,x)>0\\}}(s,x)\left(h^{i}(s,x)+f^{i}(s,x,u(s,x),\nabla
u(s,x),v(s,x))\right),\\\ &\ i=0,1,\cdots,n;\end{split}$
and
$v^{u}:=(v^{1,u},\cdots,v^{m,u}),\quad
v^{r,u}(s,x):=1_{\\{(s,x):u(s,x)>0\\}}(s,x)v^{r}(s,x),\ r=1,\cdots,m.$
The proof is very similar to that of Lemma 3.5 and is omitted here. The main
difference lies in Step 1 where we use Corollary 4.3 and Remark 4.2 instead of
Lemma 3.3 and Remark 3.2.
## 5 The maximum principles
### 5.1 The global case
###### Theorem 5.1.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}4)$ hold. Assume that
$(u,v)\in\mathcal{V}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$ is a weak solution of
(1.1). Then we have
$\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times Q}u(\omega,t,x)\,\leq
C\left\\{\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,t,x)+A_{p}(f_{0},g_{0}^{+})+\|u^{+}\|_{2;Q}\right\\}$ (5.1)
where $C$ is a constant depending on
$n,p,q,\kappa,\lambda,\beta,\varrho,\Lambda_{0},L_{0},T,|\mathcal{O}|$ and
$L$.
###### Remark 5.1.
By the inequality
$\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,t,x)\leq L_{1}$, we mean that
$(u-L_{1})^{+}\in\dot{\mathcal{V}}_{2,0}(Q)$ and with probability 1, for any
$\zeta\in C_{c}^{\infty}(\mathcal{O})$, there holds
$\lim_{t\rightarrow T_{-}}\ll\zeta,\,(u(t)-L_{1})^{+}\gg\,=0.$
###### Remark 5.2.
In Theorem 5.1, assume further that
$\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}|u(\omega,t,x)|\,\leq L_{1}<\infty.$
We have $u\in\mathcal{L}^{\infty}(Q)$ and
$\|u\|_{\infty;Q}\,\leq
C\left\\{L_{1}+A_{p}(f_{0},g_{0})+\|u\|_{2;Q}\right\\}$ (5.2)
where $C$ is a constant depending on
$n,p,q,\kappa,\lambda,\beta,\varrho,\Lambda_{0},L_{0},T,|\mathcal{O}|$ and
$L$.
We start the proof of Theorem 5.1 with borrowing the following lemma either
from [4, Lemma 1.2, Chapter 6] or from [16, Lemma 5.6, Chapter 2].
###### Lemma 5.2.
Let $\\{a_{k}:k=0,1,2,\cdots\\}$ be a sequence of nonnegative numbers
satisfying
$a_{k+1}\leq C_{0}b^{k}a_{k}^{1+\delta},\,k=0,1,2,\cdots$
where $b>1$, $\delta>0$ and $C_{0}$ is a positive constant. Then if
$a_{0}\leq\theta_{0}:=C_{0}^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^{2}}},$
we have $\lim_{k\rightarrow\infty}a_{k}=0$.
###### Sketch of the proof.
We use the induction principle. It is sufficient to prove the following
assertion:
$a_{k}\leq\frac{\theta_{0}}{\nu^{k}},\,k=0,1,2,\cdots,$ (5.3)
with the parameter $\nu>1$ waiting to be determined later. It is obvious for
$k=0$ that (5.3) holds. Assume that (5.3) holds for $k=r$. Then we have
$a_{r+1}\leq C_{0}b^{r}a_{r}^{1+\delta}\leq
C_{0}b^{r}\left(\frac{\theta_{0}}{\nu^{r}}\right)^{1+\delta}=\frac{\theta_{0}}{\nu^{r+1}}\cdot\frac{C_{0}b^{r}\theta_{0}^{\delta}}{\nu^{r\delta-1}}.$
Taking $\nu=b^{\frac{1}{\delta}}>1$, we obtain
$a_{r+1}\leq\frac{\theta_{0}}{\nu^{r+1}}\cdot
C_{0}\nu\theta_{0}^{\delta}=\frac{\theta_{0}}{\nu^{r+1}}.$
∎
###### Corollary 5.3.
Let $\phi:[r_{0},\infty)\longrightarrow\mathbb{R}^{+}$ be a nonnegative and
decreasing function. Moreover, there exist constants $C_{1}>0$, $\alpha>0$ and
$\zeta>1$ such that for any $l>r>r_{0}$,
$\phi(l)\leq\frac{C_{1}}{(l-r)^{\alpha}}\phi(r)^{\zeta}.$
Then for
$d\geq
C_{1}^{\frac{1}{\alpha}}|\phi(r_{0})|^{\frac{\zeta-1}{\alpha}}2^{\frac{\zeta}{\zeta-1}},$
we have $\phi(r_{0}+d)=0$.
###### Sketch of the proof.
Define
$r_{k}:=r_{0}+d-\frac{d}{2^{k}},k=0,1,2,\cdots.$
Then
$\phi(r_{k+1})\leq\frac{C_{1}2^{(k+1)\alpha}}{d^{\alpha}}\phi(r_{k})^{\zeta}=\frac{C_{1}2^{\alpha}}{d^{\alpha}}2^{k\alpha}\phi(r_{k})^{\zeta}.$
In view of our assumption on $d$, since
$\phi(r_{0})\leq\theta_{0}=\left(\frac{C_{1}2^{\alpha}}{d^{\alpha}}\right)^{-\frac{1}{\zeta-1}}2^{-\frac{\alpha}{(\zeta-1)^{2}}}=d^{\frac{\alpha}{\zeta-1}}C_{1}^{-\frac{1}{\zeta-1}}2^{-\frac{\alpha\zeta}{(\zeta-1)^{2}}},$
we deduce from Lemma 5.2 that $\lim_{k\rightarrow\infty}\phi(r_{k+1})=0$.
∎
###### Proof of Theorem 5.1.
Assume that $L_{0}=0$, or else we consider $\tilde{u}(t,x):=e^{L_{0}t}u(t,x)$
instead of $u$. It is sufficient to prove our theorem for the case
$\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,t,x)\,<\infty.$
Then for
$k\geq\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,t,x)$, we have
$(u-k,v)\in\mathscr{U}\times\mathscr{V}(G-k,f^{k},g^{k})$ with
$(f^{k},g^{k})(\omega,t,x,R,Y,Z):=(f,g)(\omega,t,x,R+k,Y,Z)+(0,c(\omega,t,x)k)$
for
$(\omega,t,x,R,Y,Z)\in\Omega\times[0,T]\times\mathcal{O}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{m}$.
From Proposition 4.4, we have almost surely
$\begin{split}&\int_{\mathcal{O}}|(u(t,x)-k)^{+}|^{2}\,dx+\int_{t}^{T}\|v_{k}(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\\\
=&-2\int_{t}^{T}\ll\partial_{x_{j}}(u(s)-k)^{+},\\\
&~{}~{}~{}~{}~{}~{}~{}a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}_{k}(s)+(f^{k})^{j}(s,\cdot,(u(s)-k)^{+},\nabla
u(s),v_{k}(s))\gg ds\\\ &+2\int_{t}^{T}\ll(u(s)-k)^{+},\
b^{i}\partial_{x_{i}}u(s)+c\,(u(s)-k)^{+}+\varsigma^{r}v_{k}^{r}(s)\gg ds\\\
&+2\int_{t}^{T}\ll(u(s)-k)^{+},\ g^{k}(s,\cdot,(u(s)-k)^{+},\,\nabla
u(s),v_{k}(s))\gg ds\\\ &-2\int_{t}^{T}\ll(u(s)-k)^{+},\ v_{k}^{r}(s)\gg
dW_{s}^{r},\quad\forall\ t\in[0,T]\end{split}$
with $v_{k}:=v1_{u>k}$. Therefore, we have
$\begin{split}&\int_{\mathcal{O}}|(u(t,x)-k)^{+}|^{2}\,dx+E\bigg{[}\int_{t}^{T}\|v_{k}(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
=&-2E\biggl{[}\int_{t}^{T}\ll\partial_{x_{j}}(u(s)-k)^{+},\quad
a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}_{k}(s)\\\
&\quad\quad+(f^{k})^{j}(s,\cdot,(u(s)-k)^{+},\nabla u(s),v_{k}(s))\gg
ds\big{|}\mathscr{F}_{t}\bigg{]}\\\ &+2E\bigg{[}\int_{t}^{T}\ll(u(s)-k)^{+},\
b^{i}\partial_{x_{i}}u(s)+c\,(u(s)-k)^{+}+\varsigma^{r}v_{k}^{r}(s)\gg
ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+2E\bigg{[}\int_{t}^{T}\ll(u(s)-k)^{+},\,g^{k}(s,\cdot,(u(s)-k)^{+},\nabla
u(s),v_{k}(s))\gg ds\big{|}\mathscr{F}_{t}\biggr{]},\,a.s..\end{split}$ (5.4)
Note that
$\begin{split}&\operatorname*{ess\,sup}_{\Omega}\\!\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll(u(s)-k)^{+},\,g_{0}^{k}(s)\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\ \leq&\
2\|(g_{0}^{k})^{+}\|_{\frac{p(n+2)}{n+2+p};\mathcal{O}_{t}}\|(u-k)^{+}\|_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{\frac{1}{2}-\frac{1}{p}}~{}\textrm{
(H$\ddot{\textrm{o}}$lder inequality)}\\\ \leq&\
\delta\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}+C(\delta)\|(g_{0}^{k})^{+}\|^{2}_{\frac{p(n+2)}{n+2+p};\mathcal{O}_{t}}\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\\\
\leq&\
\delta\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}+C(\delta,p,L)\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}},\end{split}$
(5.5)
$\begin{split}&\operatorname*{ess\,sup}_{\omega\in\Omega}\\!\\!\sup_{\tau\in[t,T]}\\!\\!2E\left[\int_{\tau}^{T}\\!\\!\\!\\!\\!\ll(u(s)-k)^{+},\
b^{i}\partial_{x_{i}}u(s)+c\,(u(s)-k)^{+}+\varsigma^{r}v^{r}_{k}(s)\gg\\!ds\big{|}\mathscr{F}_{\tau}\right]\\\
\leq&\
C(\varepsilon)\operatorname*{ess\,sup}_{\omega\in\Omega}\sup_{\tau\in[t,T]}\\!E\left[\int_{\tau}^{T}\\!\\!\\!\\!\ll|b(s)|^{2}+|c(s)|+|\varsigma(s)|^{2},\
\left|(u(s)-k)^{+}\right|^{2}\gg\\!ds\big{|}\mathscr{F}_{\tau}\right]\\\
&+\varepsilon\left(\|\nabla(u-k)^{+}\|_{2;\mathcal{O}_{t}}^{2}+\|v_{k}\|_{2;\mathcal{O}_{t}}^{2}\right)\\\
\leq&\
\varepsilon\left(\|\nabla(u-k)^{+}\|_{2;\mathcal{O}_{t}}^{2}+\|v_{k}\|_{2;\mathcal{O}_{t}}^{2}\right)+C(\varepsilon)\Lambda_{0}\|(u-k)^{+}\|^{2}_{\frac{2q}{q-1};\mathcal{O}_{t}}\\\
\leq&\
\varepsilon\left(\|\nabla(u-k)^{+}\|_{2;\mathcal{O}_{t}}^{2}+\|v_{k}\|_{2;\mathcal{O}_{t}}^{2}\right)+\delta\|(u-k)^{+}\|_{\frac{2(n+2)}{n};\mathcal{O}_{t}}^{2}\\\
&+C(\delta,n,q,\varepsilon,\Lambda_{0})\|(u-k)^{+}\|_{2;\mathcal{O}_{t}}^{2},\end{split}$
(5.6)
and
$\begin{split}&2E\left[\int_{t}^{T}\ll|\nabla(u(s)-k)^{+}|,\,|f_{0}^{k}(s)|\gg
ds\Big{|}\mathscr{F}_{t}\right]\\\ \leq&\ \varepsilon
E\left[\int_{t}^{T}\|\nabla(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\Big{|}\mathscr{F}_{t}\right]+C(\varepsilon)E\left[\int_{t}^{T}\|f_{0}^{k}1_{u>k}(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds\Big{|}\mathscr{F}_{t}\right]\\\
\leq&\ \varepsilon
E\left[\int_{t}^{T}\|\nabla(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\Big{|}\mathscr{F}_{t}\right]+C(\varepsilon)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\|f_{0}^{k}\|_{p;\mathcal{O}_{t}}^{2}\\\
\leq&\ \varepsilon
E\left[\int_{t}^{T}\|\nabla(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\Big{|}\mathscr{F}_{t}\right]\\\
&+C(\varepsilon,p,L)\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}},a.s.\end{split}$
(5.7)
where $\varepsilon$ and $\delta$ are two positive parameters waiting to be
determined later and
$\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}:=\operatorname*{ess\,sup}_{\Omega}\sup_{\tau\in[t,T]}E\left[|Q_{\tau}\cap\\{u>k\\}|\big{|}{\mathscr{F}_{\tau}}\right].$
In a similar way to (4.5) and (4.7) in the proof of Theorem 4.2, we obtain
from (5.5) and (5.7) that with probability 1, for all $t\in[0,T]$
$\begin{split}&-2E\bigg{[}\int_{t}^{T}\ll\partial_{x_{j}}(u(s)-k)^{+},\quad
a^{ij}\partial_{x_{i}}u(s)+\sigma^{jr}v^{r}_{k}(s)\\\
&\quad\quad+(f^{k})^{j}(s,\cdot,(u(s)-k)^{+},\,\nabla u(s),v_{k}(s))\gg
ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
\leq&-(\lambda-\kappa-\beta\theta-\varepsilon)E\bigg{[}\int_{t}^{T}\|\nabla(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+\left(\frac{1}{\varrho}+\frac{1}{\theta}\right)E\bigg{[}\int_{t}^{T}\|v_{k}(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+C(\varepsilon,L)E\bigg{[}\int_{t}^{T}\|(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+2E\bigg{[}\int_{t}^{T}(|\nabla(u(s)-k)^{+}|,|f_{0}^{k}(s)|)\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
\leq&-(\lambda-\kappa-\beta\theta-2\varepsilon)E\bigg{[}\int_{t}^{T}\|\nabla(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+\left(\frac{1}{\varrho}+\frac{1}{\theta}\right)E\bigg{[}\int_{t}^{T}\|v_{k}(s)\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+C(\varepsilon,L)E\bigg{[}\int_{t}^{T}\|(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+C(\varepsilon,p,L)\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\end{split}$
(5.8)
and
$\begin{split}&\operatorname*{ess\,sup}_{\Omega}\sup_{\tau\in[t,T]}2E\bigg{[}\int_{\tau}^{T}\ll(u(s)-k)^{+},\,g^{k}(s,\cdot,(u(s)-k)^{+},\nabla
u(s),v_{k}(s))\gg ds\big{|}\mathscr{F}_{\tau}\bigg{]}\\\
&\leq\varepsilon\|\nabla(u-k)^{+}\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v_{k}\|_{2;\mathcal{O}_{t}}^{2}+C(\varepsilon,\varepsilon_{1},L)\|(u-k)^{+}\|^{2}_{2;\mathcal{O}_{t}}\\\
&~{}~{}+\operatorname*{ess\,sup}_{\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll(u(s)-k)^{+},\,g_{0}^{k}(s)\gg
ds\big{|}\mathscr{F}_{\tau}\right]\\\
&\leq\varepsilon\|\nabla(u-k)^{+}\|_{2;\mathcal{O}_{t}}^{2}+\varepsilon_{1}\|v_{k}\|_{2;\mathcal{O}_{t}}^{2}+C(\varepsilon,\varepsilon_{1},L)\|(u-k)^{+}\|^{2}_{2;\mathcal{O}_{t}}\\\
&~{}~{}+\delta\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}+C(L,\delta,p)\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\end{split}$
(5.9)
where $\theta$, $\varepsilon$, $\varepsilon_{1}$ and $\delta$ are four
positive parameters such that
$\theta>\frac{\varrho}{\varrho-1}>1,\frac{1}{\varrho}+\frac{1}{\theta}+\varepsilon+\varepsilon_{1}<1\textrm{
and }\lambda-\kappa-\beta\theta-4\varepsilon>0.$
Combining (5.4), (5.6), (5.8) and (5.9), we have
$\begin{split}&\|(u-k)^{+}\|^{2}_{\mathcal{V}_{2}(\mathcal{O}_{t})}+\|v_{k}\|^{2}_{2;\mathcal{O}_{t}}\\\
\leq&\
C\bigg{\\{}C(\delta)\|(u-k)^{+}\|^{2}_{2;\mathcal{O}_{t}}+\delta\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\\\
&\ \ \ \
+C(\delta)\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\bigg{\\}},\end{split}$
(5.10)
where $C$ is a constant independent of $t$ and $\delta$.
By Lemma 3.1, $\mathcal{V}_{2,0}(\mathcal{O}_{t})$ is continuously embedded
into $\mathcal{M}^{\frac{2(n+2)}{n}}(\mathcal{O}_{t})$. That is
$\|(u-k)^{+}\|_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\leq
C\|(u-k)^{+}\|_{\mathcal{V}_{2}(\mathcal{O}_{t})}.$
Therefore, choosing $\delta$ to be small enough, we obtain
$\begin{split}\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\leq&C\|(u-k)^{+}\|^{2}_{2;\mathcal{O}_{t}}+C\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\\\
\leq&C(|T-t||\mathcal{O}|)^{\frac{2}{n+2}}\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}\\\
&+C\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}.\\\
\end{split}$
Choosing $t_{1}\in[0,T)$ such that
$C(|T-t_{1}||\mathcal{O}|)^{\frac{2}{n+2}}\leq\frac{1}{2}$, we get
$\begin{split}\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t_{1}}}\leq\,C\left(\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}+k^{2}\right)\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t_{1}}}^{1-\frac{2}{p}}\end{split}$
where the constant $C$ does not depend on $t_{1}$.
Define
$\psi:\mathbb{R}\longrightarrow\mathbb{R},~{}~{}~{}\psi(h)=\left|\\{u>h\\}\right|_{\infty;\mathcal{O}_{t_{1}}}.$
Since for any $h>k$,
$\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t_{1}}}\geq(h-k)^{2}\left|\\{u>h\\}\right|_{\infty;\mathcal{O}_{t_{1}}}^{\frac{n}{n+2}},$
taking $k\geq A_{p}(f_{0},g_{0}^{+})$ we have
$\begin{split}\psi(h)^{\frac{n}{n+2}}\leq\frac{Ck^{2}}{(h-k)^{2}}\psi(k)^{1-\frac{2}{p}}\end{split}$
which implies
$\begin{split}\psi(h)\leq\frac{Ck^{\alpha}}{(h-k)^{\alpha}}\psi(k)^{1+\bar{\varepsilon}}\end{split}$
(5.11)
where $\alpha=\frac{2(n+2)}{n}$ and $\bar{\varepsilon}=\frac{2(p-n-2)}{pn}>0$.
Take $k_{l}=k(2-2^{-l})$, $l=0,1,2,\cdots.$ Then from
$\psi(k_{l+1})\leq\frac{Ck_{l}^{\alpha}}{(k_{l+1}-k_{l})^{\alpha}}\psi(k_{l})^{1+\bar{\varepsilon}},$
it follows that
$\begin{split}\psi(k_{l+1})\leq\hat{C}2^{\alpha(l+1)}\psi(k_{l})^{1+\bar{\varepsilon}}.\end{split}$
By Lemma 5.2, there exists a constant
$\theta_{0}=\theta_{0}(\hat{C},\bar{\varepsilon})>0$, such that if
$\psi(k_{0})\leq\theta_{0}$, $\lim_{l\rightarrow\infty}\psi(k_{l})=0$. Note
that $k_{0}=k$ and
$\psi(k_{0})=\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t_{1}}}$.
Taking
$k=\operatorname*{ess\,sup}_{(\omega,s,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,s,x)+A_{p}(f_{0},g_{0}^{+})+{\theta_{0}^{-\frac{1}{2}}}\|u^{+}\|_{2;\mathcal{O}_{t_{1}}},$
we have
$\begin{split}k^{2}\geq\frac{1}{\theta_{0}}\|u^{+}\|^{2}_{2;\mathcal{O}_{t_{1}}}\geq\frac{1}{\theta_{0}}k^{2}\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t_{1}}}\end{split}$
which implies
$\begin{split}\psi(k_{0})=\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t_{1}}}\leq\theta_{0}.\end{split}$
Hence, $\psi(k_{\infty})=0$. Since $k_{\infty}=2k$, we obtain
$\operatorname*{ess\,sup}_{(\omega,s,x)\in\Omega\times\mathcal{O}_{t_{1}}}\\!\\!\\!u(\omega,s,x)\leq
2k=2\left\\{\\!\operatorname*{ess\,sup}_{(\omega,s,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,s,x)+A_{p}(f_{0},g_{0}^{+})+{\theta_{0}^{-\frac{1}{2}}}\|u^{+}\|_{2;Q}\right\\}.$
As $T-t_{1}$ only depends on the structure terms like
$n,\lambda,\kappa,\beta,\varrho,p,q,L,\Lambda_{0},|\mathcal{O}|$ and $T$, by
induction, we get estimate (5.1). ∎
###### Theorem 5.4.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}4)$ be satisfied and
$(u,v)\in\mathcal{V}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$ be a weak solution of
(1.1). If $L_{0}=0$ and with probability 1
$\begin{split}f(t,x,R,0,0)\equiv f(t,x,0,0)\textrm{ and }g(t,x,R,0,0)\textrm{
are decreasing in }R\in\mathbb{R}\end{split}$ (5.12)
for all $(t,x)\in[0,T]\times\mathbb{R}^{n}$, then we assert
$\begin{split}\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times
Q}u(\omega,t,x)\leq\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,t,x)+CA_{p}(f_{0},g_{0}^{+})|\mathcal{O}|^{\frac{1}{n+2}-\frac{1}{p}}\end{split}$
(5.13)
with the constant $C$ only depending on
$n,p,q,\kappa,\lambda,\beta,\varrho,T,\Lambda_{0}$ and $L$.
###### Proof.
We use De Giorgi iteration and the same notations in the proof of Theorem 5.1.
Similar to the proof of (5.5) and (5.7), under condition (5.12), we have for
each $t\in[0,T]$,
$\begin{split}&\operatorname*{ess\,sup}_{\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll(u(s)-k)^{+},\,\
g_{0}^{k}(s)\gg ds\big{|}\mathscr{F}_{\tau}\right]\\\
\leq&\operatorname*{ess\,sup}_{\Omega}\sup_{\tau\in[t,T]}2E\left[\int_{\tau}^{T}\ll(u(s)-k)^{+},\,\
g_{0}(s)\gg ds\big{|}\mathscr{F}_{\tau}\right]\\\
\leq&\delta\|(u-k)^{+}\|^{2}_{\frac{2(n+2)}{n};\mathcal{O}_{t}}+C(\delta)\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}\end{split}$
(5.14)
and almost surely
$\begin{split}&2E\left[\int_{t}^{T}\ll|\nabla(u(s)-k)^{+}|,\,|f_{0}^{k}(s)|\gg
ds\Big{|}\mathscr{F}_{t}\right]\\\
=&2E\left[\int_{t}^{T}\ll|\nabla(u(s)-k)^{+}|,\,|f_{0}(s)|\gg
ds\Big{|}\mathscr{F}_{t}\right]\\\ \leq&\varepsilon
E\left[\int_{t}^{T}\|\nabla(u(s)-k)^{+}\|^{2}_{L^{2}(\mathcal{O})}\,ds\Big{|}\mathscr{F}_{t}\right]+C(\varepsilon)\left|A_{p}(f_{0},g_{0}^{+})\right|^{2}\left|\\{u>k\\}\right|_{\infty;\mathcal{O}_{t}}^{1-\frac{2}{p}}.\end{split}$
(5.15)
Hence instead of (5.11), we obtain
$\psi(h)\leq\frac{C\left|A_{p}(f_{0},g_{0}^{+})\right|^{\alpha}}{(h-k)^{\alpha}}\psi(k)^{1+\bar{\varepsilon}}.$
By Corollary 5.3, for any $\bar{\theta}_{0}\geq
CA_{p}(f_{0},g_{0}^{+})|\mathcal{O}_{t_{1}}|^{\frac{1}{n+2}-\frac{1}{p}}$, we
have
$\begin{split}\left|\left\\{u>\operatorname*{ess\,sup}_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}(\omega,t,x)+\bar{\theta}_{0}\right\\}\right|_{\infty;\mathcal{O}_{t_{1}}}=0,\end{split}$
(5.16)
which implies
$\begin{split}\sup_{(\omega,t,x)\in\Omega\times\mathcal{O}_{t_{1}}}u\,\leq\sup_{(\omega,t,x)\in\Omega\times\partial_{\rm
p}Q}u^{+}+CA_{p}(f_{0},g_{0}^{+})|\mathcal{O}_{t_{1}}|^{\frac{1}{n+2}-\frac{1}{p}}\end{split}$
(5.17)
where the constant $C$ depends only on
$n,\lambda,p,q,\beta,\kappa,\varrho,\Lambda_{0}$ and $L$. As $T-t_{1}$ only
depends on the structure terms, by induction, we get estimate (5.13) where the
constant $C$ also depends on $T$. We complete the proof. ∎
###### Remark 5.3.
In Theorem 5.4, we can dispense with the assumptions that $L_{0}=0$ and the
function $r\mapsto g(t,x,r,0,0)$ decreases in $r$, by considering the function
$\tilde{u}:=e^{2(L+L_{0})t}u(t,x)$ instead of $u$.
###### Corollary 5.5.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}4)$ be satisfied with
$L_{0}=0$. Let the two pair $(f,g^{1})$ and $(f,g^{2})$ satisfy condition
(5.12) in Theorem 5.4. Assume that $G^{1}$ and $G^{2}$ are two random
variables in $L^{\infty}(\Omega,\mathscr{F}_{T},L^{2}(\mathcal{O}))$. Let
$(u_{i},v_{i})\in\mathscr{U}\times\mathscr{V}(G^{i},f,g^{i})$, $i=1,2$ and
$(u_{1}-u_{2})^{+}\in\dot{\mathcal{V}}_{2,0}(Q)$. Then if $G^{1}\leq G^{2}$
$dP\otimes dx$-a.e. and $g^{1}(\omega,t,x,u_{2},\nabla u_{2},v_{2})\leq
g^{2}(\omega,t,x,u_{2},\nabla u_{2},v_{2}),dP\otimes dt\otimes dx$-a.e., we
have $u_{1}(\omega,t,x)\leq u_{2}(\omega,t,x)$, $dP\otimes dt\otimes dx$-a.e..
###### Proof.
$(u_{1}-u_{2},v_{1}-v_{2})$ belongs to
$\mathscr{U}\times\mathscr{V}(\tilde{G},\tilde{f},\tilde{g})$ with
$\begin{split}\tilde{f}(s,x,R,Y,Z):=\,&f(s,x,R+u_{2}(s,x),Y+\nabla
u_{2}(s,x),Z+v_{2}(s,x))\\\ &-f(s,x,u_{2}(s,x),\nabla
u_{2}(s,x),v_{2}(s,x)),\\\
\tilde{g}(s,x,R,Y,Z):=\,&g^{1}(s,x,R+u_{2}(s,x),Y+\nabla
u_{2}(s,x),Z+v_{2}(s,x))\\\ &-g^{2}(s,x,u_{2}(s,x),\nabla
u_{2}(s,x),v_{2}(s,x))\end{split}$
and $\tilde{G}:=G^{1}-G^{2}$. Since $\tilde{G}\leq 0$, $\tilde{g}_{0}\leq 0$
and $f_{0}=0$, the assertion follows from Theorem 5.4. ∎
### 5.2 The local case
This subsection is devoted to the local regularity of weak solutions.
###### Definition 5.1.
For domain $Q^{\prime}\subset Q$, a function $\zeta(\cdot,\cdot)$ is called a
cut-off function on $Q^{\prime}$ if
(i) $\zeta\in\dot{W}_{1}^{2,2}(Q^{\prime})$, i.e. there exists a sequence
$\\{\zeta^{l},l\in\mathbb{N}\\}\subset C_{c}^{\infty}(Q^{\prime})$ such that
$\begin{split}\|\zeta^{l}-\zeta\|_{W_{1}^{2,2}}:=\bigg{\\{}\int_{Q^{\prime}}\Big{(}&|(\zeta^{l}-\zeta)(t,x)|^{2}+|\partial_{t}(\zeta^{l}-\zeta)(t,x)|^{2}\\\
&+|\nabla(\zeta^{l}-\zeta)(t,x)|^{2}+|\nabla^{2}(\zeta^{l}-\zeta)(t,x)|^{2}\Big{)}\,dxdt\bigg{\\}}^{\frac{1}{2}}\end{split}$
(5.18)
converges to zero as $l$ tends to infinity with
$\nabla^{2}(\zeta^{l}-\zeta)(t,x)$ being the Hessian matrix of the function
$(\zeta^{l}-\zeta)(t,\cdot)$ at $x$;
(ii) $0\leq\zeta\leq 1$;
(iii) there exists a domain $Q^{\prime\prime}\Subset Q^{\prime}$ and a
nonempty domain $Q^{\prime\prime\prime}\Subset Q^{\prime\prime}$ such that
$\zeta(t,x)=\left\\{\begin{array}[]{l}\begin{split}&1,\quad(t,x)\in
Q^{\prime\prime\prime},\\\ &0,\quad(t,x)\in Q^{\prime}\setminus
Q^{\prime\prime};\end{split}\end{array}\right.$
(iv) $|\nabla\zeta|,\partial_{t}\zeta\in L^{\infty}(Q^{\prime})$.
For simplicity, we denote
$\|\nabla\zeta\|_{L^{\infty}(Q^{\prime})}:=\||\nabla\zeta|\|_{L^{\infty}(Q^{\prime})}.$
First, to study the local behavior of our weak solutions, we shall generalize
the deterministic parabolic De Giorgi class (c.f. [4, 16, 17, 25]) to our
stochastic version and introduce the definition of De Giorgi class in the
backward stochastic parabolic case.
###### Definition 5.2.
We say that a function $u\in\mathcal{V}_{2,0}(Q)$ belongs to the backward
stochastic parabolic De Giorgi class (BSPDG, for short) if for any
$k\in\mathbb{R}$, $Q_{\rho,\tau}:=[t_{0}-\tau,t_{0})\times
B_{\rho}(x_{0})\subset Q$ (with $\rho,\tau\in(0,1)$) and any cut-off function
$\zeta$ on $Q_{\rho,\tau}$, we have
$\begin{array}[]{l}\begin{split}&\|\zeta(u-k)^{\pm}\|^{2}_{\mathcal{V}_{2}(Q_{\rho,\tau})}\\\
\leq&\gamma\Big{\\{}\|(u-k)^{\pm}\|^{2}_{2;Q_{\rho,\tau}}(1+\|\nabla\zeta\|_{L^{\infty}(Q_{\rho,\tau})}^{2}+\|\partial_{t}\zeta\|_{L^{\infty}(Q_{\rho,\tau})})\\\
&\quad+(k^{2}+a_{0}^{2})|\\{(u-k)^{\pm}>0\\}|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{\mu}}\Big{\\}}\end{split}\end{array}~{}~{}~{}~{}~{}~{}~{}(\mathfrak{D}^{\pm})$
for some triplet
$(a_{0},\mu,\gamma)\in[0,\infty)\times(n+2,\infty)\times[0,\infty)$. We call
$a_{0},\mu,$ and $\gamma$ the structural parameters of $BSPDG^{\pm}$. We mean
that $u\in\mathcal{V}_{2,0}(Q)$ satisfies $(\mathfrak{D}^{+})$
($(\mathfrak{D}^{-})$, respectively) by the inclusion $u\in
BSPDG^{+}(a_{0},\mu,\gamma;Q)$ ($u\in BSPDG^{-}(a_{0},\mu,\gamma;Q)$,
respectively). We say $u\in BSPDG(a_{0},\mu,\gamma;Q)$ if both inclusions
$u\in BSPDG^{+}(a_{0},\mu,\gamma;Q)$ and $u\in BSPDG^{-}(a_{0},\mu,\gamma;Q)$
hold.
###### Proposition 5.6.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}3)$ hold. Assume that
$(u,v)\in\mathcal{V}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$ is a weak solution of
(1.1). Then we assert that $u\in BSPDG(a_{0},\mu,\gamma;Q)$, with
$a_{0}:=A_{p}(f_{0},g_{0}),\mu:=\min\\{p,2q\\}$, and some parameter $\gamma$
depending on $n,p,q,\kappa,\lambda,\beta,$ $\varrho,\Lambda,\Lambda_{0}$ and
$L$.
###### Remark 5.4.
It is worth noting that in this proposition, assumption $({\mathcal{A}}4)$ is
not made.
The proof requires the following lemma.
###### Lemma 5.7.
Let assumptions $({\mathcal{A}}1)$–$({\mathcal{A}}3)$ hold, $\zeta$ be a cut-
off function on $Q_{\rho,\tau}:=[t_{0}-\tau,t_{0})\times
B_{\rho}(x_{0})\subset Q$, and
$(u,v)\in\mathcal{V}_{2,0}(Q)\times\mathcal{M}^{2}(Q)$ be a weak solution of
(1.1). Then, we have almost surely
$\begin{split}&\ll\zeta^{2}(t),\,|u^{+}(t)|^{2}\gg_{B_{\rho}(x_{0})}+\int_{t}^{t_{0}}\ll\zeta^{2}(s),\,|v^{u}(s)|^{2}\gg_{B_{\rho}(x_{0})}ds\\\
=&-\int_{t}^{t_{0}}2\ll\zeta\partial_{s}\zeta(s),\
\,|u^{+}(s)|^{2}\gg_{B_{\rho}(x_{0})}ds\\\
&+\int_{t}^{t_{0}}2\ll\zeta^{2}(s)u^{+}(s),\
\,g^{u}(s)\gg_{B_{\rho}(x_{0})}ds\\\
&+\int_{t}^{t_{0}}2\ll\zeta^{2}(s)u^{+}(s),\
\,b^{i}(s)\partial_{x_{i}}u(s)+c(s)\,u^{+}(s)+\varsigma^{r}(s)v^{r,u}(s)\gg_{B_{\rho}(x_{0})}ds\\\
&-\int_{t}^{t_{0}}\ll 2\partial_{x_{i}}(\zeta^{2}(s)u^{+}(s)),\
\,a^{ji}(s)\partial_{x_{j}}u^{+}(s)+\sigma^{ir}(s)v^{r,u}(s)+f^{i,u}(s)\gg_{B_{\rho}(x_{0})}ds\\\
&-\int_{t}^{t_{0}}2\ll\zeta^{2}(s)u^{+}(s),\
\,v^{r,u}(s)\gg_{B_{\rho}(x_{0})}dW_{s}^{r},\quad\forall\,t\in[t_{0}-\tau,t_{0}]\end{split}$
(5.19)
where
$\begin{split}&g^{u}(s,x):=1_{\\{(s,x):u(s,x)>0\\}}(s,x)g(s,x,u(s,x),\nabla
u(s,x),v(s,x));\\\
&f^{i,u}(s,x):=1_{\\{(s,x):u(s,x)>0\\}}(s,x)f^{i}(s,x,u(s,x),\nabla
u(s,x),v(s,x)),\ i=0,1,\cdots,n;\\\ \end{split}$
and
$v^{u}:=(v^{1,u},\cdots,v^{m,u}),\quad
v^{r,u}(s,x):=1_{\\{(s,x):u(s,x)>0\\}}(s,x)v^{r}(s,x),\quad r=1,\cdots,m.$
The proof of this lemma is rather standard and is sketched below.
###### Sketch of the proof.
We use approximation. By the definition of a cut-off function, all terms of
(5.19) are well defined and there is a sequence
$\\{\zeta^{l},l\in\mathbb{N}\\}\subset C_{c}^{\infty}(Q_{\rho,\tau})$ such
that
$\lim_{l\rightarrow\infty}\|\zeta^{l}-\zeta\|_{W^{2,2}_{1}(Q_{\rho,\tau})}=0$.
In view of Definition 2.2 and Remark 2.1, we verify like in Step 1 of the
proof of Lemma 3.5 that for each $l$ there holds
$\begin{split}\zeta^{l}u(t,x)=&\int_{t}^{T}\Big{[}\partial_{x_{j}}\left(a^{ij}\partial_{x_{i}}(\zeta^{l}u)(s,x)+\sigma^{jr}\zeta^{l}v^{r}(s,x)+\tilde{f}_{l}^{j}(s,x)\right)+b^{i}\partial_{x_{j}}(\zeta^{l}u)(s,x)\\\
&+c\,\zeta^{l}u(s,x)+\varsigma^{r}\zeta^{l}v^{r}(s,x)+\tilde{g}_{l}(s,x)\Big{]}\,ds-\int_{t}^{T}\zeta^{l}v^{r}(s,x)\,dW_{s}^{r},\quad
t\in[0,T]\end{split}$
in the weak sense of Definition 2.2, where
$\begin{split}\tilde{g}_{l}(s,x):=&-\partial_{s}\zeta^{l}u(s,x)+\zeta^{l}(s,x)g(s,x,u(s,x),\nabla
u(s,x),v(s,x))\\\
&-b^{i}\partial_{x_{i}}\zeta^{l}u(s,x)-\partial_{x_{j}}\zeta^{l}\bar{f}_{l}^{j}(s,x),\\\
\bar{f}_{l}(s,x):=\,&a^{i\cdot}\partial_{x_{i}}u(s,x)+\sigma^{\cdot
r}v^{r}(s,x)+f(s,x,u(s,x),\nabla u(s,x),v(s,x)),\\\
\tilde{f}_{l}(s,x):=\,&-a^{i\cdot}\partial_{x_{i}}\zeta^{l}u(s,x)+\zeta^{l}(s,x)f(s,x,u(s,x),\nabla
u(s,x),v(s,x)).\end{split}$
Thus,
$(\zeta^{l}u,\zeta^{l}v)\in\dot{\mathscr{U}}\times\dot{\mathscr{V}}(0,\tilde{f}_{l},\tilde{g}_{l})$.
From Proposition 4.4 we conclude that (5.19) holds with $\zeta$ being replaced
by $\zeta^{l}$. Passing to the limit in $L^{1}(\Omega\times Q)$ and taking
into account the path-wise continuity of $u$, we prove our assertion. ∎
###### Proof of Proposition 5.6.
Consider the cylinder
$Q_{\rho,\tau}(X)=X+[-\tau,0)\times B_{\rho}(0)\subset Q\hbox{ \rm with
}X:=(t_{0},x_{0}).$
For simplicity, we denote $Q_{\rho,\tau}(X)$ and $B_{\rho}(x_{0})$ by
$Q_{\rho,\tau}$ and $B_{\rho}$ respectively. Let $\zeta$ be a cut-off function
on $Q_{\rho,\tau}$. Denote $\bar{u}:=(u-k)^{+}$. From Lemma 5.7, it follows
that
$\begin{split}&E\bigg{[}\|\zeta(t)\bar{u}(t)\|_{L^{2}(B_{\rho})}^{2}+\int_{t}^{t_{0}}\|\zeta(s)v_{k}(s)\|_{L^{2}(B_{\rho})}^{2}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
=&E\bigg{[}\int_{t}^{t_{0}}2\ll\zeta^{2}(s)\bar{u}(s),\,g^{k}(s,\cdot,\bar{u}(s),\nabla\bar{u}(s),v_{k}(s))\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+E\left[\int_{t}^{t_{0}}2\ll\zeta^{2}(s)\bar{u}(s),\,b^{i}(s)\partial_{x_{i}}u(s)+c(s)\,\bar{u}(s)+\varsigma^{r}(s)v^{r}_{k}(s)\gg_{B_{\rho}}\big{|}\mathscr{F}_{t}\right]\\\
&-E\bigg{[}\int_{t}^{t_{0}}2\ll\zeta(s)\partial_{s}\zeta(s),\,|\bar{u}(s)|^{2}\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&-E\bigg{[}\int_{t}^{t_{0}}\ll
2\partial_{x_{j}}(\zeta^{2}(s)\bar{u}(s)),\,a^{ij}(s)\partial_{x_{i}}\bar{u}(s)+\sigma^{jr}(s)v^{r}_{k}(s)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(f^{k})^{j}(s,\cdot,\bar{u}(s),\nabla\bar{u}(s),v_{k}(s))\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\end{split}$
(5.20)
holds almost surely for all $t\in[t_{0}-\tau,t_{0})$ where $v_{k}:=v1_{u>k}$
and for
$(\omega,t,x,R,Y,Z)\in\Omega\times[t_{0}-\tau,t_{0})\times\mathcal{O}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{m}$
$(f^{k},g^{k})(\omega,t,x,R,Y,Z):=(f,g)(\omega,t,x,R+k,Y,Z)+(0,c(\omega,t,x)k).$
In view of (4.5)-(4.7) and (5.5)-(5.9), we have almost surely for all
$t\in[t_{0}-\tau,t_{0})$
$\begin{split}&E\bigg{[}\int_{t}^{t_{0}}2\ll\zeta^{2}\bar{u}(s),\,g^{k}(s,\cdot,\bar{u}(s),\nabla\bar{u}(s),v_{k}(s))\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
\leq&\
E\bigg{[}\int_{t}^{t_{0}}2\ll\zeta^{2}\bar{u}(s),\,g_{0}^{k}(s)+L(|\bar{u}(s)|+|\nabla\bar{u}(s)|+|v_{k}(s)|)\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
\leq&\
\varepsilon_{1}\|\zeta\bar{u}\|^{2}_{\mathcal{V}_{2}(Q_{\rho,\tau})}+\varepsilon_{2}E\bigg{[}\int_{t}^{t_{0}}\|\zeta(s)v_{k}(s)\|^{2}_{L^{2}(B_{\rho})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+C(L)\|\nabla\zeta\|_{L^{\infty}(Q_{\rho,\tau})}^{2}\|\bar{u}\|^{2}_{2;Q_{\rho,\tau}}+C(\varepsilon_{1},L,n,p)(\left|A_{p}(f_{0},g_{0})\right|^{2}+k^{2})|\\{u>k\\}|^{1-\frac{2}{p}}_{\infty;Q_{\rho,\tau}}\\\
&+C(\varepsilon_{1},\varepsilon_{2},L)\|\zeta\bar{u}\|^{2}_{2;Q_{\rho,\tau}}+E\bigg{[}\int_{t}^{t_{0}}\\!2\\!\ll\zeta^{2}\bar{u}(s),\,|c(s)k|\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]},\end{split}$
$\begin{split}&E\bigg{[}\int_{t}^{t_{0}}2\ll\zeta^{2}\bar{u}(s),\,\,|c(s)k|\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
\leq&E\bigg{[}\int_{t}^{t_{0}}\ll\zeta^{2}\left|\bar{u}(s)\right|^{2},\
|c(s)|\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}+k^{2}E\bigg{[}\int_{t}^{t_{0}}|\\!\ll\zeta^{2}(s),\
|c(s)1_{u>k}|\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\ \leq&\
E\bigg{[}\int_{t}^{t_{0}}\\!\ll\zeta^{2}\left|\bar{u}(s)\right|^{2},\
|c(s)|\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}+k^{2}\Lambda_{0}|\\{u>k\\}|_{\infty;Q_{\rho,\tau}}^{1-\frac{1}{q}},\end{split}$
$\begin{split}&E\left[\int_{t}^{t_{0}}2\ll\zeta^{2}(s)\bar{u}(s),\,b^{i}\partial_{x_{i}}u(s)+c\,\bar{u}(s)+\varsigma^{r}v^{r}_{k}(s)\gg_{B_{\rho}}\big{|}\mathscr{F}_{t}\right]\\\
\leq&\,\varepsilon_{1}\|\zeta\bar{u}\|^{2}_{\mathcal{V}_{2}(Q_{\rho,\tau})}+\varepsilon_{2}\|\zeta
v_{k}\|^{2}_{2;Q_{\rho,\tau}}+\varepsilon_{1}\|\nabla\zeta\|_{L^{\infty}(Q_{\rho,\tau})}^{2}\|\bar{u}\|^{2}_{2;Q_{\rho,\tau}}\\\
&\
+C(\varepsilon_{1},\varepsilon_{2},n)E\bigg{[}\int_{t}^{t_{0}}\ll\zeta^{2}\bar{u}^{2}(s),\
|b(s)|^{2}+|c(s)|+|\varsigma(s)|^{2}\gg_{B_{\rho}}ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
\leq&\,\varepsilon_{1}\|\zeta\bar{u}\|^{2}_{\mathcal{V}_{2}(Q_{\rho,\tau})}+\varepsilon_{2}\|\zeta
v_{k}\|^{2}_{2;Q_{\rho,\tau}}+\varepsilon_{1}\|\nabla\zeta\|_{L^{\infty}(Q_{\rho,\tau})}^{2}\|\bar{u}\|^{2}_{2;Q_{\rho,\tau}}+C(\varepsilon_{1},\varepsilon_{2},n)\Lambda_{0}\|\zeta\bar{u}\|_{\frac{2q}{q-1};Q_{\rho,\tau}}^{2}\\\
\leq&\,2\varepsilon_{1}\|\zeta\bar{u}\|^{2}_{\mathcal{V}_{2}(Q_{\rho,\tau})}+\varepsilon_{2}\|\zeta
v_{k}\|^{2}_{2;Q_{\rho,\tau}}+\varepsilon_{1}\|\nabla\zeta\|_{L^{\infty}(Q_{\rho,\tau})}^{2}\|\bar{u}\|^{2}_{2;Q_{\rho,\tau}}\\\
&+C(\varepsilon_{1},\varepsilon_{2},n,q,\Lambda_{0})\|\zeta\bar{u}\|_{2;Q_{\rho,\tau}}^{2}\\\
\end{split}$
and
$\begin{split}&-2E\bigg{[}\int_{t}^{t_{0}}\\!\\!\\!\ll\partial_{x_{j}}(\zeta^{2}\bar{u}(s)),\,a^{ij}\partial_{x_{i}}\bar{u}(s)+\sigma^{jr}v^{r}_{k}(s)+(f^{k})^{j}(s,\bar{u}(s),\nabla\bar{u}(s),v_{k}(s))\gg_{B_{\rho}}\\!ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&=-2E\bigg{[}\int_{t}^{t_{0}}\\!\\!\\!\\!\ll\zeta^{2}\partial_{x_{j}}\bar{u}(s),\,a^{ij}\partial_{x_{i}}\bar{u}(s)+\sigma^{jr}v^{r}_{k}(s)+(f^{k})^{j}(s,\bar{u}(s),\nabla\bar{u}(s),v_{k}(s))\gg_{B_{\rho}}\\!ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&\
-4E\bigg{[}\int_{t}^{t_{0}}\\!\\!\\!\\!\ll\bar{u}\zeta\partial_{x_{j}}\zeta(s),\,a^{ij}\partial_{x_{i}}\bar{u}(s)+\sigma^{jr}v^{r}_{k}(s)+(f^{k})^{j}(s,\bar{u}(s),\nabla\bar{u}(s),v_{k}(s))\gg_{B_{\rho}}\\!ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&\leq-\lambda_{0}E\bigg{[}\int_{t}^{t_{0}}\|\zeta\nabla\bar{u}(s)\|^{2}_{L^{2}(B_{\rho})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}+\alpha_{0}E\bigg{[}\int_{t}^{t_{0}}\|\zeta
v_{k}(s)\|^{2}_{L^{2}(B_{\rho})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+C\left(\|\zeta\bar{u}\|^{2}_{2;Q_{\rho,\tau}}+\left(\left|A_{p}(f_{0},g_{0})\right|^{2}+k^{2}\right)|\\{u>k\\}|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{p}}\right)\\\
&+CE\left[\int_{t}^{t_{0}}\ll|\bar{u}\nabla\zeta(s)|,\,|f^{k}_{0}(s)|+|\bar{u}(s)|+|\zeta\nabla\bar{u}(s)|+|\zeta
v_{k}(s)|\gg ds\big{|}\mathscr{F}_{t}\right]\\\
&\leq-(\lambda_{0}-\varepsilon)E\bigg{[}\int_{t}^{t_{0}}\|\nabla(\zeta\bar{u}(s))\|^{2}_{L^{2}(B_{\rho})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}+(\alpha_{0}+\varepsilon)E\bigg{[}\int_{t}^{t_{0}}\|\zeta
v_{k}(s)\|^{2}_{L^{2}(B_{\rho})}\,ds\big{|}\mathscr{F}_{t}\bigg{]}\\\
&+C\left\\{\|\nabla\zeta\|^{2}_{L^{\infty}(Q_{\rho,\tau})}\|\bar{u}\|^{2}_{2;Q_{\rho,\tau}}+\|\bar{u}\zeta\|^{2}_{Q_{\rho,\tau}}+\left(\left|A_{p}(f_{0},g_{0})\right|^{2}+k^{2}\right)|\\{u>k\\}|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{p}}\right\\}\end{split}$
with $C:=C(\varepsilon,p,\lambda,\beta,\varrho,\kappa,\Lambda,L)$, where
$\alpha_{0}\in(0,1)$ and $\lambda_{0}$ are two positive constants depending
only on structure terms such as $\kappa,p,\lambda,\varrho,\beta,\Lambda$ and
$L$, and the three parameters $\varepsilon,\varepsilon_{1},\varepsilon_{2}$
are waiting to be determined later. On the other hand, it is obvious that
almost surely
$-E\left[\int_{t}^{t_{0}}2\ll\zeta\partial_{s}\zeta(s),|\bar{u}(s)|^{2}\gg_{B_{\rho}}ds\Big{|}\mathscr{F}_{t}\right]\leq
2\|\partial_{s}\zeta\|_{L^{\infty}(Q_{\rho,\tau})}\|\bar{u}\|^{2}_{2;Q_{\rho,\tau}},\
\,\forall t\in[t_{0}-\tau,t_{0}).$
Therefore, combining the above estimates and (5.20) and choosing the
parameters $\varepsilon,\varepsilon_{1}$ and $\varepsilon_{2}$ to be small
enough, we obtain
$\begin{array}[]{l}\begin{split}&\|\zeta(u-k)^{+}\|^{2}_{\infty,2;Q_{\rho,\tau}}+\|\nabla(\zeta(u-k)^{+})\|^{2}_{2;Q_{\rho,\tau}}\\\
\leq&\gamma\Big{\\{}(1+\|\nabla\zeta\|_{L^{\infty}(Q_{\rho,\tau})}^{2}+\|\partial_{t}\zeta\|_{L^{\infty}(Q_{\rho,\tau})})\|(u-k)^{+}\|^{2}_{2;Q_{\rho,\tau}}\\\
&\quad+\left(k^{2}+\left|A_{p}(f_{0},g_{0})\right|^{2}\right)|\\{(u-k)^{+}>0\\}|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{p\wedge(2q)}}\Big{\\}}\end{split}\end{array}$
where $\gamma$ is a positive constant depending on the structure terms such as
$n,p,q,\kappa,\lambda,\varrho,\beta,L,\Lambda$ and $\Lambda_{0}$. Hence $u\in
BSPDG^{+}(a_{0},\mu,\gamma;Q)$.
In a similar way, we show $u\in BSPDG^{-}(a_{0},\mu,\gamma;Q)$. The proof is
complete. ∎
###### Theorem 5.8.
If $u\in BSPDG^{\pm}(a_{0},\mu,\gamma;Q)$, we assert that for any
$Q_{\rho}=[t_{0},t_{0}+\rho^{2})\times B_{\rho}(x_{0})\subset
Q,\,\rho\in(0,1),$
there holds
$\operatorname*{ess\,sup}_{\Omega\times Q_{\frac{\rho}{2}}}u^{\pm}\,\leq
C\left\\{{\rho^{-\frac{n+2}{2}}}\|u^{\pm}\|_{2;Q_{\rho}}+a_{0}\rho^{1-\frac{n+2}{\mu}}\right\\},$
(5.21)
where $C$ is a constant depending only on $a_{0},\mu,\gamma$ and $n$.
###### Proof.
Consider $u\in BSPDG^{+}(a_{0},\mu,\gamma;Q)$. Take
$R_{l}=\frac{\rho}{2}+\frac{\rho}{2^{l+1}},\,k_{l}=k(2-\frac{1}{2^{l}}),\,l=0,1,2,\cdots$
where $k$ is a parameter waiting to be determined later. Denote
$Q^{l}:=Q_{R_{l}}=[t_{0},t_{0}+R_{l}^{2})\times B_{R_{l}}(x_{0})$. Choose
$\zeta_{l}$ to be a cut-off function on $Q^{l}$ such that
$\zeta_{l}(t,x)=\left\\{\begin{array}[]{l}\begin{split}&1,\quad(t,x)\in
Q^{l+1};\\\ &0,\quad(t,x)\in Q^{l}\setminus
Q_{\frac{R_{l}+R_{l+1}}{2}}\end{split}\end{array}\right.$
and
$\left\|\nabla\zeta_{l}\right\|^{2}_{L^{\infty}(Q_{\rho})}+\left\|\partial_{t}\zeta_{l}\right\|_{L^{\infty}(Q_{\rho})}\leq\frac{C(n)}{(R_{l}-R_{l+1})^{2}}.$
From $(\mathfrak{D}^{+})$, it follows that
$\begin{split}&\|\zeta_{l}(u-k_{l+1})^{+}\|^{2}_{\mathcal{V}_{2}(Q^{l})}\\\
\leq\,\,&C2^{2l}{\rho^{-2}}\|(u-k_{l+1})^{+}\|^{2}_{2;Q^{l}}+C(k^{2}+a_{0}^{2})|\\{u>k_{l+1}\\}|^{1-\frac{2}{\mu}}_{\infty;Q^{l}}.\end{split}$
For $k\geq a_{0}\rho^{1-\frac{n+2}{\mu}}$, we obtain from Lemma 3.1 that
$\begin{split}&\|\zeta_{l}(u-k_{l+1})^{+}\|^{2}_{\frac{2(n+2)}{n};Q^{l}}\\\
\leq\ &C\|\zeta_{l}(u-k_{l+1})^{+}\|^{2}_{\mathcal{V}_{2}(Q^{l})}\\\ \leq\
&C2^{2l}{\rho^{-2}}\|(u-k_{l+1})^{+}\|^{2}_{2;Q^{l}}+Ck^{2}{\rho^{-2(1-\frac{n+2}{\mu})}}|\\{u>k_{l+1}\\}|^{1-\frac{2}{\mu}}_{\infty;Q^{l}}.\end{split}$
Setting
$\phi_{l}:=\|(u-k_{l})^{+}\|^{2}_{2;Q^{l}},$
we have
$\begin{split}\phi_{l+1}\leq\ &\|\zeta_{l}(u-k_{l+1})^{+}\|_{2;Q^{l}}^{2}\\\
\leq\
&|\\{u>k_{l+1}\\}|^{\frac{2}{n+2}}_{\infty;Q^{l}}\|\zeta_{l}(u-k_{l+1})^{+}\|^{2}_{\frac{2(n+2)}{n};Q^{l}}~{}\textrm{
(H$\ddot{\textrm{o}}$lder inequality)}\\\ \leq\
&C2^{2l}{\rho^{-2}}\phi_{l}|\\{u>k_{l+1}\\}|^{\frac{2}{n+2}}_{\infty;Q^{l}}+Ck^{2}{\rho^{-2(1-\frac{n+2}{\mu})}}|\\{u>k_{l+1}\\}|^{1-\frac{2}{\mu}+\frac{2}{n+2}}_{\infty;Q^{l}}.\end{split}$
Note that
$\begin{split}\phi_{l}=\|(u-k_{l})^{+}\|^{2}_{2;Q^{l}}\geq\
&(k_{l+1}-k_{l})^{2}|\\{u>k_{l+1}\\}|_{\infty;Q^{l}}=k^{2}{2^{-(2l+2)}}|\\{u>k_{l+1}\\}|_{\infty;Q^{l}}.\end{split}$
Hence,
$\begin{split}\phi_{l+1}\leq\
&C2^{2l(1+\frac{2}{n+2})}\left[{\rho^{-2}k^{-\frac{4}{n+2}}}{\phi_{l}^{1+\frac{2}{n+2}}}+{\rho^{-2(1-\frac{n+2}{\mu})}k^{\frac{4}{\mu}-\frac{4}{n+2}}}{\phi_{l}^{1-\frac{2}{\mu}+\frac{2}{n+2}}}\right]\\\
=\
&C2^{2l(1+\frac{2}{n+2})}{\rho^{-2(1-\frac{n+2}{\mu})}k^{\frac{4}{\mu}-\frac{4}{n+2}}}{\phi_{l}^{1-\frac{2}{\mu}+\frac{2}{n+2}}}\left[\left({k^{-2}\rho^{-n-2}}{\phi_{l}}\right)^{\frac{2}{\mu}}+1\right].\end{split}$
For $k\geq
a_{0}\rho^{1-\frac{n+2}{\mu}}+{\rho^{-\frac{n+2}{2}}}\|u^{+}\|_{2;Q_{\rho}}$,
we have ${k^{-2}\rho^{-n-2}}{\phi_{l}}\leq 1$ and therefore
$\begin{split}\phi_{l+1}\leq
C2^{2l(1+\frac{2}{n+2})}{\rho^{-2(1-\frac{n+2}{\mu})}k^{\frac{4}{\mu}-\frac{4}{n+2}}}{\phi_{l}^{1-\frac{2}{\mu}+\frac{2}{n+2}}}.\end{split}$
Setting
$\alpha_{l}:=\rho^{-n-2}k^{-2}{\phi_{l}},$
we have
$\alpha_{l+1}\leq
C_{1}2^{2l(1+\frac{2}{n+2})}\alpha_{l}^{1-\frac{2}{\mu}+\frac{2}{n+2}}.$
From Lemma 5.2, we see that the following
$\begin{split}\alpha_{0}=&{k^{-2}\rho^{-n-2}}{\|(u-k)^{+}\|^{2}_{2;Q_{\rho}}}\
\leq\ {k^{-2}\rho^{-n-2}}{\|u^{+}\|^{2}_{2;Q_{\rho}}}\ \leq\
\theta_{0}:=C_{1}^{-\frac{1}{\alpha}}2^{-\frac{2}{\alpha^{2}}(1+\frac{2}{n+2})}\end{split}$
with $\alpha:=\frac{2}{n+2}-\frac{2}{\mu}$, implies
$\lim_{l\rightarrow\infty}\alpha_{l}=0\textrm{ and thus
}\lim_{l\rightarrow\infty}\phi_{l}=0.$
In conclusion, the two inequalities
$k^{2}\geq{\theta_{0}}^{-1}\rho^{-n-2}{\|u^{+}\|_{2;Q_{\rho}}^{2}}\textrm{ and
}k\geq
a_{0}\rho^{1-\frac{n+2}{\mu}}+\rho^{-\frac{n+2}{2}}\|u^{+}\|_{2;Q_{\rho}}$
imply the following one:
$\begin{split}\|u\|_{\infty;Q_{\frac{\rho}{2}}}\leq 2k.\end{split}$ (5.22)
Hence, (5.22) holds for the following choice
$k:=a_{0}\rho^{1-\frac{n+2}{\mu}}+\left(1+{\theta_{0}^{-\frac{1}{2}}}\right){\rho^{-\frac{n+2}{2}}}{\|u^{+}\|_{2;Q_{\rho}}}.$
which implies our desired estimate.
For $u\in BSPDG^{-}(a_{0},\mu,\gamma;Q)$, the desired assertion follows in a
similar way. We complete our proof. ∎
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|
arxiv-papers
| 2011-03-05T10:21:39 |
2024-09-04T02:49:17.456339
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinniao Qiu and Shanjian Tang",
"submitter": "Jinniao Qiu",
"url": "https://arxiv.org/abs/1103.1038"
}
|
1103.1066
|
# Thermodynamics of viscous dark energy in RSII braneworld
M. R. Setare 1,2111rezakord@ipm.ir and A. Sheykhi2,3222sheykhi@mail.uk.ac.ir
1Department of Science, Payame Noor University, Bijar, Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
###### Abstract
We show that for a RSII braneworld filled with interacting viscous dark energy
and dark matter, one can always rewrite the Friedmann equation in the form of
the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, at apparent horizon. In
addition, the generalized second law of thermodynamics can fulfilled in a
region enclosed by the apparent horizon on the brane for both constant and
time variable 5-dynamical Newton s constant $G_{5}$. These results hold
regardless of the specific form of the dark energy. Our study further support
that in an accelerating universe with spatial curvature, the apparent horizon
is a physical boundary from the thermodynamical point of view.
## I Introduction
Observational data indicates that our universe is currently under accelerating
expansion 1 ; 111 . It seems that some unknown energy components (dark energy)
with negative pressure are responsible for this late-time acceleration 2 .
However, understanding the nature of dark energy is one of the fundamental
problems of modern theoretical cosmology 3 . An alternative approach to
accommodate dark energy is modifying the general theory of relativity on large
scales. Among these theories, scalar-tensor theories 4 , f(R) gravity 5 , DGP
braneworld gravity 6 and string-inspired theories 7 are studied extensively.
The cosmological models with non-viscous cosmic fluid has been studied widely
in the literature. Early treatises on viscous cosmology are given in Pad . The
viscous entropy production in the early universe and viscous fluids on the
Randall-Sundrum branes have been studied respectively in Bre0 . A special
branch of viscous cosmology is to investigate how the bulk viscosity can
influence the future singularity, commonly called the Big Rip, when the fluid
is in the phantom state corresponding to $w_{D}<-1$. A lot of works have been
done in this direction Bre1 ; Bre2 . In particular, it was first pointed out
in Bre1 that the presence of a bulk viscosity proportional to the Hubble
expansion $H$ can cause the fluid to pass from the quintessence region into
the phantom region and thereby inevitably lead to a future singularity.
In the present work we are interested to investigate the interacting viscous
dark energy and dark matter in RSII braneworld, from the thermodynamic point
of view. In particular, we desire to examine under what conditions the
underlying system obeys the generalized second law of thermodynamics, namely
the sum of entropies of the individual components, including that of the
background, to be positive. Then we extend our analysis with considering the
time variable $5$D Newton’s constant $G_{5}$. Until now, in most the
investigated dark energy models a constant Newton’s “constant” $G$ has been
considered. However, there are significant indications that $G$ can by
varying, being a function of time or equivalently of the scale factor G4com .
In particular, observations of Hulse-Taylor binary pulsar Damour ; kogan ,
helio-seismological data guenther , Type Ia supernova observations 1 and
astereoseismological data from the pulsating white dwarf star G117-B15A
Biesiada lead to $\left|\dot{G}/G\right|\lessapprox 4.10\times
10^{-11}yr^{-1}$, for $z\lesssim 3.5$ ray1 . Additionally, a varying $G$ has
some theoretical advantages too, alleviating the dark matter problem gol , the
cosmic coincidence problem jamil and the discrepancies in Hubble parameter
value ber .
There have been many proposals in the literature attempting to theoretically
justified a varying gravitational constant, despite the lack of a full,
underlying quantum gravity theory. Starting with the simple but pioneering
work of Dirac Dirac:1938mt , the varying behavior in Kaluza-Klein theory was
associated with a scalar field appearing in the metric component corresponding
to the $5$-th dimension kal and its size variation akk . An alternative
approach arises from Brans-Dicke framework bd , where the gravitational
constant is replaced by a scalar field coupling to gravity through a new
parameter, and it has been generalized to various forms of scalar-tensor
theories gen , leading to a considerably broader range of variable-$G$
theories. In addition, justification of a varying Newton’s constant has been
established with the use of conformal invariance and its induced local
transformations bek . Finally, a varying $G$ can arise perturbatively through
a semiclassical treatment of Hilbert-Einstein action 19 , non-perturbatively
through quantum-gravitational approaches within the “Hilbert-Einstein
truncation” 21 , or through gravitational holography Guberina ; 71 .
## II Basic Equations
Our starting point is the four-dimensional homogenous and isotropic FRW
universe on the brane with the metric
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of
space with $k=0,1,-1$ corresponding to open, flat, and closed universes,
respectively. A closed universe with a small positive curvature
($\Omega_{k}\simeq 0.01$) is compatible with observations wmap . The dynamical
apparent horizon, a marginally trapped surface with vanishing expansion, is
determined by the relation
$h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$, which implies
that the vector $\nabla\tilde{r}$ is null on the apparent horizon surface. The
apparent horizon was argued as a causal horizon for a dynamical spacetime and
is associated with gravitational entropy and surface gravity Hay2 ; Bak . For
the FRW universe the apparent horizon radius reads
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (2)
The associated surface gravity on the apparent horizon can be defined as
$\kappa=\frac{1}{\sqrt{-h}}\partial_{a}\left(\sqrt{-h}h^{ab}\partial_{ab}\tilde{r}\right),$
(3)
thus one can easily express the surface gravity on the apparent horizon
$\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(4)
The associated temperature on the apparent horizon can be expressed in the
form
$T_{h}=\frac{|\kappa|}{2\pi}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(5)
where $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}<1$ ensures that the
temperature is positive. Recently the Hawking radiation on the apparent
horizon has been observed in cai3 which gives more solid physical implication
of the temperature associated with the apparent horizon.
The Friedmann equation for $3$-dimensional Randall-Sundrum (RS) II brane
embedded in an $5$-dimensional AdS bulk can be written Bin
$H^{2}+\frac{k}{a^{2}}-\frac{\kappa_{5}^{2}\Lambda_{5}}{6}-\frac{\mathcal{C}}{a^{4}}=\frac{\kappa_{5}^{4}}{36}\rho^{2}.$
(6)
where
$\kappa_{5}^{2}=8\pi
G_{5}\,,\quad\Lambda_{5}=-\frac{6}{\kappa_{5}^{2}\ell^{2}},$ (7)
$\Lambda_{5}$ is the $5$-dimensional bulk cosmological constant, and $\ell$ is
the AdS radius of the bulk spacetime. Here $\rho=\rho_{m}+\rho_{D}$ where
$\rho_{m}$ and $\rho_{D}$ are, respectively, the energy density of dark matter
and dark energy confined to the brane and $H=\dot{a}/a$ is the Hubble
parameter on the brane. The constant $\mathcal{C}$ comes from the
$5$-dimensional bulk Weyl tensor. In this paper we are interested in AdS bulk
spacetimes, so the bulk Weyl tensor vanishes and thus we set $\mathcal{C}=0$
in the following discussions. The energy-momentum tensor of the matter and
energy content on the brane is as,
$T_{\mu\nu}=\rho u_{\mu}u_{\nu}+\tilde{p}_{D}(g_{\mu\nu}+u_{\mu}u_{\nu}),$ (8)
where $u_{\mu}$ is the four-velocity vector, and
$\tilde{p}_{D}={p}_{D}-3H\xi,$ (9)
is the effective pressure of dark energy and $\xi$ is the viscosity
coefficient. The condition $\xi>0$ guaranties a positive entropy production
and, in consequence, no violation of the second law of the thermodynamics Zim
. The total energy density on the brane satisfies a conservation law
$\dot{\rho}+3H(\rho+\tilde{p}_{D})=0.$ (10)
However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately, they must rather
enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (11)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=9H^{2}\xi-Q.$ (12)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of viscous
dark energy and $Q=\Gamma\rho_{D}$ denotes the interaction between the dark
components. We also assume the interaction term is positive, $Q>0$, which
means that there is an energy transfer from the dark energy to dark matter.
Hereafter we assume that the brane cosmological constant is zero (if it does
not vanish, one can absorb it in the stress-energy tensor of fluid on the
brane).
## III First law of thermodynamics in Vicous braneworld
In this section we are going to examine the first law of thermodynamics on the
brane. In particular, we show that for a closed universe filled with viscous
dark energy and dark matter the Friedmann equation can be written directly in
the form of the first law of thermodynamics at apparent horizon on the brane.
Using Eq. (7) the Friedmann equation (6) can be written as
$\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell^{2}}}=\frac{4\pi
G_{5}}{3}(\rho_{m}+\rho_{D}).$ (13)
In terms of the apparent horizon radius we have
$\rho_{m}+\rho_{D}=\frac{3}{4\pi
G_{5}}\sqrt{\frac{1}{{\tilde{r}_{A}}^{2}}+\frac{1}{\ell^{2}}}.$ (14)
Taking differential form of equation (13) and using Eqs. (11) and (12), we can
get the differential form of the Friedmann equation
$H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]dt=\frac{\ell}{4\pi
G_{5}}\frac{d\tilde{r}_{A}}{\tilde{r}_{A}^{2}\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}.$
(15)
where $u=\rho_{m}/\rho_{D}$ is the ratio of energy densities. Multiplying both
sides of the equation (22) by a factor
$4\pi\tilde{r}_{A}^{3}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$,
and using the expression (4) for the surface gravity, after some
simplification one can rewrite this equation in the form
$\displaystyle-\frac{\kappa}{2\pi}\frac{2\pi\ell}{G_{5}}\frac{\tilde{r}_{A}^{2}d\tilde{r}_{A}}{\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}$
$\displaystyle=$ $\displaystyle
4\pi\tilde{r}_{A}^{3}H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]dt$ (16)
$\displaystyle-2\pi\tilde{r}_{A}^{2}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]d\tilde{r}_{A}.$
$E=(\rho_{m}+\rho_{D})V$ is the total energy content of the universe inside a
$3$-sphere of radius $\tilde{r}_{A}$ on the brane, where
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by 3-dimensional
sphere with the area of apparent horizon $A=4\pi\tilde{r}_{A}^{2}$. Taking
differential form of the relation
$E=(\rho_{m}+\rho_{D})\frac{4\pi}{3}\tilde{r}_{A}^{3}$ for the total matter
and energy inside the apparent horizon, we get
$dE=4\pi\tilde{r}_{A}^{2}(\rho_{m}+\rho_{D})d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}(\dot{\rho}_{m}+\dot{\rho}_{D})dt.$
(17)
Using Eqs. (11) and (12), we obtain
$dE=4\pi\tilde{r}_{A}^{2}\rho_{D}(1+u)d\tilde{r}_{A}-4\pi\tilde{r}_{A}^{3}H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]dt.$
(18)
Substituting this relation into (16), after some simplifications one can
rewrite this equation in the form
$dE-
WdV=\frac{\kappa}{2\pi}\frac{2\pi\ell}{G_{5}}\frac{\tilde{r}_{A}^{2}}{\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}d\tilde{r}_{A}.$
(19)
where
$W=\frac{1}{2}\left[\rho_{m}+\rho_{D}-\tilde{p}_{D}\right]=\frac{1}{2}\rho_{D}\left[1+u-w_{D}+3H\xi\right],$
is the matter work density Hay2 . The work density term is regarded as the
work done by the change of the apparent horizon, which is used to replace the
negative pressure if compared with the standard first law of thermodynamics,
$dE=TdS-pdV$. For a pure de Sitter space, $\rho_{m}+\rho_{D}=-\tilde{p}_{D}$,
then our work term reduces to the standard $-\tilde{p}_{D}dV$. Expression (19)
is nothing, but the first law of thermodynamics at the apparent horizon on the
brane, namely $dE=T_{h}dS_{h}+WdV$. We can define the entropy associated with
the apparent horizon on the brane as
$S_{h}=\frac{2\pi\ell}{G_{5}}{\displaystyle\int^{\tilde{r}_{A}}_{0}\frac{\tilde{r}_{A}^{2}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}d\tilde{r}_{A}}.$
(20)
After the integration we have
$S_{h}=\frac{2\pi{\tilde{r}_{A}}^{3}}{3G_{5}}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right),$
(21)
where ${}_{2}F_{1}(a,b,c,z)$ is the hypergeometric function. It is worth
noticing when $\tilde{r}_{A}\ll\ell$, which physically means that the size of
the extra dimension is very large if compared with the apparent horizon
radius, one recovers the $5$-dimensional area formula for the entropy on the
brane Shey1 ; Shey2 ; Shey3 ; Shey4 . This is due to the fact that because of
the absence of the negative cosmological constant in the bulk, no localization
of gravity happens on the brane. As a result, the gravity on the brane is
still $5$-dimensional. In this way we show that for a non-flat universe filled
with viscous dark energy and dark matter the Friedmann equation can be written
in the form of the first law of thermodynamics at apparent horizon in RSII
braneworld.
## IV GSL and interacting viscous dark energy
Our aim here is to investigate the validity of the generalized second law of
thermodynamics in a region enclosed by the apparent horizon on the brane.
Taking the derivative of Eq. (14) with respect to the cosmic time and using
Eqs. (11) and (12), one gets
$\dot{\tilde{r}}_{A}=\frac{4\pi}{\ell}G_{5}H{\tilde{r}_{A}}^{2}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}.$
(22)
Next we turn to calculate $T_{h}\dot{S_{h}}$:
$\displaystyle T_{h}\dot{S_{h}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left[\frac{2\pi{\tilde{r}_{A}}^{3}}{3G_{5}}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right)\right]$
(23) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{2\pi\ell}{G_{5}}\frac{{\tilde{r}_{A}}^{2}\dot{\tilde{r}}_{A}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}.$
Using Eq. (22), after some simplification we obtain
$T_{h}\dot{S_{h}}=4\pi
H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]{\tilde{r}_{A}}^{3}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(24)
As we argued above the term
$\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$ is positive to
ensure $T_{h}>0$, however, in an accelerating universe the equation of state
parameter of dark energy may satisfy the condition
$w_{D}<-1-u+3H\xi/\rho_{D}$. This implies that the second law of
thermodynamics, $\dot{S_{h}}\geq 0$, does not hold. However, as we will see
below the generalized second law of thermodynamics,
$\dot{S_{h}}+\dot{S_{m}}+\dot{S_{D}}\geq 0$, is still fulfilled throughout the
history of the universe. The entropy of the viscous dark energy plus dark
matter inside the apparent horizon, $S=S_{m}+S_{D}$, can be related to the
total energy $E=(\rho_{m}+\rho_{D})V$ and pressure $\tilde{p}_{D}$ in the
horizon by the Gibbs equation Pavon2
$TdS=d[(\rho_{m}+\rho_{D})V]+\tilde{p}_{D}dV=V(d\rho_{m}+d\rho_{D})+\left[\rho_{D}(1+u+w_{D})-3H\xi\right]dV,$
(25)
where $T=T_{m}=T_{D}$ and $S=S_{m}+S_{D}$ are, respectively, the temperature
and the total entropy of the energy and matter content inside the horizon, and
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by the apparent
horizon. Here we assumed that the temperature of both dark components are
equal, due to their mutual interaction. We also limit ourselves to the
assumption that the thermal system bounded by the apparent horizon remains in
equilibrium so that the temperature of the system must be uniform and the same
as the temperature of its boundary. This requires that the temperature $T$ of
the viscous dark energy inside the apparent horizon should be in equilibrium
with the temperature $T_{h}$ associated with the apparent horizon, so we have
$T=T_{h}$. This expression holds in the local equilibrium hypothesis. If the
temperature of the fluid differs much from that of the horizon, there will be
spontaneous heat flow between the horizon and the fluid and the local
equilibrium hypothesis will no longer hold. This is also at variance with the
FRW geometry. In general, when we consider the thermal equilibrium state of
the universe, the temperature of the universe is associated with the apparent
horizon. Therefore from the Gibbs equation (25) we obtain
$T_{h}(\dot{S_{m}}+\dot{S_{D}})=4\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}-4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right].$ (26)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}+S_{D}$. Adding equations (24) and
(26), we get
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})=2\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}=\frac{A}{2}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}.$
(27)
where $A=4\pi\tilde{r}_{A}^{2}$ is the area of the apparent horizon on the
brane. Substituting $\dot{\tilde{r}}_{A}$ from Eq. (22) into (27) we reach
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})=\frac{2\pi}{\ell}G_{5}A{\tilde{r}_{A}}^{2}\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}\
H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]^{2}.$ (28)
The right hand side of the above equation cannot be negative throughout the
history of the universe, which means that
$\dot{S_{h}}+\dot{S_{m}}+\dot{S}_{D}\geq 0$ always holds. This indicates that
the generalized second law of thermodynamics is fulfilled in the RS II
braneworld embedded in the AdS bulk.
## V GSL and with variable $5$D Newton’s constant
In this section we would like to perform the above analysis with considering
the time variable $5$D Newton’s constant $G_{5}$. There is some evidence of a
variable $G_{5}$ through numerous astrophysical observations Ko . Models with
variable Newton’s constant can fix some of the hardest problems in cosmology
like the age problem, cosmic coincidence problem and finding the value of the
Hubble parameter Gold .
Taking the derivative of Eq. (14) with respect to the cosmic time and using
Eqs. (11) and (12), one gets
$\dot{\tilde{r}}_{A}=\left(4\pi
G_{5}H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]-P\dot{G_{5}}\right)\frac{{\tilde{r}_{A}}^{2}}{\ell}\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}.$
(29)
where we have defined
$P=\frac{\sqrt{{\tilde{r}_{A}}^{2}+\ell^{2}}}{{\tilde{r}}_{A}G_{5}\ell}.$ (30)
Next we calculate $T_{h}\dot{S_{h}}$:
$\displaystyle T_{h}\dot{S_{h}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\left[\frac{2\pi\ell}{G_{5}}\frac{{\tilde{r}_{A}}^{2}\dot{\tilde{r}}_{A}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}-\frac{\dot{G_{5}}}{G_{5}}S_{h}\right].$
(31)
Next we examine the evolution of the total entropy $S_{h}+S_{m}+S_{D}$. Adding
equations (31) and (26), and using Eq. (29) we reach
$\displaystyle T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})$ $\displaystyle=$
$\displaystyle
2\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}$
(32)
$\displaystyle-\frac{\dot{G_{5}}}{G_{5}}\left[P\tilde{r}_{A}^{3}+\frac{S_{h}}{2\pi\tilde{r}_{A}}\right]\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (29) into (32) we reach
$\displaystyle T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\ell}G_{5}A{\tilde{r}_{A}}^{2}\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}H\left[\rho_{D}(1+u+w_{D})-3H\xi\right]^{2}$
(33)
$\displaystyle-\frac{A}{2}P\dot{G_{5}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\frac{\tilde{r}_{A}^{2}}{\ell}\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}$
$\displaystyle-\frac{\dot{G_{5}}}{G_{5}}\left[P\tilde{r}_{A}^{3}+\frac{S_{h}}{2\pi\tilde{r}_{A}}\right]\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
For $\dot{G_{5}}=0$, we obtain the result of the previous section. In this
case the validity of GSL depend to the sign of $\dot{G_{5}}$, if
$\dot{G_{5}}<0$, and $\rho_{D}(1+u+w_{D})>3H\xi$, then
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})>0$.
## VI Summary and discussions
In the present, paper we have showed that the Friedmann equations on a RSII
braneworld filled with interacting viscous dark energy and dark matter can be
written directly in the form of the first law of thermodynamics at apparent
horizon. Then We examined the validity of the generalized second law of
thermodynamics, we studied the time evolution of the total entropy, including
the entropy associated with the apparent horizon and the entropy of the
viscous dark energy inside the apparent horizon. Our study have shown that the
generalized second law of thermodynamics is always protected in a RSII
braneworld filled with interacting viscous dark energy and dark matter in a
region enclosed by the apparent horizon. Then we extended our study to the
case time variable 5-dynamical Newton s constant $G_{5}$. According to the our
calculations the generalized second law of thermodynamics is valid if
$\dot{G_{5}}<0$, and $\rho_{D}(1+u+w_{D})>3H\xi$. These results hold
regardless of the specific form of the dark energy.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha.
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|
arxiv-papers
| 2011-03-05T16:40:05 |
2024-09-04T02:49:17.468056
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. R. Setare and A. Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1103.1066"
}
|
1103.1067
|
# Viscous dark energy and generalized second law of thermodynamics
M. R. Setare 1,2111rezakord@ipm.ir and A. Sheykhi2,3222sheykhi@mail.uk.ac.ir
1Department of Science, Payame Noor University, Bijar, Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
###### Abstract
We examine the validity of the generalized second law of thermodynamics in a
non-flat universe in the presence of viscous dark energy. At first we assume
that the universe filled only with viscous dark energy. Then, we extend our
study to the case where there is an interaction between viscous dark energy
and pressureless dark matter. We examine the time evolution of the total
entropy, including the entropy associated with the apparent horizon and the
entropy of the viscous dark energy inside the apparent horizon. Our study show
that the generalized second law of thermodynamics is always protected in a
universe filled with interacting viscous dark energy and dark matter in a
region enclosed by the apparent horizon. Finally, we show that the the
generalized second law of thermodynamics is fulfilled for a universe filled
with interacting viscous dark energy and dark matter in the sense that we take
into account the Casimir effect.
## I Introduction
One of the most important problems of modern cosmology is the so-called dark
energy (DE) puzzle. The type Ia supernova observations suggest that the
universe is dominated by DE with negative pressure which provides the
dynamical mechanism for the accelerating expansion of the universe Rie . This
acceleration implies that if Einstein’s theory of gravity is reliable on
cosmological scales, then our universe is dominated by a mysterious form of
energy. This unknown energy component possesses some strange features, for
example it is not clustered on large length scales and its pressure must be
negative so that can drive the current acceleration of the universe. Since the
fundamental theory of nature that could explain the microscopic physics of DE
is unknown at present, phenomenologists take delight in constructing various
models based on its macroscopic behavior. The dynamical nature of dark energy,
at least in an effective level, can originate from various fields, such is a
canonical scalar field (quintessence) quint , a phantom field, that is a
scalar field with a negative sign of the kinetic term phant , or the
combination of quintessence and phantom in a unified model named quintom
quintom .
The cosmological models with non-viscous cosmic fluid has been studied widely
in the literature. Early treatises on viscous cosmology are given in Pad . The
viscous entropy production in the early universe and viscous fluids on the
Randall-Sundrum branes have been studied respectively in Bre0 . A special
branch of viscous cosmology is to investigate how the bulk viscosity can
influence the future singularity, commonly called the Big Rip, when the fluid
is in the phantom state corresponding to $w_{D}<-1$. A lot of works have been
done in this direction Bre1 ; Bre2 ; meng . In particular, it was first
pointed out in Bre1 that the presence of a bulk viscosity proportional to the
Hubble expansion $H$ can cause the fluid to pass from the quintessence region
into the phantom region and thereby inevitably lead to a future singularity.
Since the discovery of black hole thermodynamics in $1970$, physicists have
been speculated on the thermodynamics of the cosmological models in an
accelerated expanding universe Huan ; Pavon2 ; Cai2 ; Cai3 ; CaiKim ; Fro ;
Wang ; Cai4 . Related to the present work, the first and the second laws of
thermodynamics in a flat universe were investigated for time independent and
time dependent EoS bw . For the case of a constant EoS, the first law is valid
for the apparent horizon (Hubble horizon) and it does not hold for the event
horizon as system s IR cut-off. When the EoS is assumed to be time dependent,
using a holographic model of dark energy in flat space, the same result is
gained; the event horizon, in contrast to the apparent horizon, does not
satisfy the first law. Also, while the event horizon does not respect the
second law, it hold for the universe enclosed by the apparent horizon.
In this paper we study the validity of the generalized second law of
thermodynamics for a viscous dark energy in a universe enveloped by the
apparent horizon. Recently, it was shown that for an accelerating universe the
apparent horizon is a physical boundary from the thermodynamical point of view
Jia ; Shey1 ; Shey2 ; sheywang . In particular, it was argued that for an
accelerating universe inside the event horizon the generalized second law does
not satisfy, while the accelerating universe enveloped by the apparent horizon
satisfies the generalized second law of thermodynamics Jia . Therefore, the
event horizon in an accelerating universe might not be a physical boundary
from the thermodynamical point of view. Then we extend our study to the case
where there is an interaction between viscous dark energy and pressureless
dark matter. Most discussions on dark energy rely on the assumption that it
evolves independently of dark matter. Given the unknown nature of both dark
energy and dark matter there is nothing in principle against their mutual
interaction and it seems very special that these two major components in the
universe are entirely independent. Indeed, this possibility has received a lot
of attention recently Ame ; Zim ; Seta1 ; wang1 and in particular, it has
been shown that the coupling can alleviate the coincidence problem Pav1 .
This paper is organized as follows. In section II, we examine the generalized
second law of thermodynamics in a universe filled only with viscous dark
energy. In section III, we extend our study to the case where there is an
interaction term between viscous dark energy and pressureless dark matter. In
section IV, we study the Casimir effect in viscous dark energy. The last
section is devoted to conclusions.
## II GSL and viscous dark energy
We start from a homogenous and isotropic Friedmann-Robertson-Walker (FRW)
universe which is described by the line element
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of
space with $k=0,1,-1$ corresponding to open, flat, and closed universes,
respectively. A closed universe with a small positive curvature
($\Omega_{k}\simeq 0.01$) is compatible with observations spe . The dynamical
apparent horizon, a marginally trapped surface with vanishing expansion, is
determined by the relation
$h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$, which implies
that the vector $\nabla\tilde{r}$ is null on the apparent horizon surface. The
apparent horizon was argued as a causal horizon for a dynamical spacetime and
is associated with gravitational entropy and surface gravity Hay2 ; Bak . For
the FRW universe the apparent horizon radius reads
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (2)
The Friedmann equation for a non-flat universe filled with viscous dark energy
takes the form (we neglect the dark matter)
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho_{D},$ (3)
where $\rho_{D}$ is the energy density of dark energy inside apparent horizon.
In an isotropic and homogeneous FRW universe, the dissipative effects arise
due to the presence of bulk viscosity in cosmic fluids. The theory of bulk
viscosity was initially investigated by Eckart Eck and later on pursued by
Landau and Lifshitz Lan . Dark energy with bulk viscosity has a peculiar
property to cause accelerated expansion of phantom type in the late evolution
of the universe Bre1 ; Bre2 . It can also alleviate several cosmological
puzzles like age problem, coincidence problem and phantom crossing. The
energy-momentum tensor of the viscous fluid is
$T_{\mu\nu}=\rho_{D}u_{\mu}u_{\nu}+\tilde{p}_{D}(g_{\mu\nu}+u_{\mu}u_{\nu}),$
(4)
where $u_{\mu}$ is the four-velocity vector and
$\tilde{p}_{D}={p}_{D}-3H\xi,$ (5)
is the effective pressure of dark energy and $\xi$ is the bulk viscosity
coefficient. We require $\xi>0$ to get positive entropy production in
conformity with second law of thermodynamics Z . The energy conservation
equation is
$\displaystyle\dot{\rho}_{D}+3H(\rho_{D}+\tilde{p}_{D})=0,$ (6)
which can be written
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=9H^{2}\xi,$ (7)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state parameter of viscous
dark energy. In terms of the apparent horizon radius, we can rewrite the
Friedmann equation as
$\frac{1}{\tilde{r}_{A}^{2}}=\frac{8\pi G}{3}\rho_{D}.$ (8)
The associated surface gravity on the apparent horizon can be defined as
$\kappa=\frac{1}{\sqrt{-h}}\partial_{a}\left(\sqrt{-h}h^{ab}\partial_{ab}\tilde{r}\right).$
(9)
Then one can easily show that the surface gravity at the apparent horizon of
FRW universe can be written as
$\kappa=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(10)
The associated temperature on the apparent horizon can be defined as
$T_{h}=\frac{|\kappa|}{2\pi}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(11)
where $\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}<1$ ensures that the
temperature is positive. Recently the connection between temperature on the
apparent horizon and the Hawking radiation has been observed in cao . Hawking
radiation is an important quantum phenomenon of black hole, which is closely
related to the existence of event horizon of black hole. The cosmological
event horizon of de Sitter space has the Hawking radiation with thermal
spectrum as well. Using the tunneling approach proposed by Parikh and Wilczek,
the authors of cao showed that there is indeed a Hawking radiation with a
finite temperature, for locally defined apparent horizon of the FRW universe
with any spatial curvature. This gives more solid physical implication of the
temperature associated with the apparent horizon. The entropy associated to
the apparent horizon is
$\displaystyle S_{h}=\frac{A}{4G}=\frac{\pi\tilde{r}_{A}^{2}}{G}.$ (12)
where $A=4\pi\tilde{r}_{A}^{2}$ is the area of the apparent horizon.
Differentiating Eq. (8) with respect to the cosmic time and using Eq. (7) we
get
$\dot{\tilde{r}}_{A}=4\pi
GH{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+w_{D})-3H\xi\right].$ (13)
Let us now turn to find out $T_{h}\dot{S_{h}}$:
$T_{h}\dot{S_{h}}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left(\frac{\pi\tilde{r}_{A}^{2}}{G}\right).$
(14)
After some simplification and using Eq. (13) we get
$T_{h}\dot{S_{h}}=4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+w_{D})-3H\xi\right]\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(15)
As we argued above the term
$\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)$ is positive to
ensure $T_{h}>0$, however, in an accelerating universe the equation of state
parameter of dark energy may satisfy $w_{D}<-1+3H\xi/\rho_{D}$. This indicates
that the second law of thermodynamics, $\dot{S_{h}}\geq 0$, does not hold on
the apparent horizon. Then the question arises, “will the generalized second
law of thermodynamics, $\dot{S_{h}}+\dot{S_{D}}\geq 0$, can be satisfied in a
region enclosed by the apparent horizon?” The entropy of the viscous dark
energy inside the apparent horizon, $S_{D}$, can be related to its energy
$E_{D}=\rho_{D}V$ and its pressure $\tilde{p}_{D}$ by the Gibbs equation
Pavon2
$T_{D}dS_{D}=d(\rho_{D}V)+\tilde{p}_{D}dV=Vd\rho_{D}+(\rho_{D}+p_{D}-3H\xi)dV,$
(16)
where $T_{D}$ and is the temperature of the viscous dark energy and
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by the apparent
horizon. We also limit ourselves to the assumption that the thermal system
bounded by the apparent horizon remains in equilibrium so that the temperature
of the system must be uniform and the same as the temperature of its boundary.
This requires that the temperature $T_{D}$ of the viscous dark energy inside
the apparent horizon should be in equilibrium with the temperature $T_{h}$
associated with the apparent horizon, so we have $T_{D}=T_{h}$. This
expression holds in the local equilibrium hypothesis. If the temperature of
the fluid differs much from that of the horizon, there will be spontaneous
heat flow between the horizon and the fluid and the local equilibrium
hypothesis will no longer hold. This is also at variance with the FRW
geometry. In general, when we consider the thermal equilibrium state of the
universe, the temperature of the universe is associated with the apparent
horizon. Therefore from the Gibbs equation (16) we can obtain
$T_{h}\dot{S_{D}}=4\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}-4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+w_{D})-3H\xi\right].$ (17)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{D}$. Adding equations (15) and (17),
we get
$T_{h}(\dot{S}_{h}+\dot{S}_{D})=2\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}=\frac{A}{2}\left[\rho_{D}(1+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}.$
(18)
where $A>0$ is the area of apparent horizon. Finally, substituting
$\dot{\tilde{r}}_{A}$ from Eq. (13) into (18) we reach
$T_{h}(\dot{S}_{h}+\dot{S}_{D})=2\pi
GAH{\tilde{r}_{A}}^{3}\left[\rho_{D}(1+w_{D})-3H\xi\right]^{2}.$ (19)
The right hand side of the above equation cannot be negative throughout the
history of the universe, which means that $\dot{S_{h}}+\dot{S_{D}}\geq 0$
always holds. This indicates that for a universe with spacial curvature filled
with viscous dark energy, the generalized second law of thermodynamics is
fulfilled in a region enclosed by the apparent horizon.
## III GSL and interacting viscous dark energy with non-viscous dark matter
In this section we extend our study to the case where there is an interaction
between viscous dark energy and pressureless dark matter. In this case the
Friedmann equation can be written as
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{D}\right),$ (20)
where $\rho_{m}$ and $\rho_{D}$ are the energy density of dark matter and dark
energy inside apparent horizon, respectively. Since we consider the
interaction between dark matter and dark energy, $\rho_{m}$ and $\rho_{D}$ do
not conserve separately, they must rather enter the energy balances
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (21)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=9H^{2}\xi-Q.$ (22)
where $Q=\Gamma\rho_{D}$ denotes the interaction between the dark components.
We also assume the interaction term is positive, $Q>0$, which means that there
is an energy transfer from the dark energy to dark matter. In terms of the
apparent horizon radius, we can rewrite the Friedmann equation as
$\frac{1}{\tilde{r}_{A}^{2}}=\frac{8\pi G}{3}\left(\rho_{m}+\rho_{D}\right).$
(23)
Differentiating Eq. (23) with respect to the cosmic time and using Eqs. (21)
and (22) we get
$\dot{\tilde{r}}_{A}=4\pi
GH{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right].$ (24)
where $u=\rho_{m}/\rho_{D}$ is the ratio of energy densities. Next we turn to
calculate $T_{h}\dot{S_{h}}$. It is easy to show that
$T_{h}\dot{S_{h}}=4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(25)
Again in an accelerating universe the equation of state parameter of dark
energy may satisfy the condition $w_{D}<-1-u+3H\xi/\rho_{D}$. This implies
that the second law of thermodynamics, $\dot{S_{h}}\geq 0$, does not hold on
the apparent horizon. Then we examine the validity of the generalized second
law, $\dot{S_{h}}+\dot{S_{m}}+\dot{S_{D}}\geq 0$. The entropy of the viscous
dark energy plus dark matter inside the apparent horizon, $S=S_{m}+S_{D}$, can
be related to the total energy $E=(\rho_{m}+\rho_{D})V$ and pressure
$\tilde{p}_{D}$ in the horizon by the Gibbs equation
$TdS=d[(\rho_{m}+\rho_{D})V]+\tilde{p}_{D}dV=V(d\rho_{m}+d\rho_{D})+\left[\rho_{D}(1+u+w_{D})-3H\xi\right]dV,$
(26)
where $T=T_{m}=T_{D}$ and $S=S_{m}+S_{D}$ are the temperature and the total
entropy of the energy and matter content inside the horizon, respectively.
Here we assumed that the temperature of both dark components are equal, due to
their mutual interaction. We also assume the local equilibrium hypothesis
holds, so $T=T_{h}$. Therefore from the Gibbs equation (26) we obtain
$T_{h}(\dot{S_{m}}+\dot{S_{D}})=4\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}-4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right].$ (27)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}+S_{D}$. Adding equations (25) and
(27), we get
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})=2\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}=\frac{A}{2}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}.$
(28)
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (24) into (28) we get
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D})=2\pi
GAH{\tilde{r}_{A}}^{3}\left[\rho_{D}(1+u+w_{D})-3H\xi\right]^{2},$ (29)
which cannot be negative throughout the history of the universe and hence the
general second law of thermodynamics, $\dot{S_{h}}+\dot{S_{m}}+\dot{S_{D}}\geq
0$, is always protected for a universe filled with interacting viscous dark
energy and dark matter in a region enclosed by the apparent horizon. To see
the effect on the generalized second law of thermodynamics derived from the
interaction $Q$, one can consider the $Q=0$ in Eqs. (21), (22). After this
substituation, our result (29) do not change, so we conclude that the
interaction term does not affect on the generalized second law of
thermodynamics.
## IV Casimir effects in viscous cosmology
In this section we would like to examine the GSL of thermodynamics for an
interacting viscous dark energy in the sense that we take into account the
Casimir effect. A natural way of dealing with the Casimir effect in a non-flat
universe is to relate it to the apparent horizon radius
$\tilde{r}_{A}=1/\sqrt{H^{2}+k/a^{2}}$. It means effectively that we should
put the Casimir energy $E_{c}$ inversely proportional to the apparent horizon
radius. This is consistent with the basic property of the Casimir energy,
which states that it is a measure of the stress in the region interior to the
“shell” as compared with the unstressed region on the outside. The effect is
evidently largest in the beginning of the universe’s evolution, when
$\tilde{r}_{A}$ is small. At late times, when
$\tilde{r}_{A}\rightarrow\infty$, the Casimir influence should be expected to
fade away. Therefore, we assume the Casimir energy can be written as
$\displaystyle E_{c}=\frac{c}{\tilde{r}_{A}},$ (30)
where $c$ is a constant. We also assume that $c$ is small compared with unity.
This is physically reasonable, in view of the conventional feebleness of the
Casimir force. The Casimir pressure corresponding to energy (30) is
$\displaystyle
p_{c}=\frac{-1}{4\pi\tilde{r}_{A}^{2}}\frac{\partial{E_{c}}}{\partial{\tilde{r}_{A}}}=\frac{c}{4\pi\tilde{r}_{A}^{4}}.$
(31)
Thus the Casimir energy evolves as $\rho_{c}\propto\tilde{r}_{A}^{-4}$. The
continuity equation for the Casimir energy takes the form
$\displaystyle\dot{\rho}_{c}+3H\rho_{c}(1+w_{c})=0,$ (32)
where $w_{c}=p_{c}/\rho_{c}$ is the equation of state parameter of Casimir
energy. Using Eq. (31) as well as relation
$\displaystyle\rho_{c}=\frac{E_{c}}{V}=\frac{3c}{4\pi\tilde{r}_{A}^{4}},$ (33)
we have
$\displaystyle w_{c}=\frac{p_{c}}{\rho_{c}}=\frac{1}{3}.$ (34)
The Friedmann equation now takes the form
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{D}+\rho_{c}\right),$ (35)
which can be rewritten as
$\frac{1}{\tilde{r}_{A}^{2}}=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{D}+\rho_{c}\right).$ (36)
Differentiating Eq. (36) with respect to the cosmic time and using Eqs. (21),
(22), (32) and (34) we find
$\dot{\tilde{r}}_{A}=4\pi
GH{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right],$
(37)
where $z=\rho_{c}/\rho_{D}$. Next we calculate $T_{h}\dot{S_{h}}$. It is a
matter of calculation to show
$T_{h}\dot{S_{h}}=4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right]\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(38)
From the Gibbs equation for the total energy content of the universe we have
$\displaystyle T_{h}dS$ $\displaystyle=$ $\displaystyle
d[(\rho_{m}+\rho_{D}+\rho_{c})V]+(\tilde{p}_{D}+p_{c})dV$ (39)
$\displaystyle=$ $\displaystyle
V(d\rho_{m}+d\rho_{D}+d\rho_{c})+\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right]dV,$
where $S=S_{m}+S_{D}+S_{c}$ and we have assumed that the temperature of all
the energy content are identical and equal with the apparent horizon
temperature $T_{h}$. Thus from Eq. (39) we obtain
$\displaystyle T_{h}(\dot{S_{m}}+\dot{S_{D}}+\dot{S_{c}})$ $\displaystyle=$
$\displaystyle
4\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}$
(40) $\displaystyle-4\pi
H{\tilde{r}_{A}^{3}}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right].$
Now we are in a position to examine the GSL of thermodynamics. Adding
equations (38) and (40), we get
$\displaystyle T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D}+\dot{S}_{c})$
$\displaystyle=$ $\displaystyle
2\pi{\tilde{r}_{A}^{2}}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}$
(41) $\displaystyle=$
$\displaystyle\frac{A}{2}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right]\dot{\tilde{r}}_{A}.$
Substituting $\dot{\tilde{r}}_{A}$ from Eq. (37) into (41) we reach
$T_{h}(\dot{S}_{h}+\dot{S}_{m}+\dot{S}_{D}+\dot{S}_{c})=2\pi
GAH{\tilde{r}_{A}}^{3}\left[\rho_{D}(1+u+\frac{4z}{3}+w_{D})-3H\xi\right]^{2}.$
(42)
The right hand side of the above equation cannot be negative throughout the
history of the universe, which means that
$\dot{S_{h}}+\dot{S_{m}}+\dot{S}_{D}+\dot{S}_{c}\geq 0$ always holds. This
indicates that the GSL of thermodynamics is fulfilled for a universe filled
with interacting viscous dark energy and dark matter in the sense that we take
into account the Casimir effect.
## V Conclusions
We have investigated the validity of the generalized second law of
thermodynamics in a non-flat universe with viscous dark energy. We have
examined the total entropy evolution with time, including the derived apparent
horizon entropy and the entropy of viscous dark energy inside the apparent
horizon. Then, we have extended our study to the case where there is an
interaction between viscous dark energy and pressureless dark matter. We have
shown that the generalized second law of thermodynamics is always fulfilled
for a universe filled with interacting viscous dark energy and dark matter in
a region enclosed by the apparent horizon. We have also examined the validity
of the GSL of thermodynamics for an interacting viscous dark energy in the
sense that we take into account the Casimir effect.
###### Acknowledgements.
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha.
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|
arxiv-papers
| 2011-03-05T16:50:31 |
2024-09-04T02:49:17.473221
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. R. Setare and A. Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1103.1067"
}
|
1103.1096
|
Spiraling elliptic solitons in generic nonlocal nonlinear media
Guo Liang, Qian Shou and Qi Guo∗
Laboratory of Photonic Information Technology, South China Normal University,
Guangzhou 510631,China
∗Corresponding author: guoq@scnu.edu.cn
###### Abstract
We have introduced a class of spiraling elliptic solitons in generic nonlocal
nonlinear media. The spiraling elliptic solitons carry the orbital angular
momentum. This class solitons are stable for any degree of nonlocality except
for the local case when the response function of the material is Gaussian
function.
OCIS codes: 190.6135, 190.4360,060.1810.
Optical spatial solitons in nonlocal nonlinear media are attracting increasing
attention during recent years in both theoretical [1, 2, 3, 4, 5, 6, 7, 8, 9,
10] and experimental [11, 12, 13, 14] aspects of research. The nonlocality
plays an important role in the nonlinear evolution of waves. It may
drastically modify the properties of solitons. The solitons in bulk Kerr media
may undergo catastrophic collapse [15, 16]. The nonlocality of an arbitrary
shape can eliminate collapse in all physical dimensions [6]. Nonlocality can
support vortex solitons [17, 18]and multipole solitons [19] which are unstable
in local nonlinear media.
In theoretical aspect, ellipse-shaped solitons have been reported in saturable
nonlinear media, such as elliptic incoherent solitons [20],elliptic dark
solitons [21],and spiraling elliptic solitons [22]. In experimental aspect,
coherent elliptic solitons [23] in lead glass which is nonlocal nonlinear
media and elliptic incoherent spatial solitons [24] in photorefractive
sceening nonlinear media are observed.
In this Letter, we use the variational approach to derive the analytical
spiraling elliptic solitons solution in generic nonlocal nonlinear media. We
analyze the potential function to study the stability properties of the class
of solitons.
The propagation of the optical beams in the nonlocal cubic nonlinear media can
be modeled by the following generic dimensionless nonlocal nonlinear
Schr$\ddot{o}$dinger equation(NNLSE) [3, 7],
$i\frac{\partial\psi}{\partial z}+\frac{1}{2}\nabla_{\bot}^{2}\psi+\Delta
n\psi=0,$ (1)
where $\psi=\psi(x,y,z)$ is a paraxial beam, $z$ is the longitudinal
coordinate, $\nabla_{\bot}^{2}=\partial_{x}^{2}+\partial_{y}^{2}$, $x$ and $y$
are the transverse coordinates, $\Delta n=\int\int
R(x-x^{\prime},y-y^{\prime})|\psi(x^{\prime},y^{\prime},z)|{\rm
d}x^{\prime}{\rm d}y^{\prime}$ is the normalized nonlinear perturbation of
refraction index, and $R$ is the nonlinear response of the medium which is
normalized, real and symmetric such that$\int\int R(x,y){\rm d}x{\rm d}y=1$.
We suppose the material response to be Gaussian function [7, 25],
i.e.$R(x,y)=1/(2\pi w_{m}^{2})\exp[-(x^{2}+y^{2})/2w_{m}^{2}]$, where $w_{m}$
is the normalized characteristic length of the material response function.
By the variational approach [26], Eq.(1) can be interpreted as an Euler-
Lagrange equation corresponding to a vanishing variation
$\delta\int\int\int
l(\psi,\psi^{*},\psi_{z},\psi_{z}^{*},\psi_{x},\psi_{x}^{*},\psi_{y},\psi_{y}^{*}){\rm
d}x{\rm d}y{\rm d}z=0,$ (2)
where the Lagrangian density $l$ is given by[27, 28]
$\displaystyle{l}$ $\displaystyle=$
$\displaystyle\frac{i}{2}(\psi^{*}\frac{\partial\psi}{\partial
z}-\psi\frac{\partial\psi^{*}}{\partial
z})-\frac{1}{2}(|\frac{\partial\psi}{\partial
x}|^{2}+|\frac{\partial\psi}{\partial y}|^{2})$ (3)
$\displaystyle+\frac{1}{2}|\psi|^{2}\int\int
R(x-\xi,y-\eta)|\psi(\xi,\eta)|^{2}{\rm d}\xi{\rm d}\eta.$
We introduce a trial function,
$\psi(x,y,z)=A(z)G[X/b(z)]G[Y/c(z)]\exp(i\phi),$ (4)
where the Gaussian envelope is $G(t)=\exp(-t^{2}/2)$ the phase is
$\phi=B(z)X^{2}+\Theta(z)XY+Q(z)Y^{2}+\varphi(z)$, and
$X=x\cos\beta(z)+y\sin\beta(z),Y=-x\sin\beta(z)+y\cos\beta(z)$. Corresponding
to the trial function we can obtain its power, $P=\int\int|\psi(x,y)|^{2}{\rm
d}x{\rm d}y=\pi A^{2}bc$ and orbital angular momentum(OAM),
$M=\text{Im}\int\int\psi^{*}(\textbf{r}\times\nabla\psi){\rm
d}^{2}\textbf{r}=1/2P(b^{2}-c^{2})\Theta$ with
$\textbf{r}=x\textbf{e}_{x}+y\textbf{e}_{y}$. Substituting the trial function
above to the variational principle Eq.(2), we obtain the reduced variational
equation
$\delta\int L{\rm d}z=0,$ (5)
where $L=\int\int l_{g}{\rm d}x{\rm d}y$, and $l_{g}$ denotes the result of
inserting the Gaussian ansatz (4) into the Lagrangian density (3). It also can
be shown that the Hamiltonian corresponding to Eq.(1) is of the following form
$\displaystyle H$ $\displaystyle=$
$\displaystyle\int\int\Bigg{[}\frac{1}{2}(|\frac{\partial\psi}{\partial
x}|^{2}+|\frac{\partial\psi}{\partial y}|^{2})-\frac{1}{2}|\psi|^{2}$ (6)
$\displaystyle\int\int R(x-\xi,y-\eta)|\psi(\xi,\eta)|^{2}d\xi
d\eta\Bigg{]}{\rm d}x{\rm d}y.$
After some algebraic calculations, $L$ and $H$ can be analytically determined
as
$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{A^{2}\pi}{4bc}\Bigg{[}-b^{2}-c^{2}-4b^{4}B^{2}c^{2}-4b^{2}c^{4}Q^{2}-b^{4}c^{2}\Theta^{2}$
(7)
$\displaystyle-b^{2}c^{4}\Theta^{2}+\frac{A^{2}b^{3}c^{3}\sqrt{\left(b^{2}+w_{m}^{2}\right)\left(c^{2}+w_{m}^{2}\right)}}{\left(b^{2}+w_{m}^{2}\right)\left(c^{2}+w_{m}^{2}\right)}-2b^{4}c^{2}B^{\prime}$
$\displaystyle-2b^{2}c^{4}Q^{\prime}+2b^{4}c^{2}\Theta\beta^{\prime}-2b^{2}c^{4}\Theta\beta^{\prime}-4b^{2}c^{2}\varphi^{\prime}\Bigg{]},$
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{A^{2}\pi}{4bc}\Bigg{[}b^{2}+c^{2}+4b^{4}B^{2}c^{2}+4b^{2}c^{4}Q^{2}+b^{4}c^{2}\Theta^{2}$
(8)
$\displaystyle+b^{2}c^{4}\Theta^{2}-\frac{A^{2}b^{3}c^{3}\sqrt{\left(b^{2}+w_{m}^{2}\right)\left(c^{2}+w_{m}^{2}\right)}}{\left(b^{2}+w_{m}^{2}\right)\left(c^{2}+w_{m}^{2}\right)}\Bigg{]}.$
Following the standard procedures of the variational approach [26], we have
$b^{\prime}=2bB,c^{\prime}=2cQ,\beta^{\prime}=\left(b^{2}+c^{2}\right)\Theta\left/\left(b^{2}-c^{2}\right)\right.,P^{\prime}=0,H^{\prime}=0$
and $M^{\prime}=0$. Primes indicate derivatives with respect to the evolution
variable $z$. So we can rewrite the Hamiltonian of the system as follows,
$H=\frac{P}{4}(b^{\prime 2}+c^{\prime 2}+\Pi),$ (9) $\displaystyle\Pi$
$\displaystyle=$
$\displaystyle\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{4b^{2}\sigma^{2}}{\left(b^{2}-c^{2}\right)^{2}}+\frac{4c^{2}\sigma^{2}}{\left(b^{2}-c^{2}\right)^{2}}$
(10)
$\displaystyle-\frac{P}{\pi\sqrt{\left(b^{2}+w_{m}^{2}\right)\left(c^{2}+w_{m}^{2}\right)}},$
with $\sigma\equiv M/P=1/2(b^{2}-c^{2})\Theta$.
Solitons can be found as the extrema of the potential $\Pi(b,c)$. Letting
$\partial\Pi/\partial b=0$ and $\partial\Pi/\partial c=0$, we can obtain the
critical power and the OAM.
$\displaystyle P_{c}$ $\displaystyle=$
$\displaystyle\frac{2\left(b^{2}+c^{2}\right)^{3}\pi\left[\left(b^{2}+w_{m}^{2}\right)\left(c^{2}+w_{m}^{2}\right)\right]{}^{3/2}}{b^{4}c^{4}\left[b^{4}+6b^{2}c^{2}+c^{4}+4\left(b^{2}+c^{2}\right)w_{m}^{2}\right]},$
(11) $\displaystyle\sigma_{c}^{2}$ $\displaystyle=$
$\displaystyle\frac{\left(b^{2}-c^{2}\right)^{4}\left[b^{2}c^{2}+\left(b^{2}+c^{2}\right)w_{m}^{2}\right]}{4b^{4}c^{4}\left[b^{4}+6b^{2}c^{2}+c^{4}+4\left(b^{2}+c^{2}\right)w_{m}^{2}\right]}.$
(12)
We can also obtain the rotation velocity
$\omega\equiv\beta^{\prime}=2\left(b^{2}+c^{2}\right)\sigma\left/\left(b^{2}-c^{2}\right)^{2}\right.$.
When the input power and OAM are chosen arbitrarily the spiraling elliptic
solitons can be found the semi-axis of which are determined by Eq.11 and
Eq.12. One example is shown in Fig.1 with $P_{c}=127272.4$ and
$\sigma_{c}=0.560949$ (other parameters are
$b=2.0,c=1.0,\Theta=0.373966,w_{m}=15.0$ and $\omega=0.623277$). The
isosurface of intensity of the spiraling soliton is obtained from our
variational solution. Comparing two half widths obtained from variational
solution, $w_{x}=\sqrt{b^{2}\text{cos}^{2}\omega z+c^{2}\text{sin}^{2}\omega
z}$ and $w_{y}=\sqrt{c^{2}\text{cos}^{2}\omega z+b^{2}\text{sin}^{2}\omega
z}$, with the numerical results we find an excellent agreement as is shown in
Fig.2 and Fig.3 We introduce a nonlocal parameter
$\alpha=\text{max}(w_{m}/b,w_{m}/c)$ to define the degree of nonlocality for
the beam in nonlocal nonlinear media. The larger is the nonlocal parameter,
the stronger is the degree of nonlocality. In Fig.1, Fig.2 and Fig.3 $w_{m}$
is $15.0$, and the degree of nonlocality $\alpha$ is $7.5$.
An important aspect of any family of soliton solutions is their stability
properties. We can study the stability characteristics of our analytical
soliton solution by means of the analysis of the potential function
$\Pi(b,c)$. So we search the second derivative of the potential $\Pi(b,c)$
with respect to $b$ and $c$, then substituting Eq.11 and Eq.12 into it we get
$\displaystyle\frac{\partial^{2}\Pi}{\partial b^{2}}$ $\displaystyle=$
$\displaystyle\frac{2\left(b^{2}+c^{2}\right)}{b^{4}c^{4}\left(b^{2}+w_{m}^{2}\right)\left[b^{4}+6b^{2}c^{2}+c^{4}+4\left(b^{2}+c^{2}\right)w_{m}^{2}\right]}$
(13)
$\displaystyle\Bigg{[}b^{2}c^{2}\left(b^{4}+14b^{2}c^{2}+c^{4}\right)+(b^{6}+18b^{4}c^{2}+33b^{2}c^{4}$
$\displaystyle+4c^{6})w_{m}^{2}+\left(b^{4}+5b^{2}c^{2}+16c^{4}\right)w_{m}^{4}\Bigg{]},$
$\displaystyle\frac{\partial^{2}\Pi}{\partial b\partial c}$ $\displaystyle=$
$\displaystyle-\frac{2\left(b^{2}+c^{2}\right)}{b^{3}c^{3}\left[b^{4}+6b^{2}c^{2}+c^{4}+4\left(b^{2}+c^{2}\right)w_{m}^{2}\right]}$
(14)
$\displaystyle\Bigg{[}b^{4}+14b^{2}c^{2}+c^{4}+12\left(b^{2}+c^{2}\right)w_{m}^{2}\Bigg{]},$
$\displaystyle\frac{\partial^{2}\Pi}{\partial c^{2}}$ $\displaystyle=$
$\displaystyle\frac{2\left(b^{2}+c^{2}\right)}{b^{4}c^{4}\left(c^{2}+w_{m}^{2}\right)\left[b^{4}+6b^{2}c^{2}+c^{4}+4\left(b^{2}+c^{2}\right)w_{m}^{2}\right]}$
(15)
$\displaystyle\Bigg{[}b^{2}c^{2}\left(b^{4}+14b^{2}c^{2}+c^{4}\right)+\left(4b^{6}+33b^{4}c^{2}+18b^{2}c^{4}\right.$
$\displaystyle\left.\left.+c^{6}\right)w_{m}^{2}+\left(4b^{4}+5b^{2}c^{2}+4c^{4}\right)w_{m}^{4}\right.\Bigg{]}.$
From Eq.13, Eq.14 and Eq.15 we can easily get $\partial^{2}\Pi/\partial
b^{2}>0$, $\partial^{2}\Pi/\partial c^{2}>0$ and
$\triangle\equiv(\partial^{2}\Pi/\partial b^{2})(\partial^{2}\Pi/\partial
c^{2})-(\partial^{2}\Pi/\partial b\partial c)^{2}>0$ when $w_{m}\neq 0$ (the
degree of nonlocality $\alpha$ is not zero). So $b$ and $c$ of the spiraling
elliptic solitons what we have got analytically by the variational approach
are corresponding to the minima of the potential $\Pi(b,c)$. So the soliton
solutions are stable for any degree of nonlocality except for the local case.
But we should mention that when the degree of nonlocality decreases the low-
intensity oscillating tails which indicate the appearance of dispersive waves
radiated by the soliton will occur [22, 29]. The radiative tails take a
portion of radiated OAM from the soliton, and the reduction of OAM in the main
soliton leads to the slow reduction of ellipticity of the transverse rotating
profile [22]. This can be verified in Fig.4 and Fig.5. In Fig.4 and Fig.5
$w_{m}$ is $8.0$, and the degree of nonlocality $\alpha$ is $4.0$.
In conclusion, we have obtained spiraling elliptic solitons in generic
nonlocal nonlinear media by use of the variational approach. We show that this
class of solitons are stable for any degree of nonlocality except for the
local case. Because of the appearance of dispersive waves radiated by the
soliton the ellipticity of the spiraling elliptic solitons will reduce when
the degree of nonlocality becomes lower. Our theoretical results have been
confirmed by direct numerical simulations of the NNLSE.
Fig. 1: (color online) Propagation dynamics of the spiraling elliptic soliton
in nonlocal nonlinear media. The isointensity plot is at the level $I_{m}/2$
of the elliptic soliton with $I_{m}=20256.03$ where $I_{m}$ is
$\text{max}|\psi|^{2}$.The normalized characteristic length of the material
response function $w_{m}$ is 15, and the degree of nonlocality $\alpha$ is
7.5.
Fig. 2: (color online) Evolution of the beam width of the spiraling elliptic
soliton in the direction of x axis. Numberically obtained half width $w_{x}$
(black line) is compared to the variational result (red line). The parameters
$w_{m}$ and $\alpha$ are 15 and 7.5 respectively.
Fig. 3: (color online)Same as Fig.2 but with plot corresponding to the beam
width of the spiraling elliptic soliton in the direction of y axis.
Fig. 4: (color online) Evolution of the beam width of the spiraling elliptic
soliton in the direction of x axis. Numberically obtained half width $w_{x}$
(black line) is compared to the variational result (red line). The parameters
$w_{m}$ and $\alpha$ are 8.0 and 4.0 respectively.
Fig. 5: (color online) Same as Fig.4 but with plot corresponding to the beam
width of the spiraling elliptic soliton in the direction of y axis.
This research was supported by the National Natural Science Foundation of
China (Grant Nos. 11074080 and 10904041), the Specialized Research Fund for
the Doctoral Program of Higher Education (Grant No. 20094407110008), and the
Natural Science Foundation of Guangdong Province of China (Grant No.
10151063101000017).
## References
* [1] A.W.Snyder and D.J.Mitchell, Science 276, 1538 (1997)
* [2] A.W.Snyder and Y.Kivshar, J.Opt.Soc.Am.B 11, 3025 (1997)
* [3] D.J.Mitchell and A.W.Snyder, J.Opt.Soc.Am.B 16, 236 (1999).
* [4] W.Krolikowski et al., Phys.Rev.E 64, 016612 (2001).
* [5] W.Krolikowski and O.Bang, Phys.Rev.E 63, 016610 (2000).
* [6] O.Bang et al., Phys.Rev.E 66, 046619 (2002).
* [7] Q.Guo, B.Luo, F.Yi, S.Chi and Y.Xie, Phys.Rev.E 69, 016602 (2004).
* [8] S.G.Ouyang, Q.Guo and W.Hu, Phys.Rev.E 74, 036622 (2006).
* [9] D.M.Deng and Q.Guo, Opt.Lett. 32, 3206 (2007).
* [10] D.M.Deng and Q.Guo, Opt.Lett. 34, 43 (2009).
* [11] M.Peccianti, K.A.Brzdakiewicz and G.Assanto, Opt.Lett. 27, 1460 (2002).
* [12] M.Peccianti et al., Appl.Phys.Lett. 81, 3335 (2002).
* [13] W.Hu, T.Zhang, Q.Guo, L.Xuan and S.Lan, Appl.Phys.Lett. 89, 071111 (2006).
* [14] W.Hu, S.G.Ouyang, P.B.Yang, Q.Guo and S.Lan Phys.Rev.A 77, 033842 (2008).
* [15] K.D.Moll, A.L.Gaeta and G.Fibich, Phys.Rev.Lett.90, 203902 (2003)
* [16] L.Berge,Phys.Rep.303, 259 (1998)
* [17] A.I.Yakimenko, V.M.Lashkin and O.O.Prikhodko, Phys.Rev.E 73, 066605 (2006).
* [18] D.Buccoliero, A.S.Desyatnikov, W.Krolikowski and Y.S.Kivshar, Opt.Lett. 33, 198 (2008).
* [19] D.Buccoliero, A.S.Desyatnikov, W.Krolikowski and Y.S.Kivshar, Phys.Rev.Lett. 98, 053901 (2007).
* [20] E. D.Eugenieva and D. N.Christodoulides, Opt.Lett. 25, 972 (2000).
* [21] I.E.Papacharalampous et al., Physica Scipta.,69, 7 (2004).
* [22] A. S.Desyatnikov, D. Buccoliero, M. R.Dennis, and Y. S.Kivshar, Phys.Rev.Lett 104 053902 (2010).
* [23] C.Rotschild, O.Cohen, O.Manela and M.Segev, Phys.Rev.Lett 95 213904 (2005).
* [24] O.Katz, T.Carmon, T.Schwartz, M.Segev and D.N.Christodoulides Opt.Lett. 29, 1248 (2004).
* [25] W.Krolikowski, O.Bang, N.I.Nikolov, J. Wyller, J.J.Rasmussen, and D.Edmundson, J.Opt.B 6, S288 (2004).
* [26] D.Anderson, Phys.Rev.A 27, 3135 (1983).
* [27] D.Anderson, Opt. Commun. 48, 107 (1983).
* [28] Q.Guo, B.Luo, S.Chi, Opt. Commun. 259, 336 (2006).
* [29] J.Yang, Phys.Rev.E 66 026601 (2002).
|
arxiv-papers
| 2011-03-06T03:19:45 |
2024-09-04T02:49:17.479550
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guo Liang, Qian Shou and Qi Guo",
"submitter": "Qi Guo",
"url": "https://arxiv.org/abs/1103.1096"
}
|
1103.1101
|
Large phase shift of spatial solitons in lead glass
Qian Shou, Xiang Zhang, Wei Hu and Qi Guo∗
Laboratory of Photonic Information Technology, South China Normal University,
Guangzhou, 510631, China
∗Corresponding author: guoq@scnu.edu.cn
###### Abstract
The phenomenon of the large phase shift of the strongly nonlocal spatial
optical soliton was predicted by Guo $el$ $al$. within the phenomenological
framework [Q. Guo, el al., Phys. Rev. E 69, 016602 (2004), but has not been
experimentally confirmed so far. We theoretically and experimentally
investigate the large phase shift of that propagating in the lead glass. It is
verified that the change of the optical power carried by the optical beam
about 10 mW around the critical power for the soliton can lead to a $\pi$
phase shift, which would be of its potential in the application of all-optical
switchings.
OCIS codes: 190.5940, 1909.6135, 190.4780
The extensively investigated strongly nonlocal spatial optical solitons
(SNSOS) are first predicted by Snyder and Mitchell [1]. Compared with their
local counterparts, SNSOSs can take on complex forms, such as high order
solitons [2, 4, 3, 5] and even incoherent solitons [6, 7, 8]. More
importantly, the phase shift of the SNSOS is quite large [9]. This is an
essential attribute of the SNSOSs but ignored by Snyder and Mitchell [1].
Assuming that the scalar field of the monochromatic light is
$E(x,y,z,t)=A(x,y,z)\exp[-i(\omega t-kz)],$
$A$ is the paraxial optical beam, $k=\omega n_{0}/c$, and $n_{0}$ is the
linear refractive index. $kz$ is a linear phase shift after the propagating
distance $z$ which can be called the geometrical phase shift, while the
argument of the paraxial optical beam, $\arg A$, is “the additional phase
shift” that will be abbreviated to “the phase shift” in the following. In the
linear case, the increase rate of the phase shift per unit distance is far
slower than that of the geometrical phase shift[10]. This is the reason why
the phase shift is not treasured all along. Even in the nonlinear case, the
phase shift per unit distance of the local soliton was found to be
$1/(2kw_{0}^{2})$ [11], where $w_{0}$ is the soliton width, which is the same
order with the result for the linear optical beam. Based on the
phenomenological Gaussian response function, Guo $et$ $al$. predicted that the
phase shift rate per unit propagation distance is
$(w_{m}/w_{0})^{2}/(kw_{0}^{2})$ for the SNSOSs[9], where $w_{m}$ is the width
of the response function. The phase shift is much larger than the local case
since the strong nonlocality means $w_{m}>10w_{0}$ at least. The phase shift
rate in the nematic liquid crystal, the first-found material with the strongly
nonlocal nonlinearity [12], was found to be
$\pi^{1/2}(w_{m}/w_{0})/(2kw_{0}^{2})$[13]. Though this result is slower than
that obtained based on the phenomenological model, it is 10 times faster at
least than the results for the local soliton and the linear beam. It was also
pointed out [14] that $\pi$ phase shift of the signal SNSOS can be obtained
within a very short given distance via adjusting the pump SNSOS power with the
aid of the cross modulation between the SNSOSs. Because, however, the
additional phase shift of the SNSOS is completely covered up by the
geometrical phase shift during their propagation, it is somewhat difficult to
experimentally demonstrate the large phase shift of the SNSOS. Someone even
doubted that whether the conclusion of the large phase shift would be
right[15].
In this Letter we theoretically and experimentally investigate the large phase
shift of the SNSOS in lead glass. Based on the principle of Mach-Zehnder
interferometer, we test and verify the linear modulation of the SNSOS phase by
the power of itself. A $\pi$ phase shift is obtained by changing the soliton
power about 10 mw around the critical power, which demonstrates a high
modulation sensitivity.
The medium we concern is the lead glass with an extremely large range of
nonlocality of the thermal self-focusing type nonlinearity[2, 16, 3, 8]. The
propagation behavior of the light beam in this system is governed by the
coupled equations[2, 16], which are expressed in the cylindrical coordinate
system ($R,\phi,Z$) for $Z$-axis symmetric geometry
$\displaystyle 2ik\frac{\partial A}{\partial
Z}+\frac{1}{R}\frac{\partial}{\partial R}(R\frac{\partial A}{\partial
R})+2k^{2}\frac{\Delta n}{n_{0}}A=0,$ (1a)
$\displaystyle\frac{1}{R}\frac{\partial}{\partial R}(R\frac{\partial
T}{\partial R})=-\frac{\alpha}{\kappa}I(R),$ (1b)
where $\alpha$ and $\kappa$ are respectively the absorption coefficient and
the thermal conductivity, $\Delta n=\beta\Delta T$ with $\beta$ being the
thermo-optical coefficient, and $I(R)=|A(R)|^{2}$. We rewrite Eq. (1) in a
dimensionless form:
$\displaystyle i\partial_{z}a+\frac{1}{r}\frac{\partial}{\partial
r}(r\frac{\partial a}{\partial r})+Na=0,$ (2a)
$\displaystyle\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial
N}{\partial r})=-|a|^{2},$ (2b)
where $r=R/w_{0},z=Z/(2kw_{0}^{2}),a=A/A_{0}$,
$A_{0}^{2}=n_{0}\kappa/(2\alpha\beta k^{2}w_{0}^{4})$ and
$N=2k^{2}w_{0}^{2}\Delta n/n_{0}$.
Following the Snyder’s method [1], the nonlinear index is expanded and only
kept the first two terms of Taylor series:
$N=N^{(0)}-r^{2}N^{(2)}.$ (3)
Assuming the beam of the Gaussian function form
$a=\frac{\sqrt{p_{0}}}{\sqrt{\pi}w(z)}{\rm exp}[i\theta(z)]{\rm
exp}(-\frac{r^{2}}{2w(z)^{2}}),$ (4)
where $p_{0}=\int|a(x^{\prime}-x_{c},y^{\prime})|^{2}dx^{\prime}dy^{\prime}$
is the normalized light power, we can obtain $N^{(0)}$ by directly integrating
Eq. (2b) twice
$N^{(0)}=\frac{p_{0}}{4\pi}[\Gamma(0,\frac{R_{0}^{2}}{w^{2}(z)})+\ln(\frac{R_{0}^{2}}{w^{2}(z)})+\gamma],$
(5)
where $\gamma$ is Euler’s constant which equals to $0.5772156649$, and $R_{0}$
is the diameter of the cross section of the lead glass.
$\theta$ in Eq. (4) is just the phase shift of the beam. We rewrite it into
two terms according to the two terms of the nonlinear index(in Eq. (3)):
$\theta=\theta^{(0)}+\theta^{(2)},$ (6)
where $\theta^{(0)}=N^{(0)}z$ is the zero-order term of the phase shift. By
the method in Ref. [9], inserting Eq. (3) and Eq. (4) into Eq. (2b), the beam
width and the second-order term of the phase shift can be obtained
$\displaystyle w(z)=\sigma+(1-\sigma){\rm cos}(bz)$ (7b)
$\displaystyle\theta^{(2)}=\frac{-2}{2\sigma-1}\\{\frac{(1-\sigma)\sin(bz)}{\sigma+(1-\sigma)\cos(bz)},$
$\displaystyle-\frac{2\sigma}{\sqrt{2\sigma-1}}[\arctan(\sqrt{(2\sigma-1)}\tan\frac{bz}{2})]\\},$
where $\sigma=\sqrt{p_{c}/p_{0}}$, $b=2\sqrt{2}/\sigma^{2}$ with $p_{c}=\pi$
is the critical power for the soliton propagation.
Based on Eq. (5) and Eq. (7b), the phase shift is the function of both the
propagation distance $z$ and the power $p_{0}$. Worthy of note, the zero-order
term of the phase shift in Eq. (5) is related to the size of the lead glass.
This reveals the effect of the nonlocality on the phase shift of the SNSOS. In
lead glass the nonlocality is essentially infinite [2] but cut-off by its
boundary. Therefore it could be predicted that larger size glass should owe
higher phase modulation sensitivity. Figure 1 demonstrates the phase shift of
SNSOS in function of $p_{0}/p_{c}$. We do not provide the numerical result in
the case of $w_{0}/R_{0}=1/300$ because of the computer source available.
Fig. 1: Phase shift of SNSOS versus $p_{0}/p_{c}$. The thick and thin solid
lines are respectively the analytical results in the cases of
$w_{0}/R_{0}=1/300$ and $w_{0}/R_{0}=1/60$, which are the cases in the
following experiment. The dashed line is the numerical result in the case of
$w_{0}/R_{0}=1/60$. Fig. 2: Experimental setup. TA is the tunable attenuator,
$L_{1}$, $L_{2}$, $L_{3}$, $L_{4}$ are the lens.
We carry out the large phase shift experiment in cylindrical lead glass with
two diameters of 15 mm and 3 mm but the same length of 60 mm. The heavily-
doped glass has a high absorption coefficient $\alpha$ of 0.07 ${\rm cm}^{-1}$
and a high refractive index $n_{0}$ of $1.9$. The other parameters are the
same with those in Ref. [2]. The experimental arrangement is detailed in Fig.
2. A double frequency YAG laser(Verdi 12) with the wavelength of 532 nm is
coupled into a Mach-Zehnder interferometer. The signal beam on one arm of the
interferometer is focused by the lens $L_{1}$ onto the lead glass with beam
width of 50 $\mu$ m. The output soliton is imaged by $L_{2}$ onto CCD. The
other arm contains a beam telescope, comprised by $L_{3}$ and $L_{4}$,
adjusted to give a collimated, large diameter beam to act as a phase
reference. Considering the difference of the refractive index between the lead
glass and the air, a time delay is used in the reference optical path to
compensate the optical length. The inset of Fig. 3 shows representative
interference fringes with sharp contrast.
Along with the increase of the SNSOS power, the phase shift of the SNSOS
increases considerably, while the phase of the reference beam keeps fixed.
Therefore the interference fringes move observably. The stars in Fig. 3
designate the centers of a tracked fringe. The equidistant movement of the
star indicates a linear modulation of the SNSOS phase by its power.
Fig. 3: Intensity distributions along the direction perpendicular to the
interference fringes through the center of the fringes. Inset demonstrates the
representative interference fringes between the signal beam and the reference
beam.
Considering the critical power is measured to be 260 mW, we change the input
power from 190 mW to 340 mW by turning the tunable attenuator TA with power
interval of 3.6 mW. Following the procedure in the treatment of the
interference fringes in Fig. 3, we obtain the phase shift in function of the
input power in lead glasses showed in Fig. 4. Since the linear fittings of the
experimental data have slops of 0.33 and 0.29, the $\pi$ phase shifts are
modulated by 9.5 mW power change in big glass bar and 10.5 mW power change in
small glass bar respectively. In both case, the power changes are less than
$5$% of the soliton critical power and therefore the beams almost maintain the
form of solitons when the power is slightly changed. Although the effect of
the diameter of the glass on the phase shift is less than the theoretical
predictions, the modulation sensitivity suggested by the experimental results
is far higher than that predicted by the theoretical curves in Fig. 1. Segev
$et$ $al$. were in the similar situation when they numerically calculated the
elliptic solitons [2] and measured the soliton steering driven by the boundary
force [16]. The nonlinearity in their experiment is higher and more
anisotropic than the calculated thermal response. The biggest steering data
was three times than the theoretical prediction [16]. They presumed an
additional mechanism in lead glass, the birefringence induced by thermal
stress, giving rise to an increased $\Delta n$.
Fig. 4: Phase shift versus the input power in lead glasses with different
diameters. Circles and squares are respectively the data obtained in lead
glass with diameter of 15 mm and 3 mm. Dashed lines are the linear fittings
with slops of 0.33 and 0.29.
In conclusion we investigate the large phase shift of the SNSOS in lead glass.
The experimental result verifies that an output phase shift of $\pi$ can be
linearly modulated by a power change of about 10 mW, which is less than 5% of
the soliton critical power. The effective producing of $\pi$ phase shift is
significant to realize the treatment and control of the optical signal based
on interference principle. Additionally the modulation sensitivity is higher
in bigger size glass bar. This is the manifestation that the large phase shift
of the SNSOS stems essentially from the strong nonlocality.
This research was supported by the National Natural Science Foundation of
China (Grant No. 60908003).
## References
* [1] A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
* [2] C. Rotschild, O. Cohen, O. Manela and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
* [3] C. Rotschild, M. Segev, Z. Y. Xu, Y. V. Kartashov and L. Torner, Opt. Lett. 31, 3312 (2006).
* [4] D. M. Deng and Q. Guo, Opt. Lett. 32, 3206 (2007).
* [5] D. M. Deng, X. Zhao and Q. Guo, J. Opt. Soc. Am. B 24, 2537 (2007).
* [6] W. Królikowski, O. Bang, J. Wyller, Phys. Rev. E 70, 036617 (2004).
* [7] O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer and M. Segev, Phys. Rev. E 73, 015601(R) (2006).
* [8] C. Rotschild, T. Schwartz, O. Cohen and M. Segev, Nat. Photonics 2, 371 (2008).
* [9] Q. Guo, B. Luo, F. Yi, S. Chi, Y. Xie, Phys. Rev. E 69, 016602 (2004).
* [10] H. A. Haus, $Waves$ $and$ $fields$ $in$ $optoelectronics$ (Prentice-Hall, 1984).
* [11] J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, P. W. E. Smith, Opt. Lett. 15, 471 (1999).
* [12] C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
* [13] H. Y. Ren, S. G. Ouyang, Q. Guo, W. Hu and L. G. Cao, J. Opt. A 10, 025102 (2008).
* [14] Y. Q. Xie, Q. Guo, Opt. Quant. Electron. 36, 1335 (2004).
* [15] M. Shen, Q. Wang, J. Shi, P. Hou, and Q. Kong, Phys. Rev. E 73, 056602 (2006).
* [16] B. Alfassi, C. Rotschild, O. Manela, M. Segev, Opt. Lett. 32, 154 (2007).
|
arxiv-papers
| 2011-03-06T04:42:08 |
2024-09-04T02:49:17.483514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qian Shou, Xiang Zhang, Wei Hu and Qi Guo",
"submitter": "Qi Guo",
"url": "https://arxiv.org/abs/1103.1101"
}
|
1103.1120
|
[E-]epal-mechanics [W-]eph-wheel-mi [C-]covariant-transform
# Erlangen Programme at Large 3.2
Ladder Operators in Hypercomplex Mechanics
Vladimir V. Kisil School of Mathematics
University of Leeds
Leeds LS2 9JT
UK kisilv@maths.leeds.ac.uk http://www.maths.leeds.ac.uk/~kisilv/ Dedicated
to the memory of Ian R. Porteous
###### Abstract.
We revise the construction of creation/annihilation operators in quantum
mechanics based on the representation theory of the Heisenberg and symplectic
groups. Besides the standard harmonic oscillator (the elliptic case) we
similarly treat the repulsive oscillator (hyperbolic case) and the free
particle (the parabolic case). The respective hypercomplex numbers turn out to
be handy on this occasion. This provides a further illustration to the
Similarity and Correspondence Principle.
###### Key words and phrases:
Heisenberg group, Kirillov’s method of orbits, geometric quantisation, quantum
mechanics, classical mechanics, Planck constant, dual numbers, double numbers,
hypercomplex, jet spaces, hyperbolic mechanics, interference,
Fock–Segal–Bargmann representation, Schrödinger representation, dynamics
equation, harmonic and unharmonic oscillator, contextual probability,
symplectic group, metaplectic representation, Shale–Weil representation
###### PII:
###### 2000 Mathematics Subject Classification:
Primary 81R05; Secondary 81R15, 22E27, 22E70, 30G35, 43A65.
On leave from Odessa University.
††copyright: ©:
###### Contents
1. 1 Introduction
2. 2 Heisenberg Group and Its Automorphisms
3. 3 Ladder Operators in Quantum Mechanics
1. 3.1 Ladder Operators from the Heisenberg Group
2. 3.2 Symplectic Ladder Operators
4. 4 Ladder Operators for the Hyperbolic Subgroup
1. 4.1 Complex Ladder Operators
2. 4.2 Double Ladder Operators
5. 5 Ladder Operator for the Nilpotent Subgroup
6. 6 Conclusions: Similarity and Correspondence
## 1\. Introduction
Harmonic oscillators are treated in most textbooks on quantum mechanics. This
is efficiently done through creation/annihilation (ladder) operators
[Gazeau09a] [BoyerMiller74a]. The underlying structure is the representation
theory of the the Heisenberg and symplectic groups [Lang85]*§ VI.2
[MTaylor86]*§ 8.2 [Howe80b] [Folland89]. It is also known that quantum
mechanics and field theory can benefit from the introduction of Clifford
algebra-valued group representations [Kisil93c]
[ConstalesFaustinoKrausshar11a] [CnopsKisil97a] [GuentherKuzhel10a].
The dynamics of a harmonic oscillator generates the symplectic transformation
of the phase space of the elliptic type. The respective parabolic and
hyperbolic counterparts are also of interest [Wulfman10a]*§ 3.8 [ATorre08a].
As we will see, they are naturally connected with the respective hypercomplex
numbers.
To make this correspondence explicit we recall that the symplectic group
$\mathrm{Sp}(2)$ [Folland89]*§ 1.2 consists of $2\times 2$ matrices with real
entries and the unit determinant. It is isomorphic to the group
$\mathrm{SL}_{2}(\mathbb{R}{})$ [Lang85] [HoweTan92] [Mazorchuk09a] and
provides linear symplectomorphisms of the two-dimensional phase space. It has
three types of non-isomorphic one-dimensional subgroups represented by:
(1) $\displaystyle K$ $\displaystyle=$
$\displaystyle\left\\{{\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos
t\end{pmatrix}=\exp\begin{pmatrix}0&t\\\ -t&0\end{pmatrix}},\
t\in(-\pi,\pi]\right\\},$ (2) $\displaystyle N$ $\displaystyle=$
$\displaystyle\left\\{{\begin{pmatrix}1&t\\\
0&1\end{pmatrix}=\exp\begin{pmatrix}0&t\\\ 0&0\end{pmatrix},}\
t\in\mathbb{R}{}\right\\},$ (3) $\displaystyle A$ $\displaystyle=$
$\displaystyle\left\\{\begin{pmatrix}e^{t}&0\\\
0&e^{-t}\end{pmatrix}=\exp\begin{pmatrix}t&0\\\ 0&-t\end{pmatrix},\
t\in\mathbb{R}{}\right\\}.$
We will refer to them as elliptic, parabolic and hyperbolic subgroups,
respectively.
On the other hand, there are three non-isomorphic types of commutative,
associative two-dimensional algebras known as complex, dual and double numbers
[Yaglom79]*App. C [LavrentShabat77]*§ 5. They are represented by expressions
$x+\iota y$, where $\iota$ stands for one of the hypercomplex units
$\mathrm{i}$, $\varepsilon$ or $\mathrm{j}$ with the properties:
$\mathrm{i}^{2}=-1,\qquad\varepsilon^{2}=0,\qquad\mathrm{j}^{2}=1.$
These units can also be labelled as elliptic, parabolic and hyperbolic.
In an earlier paper [Kisil10a],we considered representations of the Heisenberg
group which are induced by hypercomplex characters of its centre. The elliptic
case (complex numbers) describesthe traditional framework of quantum
mechanics, of course.
Double-valued representations, with the imaginary unit $\mathrm{j}^{2}=1$, are
atural source of hyperbolic quantum mechanics developed for a while
[Hudson66a, Hudson04a, Khrennikov03a, Khrennikov05a, Khrennikov08a]. The
representation acts on a Krein space with an indefinite inner product
[AzizovIokhvidov71a]. This aroused significant recent interest in connection
with $\mathcal{PT}$–symmetric quantum mechanics [GuentherKuzhel10a]. However,
our approach is different from the classical treatment of Krein spaces: we use
the hyperbolic unit $\mathrm{j}$ and build the hyperbolic analytic function
theory on its own basis [Kisil97c, Kisil11c]. In the traditional approach, the
indefinite metric is mapped to a definite inner product through an auxiliary
operators.
The representation with values in dual numbers provides a convenient
description of the classical mechanics. To this end we do not take any sort of
semiclassical limit, rather the nilpotency of the imaginary unit
($\varepsilon^{2}=0$) performs the task. This removes the vicious necessity to
consider the Planck _constant_ tending to zero. Mixing this with complex
numbers we get a convenient tool for modelling the interaction between quantum
and classical systems [Kisil05c, Kisil09b].
Our construction [Kisil10a] provides three different types of dynamics and
also generates the respective rules for addition of probabilities. In this
paper we analyse the three types of dynamics produced by transformations (1–3)
from the symplectic group $\mathrm{Sp}(2)$ by means of ladder operators. As a
result we obtain further illustrations to the following:
###### Principle 1 (Similarity and Correspondence).
[Kisil09c]*Principle LABEL:W-pr:simil-corr-principle
1. (1)
Subgroups $K$, $N$ and $A$ play a similar rôle in the structure of the group
$\mathrm{Sp}(2)$ and its representations.
2. (2)
The subgroups shall be swapped simultaneously with the respective replacement
of hypercomplex unit $\iota$.
Here the two parts are interrelated: without a swap of imaginary units there
can be no similarity between different subgroups.
In this paper we work with the simplest case of a particle with only one
degree of freedom. Higher dimensions and the respective group of
symplectomorphisms $\mathrm{Sp}(2n)$ may require consideration of Clifford
algebras [Porteous95].
## 2\. Heisenberg Group and Its Automorphisms
Let $(s,x,y)$, where $s$, $x$, $y\in\mathbb{R}{}$, be an element of the one-
dimensional Heisenberg group $\mathbb{H}^{1}{}$ [Folland89, Howe80b].
Consideration of the general case of $\mathbb{H}^{n}{}$ will be similar, but
is beyond the scope of present paper. The group law on $\mathbb{H}^{1}{}$ is
given as follows:
(4)
$\textstyle(s,x,y)\cdot(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}\omega(x,y;x^{\prime},y^{\prime}),x+x^{\prime},y+y^{\prime}),$
where the non-commutativity is due to $\omega$—the _symplectic form_ on
$\mathbb{R}^{2n}{}$ [Arnold91]*§ 37:
(5) $\omega(x,y;x^{\prime},y^{\prime})=xy^{\prime}-x^{\prime}y.$
The Heisenberg group is a non-commutative Lie group. The left shifts
(6) $\Lambda(g):f(g^{\prime})\mapsto f(g^{-1}g^{\prime})$
act as a representation of $\mathbb{H}^{1}{}$ on a certain linear space of
functions. For example, an action on $L_{2}{}(\mathbb{H}{},dg)$ with respect
to the Haar measure $dg=ds\,dx\,dy$ is the _left regular_ representation,
which is unitary.
The Lie algebra $\mathfrak{h}^{n}$ of $\mathbb{H}^{1}{}$ is spanned by
left-(right-)invariant vector fields
(7) $\textstyle S^{l(r)}=\pm{\partial_{s}},\quad
X^{l(r)}=\pm\partial_{x}-\frac{1}{2}y{\partial_{s}},\quad
Y^{l(r)}=\pm\partial_{y}+\frac{1}{2}x{\partial_{s}}$
on $\mathbb{H}^{1}{}$ with the Heisenberg _commutator relation_
(8) $[X^{l(r)},Y^{l(r)}]=S^{l(r)}$
and all other commutators vanishing. We will sometime omit the superscript $l$
for left-invariant field.
The group of outer automorphisms of $\mathbb{H}^{1}{}$, which trivially acts
on the centre of $\mathbb{H}^{1}{}$, is the symplectic group $\mathrm{Sp}(2)$
defined in the precious section. It is the group of symmetries of the
symplectic form $\omega$ [Folland89]*Thm. 1.22 [Howe80a]*p. 830. The
symplectic group is isomorphic to $\mathrm{SL}_{2}(\mathbb{R}{})$ [Lang85]
[MTaylor86]*Ch. 8. The explicit action of $\mathrm{Sp}(2)$ on the Heisenberg
group is:
(9) $g:h=(s,x,y)\mapsto g(h)=(s,x^{\prime},y^{\prime}),$
where
$g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{Sp}(2),\quad\text{ and
}\quad\begin{pmatrix}x^{\prime}\\\
y^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix}.$
The Shale–Weil theorem [Folland89]*§ 4.2 [Howe80a]*p. 830 states that any
representation ${\rho_{\hslash}}$ of the Heisenberg groups generates a unitary
_oscillator_ (or _metaplectic_) representation ${\rho^{\text{SW}}_{\hslash}}$
of the $\widetilde{\mathrm{Sp}}(2)$, the two-fold cover of the symplectic
group [Folland89]*Thm. 4.58.
We can consider the semidirect product
$G=\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$ with the standard group
law:
$(h,g)*(h^{\prime},g^{\prime})=(h*g(h^{\prime}),g*g^{\prime}),\qquad\text{where
}h,h^{\prime}\in\mathbb{H}^{1}{},\quad
g,g^{\prime}\in\widetilde{\mathrm{Sp}}(2),$
and the stars denote the respective group operations while the action
$g(h^{\prime})$ is defined as the composition of the projection map
$\widetilde{\mathrm{Sp}}(2)\rightarrow{\mathrm{Sp}}(2)$ and the action (9).
This group is sometimes called the Schrödinger group, and it is known as the
maximal kinematical invariance group of both the free Schrödinger equation and
the quantum harmonic oscillator [Niederer73a]. This group is of interest not
only in quantum mechanics but also in optics [ATorre10a, ATorre08a].
Consider the Lie algebra $\mathfrak{sp}_{2}$ of the group $\mathrm{Sp}(2)$.
Pick up the following basis in $\mathfrak{sp}_{2}$ [MTaylor86]*§ 8.1:
$A=\frac{1}{2}\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix},\quad B=\frac{1}{2}\
\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad Z=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}.$
The commutation relations between the elements are:
(10) $[Z,A]=2B,\qquad[Z,B]=-2A,\qquad[A,B]=\textstyle-\frac{1}{2}Z.$
Vectors $Z$, $B+Z/2$ and $-A$ are generators of the one-parameter subgroups
$K$, $N$ and $A$ (1–3) respectively.
Furthermore, we can consider the basis $\\{S,X,Y,A,B,Z\\}$ of the Lie algebra
$\mathfrak{g}$ of the Lie group
$G=\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$. All non-zero
commutators besides those already listed in (8) and (10) are:
(11) $\displaystyle[A,X]$ $\displaystyle=\textstyle\frac{1}{2}X,$
$\displaystyle[B,X]$ $\displaystyle=\textstyle-\frac{1}{2}Y,$
$\displaystyle[Z,X]$ $\displaystyle=Y;$ (12) $\displaystyle[A,Y]$
$\displaystyle=\textstyle-\frac{1}{2}Y,$ $\displaystyle[B,Y]$
$\displaystyle=\textstyle-\frac{1}{2}X,$ $\displaystyle[Z,Y]$
$\displaystyle=-X.$
The Shale–Weil theorem allows us to expand any representation
${\rho_{\hslash}}$ of the Heisenberg group to the representation
${\tilde{\rho}_{\hslash}}={\rho_{\hslash}}\oplus{\rho^{\text{SW}}_{\hslash}}$
of group $G$.
###### Example 2.
Let ${\rho_{\hslash}}$ be the Schrödinger representation [Folland89]*§ 1.3 of
$\mathbb{H}^{1}{}$ in $L_{2}{}(\mathbb{R}{})$, that is [Kisil10a]*
(LABEL:E-eq:schroedinger-rep-conf):
(13)
$[{\rho_{\chi}}(s,x,y)f\,](q)=e^{2\pi\mathrm{i}\hslash(s-xy/2)+2\pi\mathrm{i}xq}\,f(q-\hslash
y).$
Thus the action of the derived representation on the Lie algebra
$\mathfrak{h}_{1}$ is:
(14)
${\rho_{\hslash}}(X)=2\pi\mathrm{i}q,\qquad{\rho_{\hslash}}(Y)=-\hslash\frac{d}{dq},\qquad{\rho_{\hslash}}(S)=2\pi\mathrm{i}\hslash
I.$
Then the associated Shale–Weil representation of $\mathrm{Sp}(2)$ in
$L_{2}{}(\mathbb{R}{})$ has the derived action, cf. [ATorre08a]*(2.2)
[Folland89]*§ 4.3:
(15)
${\rho^{\text{SW}}_{\hslash}}(A)=-\frac{q}{2}\frac{d}{dq}-\frac{1}{4},\quad{\rho^{\text{SW}}_{\hslash}}(B)=-\frac{\hslash\mathrm{i}}{8\pi}\frac{d^{2}}{dq^{2}}-\frac{\pi\mathrm{i}q^{2}}{2\hslash},\quad{\rho^{\text{SW}}_{\hslash}}(Z)=\frac{\hslash\mathrm{i}}{4\pi}\frac{d^{2}}{dq^{2}}-\frac{\pi\mathrm{i}q^{2}}{\hslash}.$
We can verify commutators (8) and (10–12) for operators (14–15). It is also
obvious that in this representation the following algebraic relations hold:
(16) $\displaystyle\qquad{\rho^{\text{SW}}_{\hslash}}(A)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{4\pi\hslash}({\rho_{\hslash}}(X){\rho_{\hslash}}(Y)-{\textstyle\frac{1}{2}}{\rho_{\hslash}}(S))=\frac{\mathrm{i}}{8\pi\hslash}({\rho_{\hslash}}(X){\rho_{\hslash}}(Y)+{\rho_{\hslash}}(Y){\rho_{\hslash}}(X)),$
(17) $\displaystyle{\rho^{\text{SW}}_{\hslash}}(B)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{8\pi\hslash}({\rho_{\hslash}}(X)^{2}-{\rho_{\hslash}}(Y)^{2}),$
(18) $\displaystyle{\rho^{\text{SW}}_{\hslash}}(Z)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{4\pi\hslash}({\rho_{\hslash}}(X)^{2}+{\rho_{\hslash}}(Y)^{2}).$
Thus it is common in quantum optics to name $\mathfrak{g}$ as a Lie algebra
with quadratic generators, see [Gazeau09a]*§ 2.2.4.
Note that ${\rho^{\text{SW}}_{\hslash}}(Z)$ is the Hamiltonian of the harmonic
oscillator (up to a factor). Then we can consider
${\rho^{\text{SW}}_{\hslash}}(B)$ as the Hamiltonian of a repulsive
(hyperbolic) oscillator. The operator
${\rho^{\text{SW}}_{\hslash}}(B-Z/2)=\frac{\hslash\mathrm{i}}{4\pi}\frac{d^{2}}{dq^{2}}$
is the parabolic analog. A graphical representation of all three
transformations is given in Fig. 1, and a further discussion of these
Hamiltonians can be found in [Wulfman10a]*§ 3.8.
Figure 1. Three types (elliptic, parabolic and hyperbolic) of linear
symplectic transformations on the plane
An important observation, which is often missed, is that the three linear
symplectic transformations are unitary rotations in the corresponding
hypercomplex algebra. This means, that the symplectomorphisms generated by
operators $Z$, $B-Z/2$, $B$ within time $t$ coincide with the multiplication
of hypercomplex number $q+\iota p$ by $e^{\iota t}$ [Kisil09c]*§ 3, which is
just another illustration of the Similarity and Correspondence Principle 1.
###### Example 3.
There are many advantages of considering representations of the Heisenberg
group on the phase space [Howe80b]*§ 1.7 [Folland89]*§ 1.6 [deGosson08a]. A
convenient expression for Fock–Segal–Bargmann (FSB) representation on the
phase space is [Kisil10a]*(LABEL:E-eq:stone-inf):
(19) $\textstyle[{\rho_{F}}(s,x,y)f](q,p)=e^{-2\pi\mathrm{i}(\hslash
s+qx+py)}f\left(q-\frac{\hslash}{2}y,p+\frac{\hslash}{2}x\right).$
Then the derived representation of $\mathfrak{h}_{1}$ is:
(20)
$\textstyle{\rho_{F}}(X)=-2\pi\mathrm{i}q+\frac{\hslash}{2}\partial_{p},\qquad{\rho_{F}}(Y)=-2\pi\mathrm{i}p-\frac{\hslash}{2}\partial_{q},\qquad{\rho_{F}}(S)=-2\pi\mathrm{i}\hslash
I.$
This produces the derived form of the Shale–Weil representation:
(21)
$\textstyle{\rho^{\text{SW}}_{F}}(A)=\frac{1}{2}\left(q\partial_{q}-p\partial_{p}\right),\quad{\rho^{\text{SW}}_{F}}(B)=-\frac{1}{2}\left(p\partial_{q}+q\partial_{p}\right),\quad{\rho^{\text{SW}}_{F}}(Z)=p\partial_{q}-q\partial_{p}.$
Note that this representation does not contain the parameter $\hslash$, unlike
the equivalent representation (15). Thus the FSB model explicitly shows the
equivalence of ${\rho^{\text{SW}}_{\hslash_{1}}}$ and
${\rho^{\text{SW}}_{\hslash_{2}}}$ if $\hslash_{1}\hslash_{2}>0$
[Folland89]*Thm. 4.57.
As we will also see below, the FSB-type representations in hypercomplex
numbers produce almost the same Shale–Weil representations.
## 3\. Ladder Operators in Quantum Mechanics
Let ${\rho}$ be a representation of the group
$G=\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$ in a space $V$. Consider
the derived representation of the Lie algebra $\mathfrak{g}$ [Lang85]*§ VI.1
and denote $\tilde{X}={\rho}(X)$ for $X\in\mathfrak{g}$. To see the structure
of the representation ${\rho}$ we can decompose the space $V$ into eigenspaces
of the operator $\tilde{X}$ for some $X\in\mathfrak{g}$. The canonical example
is the Taylor series in complex analysis.
We are going to consider three cases corresponding to three non-isomorphic
subgroups (1–3) of $\mathrm{Sp}(2)$ starting from the compact case. Let $H=Z$
be a generator of the compact subgroup $K$. Corresponding symplectomorphisms
(9) of the phase space are given by orthogonal rotations with matrices
$\begin{pmatrix}\cos t&\sin t\\\ -sint&\cos t\end{pmatrix}$. The Shale–Weil
representation (15) coincides with the Hamiltonian of the harmonic oscillator.
Since this is a double cover of a compact group, the corresponding eigenspaces
$\tilde{Z}v_{k}=\mathrm{i}kv_{k}$ are parametrised by a half-integer
$k\in\mathbb{Z}{}/2$. Explicitly for a half-integer $k$:
(22)
$v_{k}(q)=H_{k+\frac{1}{2}}\left(\sqrt{\frac{2\pi}{\hslash}}q\right)e^{-\frac{\pi}{\hslash}q^{2}},$
where $H_{k}$ is the Hermite polynomial [Folland89]*§ 1.7
[ErdelyiMagnusII]*8.2(9).
From the point of view of quantum mechanics and the representation theory
(which may be the same), it is beneficial to introduce the ladder operators
$L^{\\!\pm}$, known as _creation/annihilation_ in quantum mechanics
[Folland89]*p. 49 or _raising/lowering_ in representation theory [Lang85]*§
VI.2 [MTaylor86]*§ 8.2 [BoyerMiller74a]. They are defined by the following
commutation relations:
(23) $[\tilde{Z},L^{\\!\pm}]=\lambda_{\pm}L^{\\!\pm}.$
In other words, $L^{\\!\pm}$ are eigenvectors for operators
$\mathop{\operator@font ad}\nolimits Z$ of the adjoint representation of
$\mathfrak{g}$ [Lang85]*§ VI.2.
###### Remark 4.
The existence of such ladder operators follows from the general properties of
Lie algebras if the Hamiltonian belongs to a Cartan subalgebra. This is the
case for vectors $Z$ and $B$, which are the only two non-isomorphic types of
Cartan subalgebras in $\mathfrak{sl}_{2}$. However, the third case considered
in this paper, the parabolic vector $B+Z/2$, does not belong to a Cartan
subalgebra, yet a sort of ladder operators is still possible with dual number
coefficients. Moreover, for the hyperbolic vector $B$, besides the standard
ladder operators an additional pair with double number coefficients will also
be described.
From the commutators (23) we deduce that if $v_{k}$ is an eigenvector of
$\tilde{Z}$ then $L^{\\!+}v_{k}$ is an eigenvector as well:
(24) $\displaystyle\tilde{Z}(L^{\\!+}v_{k})$ $\displaystyle=$
$\displaystyle(L^{\\!+}\tilde{Z}+\lambda_{+}L^{\\!+})v_{k}=L^{\\!+}(\tilde{Z}v_{k})+\lambda_{+}L^{\\!+}v_{k}=\mathrm{i}kL^{\\!+}v_{k}+\lambda_{+}L^{\\!+}v_{k}$
$\displaystyle=$ $\displaystyle(\mathrm{i}k+\lambda_{+})L^{\\!+}v_{k}.$
Thus the action of ladder operators on the respective eigenspaces $V_{k}$ can
be visualised by the diagram:
(25)
$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,V_{\mathrm{i}k-\lambda}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,V_{\mathrm{i}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,V_{\mathrm{i}k+\lambda}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$
There are two ways to search for ladder operators: in (complexified) Lie
algebras $\mathfrak{h}_{1}$ and $\mathfrak{sp}_{2}$. We will consider them in
a sequence.
### 3.1. Ladder Operators from the Heisenberg Group
Assuming $L^{\\!+}=a\tilde{X}+b\tilde{Y}$ we obtain from the relations (11–12)
and (23) the linear equations with unknown $a$ and $b$:
$a=\lambda_{+}b,\qquad-b=\lambda_{+}a.$
The equations have a solution if and only if $\lambda_{+}^{2}+1=0$, and the
raising/lowering operators are $L^{\\!\pm}=\tilde{X}\mp\mathrm{i}\tilde{Y}$.
###### Remark 5.
Here we have an interesting asymmetric response: due to the structure of the
semidirect product $\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$ it is
the symplectic group which acts on $\mathbb{H}^{1}{}$, not vise versa.
However, the Heisenberg group has a weak action in the opposite direction: it
shifts eigenfunctions of $\mathrm{Sp}(2)$.
In the Schrödinger representation (14) the ladder operators are
(26)
${\rho_{\hslash}}(L^{\\!\pm})=2\pi\mathrm{i}q\pm\mathrm{i}\hslash\frac{d}{dq}.$
The standard treatment of the harmonic oscillator in quantum mechanics, which
can be found in many textbooks, e.g. [Folland89]*§ 1.7 [Gazeau09a]*§ 2.2.3, is
as follows. The vector $v_{-1/2}(q)=e^{-\pi q^{2}/\hslash}$ is an eigenvector
of $\tilde{Z}$ with the eigenvalue $-\frac{\mathrm{i}}{2}$. In addition
$v_{-1/2}$ is annihilated by $L^{\\!+}$. Thus the chain (25) terminates to the
right and the complete set of eigenvectors of the harmonic oscillator
Hamiltonian is presented by $(L^{\\!-})^{k}v_{-1/2}$ with $k=0,1,2,\ldots$.
We can make a wavelet transform generated by the Heisenberg group with the
mother wavelet $v_{-1/2}$, and the image will be the Fock–Segal–Bargmann (FSB)
space [Howe80b] [Folland89]*§ 1.6. Since $v_{-1/2}$ is the null solution of
$L^{\\!+}=\tilde{X}-\mathrm{i}\tilde{Y}$, then by the general result
[Kisil10c]*Cor. LABEL:C-co:cauchy-riemann the image of the wavelet transform
will be null-solutions of the corresponding linear combination of the Lie
derivatives (7):
(27)
$D=\overline{X^{r}-\mathrm{i}Y^{r}}=(\partial_{x}+\mathrm{i}\partial_{y})-\pi\hslash(x-\mathrm{i}y),$
which turns out to be the Cauchy–Riemann equation on a weighted FSB-type
space.
### 3.2. Symplectic Ladder Operators
We can also look for ladder operators within the Lie algebra
$\mathfrak{sp}_{2}$, see [Kisil09c]*§ 8. Assuming
$L_{2}^{\\!+}=a\tilde{A}+b\tilde{B}+c\tilde{Z}$ from the relations (10) and
defining condition (23) we obtain the linear equations with unknown $a$, $b$
and $c$:
$c=0,\qquad 2a=\lambda_{+}b,\qquad-2b=\lambda_{+}a.$
The equations have a solution if and only if $\lambda_{+}^{2}+4=0$, and the
raising/lowering operators are
$L_{2}^{\\!\pm}=\pm\mathrm{i}\tilde{A}+\tilde{B}$. In the Shale–Weil
representation (15) they turn out to be:
(28)
$L_{2}^{\\!\pm}=\pm\mathrm{i}\left(\frac{q}{2}\frac{d}{dq}+\frac{1}{4}\right)-\frac{\hslash\mathrm{i}}{8\pi}\frac{d^{2}}{dq^{2}}-\frac{\pi\mathrm{i}q^{2}}{2\hslash}=-\frac{\mathrm{i}}{8\pi\hslash}\left(\mp
2\pi q+\hslash\frac{d}{dq}\right)^{2}.$
Since this time $\lambda_{+}=2\mathrm{i}$ the ladder operators
$L_{2}^{\\!\pm}$ produce a shift on the diagram (25) twice bigger than the
operators $L^{\\!\pm}$ from the Heisenberg group. After all, this is not
surprising since from the explicit representations (26) and (28) we get:
$L_{2}^{\\!\pm}=-\frac{\mathrm{i}}{8\pi\hslash}(L^{\\!\pm})^{2}.$
## 4\. Ladder Operators for the Hyperbolic Subgroup
Consider the case of the Hamiltonian $H=2B$, which is a repulsive (hyperbolic)
harmonic oscillator [Wulfman10a]*§ 3.8. The corresponding one-dimensional
subgroup of symplectomorphisms produces hyperbolic rotations of the phase
space. The eigenvectors $v_{\mu}$ of the operator
${\rho^{\text{SW}}_{\hslash}}(2B)v_{\nu}=-\mathrm{i}\left(\frac{\hslash}{4\pi}\frac{d^{2}}{dq^{2}}+\frac{\pi
q^{2}}{\hslash}\right)v_{\nu}=\mathrm{i}\nu v_{\nu},$
are Weber–Hermite (or parabolic cylinder) functions
$v_{\nu}=D_{\nu-\frac{1}{2}}\left(\pm
2e^{\mathrm{i}\frac{\pi}{4}}\sqrt{\frac{\pi}{\hslash}}q\right)$, see
[ErdelyiMagnusII]*§ 8.2 [SrivastavaTuanYakubovich00a] for fundamentals of
Weber–Hermite functions and [ATorre08a] for further illustrations and
applications in optics.
The corresponding one-parameter group is not compact and the eigenvalues of
the operator $2\tilde{B}$ are not restricted by any integrality condition, but
the raising/lowering operators are still important [HoweTan92]*§ II.1
[Mazorchuk09a]*§ 1.1. We again seek solutions in two subalgebras
$\mathfrak{h}_{1}$ and $\mathfrak{sp}_{2}$ separately. However, the additional
options will be provided by a choice of the number system: either complex or
double.
### 4.1. Complex Ladder Operators
Assuming $L_{h}^{\\!+}=a\tilde{X}+b\tilde{Y}$ from the commutators (11–12), we
obtain the linear equations:
(29) $-a=\lambda_{+}b,\qquad-b=\lambda_{+}a.$
The equations have a solution if and only if $\lambda_{+}^{2}-1=0$. Taking the
real roots $\lambda=\pm 1$ we obtain that the raising/lowering operators are
$L_{h}^{\\!\pm}=\tilde{X}\mp\tilde{Y}$. In the Schrödinger representation (14)
the ladder operators are
(30) $L_{h}^{\\!\pm}=2\pi\mathrm{i}q\pm\hslash\frac{d}{dq}.$
The null solutions
$v_{\pm\frac{1}{2}}(q)=e^{\pm\frac{\pi\mathrm{i}}{\hslash}q^{2}}$ to operators
${\rho_{\hslash}}(L^{\\!\pm})$ are also eigenvectors of the Hamiltonian
${\rho^{\text{SW}}_{\hslash}}(2B)$ with the eigenvalue $\pm\frac{1}{2}$.
However the important distinction from the elliptic case is, that they are not
square-integrable on the real line anymore.
We can also look for ladder operators within the $\mathfrak{sp}_{2}$, that is
in the form $L_{2h}^{\\!+}=a\tilde{A}+b\tilde{B}+c\tilde{Z}$ for the
commutator $[2\tilde{B},L_{h}^{\\!+}]=\lambda L_{h}^{\\!+}$. We will get the
system:
$4c=\lambda a,\qquad b=0,\qquad a=\lambda c.$
A solution again exists if and only if $\lambda^{2}=4$. Within complex numbers
we get only the values $\lambda=\pm 2$ with the ladder operators
$L_{2h}^{\\!\pm}=\pm 2\tilde{A}+\tilde{Z}/2$, see [HoweTan92]*§ II.1
[Mazorchuk09a]*§ 1.1. Each indecomposable $\mathfrak{h}_{1}$\- or
$\mathfrak{sp}_{2}$-module is formed by a one-dimensional chain of eigenvalues
with a transitive action of ladder operators $L_{h}^{\\!\pm}$ or
$L_{2h}^{\\!\pm}$ respectively. And we again have a quadratic relation between
the ladder operators:
$L_{2h}^{\\!\pm}=\frac{\mathrm{i}}{4\pi\hslash}(L_{h}^{\\!\pm})^{2}.$
### 4.2. Double Ladder Operators
There are extra possibilities in in the context of hyperbolic quantum
mechanics [Khrennikov03a] [Khrennikov05a] [Khrennikov08a]. Here we use the
representation of $\mathbb{H}^{1}{}$ induced by a hyperbolic character
$e^{\mathrm{j}ht}=\cosh(ht)+\mathrm{j}\sinh(ht)$, see
[Kisil10a]*(LABEL:E-eq:schroedinger-rep-conf-hyp), and obtain the hyperbolic
representation of $\mathbb{H}^{1}{}$, cf. (13):
(31)
$[{\rho^{\mathrm{j}}_{h}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{\mathrm{j}h(s^{\prime}-x^{\prime}y^{\prime}/2)+\mathrm{j}x^{\prime}q}\,\hat{f}(q-hy^{\prime}).$
The corresponding derived representation is
(32)
${\rho^{\mathrm{j}}_{h}}(X)=\mathrm{j}q,\qquad{\rho^{\mathrm{j}}_{h}}(Y)=-h\frac{d}{dq},\qquad{\rho^{\mathrm{j}}_{h}}(S)=\mathrm{j}hI.$
Then the associated Shale–Weil derived representation of $\mathfrak{sp}_{2}$
in the Schwartz space $S{}(\mathbb{R}{})$ is, cf. (15):
(33)
${\rho^{\text{SW}}_{h}}(A)=-\frac{q}{2}\frac{d}{dq}-\frac{1}{4},\quad{\rho^{\text{SW}}_{h}}(B)=\frac{\mathrm{j}h}{4}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{4h},\quad{\rho^{\text{SW}}_{h}}(Z)=-\frac{\mathrm{j}h}{2}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{2h}.$
Note that ${\rho^{\text{SW}}_{h}}(B)$ now generates a usual harmonic
oscillator, not the repulsive one like ${\rho^{\text{SW}}_{\hslash}}(B)$ in
(15). However, the expressions in the quadratic algebra are still the same (up
to a factor), cf. (16–18):
(34) $\displaystyle\qquad{\rho^{\text{SW}}_{h}}(A)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{j}}{2h}({\rho^{\mathrm{j}}_{h}}(X){\rho^{\mathrm{j}}_{h}}(Y)-{\textstyle\frac{1}{2}}{\rho^{\mathrm{j}}_{h}}(S))=-\frac{\mathrm{j}}{4h}({\rho^{\mathrm{j}}_{h}}(X){\rho^{\mathrm{j}}_{h}}(Y)+{\rho^{\mathrm{j}}_{h}}(Y){\rho^{\mathrm{j}}_{h}}(X)),$
(35) $\displaystyle{\rho^{\text{SW}}_{h}}(B)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{j}}{4h}({\rho^{\mathrm{j}}_{h}}(X)^{2}-{\rho^{\mathrm{j}}_{h}}(Y)^{2}),$
(36) $\displaystyle{\rho^{\text{SW}}_{h}}(Z)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{j}}{2h}({\rho^{\mathrm{j}}_{h}}(X)^{2}+{\rho^{\mathrm{j}}_{h}}(Y)^{2}).$
This is due to the Principle 1 of similarity and correspondence: we can swap
operators $Z$ and $B$ with simultaneous replacement of hypercomplex units
$\mathrm{i}$ and $\mathrm{j}$.
The eigenspace of the operator $2{\rho^{\text{SW}}_{h}}(B)$ with an eigenvalue
$\mathrm{j}\nu$ are spanned by the Weber–Hermite functions
$D_{-\nu-\frac{1}{2}}\left(\pm\sqrt{\frac{2}{h}}x\right)$, see
[ErdelyiMagnusII]*§ 8.2. Functions $D_{\nu}$ are generalisations of the Hermit
functions (22).
The compatibility condition for a ladder operator within the Lie algebra
$\mathfrak{h}_{1}$ will be (29) as before, since it depends only on the
commutators (11–12). Thus we still have the set of ladder operators
corresponding to values $\lambda=\pm 1$:
$L_{h}^{\\!\pm}=\tilde{X}\mp\tilde{Y}=\mathrm{j}q\pm h\frac{d}{dq}.$
Admitting double numbers, we have an extra way to satisfy $\lambda^{2}=1$ in
(29) with values $\lambda=\pm\mathrm{j}$. Then there is an additional pair of
hyperbolic ladder operators, which are identical (up to factors) to (26):
$L_{\mathrm{j}}^{\\!\pm}=\tilde{X}\mp\mathrm{j}\tilde{Y}=\mathrm{j}q\pm\mathrm{j}h\frac{d}{dq}.$
Pairs $L_{h}^{\\!\pm}$ and $L_{\mathrm{j}}^{\\!\pm}$ shift eigenvectors in the
“orthogonal” directions changing their eigenvalues by $\pm 1$ and
$\pm\mathrm{j}$. Therefore an indecomposable $\mathfrak{sp}_{2}$-module can be
parametrised by a two-dimensional lattice of eigenvalues in double numbers,
see Table 1.
$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!+}}$$\textstyle{\,V_{(n-1)+\mathrm{j}(k-1)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{n+\mathrm{j}(k-1)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{(n+1)+\mathrm{j}(k-1)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!+}}$$\textstyle{\,V_{(n-1)+\mathrm{j}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{n+\mathrm{j}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{(n+1)+\mathrm{j}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!+}}$$\textstyle{\,V_{(n-1)+\mathrm{j}(k+1)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{n+\mathrm{j}(k+1)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{(n+1)+\mathrm{j}(k+1)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$
Table 1. The action of hyperbolic ladder operators on a 2D lattice of
eigenspaces. Operators $L_{h}^{\\!\pm}$ move the eigenvalues by $1$, making
shifts in the horizontal direction. Operators $L_{\mathrm{j}}^{\\!\pm}$ change
the eigenvalues by $\mathrm{j}$, shown as vertical shifts.
The following functions
$\displaystyle v_{\frac{1}{2}}^{\pm h}(q)$ $\displaystyle=$ $\displaystyle
e^{\mp\mathrm{j}q^{2}/(2h)}=\cosh\frac{q^{2}}{2h}\mp\mathrm{j}\sinh\frac{q^{2}}{2h},$
$\displaystyle v_{\frac{1}{2}}^{\pm\mathrm{j}}(q)$ $\displaystyle=$
$\displaystyle e^{\mp q^{2}/(2h)}$
are null solutions to the operators $L_{h}^{\\!\pm}$ and
$L_{\mathrm{j}}^{\\!\pm}$, respectively. They are also eigenvectors of
$2{\rho^{\text{SW}}_{h}}(B)$ with eigenvalues $\mp\frac{\mathrm{j}}{2}$ and
$\mp\frac{1}{2}$ respectively. If these functions are used as mother wavelets
for the wavelet transforms generated by the Heisenberg group, then the image
space will consist of the null-solutions of the following differential
operators, see [Kisil10c]*Cor. LABEL:C-co:cauchy-riemann:
$\textstyle
D_{h}=\overline{X^{r}-Y^{r}}=(\partial_{x}-\partial_{y})+\frac{h}{2}(x+y),\qquad
D_{\mathrm{j}}=\overline{X^{r}-\mathrm{j}Y^{r}}=(\partial_{x}+\mathrm{j}\partial_{y})-\frac{h}{2}(x-\mathrm{j}y),$
for $v_{\frac{1}{2}}^{\pm h}$ and $v_{\frac{1}{2}}^{\pm\mathrm{j}}$,
respectively. This is again in line with the classical result (27). However
annihilation of the eigenvector by a ladder operator does not mean that the
part of the 2D-lattice becomes void, since it can be reached via alternative
routes. Instead of multiplication by a zero, as it happens in the elliptic
case, a half-plane of eigenvalues will be multiplied by the divisors of zero
$1\pm\mathrm{j}$.
We can also search ladder operators within the algebra $\mathfrak{sp}_{2}$ and
admitting double numbers we will again find two sets of them [Kisil09c]*§
LABEL:W-sec:correspondence:
$\displaystyle L_{2h}^{\\!\pm}$ $\displaystyle=$
$\displaystyle\pm\tilde{A}+\tilde{Z}/2=\mp\frac{q}{2}\frac{d}{dq}\mp\frac{1}{4}-\frac{\mathrm{j}h}{4}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{4h}=-\frac{\mathrm{j}}{4h}(L_{h}^{\\!\pm})^{2},$
$\displaystyle L_{2\mathrm{j}}^{\\!\pm}$ $\displaystyle=$
$\displaystyle\pm\mathrm{j}\tilde{A}+\tilde{Z}/2=\mp\frac{\mathrm{j}q}{2}\frac{d}{dq}\mp\frac{\mathrm{j}}{4}-\frac{\mathrm{j}h}{4}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{4h}=-\frac{\mathrm{j}}{4h}(L_{\mathrm{j}}^{\\!\pm})^{2}.$
Again the operators $L_{2h}^{\\!\pm}$ and $L_{2h}^{\\!\pm}$ produce double
shifts in the orthogonal directions on the same two-dimensional lattice in
Tab. 1.
## 5\. Ladder Operator for the Nilpotent Subgroup
Finally, we look for ladder operators for the Hamiltonian
$\tilde{B}+\tilde{Z}/2$ or, equivalently, $-\tilde{B}+\tilde{Z}/2$. It can be
identified with a free particle [Wulfman10a]*§ 3.8.
We can look for ladder operators in the representation (14–15) within the Lie
algebra $\mathfrak{h}_{1}$ in the form
$L_{\varepsilon}^{\\!\pm}=a\tilde{X}+b\tilde{Y}$. This is possible if and only
if
(37) $-b=\lambda a,\quad 0=\lambda b.$
The compatibility condition $\lambda^{2}=0$ implies $\lambda=0$ within complex
numbers. However, such a “ladder” operator produces only the zero shift on the
eigenvectors, cf. (24).
Another possibility appears if we consider the representation of the
Heisenberg group induced by dual-valued characters. On the configurational
space such a representation is [Kisil10a]*(LABEL:E-eq:schroedinger-rep-conf-
par):
(38)
$[{\rho^{\varepsilon}_{\chi}}(s,x,y)f](q)=e^{2\pi\mathrm{i}xq}\left(\left(1-\varepsilon
h(s-{\textstyle\frac{1}{2}}xy)\right)f(q)+\frac{\varepsilon
hy}{2\pi\mathrm{i}}f^{\prime}(q)\right).$
The corresponding derived representation of $\mathfrak{h}_{1}$ is
(39)
${\rho^{p}_{h}}(X)=2\pi\mathrm{i}q,\qquad{\rho^{p}_{h}}(Y)=\frac{\varepsilon
h}{2\pi\mathrm{i}}\frac{d}{dq},\qquad{\rho^{p}_{h}}(S)=-\varepsilon hI.$
However the Shale–Weil extension generated by this representation is
inconvenient. It is better to consider the FSB–type parabolic representation
[Kisil10a]*(LABEL:E-eq:dual-repres) on the phase space induced by the same
dual-valued character, cf. (19):
(40)
$[{\rho^{\varepsilon}_{h}}(s,x,y)f](q,p)=e^{-2\pi\mathrm{i}(xq+yp)}(f(q,p)+\varepsilon
h(sf(q,p)+\frac{y}{4\pi\mathrm{i}}f^{\prime}_{q}(q,p)-\frac{x}{4\pi\mathrm{i}}f^{\prime}_{p}(q,p))).$
Then the derived representation of $\mathfrak{h}_{1}$ is:
(41) ${\rho^{p}_{h}}(X)=-2\pi\mathrm{i}q-\frac{\varepsilon
h}{4\pi\mathrm{i}}\partial_{p},\qquad{\rho^{p}_{h}}(Y)=-2\pi\mathrm{i}p+\frac{\varepsilon
h}{4\pi\mathrm{i}}\partial_{q},\qquad{\rho^{p}_{h}}(S)=\varepsilon hI.$
An advantage of the FSB representation is that the derived form of the
parabolic Shale–Weil representation coincides with the elliptic one (21).
Eigenfunctions with the eigenvalue $\mu$ of the parabolic Hamiltonian
$\tilde{B}+\tilde{Z}/2=q\partial_{p}$ have the form
(42) $v_{\mu}(q,p)=e^{\mu p/q}f(q),\text{ with an arbitrary function }f(q).$
The linear equations defining the corresponding ladder operator
$L_{\varepsilon}^{\\!\pm}=a\tilde{X}+b\tilde{Y}$ in the algebra
$\mathfrak{h}_{1}$ are (37). The compatibility condition $\lambda^{2}=0$
implies $\lambda=0$ within complex numbers again. Admitting dual numbers, we
have additional values $\lambda=\pm\varepsilon\lambda_{1}$ with
$\lambda_{1}\in\mathbb{C}{}$ with the corresponding ladder operators
$L_{\varepsilon}^{\\!\pm}=\tilde{X}\mp\varepsilon\lambda_{1}\tilde{Y}=-2\pi\mathrm{i}q-\frac{\varepsilon
h}{4\pi\mathrm{i}}\partial_{p}\pm
2\pi\varepsilon\lambda_{1}\mathrm{i}p=-2\pi\mathrm{i}q+\varepsilon\mathrm{i}(\pm
2\pi\lambda_{1}p+\frac{h}{4\pi}\partial_{p}).$
For the eigenvalue $\mu=\mu_{0}+\varepsilon\mu_{1}$ with $\mu_{0}$,
$\mu_{1}\in\mathbb{C}{}$ the eigenfunction (42) can be rewritten as:
(43) $v_{\mu}(q,p)=e^{\mu
p/q}f(q)=e^{\mu_{0}p/q}\left(1+\varepsilon\mu_{1}\frac{p}{q}\right)f(q)$
due to the nilpotency of $\varepsilon$. Then the ladder action of
$L_{\varepsilon}^{\\!\pm}$ is
$\mu_{0}+\varepsilon\mu_{1}\mapsto\mu_{0}+\varepsilon(\mu_{1}\pm\lambda_{1})$.
Therefore, these operators are suitable for building
$\mathfrak{sp}_{2}$-modules with a one-dimensional chain of eigenvalues.
Finally, consider the ladder operator for the same element $B+Z/2$ within the
Lie algebra $\mathfrak{sp}_{2}$. According to the above procedure we get the
equations:
$-b+2c=\lambda a,\qquad a=\lambda b,\qquad\frac{a}{2}=\lambda c,$
which can again be resolved if and only if $\lambda^{2}=0$. There is the only
complex root $\lambda=0$ with the corresponding operators
$L_{p}^{\\!\pm}=\tilde{B}+\tilde{Z}/2$, which does not affect the eigenvalues.
However the dual number roots $\lambda=\pm\varepsilon\lambda_{2}$ with
$\lambda_{2}\in\mathbb{C}{}$ lead to the operators
$L_{\varepsilon}^{\\!\pm}=\pm\varepsilon\lambda_{2}\tilde{A}+\tilde{B}+\tilde{Z}/2=\pm\frac{\varepsilon\lambda_{2}}{2}\left(q\partial_{q}-p\partial_{p}\right)+q\partial_{p}.$
## 6\. Conclusions: Similarity and Correspondence
We wish to summarise our findings. Firstly, the appearance of hypercomplex
numbers in ladder operators for $\mathfrak{h}_{1}$ follows exactly the same
pattern as was already noted for $\mathfrak{sp}_{2}$ [Kisil09c]*Rem.
LABEL:W-re:hyper-number-necessity:
* •
the introduction of complex numbers is a necessity for the _existence_ of
ladder operators in the elliptic case;
* •
in the parabolic case, we need dual numbers to make ladder operators _useful_
;
* •
in the hyperbolic case, double numbers are not required neither for the
existence or for the usability of ladder operators, but they do provide an
enhancement.
In the spirit of the Similarity and Correspondence Principle 1 we have the
following extension of Prop. LABEL:W-pr:ladder-sim-eq from [Kisil09c]:
###### Proposition 6.
Let a vector $H\in\mathfrak{sp}_{2}$ generates the subgroup $K$, $N^{\prime}$
or $A\\!^{\prime}$, that is $H=Z$, $B+Z/2$, or $2B$, respectively. Let $\iota$
be the respective hypercomplex unit. Then the ladder operators $L^{\\!\pm}$
satisfying to the commutation relation:
$[H,L_{2}^{\\!\pm}]=\pm\iota L^{\\!\pm}$
are given by:
1. (1)
Within the Lie algebra $\mathfrak{h}_{1}$:
$L^{\\!\pm}=\tilde{X}\mp\iota\tilde{Y}.$
2. (2)
Within the Lie algebra $\mathfrak{sp}_{2}$:
$L_{2}^{\\!\pm}=\pm\iota\tilde{A}+\tilde{E}$. Here $E\in\mathfrak{sp}_{2}$ is
a linear combination of $B$ and $Z$ with the properties:
* •
$E=[A,H]$.
* •
$H=[A,E]$.
* •
Killings form $K(H,E)$ [Kirillov76]*§ 6.2 vanishes.
Any of the above properties defines the vector $E\in\mathop{\operator@font
span}\nolimits\\{B,Z\\}$ up to a real constant factor.
It is worth continuing this investigation and describing in detail hyperbolic
and parabolic versions of FSB spaces.
Acknowledgements: I am grateful to the anonymous referees for their helpful
remarks.
## References
|
arxiv-papers
| 2011-03-06T12:02:51 |
2024-09-04T02:49:17.489990
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/1103.1120"
}
|
1103.1126
|
# The Chandra Carina Complex Project View of Trumpler 16
Scott J, Wolk11affiliation: Harvard–Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138, USA , Patrick S. Broos22affiliation:
Department of Astronomy & Astrophysics, The Pennsylvania State University, 525
Davey Lab, University Park, PA 16802, USA , Konstantin V. Getman22affiliation:
Department of Astronomy & Astrophysics, The Pennsylvania State University, 525
Davey Lab, University Park, PA 16802, USA , Eric D. Feigelson22affiliation:
Department of Astronomy & Astrophysics, The Pennsylvania State University, 525
Davey Lab, University Park, PA 16802, USA , Thomas Preibisch33affiliation:
Universitäts-Sternwarte, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679
München, Germany , Leisa K. Townsley22affiliation: Department of Astronomy &
Astrophysics, The Pennsylvania State University, 525 Davey Lab, University
Park, PA 16802, USA , Junfeng Wang11affiliation: Harvard–Smithsonian Center
for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , Keivan G.
Stassun44affiliation: Department of Physics & Astronomy, Vanderbilt
University, Nashville, TN 37235, USA 55affiliation: Department of Physics,
Fisk University, 1000 17th Ave. N., Nashville, TN 37208, USA , Robert R.
King66affiliation: Astrophysics Group, College of Engineering, Mathematics,
and Physical Sciences, University of Exeter, Exeter EX4 4QL, UK , Mark J.
McCaughrean66affiliation: Astrophysics Group, College of Engineering,
Mathematics, and Physical Sciences, University of Exeter, Exeter EX4 4QL, UK
77affiliation: European Space Agency, Research & Scientific Support
Department, ESTEC, Postbus 299, 2200 AG Noordwijk, The Netherlands , Anthony
F. J. Moffat88affiliation: Département de Physique, Université de Montréal,
Succursale Centre-Ville, Montréal, QC, H3C 3J7, Canada and Hans
Zinnecker99affiliation: Deutsches SOFIA Insitute, Univ. of Stuttgart, Germany
and NASA-Ames Research Center, USA
###### Abstract
Trumpler 16 is a well–known rich star cluster containing the eruptive
supergiant $\eta$ Carinæ and located in the Carina star-forming complex. In
the context of the Chandra Carina Complex Project, we study Trumpler 16 using
new and archival X-ray data. A revised X-ray source list of the Trumpler 16
region contains 1232 X-ray sources including 1187 likely Carina members. These
are matched to 1047 near-infrared counterparts detected by the HAWK-I
instrument at the VLT allowing for better selection of cluster members. The
cluster is irregular in shape. Although it is roughly circular, there is a
high degree of sub-clustering, no noticeable central concentration and an
extension to the southeast. The high–mass stars show neither evidence of mass
segregation nor evidence of strong differential extinction. The derived power-
law slope of the X-ray luminosity function for Trumpler 16 reveals a much
steeper function than the Orion Nebula Cluster implying different ratio of
solar- to higher-mass stars. We estimate the total Trumpler 16 pre-main
sequence population to be $>6500$ Class II and Class III X-ray sources. An
overall K-excess disk frequency of $\sim$ 8.9% is derived using the X-ray
selected sample, although there is some variation among the sub-clusters,
especially in the Southeastern extension. X-ray emission is detected from 29
high–mass stars with spectral types between B2 and O3.
ISM: individual (Great Nebula in Carina) - open clusters and associations:
individual (Trumpler 16) - stars: pre-main sequence - X-Rays: stars X-ray –
Facilities: Chandra, VLT
## 1 Introduction
Trumpler 16 lies at the heart of the Carina Nebula region. It includes three
main sequence O3 stars, the Wolf-Rayet star WR 25 and $\eta$ Carinæ. $\eta$
Carinæ first became notable when it brightened significantly in the 1840s
(Herschel 1847). During that event an estimated $\gtrsim 10$ M⊙ of material
and nearly 1050 ergs of kinetic energy were injected into the host cluster
(Smith et al. 2003). Prior to the event, $\eta$ Carinæ most likely dominated
the energy budget of the cluster. Since this event however, $\eta$ Carinæ has
been essentially cut off from the cluster due to the vast opaque shell
surrounding it. This shell is surrounded by the Homunculus nebula (Gaviola
1950) which itself is a subset of a massive HII region spanning several square
degrees. The 2.3 kpc distance to the cluster comes primarily from the
expansion parallax of the Homunculus nebula around $\eta$ Carinæ (Smith 2006,
Davidson and Humphreys 1997).
The stellar content and star formation history of Trumpler 16 has been studied
in the optical band. Early studies concentrated on the massive stars. Walborn
(1971, 1973) identified 6 Henry Draper stars within Trumpler 16 and the
neighboring cluster, Trumpler 14, which were more massive than any star known
at that time, leading to the introduction of the O3 classification. This
implied both a very high mass and young age for these cluster. Levato &
Malaroda (1982) and Morrell et al. (1988) identified many more O stars in
these clusters spectroscopically. Feinstein (1982) performed deep
photomultiplier-based observations of about 70 cluster members as faint as
$V\approx 14.$ An early CCD-based photometric study by Massey & Johnson (1993)
reached about 1 $M_{\odot}$. This was followed by Degioia–Eastwood et al.
(2001) who presented optical photometry for over 560 stars in Trumpler 16.
They found clear evidence of pre-main sequence stars in the region and argued
for a mass-dependent spread in ages with intermediate mass star forming
continuously over the past 10 million years and high mass stars forming within
the last 3 million years.
An inventory of Trumpler 16 includes 42 O stars with the total radiative
luminosity equal to log ($L/L_{\odot}$) = 7.24, a total mass loss equal to
1.08$\times 10^{-3}$ $M_{\odot}$ per year and mechanical luminosity of
6.7$\times 10^{4}~{}L_{\odot}$ in the wind (Smith 2006). The region includes
12 small Bok globules including the “Keyhole Nebula”, the “Kangaroo Nebula”,
and “The Finger” (Smith & Brooks 2008). Whether or not these are sites of
ongoing star formation remains an open question. The region is dominated by
ionized gas emission. The strong 1.2 mm continuum is dominated by free–free
emission, not cool dust (Brooks et al. 2005).
In the X-ray regime, this is a very well studied cluster. Albacete–Colombo et
al. (2003) observed this region with $XMM-Newton$ for 35 ks and detected 80 of
the brightest sources, but this observation was badly limited by source
confusion. Evans et al. (2003) used 9.3 ks of early Chandra data to study the
hardness ratios of the hot stars in Trumpler 16. The luminosity limit of that
Chandra observation was about 7$\times 10^{31}{\rm ergs}\ {\rm s}^{-1}$,
typical of single O and early B stars. Albacete–Colombo re-observed the
cluster with Chandra for 90 ks. They found 1035 sources and matched 660 to
2MASS counterparts (Albacete–Colombo et al. 2008; AC08). About 15% of the
X–ray sources with near IR counterparts were found to have infrared excesses
indicative of an optically thick disk in the $K_{S}$ band. While the AC08
study covered a square 17′ by 17′ ACIS-I field which included part of Trumpler
14, the analysis presented by Feigelson et al. (2011; hereafter Paper I) shows
that Trumpler 16 occupies only a portion of the ACIS-I field studied by AC08.
In the north, Trumpler 16 is roughly circular about 11.6′ across while towards
the southeast it has an extension about 10′ by 6′ (Figure 1). The structure,
in part, is caused by an absorption lane that crosses the middle of the field.
The highly structured nature of Trumpler 16 is qualitatively different from
Trumpler 14 and Trumpler 15 which have a single central concentration (Ascenso
et al. 2007, Wang et al. 2011).
The Trumpler 16 region was not re-observed as part of the Chandra Carina
Complex Project (CCCP; Townsley et al. 2011a); instead previous observations
were re-analyzed following the prescriptions described in Broos et al. (2010
and 2011). The purpose of this paper therefore, is not a full discussion of
the data, which are presented by AC08, but instead to present the findings in
the context of the full X-ray analysis of the Carina complex. We also invoke
the new HAWK-I infrared observations (Preibisch et al. 2011) for improved
near-infrared counterpart information. For the purpose of this paper, we
define Trumpler 16 following the clustering analysis presented by Paper I, in
which Trumpler 16 is divided into seven sub clusters and a surrounding matrix
of stars.
In the next section, we will review the results of AC08 and compare those
results with the re-analysis of the X-ray data using new techniques presented
by Broos et al. (2010, 2011). We then evaluate the global extinction due to
dust, absorption due to gas, and luminosity properties of the cluster. Next,
we study the spatial distribution of the stars concentrating on the several
sub-clusters to examine whether the sub-clusters are real physical phenomena.
This paper will put little emphasis on high–mass stars except as they pertain
exclusively to Trumpler 16, as these stars are discussed elsewhere (Nazé et
al. 2011, Gagné et al. 2011). The A0-B3 stars in Tr 16 are examined in
detailed by Evans et al. (2011), candidate new OB stars are identified by
Povich et al. (2011), and Townsley et al. (2011b) discuss the diffuse emission
in the region.
## 2 The Observations and Data Reduction
The bulk of the X-ray data discussed here were taken prior to the CCCP, as
part of a guaranteed time program (2006 August 31, PI S. Murray, 88.4 ks,
ObsID 6402). A comprehensive analysis of those data was provided by AC08,
including discussion of sources in the Trumpler 14 region (to the north of
Trumpler 16) and detailed analysis of individual sources that are X-ray
bright. In this study we limit our attention to the Trumpler 16 cluster, and
we supplement the AC08 data with additional observations that partially
overlap Tr 16 (ObsIDs 9482, 9483, 9488, 6578, and 4495), which are described
and mapped by Townsley et al. (2011a; [Table 1]).
We define Trumpler 16 as the region within the lowest contour of the kernel
smoothed star surface density distribution shown in Paper I. This is
identified by a continuous contour of density to the South and East and
terminates in a narrow region separating Trumpler 14 from Trumpler 16 in the
Northwest (Figure 1). Trumpler 16 includes the matrix of stars surrounding the
7 sub- clusters identified in Paper I. About 1/8 of the field of view of ObsID
6402 covers Trumpler 14 and about one–third of the area of ObsID 6402 lay
outside of either cluster. Lower-density portions of Trumpler 16 to the South
and East (which we will refer to as the Trumpler 16 Southeastern extension)
were not included in the ObsID 6402 field of view. This includes the CCCP-
cluster 14 (Paper I) which has been previously identified by Sanchawala et al.
(2007a,b). This region has been covered by other observations as part of the
CCCP program and so is included in the analysis presented here.
The 2MASS Survey, which has a completeness limit near the galactic plane of Ks
13.3 (Skrutskie et al. 2006), was the only near IR (NIR) data available to
AC08. Deep NIR observations obtained using the HAWK-I camera at the ESO VLT
have recently become available and are used in this study to extend the NIR
catalog to a completeness limit of Ks $\sim$ 19 mag and an ultimate detection
limit of Ks $\sim$ 21 (Preibisch et al. 2011).
The data reduction, source detection, and source extraction procedures applied
to all the CCCP data are described by Broos et al. (2011a). We adopt the
statistical classification of sources as likely Carina members, likely
contaminants, or “unclassified objects” presented by Broos et al. (2011b),
understanding that the individual classifications are not guaranteed to be
correct. Column densities and absorption-corrected X-ray luminosities for
individual stars were estimated using the photometric techniques of the XPHOT
package (Getman et al. 2010).
Briefly, the CCCP source detection strategy was to nominate a liberal catalog
of candidate point sources using multiple source finding algorithms and then
iteratively extract those candidates, calculate for each a detection
significance statistic (probability of the null hypothesis that all the X-rays
found in the source aperture arose from the background), and prune candidates
found to be not significant. Our scientific goal to push for high sensitivity,
accepting a non-trivial number of spurious detections (Broos et al. 2011a).
The AC08 catalog was defined using a different algorithm (Palermo wavelet
detection code, PWdetect; Damiani et al. 1997) and more conservative
thresholds that are expected to produce only $\sim$ 10 spurious detections
within the ACIS-I field of view.
Broos et al. (2011a) discuss why estimating the number of false detections in
the CCCP catalog or in the Trumpler 16 study region is not practical, and
point out that such an estimate would be irrelevant for any analysis that
further restricts the sample of stars, e.g., by requiring a “likely member”
classification or requiring an estimate of X-ray luminosity from XPHOT.
However, an approximate lower limit on the number of legitimate X-ray sources
in any sample can be obtained by tallying the number NIR counterparts
identified. Among the 1232 CCCP sources in our study area, 1067 (85%) have NIR
counterparts detected in at least one band. Among the 1187 classified as
likely Carina members, 1047 (88%) have NIR counterparts; among the 885 with
X-ray luminosity estimates, 804 (91%) have NIR counterparts. Since we have
confidence that X-ray sources bright enough for XPHOT photometry are real
astrophysical sources, this limits the total fraction of false positives to a
few percent of the faintest X-ray sources and hence should not effect any
conclusion.
## 3 Results: Global Considerations
Structurally, Tr 16 is a roughly circular cluster about 11′ across (7.4 pc at
2.3 kpc). We also consider the Southeastern extension to be part of the
cluster (Figure 1). Table 1 enumerates the 1232 X-ray sources within the
continuous contour with $>1$ X-ray source per 30″ kernel. Of these, the X-ray
hardness and other criteria (Broos et al. 2011) classify 1187 as probable
members of the Carina complex, 11 as foreground objects, 2 as extragalactic
and 32 unknown. We find 392 sources not in the original catalog of AC08. Most
of these are faint sources: the mean number of net counts in these new sources
is 7 and the minimum is 2.3. About 50 are found in the Southeastern extension
which was not fully included in ObsID 6402. The median energy of the
previously detected sources is indistinguishable from the newly detected
sources, $MedE\simeq 1.5$ keV. The penultimate column of Table 1 lists the
class of the X-ray source following Broos et al. (2011) H0: unclassified; H1:
source is a foreground main-sequence star; H2: source is a young star, assumed
to be in the Carina complex; H3: source is a Galactic background main-sequence
star; H4: source is an extragalactic source. The final column of Table 1
indicates the sub-cluster with which the source is identified (C3 = Sub-
cluster 3 etc.) based on the nomenclature in Paper I. We also identify those
stars which are not identified with any single cluster as being part of the
‘matrix’ of Trumpler 16. The matrix stars in the Southeastern extension are
identified separately as “SEM”.
### 3.1 Disk Fraction
For the 1187 X-ray sources found to be probable members of the Carina complex,
matches are found in the HAWK-I photometric catalog for 1047. Almost all of
those (1032 sources) are detected in all three $JHK$ bands. The bulk of the
X-ray sources matched have 12 $<K_{S}<$ 15.5 with wings extending to both
brighter and fainter sources. Of the 1032 ,probable members of the Carina
complex with JHK detections, 1013 sources have errors less than 5% in all
bands. Ninety of 1013 (8.9% $\pm$0.9%), have excesses consistent with an
optically thick disk (Figure 2). This is a slightly higher rate than the 7.8%,
found for the full CCCP by Preibisch et al. (2011) or the 6.9% disk detections
they found for the 529 X-ray sources located in the Trumpler 16 sub-clusters.
The sources reported by Preibisch et al. are a sub-sample (only the sources
which reside in sub-clusters) of the whole of Trumpler 16 which we covered
here. Also, we require a KS excess to be 10%, this is higher than the
Preibisch et al. study, but is closer to the value used by AC08 who reported a
relatively high disk fraction of $\sim 15\%\pm 2\%$.
In AC08, the sample is restricted to the 339 2MASS sources with good colors in
all three bands. The true limiting factor for inclusion is the $K_{S}$ band
magnitude which needs to be brighter than about 15. Sources without a $K_{S}$
band excess are more likely to be excluded from the 339 stars sample, which
will raise the disk fraction. In the present study the IR data are more
complete than the X-ray data, so we do not expect a strong IR bias in favor of
detecting stars with disks.
### 3.2 IR magnitude and X-ray Flux
As seen in Figure 3, the X-ray fluxes of the sources are well correlated with
$J$-band luminosities for $6<J<11$. This relation appears to reverse between
$12<J<14$ and reappears in the range $14<J<18$. Below $J$ =18 the flux
distribution appears flat. Most of this can be readily interpreted as follows.
Given the 2.3 kpc distance and a roughly 3 Myr age, objects with $6<J<11$, are
primarily high–mass stars, which generate X-rays through shocks in unstable
radiatively driven winds (Lucy 1982). Both the luminosity and the shock speeds
scale with mass down to early-B stars (See the Nazé et al. 2011 in this
issue). Similarly, objects with $14<J<18$ are pre–main sequence (PMS) G, K
through mid–M stars that are known to have their bolometric luminosity
correlate with their X-ray luminosity. This is seen for example, in the
Chandra Orion Ultradeep Project (COUP) study of the Orion Nebula Cluster
(Preibisch et al. 2005). For these stars, roughly 0.02% of their luminosity is
emitted in X-rays although the overall fraction can be higher during X-ray
flares. Below $J$=18, typical PMS stars are too faint to be detected in these
observations and only those caught during strong flares are seen.
The behavior between $12<J<14$ is curious. For the 2.3 kpc distance and $\sim$
3 Myr age these are Mid-B through A stars, which are not efficient X-ray
producers since they have weak winds, but $\sim 30-60\%$ of such stars in
Trumpler 16 and COUP were detected in X-rays (Evans et al. 2011; Stelzer et
al. 2005). Both Evans et al. and Stelzer et al. concluded that high energy
emission from these sources originates in unseen companions. It has not
previously been reported that the X-ray flux became brighter as the stars
became bolometrically fainter. Previous samples were probably too small to
detect this effect. If the X-ray emission was indeed the result of unseen
companions, the implication is that the higher mass primaries (B5-A5)
typically have lower mass companions than the primaries below A5.
To quantify these effects, we performed a piecewise least-squares linear
regression and applied nonparametric correlation measures to the data in
Figure 3. The fitted slope is $m=-0.76\pm 0.05$ for $6<J<10$, $m=-0.30\pm
0.02$ for $14.5<J<17$ but reverses direction to $m=0.18\pm 0.08$ for $12<J<14$
(these lines are plotted in Figure 3). Kendall’s $\tau$ correlation
coefficient (Kendall 1938) gives $\tau=-0.86$ for the bright stars, $-0.41$
for the faint stars, and 0.21 for the intermediate brightness stars. The
probability for the correlation of the intermediate stars is $P\sim 0.5$%,
roughly equivalent to a 3$\sigma$ effect.
Because we found the effect surprising we repeated the tests, allowing the
limits on the intermediate J magnitude band to vary by up to 0.5 mag. We also
calculated probabilities in various terms including the Spearman $\rho$
correlation coefficients (Spearman 1904) for each these are -0.70 for the
bright J band sources, -0.30 for the fainter J band sources and 0.13 for the
intermediate case. In all cases, there is a weak positive correlation in the
middle range. Further, a similar pattern has been seen in the Trumpler 14 and
Trumpler 15 clusters within the $\eta$ Carina cluster (Evans et al.
$in~{}prep$). No correlation is seen when the entire CCCP sample is used,
however this sample covers relatively broad range of ages and conditions.
### 3.3 Near–IR Extinction
We originally calculated extinction to each source following
Ã${}_{V}=(H-K_{S}-0.1)\times 13.7$ (Preibisch et al. 2011).111 Following
Preibisch et al. we use the expression “ÃV” to indicate extinction calculated
using this simple scaler relation. We use “AV” to indicate extinction
measurements obtained by individually fitting a given stellar color +
extinction to a model color in this case using Siess et al. (2000). Preibisch
et al. note that ÃV calculated in this manner may not be accurate for all
masses: the first term assumes stellar photospheric color $H-K_{S}=0.1$
appropriate for 3 Myr stars between 1-2 $M_{\odot}$ (Siess et al. 2000). Below
1 $M_{\odot}$, photospheric $H-K_{S}$ exceeds 0.1 and extinction is over-
estimated, and above 2 $M_{\odot}$, extinction is underestimated. This can
lead to estimates of negative extinction. To minimize these effects, we only
estimate extinction for stars with no evidence of optically thick disks at
$K_{S}$ and $J$ magnitudes between 14.5 and 16.5 as these are likely to have
masses of 1– 2 $M_{\odot}$.
The second term is the proportionality factor characteristic of the extinction
law. We note the derived reddening law measured for this region appears to
deviate from the typical ISM value of R= 3.1, with derived values being
between 3.8 and 5 with possible spatial variations (Nazé et al. 2011, Povich
et al. 2011, Gagné et al. 2011, Smith 1987, Thé 1980, Herbst 1976, Forte 1978,
Feinstein et al. 1973). However, using only optical data, Turner & Moffat
(1980) found R = 3.2 throughout the Carina region. He we us
Ã${}_{V}/E_{(B-V)}$=4.0 (Povich et al. 2011) which leads to the constant
values $k$=13.7. A higher value of $R_{V}$ would lower the value of $k$. Given
these constraints, we find the mean Ã${}_{V}=3.8$ with 25% and 75% quartiles
at 2.9 and 5.0, respectively, for the Trumpler 16 CCCP sample of likely Carina
members.
AC08 calculated extinction by using a $K_{S}$ vs. $J-H$ color-magnitude
diagram, individually dereddening stars until the location of the star
intersected the isochrone for a 3 Myr cluster PMS (isochrones from Siess et
al. 2000). They found a mean reddening of AV=3.6$\pm$2.4 mag, acknowledging
that their estimates of AV could be in error by up to 0.7 mag due to the
relatively high photometric errors in the 2MASS data. While overall this
method is more precise than that used by Preibisch et al. (2011), the errors
may be higher than estimated due to their unconfirmed assumption that all
stars are exactly 3 Myr old. Furthermore, we expect our data to be deeper and
hence capture more highly extinquished stars than the 2MASS sample. Another
important aspect for the comparison of HAWK-I and 2MASS is the spatial
resolution: about 2 arcsec for 2MASS versus about 0.6 arcsec for HAWK-I. Many
”point-like” 2MASS sources are resolved into several components in the HAWK-I
images. Other effects such as photometric variability of the young stars,
unresolved binary companions, and small infrared excesses (that are too small
to move the star out of the main-sequence reddening band in the color–color
diagram) will all add both random and systematic errors into the extinction
measurements.
We examined a sub-sample of about 100 stars with non-degenerate222When
calculating infrared extinctions for 3 Myr stars, the reddening vector crosses
isochrones multiple times for stars with masses between about 2.2-8.0
$M_{\odot}$. For these stars, IR data alone are not conclusive and we identify
the reddening determinations as degenerate. reddening extinctions measured by
AC08. We find an offset $\Delta$ = 0.58 mag between the mean value for ÃV
obtained here and the mean AV published by AC08. The latter tends to show less
extinction. There is considerable scatter between the two ÃV estimates,
$\sigma$ = 1.0. When we restrict the sub-sample further to 65 stars with
extinction corrected magnitudes of $13<K_{S}<14$, which would place their
masses between 1 and 2 $M_{\odot}$, the difference and dispersion are reduced
to $\Delta=0.4$ and $\sigma=0.8$. This indicates that extending the simple
correction used by Preibisch et al. beyond its designed range caused some of
the scatter. We then calculated extinctions for each source, following the
methods described by AC08, but using the HAWK-I data and found that large
scatter and systematic offsets around A${}_{V}\simeq 0.5$ persist.
Considering all factors – ranges in age, measurement errors, mass limitations,
dust particle size distribution – the level of agreement between the different
methods appears reasonable. Since the assumption of intrinsic $H-K_{S}=0.1$ is
accurate to within 3% for stars with masses $2.2>M_{\odot}>0.8$ which dominate
the X-ray distribution, the extinction estimator of Preibisch et al. (2011)
seems appropriate for the CCCP Carina member sample exclusive of the O, B and
A stars.
In Figure 4 we show the $J$ vs. $J-H$ color-magnitude diagrams for Trumpler 16
and several of the sub-clusters. The solid (green) line on the left hand side
of the plot is an isochrone from Siess et al. (2000) assuming an age of 3 Myr
and a distance modulus of 11.81. We can globally fit the extinction to the
cluster, by dereddening the ensemble of stars with $13<J<17.5$ in steps of 0.1
$J$ magnitudes and measuring the $\chi^{2}$ residuals versus the 3 Myr
isochrone model. We restricted the sample to stars without evidence for
optically thick disks. The best fit was found for an extinction value of
AV=3.3 with interquartile confidence band of $2.3-3.8$. When the sample is
restricted to 1–2 $M_{\odot}$ likely Carina members, the mean becomes
$A_{V}=4.1$ with interquartile range $3.4-4.9$.
### 3.4 X–Ray Absorption
Individual X-ray luminosity and absorption were calculated for X-ray sources
in the direction of Trumpler 16 using the XPHOT package (Table 2; Getman et
al. 2010). The concept of the XPHOT method is similar to the use of
color–magnitude diagrams in optical and infrared astronomy, with X-ray median
energy replacing color index and X-ray source counts replacing magnitude.
Using non-parametric methods, one can estimate both apparent and intrinsic
broadband X-ray fluxes and soft X-ray absorption from gas along the
line–of–sight to X-ray sources. Apparent flux is estimated from the ratio of
the source count rate to the instrumental effective area averaged over the
chosen band. Absorption, intrinsic flux, and errors on these quantities are
estimated from comparison of source photometric quantities with those of high
signal–to–noise spectra that were simulated using spectral models
characteristic of low–mass pre–main sequence stars. In the original paper, the
results were compared with spectroscopic analysis of sources in M 17 (Broos et
al. 2007). For stars with median total band energy $>1.7$ keV, Getman et al.
(2010) show that fluxes measured by XPHOT agree with fluxes obtained by
spectral fitting to within a factor of $\sim$ 1.5 for sources with more than
50 counts, and within a factor of $\sim$ 4 for sources with as few as 10
counts. The total band covers the 0.5 – 8.0 keV energy range.
For the Trumpler 16 sample, 347 sources could not be assigned a flux or
temperature. Eighty percent of these had less than 10 counts and all but six
had less than 30 counts. The final six had photon distributions inconsistent
with the model of a one-temperature $\sim$ 1.5 keV corona. The remaining 885
sources had X-ray luminosity and absorption calculated. For sources with
between 10–20 counts, the median error in total flux in total flux is about
35% and the statistical error in $N_{\rm H}$ is also about 35% (for median
energy of 1.5 keV). In the lowest count bin, 5–7 counts, the median error in
total flux in total flux is about 65% and the statistical error in $N_{\rm H}$
is also about 45% (for median energy of 1.5 keV). There is also a systematic
error on the $N_{\rm H}$ value due to low effective area below about 500 eV.
The systematic error is about 15% at 1.5 keV but approaches unity for sources
below 1 keV. The converse of this is that ACIS spectral resolution is not fine
enough to discriminate log $N_{\rm H}$ values significantly below 22.0 (Getman
et al. 2010). Below log $N_{\rm H}$=21.6 systematic errors dominate over
statistical errors with a median statistical error in log $N_{\rm H}$ of about
0.3 and systematic errors as high as log $N_{\rm H}$=1.15. Above log $N_{\rm
H}$=21.6 statistical errors dominate with a median statistical error in log
$N_{\rm H}$ of about 0.2 but statistical errors still as high as log $N_{\rm
H}$=1.0. It is cautioned that log $N_{\rm H}$ values below 21.6 are less
robust than high values of $N_{\rm H}$.
The mean $N_{\rm H}$ value derived from XPHOT results here is essentially
identical (within 10%) to that found by AC08 using other methods. However, we
find a systematic bias between XPHOT and XSPEC $N_{\rm H}$ values for the
stronger X-ray sources in Trumpler 16 in the sense that for log $N_{\rm
H}$(cm-2) $\leq$ 21.75 , XPHOT gives a larger value and for log $N_{\rm
H}$(cm-2) $>$ 21.75 XSPEC gives a higher value. At the extreme values, the
differences can be 0.75 dex. This finding is consistent with the result in
Getman et al. (2010) wherein the observed median energies of stellar sources
in M 17 were found to have a simple linear relation to the log $N_{\rm H}$ for
values of log $N_{\rm H}$(cm-2) $>$ 21.6. The XPHOT derived absorptions were
deemed unreliable for 205 sources with log $N_{\rm H}$$<$ 21.6 cm-2 and below
a median energy of 1.7 keV, due to degeneracy in the median energy – log
$N_{\rm H}$ relationship used by XPHOT.
Figure 5 shows the distribution of absorption columns found by XPHOT for the
sources in the direction of Trumpler 16. The distribution is highly
structured, especially when compared to Figure 7 of AC08. About 5% of the
sources have log $N_{\rm H}$ $<$ 21; some of these may be Galactic field
foreground stars misclassified as likely Carina members. About 15% of the CCCP
likely Carina members have 21.2 $<$ log $N_{\rm H}$(cm-2) $<$ 21.6, leading to
a strong peak in the distribution of sources at log $N_{\rm H}$ $\simeq 21.7$
cm-2, with a similar shoulder around 22.0 $<$ log $N_{\rm H}$(cm-2) $<$ 22.3.
A small number of sources have higher absorptions in the range 22.3 $<$ log
$N_{\rm H}$(cm-2) $<$ 23.4; some of these may be embedded Carina protostars,
while others may be misclassified extragalactic sources.
If the very low absorption tail (log $N_{\rm H}$(cm-2) $<$ 21.0) is attributed
to foreground contaminants, then the distribution of X-ray absorptions has a
strongly peaked unimodal distribution very similar to the distribution of
absorptions derived from IR photometry (Figure 4). Using the conversion
$N_{\rm H}$ = $1.6\times 10^{21}A_{V}$ cm-2 obtained by Vuong et al. (2003),
the peak of the Trumpler 16 X-ray absorption distribution is equivalent to
$A_{V}\simeq 3$ mag, in agreement with the value $A_{V}\simeq 3-4$ mag
obtained from the near-IR color-magnitude diagrams (§ 3.3). Both the X-ray and
IR absorption distributions have a sparse tail of extinctions found going out
to A${}_{V}>30$ mag.
### 3.5 X-ray Luminosity Distribution
As most of the CCCP sources in Trumpler 16 are too faint in the X-ray band for
direct spectral modeling (Figure 6), the XPHOT technique is used to scale the
observed count rate to broad-band luminosity with a correction for soft X-ray
absorption. Getman et al. (2010) find that most XPHOT total-band fluxes are
within $\pm 20$% of values determined with XSPEC for the absorption ranges
typically seen in the Trumpler 16 region. They estimate the errors in total-
band source fluxes to be better than 60%, 50%, 30%, and 20% for net count
strata 7–10, 10–20, 20–50, and $>$50 counts, respectively, with a small
systematic bias. In Trumpler 16, nearly 500 of the 885 sources with fluxes and
luminosities measured by XPHOT have less than 7 net counts. The simulations
estimate less than 70% errors for these sources.
As noted by Feigelson et al. (2005), the X-ray luminosity function (XLF) is
the product of the initial mass function and the correlation between mass-age
and X-ray luminosity. This correlation is well-studied in the COUP and Taurus
young stellar populations (Preibisch et al. 2005, Telleschi et al. 2007). The
X-ray luminosity functions of the ONC, IC 348, NGC 1333 and other young
clusters appear similar, suggesting that the XLF shape may be roughly
“universal” (Wang et al. 2008). For the youngest clusters, differential
evolutionary effects appear minimized and the XLFs of the clusters are well
fitted by a log-normal function with $<$log L${}_{X}>$(erg s-1) = 29.3 and
$\sigma$=1.0 (Feigelson et al. 2005). Wang et al. (2008) find some deviations
from precise agreement for different clusters; for example, a steeper slope in
the XLF of M17 and a more shallow slope for the Cep OB3b cluster is seen.
Meanwhile M17 has nearly 3 times the stars of the ONC while Cep B less than
half the unobscured population of the ONC (Broos et al. 2007, Getman et al.
2005, 2006). The implication is that more massive clusters may have a steeper
slope to their luminosity function.
Figure 7 compares the XLF of Trumpler 16 (excluding the SE extension) with the
XLF of the COUP data (Getman et al. 2005). To determine the XLF of Trumpler
16, the first step is to estimate the completeness of the X-ray data. Figure
7b shows a histogram of the luminosity as derived by XPHOT of 687 X-ray
sources observed in Trumpler 16 (this is down from 885 because we are
excluding the SE extension which has different properties as will be discussed
in § 4.). Visual examination of Figure 7b shows a drop off starting at log
$L_{t,c}$= 30.5 ($L_{t,c}$ = absorption corrected luminosity in the total band
covering 0.5 - 8.0 keV). This value is consistent with the CCCP completeness
limits discussed by Broos et al. (2011a), which vary strongly with off-axis
distance and with absorption. The implication is that incompleteness in the
distribution occurs before this point, mostly before log $L_{t,c}$= 30.7.
As shown in Figure 7a, we fitted the cumulative XLF of Trumpler 16 (excluding
the SE extension) with a power-law between log $L_{t,c}$= 30.7 - 31.5 and find
a slope $\Gamma=-1.27$. This slope was sensitive to the lower luminosity
cut–off used at the level of 0.05 in slope for a 20% change in the luminosity
limits; $\Gamma=-1.13$ if we brought the lower cut–off down to $L_{t,c}$= 30.3
which is clearly incomplete. The upper cutoff of $L_{t,c}$= 31.5 is
essentially the brightest cool star. This slope is steeper than a similarly
measured slope for the COUP data which is found to be $\Gamma=$-$0.93$ for log
$L_{t,c}$ =30.2 – 31.5. The slope of the COUP data is less sensitive to the
lower luminosity cut–off and is only effected at the level of 0.01 for a 20%
change in the luminosity limits.
To estimate the total number of sources in the cluster, we simply compare the
total number of sources in Trumpler 16 to the number in the COUP sample in the
range from log $L_{t,c}$= 30.7 - 31.5 which should be dominated by cool stars.
The COUP sample contains 51 sources in this range, while the Trumpler 16
sample has 255 for a factor of 5($\pm 0.5$) more. Given an estimate for the
total population of the ONC from the COUP as 1300 X-ray sources (Getman et al.
2005) we estimate the total population of Trumpler 16 at 6500 $\pm 650$ if
observed to a similar X-ray luminosity limit. Further, Hillenbrand & Hartmann
(1998) estimate the overall ONC population to be 2800. If this is correct,
then the X-ray sources represent less than 50% of the total cluster membership
which may be as high as 14,000.
## 4 The Sub-clusters within Trumpler 16
Paper I defines Trumpler 16 as a region in which the surface density of CCCP
likely Carina members smoothed with a 30″ Gaussian kernel (FWHM=0.8 pc)
exceeds 1 source per kernel. Within this region, Paper I identified seven sub-
clusters within Trumpler 16 with sub-cluster surface density exceeding three
sources per 30″ Gaussian kernel. Overall, Paper I finds 31 small X-ray
selected groups of probable Carina members in the CCCP. Trumpler 16 contains 8
of these groups, including one in the Southeastern extension, which we refer
to as sub-clusters. No single sub-cluster exceeds 15% of the total number of
sources in the Trumpler 16 cluster. The exact number and shape of these sub-
clusters varies depending on the smoothing kernel the density factor. This is
unlike Trumpler 14 and 15 which are each dominated by a single, central
concentration.
In this section, we examine whether the sub-clusters are physically distinct
units or simply statistical fluctuations, and we compare their properties. The
sub-cluster identification for each source is given in Table 1 and Figure 8.
In the figure, we have approximated each sub-cluster as an ellipse to aid in
the calculation of geometric parameters. For each sub-cluster we have assessed
extinction as ÃV for stars of $K_{S}<$ 13\. We then applied the extinction to
the observed $K_{S}$ magnitude and the distance to produce a K-band luminosity
function (KLF). As discussed above, the actual values of individual stars are
not reliable and may be in error by up to 0.2 $K_{S}$ mag., but the structure
of the distribution should be representative. To quantify the structures, we
have identified the quartile values of absolute $K_{S}$ and ÃV for all sub-
clusters.
In order to determine if the sub-clusters truly stand out from the background
cluster population of Trumpler 16, we first look at the 506 X-ray sources
which form the matrix of Trumpler 16. These are cluster members, but not
coincident with any specific sub-cluster, nor the Southeastern extension. Of
the 438 sources with good mid-IR colors, 6.4% ($\pm$ 1.2%) show IR excesses
consistent with an optically thick disk. This is more than 2$\sigma$ less than
the cluster as a whole and may indicate that these dispersed stars are more
evolved than the global population. The mean extinction value of Ã${}_{V}=3.8$
is indistinguishable from the cluster as a whole.
### 4.1 Sub-cluster 3
Sub-cluster 3 is the westernmost of the Trumpler 16 sub-clusters. When a
larger smoothing kernel is used, it appears as a low density extension of Sub-
cluster 6. It contains 33 X-ray sources and has a spatially averaged source
density of 32 src pc-2. Twenty–seven of the X-ray sources have been matched
with HAWK-I sources; almost all of these are between 1 and 2 $M_{\odot}$.
About $15\%\pm 7$% of the IR detected X-ray sources have disks. The mean ÃV is
3.8 similar to the cluster as a whole. The massive supergiant Tr 14 Y 398
(spectral type O3 Iab) as well as WR 25 (HD 93162, WN) lie on the western edge
of the sub-cluster.
### 4.2 Sub-cluster 4
Sub-cluster 4 is the smallest of the Trumpler 16 sub-clusters with only 11
X-ray sources. The spatially averaged source density of 36 src pc-2 is about
the same as Sub-cluster 3. Of the eight X-ray sources matched to HAWK-I, two
have $K_{S}$ excesses consistent with an optically thick disk. The mean ÃV is
3.5. The B1 V star CPD$-$59o2581 lies near center of the sub cluster, and the
similar B1 V CPD$-$59o2574 at the northwestern edge of the sub-cluster.
Neither of these stars were detected in X-rays.
### 4.3 Sub-cluster 6
Sub-cluster 6 is a centrally condensed sub-cluster and the second largest in
Trumpler 16 overall with 109 sources and an average source density of 45 src
pc-2. Of the 88 sources with good mid-IR colors, 6.8% $\pm$ 2.8% show IR
excesses from an optically thick disk, consistent with the global population.
Sub-cluster 6 has a similar absorption to the previously discussed regions
with mean Ã${}_{V}=3.6$ and 75% quartile Ã${}_{V}=4.1$. There are three O
stars in this sub-cluster, the O3.5 V+ O8 V double HD 93205 and an O5 V HD
93204. These are displaced by 0.25 pc to the southwest of the cluster center.
### 4.4 Sub-cluster 9
Sub-cluster 9 is the western half of a bow-tie shaped enhancement toward the
southern part of Trumpler 16. The 53 sources appear uniformly distributed with
no internal concentrations and a high spatially averaged source density of 48
src pc-2. The X-ray sources here are relatively faint, about half of the
sources in the region required the more sensitive source detection procedure
described by Broos et al. (2010) and were not found by AC08. This is about
twice the usual fraction of newly identified sources. Of the 45 sources with
good mid-IR colors, only 2 (4%$\pm$ 3%) show IR excesses consistent with an
optically thick disk. One of these is exceptionally faint, $K_{S}=19.5$ and
hence the colors are quite dubious. The mean extinction calculated for the
region as a whole is typical of other regions with Ã${}_{V}=$3.7. There are no
high-mass stars in this region.
### 4.5 Sub-cluster 10
Sub-cluster 10 is the eastern half of the bow-tie shaped enhancement toward
the southern part of Trumpler 16. The 82 sources appear uniformly distributed
with no strong internal concentrations and a high spatially averaged source
density of 42 src pc-2. The X-ray luminosity is more typical here, 20% of the
sources required the more sensitive source detection procedure described by
Broos et al. (2010). Of the 74 sources with good mid-IR colors, only 4
(5.4%$\pm$ 2.7%) show IR excesses consistent with an optically thick disk. Two
of these are faint, $J>18.1$ and hence the colors have errors of about 35%.
Thus there are only two bona-fide disk systems in this subcluster. The mean
extinction calculated for the region as a whole is typical of other regions
with mean Ã${}_{V}=$3.7. Four late O stars in the region are all detected in
X-rays. The most massive, O7 V star CPD$-$59o2626, lies near the subcluster
center, while the others lie towards the east and southeast.
Although sub-clusters 9 and 10 are in contact, they have different mass
distribution. Sub-cluster 9 has fewer higher mass stars. Both clusters have
relatively low numbers of stars with optically thick disks in the $K_{S}$
band.
### 4.6 Sub-cluster 11
$\eta$ Carinæ and 6 other high-mass stars reside in sub-cluster 11. The 71
X-ray sources include five of the high mass stars. Overall the X-ray sources
are centrally condensed with a peak at 10:45:06 $-$59:40:25, about 0.3 pc from
$\eta$ Carinæ. The spatially averaged source density of 27 src pc-2 is among
the lowest of any region. This may be due, in part, to reduced point source
sensitivity in the immediate vicinity of $\eta$ Carinæ and the associated
X-ray nebula. Of the 57 sources with good mid-IR colors, 4 show IR excesses
consistent with an optically thick disk (7%$\pm$ 4%). The mean extinction
calculated for the region as a whole is low, $<$Ã${}_{V}>=$2.9.
### 4.7 Sub-cluster 12
Sub-cluster 12 is immediately to the south of sub-cluster 11, centered at
10:45:10.6, $-$59:42:54. It is the most centrally condensed and populous sub-
cluster of Trumpler 16 with 166 X-ray sources. It includes six of the nine
most massive stars in Trumpler 16. There are two high-mass stars within 15″ of
the center, both of which are early B stars, not O stars. The most massive
star complex in the cluster, CPD$-$59o3310, contains two high–mass stars, an
O6 V + a B2 sub-giant, and is located about 2′ to the southeast of the cluster
center. An additional candidate OB star at the center of sub-cluster 12 is
identified by Povich et al. (2011). This is the only one of 94 new OB star
candidates in the CCCP located within the main area of Tr 16 (three were
identified around the periphery and five in the Southeastern
extension).333There is also a strong selection effect as the Spitzer Vela
Carina survey was highly constrained when observing near $\eta$ Carinæ. The
spatially averaged source density is 45 src pc-2. Of the 142 sources with good
mid-IR colors, $10.6\%\pm 2.7\%$ show IR excesses consistent with an optically
thick disk. All but one of these are found at relatively high extinction
(A${}_{V}>$2.6), indicating they likely lie behind a source of extinction
within the sub-cluster. Further, the cumulative extinction distribution is
very steep, with 60% of the X-ray sources with $3.4<$ Ã${}_{V}<4.2$. All the
disked stars are among the most absorbed 15%. It appears there is a thin cloud
with Ã${}_{V}\sim 1$ near the front of this sub-cluster. Figure 9 shows the
extinction functions of several regions within Trumpler 16, with sub-cluster
12 showing the steepest distribution.
### 4.8 The Southeastern Extension and Sub-cluster 14
To the southeast of the main body of Trumpler 16, separated by a dust lane, is
a continuation of the density distribution which we identify with Trumpler 16.
This region was first recognized by Sanchawala et al. (2007a, b) who
identified 10 X-ray sources in this region using the first two $Chandra$
observations of the Carina nebula (ObsIDs 1249 and 50). The region is
distinguished by high extinction, $<$A${}_{V}>$= 4.2, and a steep KLF
indicating a young age. Using the larger CCCP mosaic, we find this extension
covers about 40% of the area of the main cluster and contains 156 X-ray
sources. The bulk of this extension was outside the field of ObsID 6402 and
has somewhat lower overall exposure time, about 60 ks rather than 90 ks.
Hence, a lower surface density of sources is expected. Even in light of the
difference in exposure times, it is clear that conditions are different in
this region. The disked fraction is much higher than the rest of the cluster,
$19\pm 4$%.
Within the Southeastern extension, there is a density enhancement identified
as sub-cluster 14 in Paper I. This corresponds with the location of the 10
X-ray sources identified by Sanchawala et al. (2007a, b). We find 40 stars
within this sub-cluster and a very high extinction, $<$Ã${}_{V}>=$7.4. Sub-
cluster 14 is identified as a more embedded region within the Southeastern
extension. This sub-cluster also has the highest disked fraction, 21$\pm 8$%,
of any sub-cluster with a significant number of stars. The eclipsing O5.5 +
O9.5 binary V662 Carinæ is located within about 15″ (0.1 pc) of the cluster
center making sub-cluster 14 the only one of the eight sub-clusters with a
dominant O star so close to its geometric center. Four of the five OB
candidates identified by Povich et al. (2011) in the Southeastern extension
lie within sub-cluster 14. Three rank among the seven most luminous of all the
OB star candidates in the CCCP.
The matrix of stars surrounding sub-cluster 14 is composed of 116 X-ray
sources in the region, including the O4 V star LS 1886 located in the eastern
portion of the extension. Removing sub-cluster 14 from the Southeastern
extension,the region still stands out with high extinction $<$Ã${}_{V}>=$4.8
and a high disked fraction of 18$\pm 4$%.
### 4.9 Discussion of Sub-clustering
The metrics for each sub-cluster are summarized in Table 3. It is clear that
the main body of Trumpler 16 (subclusters 3, 6, 9, 10, 12 and the surrounding
matrix) is different from the Southeastern extension. The extinction is nearly
3 magnitudes greater in the Southeastern extension and the disked fraction is
more than twice that of the main body. Within the Southeastern extension, sub-
cluster 14 is distinguished by an additional 2 magnitudes of extinction in the
V band and a higher disked fraction. The disk fraction can be used as a proxy
for age (Haisch et al. 2001). While the age calibration is sensitive to the
sensitivity of the survey, disked fraction appears to decrease nearly linearly
in time – indicating that the Southeastern extension is about 80% the age of
the main body of Trumpler 16.
Within the main body of Trumpler 16, the sub-clusters appear to be dynamically
distinct structures. Some have $>100$ stars with surface densities $>3$ times
that of the outer portions of the matrix, too rich to arise from statistical
fluctuations. On the other hand, except for chance difference in line-of-sight
absorptions with respect to Carina clouds, the X-ray selected stars in the
sub-clusters are similar to each other and to the stars in the matrix. Stars
in all of the sub-structures outside of the SE extension have disk fractions
consistent with $\sim 7$%. The stars in the matrix tend have a slightly lower
disk fraction than the stars in the sub-clusters but the significance is
$<2\sigma$.
The massive O stars, both main sequence and supergiant, do not lie at the
cores of the subclusters. Few young stellar clusters are documented to have
widely distributed massive stars; the best example may be NGC 2244, the
central cluster in the Rosette Nebula (Wang et al. 2008).
## 5 The Relation of the High–Mass Stars to the Sub-clusters
X-ray emission is detected from 29 stars in Trumpler 16 that were classified
with spectral types earlier than B2 (Skiff 2010). This includes $\eta$ Carinæ
itself, which was detected, but not cataloged by Broos et al. (2011) due to
its high degree of pile-up (saturation in the ACIS CCD detection) which makes
characterization difficult. The remainder include WR 25 and 19 O stars, many
of which are in multiple systems. In addition, there are eight B stars in the
range from B0.5 to B2 which were detected in X-rays, six of these with less
than 15 net counts.
There are 21 early-type stars which were not detected in the X-ray
observations. Most of the non-detections are from B0V to B2V in spectral type.
This indicates a very sharp drop off in X-ray activity across the O/B divide
as all the O stars were detected. There were also a few early–type stars not
detected in X-rays which were also not considered to be main sequence either.
All of these are in the far south and all but emission object Hen 3-480 are
considered to be part of the Southeastern extension. We list the detections in
Tables 4 and 5 and the non-detections in Table 6. More comprehensive studies
of early–type stars are the topics of separate papers in the framework of the
CCCP (e.g., Nazé et al. 2011; Gagné et al. 2011). Candidates from Povich et
al. (2011) are excluded from this discussion as well since followup is still
required to establish their spectral types and confirm their OB status.
Following the finding of AC08 that OB stars are less absorbed than their late
type counterparts (AV=2.0 for OB stars versus 3.6 for GK stars), we examine
the extinction to the X-ray detected early–type stars. Twenty-five of these
have matches in the HAWK-I photometry catalog. We calculated the extinction by
comparison to the $J-H$ and $J-K_{S}$ colors of high–mass stars which are
expected to be nearly constant at $-0.15$ and $-0.2$ respectively, in the
absence of extinction. We find a range of extinctions from $A_{V}\sim
0.85-8.9$. The average $A_{V}$ is about 2.5 but this is dominated by a single
outlier (MJ 224) with $A_{V}\sim 8.75$. The mean extinction may be the more
appropriate metric at $A_{V}\sim 2.1$. However the dispersion is high,
$\sigma=1.0$ even excluding the outlier. Hence, the extinction of the
high–mass stars is generally lower than that of the low–mass population. The
single high mass member of the Southeastern extension detected in X-rays has
$A_{V}\sim 4.5$ consistent with membership in Sub-cluster 14.
Remarkably, the spatial distribution of the high–mass stars does not follow
the sub-clustering seen in the lower mass stars. None of the seven sub-
clusters in the main body of Trumpler 16 has an O star within 0.2 pc of the
cluster center; this includes sub-cluster 11 – the host of $\eta$ Carinæ. The
centroid of the X-ray sources in this sub-cluster is located 0.7′ (0.5 pc)
from $\eta$ Carinæ. The unclustered matrix has the same fraction of high-mass
stars to lower mass stars. Only Sub-cluster 14 in the Southeast extension
appears centered on a high mass star.
## 6 Trumpler 16 in Context
In Paper I, it is shown that the Carina nebula has very complex clustering
properties including many sparse groups, a few small clusters such as Bochum
11 and the “Treasure Chest,” as well as three large clusters, Trumpler 14,
Trumpler 15, and Trumpler 16. This range of stellar groupings is seen in other
large star forming regions such as the Orion A cloud, which includes the ONC,
OMC 2/3, Lynds 1641 North, Lynds 1641 South, as well as numerous small
agglomerations. But comparison of the three largest clusters implies something
other than size matters.
In Table 7 we provide a direct comparison of the clusters. The data sets are
not analyzed in identical ways but the methods are comparable. Trumpler 14
appears to be the youngest of the three clusters as determined by a variety of
metrics including the location of the stars on the color–magnitude diagram and
the disk fraction. It is also possesses the most X-ray sources and is most
centrally condensed. In this Table we calculated the total number of stars in
Trumpler 14 based on the total mass estimate of 9000–11,000 $M_{\odot}$ of
stars by Ascenso et al. (2007) and then estimating the mean stellar mass to be
0.8 $M_{\odot}$ following Hillenbrand & Hartmann (1998). Ascenso et al.
estimate the distance to the Carina nebula to be 2.8 kpc, not the 2.3 kpc used
here, so the total number of stars should be estimated to be perhaps 10%
larger. The densities shown in Table 7 were all estimated using a 2.3 kpc
distance. Meanwhile, Trumpler 15 is the oldest, and lowest mass with a less
dense core than Trumpler 14. Trumpler 14 and Trumpler 15 are well fitted by
King models (King 1962) with a core radius of about 0.7 pc containing about
30–35% of the stars.
Trumpler 16 does not fit into any simple relationship with its two neighbors
to the north. It is nearly identical in mass (stellar numbers) to Trumpler 14.
The main body of Trumpler 16 and the southeastern extension appear to bracket
Trumpler 14 in age and the overall stellar density of the two is nearly the
same. The number of high-mass stars traces the total estimated population of
all three clusters. Of course. some high mass stars have gone supernova – more
in the older clusters. But Trumpler 16 is different. It has no identifiable
core and its high mass population is fully distributed. While Ascenso et al.
find no mass segregation in Trumpler 14 either, the high mass stars trace the
centrally condensed overall population. So overall it would appear that
Trumpler 16 is the result of a different mode of star formation.
In this mode the gas within Trumpler 16 appears to have been collected into
several lumpy groupings within which the mass clumps were not smoothly
distributed. Before a single dynamical timescale, perhaps a triggering event
occurred which caused all the clumps to collapse contemporaneously. This is in
contrast to the rather organized nature of the other two large clusters.
Nonetheless, the mode of star formation which produced Trumpler 16 created a
high-mass to low mass star ratio nearly identical to that seen in the other
clusters and an XLF nearly identical to that of Trumpler 15, which is well
described by a simple King model. This implies that the IMF and the XLF are
robust to different modes of star formation, assuming that the observed IMF
and the XLF are not significantly altered by the evolution of cluster members.
## 7 Conclusions
The Chandra ACIS observations of the Trumpler 16 cluster, portions of which
were previously studied by Albacete-Colombo et al. (2008), were presented in
the framework of the CCCP study. Our analysis shows that Tr 16 is an
irregularly shaped cluster. It is not highly centrally condensed but rather
breaks up into several sub-clusters. With the benefit of the deep HAWK-I data
to identify faint counterparts to the X-ray sources we find no mass
segregation in the cluster as a whole. Neither is there apparent mass
segregation within the individual sub-clusters. In fact, none of the sub-
clusters appear centered on a high mass star with the exception of sub-cluster
14. Other results are summarized as follows:
1. 1.
There are 1187 X-ray sources in the total CCCP sample which are classified as
likely members of the Trumpler 16 cluster. Positional coincidence matching
yields a total of 1047 HAWK-I near-IR counterparts, 1013 of these have three
band detections with errors less than 5%.
2. 2.
The Trumpler 16 cluster has a roughly circular shape about 11′ across (7.4 pc
at a distance of 2.3 kpc). Within this outline are seven sub-clusters
identified in Paper I. We also discuss a less densely populated, more embedded
and younger Southeastern extension to the cluster which is about 10′ long
running east to west and about 5′ north to south (6.7 pc $\times$3.4 pc).
3. 3.
We confirm the well-established correlations between X-ray flux and near-IR
magnitude for high-mass stars and pre-main sequence G and K stars. We also
find a weak anti–correlation between X-ray flux and near-IR luminosity for
intermediate mass stars.
4. 4.
The X-ray luminosity function for stars in Trumpler 16 is compared to the COUP
Orion Nebula Cluster XLF. Trumpler 16 shows more low X-ray luminosity and
solar mass stars and proportionally fewer high luminosity (intermediate/high
mass) stars than in ONC. We estimate the total number of X-ray sources
brighter than $L_{t,c}$= 27.5, the nominal limit of the COUP, to be about 5
times the number of Class II and Class III sources in the ONC area covered by
the COUP survey. This estimate excludes secondary companions.
5. 5.
The locations of X-ray detected stars in Trumpler 16 in the near-IR color-
magnitude diagram is consistent with a population of 1-3 Myr PMS stars. The
extinctions range from near 0 to over 20 in $A_{V}$. There is some spread with
2.0 mag of extinction at $V$ typically separating the first and fourth
quartile.
6. 6.
We derive an overall $K$-band excess disk frequency of 8.9$\pm$ 0.9% using the
X-ray selected sample. Excluding the Southeastern extension the disk frequency
is about 7%. Both rates are significantly larger than the rate found in
Trumpler 15 – the oldest of the Carina rich clusters – of 3.8 $\pm$ 0.7% (Wang
et al. 2011).
7. 7.
We study the seven density enhancements within the main body of the cluster,
some of which present unique characteristics. The stellar characteristics of
the sub-clusters are very similar. The matrix and the subclusters each contain
roughly half the stellar population of Trumpler 16. No mass segregation is
seen (i.e., the massive stars are not concentrated in the cluster cores). The
pre-main sequence disk fraction found in the subclusters is 8.4$\pm$ 1.4%
which is consistent with 6.4$\pm$ 1.2% found in the surrounding matrix of
stars. The exception is the Southeastern extension which has a disk fraction
more than a factor of 2 higher. Absorption properties differ, particularly in
the Southeastern extension that is heavily extinguished.
8. 8.
In addition to the high extinction and higher disk fraction, the Southeastern
extension was also found to possess some of the most bolometrically luminous
newly identified O star candidates in the region. Taken together, the high
disk fraction, high extinction, and luminous O stars are evidence that the
Southeastern extension is younger than the core region of Trumpler 16.
9. 9.
We detected 29 previously known high mass stars including $\eta$ Carinæ, WR
25, and main sequence stars with spectral types ranging from B2 to O3. Most B0
to B2 stars in this region were not detected. We find marginal evidence that
high–mass stars are less absorbed than lower mass stars.
10. 10.
The overall structure of Trumpler 16 differs greatly from that of Trumpler 14
and Trumpler 15. Nevertheless the XLF of Trumpler 15 and Trumpler 16 are
nearly identical.
We thank the referee for many useful comments. S.J.W. is supported by NASA
contract NAS8-03060 (Chandra). This work is supported by Chandra X-ray
Observatory grant GO8-9131X (PI: L. Townsley) and by the ACIS Instrument Team
contract SV4-74018 (PI: G. Garmire), issued by the Chandra X-ray Center, which
is operated by the Smithsonian Astrophysical Observatory for and on behalf of
NASA under contract NAS8-03060. AFJM is grateful to NSERC (Canada) and FQRNT
(Quebec) for financial aide. The near-infrared observations were collected
with the HAWK-I instrument on the VLT at Paranal Observatory, Chile, under ESO
program 60.A-9284(K). This research has made use of the SIMBAD database and
the VizieR catalogue access tool, operated at CDS, Strasbourg, France.
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Table 1: Basic Properties of Chandra ACIS Point Sources in the Trumpler 16
Region
Seq. # | Designation | R.A. | Dec. | Net | X-ray Cnt | Class | Sub-
---|---|---|---|---|---|---|---
| CXOGNCJ | J2000. | J2000 | X-ray Cts | err | | cluster
5277 | 104405.29$-$594543.4 | 161.022048 | $-$59.762068 | 24.9 | 5.8 | H2 | Matrix
5294 | 104405.79$-$594353.6 | 161.024162 | $-$59.731567 | 12.4 | 4.7 | H2 | Matrix
5409 | 104407.86$-$594315.7 | 161.032779 | $-$59.721036 | 73.0 | 9.6 | H0 | Matrix
5416 | 104408.06$-$594522.4 | 161.033601 | $-$59.756223 | 36.8 | 6.7 | H2 | Matrix
5466 | 104409.04$-$594538.7 | 161.037671 | $-$59.760751 | 49.5 | 7.4 | H2 | Matrix
5493 | 104409.75$-$594338.7 | 161.040642 | $-$59.727435 | 19.8 | 5.1 | H2 | Matrix
5496 | 104409.80$-$594448.0 | 161.040853 | $-$59.746691 | 49.6 | 7.4 | H2 | Matrix
5534 | 104410.39$-$594311.1 | 161.043323 | $-$59.719771 | 4609.4 | 162.6 | H2 | Matrix
5541 | 104410.47$-$594352.7 | 161.043650 | $-$59.731332 | 15.3 | 4.8 | H2 | Matrix
5576 | 104411.23$-$594445.7 | 161.046796 | $-$59.746034 | 12.6 | 4.0 | H2 | Matrix
5609 | 104411.89$-$594414.8 | 161.049575 | $-$59.737468 | 14.1 | 4.4 | H2 | Matrix
5629 | 104412.43$-$594212.6 | 161.051825 | $-$59.703517 | 18.9 | 4.9 | H2 | Matrix
5638 | 104412.53$-$594351.3 | 161.052242 | $-$59.730934 | 21.9 | 5.2 | H2 | C3
5647 | 104412.84$-$594333.1 | 161.053541 | $-$59.725883 | 16.3 | 4.5 | H2 | C3
5649 | 104412.86$-$594344.6 | 161.053617 | $-$59.729078 | 162.1 | 13.0 | H2 | C3
aafootnotetext: Table 1 with complete notes is published in its entirety in
the electronic edition of the Astrophysical Journal. A portion is shown here
for guidance regarding its form and content.
bbfootnotetext: Column 1: CCCP X-ray catalog sequence number (Broos et al.
2010). Column 2: IAU designation. Columns 3,4: Right ascension and declination
for epoch J2000.0 in degrees. Column 5: Net X-ray events detected in the
source extraction aperture in the full band (0.5-8 keV; Broos et al. 2011).
Column 6: Gaussian error on the net X-ray counts. Column 7: A set of mutually
exclusive classification hypotheses defined for each source in Broos et al.
(2011) H0 : unclassified; H1: source is a foreground main-sequence star; H2:
source is a young star, assumed to be in the Carina complex; H3: source is a
Galactic background main-sequence star; H4: source is an extragalactic source.
Column 8: The sub-cluster within Trumpler 16. C3, C4, C6 etc (Paper I).
’Matrix’ means no sub-cluster and not in the Southeastern extension, SEM means
no sub-cluster and in the Southeastern extension.
Table 2: XPHOT Derived Properties of Chandra ACIS Point Sources in the
Trumpler 16 Region
Seq. # | Designation | Median | error | Log flux | Statistical err | Systematic err | Log $N_{\rm H}$ | Statistical err | Systematic err
---|---|---|---|---|---|---|---|---|---
| | energy [keV] | Med. energy | [ergs cm2 sec-1] | log flux | log flux | [cm-2] | Log $N_{\rm H}$ | Log $N_{\rm H}$
5277 | 104405.29$-$594543.4 | 0.97 | 0.12 | $-$14.64 | $-$15.19 | $-$16.33 | 20.26 | 0.00 | 0.26
5294 | 104405.79$-$594353.6 | 1.08 | 0.19 | $-$14.87 | $-$15.21 | $-$16.56 | 20.26 | 0.36 | 0.26
5409 | 104407.86$-$594315.7 | 2.84 | 0.24 | $-$13.23 | $-$13.98 | $-$13.97 | 22.41 | 0.09 | 0.05
5416 | 104408.06$-$594522.4 | 1.76 | 0.16 | $-$13.95 | $-$14.56 | $-$14.50 | 21.95 | 0.15 | 0.09
5466 | 104409.04$-$594538.7 | 1.61 | 0.14 | $-$13.90 | $-$14.55 | $-$14.46 | 21.78 | 0.18 | 0.12
5493 | 104409.75$-$594338.7 | 1.26 | 0.15 | $-$14.35 | $-$14.78 | $-$14.72 | 20.98 | 0.61 | 0.50
5496 | 104409.80$-$594448.0 | 1.48 | 0.13 | $-$13.97 | $-$14.60 | $-$14.68 | 21.60 | 0.24 | 0.12
5534 | 104410.39$-$594311.1 | 1.51 | 0.03 | $-$11.24 | $-$12.63 | $-$12.21 | 21.48 | 0.06 | 0.12
5541 | 104410.47$-$594352.7 | 1.88 | 0.28 | $-$14.17 | $-$14.56 | $-$14.71 | 22.08 | 0.21 | 0.08
5576 | 104411.23$-$594445.7 | 1.35 | 0.24 | $-$14.65 | $-$14.91 | $-$15.00 | 21.48 | 0.85 | 0.22
5607 | 104411.88$-$594223.2 | 1.47 | 0.28 | $-$14.60 | $-$14.87 | $-$14.89 | 21.60 | 0.68 | 0.24
5609 | 104411.89$-$594414.8 | 1.93 | 0.44 | $-$14.23 | $-$14.54 | $-$14.48 | 22.15 | 0.31 | 0.11
5629 | 104412.43$-$594212.6 | 1.99 | 0.31 | $-$14.10 | $-$14.54 | $-$14.58 | 22.15 | 0.21 | 0.08
5638 | 104412.53$-$594351.3 | 1.29 | 0.16 | $-$14.46 | $-$14.90 | $-$14.85 | 21.30 | 0.67 | 0.35
5647 | 104412.84$-$594333.1 | 1.42 | 0.18 | $-$14.20 | $-$14.58 | $-$14.60 | 21.60 | 0.43 | 0.18
5649 | 104412.86$-$594344.6 | 2.02 | 0.12 | $-$13.18 | $-$14.11 | $-$13.80 | 22.11 | 0.07 | 0.09
aafootnotetext: Table 2 with complete notes is published in its entirety in
the electronic edition of the Astrophysical Journal. A portion is shown here
for guidance regarding its form and content.
Table 3: Summary of Metric for Each Sub-cluster Sub-cluster | No. Sources | Density | Disk Fraction | $<$ Ã${}_{V}>$ | Abs. $K_{S}$
---|---|---|---|---|---
| | | | 25/50/75 | 25/50/75
| | [src pc-2] | | percentiles | percentiles
All | 1187 | 27 | 8.9$\pm 0.9$% | 2.9/3.7/4.8 | 1.25/2.0/2.5
no sub | 506 | $\cdots$ | 6.4$\pm 1.2$% | 2.8/3.8/5.2 | 1.0/2.0/2.5
all N. sub | 525 | $\cdots$ | 8.4$\pm 1.4$% | 2.9/3.6/4.3 | 1.25/2.0/2.5
3 | 33 | 33 | 14.8$\pm 7.4$% | 3.0/3.8/4.5 | 1.75/2.0/2.25
4 | 11 | 34 | 25.0$\pm 17.7$% | 2.7/3.5/3.8 | 1.5/2.5/2.5
6 | 109 | 45 | 6.8$\pm 2.8$% | 2.9/3.6/4.1 | 1.25/2.25/2.75
9 | 53 | 48 | 4.4$\pm 3.1$% | 2.8/3.7/4.3 | 1.75/2.25/3.25
10 | 82 | 42 | 5.4$\pm 2.7$% | 3.1/3.7/4.5 | 1.0/2.0/2.5
11 | 71 | 27 | 7.0$\pm 3.5$% | 1.9/2.9/3.9 | 1.25/1.75/2.5
12 | 166 | 42 | 10.6$\pm 2.7$% | 3.4/3.8/4.3 | 1.25/2.0/2.5
SE ext | 116 | $\cdots$ | 17.8$\pm 4.2$% | 3.2/4.8/6.3 | 0.5/1.5/2.5
14 | 40 | 10 | 21.2$\pm 8.0$% | 5.1/7.4/10.5 | 0.25/0.75/1.5
SE all | 156 | 9 | 18.7$\pm 3.7$% | 4.3/5.6/8.6 | 0.5/1.25/2.25
Table 4: CCCP Detected OB Stars in the Trumpler 16 Region 1
Seq. # | sub Cl. | R.A. | Dec. | Name | SpType | V | X-ray Cts
---|---|---|---|---|---|---|---
| | (J2000.) | (J2000.) | | | (mag) | (0.5-8.0 keV)
5294 | Matrix | 10 44 05.82 | $-$59 35 11.7 | Cl* Trumpler 16 MJ 224 | B1V | 11.14 | 12.4
5534 | Matrix | 10 44 10.39 | $-$59 43 11.1 | WR 25 | WN6h + OB? | 8.1 | 24609
5665 | C3 | 10 44 13.20 | $-$59 43 10.2 | Cl* Trumpler 16 MJ 257 | O3/4If | 10.8 | 359.2
6676 | C6 | 10 44 32.34 | $-$59 44 31.0 | HD 93204 | O5.5V((f)) | 8.42 | 310.8
6773 | C6 | 10 44 33.74 | $-$59 44 15.5 | HD 93205 | O3.5V((f+)) + O8V | 7.75 | 1408.7
6955 | Matrix | 10 44 36.70 | $-$59 47 29.7 | Cl* Trumpler 16 MJ 359 | O8V | 10.89 | 68.9
6691 | Matrix | 10 44 37.17 | $-$59 40 01.3 | Cl* Trumpler 16 MJ 357 | B0.5V | 11.57 | 6.1
7224 | Matrix | 10 44 40.99 | $-$59 40 10.2 | Cl* Trumpler 16 MJ 372 | B0V | 11.4 | 14.2
7277 | Matrix | 10 44 41.80 | $-$59 46 56.4 | Cl Trumpler 16 100 | O5.5V | 8.6 | 976
7621 | Matrix | 10 44 47.31 | $-$59 43 53.2 | CD$-$59 3303 | O7V + O9.5V + B0.2IV | 8.8 | 108.4
8036 | Matrix | 10 44 54.06 | $-$59 41 29.4 | Cl* Trumpler 16 MJ 427 | B1V | 10.9 | 173.1
8380 | C12 | 10 44 59.90 | $-$59 43 14.8 | Cl Trumpler 16 26 | B1.5V | 11.66 | 9.5
8579 | C11 | 10 45 03.16 | $-$59 40 12.5 | Cl* Trumpler 16 MJ 467 | B0.5V | 10.82 | 6.6
$\cdots$ | $\cdots$ | 10 45 03.55 | $-$59 41 04.0 | $\eta$ Carinæ | pec. | 6 |
8648 | C11 | 10 45 04.78 | $-$59 40 53.5 | Cl Trumpler 16 64 | B1.5V:b | 10.7 | 281.2
8705 | C10 | 10 45 05.80 | $-$59 45 19.6 | Cl* Trumpler 16 MJ 484 | O7V | 10 | 204.9
8707 | C12 | 10 45 05.83 | $-$59 43 07.7 | Cl* Trumpler 16 MJ 481 | O9.5V | 9.77 | 102.2
8714 | C11 | 10 45 05.92 | $-$59 40 05.9 | HD 303308 | O4V((f)) | 8.17 | 1654
8758 | C12 | 10 45 06.72 | $-$59 41 56.6 | Cl* Trumpler 16 MJ 488 | O8.5V | 9.9 | 91.8
8831 | C11 | 10 45 08.23 | $-$59 40 49.4 | CPD$-$59 2628 | O9.5V + B0.3V | 9.5 | 65.5
8832 | C10 | 10 45 08.24 | $-$59 46 07.0 | Cl* Trumpler 16 MJ 496 | O8.5V | 10.93 | 1909.4
9028 | C10 | 10 45 12.22 | $-$59 45 00.4 | HD 93343 | O7V(n) | 9.7 | 204.9
9038 | C12 | 10 45 12.65 | $-$59 42 48.7 | Cl* Trumpler 16 MJ 513 | B2:V | 11.2 | 14.9
9044 | C10 | 10 45 12.72 | $-$59 44 46.2 | CPD$-$59 2635 | O8.5 | 9.3 | 384.1
9050 | Matrix | 10 45 12.87 | $-$59 44 19.3 | CPD$-$59 2636 | O8.5 | 9.3 | 579.7
9195 | C12 | 10 45 16.52 | $-$59 43 37.0 | Cl Trumpler 16 112 | O4.5((f)) | 9.3 | 624.4
9344 | C12 | 10 45 20.57 | $-$59 42 51.2 | Cl* Trumpler 16 MJ 554 | O8.5V | 10.09 | 77.1
9857 | C14 | 10 45 36.32 | $-$59 48 23.2 | Cl* Trumpler 16 MJ 596 | O5.5Vz + O9.5V | 12.1 | 69
10748 | Matrix | 10 46 05.70 | $-$59 50 49.4 | LS 1886 | O4V | 10.7 | 544.6
11footnotetext: Adapted from Skiff (2010)
Table 5: IR colors of CCCP Detected OB Stars in the Trumpler 16 Region1
Seq. # | Name | J | Jerr | H | Herr | K | Kerr | $A_{V}(J-K)$ | $A_{V}(J-H)$
---|---|---|---|---|---|---|---|---|---
5294 | Cl* Trumpler 16 MJ 224 | 15.459 | 0.002 | 14.398 | 0.001 | 13.712 | 0.001 | 8.88 | 8.69
5534 | WR 25 | 6.26 | 0.007 | 5.97 | 0.023 | 5.721 | 0.015 | 3.37 | 3.16
5665 | Cl* Trumpler 16 MJ 257 | 7.84 | 0.013 | 7.381 | 0.037 | 7.061 | 0.017 | 4.46 | 4.37
6676 | HD 93204 | 8.026 | 0.021 | 7.987 | 0.035 | 7.97 | 0.029 | 1.17 | 1.36
6773 | HD 93205 | 7.389 | 0.009 | 7.386 | 0.027 | 7.342 | 0.029 | 1.13 | 1.10
6955 | Cl* Trumpler 16 MJ 359 | 9.384 | 0.017 | 9.142 | 0.019 | 9.007 | 0.018 | 2.63 | 2.81
6691 | Cl* Trumpler 16 MJ 357 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
7224 | Cl* Trumpler 16 MJ 372 | 10.106 | 0.021 | 9.898 | 0.019 | 9.866 | 0.02 | 2.01 | 2.57
7277 | Cl Trumpler 16 100 | 7.798 | 0.015 | 7.735 | 0.035 | 7.639 | 0.025 | 1.64 | 1.53
7621 | CD$-$59 3303 | 8.343 | 0.013 | 8.344 | 0.021 | 8.286 | 0.019 | 1.17 | 1.07
8036 | Cl* Trumpler 16 MJ 427 | 10.209 | 0.047 | 10.153 | 0.069 | 10.16 | 0.039 | 1.14 | 1.48
8380 | Cl Trumpler 16 26 | 10.863 | 0.017 | 10.66 | 0.025 | 10.509 | 0.021 | 2.53 | 2.53
8579 | Cl* Trumpler 16 MJ 467 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\eta$ Carinæ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
8648 | Cl Trumpler 16 64 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
8705 | Cl* Trumpler 16 MJ 484 | 8.649 | 0.019 | 8.416 | 0.017 | 8.335 | 0.021 | 2.34 | 2.75
8707 | Cl* Trumpler 16 MJ 481 | 8.916 | 0.007 | 8.8 | 0.021 | 8.762 | 0.02 | 1.61 | 1.91
8714 | HD 303308 | 7.707 | 0.013 | 7.714 | 0.033 | 7.625 | 0.045 | 1.29 | 1.03
8758 | Cl* Trumpler 16 MJ 488 | 9.45 | 0.015 | 9.412 | 0.031 | 9.36 | 0.027 | 1.32 | 1.35
8831 | CPD$-$59 2628 | 9.249 | 0.019 | 9.136 | 0.017 | 9.253 | 0.061 | 0.89 | 1.89
8832 | Cl* Trumpler 16 MJ 496 | 9.27 | 0.017 | 8.96 | 0.021 | 8.813 | 0.019 | 3.00 | 3.30
9028 | HD 93343 | 8.681 | 0.009 | 8.543 | 0.019 | 8.434 | 0.019 | 2.04 | 2.07
9038 | Cl* Trumpler 16 MJ 513 | 9.83 | 0.019 | 9.646 | 0.025 | 9.531 | 0.023 | 2.28 | 2.40
9044 | CPD$-$59 2635 | 8.33 | 0.011 | 8.18 | 0.033 | 8.091 | 0.017 | 2.00 | 2.15
9050 | CPD$-$59 2636 | 8.076 | 0.017 | 7.894 | 0.031 | 7.756 | 0.017 | 2.37 | 2.38
9195 | Cl Trumpler 16 112 | 8.147 | 0.005 | 7.993 | 0.021 | 7.884 | 0.033 | 2.11 | 2.18
9344 | Cl* Trumpler 16 MJ 554 | 9.387 | 0.017 | 9.309 | 0.021 | 9.257 | 0.018 | 1.50 | 1.64
9857 | Cl* Trumpler 16 MJ 596 | 9.293 | 0.021 | 8.809 | 0.027 | 8.505 | 0.023 | 4.51 | 4.55
10748 | LS 1886 | 8.176 | 0.013 | 7.962 | 0.035 | 7.768 | 0.025 | 2.77 | 2.61
11footnotetext: Adapted from Skiff (2010)
Table 6: High Mass Stars in the Trumpler 16 Region - Not Detected in X-rays1
R.A. | Dec. | Name | SpType | V | sub Cl.
---|---|---|---|---|---
(J2000.) | (J2000.) | | | (mag) |
10 44 13.80 | $-$59 42 57.0 | Cl* Trumpler 16 MJ 261 | B0V | 12.1 | Matrix
10 44 14.75 | $-$59 42 51.7 | Cl* Trumpler 16 MJ 263 | B0.5V | 11.9 | Matrix
10 44 26.48 | $-$59 41 02.7 | Cl* Trumpler 16 MJ 306 | B1.5V | 9.88 | Matrix
10 44 28.98 | $-$59 42 34.2 | Cl* Trumpler 16 MJ 323 | B2V | 12.1 | Matrix
10 44 30.49 | $-$59 41 40.5 | Cl* Trumpler 16 MJ 329 | B1V | 10.88 | C4
10 44 32.89 | $-$59 40 25.9 | Cl* Trumpler 16 MJ 339 | B1V | 10.8 | Matrix
10 44 38.66 | $-$59 48 14.2 | Cl Trumpler 16 20 | B1:V | 10.2 | Matrix
10 44 40.32 | $-$59 41 48.8 | Cl* Trumpler 16 MJ 370 | B1V | 10.77 | Matrix
10 44 58.79 | $-$59 49 21.1 | Hen 3–480 | em | 11.5 | Matrix
10 45 00.24 | $-$59 43 34.4 | Cl Trumpler 16 25 | B2V | 11.88 | C12
10 45 02.19 | $-$59 42 01.1 | Cl* Trumpler 16 MJ 466 | B1V | 10.96 | C12
10 45 05.18 | $-$59 41 42.4 | Cl* Trumpler 16 MJ 477 | B1V | 12.1 | C12/11
10 45 05.88 | $-$59 44 18.9 | Cl* Trumpler 16 MJ 483 | B2V | 11.5 | C12
10 45 09.65 | $-$59 40 08.5 | Cl* Trumpler 16 MJ 499 | B2V | 12.2 | C11
10 45 09.74 | $-$59 42 57.1 | Cl* Trumpler 16 MJ 501 | B1V | 11.69 | C12
10 45 11.18 | $-$59 41 11.2 | Cl* Trumpler 16 MJ 506 | B1V | 10.78 | C11
10 45 19.42 | $-$59 39 37.3 | Cl* Trumpler 16 MJ 547 | B1.5V | 12.2 | Matrix
10 45 31.86 | $-$59 51 09.4 | BM VII 10 | S | 13.6 | SEM
10 45 44.61 | $-$59 50 41.1 | FO 16 | OB- | 12.34 | SEM
10 45 54.80 | $-$59 48 15.4 | Trumpler 16 MJ 633 | em | 13.1 | SEM
10 46 32.62 | $-$59 49 50.0 | HD 93538 | A5/8 | 9.6 | SEM
11footnotetext: Adapted from Skiff (2010)
Table 7: Comparison of the Three Massive Clusters in the Carina Nebula
| Trumpler 141 | Trumpler 152 | Trumpler 16
---|---|---|---
X-Ray Sources: | | |
Probable Members3 : | 1378 | 829 | 1187
HAWK-I Detections4 : | 1219 | 748 | 1050
XLF ($\Gamma$) | $\cdots$ | $\sim-$1.27 | $\sim-$1.27
Estimated # Stars ($\pm 10\%$) | 12,5005 | 5,900 | 14,000
High Mass Stars6 | 46 | 24 | 57
Area [ pc2] | | |
Core: | 1.4 | 1.4 | $\cdots$
Total: | 95 | 50 | 80
Density [X-ray sources pc-2] | | |
Core: | 300 | 200 | $\cdots$
Total: | 14.5 | 16.6 | 14.8
Disk Fraction [%] | 9.74 | 3.8 | 7.4 – north
| | | 17.8 – SE ext.
Age | 2-3 Myr4 | 5-10 Myr | 3-4 Myr4
11footnotetext: Reference: Acsenso et al. (2008) unless otherwise noted.
22footnotetext: Reference: Wang et al. (2011) unless otherwise noted.
33footnotetext: Reference: Broos et al. (2011)
44footnotetext: Reference: Preibisch et al. (2011)
55footnotetext: See text for this calculation.
66footnotetext: Reference: Skiff et al. (2010)
Figure 1: Trumpler 16 cluster overview. The background image is a far-red
image from the Digitized Sky Survey (DSS2-I; squared scaling). The square
indicates the field of view of ObsID 6204. The contours indicate an increase
of source density of 1 X-ray source per 30″. The outer thick contour indicates
the extent of the Trumpler 14 and 16 clusters. $\eta$ Car is the bright star
near the north-east X-ray source concentration. The dashed line indicates our
boundary between Trumpler 16 and Trumpler 14 (to the Northwest). Figure 2:
Near-infrared color-color diagram of 1013 CCCP sources in Trumpler 16
(including the Southeastern extension) using the HAWK-I data (Preibisch et al.
2011). A reddening vector of 10 visual magnitudes is indicated. The short
curve in the lower-left indicates the nominal main sequence. The parallel
lines indicate the reddening band for stars without optically thick disks in
the $K_{S}$ band. Triangles indicate stars at least 0.1 magnitudes to the
right of the reddening band; these are probable disk systems with optically
thick disks in the $K_{S}$ band. We find 9% of the X-ray sources which are
associated with probable cluster members have disks. Several of these are high
mass stars. Similar analysis was carried out separately on each subcluster
within Trumpler 16. Figure 3: The X-ray flux (in units of log photons
sec-1cm-2) compared to the J band magnitude. X-ray sources with $J<11$ (O and
early-B stars) and within$14<J<17$ (pre-main sequence stars) show well-known
correlations between X-ray and optical luminosities. The weak anti-correlation
at intermediate luminosities (12$<$ J $<$ 14) is newly reported here. Figure
4: Near-infrared color-magnitude diagrams for Trumpler 16 and several of its
sub–structures. The green solid line indicates a 3 Myr isochrone derived from
Siess et al. (2000) set at a distance modulus of 11.8. The dashed green line
is the same isochrone with 10 $A_{V}$ of extinction applied. The thick purple
line is an approximate fit of the isochrone to the data with only the
extinction being allowed as a free parameter. The short arrow in the upper
middle of each frame shows the derived extinction for each region. The
triangles indicate stars with disks as derived from the IR-color–color
diagram. All of the regions in the main part of Trumpler 16 fit well to an
extinction of $A_{V}=3.3$. The stars in the Southeastern extension average
about 150% this extinction, and its Sub-cluster 14 is even more absorbed. The
bulk of the disked stars are more absorbed than the cluster mean. The cyan
horizontal lines indicate rough color error bars. Figure 5: Histogram of log
$N_{\rm H}$ of 687 X-ray sources in Trumpler 16 as measured using the XPHOT
method. The top axis gives the approximate optical extinction (AV) using a
conversion ratio of $N_{\rm H}$/$A_{V}=1.6\times 10^{21}$ (Vuong et al. 2003).
The histogram is in grey below log $N_{\rm H}$= 21.6 to indicate the less
robust nature of these measurements (see text). Figure 6: Histogram
distribution of the net counts from the 1232 X-rays sources detected in the
Trumpler 16 region. Figure 7: Top: Power-law fits to the Trumpler 16 data
from log $L_{t,c}$= 30.7-31.5 and to the COUP data from log $L_{t,c}$=
30.2-31.5 are shown by blue line. The Trumpler 16 distribution has a slope of
$\Gamma=-1.27$, while the COUP data have a slope of $\Gamma=-0.92$ Bottom:
Histogram of absorption-corrected, total-band (0.5-8keV) X-ray luminosities of
all 687 X-ray sources in Trumpler 16 (black, solid) for which XPHOT could
calculate luminosities and which are not in the SE extension. These are
compared to the COUP sample (magenta, dotted) of 839 sources from the ONC.
Figure 8: The sub-clusters within Trumpler 16 with X-ray sources noted by
$\times$ symbols. Each sub-cluster is labeled with designations from Paper I
and shown by an approximate ellipse. In the color version the various sub-
clusters are indicated by color as well. Members of the matrix are indicated
in white and occasionally cross into region ovals as the latter are
approximated. The background image is the same as Fig. 1, but linearly scaled.
Figure 9: The extinction functions for several regions within Trumpler 16.
|
arxiv-papers
| 2011-03-06T14:22:29 |
2024-09-04T02:49:17.496465
|
{
"license": "Public Domain",
"authors": "Scott J. Wolk, Patrick S. Broos, Konstantin V. Getman, Eric D.\n Feigelson, Thomas Preibisch, Leisa K. Townsley, Junfeng Wang, Keivan G.\n Stassun, Robert R. King, Mark J. McCaughrean, Anthony F. J. Moffat and Hans\n Zinnecker",
"submitter": "Scott J. Wolk",
"url": "https://arxiv.org/abs/1103.1126"
}
|
1103.1136
|
# Deterministic spin-wave interferometer based on Rydberg blockade
Ran Wei Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China Bo Zhao bo.zhao@uibk.ac.at Institute for
Theoretical physics, University of Innsbruck, A-6020 Innsbruck, Austria
Institute for Quantum Optics and Quantum Information of the Austrian Academy
of Science,
A-6020 Innsbruck, Austria Youjin Deng yjdeng@ustc.edu.cn Hefei National
Laboratory for Physical Sciences at Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China Yu-Ao Chen Fakultät für Physik, Ludwig-Maximilian-Universität,
Schellingstrasse 4, 80798 München, Germany Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany Jian-Wei Pan
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei, Anhui
230026, China
###### Abstract
The spin-wave (SW) NOON state is an $N$-particle Fock state with two atomic
spin-wave modes maximally entangled. Attributed to the property that the phase
is sensitive to collective atomic motion, the SW NOON state can be utilized as
a novel atomic interferometer and has promising application in quantum
enhanced measurement. In this paper we propose an efficient protocol to
deterministically produce the atomic SW NOON state by employing Rydberg
blockade. Possible errors in practical manipulations are analyzed. A feasible
experimental scheme is suggested. Our scheme is far more efficient than the
recent experimentally demonstrated one, which only creates a heralded second-
order SW NOON state.
###### pacs:
42.50.-p, 42.50.Dv, 32.80.Ee, 32.80.Qk, 37.25.+k, 03.75.Dg
## I Introduction
The NOON state, an $N$-particle Fock state with two modes maximally entangled,
has attracted many interests since it has the potential to enhance the
measurement precision by employing quantum entanglement lee2002 . Attributed
to the property of superresolution and supersensitivity, the NOON state has
been experimentally realized in various photonic systems walther2004 ;
mitchell2004 ; nagata2007 ; resch2007 . Recently, a new type of NOON state -
the atomic spin wave (SW) NOON state - was proposed, and a heralded second-
order SW NOON state as well, was experimentally demonstrated yuao2010 . The
scheme yuao2010 employs Raman transitions to generate the atom-photon
entanglement and the SW NOON state is created in a herald way by detecting the
photons. The SW NOON state can be used as an atomic SW interferometer and can
in principle be implemented in a scalable way. However, owing to the
probabilistic nature, this SW interferometer works in a very low efficiency
and thus cannot be put into practical measurement.
In recent years, the Ryberg atom draw extensive concern in quantum information
processing molmer2010 . It has large size and can exhibit large electric
dipole moment. This property introduces strong interactions between two
Rydberg atoms. Consequently, in a small volume, when an atom is excited to the
Rydberg state $|r\rangle$, the energy level of state $|r\rangle$ for other
atoms will be shifted by $\Delta_{e}$. Therefore, the probability for other
atoms being excited to $|r\rangle$ is suppressed by a factor of
$1/\Delta_{e}^{2}$. In the limit $\Delta_{e}\rightarrow\infty$, only one atom
is excited to $|r\rangle$. This is the so-called Rydberg blockade mechanism.
The Rydberg blockade has been proposed to deterministically implement quantum
computer and quantum repeater jaksch2000 ; lukin2001 ; saffman2005a ;
saffman2005b ; saffman2009 ; markus2009 ; zhaobo2010 ; yanghan2010 ;
isenhower2010 .
In this paper, we propose an efficient way to implement the SW interferometer
by deterministically generating the SW NOON state with Rydberg blockade. An
elaborate error analysis shows that the $20$th-order SW NOON state can be
generated with $91\%$ fidelity under realistic parameters, and accordingly a
high fidelity SW interferometer with $F\approx 82\%$ can be realized. This
Rydberg-based SW interferometer is much more efficient than the one based on
photon detection and might be used as an inertial sensor, for measuring
position and displacement, or further, for measuring acceleration and platform
rotation. The remaining of this paper is organized as follows. Sec. II
describes an envisioned setup and presents the scheme to generate and measure
the SW NOON state. Error analysis in practical implementations is given in
Sec. III. Experimental realization is suggested in Sec. IV, and finally we
conclude in Sec. V.
## II Protocol
We envision a setup as illustrated in Fig. 1(a). An ensemble of $N$ atoms is
confined in a volume $V$, where the blockade mechanism is effective. In other
words, the scale of $V$ is smaller than the blockade radius. The working
atomic energy levels are chosen to be of the double-$\Lambda$ type, as shown
in Fig. 1(b). They are labeled as the ground state $|g\rangle$, the Rydberg
state $|r_{a}\rangle$, $|r_{b}\rangle$, and the metastable state
$|s_{a}\rangle$, $|s_{b}\rangle$. The atoms are coupled by four types of
classic light pulses propagating along two spatial modes $a,b$, whose wave
vectors are denoted as $\bm{k}_{gr_{a}}$, $\bm{k}_{r_{a}s_{a}}$,
$\bm{k}_{gr_{b}}$ and $\bm{k}_{r_{b}s_{b}}$ respectively. They will also be
used to denote the corresponding light pulses if no ambiguity arises. These
light pulses couple $|g\rangle$ and $|r_{a}\rangle$, $|r_{a}\rangle$ and
$|s_{a}\rangle$, $|g\rangle$ and $|r_{b}\rangle$, and $|r_{b}\rangle$ and
$|s_{b}\rangle$ respectively, as illustrated in Fig. 1(b).
Before giving the detailed scheme, we shall first introduce some definitions.
We define a collective ground state $|\bm{0}\rangle\equiv|g...g\rangle$, a
collective operator
$M_{\bm{k},\epsilon}^{\dagger}\equiv\frac{1}{\sqrt{N}}\sum\limits_{j}^{N}e^{i\bm{k}\cdot\bm{r}_{j}}|\epsilon_{j}\rangle\langle
g|$, and $|\bm{1},\bm{k}\rangle_{\epsilon}$
$(\epsilon=r_{a},r_{b},s_{a},s_{b})$ to describe a collective state with wave
vector $\bm{k}$,
$|\bm{1},\bm{k}\rangle_{\epsilon}\equiv\frac{1}{\sqrt{N}}\sum\limits_{j}^{N}e^{i\bm{k}\cdot\bm{r}_{j}}|g...\epsilon_{j}...g\rangle=M_{\bm{k},\epsilon}^{\dagger}|\bm{0}\rangle.$
(1)
Namely, state $|\bm{1},\bm{k}\rangle_{\epsilon}$ is a coherent superposition
of states which have a specific atom at $|\epsilon\rangle$ with the position-
dependent phase under the wave vector $\bm{k}$. The same applies to the
higher-order collective state
$|\bm{\ell},\bm{k}\rangle_{\epsilon}\equiv\frac{1}{\sqrt{\ell!}}(M_{\bm{k},\epsilon}^{\dagger})^{\ell}|0\rangle_{\epsilon}$,
with $\ell$ a positive integer. On this basis, a $\ell$th-order SW NOON state
can be written as
$\left|\mathrm{NOON}\right\rangle_{\ell}=\frac{1}{\sqrt{2}}\left(\left|\bm{\ell},\bm{k}\right\rangle_{s_{a}}+\left|\bm{\ell},\bm{k}\right\rangle_{s_{b}}\right).$
(2)
Figure 1: (Color Online) $\bm{(a)}$ An ensemble of N atoms trapped in volume
V. The atoms are coupled by four types of light pulses, propagating along two
spatial modes $a,b$. $\bm{(b)}$ The double-$\Lambda$ type energy levels. An
effective energy shift $\Delta_{e}$ is introduced because of the strong
interaction between the atoms at the Rydberg states.
We first consider the ideal case by making the following assumptions. (1), the
atom number is exactly known, i.e., $\Delta N=0$; (2), the Rydberg blockade
mechanism is perfect, i.e., $\Delta_{e}\rightarrow\infty$; (3), the lifetime
of the Rydberg state is infinite and thus no spontaneous decay occurs; (4),
the atomic cloud remains still during the whole process. On this basis, our
scheme to generate a $\ell$th-order SW NOON state can be described as
1. 1.
Prepare an ensemble at the ground state $|\bm{0}\rangle$.
2. 2.
Apply sequentially a collective $\pi$ pulse $\bm{k}_{gr_{a}}$ and a single-
atomic $\pi/2$ pulse $\bm{k}_{r_{a}s_{a}}$. The former flips one of the $N$
atoms from $|\bm{0}\rangle$ to the Rydberg state
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ and the latter flips
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ to the equal superposition of the
first-order SW state $|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$ and
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$, where
$\bm{k}_{\epsilon_{1}\epsilon_{2}\epsilon_{2}}\equiv\bm{k}_{\epsilon_{1}\epsilon_{2}}-\bm{k}_{\epsilon_{2}\epsilon_{3}}$
$(\epsilon_{1},\epsilon_{2},\epsilon_{3}=g,r_{a},r_{b},s_{a},s_{b})$.
Accordingly, one obtains
$i|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}},$
where a relative phase shift $\pi/2$ is introduced.
3. 3.
Apply successively three collective $\pi$ pulses $\bm{k}_{gr_{b}}$,
$\bm{k}_{gr_{a}}$ and $\bm{k}_{gr_{b}}$, which leads to
$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}-|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$
$\displaystyle\qquad\qquad\qquad\bm{\Downarrow}$ $\displaystyle
i|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}+|\bm{0}\rangle$
$\displaystyle\qquad\qquad\qquad\bm{\Downarrow}$ $\displaystyle
i|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}.$
4. 4.
Apply in order a collective $\pi$ pulse $\bm{k}_{gr_{a}}$ and a single-atomic
$\pi$ pulse $\bm{k}_{r_{b}s_{b}}$, and a collective $\pi$ pulse
$\bm{k}_{gr_{b}}$ and a single-atomic $\pi$ pulse $\bm{k}_{r_{a}s_{a}}$, which
results in
$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{a}}-|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\bm{\Downarrow}$
$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{a}}-i|\bm{1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\bm{\Downarrow}$
$\displaystyle|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{a}}+|\bm{1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}$
$\displaystyle\qquad\qquad\qquad\qquad\bm{\Downarrow}$ $\displaystyle
i|\bm{2},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{gr_{b}}\rangle_{r_{b}}.$
5. 5.
Repeatedly apply a sequence of four collective $\pi$ pulses $\bm{k}_{gr_{a}}$,
$\bm{k}_{r_{b}s_{b}}$, $\bm{k}_{gr_{b}}$, $\bm{k}_{r_{a}s_{a}}$ for $\ell-2$
times, and one obtains
$\displaystyle|\bm{\ell},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{\bm{\ell}-1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{r_{b}s_{b}}\rangle_{r_{b}}.$
6. 6.
Apply a collective $\pi$ pulse to flip the atom from
$|\bm{\bm{\ell}-1},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}|\bm{1},\bm{k}_{r_{b}s_{b}}\rangle_{r_{b}}$
to $|\bm{\ell},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}$ and take into account the
normalized factor, and one obtains a $\ell$th-order SW NOON state
$|\Psi\rangle_{\ell}=(|\bm{\ell},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}+|\bm{\ell},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}})/\sqrt{2}.$
(3)
According to the above procedure, the generation of a $\ell$th-order SW NOON
state needs totally $4\ell+2$ light pulses, the number of which is linear to
$\ell$. Note that one needs two $\pi$ pulses $\bm{k}_{gr_{a}}$ and
$\bm{k}_{r_{a}s_{a}}$ to produce a first-order SW state
$|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$. Accordingly, two $\ell$th-order
SW states $|\bm{\ell},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$ and
$|\bm{\ell},\bm{k}_{gr_{b}s_{b}}\rangle_{s_{b}}$ would consume $4\ell$ light
pulses. The $\ell$th-order SW NOON state is the superposition of two $\ell$th-
order SW states at the $a$ and $b$ modes. Thus, we consider the above protocol
close to being optimal, albeit the possibility of further improvement is not
entirely excluded.
Here we demonstrate how the SW NOON state can be utilized as an atomic
interferometer. Let’s assume that, after the $\ell$th-order SW NOON state is
prepared, the atomic cloud moves to a new position with a displacement
$\Delta\bm{x}$. To measure $\Delta\bm{x}$, we apply a sequence of operations
reverse to the generation procedure, until the last operation, i.e., the
collective $\pi$ pulse $\bm{k}_{gr_{a}}$. Detailed calculations show that we
obtain the superposition state
$\displaystyle|\Psi^{\prime}\rangle_{\ell}$ $\displaystyle=$
$\displaystyle(ie^{i(\bm{k}_{gr_{a}s_{a}})\cdot\Delta\bm{x}}(1+e^{i\ell\Delta\bm{k}\cdot\Delta\bm{x}})|\bm{1},\bm{k}_{gr_{a}s_{a}}\rangle_{s_{a}}$
(4) $\displaystyle+$ $\displaystyle
e^{i\bm{k}_{gr_{b}}\cdot\Delta\bm{x}}(1-e^{i\ell\Delta\bm{k}\cdot\Delta\bm{x}})|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}})/\sqrt{2},$
where $\Delta\bm{k}\equiv-\bm{k}_{gr_{a}s_{a}}-\bm{k}_{gr_{b}s_{b}}$. Note
that, by applying an ionizing electric field, the Rydberg state
$|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ will be ionized and a free electron
will fly out of the atomic ensemble. Thus, the state (4) can be measured onto
the $|\bm{1},\bm{k}_{gr_{a}}\rangle_{r_{a}}$ basis, and the average result
will reflect the phase shift $\ell\Delta\bm{k}\cdot\Delta\bm{x}$. Since the
wave vectors of the light pulses are known, this gives the displacement
$\Delta\bm{x}$. The phase shift is proportional to the order $\ell$, and thus
the larger $\ell$ would bring $\Delta\bm{x}$ the better precision.
## III Error analysis
In actual implementations, errors can always occur. For instance, the precise
number $N$ of atoms in the ensemble is normally unknown, and the atom number
$N$ also varies for different experimental trials. This leads to an
uncertainty $\Delta N$ of the atom number, which is $\Delta N\simeq\sqrt{N}$
for large $N$. Since the collective Rabi frequency $\Omega_{c}$ of the $\pi$
pulse $\bm{k}_{gr_{\lambda}}$ definition is related to the atom number $N$ as
$\Omega_{c}\propto\sqrt{N}$, $\Delta N$ would induce an imprecision in
$\Omega_{c}$ as $\Delta\Omega_{c}/\Omega_{c}\simeq 1/(2\sqrt{N})$. This means
that, when a collective $\pi$ pulse $\bm{k}_{gr_{\lambda}}$ is applied to flip
one of the atoms from $|\bm{0}\rangle$ to
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$, there exists a
probability $p\simeq\pi^{2}/(16N)$ that the flip fails. To generate a
$\ell$th-order SW NOON state, the total error introduced by $\Delta N$ is
about $\pi^{2}\ell/(8N)$. In lab, one can prepare an ensemble of $N\approx
400$ atoms, and thus the error is about $\pi^{2}\ell/(8N)\approx 6\%$ for
order $\ell=20$.
Aside from the error induced by the uncertainty of the atom number, the
imperfect blockade mechanism and the finite lifetime of the Rydberg state also
introduces errors. Attributed to these factors, each operation in our scheme
is implemented with a non-unity probability. We step by step analyze all the
operations from Step $1$ to Step $6$, and find that, these non-unity
probabilities can be categorized into five types, denoted as $P^{I}$,
$P^{II}$, $P^{III}$, $P^{IV}_{q}$, $P^{V}_{q}$, and the generated $\ell$th-
order SW NOON state should be rewritten approximately as
$\displaystyle\sqrt{\mathcal{P}_{\ell}(P^{I}P^{II}P^{III})^{\ell}}|\Psi\rangle_{\ell},$
(5)
where $\mathcal{P}_{\ell}=\prod_{q=1}^{\ell}P^{IV}_{q}P^{V}_{q}$. Symbol $q$
stands for the order of the SW state during the generation process, and it
increases from $1$ to $\ell$ as one produces the $\ell$th-order SW NOON state.
Accordingly, the probability for preparing the $\ell$th-order SW NOON state is
$\displaystyle P(\ell)=\mathcal{P}_{\ell}(P^{I}P^{II}P^{III})^{\ell}.$ (6)
The total error accumulated by these operations is the probability that one
fails to generate the $\ell$th-order SW NOON state, thus it reads
$E(\ell)=1-P(\ell)$. (The error induced by the uncertainty of atom number is
not included in $E(\ell)$.) Before evaluating $E(\ell)$, we shall first
analyze the origins of these probabilities.
The probability $P^{I}$ is introduced by the imperfect blockade that occurs
between the atoms of the same mode when the pulse $\bm{k}_{gr_{\lambda}}$
flips one of the atoms from $|\bm{0}\rangle$ to
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$. In other words, there is
an error that two atoms are excited to the Rydberg state
$|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ due to the non-infinite
energy shift. This mechanics is described by the following equations,
$\displaystyle i\dot{c}_{0}$ $\displaystyle=-\frac{\sqrt{N}\Omega}{2}c_{1},$
(7) $\displaystyle i\dot{c}_{1}$
$\displaystyle=-i\frac{\gamma}{2}c_{1}-\frac{\sqrt{N}\Omega}{2}c_{0}-\frac{\sqrt{2N}\Omega}{2}c_{2},$
(8) $\displaystyle i\dot{c}_{2}$
$\displaystyle=(\Delta_{e}-i\gamma)c_{2}-\frac{\sqrt{2N}\Omega}{2}c_{1},$ (9)
where $c_{0},c_{1},c_{2}$ stand for the amplitudes of $|\bm{0}\rangle$,
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$,
$|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$. Symbols $\gamma/2$ and
$\gamma$ are the decay rates of
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ and
$|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$. Symbol $\Delta_{e}$ is
the effective finite energy shift, and $\sqrt{N}\Omega,\sqrt{2N}\Omega$ are
the corresponding two collective Rabi frequencies, which have been assumed to
be real. Since the amplitudes for the states of more than two atoms being
excited are significantly suppressed due to the Rydberg blockade, we have
neglected them here and in the following. Besides, we have assumed the number
of atoms $N\gg 1$ and the coupling light pulses are all in resonance. The
initial condition describing this mechanics is
$c_{0}(0)=1,c_{1}(0)=0,c_{2}(0)=0$. After applying the collective $\pi$ pulse
$\bm{k}_{gr_{\lambda}}$ with the operation time $\Delta
t=\pi/(\sqrt{N}\Omega)$, one can express the probability for generating
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ from $|\bm{0}\rangle$, as
$P^{I}=|c_{1}(\Delta t)|^{2}$.
The probability $P^{II}$ characterizes the imperfect blockade that takes place
between the atoms of the different modes during $\Delta t$. That is to say,
there is an error that the pulse $\bm{k}_{gr_{\lambda}}$ would flip one of the
atoms from $|\bm{0}\rangle$ to
$|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$ when another atom has
already been excited to
$|\bm{1},\bm{k}_{gr_{\bar{\lambda}}}\rangle_{r_{\bar{\lambda}}}$. Accordingly,
this mechanics is governed by the following equations,
$\displaystyle i\dot{c}_{0}$ $\displaystyle=-\frac{\sqrt{N}\Omega}{2}c_{1},$
(10) $\displaystyle i\dot{c}_{1}$
$\displaystyle=(\Delta_{e}-i\frac{\gamma}{2})c_{1}-\frac{\sqrt{N}\Omega}{2}c_{0}.$
(11)
The initial condition describing this mechanics is $c_{0}(0)=1,c_{1}(0)=0$,
and one can express the probability for holding the atoms at the ground state,
as $P^{II}=|c_{0}(\Delta t)|^{2}$.
The probability $P^{III}$ is contributed by the decay rate of the Rydberg
state. The finite lifetime will inevitably cause some loss when the atom is
still at the Rydberg state during $\Delta t$, thus the probability for the
atom remaining at the Rydberg state is $P^{III}=e^{-\gamma\Delta t}$.
These three types ($P^{I}$, $P^{II}$, $P^{III}$) are all determined by a
shared Rabi frequency $\Omega$ or a shared operation time $\Delta t$. Note
that there is tradeoff between the imperfect Rydberg blockade and the loss
caused by the decay, and a simple argument is that if we enhance the the
magnitude of the Rabi frequency to shorten the operation time, which reduces
the loss from the Rydberg state, it will be associated with more errors from
the imperfect blockade. Therefore, there is an optimal Rabi frequency to
maximize the value of $P^{I}P^{II}P^{III}$. By numerically solving Eqs. (7-9)
and Eqs. (10-11), one can easily obtain this maximal value.
The probability $P^{IV}_{q}$ reflects an error that one of the atoms at
$|\bm{q-1},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$
would be flipped back to
$|\bm{q-2},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$
when the pulse $\bm{k}_{r_{\lambda}s_{\lambda}}$ is applied to flip the atom
from
$|\bm{q-1},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$
to $|\bm{q},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}$. This
mechanics is described by the following equations,
$\displaystyle i\dot{\widetilde{c}}_{0}$
$\displaystyle=-\frac{\sqrt{q}\widetilde{\Omega}}{2}\widetilde{c}_{1},$ (12)
$\displaystyle i\dot{\widetilde{c}}_{1}$
$\displaystyle=-i\frac{\gamma}{2}\widetilde{c}_{1}-\frac{\sqrt{q}\widetilde{\Omega}}{2}\widetilde{c}_{0}-\frac{\sqrt{2(q-1)}\widetilde{\Omega}}{2}\widetilde{c}_{2},$
(13) $\displaystyle i\dot{\widetilde{c}}_{2}$
$\displaystyle=(\Delta_{e}-i\gamma)\widetilde{c}_{2}-\frac{\sqrt{2(q-1)}\widetilde{\Omega}}{2}\widetilde{c}_{1},$
(14)
where $\widetilde{c}_{0},\widetilde{c}_{1},\widetilde{c}_{2}$ are the
amplitudes of $|\bm{q},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}$,
$|\bm{q-1},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{1},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$,
$|\bm{q-2},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}|\bm{2},\bm{k}_{gr_{\lambda}}\rangle_{r_{\lambda}}$.
Symbols $\sqrt{q}\widetilde{\Omega},\sqrt{2(q-1)}\widetilde{\Omega}$ are the
corresponding two collective Rabi frequencies, which have also been assumed to
be real. The initial condition describing this mechanics is
$\widetilde{c}_{0}(0)=0,\widetilde{c}_{1}(0)=1,\widetilde{c}_{2}(0)=0$. After
applying the collective $\pi$ pulse $\bm{k}_{r_{\lambda}s_{\lambda}}$ with the
operation time $\Delta\widetilde{t}_{q}=\pi/(\sqrt{q}\widetilde{\Omega})$, one
can express the probability for producing the $q$th-order SW state
$|\bm{q},\bm{k}_{gr_{\lambda}s_{\lambda}}\rangle_{s_{\lambda}}$, as
$P^{IV}_{q}=|\widetilde{c}_{0}(\Delta\widetilde{t}_{q})|^{2}$.
The origin of $P^{V}_{q}$ is similar to $P^{III}$, it reflects the probability
that the atom remains at the Rydberg state during $\Delta\widetilde{t}_{q}$,
and thus $P^{V}_{q}=e^{-\gamma\Delta\widetilde{t}_{q}}$. The value of
$P^{IV}_{q}P^{V}_{q}$ is determined by a shared Rabi frequency
$\widetilde{\Omega}$ or a shared operation time $\Delta\widetilde{t}_{q}$.
Likewise, one can calculate the maximal value of $P^{IV}_{q}P^{V}_{q}$ by
numerically solving Eqs. (12-14) with $q$ from $1$ to $\ell$.
To evaluate $E(\ell)$, we choose the parameters as, the atom number $N=400$,
the lifetime of the Rydberg state $\tau=1/(2\pi\gamma)=300$ $\mu s$ and $400$
$\mu s$, and the energy shift $\Delta_{e}$ varying from $20$ $MHz$ to $400$
$MHz$. Accordingly, Eq.(6) can be calculated in a numerical way. We obtain the
error $E(\ell)$ versus the energy shift $\Delta_{e}$, shown in Fig.2.
Figure 2: (Color Online) The figure demonstrates the error $E(\ell)$ versus
the energy shift $\Delta_{e}$ under various order $\ell$ after generating the
SW NOON state. The solid data and the open one respectively denote the
lifetime of the Rydberg state with $\tau=300$ $\mu s$ and $\tau=400$ $\mu s$.
From the figure, we see that the larger the energy shift, the smaller the
error, and the error vanishes as $\Delta_{e}$ tends to infinity. This is an
anticipated result since the error $E\sim\Omega^{2}/\Delta_{e}^{2}$. However,
in actual experiment, $\Delta_{e}$ cannot be unlimitedly large. An intrinsic
limitation originates from the average distance of two Rydberg atoms, which
should be larger than the radius of each Rydberg atom. In the limit of high
density where the Rydberg atoms remarkably overlap, our blockade model is
inappropriate, and a more elaborate mechanism should be taken into account.
This mechanism goes beyond the extent of our paper and will not be discussed.
Besides, as one readily expects, the figure shows that the error is suppressed
as the lifetime of the Rydberg state becomes longer, and is intensified when
the order $\ell$ of the SW NOON state increases.
## IV Experimental realization
To design an atomic interferometer with sufficiently high precision and
relatively high fidelity, we use the $20$th-order SW NOON state for the
practical application. The interferometer can be implemented by cold alkali
atoms. By choosing the suitable laser polarization, the two spacial modes $a$
and $b$ can be individually addressed. The energy shift is isotropic due to
the property of repulsive van der Waals interaction. The lifetime of the
Rydberg state with $\tau=300\sim 400$ $\mu s$ is achievable by exciting the
atoms to the Rydberg $s$ state with a principal quantum number $n=100$
saffman2005a . In our scheme, the energy shift $\Delta_{e}$ of the Rydberg
state can be expressed as $\Delta_{e}=-n^{11}(c_{0}+c_{1}n+c_{2}n^{2})/r^{6}$
singer2005 , where the terms $1/r^{8}$ and $1/r^{10}$ are neglected due to the
dominating long-range property. For Rubidium,
$c_{0}=13,c_{1}=-0.85,c_{2}=0.0034$ singer2005 , and thus an ensemble of atoms
with the radius $R=3.8$ $\mu m$ enables the energy shift $\Delta_{e}\geq 300$
$MHz$, which ensures the error $E(20)<3\%$, as illustrated in Fig.2. In a
volume of $4\pi/3R^{3}$, a density of $1.7\times 10^{12}$ $cm^{-3}$ allows
$N\approx 400$ atoms in an ensemble. Based on these estimated parameters
above, we suggest to employ the one-dimensional optical lattice as the
experimental setup, where the size of the ensemble can be controlled by tuning
the angle between the trapping light fields fallani2005 . Finally, we should
point out that, to detect the displacement of atomic cloud by the
interferometer, the reverse operations to those in the generation procedure
should be considered, and thus the total error is doubled. Fortunately, the
field ionization can be implemented with near-unity detection efficiency
guerlin2007 . Therefore, taking into account the error induced by the
uncertainty of atom number, our proposed atomic SW interferometer with a high
precision ($\ell=20$) can reach a high fidelity as $F\approx
1-2\times(6\%+3\%)=82\%$.
## V Summary
By employing Rydberg blockade, we have demonstrated an efficient scheme to
deterministically produce the atomic SW NOON state, of which, a direct
application is the atomic SW interferometer. Possible errors in practical
manipulations are analyzed, and the experimental realization also is
suggested. Our proposed atomic SW interferometer is far more efficient than
the recent experimentally demonstrated one, and holds promise in the practical
application.
## VI Acknowledgement
This work is supported by the NNSFC, the NNSFC of Anhui (under Grant No.
090416224), the CAS, the National Fundamental Research Program (under Grant
No. 2011CB921304), and the SFB FOQUS of FWF.
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ran Wei, Bo Zhao, Youjin Deng, Yu-Ao Chen, Jian-Wei Pan",
"submitter": "Ran Wei",
"url": "https://arxiv.org/abs/1103.1136"
}
|
1103.1250
|
# Computation of the structure of magnetized strange quark star
G.H. Bordbar 1,3111Corresponding author 222E-Mail: bordbar@physics.susc.ac.ir
and A. R. Peivand 2 1Department of Physics, Shiraz University, Shiraz 71454,
Iran333Permanent address,
2Department of Physics, Tafresh University, Tafresh, Iran
3Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
In this work, we have calculated some properties of the spin polarized strange
quark matter (SQM) in a strong magnetic field at zero temperature using the
MIT bag model. We have shown that the equation of state of spin polarized SQM
is stiffer than that of the unpolarized case. We have also computed the
structure properties of the spin polarized strange quark star (SQS) and have
found that the presence of magnetic field leads to a more stable SQS compared
to the unpolarized SQS.
## I Introduction
Strange quark stars (SQS) are those which are built mainly from self bound
strange quark matter (SQM). The surface density of SQS is equal to the density
of SQM at zero pressure ($\sim 10^{15}\ g/cm^{3}$), which is fourteen orders
of magnitude greater than the surface density of a normal neutron star. The
central density of these stars is about five times greater than their surface
density haensel ; glendening ; weber ; 10 . The existence of SQS which are
made of SQM was first proposed by Itoh a even before the full developments of
QCD. Later Bodmer b discussed the fate of an astronomical object collapsing
to such a state of matter. In 1970s, after the formulation of QCD, the
perturbative calculations of the equation of state of the SQM was developed,
but the region of validity of these calculations was restricted to very high
densities collins . The existence of SQS was also discussed by Witten c who
conjectured that a first order QCD phase transition in the early universe
could concentrate most of the quark excess in dense quark nuggets. He
suggested that the true state of matter was SQM. Witten proposal was that the
SQM composed of light quarks is more stable than nuclei, therefore SQM can be
considered as the ground state of matter. SQS would be the bulk SQM phase
consisting of almost equal numbers of up, down, and strange quarks plus a
small number of electrons to ensure the charge neutrality. A typical electron
fraction is less than $10^{-3}$ and it decreases from the surface to the
center of SQS haensel ; glendening ; weber ; 10 . SQM would have a lower
charge to baryon ratio compared to the nuclear matter and can show itself in
the form of SQS c ; d ; e ; f .
The collapse of a massive star may lead to the formation of a SQS. A SQS may
be also formed from a neutron star and is denser than the neutron star 2 . If
sufficient additional matter is added to a SQS, it will collapse into a black
hole. Neutron stars with masses of $1.5-1.8M_{\odot}$ with rapid spins are
theoretically the best candidates for conversion to the SQS. An extrapolation
based on this indicates that up to two quark-novae occur in the observable
universe each day. Besides, recent Chandra observations indicate that objects
RX J185635-3754 and 3C58 may be bare SQS prakash .
It is known that the compact objects such as the neutron stars, pulsars,
magnetars, and strange quark stars are under the influence of the strong
magnetic field, which typically is about $10^{15}-10^{19}\ G$ kouv1 ; kouv2 ;
haensel ; glendening ; weber ; 10 . Therefore, in astrophysics, it is of
special interest to study the effect of strong magnetic field on SQM
properties which can be found in the core of neutron stars and also in the
SQS. We note that in the presence of magnetic field, the conversion of neutron
stars to bare quark stars can not take place unless the value of magnetic
field exceeds $10^{20}\ G$ chak .
Recently, we have calculated the structure of unpolarized SQS at zero
temperature nurafshan and finite temperature zamani . In this article, we
focus on SQS which is purely composed of the spin polarized SQM, and
investigate the effects of strong magnetic field on different properties of
such an star. In section 2, we study the spin polarized SQM in the absence and
presence of the strong magnetic field. In section 3, by numerically solving
the Tolman-Oppenhaimer-Volkoff equation, we obtain the structure properties of
the spin polarized SQS. Moreover, we discuss the stability of spin polarized
SQS.
## II Energy calculation for the spin polarized SQM
We consider the spin polarized SQM composed of $u$, $d$, and $s$ quarks with
spin up ($+$) and down ($-$). We denote the number density of quark $i$ with
spin up by $\rho^{(+)}_{i}$, and spin down by $\rho^{(-)}_{i}$. We introduce
the polarization parameter $\xi_{i}$ by
$\xi_{i}=\frac{\rho^{(+)}_{i}-\rho^{(-)}_{i}}{\rho_{i}},$ (1)
where $0\leq\xi_{i}\leq 1\,$ and $\rho_{i}=\rho^{(+)}_{i}+\rho^{(-)}_{i}$.
Under the conditions of beta-equilibrium and charge neutrality, we get the
following relation for the number density,
$\rho=\rho_{u}=\rho_{d}=\rho_{s},$ (2)
where $\rho$ is the total baryonic density of the system.
Now, we calculate the energy density of spin polarized SQM. To calculate the
total energy of spin polarized SQM, we use MIT bag model in which the total
energy is the sum of kinetic energy of quarks plus a bag constant ($B_{bag}$)
chodos . The bag constant $B_{bag}$ can be interpreted as the difference
between the energy densities of the noninteracting quarks and the interacting
ones. Dynamically it acts as a pressure that keeps the quark gas in constant
density and potential. In MIT bag models, different values are considered for
the bag constant such as $55$ and $90\ \frac{MeV}{fm^{3}}$ . We calculate the
energy density of SQM in the absence and presence of the magnetic field in the
following two separate sections.
### II.1 Energy density of spin polarized SQM in the absence of magnetic
field
The total energy of the spin polarized SQM in the absence of magnetic field
($B=0$) is given by
$\varepsilon_{tot}^{(B=0)}=\varepsilon_{u}+\varepsilon_{d}+\varepsilon_{s}+{B_{bag}},$
(3)
where $\varepsilon_{i}$ is the kinetic energy per volume of quark $i$,
$\varepsilon_{i}=\sum_{p=\pm}\
\sum_{k^{(p)}}\sqrt{m_{i}^{2}c^{4}+\hbar^{2}{k^{(p)}}^{2}c^{2}}.$ (4)
We ignore the masses of quarks $u$ and $d$, while we consider $m_{s}=150\,MeV$
for quark $s$. After doing some algebra, supposing that
$\xi_{s}=\xi_{u}=\xi_{d}=\xi$, we get the following relation for the total
energy of the spin polarized SQM,
$\displaystyle\varepsilon^{(B=0)}_{tot}$ $\displaystyle=$
$\displaystyle\frac{3}{16\pi^{2}\hbar^{3}}{\large\sum_{p=\pm}}\left[\frac{\hbar}{c^{2}}\,k_{F}^{(p)}E_{F}^{(p)}\left(2\hbar^{2}k_{F}^{(p)2}c^{2}+m^{2}_{s}c^{4}\right)-m^{4}_{s}c^{5}\ln(\frac{\hbar
k_{F}^{(p)}+E_{F}^{(p)}/c}{m_{s}c})\right]$ (5) $\displaystyle+$
$\displaystyle\frac{3\,\hbar
c\pi^{2/3}}{4}\,\rho^{4/3}\left[(1+\xi)^{4/3}+(1-\xi)^{4/3}\right]+B_{bag},$
where
$k_{F}^{\pm}=(\pi^{2}\rho)^{1/3}(1\pm\xi)^{1/3},$ (6)
and
$E_{F}^{\pm}=\left(\hbar^{2}k_{F}^{(\pm)2}c^{2}+m_{s}^{2}c^{4}\right)^{1/2}.$
(7)
In Fig. 1, we have plotted the total energy density of spin polarized SQM as a
function of the density for different values of the polarization ($\xi$) in
the absence of magnetic field. Fig. 1 shows that the energy is an increasing
function of the density, however the increasing rate of energy versus density
increases by increasing polarization. For each density, we see that the energy
of spin polarized SQM increases by increasing polarization, specially at high
densities.
For the spin polarized SQM, we can also calculate the equation of state (EoS)
using the following relation,
$P(\rho)=\rho\frac{\partial\varepsilon_{tot}}{\partial\rho}-\varepsilon_{tot},$
(8)
where $P$ is the pressure and $\varepsilon_{tot}$ is the energy density which
in the absence of magnetic field, is obtained from Eq. (5). In Fig. 2, we have
shown the pressure of spin polarized SQM as a function of the density for
various values of the polarization parameter in the absence of magnetic field.
We see that for a given density, the pressure increases by increasing
polarization. This shows that the EoS of spin polarized SQM is stiffer than
that of the unpolarized case. From Fig. 2, it can be seen that by increasing
polarization, the density corresponding to zero pressure takes lower values.
### II.2 Energy density of spin polarized SQM in the presence of magnetic
field
In this section, we consider the spin polarized SQM which is under influence
of a strong magnetic field (${\bf B}$). For this system, the contribution of
magnetic energy is $E_{M}=-{\bf M\cdot B}$. If we consider the magnetic field
along $z$ direction, the contribution of magnetic energy of the spin polarized
SQM is given by
$E_{M}=-\sum_{i=u,d,s}M^{(i)}_{z}B,$ (9)
where $M^{(i)}_{z}$ is the magnetization of system corresponding to particle
$i$ which is given by
$M^{(i)}_{z}=N_{i}\mu_{i}\xi_{i}.$ (10)
In the above equation, $N_{i}$ and $\mu_{i}$ are the number and magnetic
moment of particle $i$, respectively. By some simplification, the contribution
of magnetic energy density of the spin polarized SQM,
$\varepsilon_{M}=\frac{E_{M}}{V}$, can be obtained as follows,
$\varepsilon_{M}=-\sum_{i=u,d,s}\rho_{i}\mu_{i}\xi_{i}B.$ (11)
Consequently, the total energy density of spin polarized SQM in the presence
of magnetic field can be written as
$\displaystyle\varepsilon^{(B)}_{tot}$ $\displaystyle=$
$\displaystyle\varepsilon_{tot}^{(B=0)}+\varepsilon_{M}.$ (12)
In Fig. 3, we have shown the total energy density of the spin polarized SQM as
a function of the polarization parameter ($\xi$), for $B=5\times 10^{18}G$ at
various densities. From Fig. 3, we have seen that the energy curve shows a
minimum for each relevant density. This behavior indicates that for each
density there is a metastable state. We have also seen that as the density
increases, this metastable state is shifted to lower values of the
polarization parameter. Therefore, we can conclude that the metastable state
disappears at high densities. We have also found that at high densities, the
system becomes nearly identical with the unpolarized case. These results agree
with those of reference 6 . In Fig. 4, we have plotted the total energy
density of the spin polarized SQM versus the number density in the presence of
magnetic field. We have seen that the total energy increases by increasing the
density. We have found that the energy density of the spin polarized SQM in
the presence of magnetic field is nearly identical with that of the
unpolarized case which has been clarified in panel (b) of Fig. 4. As we will
see in the next paragraph, this is due to the fact that the polarization
parameter in the presence of magnetic field is very small, especially at high
densities.
In Fig. 5, we have presented the polarization parameter corresponding to the
minimum point of energy density as a function of the number density at
$B=5\times 10^{18}\ G$. We see that the polarization parameter decreases by
increasing the number density. From Fig. 5, it can be seen that for $\rho<0.2\
fm^{-3}$, the decreasing rate of polarization versus density is substantially
higher than for $\rho>0.2\ fm^{-3}$. In Fig. 6, we have shown the polarization
parameter versus the magnetic field for different values of the number
density. For each density, we can see that the polarization increases by
increasing the magnetic field. This figure also shows that the increasing rate
of polarization versus magnetic field increases by increasing density.
We have also calculated EoS of spin polarized SQM in the presence of the
magnetic field, where the contribution of magnetic pressure
($\frac{B^{2}}{8\pi}$) should be added to Eq. (8) in which the total energy
density is obtained from Eq. (12). In Fig. 7, we have plotted EoS of spin
polarized SQM where the magnetic field is switched on. We have found that this
EoS is nearly identical with that of the unpolarized case. This is due to the
fact that polarization at minimum of energy is very low, especially at high
densities.
In Fig. 8, we have plotted the energy per baryon ($E/A$) for the spin
polarized SQM as a function of pressure at $B=5\times 10^{18}\ G$. Our results
for the case of SQM in the absence of magnetic field ($B=0$) are also given
for comparison. We have seen that the zero point of pressure in the presence
of magnetic field has a lower $E/A$ compared to the case of SQM in the absence
of magnetic field ($B=0$). This indicates that, in the presence of magnetic
field, the spin polarized SQM is more stable than that in the absence of
magnetic field.
## III Structure of the spin polarized SQS
The gravitational mass ($M$) and radius ($R$) of compact stars are of special
interests in astrophysics. In this section, we calculate the structure
properties of spin polarized SQS and compare the results of this calculation
with those of the unpolarized case. Using the EoS of spin polarized SQM, We
can obtain $M$ and $R$ by numerically integrating the general relativistic
equations of hydrostatic equilibrium, Tolman-Oppenheimer-Volkoff (TOV)
equations, which are as follows 9 ,
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\varepsilon(r),$ $\displaystyle\frac{dP}{dr}$ $\displaystyle=$
$\displaystyle-\frac{Gm(r)\varepsilon(r)}{r^{2}}\left(1+\frac{P(r)}{\varepsilon(r)c^{2}}\right)\left(1+\frac{4\pi
r^{3}P(r)}{m(r)c^{2}}\right)\left(1-\frac{2Gm(r)}{c^{2}r}\right)^{-1},$ (13)
where $\varepsilon(r)$ is the energy density, $G$ is the gravitational
constant, and
$m(r)=\int_{0}^{r}4\pi r^{\prime 2}\varepsilon(r^{\prime})dr^{\prime}$ (14)
has the interpretation of the mass inside radius $r$. By selecting a central
energy density $\varepsilon_{c}$, under the boundary conditions $P(0)=P_{c}$,
$m(0)=0$, we integrate the TOV equation outwards to a radius $r=R$, at which
$P$ vanishes. This yields the radius $R$ and mass $M=m(R)$ 9 .
Our results for the structure of spin polarized SQS in the absence and
presence of the magnetic field are given separately in two following sections.
### III.1 Structure of the spin polarized SQS in the absence of magnetic
field
In Figs. 9 and 10, we have plotted the gravitational mass and radius of the
spin polarized SQS in the absence of magnetic field versus the central energy
density $(\varepsilon_{c})$ for different values of the polarization parameter
($\xi$). From these figures, we see that for each central density, the mass
and radius of SQS decrease by increasing the polarization parameter. This is
due to the fact that by increasing the polarization parameter, the pressure of
spin polarized SQM increases, which leads to the stiffer equation of state for
this system (Fig. 2). Figs. 9 and 10 show that for a given polarization
parameter, the gravitational mass and radius of SQS increase by increasing the
central density. From Fig. 9, it can be seen that the gravitational mass of
SQS reaches a limiting value called the maximum mass. In Fig. 11, we have
plotted our results for the gravitational mass of spin polarized SQS as a
function of the radius (mass-radius relation) in the absence of magnetic
field. For this system, we see that the gravitational mass increases by
increasing the radius. It is seen that the rate of increasing mass versus
radius increases by increasing the polarization. In Table 1, the maximum mass
($M_{max}$) and the corresponding radius ($R$) of spin polarized SQS have been
given for different values of the polarization parameter ($\xi$) in the
absence of magnetic field. We can see that both maximum mass and the
corresponding radius decrease by increasing $\xi$. This shows that increasing
polarization leads to a more stable SQS.
### III.2 Structure of the spin polarized SQS in the presence of magnetic
field
In this section, we present our calculations for the structure of SQS in the
presence of the magnetic field. It should be noted that the strong magnetic
field changes the spherical symmetry of the system. However, for the magnetic
fields less than $10^{19}\ G$, this effect is negligible gonzalez ; perez ,
therefore, we can solve the TOV equations using a spherical metric, which
leads to Eq. (13). Our results for the gravitational mass and radius of the
spin polarized SQS in the presence of magnetic field versus the central energy
density $(\varepsilon_{c})$ have been shown in Figs. 12 and 13, respectively.
In these figures, our results for the unpolarized case of SQS ($B=0$) are also
given for comparison. Figs. 12 and 13 show that for all values of central
density, the mass and radius of SQS decrease when the magnetic field is
switched on. From Fig. 12, we see that as the central density increases, the
gravitational mass of SQS increases and finally reaches a limiting value
(maximum mass). In Table 2, we have given the maximum mass and the
corresponding radius of SQS for two cases $B=0$ (unpolarized SQS) and
$B=5\times 10^{18}\ G$. It is shown that the presence of magnetic field leads
to lower values for both maximum mass and the corresponding radius of SQS
showing a more stable SQS compared to the unpolarized SQS.
## IV Summary and Conclusions
We have studied the spin polarized strange quark matter (SQM) for both cases
in the absence and presence of magnetic field. We have calculated some of the
bulk properties of this system such as the energy, equation of state (EoS),
and polarization. We have shown that the energy of spin polarized SQM in the
absence of magnetic field increases by increasing polarization. Calculation of
energy in the presence of magnetic field shows that for each density, there is
a minimum point for the energy of SQM showing a metastable state. We have seen
that the EoS of spin polarized SQM becomes stiffer as the polarization
increases. We have also seen that the spin polarized SQM in the presence of
magnetic field is more stable than the unpolarized SQM. The structure
properties of spin polarized strange quark star (SQS) have been also
calculated in the absence and presence of the magnetic field. We have seen
that for each central density, the mass and radius of spin polarized SQS
decrease by increasing polarization. We have also seen that both maximum mass
and the corresponding radius of this system decrease by increasing
polarization. We have indicated that in the presence of magnetic field, the
maximum mass and the corresponding radius of the polarized SQS get lower
values than those of unpolarized SQS. Therefore, we can conclude that the
presence of magnetic field leads to a more stable SQS compared to the
unpolarized SQS.
Our results for the maximum mass and radius of SQS (Tables 1 and 2) are
consistent with those observed for the object SAX J1808.4-3658 li . We can
conclude that this object is a good candidate for SQS.
One of the other astrophysical implications of our results is calculation of
the surface redshift $(z_{s})$ of SQS. This parameter is of special interest
in astrophysics and can be obtained from the mass and radius of the star using
the following relation 10 ,
$\displaystyle z_{s}=(1-\frac{2GM}{Rc^{2}})^{-\frac{1}{2}}-1.$ (15)
Our results corresponding to the maximum mass and radius of SQS lead to
$z_{s}=0.45\ m\,s^{-1}$ in the absence of magnetic field and $z_{s}=0.44\
m\,s^{-1}$ for the magnetic field $B=5\times 10^{18}\ G$. This indicates that
the presence of magnetic field leads to the (nearly) lower values for the
surface redshift.
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University and Tafresh
University Research Councils. One of us (A. R. Peivand) also wishes to thank
M. Mirza.
## References
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* (2) Glendenning N. K., 2000, _Compact Stars: Nuclear Physics, Particle Physics, and General Relativity_ , New York: Springer.
* (3) Weber F., 1999, _Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics_ , Bristol: IOP Publishing.
* (4) Camenzind M., 2007, _Compact Objects in Astrophysics: White Dwarfs, Neutron Stars and Black Holes_ , Springer.
* (5) Itoh N., 1970, Prog. Theor. Phys. 44, 291.
* (6) Bodmer A. R., 1971, Phys. Rev. D 4, 1601.
* (7) Collins J. C., Perry M. G., 1975, Phys. Rev. Lett. 34, 1353.
* (8) Witten E., 1984, Phys. Rev. D 30, 272.
* (9) Alcock C., Farhi E., Olinto A., 1986, Astrophy. J. 310, 261.
* (10) Haensel P., Zdunik J. L., Schaeffer R., 1986, Astron. Astrophys. 160, 121.
* (11) Kettner C., Weber F., Weigel M. K., Glendenning N. K., 1995, Phys. Rev. D 51, 1440.
* (12) Bhattacharyya A. et al., 2006, Phys. Rev. C 74, 065804\.
* (13) Prakash M., Lattimer J. M., Steiner A. W., Page D., 2003, Nucl. Phys. A 715, 835.
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* (15) Kouveliotou C. et al., 1998, Nature 393, 235.
* (16) Ghosha T., Chakrabarty S., 2001, Phys. Rev. D 63, 043006.
* (17) Bordbar G. H., Nourafshan M. and Khosropour B., 2009, Iranian J. Phys. Res. 9, 237 .
* (18) Bordbar G. H., Poostforush A. and Zamani A., Astrophys. (2011) accepted for publication.
* (19) Chodos A. et al., 1974, Phys. Rev. D 9, 3471 .
* (20) Pal K., Biswas S., Dutt-Mazumder A. K., 2009, Phys. Rev. C 79, 015205\.
* (21) Shapiro Stuart L. and Teukolsky Saul. A., 1983, _Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects_ , NewYork: Wiley-Interscience, First edition.
* (22) Gonzalez Felipe R., Perez Martinez A., 2009, J. Phys. G 36, 075202\.
* (23) Perez Martinez A., Gonzalez Felipe R., Manreza Paret D., 2010, arXiv:1001.4038.
* (24) Li X.-D et al, 1999, Phys.Rev.Lett. 83, 3776-3779.
Table 1: Maximum gravitational mass ($M_{max}$) and the corresponding radius ($R$) of the spin polarized SQS for different values of the polarization parameter. $\mathbf{Star}$ | $\mathbf{M_{max}\ (M_{\odot})}$ | $\mathbf{R\ (km)}$
---|---|---
Unpolarized SQS $(\xi=0)$ | 1.35 | 7.6
Polarized SQS (${\xi=0.33}$) | 1.33 | 7.5
Polarized SQS (${\xi=0.66}$) | 1.27 | 7.2
Polarized SQS (${\xi=1}$) | 1.17 | 6.7
Table 2: Maximum gravitational mass ($M_{max}$) and the corresponding radius ($R$) of SQS for $B=0$ and $5\times 10^{18}\ G$. $\mathbf{Star}$ | $\mathbf{M_{max}\ (M_{\odot})}$ | $\mathbf{R\ (km)}$
---|---|---
Unpolarized SQS $(B=0)$ | 1.35 | 7.6
Polarized SQS (${B=5\times 10^{18}G\,}$) | 1.31 | 7.4
Figure 1: The total energy density of spin polarized SQM as a function of the
density ($\rho$) at different values of the polarization parameter ($\xi$) in
the absence of magnetic field.
Figure 2: As Fig. 1 but for the equation of state of spin polarized SQM.
Figure 3: The total energy density of polarized SQM as a function of the
polarization parameter ($\xi$) for $B=5\times 10^{18}\ G$ at different
densities ($\rho$).
Figure 4: (a) The total energy density of spin polarized SQM versus the
density ($\rho$) at $B=5\times 10^{18}G$. (b) Comparison between the total
energy for two cases of $B=5\times 10^{18}\ G$ and $B=0$.
Figure 5: The polarization parameter ($\xi$) corresponding to the minimum
points of the energy density versus the density ($\rho$) at $B=5\times
10^{18}\ G$.
Figure 6: The polarization parameter ($\xi$) corresponding to the minimum
points of the energy density versus the magnetic field ($B$) for different
values of density ($\rho$).
Figure 7: The pressure ($P$) versus density ($\rho$) for spin polarized SQM
at $B=5\times 10^{18}\ G$.
Figure 8: The energy per baryon versus the pressure ($P$) for spin polarized
SQM at $B=0$ (full curve) and $B=5\times 10^{18}\ G$ (dashed curve).
Figure 9: The gravitational mass of spin polarized SQS versus the central
density ($\varepsilon_{c}$) for different values of the polarization parameter
($\xi$) in the absence of magnetic field.
Figure 10: As Fig. 9 but for the radius of spin polarized SQS.
Figure 11: The mass-radius relation for spin polarized SQS in the absence of
magnetic field at different values of the polarization parameter ($\xi$).
Figure 12: The gravitational mass versus the central density
($\varepsilon_{c}$) for the spin polarized SQS at $B=0$ and $B=5\times
10^{18}\ G$.
Figure 13: As Fig. 12 but for the radius of spin polarized SQS.
|
arxiv-papers
| 2011-03-07T11:23:39 |
2024-09-04T02:49:17.511716
|
{
"license": "Public Domain",
"authors": "G. H. Bordbar and A. Peivand",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1103.1250"
}
|
1103.1402
|
# ppiTrim: constructing non-redundant and up-to-date interactomes
Aleksandar Stojmirović and Yi-Kuo Yu∗
National Center for Biotechnology Information,
National Library of Medicine, National Institutes of Health,
Bethesda, MD 20894, United States
stojmira@ncbi.nlm.nih.gov and yyu@ncbi.nlm.nih.gov
###### Abstract
Robust advances in interactome analysis demand comprehensive, non-redundant
and consistently annotated datasets. By non-redundant, we mean that the
accounting of evidence for every interaction should be faithful: each
independent experimental support is counted exactly once, no more, no less.
While many interactions are shared among public repositories, none of them
contains the complete known interactome for any model organism. In addition,
the annotations of the same experimental result by different repositories
often disagree. This brings up the issue of which annotation to keep while
consolidating evidences that are the same. The iRefIndex database, including
interactions from most popular repositories with a standardized protein
nomenclature, represents a significant advance in all aspects, especially in
comprehensiveness. However, iRefIndex aims to maintain all
information/annotation from original sources and requires users to perform
additional processing to fully achieve the aforementioned goals. Another issue
has to do with protein complexes. Some databases represent experimentally
observed complexes as interactions with more than two participants, while
others expand them into binary interactions using spoke or matrix model. To
avoid untested interaction information buildup, it is preferable to replace
the expanded protein complexes, either from spoke or matrix models, with a
flat list of complex members.
To address these issues and to achieve our goals, we have developed ppiTrim, a
script that processes iRefIndex to produce non-redundant, consistently
annotated datasets of physical interactions. Our script proceeds in three
stages: mapping all interactants to gene identifiers and removing all
undesired raw interactions, deflating potentially expanded complexes, and
reconciling for each interaction the annotation labels among different source
databases. As an illustration, we have processed the three largest organismal
datasets: yeast, human and fruitfly. While ppiTrim can resolve most apparent
conflicts between different labelings, we also discovered some unresolvable
disagreements mostly resulting from different annotation policies among
repositories.
URL: www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads/ppiTrim.html
## Introduction
The current decade has witnessed a significant amount of effort towards
discovering the networks of protein-protein interactions (interactomes) in a
number of model organisms. These efforts resulted in hundreds of thousands of
individual interactions between pairs of proteins being reported (1).
Repositories such as the BioGRID (2), IntAct (3), MINT (4), DIP (5), BIND (6,
7) and HPRD (8) have been established to store and distribute sets of
interactions collected from high-throughput scans as well as from curation of
individual publications. Depending on its goals, each interaction database,
maintained by a different team of curators located around the world includes
and annotates interactions differently. Consequently, while many interactions
of specific interactomes are shared among databases (1, 9), no one contains
the complete known interactome for any model organism. Constructing a full-
coverage protein-protein interaction network therefore requires retrieving and
combining entries from many databases.
This task is facilitated by several initiatives developed by the proteomics
community over the years. The IMEx consortium (10) was formed to facilitate
interchange of information between different primary databases by using a
standardized format. The Proteomics Standards Initiative Molecular Interaction
(PSI-MI) format (11) allows a standard way to represent protein interaction
information. One of its salient features is the controlled vocabulary of terms
that can be used to describe various facets of a protein-protein interaction
including source database, interaction detection method, cellular and
experimental roles of interacting proteins and others. The PSI-MI vocabulary
is organized as an ontology, a directed acyclic graph (DAG), where nodes
correspond to terms and links to relations between terms. This enables the
terms to be related in an efficient and algorithm-friendly manner.
Consistently annotated datasets are useful for development and assessment of
interaction prediction tools (12, 13, 14, 15). Furthermore, such datasets also
form the basis of interaction networks, for which numerous analysis tools have
been developed (16, 17). Depending on biological aims of a tool, different
entities (nodes) and potentially weighted interactions (edges) may be
preferred. The chance of conflicting predictions from different tools can be
reduced by starting from a consistently annotated dataset that faithfully
represents all available evidences. Such dataset ought to be comprehensive but
also non-redundant: the same experimental evidence for an interaction should
appear once and only once. To maintain a coherent development of biological
understanding, it is indispensable to keep the reference datasets up-to-date.
We examined several primary interaction databases with the aim of constructing
non-redundant (in terms of evidence), consistently annotated and up-to-date
reference datasets of physical interactions for several model organisms.
Unfortunately, the common standard format used by most primary databases still
does not allow direct compilation of full non-redundant interactomes. This
mainly results from the fact that different primary databases may use
different identifiers for interacting proteins and different conventions for
representing and annotating each interaction. Combining interaction data from
BIND (6, 7) (in two versions called ‘BIND’ and ‘BIND_Translation’), BioGRID
(2), CORUM (18), DIP (5), HPRD (8), IntAct (3), MINT (4), MPact (19), MPPI
(20) and OPHID (21), the iRefIndex (22) database represents a significant
advance towards a complete and consistent set of all publicly available
protein interactions. Apart from being comprehensive and relatively up-to-
date, the main contribution of iRefIndex is in addressing the problem of
protein identifiers by mapping the sequence of every interactant into a unique
identifier that can be used to compare interactants from different source
databases. In a further ‘canonicalization’ procedure (23), different isoforms
of the same protein are mapped to the same canonical identifier. By adhering
to the PSI-MI vocabulary and file format, iRefIndex provides largely
standardized annotations for interactants and interactions. Construction of
iRefIndex led to the development of iRefWeb, a web interface for interactive
access to iRefIndex data (23). iRefWeb allows an easy visualization of
evidence for interactions associated with user-selected proteins or
publications. Recently, the authors of iRefIndex and iRefWeb published a
detailed analysis of agreement between curated interactions within iRefIndex
that are shared between major databases (24).
However, aiming to maintain all information from original sources, iRefIndex
requires users to perform additional processing to fully achieve the
aforementioned goals. In particular, iRefIndex considers redundancy in terms
of (unordered) pairs of interactants rather than in terms of experimental
evidence associated with an interaction. Consequently, there will be features
one desires to have that may not fit well within the scope of iRefIndex. For
example, one may wish to treat interactions arising from enzymatic reactions
as directed and to be able to selectively include/exclude certain types of
reactions such as acetylation. In many cases, the information about post-
translational modifications is available directly from source databases, but
is not integrated into iRefIndex. Another issue that propagates into iRefIndex
from source databases has to do with protein complexes. Some databases
represent experimentally observed complexes as interactions with more than two
participants, while others expand them into binary interactions using spoke or
matrix model (1). Turinsky _et al._ (24) recently observed that this different
representation of complexes is responsible for a significant number of
disagreements between major databases curating the same publication. From our
earlier work (25), we found that such expanded complexes may lead to nodes
with very high degree and often introduce undesirable shortcuts in networks.
To fairly treat the information provided by protein complexes without
exaggeration, it is preferable to replace the expanded interactions, either
from spoke or matrix models, with a flat list of complex members.
Additionally, we discovered that the mapping of each protein to a canonical
group by iRefIndex would sometimes place protein sequences clearly originating
from the same gene (for example differing in one or two amino acids) into
different canonical groups.
To achieve the goal of constructing non-redundant, consistently annotated and
up-to-date reference datasets, we developed a script, called ppiTrim, that
processes iRefIndex and produces a consolidated dataset of physical protein-
protein interactions within a single organism.
## Materials and Methods
Our script, called ppiTrim, is written in the Python programming language. It
takes as input a dataset in iRefIndex PSI-MI TAB 2.6 format, with 54 TAB-
delimited columns (36 standard and 18 added by iRefIndex). After three major
processing steps, it outputs a consolidated dataset, in PSI-MI TAB 2.6 format,
containing only the 36 standard columns (Supplementary Table 1). The three
processing steps are: (i) mapping all interactants to NCBI Gene IDs and
removing all undesired raw interactions; (ii) deflating potentially expanded
complexes; and (iii) collecting all raw interactions, originated from a single
publication, that have the same interactants and compatible experimental
detection method annotations into one consolidated interaction. At each step,
ppiTrim downloads the files it requires from the public repositories and
writes its intermediate results as temporary files.
### Phase I: initial filtering and mapping interactants
In Phase I, ppiTrim takes the original iRefIndex dataset and classifies each
raw interaction (either a binary interaction corresponding to a single line in
the input file or a complex supported by several lines) into one of four
distinct categories: removed (not examined further), biochemical reaction,
complex or potentially part of a complex, and other (direct binary binding
interaction). It removes interactions marked as genetic, originating from
publications specified through a command line parameter or having interactants
from organisms other than the main species of the input dataset (the allowed
species can be explicitly provided or any interaction with interactants having
different Taxonomy IDs is removed). Additionally, ppiTrim removes all
interactions from OPHID and the ‘original’ BIND. The former is removed because
it contains either computationally predicted interactions or interactions
verified from the literature using text mining (i.e. without human curation).
The latter is removed because it processes the same original dataset as
BIND_Translation (7).
As a first step, the script seeks to map each interactant to an NCBI Entrez
Gene (26) identifier. For most interactants, it uses the mapping already
provided by iRefIndex. In the cases where iRefIndex provides only a Uniprot
(27) knowledge base accession, the script attempts to obtain a Gene ID in
three different ways. First, it searches the iRefIndex mappings.txt file
(found compressed in ftp.no.embnet.org/irefindex/data/current/Mappingfiles/
for any additional mappings. This part is optional because the mappings.txt
file is very large even compressed and it would not be feasible to perform
automatic download each time ppiTrim is run. Second, for all unmapped Uniprot
IDs, it retrieves the corresponding full Uniprot records using the dbfetch
tool from EBI (www.ebi.ac.uk/Tools/dbfetch). If a direct mapping to Gene ID is
present within the record as a part of DR field, it is used. Otherwise, the
canonical gene name (field GN) is used to query the NCBI Entrez Gene database
for a matching Gene record using an Eutils interface. If a single unambiguous
match is found, the record’s Gene ID is used for the interactant. No mapping
is performed if multiple matches are obtained. Every mapped Gene ID is checked
against the list of obsolete Gene IDs, which are no longer considered to have
a protein product existing in vivo. The interactants that cannot be mapped to
valid (non-obsolete) Gene IDs are removed along with all raw interactions they
participate in.
After assigning Gene IDs, the script considers the PSI-MI ontology terms
associated with interaction detection method, interaction type and
interactants’ biological roles. Using the full PSI-MI ontology file in Open
Biomedical Ontology (OBO) format (28), it replaces any non-standard terms in
these fields (labeled MI:0000) with the corresponding valid PSI-MI ontology
terms. The terms marked as obsolete in the PSI-MI OBO file are exchanged for
their recommended replacements (Supplementary Table 2). The single exception
are the interaction detection method terms for HPRD ‘in vitro’ (MI:0492,
translated from MI:0045 label in iRefIndex) and ‘in vivo’ (MI:0493)
interactions, which are kept throughout the entire processing.
Source interactions annotated with a descendant of the term MI:0415 (enzymatic
study) as their detection method or with a descendant of the term MI:0414
(enzymatic reaction) as their interaction type are classified as candidate
biochemical reactions. This category also includes any interactions (including
those with more than two interactants) where one of interactants has a
biological role of MI:0501 (enzyme) or MI:0502 (enzyme target). In the recent
months, the BioGRID database has started to provide additional information
about the post-translational modifications associated with the ‘biochemical
activity’ interactions, such as phosphorylation, ubiquitination etc. This
information is available from the BioGRID datasets in the new TAB2 format but
is not yet reflected in the PSI-MI terms for interaction type provided in the
PSI-MI 2.5 format or in iRefIndex. Since the post-translational modifications
annotated by the BioGRID can be directly matched to standard PSI-MI terms
(Supplementary Table 3), the script downloads the most recent BioGRID dataset
in TAB2 format, extracts this information and assigns appropriate PSI-MI terms
for interaction type to the candidate biochemical reactions from iRefIndex
that originate from the BioGRID.
Any source interaction not classified as candidate biochemical reaction is
considered for assignment to the candidate complex categories. This category
includes all true complexes (having edge type ‘C’ in iRefIndex), interactions
having a descendant of MI:0004 (affinity chromatography) as the detection
method term or MI:0403 (colocalization) as the interaction type, as well as
the interactions corresponding to the BioGRID’s ‘Co-purification’ category.
Interactions with interaction type MI:0407 (direct interaction) are never
considered candidates for complexes. All source interactions not falling into
candidate biochemical reaction or candidate complex categories are considered
ordinary binary physical interactions.
### Phase II: deflating spoke-expanded complexes
The Phase II script attempts to detect spoke-expanded complexes from
‘candidate complex’ interactions and deflate them into interactions with
multiple interactants. First, all candidate interactions are grouped according
to their publication (Pubmed ID), source database, detection method and
interaction type. Each group of source interactions is turned into a graph and
considered separately for consolidation into one or more complexes. When a
portion of a group of interactions is deflated, we replace these source
interactions by a complex containing all their participants. Each collapsed
complex is represented using bipartite representation in the output MITAB file
(the same as the original complexes from iRefIndex, but using newly generated
complex IDs) and the references to the original source interactions are
preserved (Supplementary Table 1). Two procedures are used for consolidation:
pattern detection and template matching (Fig. 1). The deflation algorithm for
each new complex is indicated in the output file through its edge type (Table
1).
Figure 1: ppiTrim uses two procedures for complex deflation: pattern detection
(top) and template matching (bottom). As an example, assume that a graph
ABCDEFG, shown on the left, could be constructed from complex candidate
interactions annotated by the BioGRID from a single publication. The arrows
indicate bait to prey relationships, with the interaction A–D being repeated
twice, once with A and once with D as a bait. Pattern detection algorithm
(top) would recognize A and D as hubs of potentially spoke-expanded complexes
and thus replace all pairwise interactions on the left with complexes ABCDEF
and ACDEFG. Suppose that the complex ACDEF was reported from the same
publication by a different database. Then, template matching procedure
(bottom) would generate the complex ACDEF (with all other annotation, such as
experimental detection method, retained from the original interactions) and
remove all original interactions except D–G and A–B. After performing both
procedures, ppiTrim consolidates the results so that the overall result would
be replacing the original interactions by complexes ACDEF, ABCDEF and ACDEFG
with edge type codes ‘R’, ‘A’ and ‘A’, respectively. The interactions A–B and
D–G would not be retained since they are contained within the deflated
complexes ABCDEF and ACDEFG.
Pattern detection procedure is used only for the interactions from the
BioGRID. Unlike the interactions from the DIP, those interactions are
inherently directed since one protein is always labeled as bait and other as
prey (in many cases this labeling is unrelated to the actual experimental
roles of the proteins). The pattern indicating a possible spoke-expanded
complex consists of a single bait being linked to many preys. Since all
interactions in the BioGRID’s ’Co-purification’ and ’Co-fractionation’
categories arise from complexes that are spoke-expanded using an arbitrary
protein as a bait (BioGRID Administration Team, private communication), a bait
linked to two or more preys can in that case always be considered an expanded
complex and deflated. Such deflated complexes are assigned the edge type code
‘G’. The remainder of the complex candidate interactions from the BioGRID were
obtained by affinity chromatography and are, in most cases, also derived from
complexes. Here we adopted a heuristic that a bait linked to at least three
preys can be considered a complex. Clearly, some experiments involve a single
bait being used with many independent preys, in which case this procedure
would generate a false complex. Therefore, complexes generated in this way are
assigned a different edge type code (‘A’) and the user is able to specify
specific publications to be excluded from consideration as well as the maximal
size of the complex.
Table 1: Edge type codes used by ppiTrim Code | Description
---|---
X | undirected binary interaction (physical binding)
D | directed binary interaction (biochemical reaction)
B | biochemical reaction without indication of directionality
C | original complex (from iRefIndex)
G | spoke-expanded complex; deflated by pattern matching from BioGRID’s ’Co-purification’ and ’Co-fractionation’ categories (reliable)
R | potential spoke-expanded complex; deflated by template matching of a ‘C’-complex
A | potential spoke-expanded complex (BioGRID only); deflated by pattern detection
N | potential spoke-expanded complex; deflated by template matching of a ‘G’- or ‘A’-complex
The second procedure is based on matching each group of candidate interactions
to the complexes indicated by other databases (templates), mostly from IntAct,
MINT, DIP and BIND. In this case, the script checks for each protein in the
group whether it, together with all its neighbors, is a superset of a template
complex. If so, all the candidate interactions between the proteins within the
complex are deflated. The neighborhood graph is undirected for all source
databases except the BioGRID. The new complexes generated in this way are
given the code ‘R’. The scripts also attempts to use complexes generated from
the BioGRID’s interactions through a pattern detection procedure as templates,
in which case the newly generated complexes have the code ‘N’. Any source
interactions that cannot be deflated into complexes are retained for Phase
III.
### Phase III: Normalizing interaction type annotation
#### Overview
The goal of the final phase of ppiTrim is to consolidate all evidence for an
interaction, obtained from a single experiment, into one _consolidated
interaction_ record. Every source publication contains descriptions of one or
more experiments that result in reported interactions. Unfortunately, distinct
experiments within each publication are not annotated in all source databases,
with the exception of the interactions from IntAct and MINT that appear to
distinguish experiments using a numbered suffix to the author’s name in the
‘Author’ field. It is therefore necessary to rely on the experimental
detection method terms to determine whether source records from different
databases, with the same interactants and source publication, represent the
evidence for the same interaction. Ideally, all such records with the same
detection method can be collapsed into one consolidated interaction, although
this may undercount multiple evidences from the same publication obtained by
distinct experiments. However, different databases have different annotation
policies and do not necessarily use the same PSI-MI term to annotate a given
experimental method. To resolve detection method term disagreements, we use
the PSI-MI ontology structure (Fig. 2). Two compatible terms assigned by
different source databases are considered to represent the same experimental
method within a publication. These annotated records are thus consolidated.
The Phase III algorithm proceeds as follows. All source interactions and
complexes (original as well as deflated in Phase II) are divided into
‘clusters’. Interactions that share the same interactants and the source
publication are placed into the same cluster. The order of interactants is
significant only for biochemical reactions, which are treated as directed
interactions (only when direction can be ascertained). Each cluster is
processed independently and divided into subclusters based on compatibility of
the PSI-MI terms for interaction detection method. Interactions from each
subcluster are collected into a single consolidated interaction, which is
output to the final dataset. The consolidated record preserves references to
all original interactions. Each consolidated interaction is assigned a single
PSI-MI term for interaction detection method that most specifically describes
the entire collection of annotation terms within the subcluster. For easier
reference, each consolidated interaction is given a unique ppiTrim ID, which
is similar to RIGID from iRefIndex. This is a SHA1 hash of a dot-separated
concatenation of its interactants (Gene IDs), publication(s), detection
method, interaction type and edge type. Every complex uses its ppiTrim ID as
its primary ID.
#### Reconciling annotation
The DAG structure of an ontology naturally induces a partial order between the
terms: for two terms $u$ and $v$, we say that $u$ refines $v$ ($u$ is smaller
$v$, $u$ precedes $v$) if there exists a directed path in the DAG from $u$ to
$v$. Two PSI-MI terms can be considered compatible if they are comparable,
that is, one refines the other. Every nonempty collection of terms $U$ can be
uniquely split into disjoint sets $U_{i}$, such that every $U_{i}$ has a
single maximal element (an element comparable to and not smaller than any
other member) and contains all members of $U$ comparable to its maximal
element. Every subcollection $U_{i}$ is then consistent because there exists
at least one term within it that can describe all its members, while any two
members from different subcollections are incomparable. The _finest consistent
term_ of a subcollection $U_{i}$ is the smallest member of $U_{i}$ that is
comparable to all its members (it can also be defined as the smallest member
of the intersection of the transitive closures of all the members of
$U_{i}$.). If $U_{i}$ is a total order, where all members are pairwise
comparable, the finest consistent term is the minimal term. On the other hand,
the minimal term need not exist (Fig. 2), so that the finest consistent term
is higher in the hierarchy and represents the most specific annotation that
can be assigned to $U_{i}$ as a whole.
To produce consolidated interactions from a single cluster, each of its
members (interactions) is identified with its PSI-MI term for information
detection method. For every cluster member, the set of all other members with
compatible annotations (‘compatible set’) is computed. As a special case, the
following detection method tags are treated as smaller than any other:
‘unspecified method’ (MI:0686), ‘in vivo’ and ‘in vitro’ (The latter two are
from HPRD only). In this way, non-specific annotations are considered as
compatible with all other, more specific evidences. Compatible sets are
further grouped according to their maximal elements. Within each group, the
union of the compatible sets produces a subcluster. The finest consistent term
for each subcluster is found by considering all PSI-MI terms on the paths from
the subcluster members to its maximum – the search is not restricted to those
terms that are within the subcluster (Fig. 2).
#### Conflicts
We consider two subclusters of the same cluster to be in an unresolvable
conflict if there is no source database shared between them. This definition
takes into account that a source database may report an interaction several
times for the same publication, using the same or different interaction
detection method. If two databases annotate the same interaction using
incompatible terms, this is most likely due to an error or specific
disagreement about the appropriate label, rather than that each database is
reporting a different experiment from the same publication. Unresolvably
conflicting interaction records, after consolidation, point to each other
using ppiTrim ID in the ‘Confidence’ field.
ppiTrim also collects statistics about resolvable conflicts in its temporary
output files. A resolvable conflict is the case where source interactions
within a single subcluster have compatible but different experimental
detection method labels.
Figure 2: The picture shows a part of the PSI-MI ontology graph for
interaction detection method associated with a hypothetical cluster of source
interactions involving the same interactants from the same publication. The
terms colored blue are associated with the source interactions within the
cluster, while those marked yellow and green are present in the ontology but
do not label any source interaction from the cluster. The entire cluster as
shown is consistent, with the term MI:0401 as the maximal element. Its finest
consistent term is MI:0004 (colored green) since the cluster members smaller
than it are not comparable between themselves. Removing the source
interactions labeled by MI:0401 from the cluster would result in three
distinct subclusters. If two subclusters contain no interaction from the same
source database, they would be reported as conflicts.
### Evaluation of the script
To test ppiTrim, we applied it to the yeast (S. cerevisiae), human (H.
sapiens) and fruitfly (D. melanogaster) datasets from iRefIndex release
8.0-beta, dated Jan 19th 2011. The script was run on June 13th 2011 and used
the then-current versions of Uniprot and NCBI Gene databases. We restricted
protein interactors to allowed NCBI Taxonomy IDs: 4932 and 559292 for yeast,
9606 for human, and 7227 for fruitfly datasets. When processing the yeast
dataset, we accounted for two special cases. First, we specifically removed
the genetic interactions reported by Tong _et al._ (29) because they were not
labeled as genetic for all source databases. Second, we excluded the dataset
by Collins _et al._ (30) from Phase II and retained all its interactions as
binary undirected. This dataset is present only in the BioGRID and can be
considered computationally derived and partially redundant. Collins _et al._
(30) reprocessed the data from Gavin _et al._ (31) and Krogan _et al._ (32) to
obtain an improved set of pairwise interactions. Collins _et al._ (30) used
hierarchical clustering to recover protein complexes, but these are not
present in the BioGRID. In spite of its redundancy, we decided not to entirely
remove this dataset but also not to attempt to deflate its potential complexes
because bait/prey assignments may not be meaningful in this case.
## Results and Discussion
The results of applying ppiTrim to process iRefIndex 8.0 are shown in Tables 2
– 5. The statistics of ID mapping (Tables 2 and 3) show that a considerable
number of interactants could be additionally mapped to Gene ID in human and
fruitfly datasets, thus enabling us to take into consideration a few thousand
of raw interactions that would otherwise be filtered. This is also evident in
terms of iRefIndex RIGIDs (Supplementary Table 4), which associate all raw
interactions with interactants with same sequences to a single record. For
yeast, the number of interactions gained by mapping to Gene IDs is small
because most of mapped IDs were not valid.
Table 2: Processing source interactions Species | Initial | Removed | Without Gene ID | Retained | With Mapped Gene ID
---|---|---|---|---|---
S. cerevisiae | 400449 | 173815 | 3608 | 223026 | 880
H. sapiens | 382094 | 148724 | 2738 | 230632 | 16187
D. melanogaster | 154770 | 32477 | 9476 | 112817 | 3427
Statistics of initial processing of raw interactions from iRefIndex. Shown are the initial number, total number removed due to filtering criteria, number removed due to missing Gene ID, total number of retained and the number retained containing at least one interactant with mapped Gene ID. Table 3: Mapping CROGID identifiers from iRefIndex into Gene IDs Species | Initial CROGIDs | Aditional Mapped | Final
---|---|---|---
total | mapped | orphans | total | valid | CROGIDs | Gene IDs
S. cerevisiae | 6159 | 5552 | 607 | 433 | 47 | 5599 | 5618
H. sapiens | 14047 | 11432 | 2615 | 1261 | 1261 | 12693 | 11786
D. melanogaster | 9379 | 7810 | 1569 | 566 | 566 | 8346 | 7846
Statistics of mapping CROGIDs into Gene IDs. Columns 2-4 show the total number
of CROGIDs considered, the number that could be directly mapped to GeneIDs and
the number of ‘orphans’ that are not associated with a Gene ID in the
iRefIndex file. Columns 5 and 6 show the numbers of CROGIDs additionally
mapped to GeneIDs, while the last two columns show the final number of CROGIDs
accepted and the corresponding number of Gene IDs. It is possible for a CROGID
to map to multiple Gene IDs (if multiple genes encode the same protein
sequence) as well as for multiple CROGIDs to map to a single GeneID (if our
additional mapping links them to the same gene).
We chose to standardize proteins using NCBI Gene identifiers rather than the
iRefIndex-provided canonical IDs (CROGIDs) for several reasons. NCBI Gene
records not only associate each gene with a set of reference sequences, but
also include a wealth of additional data (e.g. list of synonyms) and links to
other databases such as Gene Ontology (33) that are important when using the
interaction dataset in practice. In addition, Gene records are regularly
updated and their status evaluated based on new evidence. Thus, a gene record
may be split into several new records or marked as obsolete if it corresponds
to an ORF that is known not to produce a protein. For network analysis
applications, it is desirable that only the proteins actually expressed in the
cell are represented in the network and hence the gene status provided by NCBI
Gene is a valuable filtering criterion. Our results in yeast (Table 3) support
this premise: most CROGIDs without Gene ID are associated with sequences
derived from ORFs that were subsequently declassified as genes. However,
CROGIDs do have one advantage over NCBI Gene IDs in that they are protein-
based and hence identical protein products of several genes (like histones)
are clustered together.
There are several reasons that our algorithm was able to introduce many
additional associations of CROGIDs to Gene IDs. First, iRefIndex only provides
mappings to Gene IDs for interactors that have a sequence that exactly matches
a sequence in an NCBI RefSeq record (Ian Donaldson, private communication). By
a case-by-case examination of some orphaned yeast sequences that could be
mapped to Gene ID, we found that they were orphans because they differed in
one or two amino acids from that protein’s reference representative in RefSeq
but were not clustered with that representative’s Gene record. Additional
mappings can be found through database cross-reference from a Uniprot record
pointing to a Gene ID. The iRefIndex canonicalization procedure captures some
of these associations in the mappings.txt file but they are not available in
the main iRefIndex MITAB files. We have found (Supplementary Table 5) that
some CROGIDs (mostly in human) can be additionally mapped by using this
information in the mappings.txt file. Notably, ppiTrim accesses a more recent
version of Uniprot then iRefIndex and is thus able to find more mappings by
accessing Uniprot cross-references directly. Finally, there is a substantial
number of Uniprot records that do not have a cross-reference to NCBI Gene but
can be linked to a Gene record through their canonical gene names. This last
approach can be suggested as an improvement for iRefIndex canonicalization
processing.
Around 10% of CROGIDs could not be mapped to Gene IDs even after processing
with ppiTrim algorithms. A few interactors (Supplementary Table 5) have only
PDB accessions as their primary IDs since their interactions were derived from
crystal structures. In such cases, often only partial sequences of
participating proteins are available. These partial sequences cannot be fully
matched to any Uniprot or RefSeq record and hence are assigned a separate ID.
Hence, an improvement for our procedure, that would account for this case as
well as for those unmapped proteins that differ from canonical sequences only
by few amino acids, would be to use direct sequence comparison to find the
closest valid reference sequence. This task may not be technically difficult
(a similar procedure was applied by Alves _et al._ (34) to construct protein
databases for mass spectrometry data analysis) but is beyond the scope of
ppiTrim, which is intended as a relatively short standalone script. In our
opinion, such additional mappings would best be performed at the level of
reference sequence databases such as Uniprot or RefSeq, which contain curator
expertise to resolve ambiguous cases.
Table 4: Deflating spoke-expanded complexes Species | Publications | Pairs | Complexes
---|---|---|---
initial | remaining | C | G | R | A | N
S. cerevisiae | 3924 | 118819 | 28643 | 7729 | 323 | 5384 | 3190 | 1311
H. sapiens | 10317 | 56111 | 35650 | 8382 | 181 | 1143 | 1443 | 304
D. melanogaster | 398 | 1722 | 1053 | 220 | 16 | 82 | 33 | 3
Shown are the numbers of complexes obtained by deflating binary interactions
with affinity chromatography (or related) as experimental method. Types of
complexes are indicated by one letter codes described in Table 1. The counts
of pairs shown include those from publications with fewer than three
interactions (per database), which could never be deflated into complexes.
Protein complexes obtained through chromatography techniques provide
information complementary to direct binary interactions. While it is often
difficult to determine the exact layout of within-complex pairwise
interactions, an identification of an association of several proteins using
mass spectroscopy is an evidence for in vivo existence of that association.
Unfortunately, in spite of its great importance, the currently available
information within iRefIndex is deficient because of different treatments of
complexes by different source databases. Our results (Table 4) show that the
apparently inflated complexity of interaction datasets can be substantially
reduced by attempting to collapse spoke-expanded complexes. For yeast, this
results in almost three quarters reduction of the number of candidate
interactions. The majority of new complexes falls into ‘G’ and ‘R’ categories,
which can be considered most reliable. For the human dataset, reduction is
small as a proportion although in absolute terms the number of new complexes
is over 3000. The fruitfly dataset did not contain many candidate interactions
or complexes and hence not many new complexes were obtained.
In general, it is difficult to assess whether newly generated complexes from
‘A’ and ‘N’ categories are biologically justified, that is, whether they
represent a functional entity. If a bait and its preys genuinely originate
from a single experiment, they definitely form a physical association that may
be a part of or an entire functional complex. Since ppiTrim preserves the
experimental role labels and the original interaction identifiers, little
information is lost by deflating such associations into a single record. On
the other hand, for some publications, especially those involving experiments
with ubiquitin-like proteins as bait, each bait-prey association may represent
a separate experiment and it does not substantiate that different prey
proteins may be co-present in the cell. For example, BioGRID provides 158
physical associations from the paper by Hannich _et al._ (35), each involving
the yeast Smt3p (SUMO, a ubiquitin-like) protein as a bait. In this case, it
is not true that all the involved preys together form a large complex with the
bait. ppiTrim avoids this particular case by not deflating potentially too
large complexes (the maximum deflated complex size is tunable by the user with
the default of 120 proteins), but one can assume that some of deflated
‘complexes’ do not exist in vivo.
To more closely investigate the fidelity of generated complexes, we randomly
sampled 25 ‘A’ and ‘N’ deflated yeast complexes from the final output of
ppiTrim and examined their original publications. Out of these 25 complexes,
15 originated from high-throughput publications (mostly Gavin _et al._ (31)
and Krogan _et al._ (32) – Supplementary Table 6), while 10 came from small
experiments (Supplementary Table 7). In all high-throughput cases, the
deflated complex represents a true experimental association. In the cases when
authors present their own derived complexes, which in many cases can be found
separately under the ‘C’ category, our deflated complexes form parts of larger
derived complexes. Indeed, such derived complexes are obtained by assembling
the results of several bait-prey experiments, each of which forms a single
deflated complex. The results are more varied for low-throughput publications.
In most cases, deflated complexes clearly correspond to functional complexes,
although it is sometimes difficult to fully relate author’s conclusions with
their reported results. In two cases, the inferred association is incorrect
due to curation errors in the original database. We have also found a single
case where the publication authors directly state that proteins in a deflated
complex do not form a stable complex.
While our sample is extremely small, it does indicate several issues arising
from deflation of bait-prey relationships. In most cases, deflated complexes
form parts of what are believed to be functional complexes. It appears that
curation errors or ambiguities may be a more significant source of wrongly
inferred associations than our main assumption that a bait with several preys
in a single publication represents a single unit. Overall, we feel that the
benefits from reduction of interactome complexity outweigh the disadvantages
from potentially over deflating interactions. The best way to solve the
problem of different representations of protein complexes would be at the
level of source databases (BioGRID in particular), by reexamining the original
publications. Our complexes from the ‘R’ category, where deflated complexes
fully agree with an annotated complex from a different database, could serve
as a guide in this case.
Table 5: Final consolidated datasets Species | Publications | Input Pairs | Consolidated | Conflicts
---|---|---|---|---
biochem | other | complexes | directed | undirected | resolvable | unresolvable
S. cerevisiae | 6303 | 5780 | 119329 | 10778 | 5525 | 63648 | 19344 | 454
H. sapiens | 22660 | 2446 | 199094 | 6483 | 2042 | 85480 | 26478 | 1333
D. melanogaster | 564 | 51 | 111862 | 227 | 33 | 27981 | 19430 | 11
For each species, shown are the numbers of input pairs (input complexes are those from Table 4), classified as either biochemical reactions (potentially directed) or others; also shown are the final numbers of consolidated interactions (classified as complexes, directed or undirected). The ‘other’ column accounts only for those interactions that were not deflated into complexes in Phase II. The last two columns show the total numbers of resolvable and unresolvable conflicts between consolidated interactions. An unresolvable conflict is an instance where two consolidated interactions, originated from the same publication, are reported using incompatible experimental detection method labels by different databases. A resolvable conflict is the case where source interactions within a single consolidated interaction have different (but compatible) experimental detection method labels. Table 6: Most common interaction detection method PSI-MI term conflicts Term A | Sources A | Term B | Sources B | Counts
---|---|---|---|---
MI:0007 (anti tag coimmunoprecipitation) | M | MI:0676 (tandem affinity purification) | DI | 132
MI:0004 (affinity chromatography) | B | MI:0363 (inferred by author) | I | 60
MI:0018 (two hybrid) | DIMN | MI:0096 (pull down) | BI | 43
MI:0071 (molecular sieving) | DIN | MI:0096 (pull down) | B | 32
MI:0030 (cross-linking study) | DIMN | MI:0096 (pull down) | B | 22
MI:0007 (anti tag coimmunoprecipitation) | IM | MI:0676 (tandem affinity purification) | DI | 1227
MI:0018 (two hybrid) | BDHIM | MI:0096 (pull down) | BM | 17
MI:0096 (pull down) | B | MI:0107 (surface plasmon resonance) | DM | 6
MI:0008 (array technology) | I | MI:0049 (filter binding) | M | 5
MI:0019 (coimmunoprecipitation) | IM | MI:0096 (pull down) | BI | 5
Top five most common interaction detection method PSI-MI term unresolvable
conflicts for yeast (top) and human (bottom) datasets are shown. Source
databases are indicated by one letter codes B (BioGRID), D (DIP), I (IntAct),
H (HPRD), M (MINT), P (MPPI).
Overall, our processing significantly reduced the number of interactions
within each of the three datasets considered (Table 5). This indicates a
significant redundancy, particularly for protein complexes, original and
deflated (compare Table 4 with Table 5), and for binary interactions. The
directed interactions (biochemical reactions) are relatively rarer and largely
non-redundant at this stage. Given their importance in elucidating biological
function, the directed interactions are expected to be discovered more fully
with time. However, one should note that PSI-MI format can only represent a
static relationship among a set of physical entities involved in the same
event, but cannot actually represent two sides of a reaction e.g. $A+B\to
C+D$. Certain pairs of PSI-MI biological role terms can be combined to
represent interaction direction e.g. ‘enzyme’ and ‘enzyme target’, but these
are weak compared to the rich ways that pathway databases like Reactome (36)
represent events.
To demonstrate the utility of our conflict resolution method, we present the
counts for resolvable and unresolvable conflicts in Table 5. Resolvable
conflicts significantly outnumber the unresolvable ones. Examining the most
common examples of resolvable conflicts (Supplementary Table 8), one can see
that a majority of them indeed represent the same experiment. Possible
exceptions are human interactions annotated by HPRD, which have ambiguous
detection method labels. To address this and similar problems, ppiTrim
provides the maxsources confidence score (Supplementary Table 1), which is an
estimate of the maximal number of independent experiments contributing to a
consolidated interaction. An interesting example of a resolvable conflict in
Supplementary Table 8 is the 444 instances of a consolidated interaction
containing source interactions with detection method labels MI:0004 (affinity
chromatography technology), MI:0007 (anti tag coimmunoprecipitation), and
MI:0676 (tandem affinity purification). This case is very similar to the one
described in Figure 2: the last two terms are incompatible but the first
resolves the conflict as the finest consistent term.
Upon closer examination of the few unresolvable conflicts (Table 6), it can be
seen that most common conflicts arise as instances of few specific labeling
disagreements between databases. In many cases, such disagreements arise from
using different sub-terms of affinity chromatography (see Fig. 2) and can be
resolved by assigning a more general term consistent with both conflicting
terms. In many other cases, the conflicts are due to BioGRID internally using
a more restricted detection method vocabulary than the IMEx databases (DIP,
IntAct and MINT). However, in some rare cases, an unresolvable conflict arises
when different databases annotate different experiments from the same
publication. For example, each of DIP, BioGRID and IntAct report several raw
interactions from the paper by Blaiseau and Thomas (37) (pubmed:9799240),
where yeast Met4p protein interacts with each of Met28p, Met31p and Met32p in
binary interactions. The paper reports several experiments using different
techniques including northern blotting, yeast two hybrid and electrophoretic
mobility shift assays. For the interaction between Met4p and Met28p, BioGRID
and IntAct report only MI:0018 (yeast two hybrid) method, while DIP reports
only MI:0404 (comigration in non denaturing gel electrophoresis), resulting in
unresolvable conflict. Hence, in this case, each database on its own provides
incomplete evidence for this interaction.
The ppiTrim algorithms work best if accurate and fully populated fields for
interaction detection method, publication and interaction type are available
in its input dataset. This requirement is mostly fulfilled. Nevertheless, we
have noticed two minor inconsistencies. The first, which will be fixed in a
subsequent release of iRefIndex (Ian Donaldson, private communication),
involves the PSI-MI labels for interaction detection method for CORUM
interactions and complexes. These are missing from iRefIndex although they are
present in the original CORUM source files. The second issue concerns missing
or invalid Pubmed IDs for certain interactions. We found that a number of
interactions with missing Pubmed IDs come from MINT. Upon inspection of the
original MINT files, we discovered that in many cases MINT supplies a Digital
Object Identifier (DOI) for a publication as its identifier instead of a
Pubmed ID (although the corresponding Pubmed ID can be obtained from the MINT
web interface). To ensure consistency with other source databases within
iRefIndex, it would be desirable to have the Pubmed IDs available for these
interactions as well.
In this paper, we have identified the tasks needed for using combined
interaction datasets provided by iRefIndex as a basis for construction of
reference networks and developed a script to process them into consistent
consolidated datasets. We see ppiTrim as answering a temporary need for a
consolidated database and hope that most of the issues that required
processing will be eventually fixed in upstream databases and distributed
through IMEx consortium. At this stage we have not addressed the issue of
quality of interactions although such information is available in some
databases for some publications (23). Utilizing the quality information in
consolidating datasets demands a universal data-quality measure that is not
yet existent.
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## Acknowledgments
This work was supported by the Intramural Research Program of the National
Library of Medicine at the National Institutes of Health. We thank Dr.
Donaldson for his critical reading of this manuscript and for providing us
with the proprietary version of iRefIndex 7.0 dataset, which was used for
initial development of ppiTrim.
Supplementary Materials for ‘ppiTrim: constructing non-redundant and up-to-
date interactomes’
Aleksandar Stojmirović, and Yi-Kuo Yu
National Center for Biotechnology Information
National Library of Medicine
National Institutes of Health
Bethesda, MD 20894
United States
Supplementary Table 1: Description of ppiTrim MITAB 2.6 columns Column | Short Name | Description | Example
---|---|---|---
1 | uidA | Smallest Gene ID of the interactor A∗† | entrezgene/locuslink:854647
2 | uidB | Smallest Gene ID of the interactor B∗ | entrezgene/locuslink:855136
3 | altA | All gene IDs of the interactor A∗ | entrezgene/locuslink:854647
4 | altB | All gene IDs of the interactor B∗ | entrezgene/locuslink:855136
5 | aliasA | All canonical gene symbols and integer CROGIDs of interactor A | entrezgene/locuslink:BNR1| icrogid:2105284
6 | aliasB | All canonical gene symbols and integer CROGIDs of interactor B | entrezgene/locuslink:MYO5| icrogid:3144798
7 | method | PSI-MI term for interaction detection method | MI:0018(two hybrid)
8 | author | First author name(s) of the publication in which this interaction has been shown‡ | Tong AH [2002]|tong-2002a-3
9 | pmids | Pubmed ID(s) of the publication in which this interaction has been shown | pubmed:11743162
10 | taxA | NCBI Taxonomy identifier for interactor A | taxid:4932(Saccharomyces cerevisiae)
11 | taxB | NCBI Taxonomy identifier for interactor B | taxid:4932(Saccharomyces cerevisiae)
12 | interactionType | PSI-MI term for interaction type | MI:0407(direct interaction)
13 | sourcedb | PSI-MI terms for source databases‡ | MI:0000(MPACT)|MI:0463(grid)| MI:0465(dip)|MI:0469(intact)
14 | interactionIdentifier | A list of interaction identifiers⋆ | ppiTrim:tyuGkSOK231dh3YnSi6GbczJCFE=| MPACT:8233|dip:DIP-11198E|grid:147506| intact:EBI-601565|intact:EBI-601728| irigid:288990|edgetype:X
15 | confidence | A list of ppiTrim confidence scores∙ | maxsources:2|dmconsistency:full| conflicts:S3oaiXt5tA4vVrUsO1rc1TA9krk=
16 | expansion | Either ‘none’ for binary interactions or ‘bipartite’ for subunits of complexes | none
17 | biologicalRoleA | PSI-MI term(s) for the biological role of interactor A‡ | MI:0499(unspecified role)
18 | biologicalRoleB | PSI-MI term(s) for the biological role of interactor B ‡ | MI:0499(unspecified role)
19 | experimentalRoleA | PSI-MI term(s) for the experimental role of interactor A‡ | MI:0496(bait)|MI:0498(prey)| MI:0499(unspecified role)
20 | experimentalRoleB | PSI-MI term(s) for the experimental role of interactor B‡ | MI:0496(bait)|MI:0498(prey)| MI:0499(unspecified role)
21 | interactorTypeA | PSI-MI term for the type of interactor A (either ‘protein’ or ‘protein complex’) | MI:0326(protein)
22 | interactorTypeB | PSI-MI term for the type of interactor B (always ‘protein’) | MI:0326(protein)
29 | hostOrganismTaxid | NCBI Taxonomy identifier for the host organism | taxid:4932(Saccharomyces cerevisiae)
31 | creationDate | Date when ppiTrim was run | 2011/05/11
32 | updateDate | Date when ppiTrim was run | 2011/05/11
35 | checksumInteraction | ppiTrim ID for an interaction | ppiTrim:tyuGkSOK231dh3YnSi6GbczJCFE=
36 | negative | Always ‘false’ | false
The above table shows short descriptions for the columns of lines output by ppiTrim with examples. The columns that are not used by ppiTrim (- output) are omitted. List of items are always separated by the $|$ character (without any intervening spaces). This description only applies to ppiTrim output; the full PSI-MI 2.6 TAB format description can be found at http://code.google.com/p/psimi/wiki/PsimiTab26Format Notes: ∗An interactor may be associated with several Gene IDs. In that case the smallest one is written in uid columns while the entire list is shown in alt columns. †Interactor A may be used to denote a protein complex. In that case the uidA is of the form complex:$<$ppiTrim ID$>$, while altA and aliasA are left empty. ‡Multiple items are possible, originating from all source records contributing to the consolidated interaction. ⋆First ID is always the ppiTrim ID for the consolidated interaction, followed by the original IDs for all contributing interactions and their integer RIGIDs from iRefIndex. The final item is the edge type code. ∙maxsources: an estimate of the maximal number of independent experiments contributing to the consolidated interaction; dmconsistency: consistency of contributing detection method terms. Values are one of invalid (no method terms present), single (only one method term), min (minimum term found but not maximum), max (maximum term found but not minimum), and full (both minimum and maximum term present in subcluster); conflicts: ppiTrim IDs of consolidated interactions with detection method term in conflict with the current one. Supplementary Table 2: Remapping of obsolete PSI-MI terms Original Term | Mapped Term | Notes
---|---|---
MI:0021 | colocalization by fluorescent probes cloning | MI:0428 | imaging technique |
MI:0022 | colocalization by immunostaining | MI:0428 | imaging technique | $\ast$
MI:0023 | colocalization/visualisation technologies | MI:0428 | imaging technique | $\ast$
MI:0025 | copurification | MI:0401 | biochemical |
MI:0059 | gst pull down | MI:0096 | pull down |
MI:0061 | his pull down | MI:0096 | pull down |
MI:0079 | other biochemical technologies | MI:0401 | biochemical |
MI:0109 | tap tag coimmunoprecipitation | MI:0676 | tandem affinity purification |
MI:0045 | experimental interaction detection | MI:0492 | in vitro | $\dagger$
MI:0493 | in vivo | MI:0493 | in vivo | $\dagger$
MI:0000 | coip coimmunoprecipitation | MI:0019 | coimmunoprecipitation | $\star$
MI:0000 | elisa enzyme-linked immunosorbent assay | MI:0411 | enzyme linked immunosorbent assay | $\star$
$\ast$ Interaction type is also adjusted to MI:0403 as recommended in psi-mi.obo; $\dagger$ HPRD terms are treated as a special case, see main text; $\star$ MPPI interactions in the human dataset. Supplementary Table 3: Mapping PTM labels from BioGRID into PSI-MI terms Original Term | Mapped Term
---|---
Acetylation | MI:0192 | acetylation reaction
Deacetylation | MI:0197 | deacetylation reaction
Demethylation | MI:0871 | demethylation reaction
Dephosphorylation | MI:0203 | dephosphorylation reaction
Deubiquitination | MI:0204 | deubiquitination reaction
Glucosylation | MI:0559 | glycosylation reaction
Methylation | MI:0213 | methylation reaction
Nedd(Rub1)ylation | MI:0567 | neddylation reaction
No Modification | MI:0414 | enzymatic reaction
Phosphorylation | MI:0217 | phosphorylation reaction
Prenylation | MI:0211 | lipid addition
Proteolytic Processing | MI:0570 | protein cleavage
Ribosylation | MI:0557 | adp ribosylation reaction
Sumoylation | MI:0566 | sumoylation reaction
Ubiquitination | MI:0220 | ubiquitination reaction
Supplementary Table 4: Processing source interactions (RIGIDs) Species | Initial | Without Gene ID | Retained | With Mapped Gene ID
---|---|---|---|---
S. cerevisiae | 186530 | 1272 | 79931 | 591
H. sapiens | 138570 | 1917 | 84860 | 7158
D. melanogaster | 46925 | 4988 | 39200 | 2176
Statistics of initial processing of raw interactions from in terms of iRefIndex RIGIDs. A RIGID for an interaction is a unique hash derived from its interactants’ sequences (with order not significant). Thus, multiple interactions with the same interactants share the same RIGID. Shown are the initial number, number removed due to missing Gene ID, total number of retained and the number retained containing at least one interactant with mapped Gene ID. Compared to Table 2 in the main text, this table does not contain a column showing the number of removed RIGIDs due to filtering criteria. This is becuase the ppiTrim filtering routine operates on raw interactions (corresponding to a single record from a source database) and some RIGIDs would be associated with both accepted and removed raw interactions. Supplementary Table 5: Mapping CROGID identifiers from iRefIndex into Gene IDs: details Species | I | V | O | R | P | T | M | G | S | B
---|---|---|---|---|---|---|---|---|---|---
S. cerevisiae | 5552 | 0 | 0 | 607 | 95 | 461 | 0 | 26 | 21 | 386
H. sapiens | 11428 | 11 | 0 | 2615 | 155 | 2017 | 71 | 754 | 429 | 0
D. melanogaster | 7780 | 0 | 30 | 1569 | 18 | 814 | 2 | 124 | 440 | 0
Detailed statistics of mapping CROGIDs into Gene IDs. All numbers denote CROGIDs: directly mapped to valid Gene IDs in the iRefIndex file (I); directly mapped to Gene IDs but the Gene IDs were updated during validation (V); directly mapped to obsolete Gene IDs (O); not directly mapped to Gene IDs – total orphans (R); orphans with PDB accession as a primary ID (P); orphans with Uniprot accession as a primary ID (T); additionally mapped to a valid Gene ID using mapping.txt file from iRefIndex (M); additionally mapped to a valid Gene ID using a direct reference from Uniprot record (G); additionally mapped to a valid Gene ID using a gene name from Uniprot record (S); additionally mapped to a Gene ID that was not valid (B). Supplementary Table 6: Randomly sampled deflated complexes from high throughput publications ppiTrim Complex ID | Sources | Pubmed ID | Members | Comments
---|---|---|---|---
8AVRUHG76vkiFn2cZGICNZzr00Y= | grid | 14759368 | CFT2, YSH1, PTA1, MPE1 | Part of mRNA cleavage/polyadenylation complex (4/10 proteins).
9yS57j/gbRbOlNmmimsVeonoraA= | grid | 14759368 | NUT1, MED7, MED4, SIN4, SRB4 | Part of mediator complex.
JU+EOkq6ipLh9DJKRtGRLUvT7vM= | grid,mint | 14759368 | UBP6, RPT3, RPN9, RPT1, RPN8, RPN2, RPN7, RPN1 | Part of proteasome. MINT does not contain complexes from the original paper.
HtTmhGiPyfIT2vFtRZ94uWw0rsY= | grid | 16429126 | IOC3, HTB1, HTA2, HHF2, ISW1, KAP114, ITC1, RPS4A, VPS1, NAP1, RPO31, ISW2, TBF1, BRO1, MOT1 | Part of Complex # 99.
LnNzfyPGShcG7zkKynU6+fsK2eU= | grid | 16429126 | PSK1, NTH1, BMH2, RTG2, BMH1 | Part of complex # 147 (two core proteins plus three attachments).
S2I6VRjFMWC6rkkM+oYXwKCg9YQ= | grid | 16429126 | RPL4B, MNN10, MNN11, HOC1, MNN9, ANP1 | Core complex (# 111 – mannan polymerase II) + one attachment protein (RPL4B).
1fRmAapl2ruoQq202YUJg55maFo= | grid,mint | 16554755 | RSM24, RSM28, MRPS5, MRP13, MRPS35, RSM27, RSM7, RSM25, MRPS17, MRPS12, RSM19, MRP4 | Part of complex # 1.
5tBkYOmK/G1h3vaQmiOnUoBHHMQ= | grid,mint | 16554755 | CFT2, YSH1, MPE1, PAP1 | Part of complex # 18.
9f2DVj2rDGeCP53LHOnWRMwq14A= | grid,mint | 16554755 | KAP95, RTT103, VMA2, RAI1, RAT1, RPB2, SRP1 | True experimental association but not part of any derived complex.
AVawv51+6Fqe3DquygD/XfyrXxE= | grid,mint | 16554755 | RRP42, RRP45, RRP6, CSL4, MPP6, RRP4, LRP1, DDI1 | Part of complex # 19.
NOLEwovavMsFrQEdkSUt/mldeMc= | grid,mint | 16554755 | CDC3, SHS1, CDC11, CDC12 | Part of complex # 121.
WA51i87Lj1wGp/EeF1OV/YvbW1Y= | grid,mint | 16554755 | GTT2, TRX1, CRN1, SSA3, IPP1, CMD1, TRX2, TDH1, RPL40B, CDC21, OYE2 | True experimental association but not part of any derived complex.
YN/hQXQvzoB5HqrgPzVth28mGsY= | grid,mint | 16554755 | RRP43, RRP42, RRP45, RRP40, DIS3, RRP6, RRP4, LRP1 | Part of complex # 19.
1LRk+AgI8HpGOSAgkhDzNJWSvtI= | grid | 20489023 | RTG3, RTG2, TOR1, TOR2, CKA2, MYO2, MKS1, KOG1 | True experimental association.
xWzvxeJFGqjkCihjmQVf5gZhJjQ= | dip,grid,mint | 20489023 | PUF3, SAM1, GCD6, SPT16, MTC1, YGK3, LSM12 | True experimental association.
To partially investigate the fidelity of deflated complexes of type A and N, we randomly sampled 25 such complexes from the final ppiTrim yeast dataset and examined the original publications associated with them. This table contains 15 deflated complexes from high-throughput publications, while Supplemenary Table 7 contains the complexes from low-throughput publications. Most of high-throughput papers referred to in this table present both the lists of bait-prey associations and of derived complexes. The complexes delated by ppiTrim are often derived from the former and form only parts of the latter. In the last column of this table, the complex numbers referred to are labels used by the publication’s authors. Supplementary Table 7: Randomly sampled deflated complexes from low-throughput publications ppiTrim Complex ID | Sources | Pubmed ID | Members | Comments
---|---|---|---|---
15VfQtoe5gxGNwPSY3AG0sq6A2U= | grid | 9891041 | CCR4, HPR1, PAF1, SRB5, GAL11 | NOT a true complex. This is because of bad annotation of PAF1–SRB5 interaction by the BioGRID. Completely opposite interpretation was given in the paper.
d79IdtwfTAENrH8CQ+c8CpS389Y= | grid | 10329679 | YPT1, VPS21, YPT7, GDI1 | True complex. This is the only experiment in the paper.
EtS4cgphEpTqJb/FS5qxyzf0ke8= | grid | 11733989 | CDC39, CCR4, CDC36, CAF130, CAF40, CAF120, POP2, NOT5, MOT2 | True complex. CAF120 is an unusual member that could almost be left out.
2kOyGdwzWywSpN5mhK26gCcC6LQ= | grid | 14769921 | GBP2, IMD3, TEF1, KEM1, CTK2, CTK1, CTK3 | True complex, except that TEF1 should be TEF2. This is an error in the iRefIndex source file; the BioGRID website has the correct assignment.
Kd07BBUF07Sqy9NP3D0lixsS/TY= | grid | 15303280 | BUD31, RPL2B, PRP19, CDC13, ATP1, RPS4A, SNU114, MDH1, MAM33, MRPL3, MRPL17, PRP8, PRP22, PAB1, BRR2 | True association
ZAGz/IZqkEr3/NTDLzPEDAD9cKo= | grid | 16179952 | CDC40, UFD1, SSM4, UBX2 | NOT a true complex, probably due to a typo in annotation. CDC40 cannot be found anywhere in the paper and should most likely be CDC48.
RDu0dsPAN0QEadfSU5sv05Ifihw= | grid | 16286007 | SIN3, RCO1, RPD3, UME1, EAF3 | True complex.
Vqbn3dDwTPgyE9DzbatFNqzdFe0= | grid | 16615894 | VPS36, VPS25, VPS28, SNF8 | Vps28 binds the other three, which form a complex.
lmdypAN9kaHBdasLWS19x8K7KkE= | grid | 20159987 | UBI4, UFD2, PEX29, SSM4 | Biological association but indicated as ‘NOT a stable complex’ in the paper.
aakRh6qVahGxGvqHe399+faxPvA= | grid | 20655618 | PEX13, PEX10, PEX8, PEX12 | Association is correct, although mutant strain was used to obtain this particular complex.
To partially investigate the fidelity of deflated complexes of type A and N, we randomly sampled 25 such complexes from the final ppiTrim yeast dataset and examined the original publications associated with them. This table contains 10 deflated complexes from low-throughput publications, while Supplemenary Table 6 contains the complexes from high-throughput publications. Supplementary Table 8: Summary of resolvable conflicts Consolidated terms | Count
---|---
MI:0018 (two hybrid), MI:0045 (experimental interaction detection), MI:0398 (two hybrid pooling approach), MI:0399 (two hybrid fragment pooling approach) | 3959
MI:0090 (protein complementation assay), MI:0111 (dihydrofolate reductase reconstruction) | 2612
MI:0090 (protein complementation assay), MI:0112 (ubiquitin reconstruction) | 2077
MI:0004 (affinity chromatography technology), MI:0676 (tandem affinity purification) | 1840
MI:0004 (affinity chromatography technology), MI:0007 (anti tag coimmunoprecipitation) | 1408
MI:0018 (two hybrid), MI:0045 (experimental interaction detection), MI:0397 (two hybrid array) | 1231
MI:0018 (two hybrid), MI:0045 (experimental interaction detection) | 954
MI:0018 (two hybrid), MI:0397 (two hybrid array) | 914
MI:0045 (experimental interaction detection), MI:0686 (unspecified method) | 628
MI:0004 (affinity chromatography technology), MI:0019 (coimmunoprecipitation) | 598
MI:0018 (two hybrid), MI:0398 (two hybrid pooling approach) | 506
MI:0004 (affinity chromatography technology), MI:0007 (anti tag coimmunoprecipitation), MI:0676 (tandem affinity purification) | 444
MI:0018 (two hybrid), MI:0045 (experimental interaction detection), MI:0686 (unspecified method) | 320
MI:0004 (affinity chromatography technology), MI:0096 (pull down) | 217
MI:0415 (enzymatic study), MI:0424 (protein kinase assay) | 192
MI:0045 (experimental interaction detection), MI:0081 (peptide array) | 150
MI:0045 (experimental interaction detection), MI:0676 (tandem affinity purification) | 120
MI:0492 (in vitro), MI:0493 (in vivo) | 5739
MI:0018 (two hybrid), MI:0398 (two hybrid pooling approach) | 5394
MI:0018 (two hybrid), MI:0492 (in vitro), MI:0493 (in vivo) | 2796
MI:0096 (pull down), MI:0492 (in vitro), MI:0493 (in vivo) | 2760
MI:0096 (pull down), MI:0492 (in vitro) | 2134
MI:0018 (two hybrid), MI:0492 (in vitro) | 1658
MI:0018 (two hybrid), MI:0493 (in vivo) | 1193
MI:0018 (two hybrid), MI:0397 (two hybrid array) | 1045
MI:0096 (pull down), MI:0493 (in vivo) | 513
MI:0004 (affinity chromatography technology), MI:0006 (anti bait coimmunoprecipitation) | 384
MI:0004 (affinity chromatography technology), MI:0019 (coimmunoprecipitation) | 309
MI:0004 (affinity chromatography technology), MI:0007 (anti tag coimmunoprecipitation) | 195
MI:0114 (x-ray crystallography), MI:0492 (in vitro) | 166
MI:0004 (affinity chromatography technology), MI:0096 (pull down) | 161
MI:0047 (far western blotting), MI:0492 (in vitro), MI:0493 (in vivo) | 106
MI:0018 (two hybrid), MI:0398 (two hybrid pooling approach) | 17738
MI:0018 (two hybrid), MI:0399 (two hybrid fragment pooling approach) | 1426
All resolvable conflicts with counts of more than 100 for yeast (top), human
(middle) and fruitfly (bottom) datasets are shown.
|
arxiv-papers
| 2011-03-07T22:47:55 |
2024-09-04T02:49:17.520327
|
{
"license": "Public Domain",
"authors": "Aleksandar Stojmirovi\\'c and Yi-Kuo Yu",
"submitter": "Aleksandar Stojmirovi\\'c",
"url": "https://arxiv.org/abs/1103.1402"
}
|
1103.1418
|
# Integral Solutions to Linear Indeterminate Equation
Changjiang Zhu
The Hubei Key Laboratory of Mathematical Sciences,
School of Mathematics and Statistics,
Huazhong Normal University, Wuhan 430079, P.R. China
Abstract: In this paper, using Euler’s function, we give a formula of all
integral solutions to linear indeterminate equation with $s$-variables
$a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{s}x_{s}=n$. It is a explicit formula of the
coefficients $a_{1}$, $a_{2}$, $\cdots$, $a_{s}$ and the free term $n$.
Key words: Linear indeterminate equation, Euler’s function, integral solution.
2000 AMS Subject Classification: 11D04, 11D72.
## 1\. Introduction and Main Theorem
In this paper, we consider the integral solutions to linear indeterminate
equation with $s$-variables
$a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{s}x_{s}=n.$ $None$
It is well-known that there exist the integral solutions of (1.1) if and only
if
$(a_{1},a_{2},\cdots,a_{s})|n.$ $None$
Under the assumption (1.2), if we can obtain a special solution of (1.1) by
applying the mutual division, fraction, parameter methods, etc., then the all
integral solutions to (1.1) can be represented by using the special solution
obtained above and $s-1$ parameters $t_{1},\ t_{2},\ \cdots,\ t_{s-1}$.
However, the methods seeking above special solution are too complicated to
lose availability in many problems. For example, it is very difficult to
obtain a special solution to the following simple indeterminate equation with
$s=2$
$2^{m}x+3^{n}y=1,$ $None$
where $m$ and $n$ are positive integers. Therefore, it is very important and
interesting to seek a formula of all integral solutions to (1.1). In this
paper, using Euler’s function, we give a formula of all integral solutions to
(1.1), which is a explicit function of the coefficients $a_{1}$, $a_{2}$,
$\cdots$, $a_{s}$ and the free term $n$.
To state our result, let $(a_{1},a_{2},\cdots,a_{s})=d,$ $n=dn_{1}$,
$(a_{1},a_{2})=d_{2}$, $(d_{2},a_{3})=d_{3},\ \cdots$,
$(d_{s-1},a_{s})=d_{s}=d$, $a_{1}=d_{2}\bar{a}_{1}$, $a_{2}=d_{2}\bar{a}_{2},\
\cdots$, $a_{s}=d_{s}\bar{a}_{s}$, $d_{2}=d_{3}\bar{d}_{2}$,
$d_{3}=d_{4}\bar{d}_{3},\ \cdots$, and $d_{s-1}=d_{s}\bar{d}_{s-1}$.
Then
$(\bar{a}_{1},\bar{a}_{2})=1,\ \ (\bar{d}_{i},\bar{a}_{i+1})=1,\ \
i=2,3,\cdots,s-1.$
Also we appoint
$\bar{a}_{1}=\bar{d}_{1},\ \ \sum\limits_{i=j}^{k}(\cdot)=0,\ \ {\rm if}\ \
k<j$
and
$\prod\limits_{i=j}^{j-\lambda}(\cdot)=\left\\{\begin{array}[]{l}1,\ \ \ \
\lambda=1,\\\ 0,\ \ \ \ \lambda\geq 2.\end{array}\right.$
Theorem 1.1. (Main Theorem) If $(a_{1},a_{2},\cdots,a_{s})|n$, then all
integral solutions to the indeterminate equation (1.1) have the following
forms:
$\left\\{\begin{array}[]{rl}x_{1}=&\displaystyle
n_{1}\prod\limits_{i=1}^{s-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}+\sum\limits_{m=1}^{s-1}\bar{a}_{m+1}\prod\limits_{i=1}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m},\\\\[8.53581pt]
x_{k}=&\displaystyle\frac{n_{1}}{\bar{a}_{k}}\left(1-\bar{d}_{k-1}^{\phi(|\bar{a}_{k}|)}\right)\prod\limits_{i=k}^{s-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}-\bar{d}_{k-1}t_{k-1}\\\\[8.53581pt]
&\displaystyle+\sum\limits_{m=2}^{s-1}\frac{\bar{a}_{m+1}}{\bar{a}_{k}}\left(1-\bar{d}_{k-1}^{\phi(|\bar{a}_{k}|)}\right)\prod\limits_{i=k}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m},\\\\[8.53581pt]
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k=2,3,\cdots,s,\end{array}\right.$
$None$
where $t_{1},\ t_{2},\ \cdots,\ t_{s-1}$ are arbitrary integers.
## 2\. The Proof of Theorem 1.1
To prove our Theorem 1.1, we restate the following Euler’s lemma, which is
required in later analysis.
Lemma 2.1 (Euler’s Lemma [2, 3, 5]). Let $(a,b)=1$. Then
$b\left|\left(1-a^{\phi(|b|)}\right)\right.,$ $None$
and
$a\left|\left(1-b^{\phi(|a|)}\right)\right.,$ $None$
where $\phi(\cdot)$ denotes Euler’s function.
To study the indeterminate equation (1.1), we first discuss a simple case of
(1.1) with $s=2$.
Lemma 2.2. For the indeterminate equation
$ax+by=c,$ $None$
if $(a,b)|c$, then all integral solutions to (2.3) have the following forms:
$\left\\{\begin{array}[]{l}\displaystyle
x=c_{0}a_{0}^{\phi(|b_{0}|)-1}+b_{0}t,\\\\[8.53581pt] \displaystyle
y=\frac{c_{0}}{b_{0}}\left(1-a_{0}^{\phi(|b_{0}|)}\right)-a_{0}t\end{array}\right.$
$None$
or
$\left\\{\begin{array}[]{l}\displaystyle
x=\frac{c_{0}}{a_{0}}\left(1-b_{0}^{\phi(|a_{0}|)}\right)-b_{0}t,\\\\[8.53581pt]
\displaystyle y=c_{0}b_{0}^{\phi(|a_{0}|)-1}+a_{0}t,\end{array}\right.$ $None$
where $a_{0}=\frac{a}{(a,b)}$, $b_{0}=\frac{b}{(a,b)}$,
$c_{0}=\frac{c}{(a,b)}$, $t=0,\ \pm 1,\ \pm 2,\cdots$.
Proof. Without the loss of generality, we only prove (2.4). The proof of (2.5)
is similar and the details are omitted. In fact, using Lemma 2.1, it is easy
to verify that $(x,y)$ is a integral solution to (2.2). On the other hand, let
$(x_{0},y_{0})$ be a integral solution to (2.2), i.e.,
$ax_{0}+by_{0}=c.$ $None$
Then
$a_{0}x_{0}+b_{0}y_{0}=c_{0},$ $None$
where $(a_{0},b_{0})=1$, which implies
$a_{0}x_{0}\equiv c_{0}\ \ \ \ \ ({\rm mod}\ |b_{0}|).$ $None$
Therefore, $x\equiv x_{0}\ \ ({\rm mod}\ |b_{0}|)$ must be the solution of
(2.8).
Noticing (2.8) has a unique solution
$x\equiv a_{0}^{\phi(|b_{0}|)-1}c_{0}\ \ \ \ \ ({\rm mod}\ |b_{0}|),$
it follows
$x_{0}\equiv a_{0}^{\phi(|b_{0}|)-1}c_{0}\ \ \ \ \ ({\rm mod}\ |b_{0}|).$
This shows that there exists a $t_{0}\in\\{0,\pm 1,\pm 2,\cdots\\}$, such that
$x_{0}=c_{0}a_{0}^{\phi(|b_{0}|)-1}+b_{0}t_{0}.$ $None$
Substituting (2.9) into (2.7), we have
$a_{0}\left(c_{0}a_{0}^{\phi(|b_{0}|)-1}+b_{0}t_{0}\right)+b_{0}y_{0}=c_{0},$
$None$
which implies
$y_{0}=\frac{c_{0}}{b_{0}}\left(1-a_{0}^{\phi(|b_{0}|)}\right)-a_{0}t_{0}.$
$None$
(2.9) and (2.10) show that every solution $(x_{0},y_{0})$ to equation (2.3)
satisfies (2.4).
The proof of Lemma 2.2 is completed.
Now we will seek a formula of all integral solutions to (1.1). To do this,
Proof of Theorem 1.1. First, we prove that $(x_{1},x_{2},\cdots,x_{s})$
defined by (1.4) is a integer solution to (1.1). By using Lemma 1.1, we know
that $x_{1},\ x_{2},\ \cdots,\ x_{s}$ defined by (1.4) are integers. Moreover,
since
$n_{1}a_{1}=n_{1}\bar{d}_{1}d_{2}=n_{1}\bar{d}_{1}\bar{d}_{2}d_{3}=\cdots=n_{1}\bar{d}_{1}\bar{d}_{2}\cdots\bar{d}_{s-1}d_{s}=n\bar{d}_{1}\bar{d}_{2}\cdots\bar{d}_{s-1},$
$n_{1}a_{k}=n_{1}\bar{a}_{k}d_{k}=n_{1}\bar{a}_{k}\bar{d}_{k}d_{k+1}=\cdots=n_{1}\bar{a}_{k}\bar{d}_{k}\cdots\bar{d}_{s-1}d_{s}=n\bar{a}_{k}\bar{d}_{k}\cdots\bar{d}_{s-1},$
$k=2,3,\cdots,s-1,$
and
$n_{1}a_{s}=n_{1}\bar{a}_{s}d_{s}=n\bar{a}_{s},$
we have
$\begin{array}[b]{rl}&\displaystyle
a_{1}n_{1}\prod\limits_{i=1}^{s-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}+a_{2}\frac{n_{1}}{\bar{a}_{2}}\left(1-\bar{d}_{1}^{\phi(|\bar{a}_{2}|)}\right)\prod\limits_{i=2}^{s-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}+\cdots\\\\[8.53581pt]
&\displaystyle+a_{s-1}\frac{n_{1}}{\bar{a}_{s-1}}\left(1-\bar{d}_{s-2}^{\phi(|\bar{a}_{s-1}|)}\right)\prod\limits_{i=s-1}^{s-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}+a_{s}\frac{n_{1}}{\bar{a}_{s}}\left(1-\bar{d}_{s-1}^{\phi(|\bar{a}_{s}|)}\right)=n,\end{array}$
$None$
$a_{1}\bar{a}_{2}t_{1}-a_{2}\bar{d}_{1}t_{1}=\bar{d}_{1}d_{2}\bar{a}_{2}t_{1}-\bar{a}_{2}d_{2}\bar{d}_{1}t_{1}=0,$
$None$
and
$\begin{array}[b]{rl}&\displaystyle
a_{1}\bar{a}_{m+1}\prod\limits_{i=1}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m}+a_{2}\frac{\bar{a}_{m+1}}{\bar{a}_{2}}\left(1-\bar{a}_{1}^{\phi(|\bar{a}_{2}|)}\right)\prod\limits_{i=2}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m}\\\\[8.53581pt]
&\displaystyle+\cdots+a_{m-1}\frac{\bar{a}_{m+1}}{\bar{a}_{m-1}}\left(1-\bar{d}_{m-2}^{\phi(|\bar{a}_{m-1}|)}\right)\prod\limits_{i=m-1}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m}\\\\[8.53581pt]
&\displaystyle+a_{m}\frac{\bar{a}_{m+1}}{\bar{a}_{m}}\left(1-\bar{d}_{m-1}^{\phi(|\bar{a}_{m}|)}\right)t_{m}-a_{m+1}\bar{d}_{m}t_{m}=0,\\\\[8.53581pt]
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
m=2,3,\cdots,s-1.\end{array}$ $None$
Adding both sides of (2.12), (2.13) and (2.14), we have
$a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{s}x_{s}=n,$
which implies $(x_{1},x_{2},\cdots,x_{s})$ defined by (1.4) is a integral
solution to (1.1).
On the other hand, we will prove that every integral solution to (1.1) can be
represented into form (1.4) by using induction for $s$.
For $s=2$, it is true by Lemma 2.2.
Suppose that it is true for the indeterminate equation of $s-1$ variables,
i.e., the every solution of
$a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{s-1}x_{s-1}=n$
can be represented into form (1.4). Now we will show that it is true for $s$.
Since $d_{s-1}|(a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{s-1}x_{s-1})$, there exists
$y_{s-1}$ such that
$a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{s-1}x_{s-1}=d_{s-1}y_{s-1}.$ $None$
(1.1) and (2.15) show
$d_{s-1}y_{s-1}+a_{s}x_{s}=n.$ $None$
From Lemma 2.2 and the inductive assumption, we have
$x_{1}=y_{s-1}\prod\limits_{i=1}^{s-2}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}+\sum\limits_{m=1}^{s-2}\bar{a}_{m+1}\prod\limits_{i=1}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m},$
$None$
$\begin{array}[b]{rl}x_{k}=&\displaystyle\frac{y_{s-1}}{\bar{a}_{k}}\left(1-\bar{d}_{k-1}^{\phi(|\bar{a}_{k}|)}\right)\prod\limits_{i=k}^{s-2}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}-\bar{d}_{k-1}t_{k-1}\\\\[8.53581pt]
&\displaystyle+\sum\limits_{m=2}^{s-2}\frac{\bar{a}_{m+1}}{\bar{a}_{k}}\left(1-\bar{d}_{k-1}^{\phi(|\bar{a}_{k}|)}\right)\prod\limits_{i=k}^{m-1}\bar{d}_{i}^{\phi(|\bar{a}_{i+1}|)-1}t_{m},\\\\[8.53581pt]
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k=2,3,\cdots,s-1,\end{array}$ $None$
$y_{s-1}=n_{1}\bar{d}_{s-1}^{\phi(|\bar{a}_{s}|)-1}+\bar{a}_{s}t_{s-1},$
$None$
and
$x_{s}=\frac{n_{1}}{\bar{a}_{s}}\left(1-\bar{d}_{s-1}^{\phi(|\bar{a}_{s}|)}\right)-\bar{d}_{s-1}t_{s-1}.$
$None$
Substituting (2.19) into (2.17), (2.18) and noticing (2.20), we know every
integral solution to (1.1) can be represented into form (1.4).
This completes the proof of Theorem 1.1.
Remark 2.4. The formula (1.4) of all integral solutions in Theorem 2.3 was
deduced from the first group formula (2.4) of Lemma 2.2. If we use the second
group formula (2.5) to solve the indeterminate equation (1.1), we can obtain
the other formula with different form of all integral solutions to (1.1).
## 3\. Applications
In this section, we will solve the indeterminate equation (1.3) by using
Theorem 1.1. To do this, we first give Euler’s functions $\phi(2^{m})$ and
$\phi(3^{n})$ as follows:
$\left\\{\begin{array}[]{l}\displaystyle\phi(2^{m})=2^{m}-2^{m-1},\\\\[8.53581pt]
\displaystyle\phi(3^{n})=3^{n}-3^{n-1}.\end{array}\right.$ $None$
By applying Theorem 1.1, the all integral solutions to (1.3) can be
represented into the following forms:
$\left\\{\begin{array}[]{l}\displaystyle
x=2^{m(3^{n}-3^{n-1}-1)}+3^{n}t,\\\\[8.53581pt] \displaystyle
y=\frac{1}{3^{n}}\left(1-2^{m(3^{n}-3^{n-1})}\right)-2^{m}t\end{array}\right.$
$None$
or
$\left\\{\begin{array}[]{l}\displaystyle
x=\frac{1}{2^{m}}\left(1-3^{n(2^{m}-2^{m-1})}\right)-3^{n}t,\\\\[8.53581pt]
\displaystyle y=3^{n(2^{m}-2^{m-1}-1)}+2^{m}t,\end{array}\right.$ $None$
where $t=0,\ \pm 1,\ \pm 2,\cdots$.
Acknowledgement: The research was supported by the Natural Science Foundation
of China $\\#$10625105, $\\#$11071093, the PhD specialized grant of the
Ministry of Education of China $\\#$20100144110001, and the Special Fund for
Basic Scientific Research of Central Colleges $\\#$CCNU10C01001.
## References
* [1] Apostol, T.M., Introduction to Analytic Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1976.
* [2] Dudley, U., Elementary Number Theory, W.H. Freeman and Company, New York, 1978\.
* [3] Gareth, A. Jones and J. Mary Jones, Elementary Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1978.
* [4] Hardy, G.H. and Wright, E.M., An Introduction to the Thoery of Numbers, Oxford University Press, Walton Street, Oxford OX2 6DP, 1979\.
* [5] Hua, L.G., Introduction to Number Theorey, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
* [6] Kenneth Ireland and Michael Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1990\.
* [7] Mathews, G.B., Theory of Numbers, New York: Chelsea Publishing, 1961\.
* [8] Redmond, D., Number Theory: An Introduction, New York: Marcel Dekker, c1996.
* [9] Schmidt, W.M., Diophantine Approximations and Diophantine Equations, Lecture Notes in Math., Vol. 1467, Springer-Verlag, Berlin, Heidelberg, New York, 1991.
|
arxiv-papers
| 2011-03-08T02:38:05 |
2024-09-04T02:49:17.527959
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Changjiang Zhu",
"submitter": "Changjiang Zhu",
"url": "https://arxiv.org/abs/1103.1418"
}
|
1103.1421
|
# Global Classical Large Solutions to Navier-Stokes Equations for Viscous
Compressible and Heat Conducting Fluids with Vacuum
Huanyao Wen1,2, Changjiang Zhu2 1 School of Mathematical Sciences South China
Normal University, Guangzhou 510631, P.R. China 2 The Hubei Key Laboratory of
Mathematical Physics School of Mathematics and Statistics Huazhong Normal
University, Wuhan 430079, P.R. China Corresponding author. Email:
cjzhu@mail.ccnu.edu.cn
###### Abstract
In this paper, we consider the 1D Navier-Stokes equations for viscous
compressible and heat conducting fluids (i.e., the full Navier-Stokes
equations). We get a unique global classical solution to the equations with
large initial data and vacuum. Because of the strong nonlinearity and
degeneration of the equations brought by the temperature equation and by
vanishing of density (i.e., appearance of vacuum) respectively, to our best
knowledge, there are only two results until now about global existence of
solutions to the full Navier-Stokes equations with special pressure, viscosity
and heat conductivity when vacuum appears (see [13] where the viscosity
$\mu=$const and the so-called variational solutions were obtained, and see [1]
where the viscosity $\mu=\mu(\rho)$ degenerated when the density vanishes and
the global weak solutions were got). It is open whether the global strong or
classical solutions exist. By applying our ideas which were used in our former
paper [8] to get $H^{3}-$estimates of $u$ and $\theta$ (see Lemma 3.10, Lemma
3.11, Lemma 3.12 and the corresponding corollaries), we get the existence and
uniqueness of the global classical solutions (see Theorem 1.1). In fact, the
existence of strong solutions would be done obviously by our estimates if the
regularity of the initial data is assumed to be weaker. Like [8], we get
$H^{4}-$regularity of $\rho$ and $u$ (see Theorem 1.2). We do not get further
regularity of $\theta$ such as $H^{4}-$regularity, because of the degeneration
and strong nonlinearity brought by vacuum and the term $(\mu uu_{x})_{x}$ in
the temperature equation. This can be viewed the first result on global
classical solutions to the 1D Navier-Stokes equations for viscous compressible
and heat conducting fluids which may be large initial data and contain vacuum.
Key Words: Compressible Navier-Stokes equations, heat conducting fluids,
vacuum, global classical solutions.
2000 Mathematics Subject Classification. 35Q30, 35K65, 76N10.
## Contents
1\. Introduction 2
2\. Preliminaries 7
3\. Proof of Theorem 1.1 9
4\. Proof of Theorem 1.2 26
References 36
## 1 Introduction
In this paper, we consider the Navier-Stokes equations for viscous
compressible and heat conducting fluids (i.e. the full Navier-Stokes
equations). The model, describing for instance the motion of gas, plays an
important role in applied physics. Mathematically, the model in one dimension
can be written as follows in sense of Eulerian coordinates:
$\displaystyle\begin{cases}\rho_{t}+(\rho u)_{x}=0,\ \rho\geq 0,\\\ (\rho
u)_{t}+(\rho u^{2})_{x}+P_{x}=(\mu u_{x})_{x},\\\ (\rho E)_{t}+(\rho
uE)_{x}+(Pu)_{x}=(\mu uu_{x})_{x}+(\kappa\theta_{x})_{x},\end{cases}$ (1.1)
for $(x,t)\in(0,1)\times(0,+\infty)$. Here $\rho=\rho(x,t)$, $u=u(x,t)$,
$P=P(\rho,\theta)$, $E$, $\theta$ and $\kappa=\kappa(\rho,\theta)$ denote the
density, velocity, pressure, total energy, absolute temperature and
coefficient of heat conduction, respectively. The total energy
$E=e+\frac{1}{2}u^{2}$, where $e$ is the internal energy. $\mu>0$ is the
coefficient of viscosity. $P$ and $e$ satisfy the second principle of
thermodynamics:
$P=\rho^{2}\frac{\partial e}{\partial\rho}+\theta\frac{\partial
P}{\partial\theta}.$ (1.2)
In the present paper, we consider the initial and boundary conditions:
$(\rho,\ u,\ \theta)\big{|}_{t=0}=(\rho_{0},\ u_{0},\ \theta_{0})(x)\ \
{\rm{in}}\ \ [0,1],$ (1.3)
and
$(u,\ \theta_{x})\big{|}_{x=0,1}=0,\ t\geq 0.$ (1.4)
Since the model is important, lots of works on the existence, uniqueness,
regularity and asymptotic behavior of the solutions were done during the last
five decades. While, because of the stronger nonlinearity in (1.1) compared
with the Navier-Stokes equations for isentropic fluids (no temperature
equation), many known mathematical results mainly focused on the absence of
vacuum (vacuum means $\rho=0$), refer for instance to [17, 18, 24, 25, 29, 30,
34] for classical solutions. More precisely, the local classical solutions to
the Navier-Stokes equations with heat-conducting fluid in Hölder spaces was
obtained respectively by Itaya in [17] for Cauchy problem and by Tani in [34]
for IBVP with $\inf\rho_{0}>0$, where the spatial dimension $N=3$. Using
delicate energy methods in Sobolev spaces, Matsumura and Nishida showed in
[29, 30] that the global classical solutions exist provided that the initial
data is small in some sense and away from vacuum and the spatial dimension
$N=3$. For large initial data and dimension $N=1$, Kazhikhov, Shelukhi in [25]
(for polytropic perfect gas with $\mu,\kappa=$const.) and Kawohl in [24] (for
real gas with $\kappa=\kappa(\rho,\theta),\ \mu=\mathrm{const.}$) respectively
got global classical solutions to (1.1) in Lagrangian coordinates with
boundary condition (1.4) and $\inf\rho_{0}>0$. The internal energy $e$ and the
coefficient of heat conduction $\kappa$ in [24] satisfy the following
assumptions for $\rho\leq\overline{\varrho}$ and $\theta\geq 0$ (we translate
these conditions in Eulerian coordinates)
$\begin{cases}e(\rho,0)\geq 0,\ \
\nu(1+\theta^{r})\leq\partial_{\theta}e(\rho,\theta)\leq
N(\overline{\varrho})(1+\theta^{r}),\\\
\kappa_{0}(1+\theta^{q})\leq\kappa(\rho,\theta)\leq\kappa_{1}(1+\theta^{q}),\\\
|\partial_{\rho}\kappa(\rho,\theta)|+|\partial_{\rho\rho}\kappa(\rho,\theta)|\leq\kappa_{1}(1+\theta^{q}),\end{cases}$
(1.5)
where $r\in[0,1]$, $q\geq 2+2r$, and $\nu$, $N(\overline{\varrho})$,
$\kappa_{0}$ and $\kappa_{1}$ are positive constants. For the perfect gas in
the domain exterior to a ball in $\mathbb{R}^{N}$ ($N=2$ or $3$) with
$\mu,\kappa=$const., Jiang in [18] got the existence of global classical
spherically symmetric large solutions in Hölder spaces.
In fact, Kawohl in [24] also considered the case of $\mu=\mu(\rho)$ for
another boundary condition with $\inf\rho_{0}>0$, where
$0<\underline{\mu}_{0}\leq\mu(\rho)\leq\overline{\mu}_{0}$ for any $\rho\geq
0$ and $\underline{\mu}_{0}$ and $\overline{\mu}_{0}$ are positive constant.
This result was generalized to the case $\mu(\rho)=\rho^{\alpha}$ by Jiang in
[19] for $\alpha\in(0,\frac{1}{4})$, and by Qin, Yao in [31] for
$\alpha\in(0,\frac{1}{2})$, respectively.
On the existence, asymptotic behavior of the weak solutions for full Navier-
Stokes equations (including the temperature equation) with $\inf\rho_{0}>0$,
please refer for instance to [20, 21, 23] for weak solutions in 1D and for
spherically symmetric weak solutions in bounded annular domains in
$\mathbb{R}^{N}$ ($N=2$, $3$), and refer to [12] for variational solutions in
a bounded domain in $\mathbb{R}^{N}$ ($N=2$, $3$).
In the presence of vacuum (i.e. $\rho$ may vanish), to our best knowledge, the
mathematical results about global well-posedness of the full Navier-Stokes
equations are usually limited to the existence of weak solutions with special
pressure, viscosity and heat conductivity (see [1, 13]). More precisely,
Feireisl in [13] got the existence of so-called variational solutions in
dimension $N\geq 2$. The temperature equation in [13] is satisfied only as an
inequality. Anyway, this work in [13] is the very first attempt towards the
existence of weak solutions to the full compressible Navier-Stokes equations
in higher dimensions, where the viscosity $\mu$ is constant and
$\begin{cases}\kappa=\kappa(\theta)\in C^{2}[0,\infty),\
\underline{\kappa}(1+\theta^{a})\leq\kappa(\theta)\leq\overline{\kappa}(1+\theta^{a})\
\ \mathrm{for}\ \mathrm{all}\ \theta\geq 0,\\\
P=P(\rho,\theta)=\mathcal{P}_{e}(\rho)+\theta\mathcal{P}_{\theta}(\rho)\ \
\mathrm{for}\ \mathrm{all}\ \rho\geq 0\ \mathrm{and}\ \theta\geq 0,\\\
\mathcal{P}_{e},\mathcal{P}_{\theta}\in C[0,\infty)\cap C^{1}(0,\infty);\
\mathcal{P}_{e}(0)=0,\ \mathcal{P}_{\theta}(0)=0,\\\
\mathcal{P}_{e}^{\prime}(\rho)\geq a_{1}\rho^{\overline{\gamma}-1}-b_{1}\ \
\mathrm{for}\ \mathrm{all}\ \rho>0;\ \mathcal{P}_{e}(\rho)\leq
a_{2}\rho^{\overline{\gamma}}+b_{1}\ \ \mathrm{for}\ \mathrm{all}\ \rho\geq
0,\\\ \mathcal{P}_{\theta}\ \ \mathrm{is}\ \ \mbox{non-decreasing}\
\mathrm{in}\ [0,\infty);\ \mathcal{P}_{\theta}(\rho)\leq
a_{3}(1+\rho^{\Gamma})\ \ \mathrm{for}\ \mathrm{all}\ \rho\geq 0,\end{cases}$
(1.6)
where $\Gamma<\frac{\overline{\gamma}}{2}$ if $N=2$ and
$\Gamma=\frac{\overline{\gamma}}{N}$ for $N\geq 3$; $a\geq 2$,
$\overline{\gamma}>1$, and $a_{1}$, $a_{2}$, $a_{3}$, $b_{1}$,
$\underline{\kappa}$ and $\overline{\kappa}$ are positive constants. Note that
the perfect gas equation of state (i.e. $P=R\rho\theta$ for some constant
$R>0$) is not involved in (1.6). In order that the equations are satisfied as
equalities in the sense of distribution, Bresch and Desjardins in [1] proposed
some different assumptions from [13], and obtained the existence of global
weak solutions to the full Navier-Stokes equations with large initial data in
$\mathbb{T}^{3}$ or $\mathbb{R}^{3}$. In [1], the viscosity $\mu=\mu(\rho)$
may vanish when vacuum appears, and $\kappa$, $P$ and $e$ are assumed to
satisfy
$\begin{cases}\kappa(\rho,\theta)=\kappa_{0}(\rho,\theta)(\rho+1)(\theta^{a}+1),\\\
P=r\rho\theta+p_{c}(\rho),\\\ e=C_{\upsilon}\theta+e_{c}(\rho),\end{cases}$
(1.7)
where $a\geq 2$, $r$ and $C_{\upsilon}$ are two positive constants,
$p_{c}(\rho)=O(\rho^{-\ell})$ and $e_{c}(\rho)=O(\rho^{-\ell-1})$ (for some
$\ell>1$) when $\rho$ is small enough, and $\kappa_{0}(\rho,\theta)$ is
assumed to satisfy
$\underline{c}_{0}\leq\kappa_{0}(\rho,\theta)\leq\frac{1}{\underline{c}_{0}},$
for $\underline{c}_{0}>0$. We have to mention that the smooth solutions in
$C^{1}\left([0,\infty);H^{d}(\mathbb{R}^{N})\right)$ ($d>2+[\frac{N}{2}]$)
would blow up when the initial density is of nontrivial compact support (see
[36]). On the local existence and uniqueness of strong solutions in
$\mathbb{R}^{3}$, please refer to [4] for the perfect gas with
$\mu,\kappa=$const.
It is still open whether global strong (or classical) solutions exist when
vacuum appears (i.e., the density may vanish). Our main concern here is to
show the existence and uniqueness of global classical solutions to (1.1)-(1.4)
with vacuum and large initial data. In fact, the existence of the strong
solutions to this problem is obvious if the regularity of initial data is
assumed to be weaker.
For compressible isentropic Navier-Stokes equations (i.e. no temperature
equation), there are so many results about the well-posedness and asymptotic
behaviors of the solutions when vacuum appears. Refer to [14, 22, 26, 28] and
[15, 27, 35, 38, 39, 40] for global weak solutions with constant viscosity and
with density-dependent viscosity, respectively. Refer to [6, 10] and [2, 3, 5,
33] for global strong solutions and for local strong (classical) solutions
with constant viscosity, respectively. Recently, Huang, Li, Xin in [16] and
Ding, Wen, Yao, Zhu in [8, 9] independently got existence and uniqueness of
global classical solutions, where the initial energy in [16] is assumed to be
small in $\mathbb{R}^{3}$ and $\rho-\widetilde{\rho}\in
C\left([0,T];H^{3}(\mathbb{R}^{3})\right)$, $u\in
C\left([0,T];D^{1}(\mathbb{R}^{3})\cap D^{3}(\mathbb{R}^{3})\right)\cap
L^{\infty}\left([\tau,T];D^{4}(\mathbb{R}^{3})\right)$ (for $\tau>0$) which
generalized the results in [3], and the initial data in [8, 9] could be large
for dimension $N=1$ and could be large but spherically symmetric for $N\geq
2$, and $(\rho,u)\in C([0,T];H^{4}(I))$ ($I$ is bounded in [8], and is bounded
or an exterior domain in [9]).
We would like to give some notations which will be used throughout the paper.
Notations:
(1) $I=[0,1]$, $\partial I=\\{0,1\\}$, $Q_{T}=I\times[0,T]$ for $T>0$.
(2) For $p\in[1,\infty]$, $L^{p}=L^{p}(I)$ denotes the $L^{p}$ space with the
norm $\|\cdot\|_{L^{p}}$. For $k\geq 1$ and $p\in[1,\infty]$,
$W^{k,p}=W^{k,p}(I)$ denotes the Sobolev space, whose norm is denoted as
$\|\cdot\|_{W^{k,p}}$, $H^{k}=W^{k,2}(I)$.
(3) For an integer $k\geq 0$ and $0<\alpha<1$, let $C^{k+\alpha}(I)$ denote
the Schauder space of functions on $I$, whose $k$th order derivative is Hölder
continuous with exponents $\alpha$, with the norm $\|\cdot\|_{C^{k+\alpha}}$.
In this paper, our assumptions are the following:
($A_{1}$): $\int_{I}\rho_{0}>0$.
($A_{2}$): $\mu=\mathrm{const.}>0$, $e=C_{0}Q(\theta)+e_{c}(\rho)$, $P=\rho
Q(\theta)+P_{c}(\rho)$, $\kappa=\kappa(\theta)$, for some constant $C_{0}>0$.
($A_{3}$): $P_{c}(\rho)\geq 0$, $e_{c}(\rho)\geq 0$, for $\rho\geq 0$;
$P_{c}\in C^{2}[0,\infty)$; $\rho|\frac{\partial e_{c}}{\partial\rho}|\leq
C_{1}e_{c}(\rho)$, for some constant $C_{1}>0$.
$(A_{4}):$ $Q(\cdot)\in C^{2}[0,\infty)$ satisfies
$\begin{cases}C_{2}\left(\beta+(1-\beta)\theta+\theta^{1+r}\right)\leq
Q(\theta)\leq C_{3}\left(\beta+(1-\beta)\theta+\theta^{1+r}\right),\\\
C_{4}(1+\theta^{r})\leq Q^{\prime}(\theta)\leq
C_{5}(1+\theta^{r}),\end{cases}\ \ \ \ \ \ \ \ \ \ \ \ \ $
for some constants $C_{i}>0$ ($i=2$, $3$, $4$, $5$) and $r\geq 0$, $\beta=0$
or $1$.
($A_{5}$): $\kappa\in C^{2}[0,\infty)$ satisfies
$C_{6}(1+\theta^{q})\leq\kappa(\theta)\leq C_{7}(1+\theta^{q}),$
for $q\geq 2+2r$, and some constants $C_{i}>0$ ($i=6$, $7$).
($A_{6}$): $Q,P_{c}\in C^{4}[0,\infty)$, and $\kappa$ satisfies
$|\partial_{\theta}^{3}\kappa(\theta)|\leq C_{8}(1+\theta^{q-3}),$
for $\theta>0$ and some constant $C_{8}>0$.
###### Remark 1.1
$(i)$ $(A_{1})$ is needed to get the upper bounds of $\theta$ and $\theta_{t}$
in terms of some norms by using mass conservation, Lemma 2.1 and Corollary
2.1.
$(ii)$ The case for the perfect gas (i.e. $P=R\rho\theta$, $e=C_{\nu}\theta$
for constants $R>0$ and $C_{\nu}>0$) is involved in the above assumptions.
$(iii)$ As it mentioned in [24], the restriction on $\mu$ ($\mu$=const., see
other restrictions on $\kappa$ and e in (1.5)) is not physically motivated.
Physically, it seems more importantly that the state functions e, $\mu$ and
$\kappa$ usually depend on both $\rho$ and $\theta$. Particularly, the
internal energy e grows as $\theta^{1+r}$ with $r\approx 0.5$, the
conductivity $\kappa$ grows as $\theta^{q}$ with $4.5\leq q\leq 5.5$ and
viscosity $\mu$ increases like $\theta^{p}$ with $0.5\leq p\leq 0.8$ (see [24,
31] and references therein). Because of mathematical technique, in the present
paper, we assume $\mu=$const. and $\kappa=\kappa(\theta)$ as in [13] (see
(1.6)). From ($A_{2}$)–($A_{5}$), we know that e and $\kappa$ grow
respectively as $\theta^{1+r}$ and $\theta^{q}$, where $q$ can be taken as
$q\in[4.5,5.5]$, and r can be taken as $r=0.5$ if we consider $\theta>0$.
$(iv)$ The restriction on $q$ in ($A_{5}$) (i.e. $q\geq 2+2r$) is same as
(1.5), and is the same as (1.6) and (1.7) when we take $r=0$. This assumption
plays an important role in the analysis.
Main results:
###### Theorem 1.1
In addition to $(A_{1})$-$(A_{5})$, we assume $\rho_{0}\geq 0$, $\rho_{0}\in
H^{2}$, $(\sqrt{\rho_{0}})_{x}\in L^{\infty}$, $u_{0}\in H^{3}\cap H_{0}^{1}$,
$\theta_{0}\in H^{3}$, $\partial_{x}\theta_{0}|_{x=0,1}=0$, and that the
following compatible conditions are valid:
$\displaystyle\begin{cases}\mu u_{0xx}-[P(\rho_{0},\
\theta_{0})]_{x}=\sqrt{\rho}_{0}g_{1},\\\
\left[\kappa(\theta_{0})\theta_{0x}\right]_{x}+\mu|u_{0x}|^{2}=\sqrt{\rho_{0}}\
g_{2},\ x\in I,\end{cases}$ (1.8)
for some $g_{1},g_{2}\in L^{2}$, and
$\left(\sqrt{\rho_{0}}g_{1}\right)_{x},\left(\sqrt{\rho_{0}}g_{2}\right)_{x}\in
L^{2}$. Then for any $T>0$ there exists a unique global solution $(\rho,\ u,\
\theta)$ to (1.1)-(1.4) such that
$\displaystyle\rho\in C([0,T];H^{2}),\ \rho_{t}\in C([0,T];H^{1}),\
\sqrt{\rho}\in W^{1,\infty}(Q_{T}),$ $\displaystyle u\in
L^{\infty}([0,T];H^{3}),\ \sqrt{\rho}u_{t}\in L^{\infty}([0,T];L^{2}),$
$\displaystyle\rho u_{t}\in L^{\infty}([0,T];H^{1}),\ \ u_{t}\in
L^{2}([0,T];H_{0}^{1}),\ \ \sqrt{\rho}e_{t}\in L^{\infty}([0,T];L^{2}),$
$\displaystyle\rho e_{t}\in L^{\infty}([0,T];H^{1}),\ \ \theta\in
L^{\infty}([0,T];H^{3}),\ \ \theta_{t}\in L^{2}([0,T];H^{1}).$
###### Remark 1.2
(i) (1.8) was proposed by Cho and Kim in [4] to get $\mathrm{local}$
$H^{2}$-regularity of $u$ and $\theta$ for $\mathrm{the\ polytropic\ perfect\
gas}$. The detailed reasons why such conditions were needed can be found in
[4]. Roughly speaking, $g_{1}$ and $g_{2}$ are equivalent to
$\sqrt{\rho}u_{t}$ and $\sqrt{\rho}e_{t}$ at $t=0$, respectively.
(ii) By the Sobolev embedding theorems (cf. [7]) and Lemma 2.3, we know from
Theorem 1.1
$\displaystyle\rho\in C\left([0,T];C^{1+\frac{1}{2}}(I)\right)\cap
C^{1}\left([0,T];C^{\frac{1}{2}}(I)\right),$ $\displaystyle u\in
C\left([0,T];C^{2+\sigma}(I)\right),\ (\rho u)_{t}\in
C\left([0,T];C^{\sigma}(I)\right),$ $\displaystyle\theta\in
C\left([0,T];C^{2+\sigma}(I)\right),\ (\rho e)_{t}\in
C\left([0,T];C^{\sigma}(I)\right),$
for any $T>0$ and $\sigma\in(0,\frac{1}{2})$. This implies $(\rho,u,\theta)$
is the classical solution to (1.1)-(1.4).
###### Theorem 1.2
In addition to $(A_{1})$-$(A_{6})$, we assume $\rho_{0}\geq 0$, $\rho_{0}\in
H^{4}$, $(\sqrt{\rho_{0}})_{x}\in L^{\infty}$, $u_{0}\in H^{4}\cap H_{0}^{1}$,
$\theta_{0}\in H^{3}$, $\partial_{x}\theta_{0}|_{x=0,1}=0$, $q>2+2r$, and that
the following compatible conditions are valid:
$\displaystyle\begin{cases}\mu u_{0xx}-[P(\rho_{0},\
\theta_{0})]_{x}=\rho_{0}g_{3},\\\
\left[\kappa(\theta_{0})\theta_{0x}\right]_{x}+\mu|u_{0x}|^{2}=\sqrt{\rho_{0}}\
g_{2},\ x\in I,\end{cases}$ (1.9)
for some $g_{3}\in H_{0}^{1}$,
$\left(\sqrt{\rho_{0}}\partial_{x}g_{3}\right)_{x}\in L^{2}$, and
$g_{2},\left(\sqrt{\rho_{0}}g_{2}\right)_{x}\in L^{2}$. Then for any $T>0$
there exists a unique global solution $(\rho,\ u,\ \theta)$ to (1.1)-(1.4)
satisfying:
$\displaystyle\rho\in C([0,T];H^{4}),\ \rho_{t}\in C([0,T];H^{3}),\
\sqrt{\rho}\in W^{1,\infty}(Q_{T}),$ $\displaystyle u\in C([0,T];H^{4})\cap
L^{2}([0,T];H^{5}),\ u_{t}\in L^{\infty}([0,T];H_{0}^{1})\cap
L^{2}([0,T];H^{3}),$ $\displaystyle(\rho u)_{t}\in C([0,T];H^{2}),\ \
\sqrt{\rho}u_{xxt}\in L^{\infty}([0,T];L^{2}),\ \ \sqrt{\rho}e_{t}\in
L^{\infty}([0,T];L^{2}),$ $\displaystyle(\rho e)_{t}\in
L^{\infty}([0,T];H^{1}),\ \ \theta\in L^{\infty}([0,T];H^{3})\cap
L^{2}([0,T];H^{4}),\ \ \theta_{t}\in L^{2}([0,T];H^{1}).$
###### Remark 1.3
(i) (1.9)1 was proposed by Cho and Kim in [3] where they consider the local
existence of classical solutions for $\mathrm{isentropic\ fluids}$ (no
temperature equation). Roughly speaking, $g_{3}$ is equivalent to $u_{t}$ at
$t=0$.
(ii) We could not get $\theta\in C([0,T];H^{4})$ (or
$L^{\infty}([0,T];H^{4})$) even if (1.9)2 is changed similarly to (1.9)1,
because of the strong nonlinearity and degeneration brought by $(\mu
uu_{x})_{x}$ in the temperature equation and the appearance of vacuum,
respectively.
(iii) Using ideas of Cho and Kim in [3], we can also get
$u\in L^{\infty}\left([\tau,T];H^{4}\right),\ \theta\in
L^{\infty}\left([\tau,T];H^{3}\right),$
for $\tau>0$. If we can obtain our estimates in higher dimensions, it will be
useful to investigate the local (global) existence of classical solutions to
the full Navier-Stokes equations (including the temperature equation) in
$\mathbb{R}^{N}$ ($N\geq 2$). For example, to guarantee (1.4) in
$\mathbb{R}^{N}$ ($N=2$ or $3$) is valid for all $t\geq 0$, it is necessary to
get $\theta\in L^{\infty}\left([0,T];H^{3}\right)$. We will consider these
problems in the near future.
The constants $C_{0}$ in $(A_{2})$ and the viscosity $\mu$ don’t play any role
in the analysis, we assume henceforth that $C_{0}=1$ and $\mu=1$ for
simplicity.
The rest of the paper is organized as follows. In Section 2, we present some
useful lemmas which will be used in the next sections. In Section 3, we prove
Theorem 1.1 by giving the initial density and the initial temperature a lower
bound $\delta>0$, getting a sequence of approximate solutions to (1.1)-(1.4),
and taking $\delta\rightarrow 0^{+}$ after making some estimates uniformly for
$\delta$. More precisely, based on Lemma 2.1 and the one-dimensional
properties of the equations, we get $H^{2}-$estimates of the solutions. Using
our ideas in [8, 9], we obtain $H^{3}-$estimates of $u$ and $\theta$. In
Section 4, using the similar arguments as in Section 3, we prove Theorem 1.2.
## 2 Preliminaries
###### Lemma 2.1
Let $\Omega=[\overline{a},\overline{b}]$ be a bounded domain in $\mathbb{R}$,
and $\rho$ be a non-negative function such that
$0<M\leq\int_{\Omega}\rho\leq K,$
for constants $M>0$ and $K>0$. Then
$\|v\|_{L^{\infty}(\Omega)}\leq\frac{K}{M}\|v_{x}\|_{L^{1}(\Omega)}+\frac{1}{M}\left|\int_{\Omega}\rho
v\right|,$
for any $v\in H^{1}(\Omega)$.
Proof. For any $x\in\Omega$, we have
$\displaystyle|v(x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{M}\left|v(x)\int_{\Omega}\rho(y)dy\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{M}\left|\int_{\Omega}v(x)\rho(y)dy-\int_{\Omega}\rho(y)v(y)dy\right|+\frac{1}{M}\left|\int_{\Omega}\rho(y)v(y)dy\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{M}\left|\int_{\Omega}\int_{y}^{x}v_{\xi}(\xi)d\xi\rho(y)dy\right|+\frac{1}{M}\left|\int_{\Omega}\rho(y)v(y)dy\right|$
$\displaystyle\leq$
$\displaystyle\frac{K}{M}\|v_{x}\|_{L^{1}(\Omega)}+\frac{1}{M}\left|\int_{\Omega}\rho(y)v(y)dy\right|.$
$\Box$
###### Remark 2.1
The version of higher dimensions for Lemma 2.1 can be found in [12] or [13].
###### Corollary 2.1
Consider the same conditions in Lemma 2.1, and in addition assume $\Omega=I$,
and
$\|\rho v\|_{L^{1}(I)}\leq\overline{c}.$
Then for any $l>0$, there exists a positive constant $C=C(M,K,l,\overline{c})$
such that
$\|v^{l}\|_{L^{\infty}(I)}\leq C\|(v^{l})_{x}\|_{L^{2}(I)}+C,$
for any $v^{l}\in H^{1}(I)$.
Proof. By Lemma 2.1, we have
$\displaystyle\|v^{l}\|_{L^{\infty}(I)}\leq
C\|(v^{l})_{x}\|_{L^{2}(I)}+C\int_{I}\rho|v^{l}|.$
Case 1: $l\in(0,1]$.
In this case, we use the Young inequality to get
$\displaystyle\|v^{l}\|_{L^{\infty}(I)}$ $\displaystyle\leq$ $\displaystyle
C\|(v^{l})_{x}\|_{L^{2}(I)}+C\int_{I}\rho|v|+C\int_{I}\rho+C$
$\displaystyle\leq$ $\displaystyle C\|(v^{l})_{x}\|_{L^{2}(I)}+C.$
Case 2: $l\in(1,\infty)$.
In the case, we use the Young inequality again to get
$\displaystyle\|v^{l}\|_{L^{\infty}(I)}$ $\displaystyle\leq$ $\displaystyle
C\|(v^{l})_{x}\|_{L^{2}(I)}+C\|v^{l-1}\|_{L^{\infty}(I)}\int_{I}\rho|v|$
$\displaystyle\leq$ $\displaystyle
C\|(v^{l})_{x}\|_{L^{2}(I)}+\frac{1}{2}\|v^{l}\|_{L^{\infty}(I)}+C.$
This gives
$\displaystyle\|v^{l}\|_{L^{\infty}(I)}\leq C\|(v^{l})_{x}\|_{L^{2}(I)}+C.$
$\Box$
###### Lemma 2.2
For any $v\in H^{1}_{0}(I)$, we have
$\|v\|_{L^{\infty}(I)}\leq\|v_{x}\|_{L^{1}}.$
Proof. Since $v(0)=0$, we have for any $x\in I$
$\displaystyle|v(x)|=|v(x)-v(0)|=\left|\int_{0}^{x}v_{x}\right|\leq\|v_{x}\|_{L^{1}(I)}.$
This completes the proof. $\Box$
###### Lemma 2.3
([32]). Assume $X\subset E\subset Y$ are Banach spaces and
$X\hookrightarrow\hookrightarrow E$. Then the following imbedding are compact:
$\displaystyle(i)\ \ \left\\{\varphi:\varphi\in
L^{q}(0,T;X),\frac{\partial\varphi}{\partial t}\in
L^{1}(0,T;Y)\right\\}\hookrightarrow\hookrightarrow L^{q}(0,T;E),\ \ {\rm if}\
\ 1\leq q\leq\infty;$ $(ii)\ \ \left\\{\varphi:\varphi\in
L^{\infty}(0,T;X),\frac{\partial\varphi}{\partial t}\in
L^{r}(0,T;Y)\right\\}\hookrightarrow\hookrightarrow C([0,T];E),\ \ {\rm if}\ \
1<r\leq\infty.$
## 3 Proof of Theorem 1.1
In this section, we get a global solution to (1.1)-(1.4) with initial density
and initial temperature having a respectively lower bound $\delta>0$ by using
some a priori estimates of the solutions based on the local existence. Theorem
1.1 would be got after we make some a priori estimates uniformly for $\delta$
and take $\delta\rightarrow 0^{+}$.
Denote $\rho_{0}^{\delta}=\rho_{0}+\delta$ and
$\theta_{0}^{\delta}=\theta_{0}+\delta$ for $\delta\in(0,1)$. Throughout this
section, we denote $c$ to be a generic constant depending on $\rho_{0}$,
$u_{0}$, $\theta_{0}$, $T$ and some other known constants but independent of
$\delta$ for any $\delta\in(0,1)$.
Before proving Theorem 1.1, we need the following auxiliary theorem.
###### Theorem 3.1
Consider the same assumptions as in Theorem 1.1. Then for any $T>0$ and
$\delta\in(0,1)$ there exists a unique global solution $(\rho,u,\theta)$ to
(1.1)-(1.4) with initial data replaced by
($\rho_{0}^{\delta},u_{0},\theta_{0}^{\delta}$), such that
$\displaystyle\rho\in C([0,T];H^{2}),\ \ \ \rho_{t}\in C([0,T];H^{1}),\ \ \
\rho_{tt}\in L^{2}([0,T];L^{2}),\ \rho\geq\frac{\displaystyle\delta}{c}>0,$
$\displaystyle u\in C([0,T];H^{3}\cap H^{1}_{0}),\ u_{t}\in C([0,T];H^{1})\cap
L^{2}([0,T];H^{2}),\ \ \ u_{tt}\in L^{2}([0,T];L^{2}),\ $
$\displaystyle\theta\geq c_{\delta}>0,\ \theta\in C([0,T];H^{3}),\ \
\theta_{t}\in C([0,T];H^{1})\cap L^{2}([0,T];H^{2}),\ \ \theta_{tt}\in
L^{2}([0,T];L^{2}),$
where $c_{\delta}$ is a constant depending on $\delta$, but independent of
$u$.
Proof of Theorem 3.1:
The local solutions as in Theorem 3.1 can be obtained by the successive
approximations like in [4]. We omit it here for brevity. The regularities
guarantee the uniqueness (refer for instance to [4]). Based on it, Theorem 3.1
can be proved by some a priori estimates globally in time.
For any given $T\in(0,+\infty)$, let $(\rho,u,\theta)$ be the solution to
(1.1)-(1.4) as in Theorem 3.1. Then we have the following basic energy
estimate.
###### Lemma 3.1
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\displaystyle\int_{I}\rho\left(1+e_{c}(\rho)+\theta^{1+r}+u^{2}\right)(t)\leq
c.$
Proof. Integrating $(\ref{non-1.2})_{1}$ and $(\ref{non-1.2})_{3}$ over
$I\times[0,t]$, and using (1.4) , $(A_{2})$ and $(A_{4})$, we complete the
proof of Lemma 3.1. $\Box$
###### Lemma 3.2
Under the conditions of Theorem 3.1, it holds for any $(x,t)\in Q_{T}$
$\displaystyle\begin{cases}0<\rho(x,t)\leq c,\\\ \theta(x,t)>0.\end{cases}$
Proof. The proof of the upper bound of $\rho$ relies on constant viscosity
(i.e. $\mu=const.$). It is similar to [37].
Denote
$w(x,t)=\int_{0}^{t}(u_{x}-P-\rho u^{2})+\int_{0}^{x}\rho_{0}u_{0}.$ (3.1)
Differentiating (3.1) with respect to $x$, and using $(\ref{non-1.2})_{2}$, we
have
$w_{x}=\rho u.$
This together with Lemma 3.1 and the Cauchy inequality gives
$\int_{I}|w_{x}|\leq c.$
It follows from (3.1), (1.2), ($A_{2}$), ($A_{3}$), ($A_{4}$), (1.4), and
Lemma 3.1 that
$\left|\int_{I}w\right|\leq c.$
This gives for any $(x,t)\in Q_{T}$
$\displaystyle|w(x,t)|$ $\displaystyle\leq$
$\displaystyle\left|w(x,t)-\int_{I}w\right|+\left|\int_{I}w\right|$
$\displaystyle\leq$ $\displaystyle\left|\int_{I}\int_{y}^{x}w_{\xi}(\xi,t)d\xi
dy\right|+c$ $\displaystyle\leq$ $\displaystyle\int_{I}|w_{x}|+c\leq c,$
which implies
$\|w\|_{L^{\infty}(Q_{T})}\leq c.$ (3.2)
For any $(x,t)\in Q_{T}$, let $X(s;x,t)$ satisfy
$\displaystyle\begin{cases}\frac{dX(s;x,t)}{ds}=u\left(X(s;x,t),s\right),\
0\leq s<t,\\\ X(t;x,t)=x.\end{cases}$ (3.3)
Denote
$F(x,t)=\exp\left\\{w(x,t)\right\\}.$
It is easy to verify
$\displaystyle\frac{d(\rho F)\left(X(s;x,t),s\right)}{ds}$ $\displaystyle=$
$\displaystyle F\left(\rho_{s}+\frac{\partial\rho}{\partial
X}u+\rho\frac{\partial w}{\partial X}u+\rho w_{s}\right)$ (3.4)
$\displaystyle=$ $\displaystyle-\rho PF.$
Multiplying (3.4) by $\exp\left(\int_{0}^{s}P\right)$, we have
$\frac{d}{ds}\left\\{\rho F\exp\left(\int_{0}^{s}P\right)\right\\}=0.$
Integrating it over $(0,t)$, we have
$\rho(x,t)=\frac{F(X(0;x,t),0)}{F(x,t)}\rho_{0}^{\delta}\exp\left(-\int_{0}^{t}P\right),$
(3.5)
which implies
$\rho(x,t)>0,$
for any $(x,t)\in Q_{T}$.
By (3.2), (3.5) and $P\geq 0$, we get the upper bound of $\rho$. The lower
bound of $\theta$ can be got by (3.7) and the maximum principle for parabolic
equations. $\Box$
###### Lemma 3.3
Under the conditions of Theorem 3.1, it holds for any given $\alpha\in(0,1)$
$\int_{Q_{T}}\left(\frac{u_{x}^{2}}{\theta^{\alpha}}+\frac{(1+\theta^{q})\theta_{x}^{2}}{\theta^{1+\alpha}}\right)\leq
c,$
where $c$ may depend on $\alpha$.
###### Remark 3.1
$\alpha$ was usually taken as $1$ when the basic energy inequality was done
(see [1] and references therein). This depends on $\rho_{0}\log\theta_{0}\in
L^{1}$ which can not be got under the assumptions of Theorem 1.1 and Theorem
1.2, since $\theta_{0}$ may vanish.
Proof. From (1.2) and (1.1), we get
$\rho e_{\theta}\theta_{t}+\rho ue_{\theta}\theta_{x}+\theta
P_{\theta}u_{x}=u_{x}^{2}+\left(\kappa(\theta)\theta_{x}\right)_{x}.$ (3.6)
Substituting $e=Q(\theta)+e_{c}(\rho)$ and $P=\rho Q(\theta)+P_{c}(\rho)$ into
(3.6), we get
$\rho Q^{\prime}(\theta)\theta_{t}+\rho
uQ^{\prime}(\theta)\theta_{x}+\rho\theta
Q^{\prime}(\theta)u_{x}=u_{x}^{2}+\left(\kappa(\theta)\theta_{x}\right)_{x},$
(3.7)
or
$(\rho Q)_{t}+(\rho uQ)_{x}+\rho\theta
Q^{\prime}(\theta)u_{x}=u_{x}^{2}+\left(\kappa(\theta)\theta_{x}\right)_{x}.$
(3.8)
Multiplying (3.7) by $\theta^{-\alpha}$, integrating the resulting equation
over $Q_{T}$, and using integration by parts, we have
$\displaystyle\int_{Q_{T}}\left(\frac{u_{x}^{2}}{\theta^{\alpha}}+\frac{\alpha\kappa(\theta)\theta_{x}^{2}}{\theta^{1+\alpha}}\right)$
(3.9) $\displaystyle=$
$\displaystyle\int_{I}\rho\int_{0}^{\theta}\frac{Q^{\prime}(\xi)}{\xi^{\alpha}}-\int_{I}\rho_{0}\int_{0}^{\theta_{0}}\frac{Q^{\prime}(\xi)}{\xi^{\alpha}}+\int_{Q_{T}}\rho\theta^{1-\alpha}Q^{\prime}(\theta)u_{x}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\int_{0}^{\theta}\frac{1+\xi^{r}}{\xi^{\alpha}}+c\int_{I}\rho_{0}\int_{0}^{\theta_{0}}\xi^{r-\alpha}+c\int_{Q_{T}}\rho\theta^{1-\alpha}(1+\theta^{r})|u_{x}|$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho(1+\theta^{1+r})+c+\frac{1}{2}\int_{Q_{T}}\frac{u_{x}^{2}}{\theta^{\alpha}}+c\int_{Q_{T}}\rho^{2}\theta^{2-\alpha+2r}$
$\displaystyle\leq$ $\displaystyle
c+\frac{1}{2}\int_{Q_{T}}\frac{u_{x}^{2}}{\theta^{\alpha}}+c\int_{0}^{T}\max\limits_{x\in
I}\theta^{1+r-\alpha},$
where we have used ($A_{4}$), the Cauchy inequality, Lemma 3.1 and Lemma 3.2.
Now we estimate the last term of (3.9) as follows:
$\displaystyle c\int_{0}^{T}\max\limits_{x\in I}\theta^{1+r-\alpha}$
$\displaystyle\leq$ $\displaystyle
c+\int_{0}^{T}\|\theta^{r-\alpha}\theta_{x}\|_{L^{2}}$ (3.10)
$\displaystyle\leq$ $\displaystyle
c+c\int_{0}^{T}\left(\int_{I}\frac{\theta_{x}^{2}\theta^{2r-\alpha+1}}{\theta^{1+\alpha}}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c+\frac{1}{2}\int_{Q_{T}}\frac{\alpha\kappa(\theta)\theta_{x}^{2}}{\theta^{1+\alpha}},$
where we have used Corollary 2.1, Lemma 3.1, ($A_{5}$) and the Cauchy
inequality. By (3.9), (3.10) and ($A_{5}$), we complete the proof. $\Box$
###### Corollary 3.1
Under the conditions of Theorem 3.1, it holds
$\displaystyle\int_{0}^{T}\|\theta\|_{L^{\infty}}^{q-\alpha+1}\leq c.$
Proof. By Corollary 2.1 and Lemma 3.1, we have
$\displaystyle\int_{0}^{T}\|\theta\|_{L^{\infty}}^{q-\alpha+1}$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\|\theta^{\frac{q-\alpha+1}{2}}\|_{L^{\infty}}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{T}\int_{I}\left(\theta^{\frac{q-\alpha-1}{2}}\theta_{x}\right)^{2}+c$
$\displaystyle=$ $\displaystyle
c\int_{0}^{T}\int_{I}\theta^{q-\alpha-1}\theta_{x}^{2}+c$ $\displaystyle\leq$
$\displaystyle c.$
$\Box$
###### Lemma 3.4
Under the conditions of Theorem 3.1, it holds
$\int_{Q_{T}}u_{x}^{2}\leq c.$
Proof. From (1.1)1 and (1.1)2, we get
$\rho u_{t}+\rho uu_{x}+P_{x}=u_{xx}.$ (3.11)
Multiplying (3.11) by $u$, integrating it over $I$, and using integration by
parts, we have
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho u^{2}+\int_{I}u_{x}^{2}$
$\displaystyle=$ $\displaystyle\int_{I}Pu_{x}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{x}^{2}+c\int_{I}\rho^{2}Q^{2}+c\int_{I}P_{c}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{x}^{2}+c\int_{I}\theta^{2+2r}+c,$
where we have used the Cauchy inequality, ($A_{2}$), ($A_{3}$), ($A_{4}$) and
Lemma 3.2. This implies
$\displaystyle\frac{d}{dt}\int_{I}\rho u^{2}+\int_{I}u_{x}^{2}\leq
c\sup\limits_{x\in I}\theta^{q-\alpha+1}+c.$
Integrating it over $(0,t)$, and using Corollary 3.1, we complete the proof of
Lemma 3.4. $\Box$
###### Lemma 3.5
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}(u_{x}^{2}+\rho\theta^{q+2+r})+\int_{Q_{T}}\left(\rho
u_{t}^{2}+(1+\theta^{q})^{2}\theta_{x}^{2}\right)\leq c.$
Proof. Multiplying (3.11) by $u_{t}$, integrating it over $I$, and using
integration by parts, Lemma 2.2, Lemma 3.2 and the Cauchy inequality, we have
$\displaystyle\int_{I}\rho u_{t}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}u_{x}^{2}$
$\displaystyle=$ $\displaystyle\frac{d}{dt}\int_{I}Pu_{x}-\int_{I}\rho
uu_{x}u_{t}-\int_{I}P_{t}u_{x}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\rho u_{t}^{2}+\frac{1}{2}\int_{I}\rho
u^{2}u_{x}^{2}+\frac{d}{dt}\int_{I}Pu_{x}-\int_{I}P_{t}u_{x}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}\rho
u_{t}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+\frac{d}{dt}\int_{I}Pu_{x}-\int_{I}P_{t}(u_{x}-P)-\frac{1}{2}\frac{d}{dt}\int_{I}P^{2},$
which implies
$\displaystyle\int_{I}\rho u_{t}^{2}+\frac{d}{dt}\int_{I}u_{x}^{2}\leq
c\left(\int_{I}u_{x}^{2}\right)^{2}+2\frac{d}{dt}\int_{I}Pu_{x}-\frac{d}{dt}\int_{I}P^{2}-2\int_{I}P_{t}(u_{x}-P).$
(3.12)
We are going to estimate the last term of the right side of (3.12). Using
($A_{2}$), (3.8), (1.1)1 and integration by parts, we have
$\displaystyle-2\int_{I}P_{t}(u_{x}-P)$ $\displaystyle=$
$\displaystyle-2\int_{I}(\rho Q)_{t}(u_{x}-P)-2\int_{I}(P_{c})_{t}(u_{x}-P)$
$\displaystyle=$
$\displaystyle-2\int_{I}\left[(\kappa\theta_{x})_{x}+u_{x}^{2}-(\rho
uQ)_{x}-\rho\theta Q^{\prime}(\theta)u_{x}\right](u_{x}-P)$
$\displaystyle+2\int_{I}P_{c}^{\prime}(\rho)(\rho_{x}u+\rho u_{x})(u_{x}-P)$
$\displaystyle=$ $\displaystyle
2\int_{I}\kappa\theta_{x}(u_{xx}-P_{x})-2\int_{I}u_{x}^{2}(u_{x}-P)-2\int_{I}\rho
uQ(u_{xx}-P_{x})$ $\displaystyle+2\int_{I}\rho\theta
Q^{\prime}(\theta)u_{x}(u_{x}-P)-2\int_{I}P_{c}u(u_{xx}-P_{x})-2\int_{I}P_{c}u_{x}(u_{x}-P)$
$\displaystyle+2\int_{I}\rho P_{c}^{\prime}(\rho)u_{x}(u_{x}-P).$
This, together with (3.11), ($A_{2}$), ($A_{4}$), Lemma 2.2, Lemma 3.2, the
Cauchy inequality, and $W^{1,1}(I)\hookrightarrow L^{\infty}(I)$, gives
$\displaystyle-2\int_{I}P_{t}(u_{x}-P)$ $\displaystyle\leq$ $\displaystyle
2\int_{I}\kappa\theta_{x}(\rho u_{t}+\rho
uu_{x})+2\|u_{x}-P\|_{L^{\infty}}\int_{I}u_{x}^{2}-2\int_{I}\rho uQ(\rho
u_{t}+\rho uu_{x})$ (3.13) $\displaystyle+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}u_{x}^{2}+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho Q^{2}+c\sup\limits_{x\in I}\theta^{1+r}$
$\displaystyle-2\int_{I}P_{c}u(\rho u_{t}+\rho
uu_{x})+c\int_{I}u_{x}^{2}+c\int_{I}\rho Q^{2}+c$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\int_{I}\rho
u_{t}^{2}+c\int_{I}\kappa^{2}\theta_{x}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+c\left(\|u_{x}-P\|_{L^{1}}+\|\rho
u_{t}+\rho uu_{x}\|_{L^{1}}\right)\int_{I}u_{x}^{2}$
$\displaystyle+c\int_{I}u_{x}^{2}\int_{I}\rho Q^{2}+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}(u_{x}^{2}+\rho Q^{2})+c\sup\limits_{x\in
I}\theta^{1+r}+c$ $\displaystyle\leq$ $\displaystyle\frac{1}{4}\int_{I}\rho
u_{t}^{2}+c\int_{I}\kappa^{2}\theta_{x}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+\frac{1}{4}\int_{I}\rho
u_{t}^{2}+c\int_{I}u_{x}^{2}\int_{I}\rho Q^{2}$
$\displaystyle+c\sup\limits_{x\in I}(1+\theta^{1+r})\int_{I}(u_{x}^{2}+\rho
Q^{2})+c\sup\limits_{x\in I}\theta^{1+r}+c.$
Substituting (3.13) into (3.12), we have
$\displaystyle\frac{1}{2}\int_{I}\rho u_{t}^{2}+\frac{d}{dt}\int_{I}u_{x}^{2}$
(3.14) $\displaystyle\leq$ $\displaystyle
c\left(\int_{I}u_{x}^{2}\right)^{2}+2\frac{d}{dt}\int_{I}Pu_{x}-\frac{d}{dt}\int_{I}P^{2}+c\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{I}u_{x}^{2}\int_{I}\rho
Q^{2}$ $\displaystyle+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}(u_{x}^{2}+\rho Q^{2})+c\sup\limits_{x\in
I}\theta^{1+r}+c.$
Integrating (3.14) over $(0,t)$, and using ($A_{2}$)-($A_{4}$), Lemma 3.2,
Corollary 3.1, Lemma 3.4 and the Cauchy inequality, we have
$\displaystyle\frac{1}{2}\int_{0}^{t}\int_{I}\rho u_{t}^{2}+\int_{I}u_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+2\int_{I}(\rho
Q+P_{c})u_{x}+c\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{0}^{t}\int_{I}u_{x}^{2}\int_{I}\rho\theta^{2+2r}$
$\displaystyle+c\int_{0}^{t}\sup\limits_{x\in
I}\theta^{1+r}\int_{I}u_{x}^{2}+c\int_{0}^{t}\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho\theta^{2+2r}+c$ $\displaystyle\leq$
$\displaystyle
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+\frac{1}{2}\int_{I}u_{x}^{2}+c\int_{I}\rho\theta^{2+2r}+c\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{0}^{t}\int_{I}u_{x}^{2}\int_{I}\rho\theta^{2+2r}$
$\displaystyle+c\int_{0}^{t}\sup\limits_{x\in
I}\theta^{1+r}\int_{I}u_{x}^{2}+c\int_{0}^{t}\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho\theta^{2+2r}+c.$
The second term of the right side can be absorbed by the left. After that, we
have
$\displaystyle\int_{0}^{t}\int_{I}\rho u_{t}^{2}+\int_{I}u_{x}^{2}$ (3.15)
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho\theta^{q+2+r}+c\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{0}^{t}\int_{I}u_{x}^{2}\int_{I}\rho\theta^{2+2r}$
$\displaystyle+c\int_{0}^{t}\sup\limits_{x\in
I}\theta^{1+r}\int_{I}u_{x}^{2}+c\int_{0}^{t}\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho\theta^{2+2r}+c.$
Here, we have used Lemma 3.2 and the Young inequality on the second term of
the right side. Note that the terms about $\theta$ in (3.15) need to be
handled. To do this, we make use of (3.7).
Multiplying (3.7) by $\int_{0}^{\theta}\kappa(\xi)d\xi$, integrating it over
$I$, and using integration by parts, ($A_{4}$) and ($A_{5}$), we have
$\displaystyle\frac{d}{dt}\int_{I}\rho\left[\int_{0}^{\theta}Q^{\prime}(\eta)\int_{0}^{\eta}\kappa(\xi)d\xi
d\eta\right]+\int_{I}\kappa^{2}\theta_{x}^{2}$
$\displaystyle=\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi-\int_{I}\rho\theta
Q^{\prime}(\theta)u_{x}\int_{0}^{\theta}\kappa(\xi)d\xi$ $\displaystyle\leq
c\|(1+\theta^{q})\theta\|_{L^{\infty}}\int_{I}u_{x}^{2}+c\|(1+\theta^{q})\theta\|_{L^{\infty}}\int_{I}\rho(1+\theta^{1+r})|u_{x}|.$
(3.16)
By Corollary 2.1 and ($A_{5}$), we get
$\|(1+\theta^{q})\theta\|_{L^{\infty}}\leq c\|\kappa\theta_{x}\|_{L^{2}}+c.$
(3.17)
Substituting (3.17) into (3), and using the Hölder inequality, the Cauchy
inequality and Lemma 3.2, we get
$\displaystyle\frac{d}{dt}\int_{I}\rho\left[\int_{0}^{\theta}Q^{\prime}(\eta)\int_{0}^{\eta}\kappa(\xi)d\xi
d\eta\right]+\int_{I}\kappa^{2}\theta_{x}^{2}$ $\displaystyle\leq$
$\displaystyle
c\|\kappa\theta_{x}\|_{L^{2}}\int_{I}u_{x}^{2}+c\int_{I}u_{x}^{2}+c\|\kappa\theta_{x}\|_{L^{2}}\int_{I}\rho(1+\theta^{1+r})|u_{x}|+c\int_{I}\rho(1+\theta^{1+r})|u_{x}|$
$\displaystyle\leq$ $\displaystyle
c\|\kappa\theta_{x}\|_{L^{2}}\left(\int_{I}u_{x}^{2}+\|\rho(1+\theta^{1+r})\|_{L^{2}}\|u_{x}\|_{L^{2}}\right)+c\int_{I}u_{x}^{2}+c\int_{I}\rho(1+\theta^{2+2r})$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\kappa^{2}\theta_{x}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho\theta^{2+2r}\int_{I}u_{x}^{2}+c\int_{I}\rho\theta^{2+2r}+c,$
which implies
$\displaystyle\frac{d}{dt}\int_{I}\rho\left[\int_{0}^{\theta}Q^{\prime}(\eta)\int_{0}^{\eta}\kappa(\xi)d\xi
d\eta\right]+\frac{1}{2}\int_{I}\kappa^{2}\theta_{x}^{2}$ $\displaystyle\leq$
$\displaystyle
c\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho\theta^{2+2r}\int_{I}u_{x}^{2}+c\int_{I}\rho\theta^{2+2r}+c.$
Integrating it over $(0,t)$, and using ($A_{4}$), ($A_{5}$), Lemma 3.1 and
Corollary 3.1, we get
$\int_{I}\rho\theta^{q+2+r}+\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}\leq
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{0}^{t}\left(\int_{I}\rho\theta^{2+2r}\int_{I}u_{x}^{2}\right)+c.$
(3.18)
By (3.15), (3.18), Corollary 3.1, Lemma 3.4, and the Gronwall inequality, we
complete the proof. $\Box$
###### Lemma 3.6
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}(\rho_{x}^{2}+\rho_{t}^{2})+\int_{Q_{T}}u_{xx}^{2}\leq c.$
Proof. Differentiating $(\ref{non-1.2})_{1}$ with respect to $x$, we have
$\rho_{xt}+\rho_{xx}u+2\rho_{x}u_{x}+\rho u_{xx}=0.$ (3.19)
Multiplying (3.19) by $2\rho_{x}$, integrating it over $I$ and using
integration by parts, we have
$\displaystyle\frac{d}{dt}\int_{I}\rho_{x}^{2}$ $\displaystyle=$
$\displaystyle-3\int_{I}\rho_{x}^{2}u_{x}-2\int_{I}\rho\rho_{x}u_{xx}$ (3.20)
$\displaystyle=$
$\displaystyle-3\int_{I}\rho_{x}^{2}(u_{x}-P)-3\int_{I}\rho_{x}^{2}P-2\int_{I}\rho\rho_{x}u_{xx}$
$\displaystyle\leq$ $\displaystyle
c\left(\|u_{x}-P\|_{L^{2}}+\|u_{xx}-P_{x}\|_{L^{2}}\right)\int_{I}\rho_{x}^{2}+c\int_{I}u_{xx}^{2}+c\int_{I}\rho_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\left(1+\|\sqrt{\rho}u_{t}\|_{L^{2}}\right)\int_{I}\rho_{x}^{2}+c\int_{I}u_{xx}^{2},$
where we have used (3.11), the Sobolev inequality, ($A_{2}$)-($A_{4}$), the
Cauchy inequality, Lemma 2.2, Lemma 3.2 and Lemma 3.5.
It follows from (3.11), Lemma 2.2, Lemma 3.2, ($A_{2}$)-($A_{4}$), Lemma 3.5
and the Cauchy inequality that
$\displaystyle\int_{I}u_{xx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho
u_{t}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho_{x}^{2}Q^{2}+c\int_{I}\rho^{2}\left[Q^{\prime}(\theta)\right]^{2}\theta_{x}^{2}+c\int_{I}\rho_{x}^{2}+c$
(3.21) $\displaystyle\leq$ $\displaystyle c\int_{I}\rho
u_{t}^{2}+c\sup\limits_{x\in
I}(1+\theta^{2+2r})\int_{I}\rho_{x}^{2}+c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c$
$\displaystyle\leq$ $\displaystyle c\int_{I}\rho u_{t}^{2}+c\sup\limits_{x\in
I}(1+\theta^{q-\alpha+1})\int_{I}\rho_{x}^{2}+c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c.$
Substituting (3.21) into (3.20), and using the Gronwall inequality, Corollary
3.1 and Lemma 3.5, we get
$\int_{I}\rho_{x}^{2}\leq c.$ (3.22)
By (3.21), (3.22), Corollary 3.1 and Lemma 3.5, we have
$\displaystyle\int_{Q_{T}}u_{xx}^{2}\leq c.$
It follows from (1.1)1, (3.22), Lemma 2.2, Lemma 3.2 and Lemma 3.5 that
$\int_{I}\rho_{t}^{2}\leq c.$
The proof of the lemma is complete. $\Box$
###### Lemma 3.7
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}\left(\rho
u_{t}^{2}+\theta_{x}^{2}\right)+\int_{Q_{T}}\left(u_{xt}^{2}+\rho\theta_{t}^{2}\right)\leq
c.$
Proof. Differentiating (3.11) with respect to $t$, we have
$\rho u_{tt}+\rho_{t}u_{t}+\rho_{t}uu_{x}+\rho u_{t}u_{x}+\rho
uu_{xt}+P_{xt}=u_{xxt}.$ (3.23)
Multiplying (3.23) by $u_{t}$, integrating the resulting equation over $I$, we
have
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
u_{t}^{2}+\int_{I}u_{xt}^{2}$ $\displaystyle=$ $\displaystyle-2\int_{I}\rho
uu_{t}u_{xt}-\int_{I}\rho_{t}uu_{x}u_{t}-\int_{I}\rho
u_{t}^{2}u_{x}+\int_{I}P_{t}u_{tx}$ $\displaystyle\leq$ $\displaystyle
2\|\sqrt{\rho}u_{t}\|_{L^{2}}\|\sqrt{\rho}u\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}+\|u_{t}\|_{L^{\infty}}\|u\|_{L^{\infty}}\|\rho_{t}\|_{L^{2}}\|u_{x}\|_{L^{2}}+\|u_{x}\|_{L^{\infty}}\int_{I}\rho
u_{t}^{2}$
$\displaystyle+\|P_{c}^{\prime}(\rho)\|_{L^{\infty}}\|\rho_{t}\|_{L^{2}}\|u_{xt}\|_{L^{2}}+\|Q(\theta)\|_{L^{\infty}}\|\rho_{t}\|_{L^{2}}\|u_{xt}\|_{L^{2}}+\|\rho
Q^{\prime}(\theta)\theta_{t}\|_{L^{2}}\|u_{xt}\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{xt}^{2}+c\int_{I}\rho
u_{t}^{2}+c+c\int_{I}u_{xx}^{2}\int_{I}\rho u_{t}^{2}+c\sup\limits_{x\in
I}\theta^{2+2r}+c\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$
Here, we have used $(\ref{non-1.2})_{1}$, integration by parts, the Hölder
inequality, the Cauchy inequality, the Sobolev inequality,
($A_{2}$)-($A_{4}$), Lemma 2.2, Lemma 3.2, Lemma 3.5 and Lemma 3.6.
The first term of the right side can be absorbed by the left. This implies
$\displaystyle\frac{d}{dt}\int_{I}\rho u_{t}^{2}+\int_{I}u_{xt}^{2}$ (3.24)
$\displaystyle\leq$ $\displaystyle c\int_{I}\rho
u_{t}^{2}+c+c\int_{I}u_{xx}^{2}\int_{I}\rho u_{t}^{2}+c\sup\limits_{x\in
I}\theta^{2+2r}+c\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$
Integrating (3.24) over $(0,t)$, and using Corollary 3.1 and Lemma 3.5, we
have
$\displaystyle\int_{I}\rho
u_{t}^{2}+\int_{0}^{t}\int_{I}u_{xt}^{2}\leq\int_{I}\rho
u_{t}^{2}(0)+c+c\int_{0}^{t}\int_{I}u_{xx}^{2}\int_{I}\rho
u_{t}^{2}+c\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$
(3.25)
Multiplying (3.11) by $\frac{1}{\sqrt{\rho}}$, taking $t\rightarrow 0^{+}$ and
using (1.8)1, we have
$\displaystyle|\sqrt{\rho}u_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|u_{0xx}-P(\rho_{0}^{\delta},\theta_{0}^{\delta})_{x}\right|}{\sqrt{\rho_{0}^{\delta}}}+\sqrt{\rho_{0}^{\delta}}|u_{0}u_{0x}|$
$\displaystyle\leq$
$\displaystyle\frac{\left|u_{0xx}-P(\rho_{0},\theta_{0})_{x}\right|}{\sqrt{\rho_{0}^{\delta}}}+\frac{|P(\rho_{0},\theta_{0})_{x}-P(\rho_{0}^{\delta},\theta_{0}^{\delta})_{x}|}{\sqrt{\rho_{0}^{\delta}}}+\sqrt{\rho_{0}^{\delta}}|u_{0}u_{0x}|$
$\displaystyle\leq$
$\displaystyle|g_{1}|+c\frac{\delta}{\sqrt{\rho_{0}^{\delta}}}(|\rho_{0x}|+|\theta_{0x}|)+c,$
which implies
$\int_{I}\rho u_{t}^{2}(0)\leq c.$ (3.26)
Substituting (3.26) into (3.25), we have
$\displaystyle\int_{I}\rho u_{t}^{2}+\int_{0}^{t}\int_{I}u_{xt}^{2}\leq
c+c\int_{0}^{t}\int_{I}u_{xx}^{2}\int_{I}\rho
u_{t}^{2}+c\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$
(3.27)
Multiplying (3.7) by $\left(\int_{0}^{\theta}\kappa(\xi)d\xi\right)_{t}$(i.e.
$\kappa(\theta)\theta_{t}$), integrating the resulting equation over $I$, and
using integration by parts, ($A_{4}$), ($A_{5}$), Lemma 2.2, Lemma 3.2, Lemma
3.5 and the Cauchy inequality, we have for any $\varepsilon>0$
$\displaystyle\int_{I}\rho
Q^{\prime}(\theta)\kappa(\theta)\theta_{t}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\kappa^{2}\theta_{x}^{2}$
$\displaystyle=$ $\displaystyle-\int_{I}\rho
uQ^{\prime}\kappa\theta_{x}\theta_{t}-\int_{I}\rho\theta Q^{\prime}\kappa
u_{x}\theta_{t}+\int_{I}u_{x}^{2}\left(\int_{0}^{\theta}\kappa(\xi)d\xi\right)_{t}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+c\int_{I}\rho
u^{2}Q^{\prime}\kappa\theta_{x}^{2}+c\int_{I}\rho Q^{2}Q^{\prime}\kappa
u_{x}^{2}$
$\displaystyle+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)-2\int_{I}u_{x}u_{xt}\int_{0}^{\theta}\kappa(\xi)d\xi$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c\left(1+\int_{I}u_{xx}^{2}\right)\int_{I}\rho(1+\theta^{q+r+2})$
$\displaystyle+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)+\varepsilon\int_{I}u_{xt}^{2}+c_{\varepsilon}\sup\limits_{x\in
I}(1+\theta^{q})^{2}\theta^{2},$
which combining Lemma 2.2, Lemma 3.2, Lemma 3.5 implies
$\displaystyle\int_{I}\rho
Q^{\prime}(\theta)\kappa(\theta)\theta_{t}^{2}+\frac{d}{dt}\int_{I}\kappa^{2}\theta_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c\int_{I}u_{xx}^{2}+c+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)+\varepsilon\int_{I}u_{xt}^{2}+c_{\varepsilon}\sup\limits_{x\in
I}(1+\theta^{q})^{2}\theta^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}u_{xx}^{2}+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)+\varepsilon\int_{I}u_{xt}^{2}+c_{\varepsilon}\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c_{\varepsilon}.$
Integrating it over $(0,t)$, and using ($A_{4}$) and ($A_{5}$), Lemma 2.2,
Lemma 3.5, Lemma 3.6 and the Cauchy inequality, we obtain
$\displaystyle\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}+\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}$
$\displaystyle\leq$ $\displaystyle c\sup\limits_{x\in
I}(1+\theta^{q})\theta+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}$
$\displaystyle\leq$ $\displaystyle
c\|(1+\theta^{q})\theta_{x}\|_{L^{2}}+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}.$
After the first term of the right side is absorbed by the left, we get
$\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}+\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}\leq
c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}.$ (3.28)
Multiplying (3.28) by $2c$, adding the resulting inequality to (3.25), taking
$\varepsilon=\frac{1}{4c^{2}}$, and using the Gronwall inequality and Lemma
3.6, we complete the proof of Lemma 3.7. $\Box$
From Corollary 2.1, Lemma 3.1 and Lemma 3.7, we get the following corollary
immediately.
###### Corollary 3.2
Under the conditions of Theorem 3.1, it holds
$\|\theta\|_{L^{\infty}(Q_{T})}\leq c.$
###### Corollary 3.3
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\|u\|_{W^{1,\infty}(Q_{T})}+\int_{I}u_{xx}^{2}+\int_{Q_{T}}\theta_{xx}^{2}\leq
c.$
Proof. It follows from (3.21), Lemma 3.6, Lemma 3.7 and Corollary 3.2 that
$\int_{I}u_{xx}^{2}\leq c,$
which, combining Lemma 2.2, Lemma 3.5 and the Sobolev inequality, gives
$\|u\|_{W^{1,\infty}(Q_{T})}\leq c.$ (3.29)
By (3.7), Corollary 3.2, ($A_{4}$), ($A_{5}$), Lemma 2.2, Lemma 3.2, (3.29),
Lemma 3.7, the Hölder inequality, Sobolev inequality and Cauchy inequality, we
have
$\displaystyle\int_{I}\theta_{xx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\theta_{x}^{4}+c\int_{I}u_{x}^{4}+\int_{I}\rho\theta_{t}^{2}+c\int_{I}u^{2}\theta_{x}^{2}+c\int_{I}\theta^{2}u_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{x}\theta_{xx}\|_{L^{1}}\int_{I}\theta_{x}^{2}+\int_{I}\rho\theta_{t}^{2}+c$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{xx}\|_{L^{2}}+\int_{I}\rho\theta_{t}^{2}+c$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\theta_{xx}^{2}+\int_{I}\rho\theta_{t}^{2}+c.$
After the first term of the right side is absorbed by the left, we get
$\int_{I}\theta_{xx}^{2}\leq\int_{I}\rho\theta_{t}^{2}+c.$ (3.30)
Integrating (3.30) over $[0,T]$, and using Lemma 3.7, we get
$\int_{Q_{T}}\theta_{xx}^{2}\leq c.$
This proves Corollary 3.3. $\Box$
###### Lemma 3.8
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\|\rho\|_{W^{1,\infty}(Q_{T})}+\|\rho_{t}\|_{L^{\infty}(Q_{T})}+\int_{I}(\rho_{xx}^{2}+\rho_{xt}^{2})+\int_{Q_{T}}(\rho_{tt}^{2}+u_{xxx}^{2})\leq
c.$
Proof. Differentiating (3.19) w.r.t. $x$, we have
$\rho_{xxt}=-\rho_{xxx}u-3\rho_{xx}u_{x}-3\rho_{x}u_{xx}-\rho u_{xxx}.$ (3.31)
Multiplying (3.31) by $2\rho_{xx}$, integrating it over $I$, and using
integration by parts and the Hölder inequality, we have
$\displaystyle\frac{d}{dt}\int_{I}\rho_{xx}^{2}$ $\displaystyle=$
$\displaystyle-5\int_{I}\rho_{xx}^{2}u_{x}-6\int_{I}\rho_{x}\rho_{xx}u_{xx}-2\int_{I}\rho\rho_{xx}u_{xxx}$
$\displaystyle\leq$ $\displaystyle
5\|u_{x}\|_{L^{\infty}}\int_{I}\rho_{xx}^{2}+6\|\rho_{x}\|_{L^{\infty}}\|\rho_{xx}\|_{L^{2}}\|u_{xx}\|_{L^{2}}+2\|\rho\|_{L^{\infty}}\|\rho_{xx}\|_{L^{2}}\|u_{xxx}\|_{L^{2}}.$
By the Sobolev inequality, Cauchy inequality, Lemma 3.2, Lemma 3.6, and
Corollary 3.3, we have
$\displaystyle\frac{d}{dt}\int_{I}\rho_{xx}^{2}\leq
c\int_{I}\rho_{xx}^{2}+c\int_{I}u_{xxx}^{2}+c.$ (3.32)
The next step is to estimate the term $\int_{I}u_{xxx}^{2}$. Differentiating
(3.11) with respect to $x$, we have
$\displaystyle u_{xxx}=\rho_{x}u_{t}+\rho u_{xt}+\rho_{x}uu_{x}+\rho
u_{x}^{2}+\rho uu_{xx}+(P_{c})_{xx}+(\rho Q)_{xx}.$ (3.33)
By ($A_{3}$), ($A_{4}$), Lemma 2.2, Lemma 3.2, Lemma 3.6, Corollary 3.2,
Corollary 3.3 and the Sobolev inequality, we get
$\displaystyle\int_{I}u_{xxx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}u_{xt}^{2}+c\int_{I}\rho_{x}^{2}u_{t}^{2}+c\int_{I}\rho_{xx}^{2}+c\int_{I}\theta_{xx}^{2}+c$
(3.34) $\displaystyle\leq$ $\displaystyle
c\int_{I}u_{xt}^{2}+c\int_{I}\rho_{xx}^{2}+c\int_{I}\theta_{xx}^{2}+c.$
Substituting (3.34) into (3.32), and using the Gronwall inequality, Lemma 3.7
and Corollary 3.3, we get
$\int_{I}\rho_{xx}^{2}\leq c.$ (3.35)
By (3.35), Lemma 3.2, Lemma 3.6 and the Sobolev inequality, we have
$\|\rho\|_{W^{1,\infty}(Q_{T})}\leq c.$ (3.36)
By (3.34), (3.35), Lemma 3.7 and Corollary 3.3, we get
$\int_{Q_{T}}u_{xxx}^{2}\leq c.$
The estimates of $\rho_{xt}$ and $\rho_{tt}$ can be obtained directly by
(3.19), (1.1)1, (3.35), (3.36), Lemma 2.2, Lemma 3.2, Lemma 3.6, Lemma 3.7,
and Corollary 3.3. The proof of Lemma 3.8 is complete. $\Box$
###### Lemma 3.9
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}\rho\theta_{t}^{2}+\int_{Q_{T}}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c.$
Differentiating (3.7) w.r.t. $t$, we have
$\displaystyle\rho Q^{\prime}\theta_{tt}+\rho
Q^{\prime\prime}\theta_{t}^{2}+\rho_{t}Q^{\prime}\theta_{t}+(\rho
uQ^{\prime}\theta_{x})_{t}+(\rho\theta
Q^{\prime}u_{x})_{t}=2u_{x}u_{xt}+(\kappa\theta_{x})_{xt}.$ (3.37)
Multiplying (3.37) by $(\int_{0}^{\theta}\kappa(\xi)d\xi)_{t}$ (i.e.
$\kappa(\theta)\theta_{t}$), integrating it over $I$, and using integration by
parts, (1.1)1, ($A_{4}$), ($A_{5}$), Corollary 3.2, Lemma 3.2, Corollary 3.3
and the Hölder inequality, we have
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{I}\rho_{t}Q^{\prime}\kappa\theta_{t}^{2}-\frac{1}{2}\int_{I}\rho
Q^{\prime\prime}\theta_{t}^{3}\kappa+\frac{1}{2}\int_{I}\rho
Q^{\prime}\kappa^{\prime}\theta_{t}^{3}-\int_{I}(\rho
uQ^{\prime}\theta_{x})_{t}\kappa\theta_{t}$ $\displaystyle-\int_{I}(\rho\theta
Q^{\prime}u_{x})_{t}\kappa\theta_{t}+2\int_{I}u_{x}u_{xt}\kappa\theta_{t}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}(\rho
u)_{x}Q^{\prime}\kappa\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}-\int_{I}\rho
uQ^{\prime}(\kappa\theta_{x})_{t}\theta_{t}-\int_{I}\rho
u(Q^{\prime\prime}\kappa-Q^{\prime}\kappa^{\prime})\theta_{t}^{2}\theta_{x}$
$\displaystyle-\int_{I}(\rho
u)_{t}Q^{\prime}\theta_{x}\kappa\theta_{t}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}-\int_{I}\rho_{t}\theta
Q^{\prime}u_{x}\kappa\theta_{t}+c\|\kappa\theta_{t}\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}\|u_{x}\|_{L^{2}}.$
This, combining integration by parts, ($A_{4}$), ($A_{5}$), Lemma 2.1, Lemma
2.2, Corollary 3.2, Corollary 3.3, Lemma 3.7, Lemma 3.8 and the Cauchy
inequality, gives
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}$
$\displaystyle\leq$ $\displaystyle-\frac{1}{2}\int_{I}\rho
uQ^{\prime\prime}\theta_{x}\kappa\theta_{t}^{2}-\frac{1}{2}\int_{I}\rho
uQ^{\prime}\kappa^{\prime}\theta_{x}\theta_{t}^{2}-\int_{I}\rho
uQ^{\prime}\kappa\theta_{t}\theta_{xt}+c\|\kappa\theta_{t}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}$
$\displaystyle+c\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\|(\kappa\theta_{x})_{t}\|_{L^{2}}+c\|\theta_{x}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}$
$\displaystyle+c\|\kappa\theta_{t}\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{xx}\|_{L^{2}}\int_{I}\rho\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}+c\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\|(\kappa\theta_{x})_{t}\|_{L^{2}}+c\|\kappa\theta_{t}\|_{L^{\infty}}$
$\displaystyle+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{xx}\|_{L^{2}}\int_{I}\rho\theta_{t}^{2}+c\left(\|(\kappa\theta_{t})_{x}\|_{L^{2}}+\int_{I}\rho\kappa|\theta_{t}|\right)\int_{I}\rho\theta_{t}^{2}+c\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\|(\kappa\theta_{x})_{t}\|_{L^{2}}$
$\displaystyle+c\|(\kappa\theta_{t})_{x}\|_{L^{2}}+c\int_{I}\rho\kappa|\theta_{t}|+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\left(\|(\kappa\theta_{t})_{x}\|_{L^{2}}+\int_{I}\rho\kappa|\theta_{t}|\right)\|u_{xt}\|_{L^{2}},$
which together with the Cauchy inequality, Lemma 3.2, ($A_{5}$) and Corollary
3.2 gives
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq\frac{1}{2}\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\int_{I}\theta_{xx}^{2}+c\left(\int_{I}\rho\theta_{t}^{2}\right)^{2}+c\int_{I}u_{xt}^{2}+c,$
where we have used $(\kappa\theta_{t})_{x}=(\kappa\theta_{x})_{t}$. This gives
$\displaystyle\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c\int_{I}\theta_{xx}^{2}+c\left(\int_{I}\rho\theta_{t}^{2}\right)^{2}+c\int_{I}u_{xt}^{2}+c.$
Integrating it over $(0,t)$, and using ($A_{4}$), ($A_{5}$), Lemma 3.7 and
Corollary 3.3, we obtain
$\int_{I}\rho\theta_{t}^{2}+\int_{0}^{t}\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c\int_{I}\rho\theta_{t}^{2}(0)+c\int_{0}^{t}\left(\int_{I}\rho\theta_{t}^{2}\right)^{2}+c.$
(3.38)
Multiplying (3.7) by $\displaystyle\frac{1}{Q^{\prime}(\theta)\sqrt{\rho}}$,
taking $t\rightarrow 0^{+}$, and using (1.8)2, we have
$\displaystyle|\sqrt{\rho}\theta_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|u_{0x}^{2}+\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+|\sqrt{\rho_{0}^{\delta}}u_{0}\theta_{0x}|+|\sqrt{\rho_{0}^{\delta}}\theta_{0}^{\delta}u_{0x}|$
$\displaystyle\leq$
$\displaystyle\frac{\left|u_{0x}^{2}+\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+\frac{\left|\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}-\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+c$
$\displaystyle\leq$ $\displaystyle
c|g_{2}|+\frac{c\delta}{\sqrt{\rho_{0}^{\delta}}}(1+|\theta_{0xx}|)+c,$
which implies
$\displaystyle\int_{I}\rho\theta_{t}^{2}(0)$ $\displaystyle\leq$
$\displaystyle c\int_{I}g_{2}^{2}+c\int_{I}\theta_{0xx}^{2}+c$ (3.39)
$\displaystyle\leq$ $\displaystyle c.$
Substituting (3.39) into (3.38), using the Gronwall inequality and Lemma 3.7,
we complete the proof. $\Box$
###### Corollary 3.4
Under the conditions of Theorem 3.1, it holds
$\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}\leq c.$
Proof. By Lemma 3.2, ($A_{5}$), Corollary 2.1, Corollary 3.2, Lemma 3.9, and
$(\kappa\theta_{t})_{x}=(\kappa\theta_{x})_{t}$, we get
$\displaystyle\int_{0}^{T}\|\kappa\theta_{t}\|_{L^{\infty}}^{2}\leq
c\int_{0}^{T}\|(\kappa\theta_{t})_{x}\|_{L^{2}}^{2}+c\leq c.$
This combining ($A_{5}$) completes the proof. $\Box$
###### Corollary 3.5
Under the conditions of Theorem 3.1, it holds
$\int_{Q_{T}}\theta_{xt}^{2}\leq c.$
Proof. Since
$\kappa\theta_{xt}=(\kappa\theta_{x})_{t}-\kappa^{\prime}\theta_{t}\theta_{x},$
we obtain
$\displaystyle\int_{Q_{T}}\theta_{xt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{Q_{T}}\kappa^{2}\theta_{xt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{Q_{T}}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\int_{Q_{T}}(\kappa^{\prime})^{2}\theta_{t}^{2}\theta_{x}^{2}$
$\displaystyle\leq$ $\displaystyle c+c\int_{0}^{T}\sup\limits_{x\in
I}\theta_{t}^{2}\int_{I}\theta_{x}^{2}$ $\displaystyle\leq$ $\displaystyle c,$
where we have used ($A_{5}$), Lemma 3.7, Lemma 3.9, Corollary 3.2 and
Corollary 3.4. $\Box$
###### Corollary 3.6
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\|\theta\|_{W^{1,\infty}(Q_{T})}+\int_{I}\theta_{xx}^{2}+\int_{Q_{T}}\theta_{xxx}^{2}\leq
c.$
Proof. From (3.30) and Lemma 3.9, we have
$\displaystyle\int_{I}\theta_{xx}^{2}\leq c,$ (3.40)
which, combining Corollary 3.2, Lemma 3.7 and the Sobolev inequality, gives
$\|\theta\|_{W^{1,\infty}(Q_{T})}\leq c.$ (3.41)
Differentiating (3.7) w.r.t. $x$, we have
$\displaystyle\kappa\theta_{xxx}=-3\kappa^{\prime}\theta_{x}\theta_{xx}-\kappa^{\prime\prime}\theta_{x}^{3}-2u_{x}u_{xx}+\rho
Q^{\prime}\theta_{xt}+\rho_{x}Q^{\prime}\theta_{t}+\rho
Q^{\prime\prime}\theta_{x}\theta_{t}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+(\rho uQ^{\prime}\theta_{x})_{x}+(\rho\theta Q^{\prime}u_{x})_{x}.$ (3.42)
By (3.40), (3.41), (3), ($A_{4}$), ($A_{5}$), Lemma 3.8, Lemma 3.9 and
Corollary 3.3, we have
$\displaystyle\int_{I}\theta_{xxx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\theta_{x}^{2}\theta_{xx}^{2}+c\int_{I}\theta_{x}^{6}+c\int_{I}u_{x}^{2}u_{xx}^{2}+c\int_{I}\rho^{2}\theta_{xt}^{2}+c\int_{I}\rho_{x}^{2}\theta_{t}^{2}+c\int_{I}\rho^{2}\theta_{x}^{2}\theta_{t}^{2}$
(3.43) $\displaystyle+c\int_{I}\left|(\rho
uQ^{\prime}\theta_{x})_{x}\right|^{2}+c\int_{I}\left|(\rho\theta
Q^{\prime}u_{x})_{x}\right|^{2}+c$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}\theta_{xt}^{2}+c\int_{I}\rho_{x}^{2}\theta_{t}^{2}+c$
$\displaystyle\leq$ $\displaystyle c\int_{I}\theta_{xt}^{2}+c\sup\limits_{x\in
I}\theta_{t}^{2}+c.$
By (3.43), Corollary 3.4 and Corollary 3.5, we obtain
$\int_{Q_{T}}\theta_{xxx}^{2}\leq c.$
$\Box$
The next lemma, which we used in [8] to get $H^{4}-$estimates of velocity,
plays an important role in getting $H^{3}-$estimates of $\theta$ in the
following.
###### Lemma 3.10
Under the conditions of Theorem 3.1, it holds
$\|(\sqrt{\rho})_{x}\|_{L^{\infty}(Q_{T})}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}(Q_{T})}\leq
c.$
Proof. Multiplying $(\ref{non-1.2})_{1}$ by
$\displaystyle\frac{1}{2\sqrt{\rho}}$, we have
$(\sqrt{\rho})_{t}+(\sqrt{\rho})_{x}u+\frac{1}{2}\sqrt{\rho}u_{x}=0.$ (3.44)
Differentiating (3.44) with respect to $x$, we get
$(\sqrt{\rho})_{xt}+(\sqrt{\rho})_{xx}u+\frac{3}{2}(\sqrt{\rho})_{x}u_{x}+\frac{1}{2}\sqrt{\rho}u_{xx}=0.$
Denote $h=(\sqrt{\rho})_{x}$, we have
$h_{t}+h_{x}u+\frac{3}{2}hu_{x}+\frac{1}{2}\sqrt{\rho}u_{xx}=0,$
which implies
$\frac{d}{ds}\left\\{h\exp\left(\frac{3}{2}\int_{0}^{s}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)\right\\}=-\frac{1}{2}\sqrt{\rho}(\partial_{X}^{2}u)\exp\left(\frac{3}{2}\int_{0}^{s}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right),$
(3.45)
where $X(s;x,t)$ is the solution to (3.3).
Integrating (3.45) over $(0,t)$, we get
$\begin{array}[]{rl}h(x,t)=&\displaystyle\exp\left(-\frac{3}{2}\int_{0}^{t}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)h\left(X(0;x,t),0\right)\\\\[14.22636pt]
&\displaystyle-\frac{1}{2}\exp\left(-\frac{3}{2}\int_{0}^{t}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)\int_{0}^{t}\sqrt{\rho}(\partial_{X}^{2}u)\exp\left(\frac{3}{2}\int_{0}^{s}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)ds.\end{array}$
This together with Corollary 3.3, Lemma 3.8 and the Sobolev inequality,
implies
$\|(\sqrt{\rho})_{x}\|_{L^{\infty}(Q_{T})}\leq c.$ (3.46)
From (3.44), (3.46), Lemma 3.8 and Corollary 3.3, we get
$\|(\sqrt{\rho})_{t}\|_{L^{\infty}(Q_{T})}\leq c.$
This proves Lemma 3.10. $\Box$
The next lemma will be used to get $H^{3}-$estimates of $\theta$.
###### Lemma 3.11
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}+\int_{Q_{T}}\rho^{3}\theta_{tt}^{2}\leq
c.$
Proof. Multiply $(\ref{non-3.36})$ by
$\rho^{\gamma_{1}}(\kappa\theta_{t})_{t}$ (i.e.
$\rho^{\gamma_{1}}\kappa\theta_{tt}+\rho^{\gamma_{1}}\kappa^{\prime}\theta_{t}^{2}$,
where $\gamma_{1}$ is to be decided later), and using integration by parts, we
have
$\displaystyle\int_{I}\rho^{\gamma_{1}+1}\kappa
Q^{\prime}\theta_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{\gamma_{1}}\left|(\kappa\theta_{x})_{t}\right|^{2}$
(3.47) $\displaystyle=$
$\displaystyle\frac{\gamma_{1}}{2}\int_{I}\rho^{\gamma_{1}-1}\rho_{t}\left|(\kappa\theta_{x})_{t}\right|^{2}-\gamma_{1}\int_{I}\rho^{\gamma_{1}-1}\rho_{x}\kappa\theta_{tt}(\kappa\theta_{x})_{t}-\gamma_{1}\int_{I}\rho^{\gamma_{1}-1}\rho_{x}\kappa^{\prime}\theta_{t}^{2}(\kappa\theta_{x})_{t}$
$\displaystyle+2\int_{I}u_{x}u_{xt}(\rho^{\gamma_{1}}\kappa\theta_{tt}+\rho^{\gamma_{1}}\kappa^{\prime}\theta_{t}^{2})-\int_{I}\rho^{\gamma_{1}+1}Q^{\prime}\kappa^{\prime}\theta_{t}^{2}\theta_{tt}$
$\displaystyle-\int_{I}\left(\rho
Q^{\prime\prime}\theta_{t}^{2}+\rho_{t}Q^{\prime}\theta_{t}+(\rho
uQ^{\prime}\theta_{x})_{t}+(\rho\theta
Q^{\prime}u_{x})_{t}\right)\left(\rho^{\gamma_{1}}\kappa\theta_{tt}+\rho^{\gamma_{1}}\kappa^{\prime}\theta_{t}^{2}\right).$
We are going to look for the minimal of $\gamma_{1}$. It seems that the second
term of the right side plays an important role.
$\displaystyle-\gamma_{1}\int_{I}\rho^{\gamma_{1}-1}\rho_{x}\kappa\theta_{tt}(\kappa\theta_{x})_{t}$
$\displaystyle=$
$\displaystyle-2\gamma_{1}\int_{I}\rho^{\gamma_{1}-\frac{1}{2}}(\sqrt{\rho})_{x}\kappa\theta_{tt}(\kappa\theta_{x})_{t}$
(3.48) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\int_{I}\rho^{\gamma_{1}+1}\kappa
Q^{\prime}\theta_{tt}^{2}+c\int_{I}\rho^{\gamma_{1}-2}|(\kappa\theta_{x})_{t}|^{2},$
where we have used Lemma 3.10, ($A_{4}$), ($A_{5}$), Corollary 3.2 and the
Cauchy inequality.
From Lemma 3.9, we know that $\int_{Q_{T}}|(\kappa\theta_{x})_{t}|^{2}\leq c$.
This implies that the minimal of $\gamma_{1}$ should be 2. Substituting
$\gamma_{1}=2$ into (3.47) and (3.48), and then substituting (3.48) into
(3.47), we have
$\displaystyle\frac{3}{4}\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\int_{I}\rho\theta_{t}^{4}+c\int_{I}u_{xt}^{2}+c\int_{I}\theta_{xt}^{2}+c$
$\displaystyle+c\left(\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}\right)^{\frac{1}{2}}\left\\{1+\left(\int_{I}\rho\theta_{t}^{4}\right)^{\frac{1}{2}}+\|\theta_{xt}\|_{L^{2}}+\|\sqrt{\rho}u_{t}\|_{L^{2}}+\|u_{xt}\|_{L^{2}}+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\right\\}$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}+c\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c\int_{I}u_{xt}^{2}+c,$
where we have used ($A_{4}$), ($A_{5}$), Lemma 3.7, Lemma 3.8, Lemma 3.9,
Corollary 3.3, Corollary 3.6 and the Cauchy inequality. This implies
$\displaystyle\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}+\frac{d}{dt}\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c\int_{I}u_{xt}^{2}+c.$
Integrating it over $(0,t)$, and using ($A_{4}$), ($A_{5}$), Lemma 3.7, Lemma
3.9, Corollary 3.4 and Corollary 3.5, we have
$\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}+\int_{0}^{t}\int_{I}\rho^{3}\theta_{tt}^{2}\leq
c\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}(0)+c.$ (3.49)
By ($A_{5}$), (3.39), (3) and Corollary 3.2, we get
$\displaystyle\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}(0)$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}\theta_{xt}^{2}(0)+c\int_{I}\rho\theta_{t}^{2}(0)$ (3.50)
$\displaystyle\leq$ $\displaystyle
c\|\theta_{0}\|_{H^{3}}^{2}+c\|u_{0}\|_{H^{2}}^{2}+c\int_{I}\rho_{x}^{2}u_{t}^{2}(0)+c$
$\displaystyle\leq$ $\displaystyle c+c\int_{I}|(\sqrt{\rho})_{x}|^{2}\rho
u_{t}^{2}(0)$ $\displaystyle\leq$ $\displaystyle c.$
Substituting (3.50) into (3.49), we complete the proof. $\Box$
###### Corollary 3.7
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}\left(\theta_{xxx}^{2}+\rho^{2}\theta_{xt}^{2}\right)\leq c.$
Proof. A direct calculation gives
$\rho\kappa\theta_{xt}=\rho(\kappa\theta_{x})_{t}-\rho\kappa^{\prime}\theta_{t}\theta_{x},$
which implies
$\displaystyle\int_{I}\rho^{2}\theta_{xt}^{2}$ $\displaystyle\leq$
$\displaystyle
c\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\|\theta_{x}\|_{L^{\infty}}^{2}\int_{I}\rho\theta_{t}^{2}$
(3.51) $\displaystyle\leq$ $\displaystyle c.$
Here we have used $(A_{5})$, Lemma 3.8, Lemma 3.9, Lemma 3.11 and Corollary
3.6.
From the second inequality of (3.43), we obtain
$\displaystyle\int_{I}\theta_{xxx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}\theta_{xt}^{2}+c\int_{I}\rho_{x}^{2}\theta_{t}^{2}+c$
$\displaystyle\leq$ $\displaystyle
c\int_{I}|(\sqrt{\rho})_{x}|^{2}\rho\theta_{t}^{2}+c$ $\displaystyle\leq$
$\displaystyle c,$
where we have used Lemma 3.9, (3.51) and Lemma 3.10. $\Box$
The next lemma will be used to get $H^{3}-$estimates of $u$.
###### Lemma 3.12
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}\rho^{2}u_{xt}^{2}+\int_{Q_{T}}\rho^{3}u_{tt}^{2}\leq c.$
Proof. Similarly to Lemma 3.11, multiplying (3.23) by $\rho^{2}u_{tt}$, and
integrating it over $I$, we have
$\displaystyle\int_{I}\rho^{3}u_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{2}u_{xt}^{2}$
$\displaystyle=$
$\displaystyle\int_{I}\rho\rho_{t}u_{xt}^{2}-2\int_{I}\rho\rho_{x}u_{xt}u_{tt}-\int_{I}\rho^{2}u_{tt}(\rho_{t}u_{t}+\rho_{t}uu_{x}+\rho
u_{t}u_{x}+\rho uu_{xt}+P_{xt})$ $\displaystyle\leq$ $\displaystyle
c\int_{I}u_{xt}^{2}-4\int_{I}\rho^{\frac{3}{2}}(\sqrt{\rho})_{x}u_{xt}u_{tt}+\frac{1}{4}\int_{I}\rho^{3}u_{tt}^{2}+c\int_{I}\rho
u_{t}^{2}+c\int_{I}|(\rho Q)_{xt}|^{2}+c\int_{I}|(P_{c})_{xt}|^{2}+c$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\rho^{3}u_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c.$
Here, we have used integration by parts, Lemma 3.7, Lemma 3.8, Lemma 3.10,
Corollary 3.3, Corollary 3.6 and the Cauchy inequality.
The first term of the right side can be absorbed by the left. After that, we
have
$\displaystyle\int_{I}\rho^{3}u_{tt}^{2}+\frac{d}{dt}\int_{I}\rho^{2}u_{xt}^{2}\leq
c\int_{I}u_{xt}^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c.$
Integrating this inequality on both side over $(0,t)$, and using Lemma 3.7,
Corollary 3.4 and Corollary 3.5, we have
$\int_{0}^{t}\int_{I}\rho^{3}u_{tt}^{2}+\int_{I}\rho^{2}u_{xt}^{2}\leq\int_{I}\rho^{2}u_{xt}^{2}(0)+c.$
(3.52)
Similarly to (3.50), we use (3.26), (3.33), ($A_{3}$) and ($A_{4}$) to get
$\displaystyle\int_{I}\rho^{2}u_{xt}^{2}(0)$ $\displaystyle\leq$
$\displaystyle
c\|u_{0}\|_{H^{3}}^{2}+c\|\theta_{0}\|_{H^{2}}^{2}+c\|\rho_{0}\|_{H^{2}}^{2}+c\int_{I}\rho
u_{t}^{2}(0)+c$ (3.53) $\displaystyle\leq$ $\displaystyle c.$
Substituting (3.53) into (3.52), we complete the proof. $\Box$
By (3.34), Lemma 3.7, Lemma 3.8, Lemma 3.10, Lemma 3.12 and Corollary 3.6, we
get the following corollary.
###### Corollary 3.8
Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$
$\int_{I}u_{xxx}^{3}\leq c.$
From the above estimates, we get
$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{2}}+\|\rho_{t}\|_{H^{1}}+\displaystyle\|u\|_{H^{3}}+\|\rho
u_{t}\|_{H^{1}}+\|\sqrt{\rho}u_{t}\|_{L^{2}}+\|\theta\|_{H^{3}}$
$\displaystyle+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{t}\|_{H^{1}}+\int_{Q_{T}}\left(u_{xt}^{2}+\rho_{tt}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\rho^{3}u_{tt}^{2}+\rho^{3}\theta_{tt}^{2}\right)\leq
c.$ (3.54)
###### Corollary 3.9
Under the conditions of Theorem 3.1, there exists a positive constant
$c_{\delta}$ depending on $\delta$ such that for any $(x,t)\in Q_{T}$, it
holds
$\begin{cases}\rho(x,t)\geq\displaystyle\frac{\delta}{c}>0,\\\ \theta(x,t)\geq
c_{\delta}>0.\end{cases}$ (3.55)
Proof. By (3.5), ($A_{3}$), ($A_{4}$), Lemma 3.8 and Corollary 3.6, we have
for any $(x,t)\in Q_{T}$
$\displaystyle\rho(x,t)\geq\frac{\delta}{c}.$
This gets (3.55)1. (3.55)2 can be got by (3.55)1, (3), (3.7) and the maximum
principle for parabolic equation. $\Box$
From (3), (3.55), (3.23) and (3.37), we obtain
$\displaystyle\|\rho\|_{H^{2}}+\|\rho_{t}\|_{H^{1}}+\|u\|_{H^{3}}+\|u_{t}\|_{H^{1}}+\|\theta\|_{H^{3}}+\|\theta_{t}\|_{H^{1}}$
$\displaystyle+\int_{Q_{T}}\left(u_{xt}^{2}+u_{xxt}^{2}+\rho_{tt}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\theta_{xxt}^{2}+u_{tt}^{2}+\theta_{tt}^{2}\right)\leq
c.$
This proves Theorem 3.1. $\Box$
Proof of Theorem 1.1:
Consider (1.1)-(1.4) with initial data replaced by ($\rho_{0}^{\delta}$,
$u_{0}$, $\theta_{0}^{\delta}$), we obtain from Theorem 3.1 that there exists
a unique solution ($\rho^{\delta}$, $u^{\delta}$, $\theta^{\delta}$), such
that (3) and (3.55) are valid when we replace ($\rho,$ $u$, $\theta$) by
($\rho^{\delta}$, $u^{\delta}$, $\theta^{\delta}$). With the estimates uniform
for $\delta$, we take $\delta\rightarrow 0^{+}$ (take subsequence if
necessary) to get a solution to (1.1)-(1.4) still denoted by ($\rho$, $u$,
$\theta$) which satisfies (3) by the lower semi-continuity of the norms. This
proves the existence of the solutions as in Theorem 1.1. The uniqueness of the
solutions can be proved by the standard method like in [4], we omit it for
brevity. The proof of Theorem 1.1 is complete. $\Box$
## 4 Proof of Theorem 1.2
In this section, we use the similar arguments as in Section 3 to prove Theorem
1.2. Throughout this section, we denote $c$ to be a generic constant depending
on $\rho_{0}$, $u_{0}$, $\theta_{0}$, $T$ and some other known constants but
independent of $\delta$ for any $\delta\in(0,1)$.
Denote $\rho_{0}^{\delta}=\rho_{0}+\delta$,
$\theta_{0}^{\delta}=\theta_{0}+\delta$ and
$P^{\delta}_{0}=P(\rho_{0}^{\delta},\ \theta_{0}^{\delta})$, where $\rho_{0}$
and $\theta_{0}$ satisfy the same conditions as those in Theorem 1.2. Note
that $\rho_{0}^{\delta}\in H^{4}(I)$, $\rho_{0}^{\delta}\geq\delta>0$,
$\theta_{0}^{\delta}\in H^{3}(I)$,
$\partial_{x}\theta_{0}^{\delta}|_{x=0,1}=\partial_{x}\theta_{0}|_{x=0,1}=0$,
and
$\begin{cases}\|\rho_{0}^{\delta}\|_{H^{4}}\leq c,\\\
\|\big{(}\sqrt{\rho_{0}^{\delta}}\big{)}_{x}\|_{L^{\infty}}\leq c,\\\
\|\theta_{0}^{\delta}\|_{H^{3}}\leq c.\end{cases}$ (4.1)
Different from Section 3, we need to mollify $g_{3}$. Denote
$g_{3}^{\delta}=J_{\delta}*\overline{g}_{3}$, then $g_{3}^{\delta}\in
C^{\infty}(I)$, where
$\overline{g}_{3}(x)=\begin{cases}-g_{3}(-x),\ x\in[-1,0),\\\ g_{3}(x),\ \ \ \
\ x\in I,\\\ -g_{3}(2-x),\ x\in(1,2],\end{cases}$
and $J_{\delta}(\cdot)=\frac{1}{\sqrt{\delta}}J(\frac{\cdot}{\sqrt{\delta}})$,
and $J$ is the usual mollifier such that $J\in C_{0}^{\infty}(\mathbb{R})$,
supp$J\in(-1,1)$, and $\int_{\mathbb{R}}J(x)dx=1$. Since $g_{3}\in
H_{0}^{1}(I)$, we have $\overline{g}_{3}\in H_{0}^{1}([-1,2])$ and
$\partial_{x}\overline{g}_{3}(x)=\begin{cases}g_{3}^{\prime}(-x),\
x\in[-1,0),\\\ g_{3}^{\prime}(x),\ \ \ \ x\in I,\\\ g_{3}^{\prime}(2-x),\
x\in(1,2].\end{cases}$
Claim:
$\displaystyle\begin{cases}g_{3}^{\delta}\rightarrow g_{3}\ \ \mathrm{in}\
H^{1}(I),\ \mathrm{as}\ \delta\rightarrow 0,\\\
\|g_{3}^{\delta}\|_{H^{1}(I)}\leq c\|\overline{g}_{3}\|_{H^{1}([-1,2])}\leq
c\|g_{3}\|_{H^{1}(I)},\ \mathrm{for}\ \mathrm{any}\ \delta\in(0,1),\\\
\|\sqrt{\rho^{\delta}_{0}}(g_{3}^{\delta})_{xx}\|_{L^{2}(I)}\leq c,\
\mathrm{for}\ \mathrm{any}\ \delta\in(0,1).\end{cases}$ (4.2)
In fact, the proof of (4.2)1 and (4.2)2 can be found in [7]. We are going to
prove (4.2)3.
$\displaystyle\sqrt{\rho_{0}^{\delta}}(g_{3}^{\delta})_{xx}$ $\displaystyle=$
$\displaystyle(\sqrt{\rho_{0}^{\delta}}-\sqrt{\rho_{0}})(g_{3}^{\delta})_{xx}+\sqrt{\rho_{0}}(g_{3}^{\delta})_{xx}$
(4.3) $\displaystyle=$
$\displaystyle\frac{\delta(g_{3}^{\delta})_{xx}}{\sqrt{\rho_{0}^{\delta}}+\sqrt{\rho}}+\sqrt{\rho_{0}}(g_{3}^{\delta})_{xx}$
$\displaystyle=$ $\displaystyle A_{1}+A_{2}.$
Recall
$J_{\delta}(\cdot)=\frac{1}{\sqrt{\delta}}J(\frac{\cdot}{\sqrt{\delta}})$, we
conclude
$\displaystyle\|A_{1}\|_{L^{2}(I)}$ $\displaystyle\leq$
$\displaystyle\sqrt{\delta}\|(g_{3}^{\delta})_{xx}\|_{L^{2}(I)}$ (4.4)
$\displaystyle\leq$ $\displaystyle
c\|(\overline{g}_{3})_{x}\|_{L^{2}([-1,2])}$ $\displaystyle\leq$
$\displaystyle c\|(g_{3})_{x}\|_{L^{2}(I)}.$
A direct calculation combining $\left(\sqrt{\rho_{0}}(g_{3})_{x}\right)_{x}\in
L^{2}(I)$ gives
$\displaystyle\int_{I}|A_{2}|^{2}\leq c.$ (4.5)
By (4.3), (4.4) and (4.5), we get (4.2)3.
Let $u_{0}^{\delta}$ be the solution to the following elliptic problem for
each $\delta\in(0,1)$:
$\displaystyle\begin{cases}u^{\delta}_{0xx}-(P_{0}^{\delta})_{x}=\rho^{\delta}_{0}g^{\delta}_{3},\\\
u_{0}^{\delta}|_{x=0,1}=0.\end{cases}$ (4.6)
Since $\rho_{0}^{\delta}=\rho_{0}+\delta\in H^{4}(I)$,
$\theta_{0}^{\delta}=\theta_{0}+\delta\in H^{3}(I)$, and $g_{3}^{\delta}\in
C^{\infty}(I)$, we obtain from the elliptic theory (see [7]), (4.1), (4.2) and
(4.6) that $u_{0}^{\delta}\in H^{4}(I)\cap H^{1}_{0}(I)$ with the following
properties:
$\displaystyle\begin{cases}u_{0}^{\delta}\rightarrow u_{0}\ \mathrm{in}\
H^{3}(I),\ \mathrm{as}\ \delta\rightarrow 0,\\\
\|u_{0}^{\delta}\|_{H^{4}(I)}\leq c\ \ \mathrm{for}\ \mathrm{any}\
\delta\in(0,1).\end{cases}$ (4.7)
###### Theorem 4.1
Consider the same assumptions as in Theorem 1.2. Then for any $T>0$ and
$\delta\in(0,1)$ there exists a unique global solution $(\rho,u,\theta)$ to
(1.1)-(1.4) with initial data replaced by
($\rho_{0}^{\delta},u_{0}^{\delta},\theta_{0}^{\delta}$), such that
$\displaystyle\rho\in C([0,T];H^{4}),\ \ \ \rho_{t}\in C([0,T];H^{3}),\ \ \
\rho_{tt}\in C([0,T];H^{1})\cap L^{2}([0,T];H^{2}),$
$\displaystyle\rho_{ttt}\in L^{2}(Q_{T}),\
\rho\geq\frac{\displaystyle\delta}{c}>0,\ u\in C([0,T];H^{4}\cap
H^{1}_{0})\cap L^{2}([0,T];H^{5}),$ $\displaystyle u_{t}\in C([0,T];H^{2})\cap
L^{2}([0,T];H^{3}),\ \ \ u_{tt}\in C([0,T];L^{2})\cap L^{2}([0,T];H_{0}^{1}),\
$ $\displaystyle\theta\in C([0,T];H^{3})\cap L^{2}([0,T];H^{4}),\
\theta_{t}\in C([0,T];H^{1})\cap L^{2}([0,T];H^{2}),\ $
$\displaystyle\theta_{tt}\in L^{2}([0,T];L^{2}),\ \theta\geq c_{\delta}>0,\ $
where $c_{\delta}$ is a constant depending on $\delta$, but independent of
$u$.
Proof of Theorem 4.1:
Similarly to the proof of Theorem 3.1, Theorem 4.1 can be proved by some a
priori estimates globally in time.
For any given $T\in(0,+\infty)$, let $(\rho,u,\theta)$ be the solution to
(1.1)-(1.4) as in Theorem 4.1. Then we have the following estimates.
###### Lemma 4.1
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{2}}+\|\rho_{t}\|_{H^{1}}+\displaystyle\|u\|_{H^{3}}+\|\rho
u_{t}\|_{H^{1}}+\|\sqrt{\rho}u_{t}\|_{L^{2}}+\|\theta\|_{H^{3}}$
$\displaystyle+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{t}\|_{H^{1}}+\int_{Q_{T}}\left(u_{xt}^{2}+\rho_{tt}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\rho^{3}u_{tt}^{2}+\rho^{3}\theta_{tt}^{2}\right)\leq
c.$
Proof. Though the initial velocity in Theorem 4.1 (i.e. $u_{0}^{\delta}$) is
different from that in Theorem 3.1 (i.e. $u_{0}$), both of them are bounded in
$H^{3}$. It suffices to check if (3.26) and (3.39) work here. If do, Lemma 4.1
will be obtained from (3).
By (3.11) and (4.6)
$\displaystyle|\sqrt{\rho}u_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|u_{0xx}^{\delta}-P(\rho_{0}^{\delta},\theta_{0}^{\delta})_{x}\right|}{\sqrt{\rho_{0}^{\delta}}}+\sqrt{\rho_{0}^{\delta}}|u_{0}^{\delta}u_{0x}^{\delta}|$
$\displaystyle=$
$\displaystyle\sqrt{\rho_{0}^{\delta}}|g_{3}^{\delta}|+\sqrt{\rho_{0}^{\delta}}|u_{0}^{\delta}u_{0x}^{\delta}|.$
This gives
$\int_{I}\rho u_{t}^{2}(0)\leq c.$
Therefore, (3.26) is valid here.
Multiplying (3.7) by $\displaystyle\frac{1}{Q^{\prime}(\theta)\sqrt{\rho}}$,
taking $t\rightarrow 0^{+}$, and using (1.9)2, we have
$\displaystyle|\sqrt{\rho}\theta_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|(u^{\delta}_{0x})^{2}+\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+|\sqrt{\rho_{0}^{\delta}}u_{0}^{\delta}\theta_{0x}|+|\sqrt{\rho_{0}^{\delta}}\theta_{0}^{\delta}u_{0x}^{\delta}|$
$\displaystyle\leq$
$\displaystyle\frac{\left|u_{0x}^{2}+\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+\frac{c\left|u_{0x}^{\delta}-u_{0x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+\frac{\left|\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}-\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+c$
$\displaystyle\leq$ $\displaystyle
c|g_{2}|+\frac{c\delta}{\sqrt{\rho_{0}^{\delta}}}(1+|\theta_{0xx}|)+\frac{c\left|u_{0x}^{\delta}-u_{0x}\right|}{\sqrt{\delta}}.$
Note that $\|u_{0x}^{\delta}-u_{0x}\|_{L^{2}(I)}\leq c\sqrt{\delta}$ by (1.9)1
and (4.6). This gives
$\displaystyle\int_{I}\rho\theta_{t}^{2}(0)\leq
c\int_{I}g_{2}^{2}+c\int_{I}\theta_{0xx}^{2}+c\leq c.$
Therefore, (3.39) is valid here. $\Box$
###### Lemma 4.2
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{I}u_{xt}^{2}+\int_{Q_{T}}\rho u_{tt}^{2}\leq c.$
Proof. Multiplying (3.23) by $u_{tt}$, integrating it over $I$, and using
integration by parts, Lemma 2.2, Lemma 4.1 and the Cauchy inequality, we have
$\displaystyle\int_{I}\rho
u_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}u_{xt}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{d}{dt}\int_{I}\rho_{t}u_{t}^{2}+\frac{1}{2}\int_{I}\rho_{tt}u_{t}^{2}-\frac{d}{dt}\int_{I}\rho_{t}uu_{x}u_{t}+\int_{I}\rho_{tt}uu_{x}u_{t}+\int_{I}\rho_{t}u_{t}^{2}u_{x}$
$\displaystyle+\int_{I}\rho_{t}uu_{xt}u_{t}-\int_{I}\rho
u_{t}u_{x}u_{tt}-\int_{I}\rho
uu_{xt}u_{tt}+\frac{d}{dt}\int_{I}P_{t}u_{xt}-\int_{I}P_{tt}u_{xt}$
$\displaystyle\leq$
$\displaystyle\frac{d}{dt}\int_{I}\left(P_{t}u_{xt}-\frac{1}{2}\rho_{t}u_{t}^{2}-\rho_{t}uu_{x}u_{t}\right)+c\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}$
$\displaystyle+c\int_{I}u_{xt}^{2}+c\int\rho_{tt}^{2}+\frac{1}{2}\int_{I}\rho
u_{tt}^{2}-\int_{I}P_{tt}u_{xt}+c.$
This gives
$\displaystyle\int_{I}\rho u_{tt}^{2}+\frac{d}{dt}\int_{I}u_{xt}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{d}{dt}\int_{I}\left(2P_{t}u_{xt}-\rho_{t}u_{t}^{2}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}$
(4.8)
$\displaystyle+c\int\rho_{tt}^{2}-\frac{d}{dt}\int_{I}P_{t}^{2}-2\int_{I}P_{tt}(u_{xt}-P_{t})+c.$
We are going to estimate the last term of the right side of (4.8). By
($A_{2}$)-($A_{4}$), integration by parts, Lemma 4.1 and the Cauchy
inequality, we have
$\displaystyle-2\int_{I}P_{tt}(u_{xt}-P_{t})$ $\displaystyle=$
$\displaystyle-2\int_{I}(\rho
Q)_{tt}(u_{xt}-P_{t})-2\int_{I}(P_{c})_{tt}\left[u_{xt}-(\rho
Q)_{t}-(P_{c})_{t}\right]$ $\displaystyle\leq$
$\displaystyle-2\int_{I}\left[(\kappa\theta_{x})_{x}+u_{x}^{2}-(\rho
uQ)_{x}-\rho\theta Q^{\prime}u_{x}\right]_{t}(u_{xt}-P_{t})$
$\displaystyle+c\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c$
$\displaystyle=$ $\displaystyle
2\int_{I}(\kappa\theta_{x})_{t}(u_{xx}-P_{x})_{t}-4\int_{I}u_{x}u_{xt}(u_{xt}-P_{t})-2\int_{I}(\rho
uQ)_{t}(u_{xx}-P_{x})_{t}$ $\displaystyle+2\int_{I}\left[\rho\theta
Q^{\prime}u_{x}\right]_{t}(u_{xt}-P_{t})+c\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c.$
This, combining (3.11), ($A_{2}$)–($A_{4}$), Lemma 4.1 and the Cauchy
inequality, concludes
$\displaystyle-2\int_{I}P_{tt}(u_{xt}-P_{t})$ (4.9) $\displaystyle\leq$
$\displaystyle c+2\int_{I}(\kappa\theta_{x})_{t}(\rho u_{t}+\rho
uu_{x})_{t}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}$
$\displaystyle-2\int_{I}(\rho uQ)_{t}(\rho u_{t}+\rho
uu_{x})_{t}+2\int_{I}(\rho\theta
Q^{\prime}u_{x})_{t}(u_{xt}-P_{t})+c\int_{I}\rho_{tt}^{2}$ $\displaystyle\leq$
$\displaystyle c\int_{I}|(\kappa\theta_{x})_{t}|^{2}+\frac{1}{2}\int_{I}\rho
u_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\int_{I}\rho_{tt}^{2}+c.$
Substituting (4.9) into (4.8), we get
$\displaystyle\frac{1}{2}\int_{I}\rho
u_{tt}^{2}+\frac{d}{dt}\int_{I}u_{xt}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{d}{dt}\int_{I}\left(2P_{t}u_{xt}-\rho_{t}u_{t}^{2}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}$
(4.10)
$\displaystyle+c\int\rho_{tt}^{2}-\frac{d}{dt}\int_{I}P_{t}^{2}+c\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c\int_{I}\rho\theta_{t}^{2}+c.$
Integrating (4.10) over $(0,t)$, and using (1.1)1, integration by parts,
(3.8), (3.11), (4.2)2, (4.6), and Lemma 4.1, we have
$\displaystyle\frac{1}{2}\int_{0}^{t}\int_{I}\rho
u_{tt}^{2}+\int_{I}u_{xt}^{2}$ $\displaystyle\leq$
$\displaystyle\int_{I}\left(2P_{t}u_{xt}+(\rho
u)_{x}u_{t}^{2}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c$
$\displaystyle=$ $\displaystyle\int_{I}\left(2P_{t}u_{xt}-2\rho
uu_{t}u_{xt}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\int_{I}\rho^{2}u^{2}u_{t}^{2}+c\int_{I}\rho_{t}^{2}u^{2}u_{x}^{2}+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}++c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c,$
which implies
$\displaystyle\int_{0}^{t}\int_{I}\rho u_{tt}^{2}+\int_{I}u_{xt}^{2}\leq
c\int_{I}\rho\theta_{t}^{2}+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c.$
(4.11)
Using the Gronwall inequality and Lemma 4.1, we complete the proof of the
lemma. $\Box$
###### Corollary 4.1
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{Q_{T}}u_{xxt}^{2}\leq c.$
Proof. It follows from (3.23), Lemma 4.1 and ($A_{2}$)–($A_{4}$) that
$\displaystyle\int_{Q_{T}}u_{xxt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{Q_{T}}\rho
u_{tt}^{2}+c\int_{Q_{T}}\rho_{t}^{2}u_{t}^{2}+c\int_{Q_{T}}\rho_{t}^{2}u^{2}u_{x}^{2}+c\int_{Q_{T}}\rho^{2}u_{t}^{2}u_{x}^{2}$
$\displaystyle+c\int_{Q_{T}}\rho^{2}u^{2}u_{xt}^{2}+c\int_{Q_{T}}\left|(\rho
Q)_{xt}\right|^{2}+c\int_{Q_{T}}\left|(P_{c})_{xt}\right|^{2}$
$\displaystyle\leq$ $\displaystyle
c+c\int_{0}^{T}\|u_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}u_{xt}^{2}+c\int_{Q_{T}}\rho_{xt}^{2}+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}\theta_{xt}^{2}+c$
$\displaystyle\leq$ $\displaystyle c.$
This proves Corollary 4.1. $\Box$
###### Lemma 4.3
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{I}\left(\rho_{xxx}^{2}+\rho_{xxt}^{2}+\rho_{tt}^{2}\right)+\int_{Q_{T}}\left(\rho_{xtt}^{2}+u_{xxxx}^{2}\right)\leq
c.$
Proof. Differentiating (3.31) with respect to $x$, we have
$\rho_{xxxt}=-\rho_{xxxx}u-4\rho_{xxx}u_{x}-6\rho_{xx}u_{xx}-4\rho_{x}u_{xxx}-\rho
u_{xxxx}.$ (4.12)
Multiplying (4.12) by $2\rho_{xxx}$, integrating the resulting equation over
$I$, and using integration by parts and the Hölder inequality, we have
$\displaystyle\frac{d}{dt}\int_{I}\rho_{xxx}^{2}$ $\displaystyle=$
$\displaystyle-7\int_{I}\rho_{xxx}^{2}u_{x}-12\int_{I}\rho_{xx}\rho_{xxx}u_{xx}-8\int_{I}\rho_{x}\rho_{xxx}u_{xxx}-2\int_{I}\rho\rho_{xxx}u_{xxxx}$
$\displaystyle\leq$ $\displaystyle
7\|u_{x}\|_{L^{\infty}}\int_{I}\rho_{xxx}^{2}+12\|u_{xx}\|_{L^{\infty}}\|\rho_{xx}\|_{L^{2}}\|\rho_{xxx}\|_{L^{2}}$
$\displaystyle+8\|\rho_{x}\|_{L^{\infty}}\|\rho_{xxx}\|_{L^{2}}\|u_{xxx}\|_{L^{2}}+2\|\rho\|_{L^{\infty}}\|\rho_{xxx}\|_{L^{2}}\|u_{xxxx}\|_{L^{2}}.$
By Lemma 4.1 and the Cauchy inequality, we get
$\frac{d}{dt}\int_{I}\rho_{xxx}^{2}\leq
c\int_{I}\rho_{xxx}^{2}+c\int_{I}u_{xxxx}^{2}+c.$ (4.13)
Differentiating (3.33) with respect to $x$, we have
$\displaystyle u_{xxxx}=\rho_{xx}u_{t}+2\rho_{x}u_{xt}+\rho
u_{xxt}+(\rho_{x}uu_{x})_{x}+(\rho u_{x}^{2})_{x}+(\rho uu_{xx})_{x}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ +(P_{c})_{xxx}+(\rho Q)_{xxx}.$ (4.14)
By (4), ($A_{6}$) and Lemma 4.1, we have
$\int_{I}u_{xxxx}^{2}\leq c\int_{I}\rho
u_{xxt}^{2}+c\int_{I}\rho_{xxx}^{2}+c.$ (4.15)
By (4.13), (4.15), Corollary 4.1 and the Gronwall inequality, we get
$\int_{I}\rho_{xxx}^{2}\leq c.$ (4.16)
It follows from (4.15), (4.16) and Corollary 4.1 that
$\int_{Q_{T}}u_{xxxx}^{2}\leq c.$
A direct calculation, combining (1.1)1, (3.31), (4.16), Lemma 4.1, Corollary
4.1 and Lemma 4.2, implies
$\int_{I}\left(\rho_{xxt}^{2}++\rho_{tt}^{2}\right)+\int_{Q_{T}}\rho_{xtt}^{2}\leq
c.$
The proof of Lemma 4.3 is complete. $\Box$
The next lemma play the most important role in getting $H^{4}$ estimates of
$u$.
###### Lemma 4.4
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{I}\rho^{3}u_{tt}^{2}+\int_{Q_{T}}\rho^{2}u_{xtt}^{2}\leq c.$
Proof. Differentiating (3.23) with respect to $t$, we have
$(\rho u_{tt})_{t}+\rho_{tt}u_{t}+\rho_{t}u_{tt}+(\rho_{t}uu_{x}+\rho
u_{t}u_{x}+\rho uu_{xt})_{t}+P_{xtt}=u_{xxtt}.$ (4.17)
Multiplying (4.17) by $\rho^{\gamma_{2}}u_{tt}$ ($\gamma_{2}$ is to be decided
later), and integrating the resulting equation over $I$, we have
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}+\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}$
$\displaystyle=$
$\displaystyle\frac{\gamma_{2}-3}{2}\int_{I}\rho^{\gamma_{2}}\rho_{t}u_{tt}^{2}-\int_{I}[\rho_{tt}u_{t}+\rho_{tt}uu_{x}+2\rho_{t}u_{t}u_{x}+2\rho_{t}uu_{xt}+2\rho
u_{t}u_{xt}+P_{xtt}](\rho^{\gamma_{2}}u_{tt})$
$\displaystyle-\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}u_{x}-\int_{I}\rho^{\gamma_{2}+1}uu_{tt}u_{xtt}-\gamma_{2}\int_{I}\rho^{\gamma_{2}-1}\rho_{x}u_{tt}u_{xtt}$
$\displaystyle\leq$ $\displaystyle
c\|(\sqrt{\rho})_{t}\|_{L^{\infty}}\int_{I}\rho^{\gamma_{2}+\frac{1}{2}}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}-\int_{I}\rho^{\gamma_{2}}u_{tt}(\rho
Q)_{xtt}-\int_{I}\rho^{\gamma_{2}}u_{tt}(P_{c})_{xtt}$
$\displaystyle+c\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}-\int_{I}\rho^{\gamma_{2}+1}uu_{tt}u_{xtt}-2\gamma_{2}\int_{I}\rho^{\gamma_{2}-\frac{1}{2}}u_{xtt}u_{tt}(\sqrt{\rho})_{x}+c$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{\gamma_{2}+\frac{1}{2}}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}+\int_{I}\rho^{\gamma_{2}}u_{xtt}(\rho
Q)_{tt}+\gamma_{2}\int_{I}\rho^{\gamma_{2}-1}\rho_{x}u_{tt}(\rho Q)_{tt}$
$\displaystyle+c\|\rho_{xtt}\|_{L^{2}}^{2}+\frac{1}{4}\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}+c\int_{I}\rho^{\gamma_{2}+2}u_{tt}^{2}+c\int_{I}\rho^{\gamma_{2}-1}u_{tt}^{2}+c$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{\gamma_{2}-1}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}+\frac{1}{2}\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}+c\int_{I}\rho^{\gamma_{2}}\left|(\rho
Q)_{tt}\right|^{2}$
$\displaystyle+c\int_{I}\rho^{2\gamma_{2}-2}u_{tt}^{2}+c\int_{I}\rho\left|(\rho
Q)_{tt}\right|^{2}+c\|\rho_{xtt}\|_{L^{2}}^{2}+c.$
Here, we have used integration by parts, the Cauchy inequality, ($A_{2}$),
($A_{3}$), Lemma 2.2, Lemma 4.1, Lemma 4.2 and Lemma 4.3.
After the third term of the right side is absorbed by the left, we have
$\displaystyle\frac{d}{dt}\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}+\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{\gamma_{2}-1}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}+c\int_{I}\rho^{\gamma_{2}}\left|(\rho
Q)_{tt}\right|^{2}$ (4.18)
$\displaystyle+c\int_{I}\rho^{2\gamma_{2}-2}u_{tt}^{2}+c\int_{I}\rho\left|(\rho
Q)_{tt}\right|^{2}+c\|\rho_{xtt}\|_{L^{2}}^{2}+c.$
By Lemma 4.2, we know $\int_{Q_{T}}\rho u_{tt}^{2}\leq c$. This implies that
the minimum of $\gamma_{2}$ we should take in (4.18) is $2$. Substituting
$\gamma_{2}=2$ into (4.18), we have
$\frac{d}{dt}\int_{I}\rho^{3}u_{tt}^{2}+\int_{I}\rho^{2}u_{xtt}^{2}\leq
c\int_{I}\rho u_{tt}^{2}+c\int_{I}\rho\left|(\rho
Q)_{tt}\right|^{2}+c\int_{I}\rho_{xtt}^{2}+c.$ (4.19)
We are going to estimate $\int_{I}\rho\left|(\rho Q)_{tt}\right|^{2}$. Using
Lemma 4.1, ($A_{4}$) and Lemma 4.3, we have
$\displaystyle\int_{I}\rho\left|(\rho Q)_{tt}\right|^{2}$ $\displaystyle=$
$\displaystyle\int_{I}\rho\left|\rho_{tt}Q+2\rho_{t}Q^{\prime}\theta_{t}+\rho
Q^{\prime\prime}\theta_{t}^{2}+\rho Q^{\prime}\theta_{tt}\right|^{2}$ (4.20)
$\displaystyle\leq$ $\displaystyle
c+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\rho^{3}\theta_{tt}^{2}.$
Substituting (4.20) into (4.19), integrating the resulting inequality over
$(0,t)$, and using Lemma 4.1, Lemma 4.2 and Lemma 4.3, we get
$\int_{I}\rho^{3}u_{tt}^{2}+\int_{0}^{t}\int_{I}\rho^{2}u_{xtt}^{2}\leq\int_{I}\rho^{3}u_{tt}^{2}(x,0)+c.$
(4.21)
Using (4.1), (4.2), (4.6), (4.7), (3.8), (3.23) and (4.2)3, we have
$\int_{I}\rho^{3}u_{tt}^{2}(x,0)\leq c,$
which combining (4.21) completes the proof. $\Box$
###### Lemma 4.5
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\displaystyle\int_{I}\rho
u_{xxt}^{2}+\int_{Q_{T}}\left(u_{xxxt}^{2}+\rho\theta_{xxt}^{2}\right)\leq c.$
Proof. By (3.23), we have
$\displaystyle\int_{I}\rho u_{xxt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{3}u_{tt}^{2}+c\int_{I}\rho\rho_{t}^{2}u_{t}^{2}+c\int_{I}\rho\rho_{t}^{2}u^{2}u_{x}^{2}+\int_{I}\rho^{3}u_{t}^{2}u_{x}^{2}$
$\displaystyle+c\int_{I}\rho^{3}u^{2}u_{xt}^{2}+c\int_{I}\rho|(\rho
Q)_{xt}|^{2}+c\int_{I}\rho|(P_{c})_{xt}|^{2}$ $\displaystyle\leq$
$\displaystyle c,$
where we have used ($A_{2}$)-($A_{4}$), Lemma 2.2, Lemma 4.1, Lemma 4.2 and
Lemma 4.4.
It follows from (3.37) and ($A_{5}$) that
$\displaystyle\int_{Q_{T}}\rho\theta_{xxt}^{2}$ $\displaystyle\leq$
$\displaystyle
c\int_{Q_{T}}\rho|\kappa^{\prime}|^{2}\theta_{t}^{2}\theta_{xx}^{2}+c\int_{Q_{T}}\rho|\kappa^{\prime\prime}|^{2}\theta_{t}^{2}\theta_{x}^{4}+c\int_{Q_{T}}\rho|\kappa^{\prime}|^{2}\theta_{x}^{2}\theta_{xt}^{2}$
$\displaystyle+c\int_{Q_{T}}\rho^{3}|Q^{\prime}|^{2}\theta_{tt}^{2}+c\int_{Q_{T}}\rho^{3}|Q^{\prime\prime}|^{2}\theta_{t}^{4}+c\int_{Q_{T}}\rho\rho_{t}^{2}|Q^{\prime}|^{2}\theta_{t}^{2}$
$\displaystyle+c\int_{Q_{T}}\rho\left|(\rho
uQ^{\prime}\theta_{x})_{t}\right|^{2}+c\int_{Q_{T}}\rho\left|(\rho\theta
Q^{\prime}u_{x})_{t}\right|^{2}+c\int_{Q_{T}}\rho u_{x}^{2}u_{xt}^{2},$
which, combining ($A_{4}$), ($A_{5}$) and Lemma 4.1, gives
$\displaystyle\int_{Q_{T}}\rho\theta_{xxt}^{2}$ $\displaystyle\leq$
$\displaystyle
c\int_{Q_{T}}\theta_{xt}^{2}+c\int_{Q_{T}}\rho^{3}\theta_{tt}^{2}+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+\int_{Q_{T}}u_{xt}^{2}+c$
(4.22) $\displaystyle\leq$ $\displaystyle c.$
Differentiating (3.33) with respect to $t$, we get
$\displaystyle u_{xxxt}$ $\displaystyle=$ $\displaystyle
2(\sqrt{\rho})_{x}\sqrt{\rho}u_{tt}+\rho
u_{xtt}+\rho_{xt}u_{t}+\rho_{t}u_{xt}+\rho_{xt}uu_{x}+\rho_{t}u_{x}^{2}+\rho_{t}uu_{xx}+\rho_{x}u_{t}u_{x}$
$\displaystyle+2\rho u_{xt}u_{x}+\rho u_{t}u_{xx}+\rho_{x}uu_{xt}+\rho
uu_{xxt}+(\rho Q)_{xxt}+(P_{c})_{xxt}.$
This, together with (4.22), ($A_{6}$), Lemma 2.2, Lemma 4.1, Lemma 4.2, Lemma
4.3, Lemma 4.4 and Corollary 4.1, implies
$\displaystyle\int_{Q_{T}}u_{xxxt}^{2}\leq
c+c\int_{Q_{T}}\left(\rho\theta_{xxt}^{2}+\theta_{xt}^{2}\right)+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}\rho_{xxt}^{2}\leq
c.$
This completes the proof. $\Box$
From (4.15), Lemma 4.3 and Lemma 4.5, we get the following corollary
immediately.
###### Corollary 4.2
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{I}u_{xxxx}^{2}\leq c.$
###### Corollary 4.3
Under the conditions of Theorem 4.1, it holds
$\int_{Q_{T}}\theta_{xxxx}^{2}\leq c.$
Proof. Differentiating (3) with respect to $x$, we have
$\displaystyle\kappa\theta_{xxxx}=-4\kappa^{\prime}\theta_{x}\theta_{xxx}-3\kappa^{\prime}\theta_{xx}^{2}-3\kappa^{\prime\prime}\theta_{x}^{2}\theta_{xx}-\left(\kappa^{\prime\prime}\theta_{x}^{3}\right)_{x}-2\left(u_{x}u_{xx}\right)_{x}+\left(\rho
Q^{\prime}\theta_{xt}\right)_{x}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\left(\rho_{x}Q^{\prime}\theta_{t}\right)_{x}+\left(\rho
Q^{\prime\prime}\theta_{x}\theta_{t}\right)_{x}+(\rho
uQ^{\prime}\theta_{x})_{xx}+(\rho\theta Q^{\prime}u_{x})_{xx}.$
This, combining ($A_{5}$), ($A_{6}$), Lemma 3.3, Lemma 4.1 and Lemma 4.5,
implies
$\displaystyle\int_{Q_{T}}\theta_{xxxx}^{2}\leq
c+c\int_{Q_{T}}\left(\rho\theta_{xxt}^{2}+\theta_{xt}^{2}\right)+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}|\kappa^{\prime\prime\prime}|^{2}\theta_{x}^{8}\leq
c.$
This proves Corollary 4.3. $\Box$
###### Lemma 4.6
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{I}\rho_{xxxx}^{2}+\int_{Q_{T}}u_{xxxxx}^{2}\leq c.$
Proof. Differentiating (4.12) with respect to $x$, multiplying the resulting
equation by $2\rho_{xxxx}$, integrating over $I$, and using integration by
parts, Lemma 4.1, Lemma 4.3, Corollary 4.2 and the Cauchy inequality, we get
$\displaystyle\frac{d}{dt}\int_{I}\rho_{xxxx}^{2}$ $\displaystyle=$
$\displaystyle-9\int_{I}\rho_{xxxx}^{2}u_{x}-20\int_{I}\rho_{xxx}\rho_{xxxx}u_{xx}-20\int_{I}\rho_{xx}\rho_{xxxx}u_{xxx}$
(4.23)
$\displaystyle-10\int_{I}\rho_{x}\rho_{xxxx}u_{xxxx}-2\int_{I}\rho\rho_{xxxx}u_{xxxxx}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho_{xxxx}^{2}+c\int_{I}u_{xxxxx}^{2}+c.$
Now we estimate the second term of the right-hand side of (4.23).
Differentiating (4) with respect to $x$, we have
$\displaystyle u_{xxxxx}$ $\displaystyle=$
$\displaystyle\rho_{xxx}u_{t}+3\rho_{xx}u_{xt}+3\rho_{x}u_{xxt}+\rho
u_{xxxt}+(\rho_{x}uu_{x})_{xx}+(\rho u_{x}^{2})_{xx}$ $\displaystyle+(\rho
uu_{xx})_{xx}+(\rho Q)_{xxxx}+(P_{c})_{xxxx}.$
This, combining $(A_{6})$, Lemma 2.2, Lemma 4.1, Lemma 4.3 and Corollary 4.2,
concludes
$\displaystyle\int_{I}u_{xxxxx}^{2}\leq
c\int_{I}u_{xxt}^{2}+c\int_{I}u_{xxxt}^{2}+c\int_{I}\rho_{xxxx}^{2}+c\int_{I}\theta_{xxxx}^{2}+c.$
(4.24)
Substituting (4.24) into (4.23), and using Corollary 4.1, Corollary 4.3, Lemma
4.5 and the Gronwall inequality, we obtain
$\int_{I}\rho_{xxxx}^{2}\leq c.$ (4.25)
It follows from (4.24), (4.25), Corollary 4.1, Corollary 4.3 and Lemma 4.5
that
$\int_{Q_{T}}u_{xxxxx}^{2}\leq c.$
This completes the proof of Lemma 4.6. $\Box$
###### Corollary 4.4
Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$
$\int_{I}\left(\rho_{xtt}^{2}+\rho_{xxxt}^{2}\right)+\int_{Q_{T}}\left(\rho_{ttt}^{2}+\rho_{xxtt}^{2}\right)\leq
c.$
Here we have used the following inequality when we get the upper bound of
$\rho_{ttt}$:
$\rho_{x}^{2}u_{tt}^{2}=2\left[(\sqrt{\rho})_{x}\sqrt{\rho}\right]^{2}u_{tt}^{2}\leq
c\rho u_{tt}^{2}.$
From the above estimates, we get
$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{4}}+\|\rho_{t}\|_{H^{3}}\displaystyle+\|\rho_{tt}\|_{H^{1}}+\|u\|_{H^{4}}$
$\displaystyle+\|u_{t}\|_{H^{1}}+\|\rho^{\frac{3}{2}}u_{tt}\|_{L^{2}}+\|\sqrt{\rho}u_{xxt}\|_{L^{2}}+\|\theta\|_{H^{3}}+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{xt}\|_{L^{2}}$
$\displaystyle+\int_{Q_{T}}\left(\rho^{2}u_{xtt}^{2}+\rho
u_{tt}^{2}+u_{xxt}^{2}+u_{xxxt}^{2}+u_{xxxxx}^{2}\right)$
$\displaystyle+\int_{Q_{T}}\left(\rho_{ttt}^{2}+\rho_{xxtt}^{2}+\theta_{xxxx}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\rho\theta_{xxt}^{2}+\rho^{3}\theta_{tt}^{2}\right)\leq
c.$ (4.26)
From (4) and (3.55), we get
$\displaystyle\|\rho\|_{H^{4}}+\|\rho_{t}\|_{H^{3}}+\|\rho_{tt}\|_{H^{1}}+\|u\|_{H^{4}}+\|u_{t}\|_{H^{2}}+\|u_{tt}\|_{L^{2}}+\|\theta\|_{H^{3}}+\|\theta_{t}\|_{H^{1}}$
$\displaystyle+\int_{Q_{T}}\left(u_{xtt}^{2}+u_{xxxt}^{2}+u_{xxxxx}^{2}\right)+\int_{Q_{T}}\left(\rho_{ttt}^{2}+\rho_{xxtt}^{2}+\theta_{xxxx}^{2}+\theta_{xxt}^{2}+\theta_{tt}^{2}\right)\leq
c(\delta),$
where $c(\delta)$ is a positive constant, and may depend on $\delta$.
The proof of Theorem 4.1 is complete. $\Box$
Proof of Theorem 1.2:
Consider (1.1)-(1.4) with initial data replaced by ($\rho_{0}^{\delta}$,
$u_{0}^{\delta}$, $\theta_{0}^{\delta}$), we obtain from Theorem 4.1 that
there exists a unique solution ($\rho^{\delta}$, $u^{\delta}$,
$\theta^{\delta}$) such that (4) and (3.55) are valid when we replace ($\rho,$
$u$, $\theta$) by ($\rho^{\delta}$, $u^{\delta}$, $\theta^{\delta}$). With
this estimates uniform for $\delta$, we take $\delta\rightarrow 0^{+}$ ( take
subsequence if necessary) to get a solution to (1.1)-(1.4) still denoted by
($\rho$, $u$, $\theta$). By the lower semi-continuity of the norms, we have
$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{4}}+\|\rho_{t}\|_{H^{3}}\displaystyle+\|\rho_{tt}\|_{H^{1}}+\|u\|_{H^{4}}+\|u_{t}\|_{H^{1}}$
$\displaystyle+\|\sqrt{\rho}u_{xxt}\|_{L^{2}}+\|\theta\|_{H^{3}}+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{xt}\|_{L^{2}}+\int_{Q_{T}}\left({u}_{xxt}^{2}+{u}_{xxxt}^{2}+{u}_{xxxxx}^{2}\right)$
$\displaystyle+\int_{Q_{T}}\left({\rho}_{ttt}^{2}+{\rho}_{xxtt}^{2}+{\theta}_{xxxx}^{2}+{\theta}_{t}^{2}+{\theta}_{xt}^{2}\right)\leq
c,$
which proves the existence of the solutions as in Theorem 1.2. The uniqueness
of the solutions can be proved by the standard method like in [4], we omit it
for brevity. The proof of Theorem 1.2 is complete. $\Box$
Acknowledgment.
The first author was supported by the National Basic Research Program of China
(973 Program) No. 2010CB808002, and by the National Natural Science Foundation
of China No. 11071086. The second author was supported by the National Natural
Science Foundation of China $\\#$10625105, $\\#$11071093, the PhD specialized
grant of the Ministry of Education of China $\\#$20100144110001 and the self-
determined research funds of CCNU from the colleges’basic research and
operation of MOE.
## References
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|
arxiv-papers
| 2011-03-08T02:53:05 |
2024-09-04T02:49:17.533192
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huanyao Wen, Changjiang Zhu",
"submitter": "Changjiang Zhu",
"url": "https://arxiv.org/abs/1103.1421"
}
|
1103.1434
|
# Phase-stable limited relativistic acceleration or unlimited relativistic
acceleration in the laser-thin-foil interactions
Yongsheng Huang http://www.anianet.com/adward huangyongs@gmail.com China
Institute of Atomic Energy, Beijing 102413, China. Naiyan Wang Xiuzhang Tang
China Institute of Atomic Energy, Beijing 102413, China. Yan Xueqing State
Key Laboratory of Nuclear Physics and Technology, Institute of Heavy Ion
Physics, Peking University, Beijing 100871, China
###### Abstract
To clarify the relationship between phase-stable acceleration (PSA) and
unlimited relativistic acceleration (URA) (Phys. Rev. Lett. 104, 135003
(2010)), an analytical relativistic model is proposed in the interactions of
the ultra-intense laser and nanometer foils, based on hydrodynamic equations.
The dependence of the ion momentum on time is consistent with the previous
results and checked by PIC simulations. Depending on the initial ion momentum,
relativistic RPA contains two acceleration processes: phase-stable limited
relativistic acceleration (PS-LRA) and URA. In PS-LRA, the potential is a deep
well trapping the ions. The ion front, i.e., the bottom, separates it into two
parts: the left half region is PSA region; the right half region is PSD
region, where the ions climb up and are decelerated to return back. In PS-LRA,
the maximum ion energy is limited. If the initial ion momentum large enough,
ions will experience a potential downhill and drop into a bottomless abyss,
which is called phase-lock-like position. URA is not phase-stable any more. At
the phase-lock-like position the ions can obtain unlimited energy gain and the
ion density is non-zero. You cannot have both PSA and URA.
###### pacs:
52.38.Kd,41.75.Jv,52.40.Kh,52.65.-y
Laser-ion acceleration has been an international research focusMakoTajima ;
Machnisms ; Esirkepov ; Yin , however it is still a challenge to obtain mono-
energetic proton beams larger than $100\mathrm{MeV}$. As a promising method to
generate relativistic mono-energetic protons, radiation pressure acceleration
(RPA) has attracted more attentionEsirkepov ; Yin ; EsirkepovPRL96 ; Henig
and becomes dominant in the interaction of the ultra-intense laser pulse with
nanometer foils. The phase-stable accelerationYanxqPRL predicted that the
energy spread can be improved in the interactions of nanometer-foils with
circular-polarized laser pulses. With thin-shell modelunlimitedRPA , Bulanov
and coworkersunlimitedRPA pointed out that the ions can obtain unlimited
energy gain by RPA in the relativistic limit. Yan and coworkers tried to
predict the ion energy distribution with a self-similar hydrodynamic
theoryYanxq . However, it is nonrelativistic and under the plasma
approximation which allows $\nabla\bullet E\neq 0$ and $n_{i}-n_{e}=0$ satisfy
together, where $E$ is the acceleration field and $n_{i}$ (or $n_{e}$) is the
ion (or electron) density. However, can the ions obtain unlimited energy gain
in the phase-stable region or is the phase-stable acceleration still possible
in the relativistic limit? How about the relationship between them or what are
the critical conditions for them? To give clear answers of the questions, an
analytic self-consistent relativistic fluid model is proposed to describe the
relativistic radiation pressure acceleration and to recheck the unlimited ion-
acceleration region.
The ion acceleration in the interaction of ultra-intense laser and nanometer
foils contains two stages: the hole-boring process and the radiation pressure
acceleration. Here the transition of them is assumed steady and the
instability of the acceleration sheath is suppressed well, which can be
realized for specially designed targets stableRPA1 ; stableRPA . In the hole-
boring process, the ion velocity can reach $u_{hb}$Qiao2009 ; breaktime ,
which is the hole-boring velocity and also the initial velocity of the ions in
the radiation pressure acceleration. The initial time, $t_{0}$ is when the
compression layer is detached from the foil. It is decided by the target
thickness and $u_{hb}$. With the initial conditions: $u_{hb}$ and $t_{0}$, the
dependence of the ion energy on time can be obtained and consists with the
results of thin-shell modelEsirkepov and has been checked by PIC
simulationsEsirkepov . Depending on the laser intensity, the initial ion
momentum will determine two different acceleration processes: the phase-stable
limited relativistic acceleration (PS-LRA) and the unlimited relativistic
acceleration (URA). When the initial ion momentum is smaller than the critical
one, the potential is a deep well trapping the ions. The acceleration mode is
PS-LRA and contains the phase-stable acceleration region (PSA) and the phase-
stable deceleration region (PSD). The well is separated into two half-regions:
the left half-region and the right half-region by the bottom, which is the ion
front. The left half-region is PSA region, where the electric field is
positive and ions coast down to the bottom. While the right half-region is PSD
region, where the electric field is negative and ions climb up the potential
uphill and are decelerated to return to the bottom. The deceleration is also
phase-stable. No matter in PSA or PSD region, the maximum ion energy is
limited and ascertained by an analytical formulation. Since PSA and PSD are
separated by the ion front where the ion density is zero, the ions in two
regions can not exchange from each other. If the initial ion momentum is large
enough, the ions can get across the potential uphill and experience a
potential downhill and then drop into the bottomless abyss which is the phase-
lock-like position. If the ions can reach the phase-lock-like position, they
can obtain unlimited energy gain as Bulanov and coworkers pointed
outunlimitedRPA . The acceleration mode is URA and not phase-stable any more.
The electron density is smaller than the ion density and the electron front
increases with time. The phase-lock-like position of URA is the limiting ion
front and the ion density is non-zero as time tends to zero. You cannot have
both PSA and URA in the same acceleration process. Therefore, the maximum ion
energy is finite in the phase-stable region and URA is not phase-stable any
more. However, no matter in PS-LRA or URA, the plasma tends to neutral as time
tends to infinite.
For convenience, the physical parameters: the time, $t$, the ion position,
$x$, the ion velocity, $v$, the electron field, $E$, the electric potential,
$\varphi$, the plasma density, $n$, and the light speed, $c$, are normalized
as follows: ${\tau}={\omega
t},\hat{x}=xk,u=v/c,\hat{E}={E}/{E_{0}},{\phi}=\varphi/\varphi_{0},\hat{n}={n}/{n_{0}},$
where $n$ represents $n_{i}$ (or $n_{e}$), $n_{0}$ is the reference density,
$\omega$ is the light frequency, $k=\omega/c$ is the wave number, $c$ is the
light speed, $E_{0}={k\varphi_{0}}$, $e\varphi_{0}=\gamma_{em}m_{e}c^{2}$ and
$\gamma_{em}$ is the maximum electron energy. Here $e$ is the elemental
charge.
With reference to the results given by Mako and Tajima in Ref. MakoTajima , in
the self-similar state, the density distribution of ions is assumed as:
$\hat{n}_{k}=\frac{1}{\Sigma Q_{k}}(1+\phi)^{\alpha},k=1,...,N,$ (1)
where the subscribe k stands for the ion species, $Q_{k}$ is the charge number
of the $k$th species ion, the index $\alpha$ depends on the laser intensity
and the target thickness and discussed in the Ref. Yanxq .
With the transformation: $\xi=\hat{x}/\tau$, the normalized continuity and
motion equation of ions and Poisson’s equation are given as:
$\displaystyle\left(u_{k}-\xi\right)\frac{\partial\ln\hat{n}_{k}}{\partial\xi}=-\frac{\partial
u_{k}}{\partial\xi},$ (2)
$\displaystyle\left(u_{k}-\xi\right)\frac{\partial\gamma_{k}u_{k}}{\partial\xi}=-\beta_{k}\frac{\partial\phi}{\partial\xi},$
$\frac{1}{\tau^{2}}\frac{\partial^{2}\phi}{\partial\xi^{2}}=-\rho\left(\Sigma
Q_{k}\hat{n}_{k}-\hat{n}_{e}\right)$ (3)
where $\beta_{k}=\frac{Q_{k}\gamma_{em}m_{e}}{M_{k}}$, $M_{k}$ is the mass of
certain ions, $\gamma_{k}=(1-u_{k}^{2})^{-1/2}$,
$\rho=\frac{\omega_{pe}^{2}}{\gamma_{em}\omega^{2}}$, and
$\omega_{pe}^{2}=\frac{n_{0}e^{2}}{\epsilon_{0}m_{e}}$.
Solving Eq. (2), the ion velocity satisfies:
$\xi+\left(\frac{\gamma_{k}}{\gamma_{k,0}}\right)^{-3/2}u_{k,0}=u_{k}+\frac{\gamma_{k}^{-3/2}}{2\alpha}\left(\chi-\chi_{0}\right),$
(4)
and the potential in the ion region is given by:
$\phi_{1}=\frac{\left(\chi-\chi_{0}-2\alpha
u_{k,0}\gamma_{k,0}^{3/2}\right)^{2}}{4\alpha\beta_{k}}-1,$ (5)
where $\gamma_{k,0}=(1-u_{k,0}^{2})^{-1/2}$, $\chi_{0}=\chi(u_{k,0})$,
$\chi=\int^{u_{k}}_{0}\gamma_{k}^{3/2}du_{k},$ (6)
and $u_{k,0}$ is the hole-boring velocity given byQiao2009 ; breaktime :
$u_{k,0}=\frac{u_{hb}}{c}=\sqrt{\frac{Z}{A}\frac{m_{e}}{\gamma_{hb}M_{k}}\frac{n_{c}}{2n_{0}}}a$
(7)
where $a^{2}=0.732I_{10^{18}\mathrm{W/cm^{2}}}\lambda^{2}_{\mathrm{\mu m}}$,
$I_{10^{18}\mathrm{W/cm^{2}}}$ is the laser intensity in unit of
$10^{18}\mathrm{W/cm^{2}}$, and $\gamma_{hb}={(1-(u_{hb}/c)^{2})^{-1/2}}$. The
beginning time $\tau_{0}$ is when the compressed ion and electron layer is
detached from the foil and given bybreaktime :
$\tau_{0}=\frac{d}{\lambda}\frac{2\pi}{u_{k,0}},$ (8)
at $\xi=0$.
In order to obtain the dependence of the ion velocity on time, it is need to
solve Eq. (4). From Eq. (4), the following differential equations of two
variables are given:
$\left\\{\begin{aligned} \frac{dp_{k}}{dt}=\frac{2\alpha
V_{k}(1+p_{k}^{2})^{3/2}}{t\left[3\alpha
p_{k}V_{k}\sqrt{1+p_{k}^{2}}-(2\alpha+1)t\right]},\\\
\frac{dV_{k}}{dt}=\frac{-2\alpha V_{k}}{3\alpha
p_{k}V_{k}\sqrt{1+p_{k}^{2}}-(2\alpha+1)t}.\end{aligned}\right.$ (9)
where $p_{k}=u_{k}\gamma_{k}$ is the normalized ion momentum and $V_{k}$
satisfies:
$V_{k}=\int^{t}_{0}\frac{p_{k}}{\sqrt{1+p_{k}^{2}}}dt-t\frac{p_{k}}{\sqrt{1+p_{k}^{2}}}.$
(10)
Using matlab function $ode113$ to solve Eq. (9) with the initial time
$\tau_{0}$ and initial velocity, $u_{k,0}$, the dependence of the ion energy
on time has been calculated matlabfile . As an example, Figure 1 shows the
comparison of our analytical fluid model with the thin-shell model and PIC
simulationsEsirkepov . The results of our model consist well with that of the
PIC simulations. When $\tau$ is larger than $40$ times of the laser cycle, our
results are a little larger than that of the PIC simulations. One of the
reason is the loss of laser energy and the decreasing of the electron
temperature for large $\tau$ in the simulations, while the electron
temperature is assumed to be a constant in our model.
Figure 1: (Color online) Comparison of our analytical fluid model with thin-
shell model and Esirkepov et al.’s simulations for $\sigma/a\approx 0.1$,
$a=316$, $d=\lambda=1\mu m$, $n_{0}=49n_{c}=5.5\times 10^{22}/cm^{3}$, and
$\alpha=1.8$Yanxq .
Combing Eqs. (1) and (5), it is obtained:
$\hat{n}_{k}=\frac{1}{\Sigma Q_{k}}\frac{\left(\chi-\chi_{0}-2\alpha
u_{k,0}\gamma_{k,0}^{3/2}\right)^{2\alpha}}{\left(4\alpha\beta_{k}\right)^{\alpha}},$
(11)
With Eqs. (3), (4) and (5), the electron density in the ion region is written
as:
$\hat{n}_{e}=\left(1+\phi_{1}\right)^{\alpha}+\frac{1}{\rho\tau^{2}}\frac{\partial^{2}\phi_{1}}{\partial\xi^{2}},$
(12)
Equation (12) shows the plasma can not be quasi-neutral at a finite time.
However, the plasma tends to neutral as the time tends to infinite. It is
obtained:
$\lim_{\tau\rightarrow+\infty}n_{e}=\Sigma Q_{k}n_{k},$ (13)
With Eq. (11) and $\hat{n}_{k}(\xi=\xi_{i,f})=0$, the possible maximum ion
energy at the ion front satisfies:
$\int_{0}^{p_{k,m}}\gamma_{k}^{-\frac{3}{2}}dp_{k}=\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}},$ (14)
where $\gamma_{k}^{2}=1+p_{k}^{2}$, $p_{k,m}=u_{k,m}/\sqrt{1-u_{k,m}^{2}}$ is
the normalized limiting momentum of ions at the ion front: $\xi=\xi_{i,f}$ and
$p_{k,0}=u_{k,0}/\sqrt{1-u_{k,0}^{2}}$.
Different from the nonrelativistic case, it contains two acceleration modes
and depends on the initial conditions: $u_{k,0}$ and $\alpha$ in the
relativistic case. The critical condition is decided by:
$\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}}=\int_{0}^{+\infty}(1+x^{2})^{-3/4}dx\approx
2.622.$ (15)
First, $\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}}\lneq 2.622$ for $p_{k,0}\lneq 0.5064$ and
$\alpha=2$, which requires $a\lneq 203$ for $n_{0}=49n_{c}$, $d=\lambda=1\mu
m$. Figure 2 shows the dependence of the plasma density, electric field and
potential on $\xi$ for $n_{0}=10^{22}\mathrm{/cm^{3}}$, $\alpha=2$ and $a=70$.
In this case, it is divided into two regions: the phase-stable acceleration
region for $0\leq\xi\leq\xi_{i,f}$ and the phase-stable deceleration region
$\xi_{i,f}\leq\xi\leq 1$.
(I) In the phase-stable acceleration region, $0\leq\xi\leq\xi_{i,f}$, the
electric field $E\geq 0$ and the electron density is larger than the ion
density. The maximum ion momentum $p_{k,m1}$ is limited and given Eq. (14) at
the ion front $\xi_{i,f}$ ascertained by Eq. (4) with $u_{k}=u_{k,m1}$, where
$u_{k,m1}=p_{k,m1}/\sqrt{1+p_{k,m1}^{2}}$. At the ion front, the ion density
is zero. The difference of the electron density and ion density decreases with
time and tends to zero.
The potential shown by Figure 2 (b) in $0\leq\xi\leq\xi_{i,f}$ gives an
intuitionistic explanation of PSA. The ions coast down the slope of the
potential, and the gradient, i.e., the electric field, becomes gently as the
ions come to the bottom of the potential although few can reach there.
Therefore the ions at higher potential will obtain more acceleration and the
energy spread is improved. That is PSA. Different from the real gliding
process, the ions can not pass through the bottom and climb up since the ion
front is the limiting point and the ion density tends to zero at the bottom.
(II) In the phase-stable deceleration region, $\xi_{i,f}\leq\xi\leq 1$, the
eletric field $E\leq 0$ and the electron density is larger than the ion
density too as shown in Figure 2. In this region, the ion momentum $p_{k}$
satisfies: $p_{k,m1}\leq p_{k}\leq p_{k,m2}$, where $p_{k,m2}=p_{k}(\xi=1)$
and given by Eq. (4) with $\xi=1$. All the ions in this region are decelerated
to $\xi=\xi_{i,f}$. The absolute value of the electric field decreases with
the decreasing $\xi$ and is zero at the ion front. Therefore the deceleration
is also phase-stable.
With Figure 2 (b), for ions, the potential in $\xi_{i,f}\leq\xi\leq 1$ is a
mountain with height of about $35\mathrm{GeV}$ which is far larger than the
maximum ion energy of about $\sqrt{p_{m,2}^{2}+1}-1\approx 7\mathrm{GeV}$ at
$\xi=1$. Therefore, the ions will be decelerated at the potential uphill.
Since the gradient, i.e., the value of electric field increases with the
potential height, the deceleration is also phase-stable. As point out above,
the limiting point is still the ion front $\xi=\xi_{i,f}$, and the ions in PSD
region can also not pass through the bottom into PSA region. The ions in PSA
and PSD region can not exchange from each other because of the zero-density
dividing point $\xi=\xi_{i,f}$.
In this case, the maximum ion momentum is $p_{k,m1}$ at the ion front
$\xi_{i,f}\lneq 1$. Therefore, it is called phase-stable limited relativistic
acceleration (PS-LRA). The ions in the two phase-stable regions can not
exchange from each other and the ion front $\xi_{i,f}$ is the dividing line.
Combing the condition of PS-LRA and the above discussion about PSA and PSD,
the ions with an initial momentum of $p_{k,0}$ not large enough to get across
the potential at $\xi=1$ will drop in PSA region or PSD region and obtain a
finite maximum energy ascertained by Eq. (14).
Figure 2: (Color online) Phase-stable limited relativistic acceleration (PS-
LRA) contains two regions: phase-stable acceleration region for
$0\leq\xi\leq\xi_{i,f}\approx 0.87$ and phase-stable deceleration region for
$0.87\leq\xi\leq 1$. (a)The density of ions and electrons for different time
and the ion momentum VS the self-similar variable $\xi$. At the ion front
$\xi_{i,f}=0.87$, the ion density is zero. (b)The potential VS $\xi$. (c) and
(d) The electric field $\hat{E}$ VS $\xi$. $\hat{E}\geq 0$ for
$0\leq\xi\leq\xi_{i,f}$ and $\hat{E}\leq 0$ for $\xi_{i,f}\leq\xi\leq 1$.
Here, $n_{0}=10^{22}\mathrm{/cm^{3}}$, $\alpha=2$ and $a=70$.
If the initial ion momentum satisfies:
$\int_{0}^{p_{k,0}}\gamma_{k}^{-\frac{3}{2}}dp_{k}+2\alpha
p_{k,0}\gamma_{k,0}^{\frac{1}{2}}\geq 2.622.$ (16)
the ions will not experience a potential well in PS-LRA and will coast down
from the potential slope and drop into the bottomless abyss at $\xi=1$ as
shown by Figure 3 (d). $\xi=1$ is called phase-lock-like position and the ions
can obtain unlimited energy gain as shown in Figure 3 (b). It is called
unlimited relativistic acceleration (URA), which is not phase-stable any more.
Eq. (16) satisfies for $p_{k,0}\geq 0.5064$, i.e., $a\geq 203$ from Eq. (7)
for $\alpha=2$, $n_{0}=49n_{c}=5.5\times 10^{22}\mathrm{/cm^{3}}$ and
$d=\lambda=1\mathrm{\mu m}$. Figure 3 shows the dependence of the plasma
density, electric field and potential on $\xi$ for $a=316$, $d=\lambda=1\mu
m$, $n_{0}=49n_{c}=5.5\times 10^{22}/cm^{3}$, and $\alpha=2$. In this case,
the electron density is smaller than the ion density and the acceleration is
not phase-stable any more. In all the region, the electric field increases
with $\xi$ and is larger than zero. At a finite time, the electron front,
where $n_{e}=0$, $\xi_{e,f}\lneq 1$, which is 0.971, 0.994, 0.99997 at
$\tau=5,20,2000$ separately. Therefore the possible maximum the ion momentum
is shown by Figure 3 (a) and (b).
As discussed above, due to the initial ion momentum $p_{k,0}$ large enough,
the ions can reach URA region. Therefore the ion momentum and the field have a
sharp increase and tend to infinite, the ion density becomes a non-zero
constant in $\xi\in(1-\delta,1]$ as $\tau\rightarrow\infty$, where $\delta$ is
an infinitesimal. As shown in Figure 3 (b), at the phase-lock-like position
and the limiting ion front $\xi=1$, the ion can obtain unlimited energy gain
and the ion density is non-zero. It is similar with the unlimited phase-lock
ion acceleration as pointed out by Bulanov and coworkersunlimitedRPA in the
relativistic limit. In URA region, it is found that (I) the unlimited ion
acceleration requires the initial ion momentum is large enough and should meet
Equation (16); (II) it is not phase-stable any more; (III) the phase-lock-like
position is $\xi=1$, the limiting ion front; (IV) the ion density at the
limiting ion front is non-zero.
Figure 3: (Color online) Unlimited relativistic acceleration (URA) with the
phase-lock-like position $\xi=1$.(a)The density of ions and electrons for
different time and the ion momentum VS $\xi$. The ion density is no-zero at
the limiting ion front: $\xi=1$. The electron density is smaller than that of
ions at any finite time. (b)The enlargement of (a) near $\xi=1$. $\xi=1$ is
the phase-lock-like position where the ion momentum tends to infinity. (c) and
(d) The electric field $\hat{E}$ and the potential $\phi$ VS $\xi$.
$\hat{E}\geq 0$ for all the region and the acceleration is not phase-stable
any more. Here $a=316$, $d=\lambda=1\mathrm{\mu m}$, $n_{0}=49n_{c}=5.5\times
10^{22}\mathrm{/cm^{3}}$, and $\alpha=2$.
In the conclusion, it has been given an analytical relativistic fluid model to
describe the relativistic radiation pressure acceleration with the initial
parameters from the hole-boring stage. The dependence of the ion velocity on
the acceleration time can be obtained and is consistent with that of thin-
shell model and PIC simulations. There are two acceleration modes: PS-LRA and
URA with a critical initial ion momentum ascertained by an explicit
formulation. In PS-LRA, the ions are trapped in a deep potential well and the
maximum ion energy is limited and the ion front is the well bottom and
$\xi_{i,f}\lneq 1$. URA is not phase-stable any more and there is a phase-
lock-like position in it. At the phase-lock-like position, corresponding to
the relativistic limit, the ions can obtain unlimited energy gain and the ion
density is non-zero as time tends infinite. Although the unlimited ion
acceleration can not be reached at any finite time, the ions can be
accelerated to any large energy if the laser pulse is long enough. As an
important result, you cannot obtain both PSA and URA. Therefore, if the laser
parameters are large enough to obtain URA, the energy spread must be lost. If
one wants to improve the energy spread with PSA, the maximum ion energy is
limited.
###### Acknowledgements.
This work was supported by the Key Project of Chinese National Programs for
Fundamental Research (973 Program) under contract No. $2011CB808104$ and the
Chinese National Natural Science Foundation under contract No. $10834008$.
## References
* (1) F. Mako and T. Tajima, Phys. Fluids 27, 1815 (1984).
* (2) Y. Oishi, T. Nayuki, T. Fujii, Y. Takizawa, X. Wang, T. Yamazaki, K. Nemoto, T. Kayoiji, T. Sekiya, K. Horioka, Y. Okano, Y. Hironaka, K. G. Nakamura, K. Kondo, A. A. Andreev, Phys. Plasmas 12, 073102 (2005);H. Schwoerer, S. Pfotenhauer, O. Jackel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, T. Esirkepov, Nature 439, 445 (2006); M. Murakami and M. M. Basko, Phys. Plasmas 13, 012105 (2006).
* (3) T. Esirkepov, M. Borghesi, S. V. Bulanov, G. Mourou, and T. Tajima, Phys. Rev. Lett. 92, 175003 (2004).
* (4) L. Yin, B. J. Albright, B. M. Hegelich and J. C. Fernandez, Laser and Particle Beams 24(2), 291-298 (2006).
* (5) T. Esirkepov, M. Yamagiwa, and T. Tajima, Phys. Rev. Lett. 96, 105001 (2006).
* (6) A. Henig, S. Steinke, M. Schn rer, T. Sokollik, R. H orlein, D. Kiefer, D. Jung, J. Schreiber, B. M. Hegelich, X. Q. Yan, T. Tajima, P. V. Nickles, W. Sandner and D. Habs, arXiv:0908.4057v1 (2009).
* (7) X. Q. Yan, C. Lin, Z.M. Sheng, Z.Y. Guo, B.C. Liu, Y.R. Lu, J.X. Fang, and J.E. Chen, Phys. Rev. Lett. 100, 135003 (2008).
* (8) S. V. Bulanov, E. Yu. Echkina, T. Zh. Esirkepov, I. N. Inovenkov, M. Kando, F. Pegoraro, and G. Korn, Phys. Rev. Lett. 104, 135003 (2010).
* (9) X. Q. Yan, T. Tajima, M. Hegelich, L. Yin and D. Habs, Appl. Phys. B, 98 711-721 (2010).
* (10) X. R. Hong, B. S. Xie, S. Zhang, H. C. Wu, A. Aimidula, X. Y. Zhao, and M. P. Liu, Phys. Plasmas 17, 103107 (2010).
* (11) T. P. Yu, A. Pukhov, G. Shvets, and M. Chen, Phys. Rev. Lett. 105, 065002 (2010).
* (12) B. Qiao, M. Zepf, M. Borghesi, and M. Geissler, Phys. Rev. Lett. 102, 145002 (2009).
* (13) A. Macchi, F. Cattani, T. V. Liseykina and F. Cornolti, Phys. Rev. Lett. 94, 165003 (2005).
* (14) Supplement file.
|
arxiv-papers
| 2011-03-08T04:44:18 |
2024-09-04T02:49:17.542989
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huang Yongsheng and Wang Naiyan and Tang Xiuzhang and Yan Xueqing",
"submitter": "Yongsheng Huang",
"url": "https://arxiv.org/abs/1103.1434"
}
|
1103.1455
|
author1]ltsmechanic@zju.edu.cn author2]guoyimu@zju.edu.cn
# Application of Explicit Symplectic Algorithms to Integration of Damping
Oscillators
Tianshu Luo Yimu Guo Institute of Applied Mechanics, Department of
Mechanics, Zhejiang University, Hangzhou, Zhejiang, 310027, P.R.China
Institute of Applied Mechanics, Department of Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China [ [
(Received: date / Accepted: date)
###### Abstract
In this paper an approach is outlined. With this approach some explicit
algorithms can be applied to solve the initial value problem of
$n-$dimensional damped oscillators. This approach is based upon following
structure: for any non-conservative classical mechanical system and arbitrary
initial conditions, there exists a conservative system; both systems share one
and only one common phase curve; and, the value of the Hamiltonian of the
conservative system is, up to an additive constant, equal to the total energy
of the non-conservative system on the aforementioned phase curve, the constant
depending on the initial conditions. A key way applying explicit symplectic
algorithms to damping oscillators is that by the Newton-Laplace principle the
nonconservative force can be reasonably assumed to be equal to a function of a
component of generalized coordinates $q_{i}$ along a phase curve, such that
the damping force can be represented as a function analogous to an elastic
restoring force numerically in advance. Two numerical examples are given to
demonstrate the good characteristics of the algorithms.
###### keywords:
Hamiltonian, dissipation, non-conservative system, damping, explicit
symplectic algorithm
††journal: Communications in Nonlinear Science and Numerical Simulation
## 1 Introduction
Feng[1, 2, 3, 4],Marsden[5],Neri[6] and Yoshida[7]had developed a series of
symplectic algorithms for Hamiltonian systems. These algorithms possess some
advantages. But it is difficult to apply these algorithms to damping dynamical
systems, because it has been stated in most classical textbooks that the
Hamiltonian formalism focuses on solving conservative problems. Damping
phenomena is very important in the modeling of dynamical systems, and can not
be avoided. Our aim is to apply some explicit canonical algorithms to
nonlinear damping dynamical systems, which is generated generally by FE-
method. These canonical algorithms reported in this paper can be readily
utilized for computing large-scale nonlinear damped dynamical systems.
Betch[8][9][10] attempted to apply directly some implicit algorithms to
damping systems. The implicit symplectic algorithms utilized by Betch[8]
possess a few good characteristic, e.g. energy-conservation, momentum-
consistence, etc… In terms of energy-conservation, implicit symplectic
algorithms might be better than explicit symplectic ones. But explicit
symplectic schemes might be more suitable for nonlinear problems.
If one needs to apply symplectic algorithms to a dissipative system, one must
convert the dissipative system into a Hamiltonian system or find some
relationship between the dissipative system and a conservative one.
In the literature[11], we have stated a proposition describing a relation
among a damping dynamical system and conservative ones:
###### Proposition 1.1
For any non-conservative classical mechanical system and arbitrary initial
condition, there exists a conservative system; both systems sharing one and
only one common phase curve; and the value of the Hamiltonian of the
conservative system is equal to the sum of the total energy of the non-
conservative system on the aforementioned phase curve and a constant depending
on the initial condition.
In other words, a dissipative ordinary equation and a conservative equation
may possess a common particular solution. In the next section, an analytical
examples are given to explain this proposition. Readers can find the detailed
proof of Proposition 1.1 in the reference[11]
In the Literature [12] a basic explicit canonical integrator is proposed.
Based on this basic scheme, Neri[6] constructed 4-order explicit canonical
integrator, and then Yoshida [7] proposed a general method to construct higher
order explicit symplectic integrator. Utilizing the Proposition 1.1, we apply
this class of explicit canonical integrators to damping dynamical systems.
This point will be in detail stated in sec. 3.
## 2 One-dimensional Analytical Example
Consider a special one-dimensional simple mechanical system:
$\ddot{x}+c\dot{x}=0,$ (1)
where $c$ is a constant. The exact solution of the equation above is
$x=A_{1}+A_{2}e^{-ct},$ (2)
where $A_{1},A_{2}$ are constants. Differentiation gives the velocity:
$\dot{x}=-cA_{2}e^{-ct}.$ (3)
From the initial condition $x_{0},\dot{x}_{0}$, we find
$A_{1}=x_{0}+\dot{x}_{0}/c,A_{2}=-\dot{x}_{0}/c$. Inverting Eq. (2) yields
$t=-\frac{1}{c}\ln\frac{x-A_{1}}{A_{2}}$ (4)
and by substituting into Eq. (3), such we have
$\dot{x}=-c(x-A_{1})$ (5)
The dissipative force $F$ in the dissipative system (1) is
$F=c\dot{x}.$ (6)
Substituting Eq. (5) into Eq. (6), the conservative force $\mathcal{F}$ is
expressed as
$\mathcal{F}=-c^{2}(x-A_{1});$ (7)
Clearly, the conservative force $\mathcal{F}$ depends on the initial condition
of the dissipative system (1), in other words, an initial condition determines
a conservative force. Consequently, a new conservative system yields
$\ddot{x}+\mathcal{F}=0\rightarrow\ddot{x}-c^{2}(x-A_{1})=0.$ (8)
The stiffness coefficient in this equation must be negative. One can readily
verify that the particular solution (2) of the dissipative system can satisfy
the conservative one (8). This point agrees with Proposition (1.1).
The potential of the conservative system (8) is
$V=\int_{0}^{x}\left[-c^{2}(x-A_{1})\right]\mathrm{d}x=-\frac{c^{2}}{2}x^{2}+c^{2}A_{1}x$
Therefore the Hamiltonian is
$\hat{H}=T+V=\frac{1}{2}p^{2}-\frac{c^{2}}{2}x^{2}+c^{2}A_{1}x,$
where $p=\dot{x}$.
Furthermore, Proposition (1.1) can be depicted by Fig. 1. The phase flow of
conservative system (2) transforms the red area in phase space to the purple
area; the phase flow of conservative system (8) transforms the red area to the
green area. The blue curve in Fig. 1 illustrates the common phase curve. If
one draws more common phase curves and phase flows, the picture will like a
flower, the phase flow of the nonconservative system likes a pistil and phase
flows conservative systems like petals.
Figure 1: Relationship between nonconservative system (1) and conservative one
(8)
## 3 Modification Symplectic Numerical Schemes
### 3.1 Basic Explicit Symplectic Numerical Schemes
In the paper[12][6][7] a symplectic algorithm based second kind generation
function was stated：
$\begin{array}[]{l}{{\bm{p}}^{i+1}}={{\bm{p}}^{i}}-\tau{H_{q}}({{\bm{p}}^{i+1}},{{\bm{q}}^{i}})\\\
{{\bm{q}}^{i+1}}={{\bm{q}}^{i}}+\tau{H_{p}}({{\bm{p}}^{i+1}},{{\bm{q}}^{i}}),\\\
\end{array}$ (9)
where the superscript $i$ denotes the $i$-th time node, $\bm{q}$ denotes
coordinates and $\bm{p}$ denotes canonical momenta, and $H$ denotes
Hamiltonian quantity, $H_{q}=\partial H/\partial\bm{q},\ \ H_{p}=\partial
H/\partial\bm{p}$. If the Hamiltonian is seperable, i.e.
$H=U(\bm{p})+V(\bm{q}),V_{q}=H_{q},U_{p}=H_{p}$, then the symplectic scheme(9)
above becomes an explicit symplectic scheme:
$\begin{array}[]{l}{{\bm{p}}^{i+1}}={{\bm{p}}^{i}}-\tau{V_{q}}({{\bm{q}}^{i}})\\\
{{\bm{q}}^{i+1}}={{\bm{q}}^{i}}+\tau{U_{p}}({{\bm{p}}^{i+1}}).\\\ \end{array}$
(10)
For some nonlinear vibration mechanical system,
$V_{q}=\mathsfsl{K}(\bm{q})\bm{q}$.
Let us consider an $n-$dimensional nonlinear oscillator:
$\ddot{\bm{q}}+\mathsfsl{C}\dot{\bm{q}}+\mathsfsl{K}\bm{q}=0,$ (11)
where $\mathsfsl{C}$ denotes a non-linear damping coefficient matrix which
depends on $\bm{q}$, and $\mathsfsl{K}$ denotes a non-linear stiffness matrix
which depends on $\bm{q}$ and consists of two parts
$\mathsfsl{K}=\mathsfsl{\check{K}}+\mathsfsl{\hat{K}}$($\mathsfsl{\check{K}}$
is a diagonal matrix).
In accordance with Proposition 1.1, a conservative mechanical system was found
associated with the dissipative system (11) in addition to its initial
conditions. Subject to these initial conditions, the dissipative system (11)
possesses a common phase curve $\gamma$ with the conservative system. As in
Eq. (7), we can consider that the components of the damping force
$\mathsfsl{C}\dot{\bm{q}}$ determine the components of a conservative force on
the phase curve $\gamma$
$\begin{array}[]{ccc}c_{11}\dot{q}_{1}=\varrho_{11}(q_{1})&\dots&c_{1n}\dot{q}_{n}=\varrho_{1n}(q_{1})\\\
\vdots&\ddots&\vdots\\\
c_{n1}\dot{q}_{1}=\varrho_{21}(q_{n})&\dots&c_{nn}\dot{q}_{n}=\varrho_{nn}(q_{n}).\end{array}$
(12)
For convenience, this conservative force is assumed to be an elastic restoring
force:
$\begin{array}[]{ccc}\varrho_{11}(q_{1})=\kappa_{11}(q_{1})q_{1}&\dots&\varrho_{1n}(q_{1})=\kappa_{1n}(q_{1})q_{1}\\\
\vdots&\ddots&\vdots\\\
\varrho_{n1}(q_{1})=\kappa_{n1}(q_{n})q_{n}&\dots&\varrho_{nn}(q_{n})=\kappa_{nn}(q_{n})q_{n}.\end{array}$
(13)
In a similar manner, the components of the non-conservative force
$\mathsfsl{\hat{K}}\bm{q}$ are equal to the components of a conservative force
on the phase curve $\gamma$
$\begin{array}[]{ccc}\hat{K}_{11}q_{1}=\chi_{11}(q_{1})&\dots&\hat{K}_{1n}q_{n}=\chi_{1n}(q_{1})\\\
\vdots&\ddots&\vdots\\\
\hat{K}_{n1}q_{1}=\chi_{21}(q_{n})&\dots&\hat{K}_{nn}q_{n}=\chi_{nn}(q_{n}).\end{array}$
(14)
The conservative force can likewise be assumed to an elastic restoring force:
$\begin{array}[]{ccc}\chi_{11}(q_{1})=\lambda_{11}(q_{1})q_{1}&\dots&\chi_{1n}(q_{1})=\lambda_{1n}(q_{1})q_{1}\\\
\vdots&\ddots&\vdots\\\
\chi_{n1}(q_{1})=\lambda_{n1}(q_{n})q_{n}&\dots&\chi_{nn}(q_{n})=\lambda_{nn}(q_{n})q_{n}.\end{array}$
(15)
By an appropriate transformation, an equivalent stiffness matrix
$\mathsfsl{\tilde{K}}$ that is diagonal in form can be obtained
$\mathsfsl{\tilde{K}}_{ii}=\sum_{l=1}^{n}\kappa_{il}(q_{l})+\lambda_{il}(q_{l}).$
(16)
Consequently, an $n$-dimensional conservative system is obtained
$\bm{\ddot{q}}+(\mathsfsl{\check{K}}+\mathsfsl{\tilde{K}})\bm{q}=0$ (17)
which shares the common phase curve $\gamma$ with the $n$-dimensional damping
system described by (11). In this paper, the conservative system is called the
’substitute’ conservative system. The Lagrangian of Eqs.(17) is
$\hat{L}=\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\int_{\bm{0}}^{\bm{q}}(\mathsfsl{\check{K}}\bm{q})^{T}\mathrm{d}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(18)
with the Hamiltonian
$\hat{H}=\frac{1}{2}\bm{p}^{T}\bm{p}+\int_{\bm{0}}^{\bm{q}}(\mathsfsl{\check{K}}\bm{q})^{T}\mathrm{d}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(19)
where $\bm{0}$ is the zero vector, and $\bm{p}=\dot{\bm{q}}$. Here $\hat{H}$
in Eq. (19) is the mechanical energy of the conservative system (17), because
$\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}$ is a
potential function such that $\hat{H}$ is independent of the path taken in
phase space.
Subject to a certain initial condition, one need merely to solve the
conservative system(17). But one must in advance obtain the numerical
approximation of the matrix $\mathsfsl{\tilde{K}}$ for a time step, such that
one can utilize the algorithm (10) to integrate the conservative system (17)
for a time step. One can repeat this process above up to the end. In this way
one obtains the numerical particular solution of the conservative system (17),
which is exactly the numerical particular solution of the damping one. The he
numerical approximation of the matrix $\mathsfsl{\check{K}}$ can be assumed
as:
$\displaystyle\mathsfsl{\tilde{K}}=\left[\begin{array}[]{ccc}\tilde{K}_{11}&\dots&0\\\
\vdots&\ddots&\vdots\\\ 0&\dots&\tilde{K}_{nn}\end{array}\right]$ (23)
$\displaystyle{\tilde{K}_{j}}({q_{j}}^{i})={c_{jl}}\dot{q}_{l}^{i}/{q_{j}}^{i}+\hat{K}_{jl}q_{l}^{i}/{q_{j}}^{i}$
Hence the explicit canonical scheme (10) can be modified into
$\begin{array}[]{l}\tilde{K}_{j}^{i}({q_{j}}^{i})={c_{jl}}\dot{q}_{l}^{i}/{q_{j}}^{i}+\hat{K}_{jl}q_{l}^{i}/{q_{j}}^{i}\\\
{p_{j}}^{i+1}={p_{j}}^{i}-\tau[{K_{j}}+\tilde{K}_{j}^{i}({q_{j}}^{i})]{q^{i}})\\\
{q_{j}}^{i+1}={q^{i}}+\tau{p_{j}}^{i+1}\\\ \end{array}$ (24)
The scheme above is a one order scheme. Furthermore one can construct higher
order explicit canonical schemes utilizing the method reported in the
literatures[6][7]. Now consider a map from $\bm{z}=\bm{z}(0)$ to
$\bm{z}^{\prime}=\bm{z}(\tau)$:
${\bm{z^{\prime}}}\approx(\prod\limits_{i=1}^{h}{{e^{{r_{i}}t{\mathsfsl{E}}}}}{e^{{s_{i}}\tau{\mathsfsl{F}}}}+O({\tau^{n+1}})){\mathsfsl{z}},$
(25)
where
$\displaystyle\bm{z}=\left[\begin{array}[]{l}\bm{p},\\\
\bm{q}\end{array}\right],\bm{z}^{\prime}=\left[\begin{array}[]{l}\bm{p}^{\prime},\\\
\bm{q}^{\prime}\end{array}\right],$
$\displaystyle{\mathsfsl{E}}=\left[{\begin{array}[]{*{20}{c}}0&0\\\ 1&0\\\
\end{array}}\right]{\mathsfsl{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-({\mathsfsl{K}}+{\mathsfsl{\tilde{K}}})}\\\
0&0\\\ \end{array}}\right].$
In fact Eq.(25) is the succession of the following mappings,
$\begin{array}[]{l}{{\mathsfsl{p}}^{j+1}}={{\mathsfsl{p}}^{j}}-{s_{i}}\tau{V_{q}}({{\mathsfsl{q}}^{j}})\\\
{{\mathsfsl{q}}^{j+1}}={{\mathsfsl{q}}^{j}}+{r_{i}}\tau{U_{p}}({{\mathsfsl{p}}^{j+1}})\\\
\end{array}.$ (28)
In reality the difference between the equations above and Eq.(24) is that the
coefficients $s_{i},r_{i}$ before the time step $\tau$. In the literature [7]
a generalized method to determine $s_{i},r_{i}$ were given. Therefore, the
higher order explicit canonical scheme can be represented as:
$\begin{array}[]{l}{\mathsfsl{\tilde{K}}}({q^{j}})=\left[{\begin{array}[]{*{20}{c}}{{\tilde{K}_{1}}(q_{1}^{j})}&{}\hfil&0\\\
{}\hfil&\ddots&{}\hfil\\\ 0&{}\hfil&{{\tilde{K}_{n}}(q_{n}^{j})}\\\
\end{array}}\right]{\tilde{K}_{\alpha}}(q_{\alpha}^{j})=\sum\limits_{l=1}^{n}{{c_{\alpha
l}}\dot{q}_{l}^{j}/q_{\alpha}^{j}}+\hat{K}_{\alpha
l}q_{l}^{i}/q_{\alpha}^{i}\\\
{\mathsfsl{E}}=\left[{\begin{array}[]{*{20}{c}}0&0\\\ 1&0\\\
\end{array}}\right]\;\;\;\;{\mathsfsl{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-({\mathsfsl{K}}+{\mathsfsl{\tilde{K}}})}\\\
0&0\\\ \end{array}}\right]\\\
{{\bm{z}}^{j+1}}=(\prod\limits_{i=1}^{h}{{e^{{s_{i}}\tau{\mathsfsl{F}}}}{e^{{r_{i}}\tau{\mathsfsl{E}}}}}){{\bm{z}}^{j}}\\\
\end{array}\ $ (29)
## 4 Numerical Examples
Two examples will be given to shown this numerical method29.
### 4.1 The First Example
To begin, we consider a Van Der Pol’s oscillator
$\ddot{x}+\mu\dot{x}({x^{2}}-1)+x=0,$ (30)
where $\mu=10$. The initial conditions are given by ${x_{0}}=1,\;\;\dot{x}=0$.
We employee the $4-$order explicit symplectic method (29) with coefficients
$\displaystyle s_{1}=s_{4}=[2+(\sqrt[3]{2}+1/\sqrt[3]{2})]/6,\ \
s_{2}=s_{3}=[1-(\sqrt[3]{2}+1/\sqrt[3]{2})]/6,$ $\displaystyle\ \
r_{1}=r_{3}=[2+(\sqrt[3]{2}+1/\sqrt[3]{2})]/3,\ \
r_{2}=-[2+(\sqrt[3]{2}+1/\sqrt[3]{2})]/3,\ \ r_{4}=0,$
and classical explicit $4-$order Runge-Kutta method to compute the resonance
of the Van Der Pol’s oscillator (31) respectively, then employ a same method
to integrate the results to the total energy, which is the sum of the
mechanical energy and the work done by damping forces in the system (30). The
both methods are run with the same step size $\tau=0.01$. The resonance is
shown in Fig. 2, and the total energy is shown in Fig. 3.
Figure 2: The resonance of the Van Der Pol’s oscillator
Figure 3: The total energy of the Van Der Pol’s oscillator
It is aparent from Fig. 3 that the explicit symplectic method (29) has
qualitatively different behavior to the Runge-Kutta method. The energy
divergence between the explicit symplectic method and the exact solution is
smaller than that between Runge-Kutta method and the exact solution. The
energy divergence between the explicit symplectic method and Runge-Kutta
method increases with the time evolution. Due to the increasement of the
energy, the phase difference between both the results in Fig. 2 increases also
with the time evolution.
### 4.2 The Second Example
In the second example, we consider a $2-$dimensional damped nonlinear Duffing
oscillator
$\begin{array}[]{l}2\ddot{q}_{1}+0.1\dot{q}_{1}+(2+0.1q_{1}^{2})q_{1}+q_{2}=0\\\
3\ddot{q}_{2}+0.2\dot{q}_{2}+q_{1}+(2+0.2q_{2}^{2})q_{2}=0,\end{array}$ (31)
with the initial conditions $q_{1}=0,\ \ q_{2}=0,\ \ \dot{q}_{1}=0,\ \
\dot{q}_{2}=1$. The program of the both methods with the step size $\tau=0.01$
are carried out to simulate Eq. (31). The resonance is shown in Fig. 2, the
numerical solution of the total energy is shown in Fig.5.
There is only tiny difference between resonance results of the two methods,
correspondingly, the difference among the total energy obtained by the
numerical methods and anlytical methods is very tiny. As numerical examples in
the other literatures[13], that explicit Runge-Kutta method must cause
numerical pseudo dissipation which might be positive or negative. The
difference between our numerical examples and the examples in the
literature[13] is the total energy in our examples and the mechanical energy
in their examples111Fig.6.1 in the literature[13].
Figure 4: The $1$-th displacement of the damped Duffing oscillator
Figure 5: Total energy of the damped dissipative oscillator
## 5 Conclusions
We have introduced a class of explicit symplectic algorithms to dissipative
mechanical systems successfully, by changing these algorithms into the
scheme.(29). Because the algorithms (29) are explicit and possess good energy
preserving characteristics, the explicit symplectic algorithms (29) is quite
suitable for long term integration of arbitrary dimensional nonlinear
dissipative mechanical systems.
## References
* Feng [1985] K. Feng, On difference schemes and symplectic geometry, in: Ed. Feng Kang Proceeding of the 1984 Beijing Symposium on differential geometry and differential equations-computation of partial differential equations, Science Press, Beijing, 1985, pp. 42–58.
* Wu et al. [1989] H. Wu, M. Qin, K. Feng, Construction of canonical difference schemes for hamiltonian formalism via generating functions, JCM 7 (1989) 71–96.
* Wu et al. [1990] H. Wu, M. Qin, K. Feng, Symplectic difference schemes for the linear hamiltonian canonical systems, JCM 8 (1990) 371–380.
* Feng [1991] K. Feng, The hamiltonian way for computing hamiltonian dynamics, Math. Appl. 56 (1991) 17–35.
* Marsden et al. [1998] J. E. Marsden, G. W. Patrick, S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear pdes, Communications in Mathematical Physics 199 (1998) 351–395. Cited By (since 1996): 129.
* Neri [1988] F. Neri, Lie algebras and canonical integration, Technical Report, Department of Physics,University of Maryland, 1988.
* Yoshida [1990] H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A 150 (1990) 262–268.
* Uhlar and Betsch [2010] S. Uhlar, P. Betsch, On the derivation of energy consistent time stepping schemes for friction afflicted multibody systems, Computers & Structures 88 (2010) 737 – 754.
* Leyendecker et al. [2004] S. Leyendecker, P. Betsch, P. Steinmann, Energy-conserving integration of constrained hamiltonian systems – a comparison of approaches, Computational Mechanics 33 (2004) 174–185. 10.1007/s00466-003-0516-2.
* Betsch [2006] P. Betsch, Energy-consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints, Computer Methods in Applied Mechanics and Engineering 195 (2006) 7020 – 7035. Multibody Dynamics Analysis.
* Luo and Guo [2009] T. Luo, Y. Guo, Infinite-dimensional Hamiltonian description of a class of dissipative mechanical systems, ArXiv e-prints (2009).
* Feng and Qin [1987] K. Feng, M. Qin, The symplectic methods for the computation of hamiltonian equations, in: Numerical Methods for Partial Differential Equations, Springer, Berlin, 1987, pp. 17–35.
* Kane et al. [2000] C. Kane, J. E. Marsden, M. Ortiz, M. West, Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems, International Journal for Numerical Methods in Engineering 49 (2000) 1295–1325.
|
arxiv-papers
| 2011-03-08T08:03:21 |
2024-09-04T02:49:17.547919
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tianshu Luo and Yimu Guo",
"submitter": "Tianshu Luo",
"url": "https://arxiv.org/abs/1103.1455"
}
|
1103.1499
|
# The effect of antisymmetric tensor unparticle mediation on the charged
lepton electric dipole moment
E. O. Iltan
Physics Department, Middle East Technical University
Ankara, Turkey
E-mail address: eiltan@newton.physics.metu.edu.tr
###### Abstract
We study the contribution of antisymmetric tensor unparticle mediation to the
charged lepton electric dipole moments and restrict the free parameters of the
model by using the experimental upper bounds. We observe that the charged
lepton electric dipole moments are strongly sensitive to the the scaling
dimension $d_{U}$ and the fundamental scales $M_{U}$ and $\Lambda_{U}$. The
experimental current limits of electric dipole moments are reached for the
small values of the scaling dimension $d_{U}$.
The CP violation which leads to the unequal amounts of matter and antimatter
in the universe needs more accurate theoretical explanation. The electric
dipole moments (EDMs) of fermions are driven by the CP violating interaction
and, therefore, their search, especially the charged lepton EDMs111They are
clean theoretically since they are free from strong interactions., is
worthwhile in order to understand the CP violation mechanism. The current
experimental limits of the electron, muon and tau EDMs are $d_{e}=(0.7\pm
0.7)\times 10^{-27}e\,cm$ [1] $d_{\mu}=(3.7\pm 3.4)\times 10^{-19}e\,cm$ [2]
and Re[$d_{\tau}$]$=-0.22$ to $0.45\times 10^{-16}e\,cm$;
Im[$d_{\tau}$]$=-0.25$ to $0.008\times 10^{-16}e\,cm$ [3], respectively. These
experimental results stimulate the search of the lepton EDMs in the framework
of various theoretical models. In the standard model (SM) the source of the CP
violation and, therefore the EDM, is the complex Cabibo Kobayashi Maskawa
(CKM) matrix in the quark sector and the lepton mixing matrix in the lepton
sector. However the EDM predictions in the SM are negligible and far from
their current experimental limits. Therefore one goes beyond the SM such as
multi Higgs doublet models (MHDM), supersymmetric model (SUSY) [4], left-right
symmetric model, the seesaw model, the models including the extra dimensions
and noncommutative effects,… etc., in order to get the additional CP violating
phase (see for example [5]-[9]). Another possibility for a new CP violating
phase is to consider the recent unparticle idea which is proposed by Georgi
[10, 11]. Unparticles are new degrees of freedom arising from the SM-
ultraviolet sector interaction at some scale $M_{U}$ and, because of the scale
invariance, they are massless and have non integral scaling dimension $d_{U}$,
around the scale $\Lambda_{U}\sim 1.0\,TeV$. The effective interaction of the
SM-ultraviolet (UV) sector at the scale $M_{U}$ reads
${\cal{L}}_{eff}=\frac{C_{n}}{M_{U}^{d_{UV}+n-4}}\,O_{SM}\,O_{UV}\,,$ (1)
with the scaling dimension $d_{UV}$ of the UV operator [13] and, around the
scale $\Lambda_{U}$, it appears as (see [14], [15] and references therein)
${\cal{L}}_{eff}=\frac{C^{i}_{n}}{\Lambda_{n}^{d_{U}+n-4}}\,O_{SM,i}\,O_{U}\,,$
(2)
where
$\Lambda_{n}=\Bigg{(}\frac{M_{U}^{d_{UV}+n-4}}{\Lambda_{U}^{d_{UV}-d_{U}}}\Bigg{)}^{\frac{1}{d_{U}+n-4}}\,,$
(3)
and $n$ is the scaling dimension of SM operator of type $i$. Here the scale
$\Lambda_{n}$ is sensitive to the scaling dimension $n$ of the SM operator
$O_{SM,i}$ [14, 15] and depends on the fundamental scales $M_{U}$,
$\Lambda_{U}$222$\Lambda_{2}<M_{U}<\Lambda_{4}<\Lambda_{3}$ with the choice
$1<d_{U}<2<d_{UV}$ (see [14])..
In the present work, we consider that the new CP violating phase is coming
from the effective unparticle fermion interaction and we predict the charged
lepton EDMs (see [16] for the scalar unparticle contribution to the charged
lepton EDM). Here we assume that the antisymmetric tensor unparticle mediation
gives the contribution to the lepton EDM333The contribution of the
antisymmetric tensor unparticle mediation to the muon anomalous magnetic
dipole moment and its effects in $Z$ invisible decays and the electroweak
precision observable $S$ has been predicted in [15]. by respecting the
following conditions:
* •
The scale $\Lambda_{n}$ in the effective Lagrangian depends on the dimension
of the SM operator $O_{SM,i}$,
* •
antisymmetric tensor unparticle-lepton couplings are complex,
* •
the scale invariance is broken at some scale $\mu$ after the electroweak
symmetry breaking due to the additional interaction
$\sim\frac{\lambda_{2}}{\Lambda_{2}^{du-2}}\,O_{S}\,H^{\dagger}\,H$ where $H$
($O_{S}$) is the SM Higgs (scalar unparticle operator which exists with the
antisymmetric tensor unparticle) [17, 18].
The two point function of antisymmetric tensor unparticle reads (see Appendix
for details)
$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-p^{2}-i\epsilon)^{d_{U}-2}\,,$
(4)
where the factor $A_{d_{U}}$ is
$\displaystyle
A_{d_{U}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{U}}}\,\frac{\Gamma(d_{U}+\frac{1}{2})}{\Gamma(d_{U}-1)\,\Gamma(2\,d_{U})}\,.$
(5)
Here $\Pi^{\mu\nu\alpha\beta}$ is the projection operator
$\displaystyle\Pi_{\mu\nu\alpha\beta}=\frac{1}{2}(g_{\mu\alpha}\,g_{\nu\beta}-g_{\nu\alpha}\,g_{\mu\beta})\,,$
(6)
and it can be divided into the transverse and the longitudinal parts as
$\displaystyle\Pi^{T}_{\mu\nu\alpha\beta}=\frac{1}{2}(P^{T}_{\mu\alpha}\,P^{T}_{\nu\beta}-P^{T}_{\nu\alpha}\,P^{T}_{\mu\beta})\,,\,\,\,\,\,\,\Pi^{L}_{\mu\nu\alpha\beta}=\Pi_{\mu\nu\alpha\beta}-\Pi^{T}_{\mu\nu\alpha\beta}\,,$
(7)
with $P^{T}_{\mu\nu}=g_{\mu\nu}-p_{\mu}\,p_{\nu}/{p^{2}}$ (see for example
[15] and references therein). Furthermore, the scale invariance breaking at
the scale $\mu$ results in that the antisymmetric tensor unparticle propagator
is modified. The propagator is model dependent (see for example [19] for the
scalar unparticle case) and we consider the one in the simple model [17, 20]:
$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-(p^{2}-\mu^{2})-i\epsilon)^{d_{U}-2}\,.$
(8)
Here $\mu$ is the scale where unparticle sector changes in to the particle
sector.
Now we start with the effective Lagrangian responsible for the EDM of charged
leptons444Here we used the effective Lagrangian given in [15] and choose the
unparticle-lepton coupling complex in order to switch on the CP violation. In
this equation $H$ is the Higgs doublet, $g$ and $g^{\prime}$ are weak
couplings, $\lambda_{B}$ and $\lambda_{W}$ are the unparticle-field tensor
couplings, $B_{\mu\nu}$ is the field strength tensor of the $U(1)_{Y}$ gauge
boson $B_{\mu}=c_{W}\,A_{\mu}+s_{W}\,Z_{\mu}$ and $W^{a}_{\mu\nu}$, $a=1,2,3$,
are the field strength tensors of the $SU(2)_{L}$ gauge bosons with
$W^{3}_{\mu}=s_{W}\,A_{\mu}-c_{W}\,Z_{\mu}$ where $A_{\mu}$ and $Z_{\mu}$ are
photon and Z boson fields respectively. :
$\displaystyle{\cal{L}}_{eff}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime}\,\lambda_{B}}{\Lambda_{2}^{d_{U}-2}}\,B_{\mu\nu}\,O^{\mu\nu}_{U}+\frac{g\,\lambda_{W}}{\Lambda_{4}^{d_{U}}}\,(H^{\dagger}\,\tau_{a}\,H)\,W^{a}_{\mu\nu}\,O^{\mu\nu}_{U}$
(9) $\displaystyle+$
$\displaystyle\frac{y_{l}}{\Lambda_{4}^{d_{U}}}\Big{(}\lambda_{l}\,\bar{l}_{L}\,H\,\sigma_{\mu\nu}\,l_{R}+\lambda_{l}^{*}\,\bar{l}_{R}\,H^{\dagger}\,\sigma_{\mu\nu}\,l_{L}\Big{)}\,O^{\mu\nu}_{U}\,,$
with the lepton field $l$ and the complex coupling
$\lambda_{l}=|\lambda_{l}|\,e^{i\,\theta_{l}}$ where $\theta_{l}$ is the CP
violating parameter.
The effective EDM interaction for a charged lepton $l$ reads
$\displaystyle{\cal
L}_{EDM}=id_{l}\,\bar{l}\,\gamma_{5}\,\sigma^{\mu\nu}\,l\,F_{\mu\nu}\,\,,$
(10)
where $F_{\mu\nu}$ is the electromagnetic field tensor and ’$d_{l}$’, which is
a real number by hermiticity, is the EDM of the charged lepton. Finally, the
effective Lagrangian in eq.(9) leads to the EDM of charged leptons $l$ after
electroweak breaking as (see Appendix for details):
$\displaystyle
d_{l}=-i(\lambda_{l}-\lambda^{*}_{l})\,\frac{e\,\mu^{2\,(d_{U}-2)}\,\,A_{d_{U}}\,m_{l}\,}{2\,\,sin\,(d_{U}\pi)\,\Lambda_{4}^{d_{U}}}\,\Bigg{(}\frac{\lambda_{B}}{\Lambda_{2}^{d_{U}-2}}-\frac{v^{2}\,\lambda_{W}}{4\,\Lambda_{4}^{d_{U}}}\Bigg{)}\,,$
(11)
where $v$ is the vacuum expectation value of the SM Higgs $H^{0}$.
Discussion
In this section we predict the intermediate antisymmetric tensor unparticle
contribution (see Fig.1) to the charged lepton EDMs by considering that the CP
violating phase is carried by the tensor unparticle-charged lepton couplings
and try to restrict the free parameters of the model by using the experimental
upper bounds of the charged lepton EDMs. The scaling dimension of UV operator
$O_{UV}$ (the unparticle operator $O_{U}$) $d_{UV}$ ($d_{U}$), the fundamental
scales of the model, namely the interaction scale $M_{U}$ of the SM-
ultraviolet sector and interaction scale $\Lambda_{U}$ of the SM-unparticle
sector and the scale $\mu$ which is responsible for the flow of unparticle
sector in to the particle one are among the free parameters. In our numerical
calculations we choose the scale dimension $d_{U}$ in the range555For
antisymmetric tensor unparticle the scale dimension should satisfy $d_{U}>2$
not to violate the unitarity (see [21]). Here we assumed that the scale
invariance is broken at some scale $\mu$ and the restriction on the values of
$d_{U}$ is more relaxed. We used the simple model [17, 20] to define the new
propagator. Since this model ensures a connection with the particle sector, we
choose $d_{U}$ in the range $1<d_{U}<2$ and when $d_{U}$ tends to one one
reaches the particle sector and the connection is established. Since this
choice brings a rough connection between two sectors, unparticle and particle
sectors, we believe that it is worthwhile to study even if it needs more
careful analysis whether its is consistent with the QFT. $1<d_{U}<2$ and
$d_{UV}>d_{U}=3$ (see [14] and [15]) and we choose $\mu\sim 1.0\,GeV$. The
couplings $\lambda_{B}$, $\lambda_{W}$ and $\lambda_{l}$ are other free
parameters which should be restricted. We take $\lambda_{B}=\lambda_{W}=1$ and
choose complex $\lambda_{l}$, $\lambda_{l}=|\lambda_{l}|\,e^{i\,\theta_{l}}$
with the CP violating parameter $\theta_{l}$, in order to create the EDM. Here
we assume that the couplings $|\lambda_{l}|$ obey the mass hierarchy of
charged leptons, $|\lambda_{\tau}|>|\lambda_{\mu}|>|\lambda_{e}|$ and we take
$|\lambda_{\tau}|=1$, $|\lambda_{\mu}|=0.1$ and $|\lambda_{e}|=0.005$.
In the first part of the calculation we restrict the CP violating parameter
$\theta_{\mu}$ by assuming that the antisymmetric unparticle tensor
contribution to muon anomalous magnetic moment reaches to the experimental
upper limit $a_{\mu}=10^{-9}$ and we study its contribution to the EDM of muon
$d_{\mu}$. Furthermore we predict the EDMs of electron and tau lepton and
estimate the acceptable values of the free parameters by taking the
intermediate numerical value of the CP violating parameter, namely
$sin\theta_{e}=sin\theta_{\tau}=0.5$. Finally we study the CP violating
parameter dependence of EDMs.
In Fig.2, we present $M_{U}$ dependence of the EDM $d_{\mu}$ for
$a^{U}_{\mu}=10^{-9}$ and different values of the scale parameter $d_{U}$ and
the ratio $r_{U}=\frac{\Lambda_{U}}{M_{U}}$. Here upper-lower-the lowest solid
(dashed-long dashed; dotted) line represents the EDM for $d_{U}=1.1$,
$r_{U}=0.40-0.10-0.05$ ($d_{U}=1.3$, $r_{U}=0.40-0.10$; $d_{U}=1.5$,
$r_{U}=0.40$). It is observed that $d_{\mu}$ is strongly sensitive to the
ratio $r_{U}$ and the increasing values of $r_{U}$ causes the enhancement in
$d_{\mu}$. To reach the current experimental limit $r_{U}$ must be at least of
the order of $r_{U}\sim 10^{-1}$ if the scaling dimension satisfies
$d_{U}>1.1$. For larger values of $d_{U}$ the higher values of $r_{U}$ are
accepted. The dependence of $d_{\mu}$ to the mass scale $M_{U}$ is also strong
especially for the large values of the scaling dimension and it decreases more
than one order in the range $10^{3}\,GeV<M_{U}<10^{4}\,GeV$ for $d_{U}\sim
1.5$ and more.
Fig.3 and Fig.4 are devoted to $d_{\mu}$ with respect to the scale parameter
$d_{U}$ for $a^{U}_{\mu}=10^{-9}$ and $a^{U}_{\mu}=10^{-10}$, respectively.
Here upper-lower solid (long dashed; dashed; dotted) line represents the EDM
for $r_{U}=0.05$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.05$, $M_{U}=10^{4}\,GeV$
($r_{U}=0.1$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.1$, $M_{U}=10^{4}\,GeV$;
$r_{U}=0.4$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.4$, $M_{U}=10^{4}\,GeV$;
$r_{U}=0.5$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.5$, $M_{U}=10^{4}\,GeV$). For the
decreasing values of the ratio $r_{U}$ $d_{U}$ becomes more restricted and
with its the increasing values the current experimental value can be reached.
If the contribution of the antisymmetric tensor unparticle to the anomalous
magnetic moment of muon is taken as $a^{U}_{\mu}=10^{-10}$ (see Fig.4) the
restriction of $d_{U}$ is more relaxed and for higher values of the ratio
$r_{U}$ it would be possible to reach the current experimental value of
$d_{\mu}$ similar to the previous case.
Fig.5 (6) represents $M_{U}$ dependence of the EDM $d_{e}$ ($d_{\tau}$) for
$sin\theta_{e}=0.5$ ($sin\theta_{\tau}=0.5$) and for different values of the
scale parameter $d_{U}$ and the ratio $r_{U}$. Here the upper most-upper-
lower-the lowest solid; dashed line represents the $d_{e}$ ($d_{\tau}$) for
$d_{U}=1.1-1.3-1.5-1.8$, $r_{U}=0.05$; $r_{U}=0.10$. We see that the
increasing values of $M_{U}$ ($r_{U}$) cause the decrease (increase) in the
EDM. The current experimental limit of $d_{e}$ is reached for $r_{U}$ which is
at the order of the magnitude of $10^{-2}$ in the case of small values of the
scaling dimension $d_{U}$. $r_{U}$ can take the values of the order of
$10^{-1}$ for $1.3<d_{U}<1.5$. This can be seen also in Fig.7 which represents
$d_{U}$ dependence of $d_{e}$ where upper-lower solid (long dashed; dashed;
dotted) line represents the EDM for $r_{U}=0.05$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.05$, $M_{U}=10^{4}\,GeV$ ($r_{U}=0.1$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.1$, $M_{U}=10^{4}\,GeV$; $r_{U}=0.4$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.4$, $M_{U}=10^{4}\,GeV$; $r_{U}=0.5$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.5$, $M_{U}=10^{4}\,GeV$). For the large values of
the ratio $r_{U}$ the scaling dimension $d_{U}$ must be near $d_{U}\sim 2.0$
in order to get the current experimental value of $d_{e}$. On the other hand
Fig.6 shows that one needs the ratio $r_{U}\sim 0.5$ and the small values of
the scaling dimension, $d_{U}\sim 1.1$ in order to reach the current
experimental value of $d_{\tau}$ (see also Fig.8 which is the same as the
Fig.7 but for $d_{\tau}$).
Finally we plot the EDM $d_{e}$ ($d_{\tau}$) with respect to the CP violating
parameter $sin\theta_{e}$ ($sin\theta_{\tau}$) in Fig.9 (10). For both figures
upper-lower solid; long dashed; dashed; dotted line represents666Notice that
the dotted line which represents $r_{U}=0.1$, $M_{U}=10^{3}\,GeV$, $d_{U}=1.3$
almost coincides with the one which represents $r_{U}=0.05$,
$M_{U}=10^{3}\,GeV$, $d_{U}=1.1$ and it is not observed in the figure the
$d_{e}$ ($d_{\tau}$) for $M_{U}=10^{3}\,GeV$-$M_{U}=10^{4}\,GeV$,
$r_{U}=0.05$, $d_{U}=1.1$; $r_{U}=0.05$, $d_{U}=1.3$; $r_{U}=0.1$,
$d_{U}=1.1$; $r_{U}=0.1$, $d_{U}=1.3$. These figures show that $d_{e}$ and
$d_{\tau}$ are enhanced at least one order in the range of the CP violating
parameter, $0.1<sin\theta_{\tau}<0.9$
Now we would like to summarize our results: The charged lepton EDMs are
strongly sensitive to the parameters used, namely the scaling dimension
$d_{U}$, the ratio $r_{U}$ and the mass scale $M_{U}$. We observe that the
experimental current limits of $d_{e}$ and $d_{\mu}$ are reached in the case
that the ratio $r_{U}$ lies in the range of $0.05-0.20$ and the scaling
dimension $d_{U}$ is near $1.1-1.2$. However for the current experimental
value of $d_{\tau}$ the ratio must reach to the values $r_{U}\sim 0.5$ for the
small values of the scaling dimension, $d_{U}\sim 1.1$.
For completeness, we compare the theoretical framework and the numerical
results of the present work with the study [16] which is related to the
contribution of scalar unparticle on the charged lepton EDM. In the present
case the tensor unparticle contribution is in the tree level, however in [16]
the scalar unparticle contribution is at one loop level. In addition to this,
in the present work, we assume that the scale invariance is broken at some
scale $\mu$ after the electroweak symmetry breaking and, therefore, the
antisymmetric tensor unparticle propagator is modified. In [16] the scale
invariance is intact and the propagator is the original one. In both cases the
charged lepton EDMs are strongly sensitive to the scaling dimension $d_{U}$
and the experimental current limit of $d_{e}$ can be reached in the range
$1.6\leq d_{U}\leq 1.8$ (near $1.1-1.2$) for scalar unparticle mediation
(tensor unparticle mediation). For $d_{\mu}$ and $d_{\tau}$ the current limits
are reached for the small values of the scale $d_{U}$, $d_{U}\leq 1.1$, for
both cases.
Hopefully, with in future more accurate measurements of the lepton EDMs it
would be possible to eliminate this discrepancy. These new measurements will
give strong information about the role of unparticle scenario on the CP
violation mechanism and the nature of unparticles.
Appendix
Here we would like to present the calculation of the charged lepton EDM (see
eq.(11)) by using the effective lagrangian given in eq.(9). The first (second)
term in the effective lagrangian drives the $O^{\mu\nu}_{U}\rightarrow
A_{\nu}$ transition which is carried by the vertex
$\displaystyle
2\,i\,\frac{g^{\prime}\,c_{W}\,\lambda_{B}}{\Lambda_{2}^{d_{U}-2}}\,k_{\mu}\,\epsilon_{\nu}O^{\mu\nu}_{U}\,\,(-i\,\frac{g\,v^{2}\,s_{W}\,\lambda_{W}}{2\,\Lambda_{4}^{d_{U}}}\,k_{\mu}\,\epsilon_{\nu}O^{\mu\nu}_{U})\,,$
where $\epsilon_{\nu}$ is the outgoing photon four polarization vector. On the
other hand the third term in the effective lagrangian results in the vertex
$\displaystyle\frac{y_{l}\,v}{\sqrt{2}\,\Lambda_{4}^{d_{U}}}\,(\lambda_{l}-\lambda_{l}^{*})\,\bar{l}\,\gamma_{5}\,\sigma_{\mu\nu}\,l\,,$
which creates the EDM interaction. Finally these two vertices are connected by
the tensor unparticle propagator (see eq.(8)) and, by extracting the
coefficient of $i\,\bar{l}\,\gamma_{5}\,\sigma^{\mu\nu}\,l\,F_{\mu\nu}$, one
gets the EDM of charged leptons as in eq.(11). Now we give a brief explanation
how to obtain the tensor unparticle propagator. The starting point is the
scalar unparticle propagator which is obtained by respecting the scale
invariance. The two point function of scalar unparticle operators reads
$\displaystyle<0|\Big{(}O_{U}(x)\,O_{U}(0)\Big{)}0>=\int\,\frac{d^{4}P}{(2\,\pi)^{4}}\,e^{-iP.x}\,\rho(P^{2})\,,$
(12)
where $\rho(P^{2})$ is the spectral density:
$\displaystyle\rho(P^{2})=A_{d_{U}}\,\theta(P^{0})\,\theta(P^{2})\,(P^{2})^{\xi}\,.$
(13)
The scale invariance777The spectral density is invariant under the scale
transformation $x\rightarrow s\,x$ and $O_{U}(s\,x)\rightarrow
s^{-d_{U}}\,O_{U}(x)$. requires a restriction on the parameter $\xi$,
$\xi=d_{U}-2$, and, therefore, $\rho(P^{2})$ becomes
$\displaystyle\rho(P^{2})=A_{d_{U}}\,\theta(P^{0})\,\theta(P^{2})\,(P^{2})^{d_{U}-2}\,.$
(14)
Here the factor $A_{d_{U}}$ reads
$\displaystyle
A_{d_{U}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{U}}}\,\frac{\Gamma(d_{U}+\frac{1}{2})}{\Gamma(d_{U}-1)\,\Gamma(2\,d_{U})}\,,$
in order to get the phase space of $d_{U}$ massless particles, i.e.,
unparticle stuff having the scale dimension $d_{U}$ can be represented as non-
integral number $d_{U}$ of invisible particles [10, 11, 12]. Finally, by using
spectral formula, the scalar unparticle propagator is obtained as [11, 12]
$\displaystyle\int\,d^{4}x\,e^{iP.x}<0|T\Big{(}O_{U}(x)\,O_{U}(0)\Big{)}0>=i\frac{A_{d_{U}}}{2\,\pi}\,\int_{0}^{\infty}\\!\\!\\!ds\,\frac{s^{d_{U}-2}}{P^{2}-s+i\epsilon}\\!=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,(-P^{2}-i\epsilon)^{d_{U}-2}.$
(15)
Notice that for $P^{2}>0$, the function
$\frac{1}{(-P^{2}-i\epsilon)^{2-d_{U}}}$ in eq. (15) reads
$\displaystyle\frac{1}{(-P^{2}-i\epsilon)^{2-d_{U}}}\rightarrow\frac{e^{-i\,d_{U}\,\pi}}{(P^{2})^{2-d_{U}}}\,,$
(16)
which shows that there exists a non-trivial phase due to the non-integral
scaling dimension. In the case of tensor unparticle one needs a projection
operator
$\Pi_{\mu\nu\alpha\beta}=\frac{1}{2}(g_{\mu\alpha}\,g_{\nu\beta}-g_{\nu\alpha}\,g_{\mu\beta})$
which contains the transverse and longitudinal parts and one gets the
propagator of antisymmetric tensor unparticle as
$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-p^{2}-i\epsilon)^{d_{U}-2}\,.$
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Figure 1: Tree level diagram contributing to the EDM of charged lepton due to
tensor unparticle. Wavy (solid) line represents the electromagnetic field
(lepton field) and double dashed line the tensor unparticle field.
Figure 2: $d_{\mu}$ with respect to $M_{U}$ for $a^{U}_{\mu}=10^{-9}$. Upper-
lower-the lowest solid (dashed-long dashed; dotted) line represents the EDM
for $d_{U}=1.1$, $r_{U}=0.40-0.10-0.05$ ($d_{U}=1.3$, $r_{U}=0.40-0.10$;
$d_{U}=1.5$, $r_{U}=0.40$). Figure 3: $d_{\mu}$ with respect to the scale
parameter $d_{U}$ for $a^{U}_{\mu}=10^{-9}$. Here upper-lower solid (long
dashed; dashed; dotted) line represents the EDM for $r_{U}=0.05$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.05$, $M_{U}=10^{4}\,GeV$ ($r_{U}=0.1$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.1$, $M_{U}=10^{4}\,GeV$; $r_{U}=0.4$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.4$, $M_{U}=10^{4}\,GeV$; $r_{U}=0.5$,
$M_{U}=10^{3}\,GeV$-$r_{U}=0.5$, $M_{U}=10^{4}\,GeV$). Figure 4: The same as
Fig. 3 but for $a^{U}_{\mu}=10^{-10}$. Figure 5: $d_{e}$ with respect to
$M_{U}$ for $sin\theta_{e}=0.5$. Here the upper most-upper-lower-the lowest
solid; dashed line represents $d_{e}$ for $d_{U}=1.1-1.3-1.5-1.8$,
$r_{U}=0.05$; $r_{U}=0.10$. Figure 6: The same as Fig. 5 but for $d_{\tau}$
and $sin\theta_{\tau}=0.5$. Figure 7: $d_{e}$ with respect to the scale
parameter $d_{U}$. Here upper-lower solid (long dashed; dashed; dotted) line
represents $d_{e}$ for $r_{U}=0.05$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.05$,
$M_{U}=10^{4}\,GeV$ ($r_{U}=0.1$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.1$,
$M_{U}=10^{4}\,GeV$; $r_{U}=0.4$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.4$,
$M_{U}=10^{4}\,GeV$; $r_{U}=0.5$, $M_{U}=10^{3}\,GeV$-$r_{U}=0.5$,
$M_{U}=10^{4}\,GeV$). Figure 8: The same as the Fig.7 but for $d_{\tau}$.
Figure 9: $d_{e}$ with respect to $sin\theta_{e}$. Here upper-lower solid;
long dashed; dashed; dotted line represents $d_{e}$ for
$M_{U}=10^{3}\,GeV$-$M_{U}=10^{4}\,GeV$, $r_{U}=0.05$, $d_{U}=1.1$;
$r_{U}=0.05$, $d_{U}=1.3$; $r_{U}=0.1$, $d_{U}=1.1$; $r_{U}=0.1$, $d_{U}=1.3$.
. Figure 10: The same as Fig. 9 but for $d_{\tau}$ and with respect to
$sin\theta_{\tau}$.
|
arxiv-papers
| 2011-03-08T11:45:19 |
2024-09-04T02:49:17.552559
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. O.Iltan",
"submitter": "Erhan Iltan",
"url": "https://arxiv.org/abs/1103.1499"
}
|
1103.1561
|
# The Sun’s Shallow Meridional Circulation
David H. Hathaway NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
david.hathaway@nasa.gov
###### Abstract
The Sun’s global meridional circulation is evident as a slow poleward flow at
its surface. This flow is observed to carry magnetic elements poleward -
producing the Sun’s polar magnetic fields as a key part of the 11-year sunspot
cycle. Current theories for the sunspot cycle assume that this surface flow is
part of a circulation which sinks inward at the poles and turns equatorward at
depths below 100 Mm. Here we use the advection of the Sun’s convection cells
by the meridional flow to map the flow velocity in latitude and depth. Our
measurements show the largest cells clearly moving equatorward at depths below
35 Mm - the base of the Sun’s surface shear layer. This surprisingly shallow
return flow indicates the need for substantial revisions to solar/stellar
dynamo theory.
Sun: dynamo, Sun: rotation, Sun: surface magnetism
## 1 INTRODUCTION
The Sun’s meridional circulation has been a part of solar magnetic dynamo
theory for half a century. A poleward meridional flow from the Sun’s mid
latitudes was invoked in the earliest models (long before the flow was
actually measured) to transport magnetic elements from decaying sunspot
regions to the poles where they would erode the opposite polarity magnetic
field from the old sunspot cycle and build up the polar fields of the new
sunspot cycle (Babcock, 1961). This surface meridional flow, along with its
latitudinal structure and variation in time, is now well observed (Topka et
al., 1982; Komm et al., 1993; Hathaway et al., 1996; Hathaway & Rightmire,
2010, 2011) and its role in the surface magnetic flux transport is well
established (DeVore & Sheeley, 1987; van Ballegooijen et al., 1998; Schrijver
& Title, 2001). Dynamo theories over the last decade and a half have assumed
that the mass traveling poleward in the surface layers sinks inward at the
poles and returns to the equator along the base of the Sun’s convection zone
at a depth of 200 Mm. In these theories this slow, dense, equatorward flow is
responsible for the equatorward drift of sunspot activity (Dikpati &
Choudhuri, 1994; Dikpati & Charbonneau, 1999; Nandy & Choudhuri, 2001). Dynamo
models based on this deep meridional circulation have recently been used to
predict the currently emerging sunspot cycle, albeit with disparate results
(Dikpati et al., 2006; Choudhuri et al., 2007).
The meridional flow is difficult to measure. Its amplitude ($\sim 10-20\rm{\
m\ s}^{-1}$) is more than an order of magnitude weaker than that of the other
major flows observed on the surface of the Sun. The axisymmetric longitudinal
flow, differential rotation, has a dynamic range of $\sim 200\rm{\ m\ s}^{-1}$
and the non-axisymmetric cellular convection flows have typical velocities of
several hundred $\rm{\ m\ s}^{-1}$. The meridional flow is observed to be
poleward from the equator with peak flow speeds in the mid latitudes. The flow
amplitude measured from Doppler shifts of spectral lines formed at the surface
is $\sim 20\rm{\ m\ s}^{-1}$ (Hathaway et al., 1996; Ulrich, 2010) while the
amplitude found by measuring the motions of the small magnetic features is
$\sim 12\rm{\ m\ s}^{-1}$ (Komm et al., 1993; Hathaway & Rightmire, 2010,
2011). The meridional flow can also be measured using the methods of local
helioseismology which yield a peak velocity of $\sim 20\rm{\ m\ s}^{-1}$ that
appears to be constant with depth down to $\sim 26$ Mm (Giles et al., 1997;
Schou & Bogart, 1998). Recently, however, two new measurement methods have
indicated a decrease in amplitude with depth. A method using global
helioseismology (Mitra-Kraev & Thompson, 2007) found a meridional flow that
decreased with depth and became equatorward at a depth of only 40 Mm - but
with a large range of error. A method using the movement of the larger solar
convection cells, supergranules, also found a meridional flow that decreased
with depth but without precise depth information and without detection of a
return flow (Hathaway et al., 2010). While some helioseismic studies indicate
a poleward meridional flow at depths well below 26 Mm, Duvall & Hanasoge
(2009) found that those methods are prone to systematic errors and Gough &
Hindman (2010) conclude that the flow below 30 Mm remains unknown. Furthermore
Beckers (2007) has suggested that projection effects may have also compromised
some of the local helioseismology results and concludes that the meridional
flow velocity may decrease with depth.
Here we measure the meridional flow by tracking the motions of supergranules,
but extend the analysis to include larger cells with deeper roots. Numerical
models of compressible convection with radiative transfer in the near surface
layers (Stein & Nordlund, 2000) clearly show that small cells dominate at the
surface and larger structures are found at increasing depth. Supergranules
cover the surface of the Sun and have a broad range of sizes that sample a
corresponding range of depths. We measure the motion of the pattern of
supergranules by analyzing data acquired by the Michelson Doppler Imager (MDI)
(Scherrer et al., 1995) on the ESA/NASA Solar and Heliospheric Observatory
(SOHO) satellite in 1996 and 1997.
## 2 DATA PREPARATION
The data consist of 1024x1024 pixel images of the line-of-sight velocity
determined from the Doppler shift of a spectral line due to the trace element
nickel in the solar atmosphere. The images are acquired at a 1 min cadence. We
average them over 31 min with a Gaussian weighting function which filters out
any velocity components that vary on time scales less than about 16 min. We
then map these temporally filtered images onto a 1024x1024 grid in
heliographic latitude from pole to pole and in longitude $\pm 90\arcdeg$ from
the central meridian (Figure 1). This mapping accounts for the position angle
of the Sun’s rotation axis relative to the imaging CCD and the tilt angle of
the Sun’s rotation axis toward or away from the spacecraft. Both of these
angles include modification for the most recent determinations of the
orientation of the Sun’s rotation axis (Beck & Giles, 2005; Hathaway &
Rightmire, 2010). We analyze data obtained during two 60+ day periods of
continuous coverage - one in 1996 from May 24 to July 24 and another in 1997
from April 14 to June 18.
We also generate and analyze simulated data to assist in our determination of
the representative depths. We construct the simulated data from an evolving
spectrum of vector spherical harmonics in such a manner as to reproduce the
spatial, spectral, and temporal behavior of the observed cellular flows
(Hathaway et al., 2010). The cells are advected in longitude by differential
rotation and in latitude by meridional flow, both of which vary with cell
size.
Figure 1: Heliographic map details of the line-of-sight (Doppler) velocity
from SOHO/MDI (top) and from the data simulation (bottom). Each map detail
extends $90\arcdeg$ in longitude from the central meridian on the left and
about $35\arcdeg$ in latitude from the equator (the thick horizontal line).
The mottled pattern is the Doppler signal (blue for blue shifts and red for
red shifts) due to the supergranule convection cells. The latitudinal movement
of these supergranules yields a measurement of the Sun’s meridional
circulation.
## 3 DATA ANALYSIS AND RESULTS
We determine the motions of the cellular patterns in longitude and latitude
for strips of data by finding the displacement of the maximum in the cross-
correlation with similar strips from images acquired at time lags of 2, 4, 8,
16, 24, and 32 hr. Each strip is 11 pixels or $\sim 2\arcdeg$ high in latitude
and 600 pixels or $\sim 105\arcdeg$ long in longitude. We repeat this
procedure for 796 latitude positions between $\pm 70\arcdeg$ latitude and for
each hour over the 60 days of each MDI dataset and 60 days of simulated data.
This cross-correlation method was first used to determine the equatorial
rotation rate by Duvall (1980) who concluded that larger cells live longer and
rotated faster. Beck & Schou (2000) used a 2D Fourier transform method and
found a rotation rate that increased with the wavelength of the features.
We calculate the average differential rotation and meridional flow profiles
and fit them with 4th order polynomials in $\sin\theta$ where $\theta$ is the
heliographic latitude (Figure 2). The rotational velocity increases with
increasing time lags to a maximum at 24 hr but then decreases at 32 hr. The
meridional flow velocity decreases with time lag and, at the 32 hr time lag,
reverses sign. The cellular flows that live long enough to be positively
correlated 32 hrs later are moving equatorward. This is a clear detection of
the meridional return flow. The individual data points have a standard error
of $\sim 1\rm{\ m\ s}^{-1}$ but the vast majority of points indicate an
equatorward flow significantly bigger than this. The curves fit through the
data points indicate an equatorward return flow of $1.8\rm{\ m\ s}^{-1}$ with
a standard error of $<0.2\rm{\ m\ s}^{-1}$.
Figure 2: Flow profiles as functions of latitude from the 1996 SOHO/MDI data
(top row), the 1997 SOHO/MDI data (middle row), and the simulated data (bottom
row). The flow velocities measured at each latitude are shown with colored
dots for each time lag as indicated in the figure. The solid lines with the
same color coding represent the 4th order polynomial fits to each profile. The
meridional flow decreases in amplitude with increasing time lag and reverses
direction for 32 hr lags. The rotation rate increases as the time lag
increases up to 24 hr then drops at 32 hr.
We determine the characteristic convection cell wavelengths for the different
time lags using the wavelength dependence of the differential rotation and
meridional flow profiles used in the simulation. The differential rotation
measurements are largely reproduced with a relatively simple rotation
velocity, $u$, relative to an inertial (sidereal) frame of reference given by
$\displaystyle u(\theta,\lambda)$ $\displaystyle=$
$\displaystyle[(1980-246\sin^{2}\theta-365\sin^{4}\theta)\cos\theta]$ (1)
$\displaystyle[1.0+0.046\tanh(\lambda/35)]\rm{\ m\ s}^{-1}$
where the wavelength, $\lambda$, is given in Mm. (Note that the prograde
velocities plotted in Figure 2 are relative to a frame of reference rotating
at the Carrington rotation rate with $u_{C}(\theta)=1991\cos\theta$.) The
meridional flow measurements are largely reproduced with a northward velocity,
$v$, given by
$\displaystyle v(\theta,\lambda)$ $\displaystyle=$
$\displaystyle[(65\sin\theta-78\sin^{3}\theta)\cos\theta]$ (2)
$\displaystyle[\tanh((35-\lambda)/20)]\rm{\ m\ s}^{-1}$
where the reproduction of the return flow is particularly sensitive to the
zero crossing occurring at $\sim 35$ Mm.
Figure 3 shows the equatorial differential rotation velocity relative to the
surface and the meridional flow velocity at $30\arcdeg$ latitude along with
the data points from the MDI (averaged for the two years) and simulation data
analyses. The data points from the simulation virtually coincide with those
from the MDI data except at the 32 hr time lag and for the northward velocity
at the 16 hr time lag. No doubt, better fits could be obtained with more
complicated flow profiles. It is apparent, however, that the differential
rotation velocity must decrease for wavelengths greater than $\sim 35$ Mm and
that the reversal in the meridional flow direction must be more abrupt.
Figure 3 also shows that the points do not fall right on the curves for the
input flow profiles. We attribute this to two different processes. The two
latitudes chosen for this figure represent the latitudes at which each flow
reaches its maximum. Since the convection cells span a finite range of
latitudes the measured values should be less than these maxima. However, the
differential rotation signal is subject to an additional line-of-sight
projection effect (Hathaway et al., 2006) which makes the Doppler features
appear to rotate faster. This effect raises the measured values to
increasingly higher values with increasing wavelength for the prograde
velocity.
Figure 3: The simulation meridional flow speed at $30\arcdeg$ latitude (top)
and equatorial differential rotation relative to the surface (bottom) as
functions of wavelength are shown by the solid lines. The observed values from
the cross-correlation analysis with the MDI data and the simulated data
(filled circles and crosses respectively) are shown at their characteristic
wavelengths. Error bars centered on each symbol represent $2\sigma$ errors.
The dashed line shows the theoretical limit to the rate of increase in
rotation rate at the equator and the open circles show the rotation velocity
determined from global helioseismology by Schou et al. (1998) both assuming
that the wavelength equals the anchoring depth of the cells.
## 4 CONCLUSIONS
The relationship between the wavelength of a convection cell and the depth at
which it is anchored or steered is well constrained by the stability of the
surface shear layer and observations of the rotation rate with depth from
global helioiseismology. An increase in rotation rate with depth has long been
suggested by observations and is attributed to the conservation of angular
momentum for fluid elements moving inward and outward in the near surface
layers (Foukal & Jokipii, 1975; Gilman & Foukal, 1979; Hathaway, 1982).
However, a rotation rate which increases inward faster than that given by the
conservation of angular momentum is dynamically unstable (Chandrasekhar,
1961). Measurements of this rotation rate increase from helioseismology (Schou
et al., 1998; Corbard & Thompson, 2002) indicate that it follows this critical
gradient to depths of 10-15 Mm. This gradient (the dashed line in Figure 3)
and the helioseismicly determined rotation rates (open circles in Figure 3)
are matched by the simulation input rotation profile if we assume that the
convection cells are anchored at depths equal to their widths. Shallower cells
would give unstable gradients. The mass density increases nearly quadratically
with depth so it is reasonable to expect the cells to be advected by the flows
near their deepest extent.
This association between cell wavelength and depth indicates that the poleward
meridional flow seen at the surface reverses at a depth of 35 Mm - the base of
the surface shear layer where the rotation rate reaches its maximum. Although
this shallow return flow violates the assumptions of flux transport dynamos
(Dikpati & Choudhuri, 1994; Dikpati & Charbonneau, 1999; Nandy & Choudhuri,
2001; Dikpati et al., 2006; Choudhuri et al., 2007), it was predicted by
numerical simulations of the effects of rotation on supergranules (Hathaway,
1982), it is in agreement with global helioseismology, and it helps to
reconcile other observations.
The surface has the slowest rotation and the fastest meridional flow. Small
magnetic elements rotate faster than the surface and have poleward meridional
flow which is slower than the surface (Komm et al., 1993; Hathaway &
Rightmire, 2010, 2011). Both velocity components for the small magnetic
elements are matched at a depth of about 15 Mm. While supergranules do have a
broad range of sizes, their spectrum exhibits an excess of power at
wavelengths of 30-35 Mm (Hathaway et al., 2000). Our results indicate that
cells this size have depths roughly equal to the depth of the surface shear
layer. This hardly seems coincidental but rather suggests an intimate
connection between the characteristic size of supergranules and the depth of
the surface shear layer.
A meridional circulation confined to the surface shear layer would also
explain why numerical simulations of the solar convection zone below this
surface shear layer have had difficulty producing the observed flows (Miesch
et al., 2000). In particular, the meridional circulations in these simulations
are highly structured in latitude and highly variable in time. The source of
this structure and variability can be attributed to the small number ($\sim
100$) of convection cells that populate the simulated volume. A meridional
circulation driven by ($\sim 10000$) supergranules - the convection cells that
populate the surface shear layer - is far more likely to be less structured
and variable.
This detection of a shallow equatorward return flow for the Sun’s meridional
circulation indicates the need for a reassessment of solar dynamo theory. The
flux transport dynamo models, all of which assume and require a deep
meridional flow, apparently cannot be correct. While other dynamo models
exist, the majority of these place the dynamo action at the base of the
convection zone with a possible secondary and less organized dynamo in the
surface shear layer. Upon reassessment we may find that the surface shear
layer plays a far more important role in the global dynamo.
The author would like to thank NASA for its support of this research through a
grant from the Heliophysics Causes and Consequences of the Minimum of Solar
Cycle 23/24 Program to NASA Marshall Space Flight Center. He is also indebted
to Ron Moore and Lisa Rightmire who read and commented on the manuscript.
SOHO, is a project of international cooperation between ESA and NASA.
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|
arxiv-papers
| 2011-03-08T15:37:07 |
2024-09-04T02:49:17.557796
|
{
"license": "Public Domain",
"authors": "David H. Hathaway",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1103.1561"
}
|
1103.2144
|
apsrev4-1
# Sideband Cooling Micromechanical Motion to the Quantum Ground State
J. D. Teufel National Institute of Standards and Technology, Boulder, CO
80305, USA T. Donner JILA, National Institute of Standards and Technology
and the University of Colorado, Boulder, CO 80309, USA Dale Li National
Institute of Standards and Technology, Boulder, CO 80305, USA J. W. Harlow
JILA, National Institute of Standards and Technology and the University of
Colorado, Boulder, CO 80309, USA Department of Physics, University of
Colorado, Boulder, Colorado 80309, USA M. S. Allman National Institute of
Standards and Technology, Boulder, CO 80305, USA K. Cicak National Institute
of Standards and Technology, Boulder, CO 80305, USA A. J. Sirois National
Institute of Standards and Technology, Boulder, CO 80305, USA J. D. Whittaker
National Institute of Standards and Technology, Boulder, CO 80305, USA K. W.
Lehnert JILA, National Institute of Standards and Technology and the
University of Colorado, Boulder, CO 80309, USA Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA R. W. Simmonds National
Institute of Standards and Technology, Boulder, CO 80305, USA
The advent of laser cooling techniques revolutionized the study of many
atomic-scale systems. This has fueled progress towards quantum computers by
preparing trapped ions in their motional ground state Diedrich1989 , and
generating new states of matter by achieving Bose-Einstein condensation of
atomic vapors Anderson1995 . Analogous cooling techniques Braginsky1992 ;
Kippenberg2008 provide a general and flexible method for preparing
macroscopic objects in their motional ground state, bringing the powerful
technology of micromechanics into the quantum regime. Cavity opto- or electro-
mechanical systems achieve sideband cooling through the strong interaction
between light and motion Braginsky1970 ; Blair1995 ; Teufel2008 ; Thompson2008
; Groblacher2009a ; Park2009 ; Lin2009 ; Schliesser2009 ; Rocheleau2010 ;
Riviere2010 ; Li2011 . However, entering the quantum regime, less than a
single quantum of motion, has been elusive because sideband cooling has not
sufficiently overwhelmed the coupling of mechanical systems to their hot
environments. Here, we demonstrate sideband cooling of the motion of a
micromechanical oscillator to the quantum ground state. Entering the quantum
regime requires a large electromechanical interaction, which is achieved by
embedding a micromechanical membrane into a superconducting microwave resonant
circuit. In order to verify the cooling of the membrane motion into the
quantum regime, we perform a near quantum-limited measurement of the microwave
field, resolving this motion a factor of 5.1 from the Heisenberg limit
Braginsky1992 . Furthermore, our device exhibits strong-coupling allowing
coherent exchange of microwave photons and mechanical phonons Teufel2010 .
Simultaneously achieving strong coupling, ground state preparation and
efficient measurement sets the stage for rapid advances in the control and
detection of non-classical states of motion Bose1997 ; Mancini1997 , possibly
even testing quantum theory itself in the unexplored region of larger size and
mass Marshall2003 . The universal ability to connect disparate physical
systems through mechanical motion naturally leads towards future methods for
engineering the coherent transfer of quantum information with widely different
forms of quanta.
Mechanical oscillators that are both decoupled from their environment (high
quality factor $Q$) and placed in the quantum regime could allow us to explore
quantum mechanics in entirely new ways Bose1997 ; Mancini1997 ; Marshall2003 ;
Akram2010 ; Regal2011 . For an oscillator to be in the quantum regime, it must
be possible to prepare it in its ground state, to arbitrarily manipulate its
quantum state, and to detect its state near the Heisenberg limit. In order to
prepare an oscillator in its ground state, its temperature $T$ must be reduced
such that $k_{\mathrm{B}}T<\hbar\Omega_{\mathrm{m}}$, where
$\Omega_{\mathrm{m}}$ is the resonance frequency of the oscillator,
$k_{\mathrm{B}}$ is Boltzmann’s constant, and $\hbar$ is the reduced Planck’s
constant. While higher resonance frequency modes ($>1$ GHz) can meet this
cooling requirement with conventional refrigeration ($T<50$ mK), these stiff
oscillators are difficult to control and to detect within their short
mechanical lifetimes. One unique approach using passive cooling has
successfully overcome these difficulties by using a piezoelectric dilatation
oscillator coupled to a superconducting qubit OConnell2010 . Unfortunately,
this method is incompatible with the broad range of lower frequency, high Q,
flexural mechanical modes. In order to take advantage of the attractive
mechanical properties of these oscillators, an alternative active cooling
method is required, one that can reduce the oscillator’s temperature below
that of the surrounding environment.
Cavity opto- or electro-mechanical systems Kippenberg2008 naturally offer a
method for both detecting mechanical motion and cooling a mechanical mode to
its ground state Marquardt2007 ; Wilson2007 . An object whose motion alters
the resonance frequency $\omega_{\mathrm{c}}$ of an electromagnetic cavity
experiences a radiation pressure force governed by the parametric interaction
Hamiltonian: $\hat{H}_{\mathrm{int}}=\hbar G\hat{n}\hat{x}$, where
$G=d\omega_{\mathrm{c}}/dx$, $\hat{n}$ is the cavity photon number, and
$\hat{x}$ is the displacement of the mechanical oscillator. By driving the
cavity at a frequency $\omega_{\mathrm{d}}$, the oscillator’s motion produces
upper and lower sidebands at $\omega_{\mathrm{d}}\pm\Omega_{\mathrm{m}}$.
Because these sideband photons are inelastically scattered from the drive
field, they provide a way to exchange energy with the oscillator. If the drive
field is optimally detuned below the cavity resonance
$\Delta\equiv\omega_{\mathrm{d}}-\omega_{\mathrm{c}}=-\Omega_{\mathrm{m}}$,
photons will be preferentially up-converted to $\omega_{\mathrm{c}}$ because
the photon density of states is maximal there (Fig 1b). When an up-converted
photon leaves the cavity, it removes the energy of one mechanical quantum (one
phonon) from the motion. Thus, the mechanical oscillator is damped and cooled
via this radiation-pressure force. Because the mechanical motion is encoded in
scattered photons exiting the cavity, a quantum-limited measurement of this
photon field provides a near Heisenberg-limited detection of mechanical motion
Clerk2010 .
While there has been substantial progress in cooling mechanical oscillators
with radiation pressure forces, sideband cooling to the quantum mechanical
ground state has been an outstanding challenge. Cavity optomechanical systems
have realized very large sideband cooling rates Thompson2008 ; Groblacher2009a
; Park2009 ; Lin2009 ; Schliesser2009 ; Riviere2010 ; Li2011 ; however, these
rates are not sufficient to overcome the larger thermal heating rates of the
mechanical modes. Because electromechanical experiments use much lower-energy
photons Braginsky1970 ; Blair1995 ; Teufel2008 ; Rocheleau2010 , they are
naturally compatible with operation below $100$ mK, but have consequently
suffered from weak electromechanical interactions and inefficient detection of
the photon fields.
Here, we present a cavity electromechanical system where a flexural mode of a
thin aluminum membrane is parametrically coupled to a superconducting
microwave resonant circuit. Unlike previous microwave systems, this device
achieves large electromechanical coupling by concentrating nearly all the
microwave electric fields near the mechanical oscillator Teufel2010 . The
oscillator is a $100$ nm thick aluminum membrane with a diameter of 15 µm,
suspended $50$ nm above a second aluminum layer on a sapphire substrate
Cicak2010 (see Fig. 1). These two metal layers form a variable parallel-plate
capacitor that is shunted by a $12$ nH spiral inductor. This combination of
capacitor and inductor creates a microwave cavity whose resonance frequency
depends on the mechanical displacement of the membrane and is centered at
$\omega_{\mathrm{c}}=2\pi\times 7.54$ GHz. The device is operated in a
dilution refrigerator at $15$ mK, where aluminum is superconducting, and the
microwave cavity has a total energy decay rate of $\kappa\approx 2\pi\times
200$ kHz. As expected from the dimensions of the membrane,
$\Omega_{\mathrm{m}}=2\pi\times 10.56$ MHz, and we find an intrinsic damping
rate of $\Gamma_{\mathrm{m}}=2\pi\times 32$ Hz, resulting in a mechanical
quality factor
$Q_{\mathrm{m}}=\Omega_{\mathrm{m}}/\Gamma_{\mathrm{m}}=3.3\times 10^{5}$. The
oscillator mass $m=48$ pg implies that the zero-point motion is
$x_{\mathrm{zp}}=\sqrt{\hbar/(2m\Omega_{\mathrm{m}})}=4.1$ fm. With a ratio of
$\Omega_{\mathrm{m}}/\kappa>50$, our system is deep in the resolved-sideband
regime and perfectly suited for sideband cooling to the mechanical ground
state Marquardt2007 ; Wilson2007 .
To measure the mechanical displacement, we apply a microwave field, which is
detuned below the cavity resonance frequency by $\Delta=-\Omega_{\mathrm{m}}$,
through heavily attenuated coaxial lines to the feed line of our device. The
upper sideband at $\omega_{\mathrm{c}}$ is amplified with a custom-built
Josephson parametric amplifier (JPA) Castellanos-Beltran2008 ; Teufel2009
followed by a low-noise cryogenic amplifier, demodulated at room temperature,
and finally monitored with a spectrum analyzer. The thermal motion of the
membrane creates an easily resolvable peak in the microwave noise spectrum. As
described previouslyTeufel2009 , this measurement scheme constitutes a nearly
shot-noise-limited microwave interferometer with which we can measure
mechanical displacement with minimum added noise close to fundamental limits.
In order to calibrate the demodulated signal to the membrane’s motion, we
measure the thermal noise spectrum while varying the cryostat temperature
(Fig. 1c). Here a weak microwave drive ($\sim 3$ photons in the cavity) is
used in order to ensure that radiation pressure damping and cooling effects
are negligible. When $\Omega_{\mathrm{m}}\gg\kappa\gg\Gamma_{\mathrm{m}}$ and
$\Delta=-\Omega_{\mathrm{m}}$, the displacement spectral density $S_{x}$ is
related to the observed microwave noise spectral density $S$ by:
$S_{x}=2(\kappa\Omega_{\mathrm{m}}/G\kappa_{\mathrm{ex}})^{2}S/P_{\mathrm{o}}$,
where $\kappa_{\mathrm{ex}}$ is the coupling rate between the cavity and the
feed line, and $P_{\mathrm{o}}$ is the power of the microwave drive at the
output of the cavity. According to equipartition, the area under the resonance
curve of displacement spectral density $S_{x}$ must be proportional to the
effective temperature of the mechanical mode. This calibration procedure
allows us to convert the sideband in the microwave power spectral density to a
displacement spectral density and to extract the thermal occupation of the
mechanical mode. In Fig. 1c we show the number of thermal quanta in the
mechanical resonator as a function of $T$. The linear dependence of the
integrated power spectral density with temperature shows that the mechanical
mode equilibrates with the cryostat even for the lowest achievable temperature
of $15$ mK. This temperature corresponds to a thermal occupancy
$n_{\mathrm{m}}=30$, where
$n_{\mathrm{m}}=[\exp(\hbar\Omega_{\mathrm{m}}/k_{\mathrm{B}}T)-1]^{-1}$. The
calibration determines the electromechanical coupling strength $G/2\pi=49\pm
2$ MHz/nm. With the device parameters, we can investigate both the fundamental
sensitivity of our measurement as well as the effects of radiation pressure
cooling.
The total measured displacement noise results from two sources: the membrane’s
actual mean-square motion $S_{x}^{\mathrm{th}}$ and the _apparent_ motion
$S_{x}^{\mathrm{imp}}$ due to imprecision of the measurement. Fig. 2a
demonstrates how the use of low-noise parametric amplification significantly
lowers $S_{x}^{\mathrm{imp}}$, resulting in a reduction in the white-noise
background by a factor of more than $30$. This greatly increases the signal-
to-noise ratio of the membrane’s thermal motion, reducing the required
integration time to resolve the thermal peak by a factor of $1000$. To
investigate the measurement sensitivity in the presence of dynamical
backaction, we regulate the cryostat temperature at $20$ mK and increase the
amplitude of the detuned microwave drive while observing modifications in the
displacement spectral density. We quantify the strength of the drive by the
resulting number of photons $n_{\mathrm{d}}$ in the microwave cavity. As shown
in Fig. 2b, the measurement imprecision $S_{\mathrm{x}}^{\mathrm{imp}}$ is
inversely proportional to $n_{\mathrm{d}}$. At the highest drive power
($n_{\mathrm{d}}\approx 10^{5}$), the absolute displacement sensitivity is
$5.5\times 10^{-34}$ m2/Hz.
As expected, the increased drive power also damps and cools the mechanical
oscillator Braginsky1992 ; Marquardt2007 ; Wilson2007 . The total mechanical
dissipation rate $\Gamma_{\mathrm{m}}^{\prime}$ is the sum of the intrinsic
dissipation $\Gamma_{\mathrm{m}}$ and the radiation-pressure-induced damping
resulting from scattering photons to the upper/lower sideband
$\Gamma=\Gamma_{\mathrm{+}}-\Gamma_{\mathrm{-}}$, where
$\Gamma_{\mathrm{\pm}}=4g^{2}\kappa/[\kappa^{2}+4(\Delta\pm\Omega_{\mathrm{m}})^{2}]$.
Here, $g$ is the coupling rate between the cavity and the mechanical mode,
which depends on the amplitude of the drive:
$g=Gx_{\mathrm{zp}}\sqrt{n_{\mathrm{d}}}$. Fig. 2c shows the measured values
of $\kappa$, $g$ and $\Gamma_{\mathrm{m}}^{\prime}$ as the drive increases.
The radiation-pressure damping of the mechanical oscillator becomes pronounced
above a cavity drive amplitude of approximately 75 photons, at which point
$\Gamma=\Gamma_{\mathrm{m}}$ and the mechanical linewidth has doubled.
While the absolute value of the displacement imprecision decreases with
increasing power, the visibility of the thermal mechanical peak no longer
improves once the radiation-pressure force becomes the dominant dissipation
mechanism for the membrane. By expressing the imprecision as equivalent
thermal quanta of the oscillator
$n_{\mathrm{imp}}=\Gamma_{\mathrm{m}}^{\prime}S_{x}^{\mathrm{imp}}/8x_{\mathrm{zp}}^{2}$,
we see that the visibility of the thermal noise above the imprecision no
longer improves once the drive is much greater than $n_{\mathrm{d}}\approx
100$ (Fig. 2d). This is because a linear decrease in $S_{x}^{\mathrm{imp}}$ is
balanced by a linear increase in $\Gamma_{\mathrm{m}}^{\prime}$ due to
radiation-pressure damping. The asymptotic value of $n_{\mathrm{imp}}$ is a
direct measure of the efficiency of the microwave measurement. Ideally, for a
lossless circuit, a quantum-limited microwave measurement would imply
$n_{\mathrm{imp}}=1/4$. The incorporation of the low-noise JPA improves
$n_{\mathrm{imp}}$ close to this ideal limit, reducing the asymptotic value of
$n_{\mathrm{imp}}$ from $70$ to $1.9$ quanta. This level of sensitivity is
crucial, as we will now use this measurement to resolve the residual thermal
motion of the membrane as it is cooled into the quantum regime.
Beginning from a cryostat temperature of $20$ mK and a thermal occupation of
$n_{\mathrm{m}}^{\mathrm{T}}=40$ quanta, the fundamental mechanical mode of
the membrane is cooled by the radiation-pressure forces. Figure 3a shows the
displacement spectral density of the motional sideband as $n_{\mathrm{d}}$ is
increased from 18 to 4,500 photons along with fits to a Lorentzian lineshape
(shaded area). As described above, this increased drive results in three
effects on the spectra: lower noise floor, wider resonances and smaller area.
As it is the area that corresponds to the mean-square motion of the membrane,
it directly measures the effective temperature of the mode. At a drive
intensity that corresponds to 4,000 photons in the cavity, the thermal
occupation is reduced below one quantum of mechanical motion, entering the
quantum regime.
Observing the noise spectrum over a broader frequency range reveals that there
is also a second Lorentzian peak with linewidth $\kappa$ whose area
corresponds to the finite thermal occupation $n_{\mathrm{c}}$ of the cavity.
Over a broad frequency range it is no longer valid to evaluate the cavity
parameters at a single frequency to infer the spectrum in units of $S_{x}$.
Instead, Fig. 3b shows the noise spectrum in units of sideband power
normalized by the power at the drive frequency, $S/P_{\mathrm{o}}$. These two
sources of noise originating from either the mechanical or the electrical mode
interfere with each other and result in noise squashing Rocheleau2010 and
eventually normal-mode splitting Dobrindt2008 once $2g>\kappa/\sqrt{2}$.
Using a quantum-mechanical description applied to our circuit Rocheleau2010 ;
Clerk2010 , the expected noise spectrum is
$S/\hbar\omega=\frac{1}{2}+n_{\mathrm{add}}+\frac{2\kappa_{\mathrm{ex}}\left[\kappa
n_{\mathrm{c}}(\Gamma_{\mathrm{m}}^{2}+4\delta^{2})+4\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}g^{2}\right]}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}}\
$ (1)
where $\delta=\omega-\Omega_{\mathrm{m}}$,
$\widetilde{\Delta}=\omega_{\mathrm{d}}+\Omega_{\mathrm{m}}-\omega_{\mathrm{c}}$,
and $n_{\mathrm{add}}$ is added noise of the microwave measurement expressed
as an equivalent number of microwave photons. Fig. 3b shows the measured
spectra and corresponding fits (shaded region) to Eq. 1 as the
electromechanical system evolves first into the quantum regime
($n_{\mathrm{m}},n_{\mathrm{c}}<1$) and then into the strong-coupling regime
($2g>\kappa/2$). The results are summarized in Fig. 3c, where the thermal
occupancy of both the mechanical and electrical modes are shown as a function
of $n_{\mathrm{d}}$. For low drive power, the cavity shows no resolvable
thermal population (to within our measurement uncertainty of 0.05 quanta) as
expected for a $7.5$ GHz mode at $20$ mK. While it is unclear whether the
observed population at higher drive power is a consequence of direct heating
of the substrate, heating of the microwave attenuators preceding the circuit,
or intrinsic cavity frequency noise, we have determined that it is not the
result of frequency or amplitude noise of our microwave generator, as this
noise is reduced far below the microwave shot-noise level with narrow-band
filtering and cryogenic attenuation (see Supplementary Information). Sideband
cooling can never reduce the occupancy of the mechanical mode below that of
the cavity. Therefore, in order for the system to access the quantum regime,
the thermal population of the cavity must remain less than one quantum.
Assuming $\Omega_{\mathrm{m}}\gg\kappa$, the final occupancy of a mechanical
mode is Dobrindt2008
$n_{\mathrm{m}}=n_{\mathrm{m}}^{\mathrm{T}}\left(\frac{\Gamma_{\mathrm{m}}}{\kappa}\frac{4g^{2}+\kappa^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right)+n_{\mathrm{c}}\left(\frac{4g^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right).$
(2)
This equation shows that for moderate coupling
($\sqrt{\kappa\Gamma_{\mathrm{m}}}\ll g\ll\kappa$) the cooling of the
mechanical mode is linear in the number of drive photons. Beyond this regime,
the onset of normal-mode splitting abates further cooling. Here the mechanical
cooling rate becomes limited not by the coupling between the mechanical mode
and the cavity, but instead by the coupling rate $\kappa$ between the cavity
and its environment Dobrindt2008 . Thus, the final occupancy of the mechanical
mode can never be reduced to lower than
$n_{\mathrm{m}}^{\mathrm{T}}\Gamma_{\mathrm{m}}/\kappa$, and a stronger
parametric drive will only increase the rate at which the thermal excitations
Rabi oscillate between the cavity and mechanical modes. For our device we
achieve the desired hierarchy: as the coupling is increased, we first cool to
the ground state and then enter the strong-coupling regime
($n_{\mathrm{m}}^{\mathrm{T}}\Gamma_{\mathrm{m}}<\kappa<g$). Once
$n_{\mathrm{d}}$ exceeds $2\times 10^{4}$, the mechanical occupancy converges
toward the cavity population, reaching a minimum of $0.34\pm 0.05$ quanta. At
the highest power drive power ($n_{\mathrm{d}}=2\times 10^{5}$) the mechanical
mode has hybridized with the cavity, resulting in the normal-mode splitting
characteristic of the strong-coupling regime Teufel2010 . This level of
coupling is required to utilize the hybrid system for quantum information
processing, as it is only in the strong-coupling regime that a quantum state
may be manipulated faster than it decoheres from the coupling of either the
electromagnetic or mechanical modes to the environment.
Together the measurements shown in Fig. 2 and 3 quantify the overall
measurement efficiency of the system. The Heisenberg limit requires that a
continuous displacement measurement is necessarily accompanied by a backaction
force Braginsky1992 ; Clerk2010 ; Schliesser2009 , such that
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}}\geq\hbar$, where $S_{F}$ is the force noise
spectral density. From the thermal occupancy and damping rate of the
mechanical mode, we extract the total force spectral density
$S_{F}=4\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime}(n_{\mathrm{m}}+1/2)$.
This places a conservative upper bound on the quantum backaction by assuming
that it alone is responsible for the finite occupancy of the mechanical mode.
This experiment achieves the closest approach to Heisenberg-limited
displacement detection to date Clerk2010 ; Riviere2010 with a lowest
imprecision-backaction product
$\sqrt{S_{\mathrm{x}}^{\mathrm{imp}}S_{\mathrm{F}}}=4\hbar\sqrt{n_{\mathrm{imp}}(n_{\mathrm{m}}+1/2)}=(5.1\pm
0.4)\hbar$. Thus, this mechanical device simultaneously demonstrates ground-
state preparation, strong-coupling and near quantum-limited detection.
Looking forward, this technology offers a feasible route to achieve many of
the longstanding goals for quantum _mechanical_ systems. These prospects
include a direct measurement of the zero-point motion, observation of the
fundamental asymmetry between the rate of emission and absorption of phonons
Diedrich1989 , quantum nondemolition measurements Braginsky1992 and
generation of entangled states of mechanical motion Bose1997 ; Mancini1997 .
Furthermore, combining this device with a single-photon source and detector
(such as a superconducting qubit Hofheinz2009 ; OConnell2010 ) would enable
preparation of arbitrary quantum states of mechanical motion as well as
observation of a single excitation as it Rabi oscillates between a $7$ GHz
photon and a $10$ MHz phonon Akram2010 . Because the interaction between the
mechanical mode and the cavity is parametric, the coupling strength is
inherently tunable and can be turned on and off quickly. Thus, once a quantum
state is transfered into the mechanical mode, it can be stored there for a
time
$\tau_{\mathrm{th}}=1/(n_{\mathrm{m}}^{\mathrm{T}}\Gamma_{\mathrm{m}})>100$ µs
before absorbing one thermal phonon from its environment. As this timescale is
much longer than typical coherence times of superconducting qubits, mechanical
modes offer the potential for delay and storage of quantum information.
Lastly, because mechanical oscillators can couple to light of any frequency,
they could serve as a unique intermediary that transfers quantum information
between the microwave and optical domains Regal2011 .
These measurements demonstrate the power of sideband techniques to cool a
macroscopic ($\sim 10^{12}$ atoms) mechanical mode, beyond what is feasible
with conventional refrigeration techniques, into the quantum regime. These
broadly applicable methods for state preparation, manipulation and detection,
pave the way to access the quantum nature of a wide class of long-lived
mechanical oscillators. Through the strong interaction between photons and
phonons, mechanical systems can now inherit the experimental and theoretical
power of quantum optics, opening the field of quantum acoustics.
## I Acknowledgements
We thank A. W. Sanders for taking the micrograph in Fig. 1a and thank the JILA
instrument shop for fabrication and design of the cavity filter. This work was
financially supported by NIST and the DARPA QuASAR program. T.D. acknowledges
support from the Deutsche Forschungsgemeinschft (DFG). Contribution of the
U.S. government, not subject to copyright.
## II Author Information
Reprints and permissions information is available at www.nature.com/reprints.
The authors declare no competing financial interests. Correspondence and
requests for materials should be addressed to J.D.T (john.teufel@nist.gov).
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Figure 1: Schematic description of the experiment. a, Colorized scanning
electron micrograph showing the aluminum (grey) electromechanical circuit
fabricated on a sapphire (blue) substrate, in which a 15 µm diameter membrane
is lithographically suspended 50 nm above a lower electrode. The membrane’s
motion modulates the capacitance, and hence, the resonance frequency of the
superconducting microwave circuit. b, A coherent microwave drive inductively
coupled to the circuit acquires modulation sidebands due to the thermal motion
of the membrane. The upper sideband is amplified with a nearly quantum-limited
Josephson parametric amplifier within the cryostat. c, The microwave power in
the upper sideband provides a direct measurement of the thermal occupancy of
the mechanical mode, which may be calibrated _in situ_ by varying the
temperature of the cryostat. The mechanical mode shows thermalization with the
cryostat at all temperatures, yielding a minimum thermal occupancy of $30$
mechanical quanta without employing sideband-cooling techniques. The inset
illustrates the concept of sideband cooling. When the circuit is excited with
a detuned microwave drive such that $\Delta=-\Omega_{\mathrm{m}}$, the narrow
line shape of the electrical resonance ensures that photons are preferentially
scattered to higher energy, providing a cooling mechanism for the membrane.
Figure 2: Displacement sensitivity in the presence of radiation-pressure
damping. a, The displacement spectral density measured with (red) and without
(blue) the Josephson parametric amplifier. As the parametric amplifier greatly
reduces the total noise of the microwave measurement, the time required to
resolve the thermal motion is reduced by a factor of $1000$. b, As the
microwave drive power is increased, the absolute displacement sensitivity,
$S_{\mathrm{x}}^{\mathrm{imp}}$ improves, reaching a minimum of $5.5\times
10^{-34}$ m2/Hz at the highest power. c, The parametric coupling $g$ between
the microwave cavity and the mechanical mode increases as
$\sqrt{n_{\mathrm{d}}}$. This coupling damps the mechanical mode from its
intrinsic linewidth of $\Gamma_{\mathrm{m}}=2\pi\times 32$ Hz until it is
increased to that of the microwave cavity $\kappa$. d, The relative
measurement imprecision, in units of mechanical quanta, depends on the product
of $S_{\mathrm{x}}^{\mathrm{imp}}$ and $\Gamma_{\mathrm{m}}^{\prime}$. Thus,
once the power is large enough that radiation-pressure damping overwhelms the
intrinsic mechanical dissipation, $n_{\mathrm{imp}}$ asymptotically approaches
a constant value ($n_{\mathrm{imp}}=1.9$), which is a direct measure of the
overall efficiency of the photon measurement. Figure 3: Sideband cooling the
mechanical mode to the ground state. a, The displacement noise spectra and
Lorentzian fits (shaded region) for five different drive powers. With higher
power, the mechanical mode is both damped (larger linewidth) and cooled
(smaller area) by the radiation pressure forces. b, Over a broader frequency
span, the normalized sideband noise spectra clearly show both the narrow
mechanical peak and a broader cavity peak due to finite occupancy of the
mechanical and electrical modes, respectively. A small, but resolvable,
thermal population of the cavity appears as the drive power increases, setting
the limit for the final occupancy of the coupled optomechanical system. At the
highest drive power, the coupling rate between the mechanical oscillator and
the microwave cavity exceeds the intrinsic dissipation of either mode, and the
system hybridizes into optomechanical normal modes. c, Starting in thermal
equilibrium with the cryostat at $T=20$ mK, sideband cooling reduces the
thermal occupancy of the mechanical mode from $n_{\mathrm{m}}=40$ into the
quantum regime, reaching a minimum of $n_{\mathrm{m}}=0.34\pm 0.05$. These
data demonstrate that the parametric interaction between photons and phonons
can initialize the strongly coupled, electromechanical system in its quantum
ground state.
Supplementary Information for “Sideband Cooling Micromechanical Motion to the
Quantum Ground State”
## III Noise spectrum of an optomechanical system
A mechanical degree of freedom that parametrically couples to the cavity
resonance frequency modifies the power emerging from the cavity by scattering
photons to the upper or lower mechanical sidebands. To calculate the full
noise spectrum of the optomechanical system, we follow the general method of
input-output theory S_Walls1994 . We define
$g=Gx_{\mathrm{zp}}\sqrt{n_{\mathrm{d}}}$, where $G=d\omega_{\mathrm{c}}/dx$,
$x_{\mathrm{zp}}=\sqrt{\hbar/2m\Omega_{\mathrm{m}}}$, $m$ is the mass,
$\omega_{\mathrm{c}}$ is the cavity resonance frequency, $\Omega_{\mathrm{m}}$
is the mechanical resonance frequency and $n_{\mathrm{d}}$ is the number of
photons in the cavity due to a drive at frequency $\omega_{\mathrm{d}}$.
Furthermore, we define the response functions of the mechanical and cavity
modes as $\chi_{\mathrm{c}}^{-1}=\kappa/2+j(\delta+\widetilde{\Delta})$ and
$\chi_{\mathrm{m}}^{-1}=\Gamma_{\mathrm{m}}/2+j\delta$, where
$\Gamma_{\mathrm{m}}$ is the mechanical dissipation rate, $\kappa$ is the
cavity dissipation rate, $\delta=\omega-\Omega_{\mathrm{m}}$,
$\widetilde{\Delta}=\omega_{\mathrm{d}}-\omega_{\mathrm{c}}+\Omega_{\mathrm{m}}$
and $j=\sqrt{-1}$. $\kappa$ is total cavity dissipation rate due to both the
intentional coupling to the transmission line $\kappa_{\mathrm{ex}}$ and the
intrinsic losses $\kappa_{0}$. From these parameters, we define the
optomechanical self-energy S_Marquardt2007 ; S_Clerk2010 as a function of
$\delta$:
$\displaystyle\Sigma(\delta)$
$\displaystyle=-jg^{2}\left[\chi_{\mathrm{c}}(\delta)-\chi_{\mathrm{c}}^{*}(\delta+2\Omega_{\mathrm{m}})\right]$
($\mathrm{S}$1) $\displaystyle\approx-jg^{2}\chi_{\mathrm{c}}(\delta)$
($\mathrm{S}$2)
The approximation assumes that the drive is near the optimal detuning for
cooling ($|\widetilde{\Delta}|\ll\Omega_{\mathrm{m}}$) and the system is
sufficiently in the good-cavity limit ($\Omega_{\mathrm{m}}\gg\kappa$) such
that the cavity response at $(\delta+2\Omega_{\mathrm{m}})$ may be neglected.
Now the effective mechanical response function $\widetilde{\chi}_{\mathrm{m}}$
including the optomechanical effects is:
$\displaystyle\widetilde{\chi}_{\mathrm{m}}$
$\displaystyle=\frac{\chi_{\mathrm{m}}}{1+j\chi_{\mathrm{m}}\Sigma}$
($\mathrm{S}$3)
$\displaystyle\approx\frac{\chi_{\mathrm{c}}^{-1}}{g^{2}+\chi_{\mathrm{m}}^{-1}\chi_{\mathrm{c}}^{-1}}$
($\mathrm{S}$4)
The noise at the output of the cavity is characterized by the noise operator
$\hat{b}_{\mathrm{out}}$, which is related to the cavity field operator
$\hat{a}$ by $\hat{b}_{\mathrm{out}}=\sqrt{\beta\kappa_{\mathrm{ex}}}\hat{a}$.
$\beta$ is a dimensionless factor that depends on the geometry. Our circuit
(shown schematically in Fig. S1) couples power from the cavity equally to the
output and back to the input so here $\beta=1/2$. In principle, this fraction
could be engineered by coupling asymmetrically to the input and the output, or
by using a single port cavity ($\beta=1$).
Figure S1: Cavity coupling block diagram. Figure S2: Detailed schematic
diagram. A microwave generator creates a tone at the drive frequency. This
signal is filtered with a resonant cavity at room temperature and split into
two arms. The first arm excites the cavity through approximately 53 dB of
cryogenic attenuation. In order to avoid saturating the low-noise amplifier
with the microwave drive tone, the second arm is used to cancel the drive
before amplification. A computer-controlled variable attenuator and phase
shifter are run in a feedback loop to maintain cancellation at the part per
million level. A second microwave generator is used to provide the pump tone
for the Josephson parametric amplifier (JPA) as well as the reference
oscillator for the mixer. This pump tone is $1.3$ MHz above
$\omega_{\mathrm{c}}$ so that the JPA is operated as a non-degenerate
parametric amplifier, which measures both quadratures of the electromagnetic
filed at the upper sideband frequency. The last stage of attenuation on all
lines occurs inside a $20$ dB directional coupler, which allows us to minimize
the microwave power dissipated on the cold stage of the cryostat. The JPA is a
reflection amplifier; a signal incident on the strongly coupled port of the
JPA is reflected and amplified. A cryogenic circulator is used to separate the
incident and reflected waves, defining the input and output ports of the JPA.
The other circulators are used to isolate the cavity from the noise emitted
from the amplifier’s input.
Following directly the theoretical analysis of previous work S_Rocheleau2010 ;
S_Clerk2010 , we consider the noise operators $\hat{\eta}_{\mathrm{m}}$ and
$\hat{\eta}_{\mathrm{c}}$ associated with the mechanical and cavity modes
respectively, which satisfy the relations
$\langle\hat{\eta}_{\mathrm{m}}^{\dagger}\hat{\eta}_{\mathrm{m}}\rangle=n_{\mathrm{m}}^{T}$
and
$\langle\hat{\eta}_{\mathrm{c}}^{\dagger}\hat{\eta}_{\mathrm{c}}\rangle=n_{\mathrm{c}}$.
Thus, the output noise is S_Rocheleau2010
$\displaystyle\hat{b}_{\mathrm{out}}=$
$\displaystyle-\sqrt{\beta\kappa_{\mathrm{ex}}}\chi_{\mathrm{c}}\sqrt{\kappa}\left(1-g^{2}\widetilde{\chi}_{\mathrm{m}}\chi_{\mathrm{c}}\right)\hat{\eta}_{\mathrm{c}}$
$\displaystyle-\sqrt{\beta\kappa_{\mathrm{ex}}}\chi_{\mathrm{c}}\sqrt{\Gamma_{\mathrm{m}}}\left(jg\widetilde{\chi}_{\mathrm{m}}\right)\hat{\eta}_{\mathrm{m}}.$
In the frequency domain, the power spectral density of the noise at the output
(in units of W/Hz) is
$S=\hbar\omega\langle\hat{b}_{\mathrm{out}}^{\dagger}\hat{b}_{\mathrm{out}}\rangle$,
$\displaystyle S$
$\displaystyle=\frac{4\hbar\omega\beta\kappa_{\mathrm{ex}}(\Gamma_{\mathrm{m}}^{2}+4\delta^{2})\kappa
n_{\mathrm{c}}}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}}$
$\displaystyle+\frac{16\hbar\omega\beta\kappa_{\mathrm{ex}}g^{2}\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}}.$
The first term simply represents the thermal noise of a cavity with occupancy
$n_{\mathrm{c}}$ whose spectral weight is distributed over the ‘dressed’
cavity mode. The ‘dressed’ cavity mode includes the effect of optomechanically
induced transparency S_Agarwal2010; S_Weis2010; S_Teufel2010 and reduces to a
single Lorentzian lineshape in the limit of weak coupling
($g\ll\sqrt{\kappa\Gamma_{\mathrm{m}}}$). The second term is the thermal noise
of the mechanical mode with its modified mechanical susceptibility. Unlike
previous derivations S_Rocheleau2010 , we have not assumed the weak-coupling
regime. Thus, as this equation is valid in both the weak- and strong-coupling
regimes, it gives a unified description of the thermal noise spectrum even in
the presence of normal-mode splitting. Finally, the total noise at the output
of the measurement including the vacuum noise of the photon field and the
added noise of the measurement is
$\frac{S}{\hbar\omega}=\frac{1}{2}+n_{\mathrm{add}}^{\prime}+\frac{4\beta\kappa_{\mathrm{ex}}\left[\kappa
n_{\mathrm{c}}(\Gamma_{\mathrm{m}}^{2}+4\delta^{2})+4\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}g^{2}\right]}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}},$
($\mathrm{S}$5)
where $n_{\mathrm{add}}^{\prime}$ is the total added noise of the measurement
in units of equivalent number of photons. For an ideal measurement (_i.e._ for
a quantum-limited measurement of both quadratures of the light field),
$n_{\mathrm{add}}^{\prime}=1/2$.
Before the onset of normal-mode splitting, one can directly relate the
measured microwave power spectrum $S$ to the displacement spectral density
$S_{x}$. Assuming $\widetilde{\Delta}=0$, $n_{\mathrm{c}}\ll n_{\mathrm{m}}$
and $g,\delta\ll\kappa$,
$\displaystyle\frac{S}{\hbar\omega}$
$\displaystyle=\frac{1}{2}+n_{\mathrm{add}}^{\prime}+4\beta\frac{\kappa_{\mathrm{ex}}}{\kappa}\Gamma\frac{\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}}{\left(\Gamma_{\mathrm{m}}+\Gamma\right)^{2}+4\delta^{2}}$
($\mathrm{S}$6)
$\displaystyle=\frac{1}{2}+n_{\mathrm{add}}^{\prime}+\frac{2\beta
G^{2}n_{\mathrm{d}}}{\kappa}\frac{\kappa_{\mathrm{ex}}}{\kappa}S_{x},$
($\mathrm{S}$7)
where $\Gamma=4g^{2}/\kappa$ is the optomechanical damping rate.
## IV Microwave measurement and calibration
The detailed circuit diagram for our measurements is shown in Fig. S2. In
order to calibrate the value of $g_{0}=Gx_{\mathrm{zp}}$ for this device, we
applied a microwave drive optimally red-detuned ($\widetilde{\Delta}=0$) and
measured the thermal noise spectrum of the mechanical oscillator as a function
of cryostat temperature. Here we restricted $n_{\mathrm{d}}\approx 3$ in order
to ensure that radiation pressure effects are negligible. With the value of
$g_{0}$ now determined, we increase the drive amplitude and measure the
thermal noise spectrum at each drive power. The noise spectra are recorded and
averaged with commercial FFT spectrum analyser. Each spectrum is typically an
average of 500 traces with a measurement time of 0.5 s per trace. The cavity
response is then measured with a weak probe tone with a vector network
analyser to determine precise cavity parameters at each microwave drive power,
including the precise detuning and $\kappa$. For larger microwave drive powers
where the cavity spectrum exhibits optomechanically induced transparency
effects S_Agarwal2010; S_Weis2010; S_Teufel2010 , this spectrum also serves as
a direct measure of $g$. Finally, using additional calibration tones, each
noise spectrum is calibrated in units of absolute microwave noise quanta and
fit with Eq. 5 to determine the occupancy of both the cavity and mechanical
modes.
For our measurements, we infer that our entire measurement chain has an
effective added noise of $n_{\mathrm{add}}^{\prime}=2.1$. This value is
consistent with the independently measured value for the added noise of the
JPA ($n_{\mathrm{add}}=0.8$) and the $2.5$ dB of loss between the output of
the cavity and the JPA S_Castellanos-Beltran2008 ; S_Teufel2009 .
Figure S3: Measured transmission of filter cavity. A tunable resonant cavity
was implemented at room temperature in order to suppress noise $\sim 10$ MHz
above the drive frequency. As shown here, this cavity reduces the noise at the
cavity frequency by more than 40 dB, ensuring that the phase or amplitude
noise of the generator is not responsible for the finite occupancy of the
cavity at large drive power.
In order to ensure that the amplitude or phase noise of the signal generator
was not responsible for the finite occupancy of the cavity at high drive
power, we designed and built a custom filter cavity S_Rocheleau2010 . As shown
in Fig. S3, when the filter cavity is tuned to precisely the frequencies of
our circuit, it provides an addition 40 dB of noise suppression at the cavity
resonance frequency. The phase and amplitude noise of our signal generator
alone are specified by the manufacturer to be less than -150 dBc at Fourier
frequencies $10$ MHz away from the drive. With the addition of filter cavity,
we lower this noise to well below the shot-noise level of our microwave drive.
Furthermore, even without the filter cavity, we could not resolve an
appreciable difference in the cavity occupation. Thus, while we do not know
the precise mechanism for this occupancy, we conclusively determine the
generator noise is not the cause.
## V Inferring cavity parameter and number of drive photons
The measured microwave cavity parameters may be inferred from the transmitted
power spectrum. The power at the output of the cavity $P_{\mathrm{o}}$ is
related to the input power $P_{\mathrm{i}}$ by S_Teufel2010
$P_{\mathrm{o}}=P_{\mathrm{i}}\left(\frac{\kappa_{\mathrm{0}}^{2}+4\Delta^{2}}{\kappa^{2}+4\Delta^{2}}\right),$
($\mathrm{S}$8)
where $\Delta=\omega_{\mathrm{d}}-\omega_{\mathrm{c}}$ is the difference
between the frequency of the drive $\omega_{\mathrm{d}}$ and the cavity
resonance frequency $\omega_{\mathrm{c}}$. $\kappa$ is the total intensity
decay rate of the cavity (full width at half maximum) with
$\kappa=\kappa_{\mathrm{0}}+\kappa_{\mathrm{ex}}$. $\kappa_{\mathrm{0}}$ is
the coupling rate to the dissipative environment, and $\kappa_{\mathrm{ex}}$
is the coupling rate to the transmission line used to excite and monitor the
cavity.
The number of photons in the cavity due to a coherent input drive at detuning
$\Delta$ may be calculated from the stored energy $E$ in the cavity.
$n_{\mathrm{d}}=\frac{E}{\hbar\omega_{\mathrm{d}}}=\frac{2P_{\mathrm{i}}}{\hbar\omega_{\mathrm{d}}}\frac{\kappa_{\mathrm{ex}}}{\kappa^{2}+4\Delta^{2}}$
($\mathrm{S}$9)
For our circuit, $\kappa_{\mathrm{ex}}=2\pi\times 133$ kHz. Thus, when the
drive is optimally detuned such that $\Delta=-\Omega_{\mathrm{m}}$, the input
power required to excite the cavity with one photon is $P_{\mathrm{i}}\approx
2\hbar\omega_{\mathrm{d}}\Omega_{\mathrm{m}}^{2}/\kappa_{\mathrm{ex}}\approx
50$ fW.
## VI Fundamental limits of sideband cooling
Equation 2 in the main text gives an expression for the final occupancy of a
mechanical mode, assuming that the microwave drive is optimally detuned
($\Delta=-\Omega_{\mathrm{m}}$). This expression is only the lowest order
approximation in the small quantities $g/\Omega_{\mathrm{m}}$ and
$\kappa/\Omega_{\mathrm{m}}$. Up to second order, the final occupancy is
S_Dobrindt2008
$\displaystyle n_{\mathrm{m}}$
$\displaystyle=n_{\mathrm{m}}^{\mathrm{T}}\left(\frac{\Gamma_{\mathrm{m}}}{\kappa}\frac{4g^{2}+\kappa^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right)\left[1+\frac{g^{2}}{\Omega_{\mathrm{m}}^{2}}\frac{4g^{2}+\kappa\Gamma_{\mathrm{m}}}{4g^{2}+\kappa^{2}}\right]$
$\displaystyle+n_{\mathrm{c}}\left(\frac{4g^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right)\left[1+\frac{8g^{2}+\kappa^{2}}{8\Omega_{\mathrm{m}}^{2}}\frac{4g^{2}+\kappa\Gamma_{\mathrm{m}}}{4g^{2}}\right]$
$\displaystyle+\frac{8g^{2}+\kappa^{2}}{16\Omega_{\mathrm{m}}^{2}}.$
The last term represents the fundamental limit for sideband cooling and
demonstrates the importance of the resolved-sideband regime. For our system,
$\Omega_{\mathrm{m}}\gg\kappa,g$; and hence this last term only contributes
negligibly to the final occupancy of the mechanical mode ($<10^{-4}$ quanta).
### VI.1 Measurement imprecision and backaction
Throughout the main text and this supplementary information, we use the
“single-sided” convention for all spectral densities in which for any quantity
$A$, the mean-square fluctuations are
$\left<A^{2}\right>=\int_{0}^{\infty}S_{A}(\omega)\frac{d\omega}{2\pi}$. This
yields the familiar classical result that an oscillator coupled to a thermal
bath of temperature $T$ will experience a random force characterized by the
force spectral density $S_{F}=4k_{B}Tm\Gamma_{\mathrm{m}}$. More generally,
$S_{F}=4\hbar\Omega_{\mathrm{m}}\left(n_{\mathrm{m}}^{\mathrm{T}}+\frac{1}{2}\right)m\Gamma_{\mathrm{m}}\,,$
($\mathrm{S}$10)
where $n_{\mathrm{m}}^{\mathrm{T}}$ is the Bose-Einstein occupancy factor
given by
$n_{\mathrm{m}}^{\mathrm{T}}=[\exp(\hbar\Omega_{\mathrm{m}}/k_{B}T)-1]^{-1}$.
Independent of any convention for defining the spectral density, the
visibility of a thermal mechanical peak of given mechanical occupancy above
the noise floor of the measurement represents a direct measure of the overall
efficiency of the detection. As shown in Fig. S4, ratio of the peak height to
the white-noise background allows us to quantify the imprecision of the
measurement in units of mechanical quanta S_Teufel2009 ,
$n_{\mathrm{imp}}\equiv
S_{x}^{\mathrm{imp}}m\Omega_{\mathrm{m}}\Gamma_{\mathrm{m}}^{\prime}/(4\hbar)$.
Inspection of Eq. S6 implies
$n_{\mathrm{imp}}=\frac{1}{4\beta}\frac{\kappa}{\kappa_{\mathrm{ex}}}\frac{4g^{2}+\kappa\Gamma_{\mathrm{m}}}{4g^{2}}\left(\frac{1}{2}+n_{\mathrm{add}}^{\prime}\right)\\\
$ ($\mathrm{S}$11)
Figure S4: Measurement imprecision in units of mechanical quanta.
Once the drive is strong enough that $g\gg\sqrt{\kappa\Gamma_{\mathrm{m}}}$),
$n_{\mathrm{imp}}$ no longer decreases with increasing drive. It is precisely
because we are measuring with a detuned drive that also damps the mechanical
motion, that $n_{\mathrm{imp}}$ asymptotically approaches a constant value
S_Clerk2010 ; S_Schliesser2009 . For an ideal measurement
($\beta=1,\kappa=\kappa_{\mathrm{ex}}$, and $n_{\mathrm{add}}^{\prime}=1/2$),
$n_{\mathrm{imp}}\rightarrow 1/4$. Implicit in obtaining this optimal value
for $n_{\mathrm{add}}^{\prime}$ and hence $n_{\mathrm{imp}}$ is that all the
photons exiting the cavity are measured. Any losses between the cavity and the
detector can be modeled as a beam-splitter that only transmits a fraction
$\eta$ of the photons to the detector and adds a fraction $(1-\eta)$ of vacuum
noise. So the effective added noise $n_{\mathrm{add}}^{\prime}$ accounting for
these losses becomes
$n_{\mathrm{add}}^{\prime}=\frac{n_{\mathrm{add}}}{\eta}+\left(\frac{1-\eta}{\eta}\right)\frac{1}{2},$
($\mathrm{S}$12)
Thus, shot-noise limited detection of the photons ($n_{\mathrm{add}}=1/2$) is
a necessary, but not sufficient, condition for reaching the best possible
level of precision.
Quantum mechanics also requires that a continuous displacement measurement
must necessarily impart a force back on the measured object. For an optimally
detuned drive ($\widetilde{\Delta}=0$) in the resolved-sideband regime, this
backaction force spectral density $S_{F}^{\mathrm{ba}}$ approaches a constant
value as a function of increasing drive strength and asymptotically approaches
$S_{F}^{\mathrm{ba}}=2\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime}$.
Again, expressing the spectral density in units of mechanical quanta gives
$n_{\mathrm{ba}}\equiv
S_{F}^{\mathrm{ba}}/(4\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime})\rightarrow
1/2$.
Fundamentally, the Heisenberg limit does not restrict the imprecision
$S_{x}^{\mathrm{imp}}$ or the backaction $S_{F}^{ba}$ alone, but rather it
requires their product has a minimum value S_Braginsky1992 ; S_Clerk2010
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}^{ba}}=4\hbar\sqrt{n_{\mathrm{imp}}n_{\mathrm{ba}}}\geq\hbar.$
($\mathrm{S}$13)
An ideal cavity optomechanical system can achieve this lower limit for a
continuous measurement with a drive applied at the cavity resonance frequency.
When considering the case where the drive is instead applied detuned below the
cavity resonance ($\widetilde{\Delta}=0$), this product never reaches this
lower limit S_Clerk2010 ; S_Schliesser2009 and is at minimum
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}^{ba}}=\hbar\sqrt{2}$.
To estimate these quantities for our measurements, we can infer the total
force spectral density experienced by our oscillator as
$S_{F}^{\mathrm{total}}=4\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime}(n_{\mathrm{m}}+1/2)$.
As this total necessarily includes the backaction, we may make the most
conservative assumption that it was solely due to backaction that our
oscillator remained at finite occupancy. Hence, $n_{\mathrm{ba}}\leq
n_{\mathrm{m}}+1/2$. The low thermal occupancies attained in this work allow
us to place an upper bound on how large the backaction could possibly be, and
hence quantify our measurement in terms of approach to the Heisenberg limit.
Thus,
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}^{ba}}=4\hbar\sqrt{n_{\mathrm{imp}}n_{\mathrm{ba}}}\leq
4\hbar\sqrt{n_{\mathrm{imp}}(n_{\mathrm{m}}+1/2)}$. At $n_{\mathrm{d}}=3\times
10^{4}$, we simultaneously achieve $n_{\mathrm{m}}=0.36$ and
$n_{\mathrm{imp}}=1.9$ ($S_{F}^{\mathrm{total}}=1.6\times 10^{-34}$ N2/Hz and
$S_{x}^{\mathrm{imp}}=1.7\times 10^{-33}$ m2/Hz) yielding an upper limit on
the measured product of backaction and imprecision of $5.1~{}\hbar$. As stated
above, the best possible backaction-imprecision product is $\hbar\sqrt{2}$
when using red-detuned excitation; thus our measurement is only a factor of
$3.6$ above this limit. It may also be noted that this factor would have been
$1.8$ except that our chosen geometry losses half of the signal back to the
input ($\beta=1/2$). In future experiments, using a single-port geometry
($\beta=1$) will improve this inefficiency.
## References
* (1) Walls, D. F. & Milburn, G. J. _Quantum Optics_ (Springer, Berlin, 1994).
* (2) Marquardt, F., Chen, J. P., Clerk, A. A. & Girvin, S. M. Quantum theory of cavity-assisted sideband cooling of mechanical motion. _Phys. Rev. Lett._ 99, 093902 (2007).
* (3) Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J. Introduction to quantum noise, measurement, and amplification. _Rev. Mod. Phys._ 82, 1155–1208 (2010).
* (4) Rocheleau, T. _et al._ Preparation and detection of a mechanical resonator near the ground state of motion. _Nature_ 463, 72–75 (2010).
* (5) Agarwal, G. A. & Huang, S. Electromagnetically induced transparency in mechanical effects of light. _Phys. Rev. A_ 81, 041803 (2010).
* (6) Weis, S. _et al._ Optomechanically Induced Transparency. _Science_ 330, 1520–1523 (2010).
* (7) Teufel, J. D. _et al._ Circuit cavity electromechanics in the strong-coupling regime. _Nature_ 471, 204–208 (2011).
* (8) Castellanos-Beltran, M. A., Irwin, K. D., Hilton, G. C., Vale, L. R. & Lehnert, K. W. Amplification and squeezing of quantum noise with a tunable Josephson metamaterial. _Nature Physics_ 4, 929–931 (2008).
* (9) Teufel, J. D., Donner, T., Castellanos-Beltran, M. A., Harlow, J. W. & Lehnert, K. W. Nanomechanical motion measured with an imprecision below that at the standard quantum limit. _Nature Nanotechnology_ 4, 820–823 (2009).
* (10) Dobrindt, J. M., Wilson-Rae, I. & Kippenberg, T. J. Parametric normal-mode splitting in cavity optomechanics. _Phys. Rev. Lett._ 101, 263602 (2008).
* (11) Schliesser, A., Arcizet, O., Riviere, R., Anetsberger, G. & Kippenberg, T. J. Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit. _Nature Physics_ 5, 509–514 (2009).
* (12) Braginsky, V. B. & Khalili, F. Y. _Quantum Measurement_ (Cambridge University Press, 1992).
|
arxiv-papers
| 2011-03-10T21:28:33 |
2024-09-04T02:49:17.578601
|
{
"license": "Public Domain",
"authors": "J. D. Teufel, T. Donner, Dale Li, J. H. Harlow, M. S. Allman, K.\n Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, R. W. Simmonds",
"submitter": "John Teufel",
"url": "https://arxiv.org/abs/1103.2144"
}
|
1103.2327
|
# Device calibration impacts security of quantum key distribution
Nitin Jain nitin.jain@mpl.mpg.de Max Planck Institute for the Science of
Light, Günther-Scharowsky-Str. 1, Bau 24, 91058 Erlangen, Germany Institut
für Optik, Information und Photonik, University of Erlangen-Nuremberg,
Staudtstraße 7/B2, 91058 Erlangen, Germany Christoffer Wittmann Max Planck
Institute for the Science of Light, Günther-Scharowsky-Str. 1, Bau 24, 91058
Erlangen, Germany Institut für Optik, Information und Photonik, University of
Erlangen-Nuremberg, Staudtstraße 7/B2, 91058 Erlangen, Germany Lars Lydersen
Department of Electronics and Telecommunications, Norwegian University of
Science and Technology, NO-7491 Trondheim, Norway University Graduate Center,
NO-2027 Kjeller, Norway Carlos Wiechers Max Planck Institute for the Science
of Light, Günther-Scharowsky-Str. 1, Bau 24, 91058 Erlangen, Germany Institut
für Optik, Information und Photonik, University of Erlangen-Nuremberg,
Staudtstraße 7/B2, 91058 Erlangen, Germany Departamento de Física, Campus
León, Universidad de Guanajuato, Lomas del Bosque 103, Fracc. Lomas del
Campestre, 37150, León, Gto, México Dominique Elser Max Planck Institute for
the Science of Light, Günther-Scharowsky-Str. 1, Bau 24, 91058 Erlangen,
Germany Institut für Optik, Information und Photonik, University of Erlangen-
Nuremberg, Staudtstraße 7/B2, 91058 Erlangen, Germany Christoph Marquardt
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1, Bau
24, 91058 Erlangen, Germany Institut für Optik, Information und Photonik,
University of Erlangen-Nuremberg, Staudtstraße 7/B2, 91058 Erlangen, Germany
Vadim Makarov Department of Electronics and Telecommunications, Norwegian
University of Science and Technology, NO-7491 Trondheim, Norway University
Graduate Center, NO-2027 Kjeller, Norway Gerd Leuchs Max Planck Institute
for the Science of Light, Günther-Scharowsky-Str. 1, Bau 24, 91058 Erlangen,
Germany Institut für Optik, Information und Photonik, University of Erlangen-
Nuremberg, Staudtstraße 7/B2, 91058 Erlangen, Germany
###### Abstract
Characterizing the physical channel and calibrating the cryptosystem hardware
are prerequisites for establishing a quantum channel for quantum key
distribution (QKD). Moreover, an inappropriately implemented calibration
routine can open a fatal security loophole. We propose and experimentally
demonstrate a method to induce a large temporal detector efficiency mismatch
in a commercial QKD system by deceiving a channel length calibration routine.
We then devise an optimal and realistic strategy using faked states to break
the security of the cryptosystem. A fix for this loophole is also suggested.
###### pacs:
03.67.Hk, 03.67.Dd, 03.67.Ac, 42.50.Ex
Quantum key distribution (QKD) offers unconditionally secure communication as
eavesdropping disturbs the transmitted quantum states, which in principle
leads to the discovery of the eavesdropper Eve qc . However, practical QKD
implementations may suffer from technological and protocol-operational
imperfections that Eve could exploit in order to remain concealed revw1 ;
blckppr .
Until now, a variety of eavesdropping strategies have utilized differences
between the theoretical model and the practical implementation, arising from
(technical) imperfections or deficiencies of the components. Ranging from
photon number splitting and Trojan-horse, to leakage of information in a side
channel, time-shifting and phase-remapping, several attacks have been proposed
and experimentally demonstrated pns ; trojanH ; sidechans07 ; zhao08 ; phrmp10
. Recently, proof-of-principle attacks larsnp10 ; carlosNlars10 ; gerhardt10
based on the concept of faked states makarov05 have been presented. Eve
targets imperfections of avalanche photodiode (APD) based single-photon
detectors props that allow her to control them remotely.
Another important aspect of QKD security not yet investigated, however, is the
calibration of the devices. A QKD protocol requires a classical and a quantum
channel; while the former must be authenticated, the latter is merely required
to preserve certain properties of the quantum signals revw1 ; revw2 . The
establishment of the quantum channel remains an implicit assumption in
security proofs: channel characterization (e.g. channel length) and
calibration of the cryptosystem hardware, especially the steps involving two-
party communication, haven’t yet been taken into account. As we show, the
calibration of the QKD devices must be carefully implemented, otherwise it is
prone to hacks that may strengthen existing, or create new eavesdropping
opportunities for Eve.
Figure 1: Typical detection system in a Mach-Zehnder interferometer based QKD
implementation: The bit and basis choices of Alice and Bob (phases
$\varphi_{\rm Alice}$ and $\varphi_{\rm Bob}$) determine the interference
result at the 50:50 beam splitter (BS), or which of the two detectors D0 or D1
would click. It is thus crucial that D0 and D1 are indistinguishable to the
outside world (i.e. Eve). If gated mode APDs are employed, the detector
control board ensures that the activation of D0 and D1 (via voltage pulses
$V_{0}(t)$ and $V_{1}(t)$) happens almost simultaneously, to nullify any
existing temporal efficiency mismatch.
In this Letter, we propose and experimentally demonstrate the hacking of a
vital calibration sequence during the establishment of the quantum channel in
the commercial QKD system Clavis2 from ID Quantique clavis2guide . Eve induces
a parameter mismatch makarov06 between the detectors that can break the
security of the QKD system. Specifically, she causes a temporal separation of
the order of $450$ ps of the detection efficiencies by deceiving the detection
system, shown in Fig. 1. This allows her to control Bob’s detection outcomes
using time, a parameter already shown to be instrumental in applying a time-
shift attack (TSA) zhao08 . Alternatively, she could launch a faked-state
attack (FSA) makarov06 for which we calculate the quantum bit error rate
(QBER) under realistic conditions. Since FSA is an intercept-resend attack,
Eve has full information-theoretic knowledge about the key as long as Alice
and Bob accept the QBER at the given channel transmission $T$, and do not
abort key generation gllp . Constricting our FSA to match the raw key rate
expected by Bob and Alice, i.e. maintaining $T$ at nearly the exact pre-attack
level, we find that the security of the system is fully compromised. Our hack
has wide implications: most practical QKD schemes based on gated APDs, in both
plug-and-play and one-way configurations pnp ; pnpvar ; onewayqkd , need to
perform channel characterization and hardware calibration regularly. A careful
implementation of these steps is required to avoid leaving inadvertent
backdoors for Eve.
Figure 2: Manipulation of the calibration routine: (a) Simplified version of
Alice and Bob devices and Eve (in italic) gearing for the hack. FM: Faraday
mirror, CD: classical photodiode, DLs: delay loops, VOA: variable optical
attenuator, CR: coupler, BS: 50:50 beam splitter, PBS: polarizing beam
splitter, C: optical circulator. The hexagonal-shaped objects are phase
modulators (PMs); $\varphi_{\rm X}$, where X is Bob, Alice or Eve, represents
the applied modulation. (b) Timeline for a cycle of the hacked LLM. $V_{\pi}$:
PM voltage for a $\pi$ phase shift.
The optical setup of Clavis2 is based on the plug-and-play QKD scheme
clavis2guide ; pnp . An asymmetric Mach-Zehnder interferometer operates in a
double pass over the quantum channel by using a Faraday mirror; see Fig. 2(a)
without Eve. The interference of the paths taken by two pulses travelling from
Bob to Alice and back is determined by their relative phase modulation
($\varphi_{\rm Bob}-\varphi_{\rm Alice}$), and forms the principle for
encoding the key. Any birefringence effects of the quantum channel are
passively compensated. As a prerequisite to the key exchange, Clavis2
calibrates its detectors in time via a sequence named Line Length Measurement
(LLM). Bob emits a pair of _bright_ pulses and applies a series of detector
gates around an initial estimate of their return. The timing of the gates is
electronically scanned (while monitoring detector clicks) to refine the
estimation of the channel length and relative delay between the time of
arrival of the pulses at D0 and D1. Alice keeps her phase modulator (PM)
switched off, while Bob applies a uniform phase of $\pi/2$ to one of the
incoming pulses. Therefore, both detectors are equally illuminated and their
detection efficiencies, denoted by $\eta_{0}(t)$ and $\eta_{1}(t)$, can be
resolved in time. Any existing mismatch can thus be minimized by changing the
gate-activation times (see Fig. 1).
However, the calibration routine does not always succeed; as reported in
zhao08 , a high detector efficiency mismatch (DEM) is sometimes observed after
a normal run of LLM. For example, we have noticed a temporal mismatch as high
as $400$ ps in Clavis2. This physical limitation of the system – arising due
to fast and uncontrollable fluctuations in the quantum channel or
electromagnetic interference in the detection circuits – is the vulnerability
that the TSA exploits. However, the attack has some limitations: it is
applicable only when the temporal mismatch happens to exceed a certain
threshold value, which is merely $4\%$ of all the instances zhao08 . Also, Eve
can neither control the mismatch (as it occurs probabilistically), nor extract
its value (as it is not revealed publicly).
We exploit a weakness of the calibration routine to induce a large and
deterministic DEM without needing to extract any information from Bob. As
depicted in Fig. 2(a), Eve installs her equipment in the quantum channel such
that the laser pulse pair coming out of Bob’s short and long arm passes
through her PM. Eve’s modulation pattern is such that a rising edge in the PM
voltage flips the phase in the second (long arm) optical pulse from $-\pi/2$
to $\pi/2$, as shown in Fig. 2(b). As a result of this hack, when the pulse
pair interferes at Bob’s 50:50 beam splitter, the two temporal halves have a
relative phase difference ($\varphi_{\rm Bob}-\varphi_{\rm Eve}$) of $\pi$ and
$0$, respectively. This implies that photons from the first (second) half of
the interfering pulses yield clicks in D1 (D0) deterministically. As the LLM
localizes the detection efficiency peak corresponding to the optical power
peak, an _artificial_ temporal displacement in the detector efficiencies is
induced. An inverse displacement can be obtained by simply inverting the
polarity of Eve’s phase modulation.
In the supplementary section suppref , we describe a proof-of-principle
experiment to deceive the calibration routine. With this setup, we record the
temporal separation $\Delta_{01}$, i.e. the difference between the delays for
electronically gating D0 and D1, for several runs of LLM. Relative to the
statistics from the normal runs (denoted by $\Delta^{\rm{no\,Eve}}_{01}$), the
hacked runs yield an average shift,
$\Delta^{\rm{Eve}}_{01}-\Delta^{\rm{no\,Eve}}_{01}$ = $459$ ps with a standard
deviation of $105$ ps. Figure 3 shows the detection efficiencies $\eta_{0}(t)$
and $\eta_{1}(t)$ (measurement method explained in suppref ) for the normal
and hacked cases. It also provides a quantitative comparison between the usual
and induced mismatch. Note that a larger mismatch can be obtained by modifying
the shape of laser pulses coming from Bob.
After inducing this substantial efficiency mismatch, Eve can use an intercept-
resend strategy employing ‘faked states’ makarov05 to impose her will upon
Bob (and Alice). Compared to her intercepted measurements, she prepares the
opposite bit value in the opposite basis and sends it with such a timing that
the detection of the opposite bit value is suppressed due to negligible
detection efficiency. As an example, assume that Eve measures bit $0$ in the
$Z$ basis [in a phase-coded scheme, measuring in $Z$ $(X)$ basis
$\Leftrightarrow\text{applying }\varphi=0\,\,\left(\pi/2\right)$]. Then, she
resends bit $1$ in the $X$ basis, timed to be detected at $t=t_{0}$ (see Fig.
3), where D1 is almost blind. Using the numerical data on the induced
mismatch, Eq. $3$ from makarov06 yields a QBER $<0.5\%$ if the FSA is
launched at times $t_{0}$ and $t_{1}$ where the efficiency mismatch is high.
Figure 3: Induced temporal mismatch: Efficiencies $\eta_{0}(t)$ (dotted) and
$\eta_{1}(t)$ (dashed) from normal LLMs, on the left, and after Eve’s hack
that induced a separation of $459$ ps, on the right. The logarithm of their
ratio, quantifying the degree of mismatch (solid line), is at least an order
of magnitude higher in the flanks after Eve’s hack: the dash-dot line
indicates zero mismatch. To eavesdrop successfully, Eve times the arrival of
‘‘appropriately bright’’ faked states at $t=t_{0}$ or $t_{1}$ in Bob.
However, it can be observed that the detection probabilities for D0 and D1 are
quite low in this case. A considerable decrease in the rate of detection
events in Bob could ensue an alarm. Also, the (relatively increased) dark
counts would add significantly to the QBER. In fact, Eve needs to _match_ the
channel transmission $T$ that Alice and Bob expect, without exceeding the QBER
threshold at which they abort key generation gllp . Experimentally, we find
that the abort threshold depends on the channel loss seen by Clavis2; for an
optical loss of $1\>\\!$–$\>\\!6\>\,\text{dB}$ (corresponding to
$0.79\\!\;\\!>\\!\\!\;T\;\\!\\!>\;\\!\\!0.25$), it lies between
$5.94\>\\!$–$\>\\!8.26\%$.
$\rightarrow$Eve | Eve$\rightarrow$ | Bob’s result | Detection probability
---|---|---|---
$Z,0$ | $\,X,1,\mu_{0},t_{0}\,$ | $0$ | $\mathbf{q}_{0}=d_{0}+\left(1-d_{0}\right)\times$
| | | $\left(1-\text{exp}\left(-\mu_{0}\eta_{0}(t_{0})/2\right)\right)$
| | $1$ | $\mathbf{q}_{1}=d_{1}+\left(1-d_{1}\right)\times$
| | | $\left(1-\text{exp}\left(-\mu_{0}\eta_{1}(t_{0})/2\right)\right)$
| | $0\cap 1$ | $\mathbf{q}_{0}\mathbf{q}_{1}$
| | loss | $1-\left(\mathbf{q}_{0}+\mathbf{q}_{1}-\mathbf{q}_{0}\mathbf{q}_{1}\right)$
$X,0$ | $\,Z,1,\mu_{0},t_{0}\,$ | $0$ | $\mathbf{r}_{0}=d_{0}$
| | $1$ | $\mathbf{r}_{1}=d_{1}+\left(1-d_{1}\right)\times$
| | | $\left(1-\text{exp}\left(-\mu_{0}\eta_{1}(t_{0})\right)\right)$
| | $0\cap 1$ | $\mathbf{r}_{0}\mathbf{r}_{1}$
| | loss | $1-\left(\mathbf{r}_{0}+\mathbf{r}_{1}-\mathbf{r}_{0}\mathbf{r}_{1}\right)$
$X,1$ | $\,Z,0,\mu_{1},t_{1}\,$ | $0$ | $\mathbf{s}_{0}=d_{0}+\left(1-d_{0}\right)\times$
| | | $\left(1-\text{exp}\left(-\mu_{1}\eta_{0}(t_{1})\right)\right)$
| | $1$ | $\mathbf{s}_{1}=d_{1}$
| | $0\cap 1$ | $\mathbf{s}_{0}\mathbf{s}_{1}$
| | loss | $1-\left(\mathbf{s}_{0}+\mathbf{s}_{1}-\mathbf{s}_{0}\mathbf{s}_{1}\right)$
Table 1: Faked-state attack, given that Alice prepared bit $0$ in the $Z$
basis and that Bob measured in the $Z$ basis (only matching basis at Alice and
Bob remains after sifting). The first column contains the basis chosen by Eve
and her measurement result. The second column shows parameters of the faked
state resent by Eve: basis, bit, mean photon number, timing. The third column
shows Bob’s measurement result; $0\cap 1$ denotes a double click. The last
column shows the corresponding click probabilities (ignoring possible
superlinearity effect in gated detectors lydersen11 ). Note: The first result
$\left(\rightarrow\text{Eve}\equiv Z,0\right)$ is twice as likely to occur as
the other two.
Eve solves these problems by increasing the mean photon number of her faked
states. To evaluate her QBER, we elaborate the approach of makarov06 by
generalizing table I from this reference. Our attack strategy, carefully
accounting for all the involved factors, is summarized in Table 1. For
instance, in the first row we replace the probability of detection
$\eta_{0}(t_{0})/2$ by $1-\text{exp}\left(-\mu_{0}\eta_{0}(t_{0})/2\right)$
for a coherent-state pulse of mean photon number $\mu_{0}$ impinging on Bob’s
detectors at time $t_{0}$. Including the effect of the dark counts into this
expression, Bob’s probability to register $0$ becomes
$\mathbf{q}_{0}=d_{0}+\left(1-d_{0}\right)\left(1-\text{exp}\left(-\mu_{0}\eta_{0}(t_{0})/2\right)\right)$,
where $d_{0}$ is the dark count probability in detector D0. A row for double
clicks, i.e. simultaneous detection events in D0 and D1, is added for every
(re-sent) state.
Due to the FSA, the D0/1 click probability at time $t$ no longer depends
solely upon $\eta_{0/1}(t)$. Summing over all the states sent by Alice (by
extending Table 1), the total detection probabilities in D0 and D1 when the
attack is launched at specific times $t_{0}$ and $t_{1}$ are
$\displaystyle p$ ${}_{0}(\mu_{0},\mu_{1})=0.75+0.25d-0.25(1-d)\times$
$\displaystyle(e^{-0.5\mu_{0}\eta_{00}}+e^{-0.5\mu_{1}\eta_{01}}+e^{-\mu_{1}\eta_{01}})\,,$
(1) $\displaystyle p$ ${}_{1}(\mu_{0},\mu_{1})=0.75+0.25d-0.25(1-d)\times$
$\displaystyle(e^{-0.5\mu_{0}\eta_{10}}+e^{-0.5\mu_{1}\eta_{11}}+e^{-\mu_{0}\eta_{10}})\,.$
(2)
Here $\eta_{jk}=\eta_{j}(t_{k})$ with $j,k\in\\{0,1\\}$ and
$d=\text{mean}\left(d_{0},d_{1}\right)$ are used to simplify the expressions.
Similarly, one can compute the expression for $p_{0\cap 1}$, the total double-
click probability. Eve’s error probability, the arrival probability of the
optical signals in Bob, and the QBER are
$\displaystyle p$ ${}_{\rm error}(\mu_{0},\mu_{1})=0.75+0.25d-0.5p_{0\cap
1}-0.125\times$ (3)
$\displaystyle(1-d)\left(e^{-\mu_{0}\eta_{10}}+2e^{-0.5\mu_{0}\eta_{10}}+e^{-\mu_{1}\eta_{01}}+2e^{-0.5\mu_{1}\eta_{01}}\right),$
$\displaystyle p$ ${}_{\rm arrive}(\mu_{0},\mu_{1})=p_{0}+p_{1}-p_{0\cap
1}\,,$ (4) Q
$\displaystyle\text{BER}(\mu_{0},\mu_{1})=p_{\text{error}}(\mu_{0},\mu_{1})/p_{\text{arrive}}(\mu_{0},\mu_{1})\,.$
(5)
Here double clicks are assumed to be assigned a random bit value by Bob dblclk
, causing an error in half the cases.
Figure 4: Minimum QBER versus click probabilities in D0 and D1: Eve minimizes
the error with a suitable choice of the mean photon number of the faked states
(for this plot, $1<\mu_{0}<100$ and $21<\mu_{1}<120$ at Bob’s detectors). The
thick shaded line indicates Bob’s detection probabilities. The QBER introduced
by Eve stays below 7% for $T\gtrsim 0.25$.
If Alice and Bob are connected back-to-back (channel transmission $T\approx
1$), the click probabilities in Bob should be slightly less than half of the
peak values in Fig. 3. This is owing to optical losses ($\gtrsim
3\,\text{dB}$) in Bob’s apparatus. Eve’s constraints can now be formalized as:
starting in the vicinity of $p_{0}=0.038$ and $p_{1}=0.032$, not only does she
have to match Bob’s expected detection rate for any given $T<1$, but also keep
the resultant QBER below the threshold at which Clavis2 aborts the key
exchange. We assume Eve detects photons at Alice’s exit using a perfect
apparatus, and resends perfectly aligned faked states.
Substituting $t_{1}=-1.32$ ns, $t_{0}=1.90$ ns (marked in Fig. 3) and
$d=2.4\times 10^{-4}$ in Eqns. 1–5, Eve collects tuples
$\left[p_{0},\,p_{1},\>\text{QBER}\right]$ by varying $\mu_{0}$ and $\mu_{1}$
in a suitable range. Out of all tuples that feature the same detection
probabilities (arising from different combinations of $\mu_{0}$ and
$\mu_{1}$), Eve chooses the one having the lowest QBER. A contour plot in Fig.
4 displays this minimized error
$\min_{\mu_{0},\mu_{1}}\text{QBER}\left((\mu_{0},\mu_{1})|\left(p_{0},p_{1}\right)\right)$.
The thick shaded line shows that for $T>0.25$, Eve not only maintains the
detection rates within $5\%$ of Bob’s expected values, but also keeps the QBER
below 7% 111The QBER can be reduced even further if Bob checks only the
_overall_ detection probability $p_{0}+p_{1}$.; thus breaking the security of
the system. Note that the simulation assumes a lossless Eve, but in principle
she can cover loss from her realistic detection apparatus by increasing
$\mu_{0}$ and $\mu_{1}$ further and/or including $t_{0}$ and $t_{1}$ in the
minimization.
To counter this hack, Bob should randomly apply a phase of $0$ or $\pi$
(instead of $\pi/2$ uniformly) while performing LLM. This modification is
implementable in software and has already been proposed to ID Quantique. More
generally, a method to shield QKD systems from attacks that exploit DEM is
described in Ref. larspra11 .
In conclusion, we report a proof-of-principle experiment to induce a large
detector efficiency mismatch in a commercial QKD system by deceiving a vital
calibration routine. An optimized faked-state attack on such a compromised
system would not alarm Alice and Bob as it would introduce a QBER $<7\%$ for a
large range of expected channel transmissions. Thus, the overall security of
the system is broken. With initiatives for standardizing QKD qkdstdzn
underway, we believe this report is timely and shall facilitate elevating the
security of practical QKD systems.
Acknowledgments: We thank M. Legré from ID Quantique and N. Lütkenhaus for
helpful discussions; Q. Liu, L. Meier and A. Käppel for technical assistance.
This work was supported by the Research Council of Norway (grant no.
180439/V30), DAADppp mobility program financed by NFR (project no. 199854) and
DAAD (project no. 50727598), and BMBF CHIST-ERA (project HIPERCOM). Ca. Wi.
acknowledges support from FONCICYT project no. 94142.
## References
* (1) C. H. Bennett and G. Brassard, Proc. IEEE Int. Conf. on Computers, Systems, and Signal Processing (IEEE, New York, 1984), pp. 175–179; P. Shor and J. Preskill, Phys. Rev. Lett. 85, 441 (2000) and references therein.
* (2) V. Scarani et al., Rev. Mod. Phys. 81, 1301 (2009).
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* (5) N. Gisin et al., Phys. Rev. A 73, 022320 (2006); A. Vakhitov et al., J. Mod. Opt. 48, 2023 (2001).
* (6) A. Lamas-Linares and C. Kurtsiefer, Opt. Express 15, 9388 (2007); S. Nauerth et al., New J. Phys. 11, 065001 (2009).
* (7) Y. Zhao et al., Phys. Rev. A 78, 042333 (2008).
* (8) C.-H. F. Fung et al., Phys. Rev. A 75, 032314 (2007); F. Xu et al., New J. Phys. 12, 113026 (2010).
* (9) L. Lydersen et al., Nat. Photonics 4, 686 (2010).
* (10) L. Lydersen et al., Opt. Express 18, 27938 (2010); C. Wiechers et al., New J. Phys. 13, 013043 (2011).
* (11) I. Gerhardt et al., Nat. Comm. 2, 349 (2011).
* (12) V. Makarov and D. R. Hjelme, J. Mod. Opt. 52, 691 (2005).
* (13) V. Makarov, New J. Phys. 11, 065003 (2009).
* (14) N. Gisin et al., Rev. Mod. Phys. 74, 145 (2002).
* (15) Datasheet of Clavis2, available at ID Quantique website http://www.idquantique.com.
* (16) V. Makarov et al., Phys. Rev. A 74, 022313 (2006).
* (17) D. Gottesman et al., Quant. Inf. Comput. 4, 325 (2004); H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
* (18) L. Lydersen et al., arXiv:1106.2119.
* (19) D. Stucki et al., New J. Phys. 4, 41 (2002).
* (20) D. S. Bethune and W. P. Risk, IEEE J. Quantum Electron. 36, 340 (2000); M. Bourennane et al., Opt. Express 4, 383 (1999).
* (21) Z. Yuan and A. Shields, Opt. Express 13, 660 (2005).
* (22) See Page 5 for experimental details.
* (23) L. Lydersen et al., Phys. Rev. A 83, 032306 (2011).
* (24) ETSI GS QKD 005 V1.1.1: “Quantum key distribution (QKD); Security proofs” (ETSI, 2010); T. Länger and G. Lenhart, New J. Phys. 11, 055051 (2009).
* (25) N. Lütkenhaus, Phys. Rev. A 59, 3301 (1999).
## Appendix A Device calibration impacts security of quantum key
distribution: Technical appendix
Figure 5: Eve’s implementation ($m$Alice) by modifying Alice’s module: The
onboard pulser driving the phase modulator (PM) is disconnected, and the PM
itself is positioned _before_ the 23.5 km delay loops (DLs). The trigger
conditioner circuit allows (prevents) the pulse & delay generator to be
triggered by the short arm (long arm) optical pulses. Newly added components
to the original Alice module are labeled in italic. VOA: variable optical
attenuator, FM: Faraday mirror.
Implementation of the hack: Here, we explain our experimental implementation
of the scheme outlined in the Letter for deceiving Line Length Measurement
(LLM), the calibration routine of the Clavis2 QKD system clavis2guide . For
this purpose, we rig the module of Alice as shown in Fig. 5. From now on, we
call this manipulated device $m$Alice. An electronic tap placed on the
classical detector (normally used by Alice for measuring the incoming optical
power trojanH ) is conditioned appropriately with a homemade circuit. The
output of this circuit provides the trigger for the pulse & delay generator
(PDG, Highland Technology P400), which essentially drives the phase modulator
(PM) in $m$Alice.
For experimental convenience, we also change the settings in the Clavis2
firmware (Bob’s EEPROM specifically) such that during the execution of LLM,
$\varphi_{\rm Bob}=0$ is applied instead of the usual $\pi/2$. This relaxes
the requirement on Eve’s modulation pattern: in comparison to the waveform in
Fig. 2(b) in the Letter, the PDG needs to switch simply from $0$ to $V_{\pi}$
through the center of the optical pulse. This is in principle equivalent to
the scheme in Fig. 2(b) in the Letter, while easier to implement. In other
words, it does not affect a full implementation of Eve. Normally, Alice
applies the phase modulation in a double pass by making use of the Faraday
mirror. However, the PM in $m$Alice is shifted closer to Alice’s entrance
(i.e. before the delay loops) to enable a precise synchronization of the PDG.
To ensure that the photons passing through the PM (in a single pass now) pick
up the requisite ‘$\pi$’ modulation, a polarization controller is deployed
before the PM.
Finally, the synchronization of the rising edge of Eve’s modulation to the
center of the optical pulse is performed by scanning the delay in the PDG (in
steps of 5 ps) while monitoring the interference visibility clavis2guide . As
Eve’s modulation flips the phase of the optical pulse through the center, the
visibility reduces to zero. The corresponding delay setting of the PDG can
then be used to induce the temporal efficiency mismatch between Bob’s
detectors D0 and D1, during the execution of LLM.
We emphasize that the $m$Alice module serves as a proof-of-principle
implementation _only_ for inducing the detector efficiency mismatch during the
LLM. It should not be confused with Eve’s intercept or resend modules, needed
in the subsequent faked-state attack. Finally, note that Eve is free to modify
Bob’s pulses or replace them by her suitably-prepared pulses, and thus
effectively control the amount of detection efficiency mismatch that can be
induced.
Measurement of efficiency curves: Detection efficiencies $\eta_{0}(t)$ and
$\eta_{1}(t)$ are estimated at single-photon level by scanning the detector
gates in steps of 20 ps with an external laser (optical pulse-width $\sim 200$
ps). We average the click probability per gate and subtract $d_{0/1}$ (the
dark count rate in D0/1) from it. This gives a more accurate estimate of the
efficiencies, especially in the flanks (see Fig. 3 in the Letter).
|
arxiv-papers
| 2011-03-11T18:00:59 |
2024-09-04T02:49:17.590366
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nitin Jain, Christoffer Wittmann, Lars Lydersen, Carlos Wiechers,\n Dominique Elser, Christoph Marquardt, Vadim Makarov, Gerd Leuchs",
"submitter": "Nitin Jain",
"url": "https://arxiv.org/abs/1103.2327"
}
|
1103.2403
|
# Effects of a weakly interacting light U boson on the nuclear equation of
state and properties of neutron stars in relativistic models
Dong-Rui Zhang1, Ping-Liang Yin1, Wei Wang1, Qi-Chao Wang1, Wei-Zhou
Jiang1,2***wzjiang@seu.edu.cn 1 Department of Physics, Southeast University,
Nanjing 211189, China
2 National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
###### Abstract
We investigate the effects of the light vector U-boson that couples weakly to
nucleons in relativistic mean-field models on the equation of state and
subsequently the consequence in neutron stars. It is analyzed that the U-boson
can lead to a much clearer rise of the neutron star maximum mass in models
with the much softer equation of state. The inclusion of the U-boson may thus
allow the existence of the non-nucleonic degrees of freedom in the interior of
large mass neutron stars initiated with the favorably soft EOS of normal
nuclear matter. In addition, the sensitive role of the U-boson in the neutron
star radius and its relation to the test of the non-Newtonian gravity that is
herein addressed by the light U-boson are discussed.
U boson, equation of state, relativistic mean-field models, neutron stars
###### pacs:
26.60.Kp, 21.60.Jz, 97.60.Jd
††preprint:
## I Introduction
Confronting nuclear physics, we should highlight the great importance of the
equation of state (EOS), for it being significantly important to study the
structure of nuclei, the reaction dynamics of heavy-ion collisions, and many
issues in astrophysics lat00 ; Hor01 ; Ste05 ; Li08 . The nuclear EOS consists
usually of two ingredients: the energy density for symmetric matter and the
density dependence of the symmetry energy. For the former, the saturation
properties are quite clear nowadays, though its high-density behavior remains
to be revealed in more details. However, the density dependence of the
symmetry energy is still poorly known especially at high densities Li08 ;
Brown00 ; Ku03 ; Die03 , and even the trend of the density dependence of the
symmetry energy can be predicted to be contrary. While most relativistic
theories Hor01 ; Ste05 ; Lee98 ; to03 ; ji05 ; ji07 ; Chen07 and some non-
relativistic theories Brown00 ; Die03 ; Chen05 ; Li06 predict that the
symmetry energy increases continuously at all densities, many other non-
relativistic theories (for instance, see Brown00 ; St03 ; Chen05 ; Roy09 ), in
contrast, predict that the symmetry energy first increases, then decreases
above certain supra-saturation densities, and even in some predictions Li08 ;
Brown00 ; Ku03 becomes negative at high densities, referred as the super-soft
symmetry energy. Therefore, the experimental extraction is of necessity.
Recently, by analyzing the FOPI/GSI data on the $\pi^{-}/\pi^{+}$ radio in
relativistic heavy-ion collisions Re07 , the evidence for a super-soft
symmetry energy was found Xiao09 . This finding can result in many
consequences, while a direct challenge is how to stabilize a normal neutron
star with the super-soft symmetry energy. Conventionally, a mechanical
instability may occur if the symmetry energy starts decreasing quickly above
the certain supra-saturation density St03 ; Gle00 ; Wen09 . To solve this
problem, one possible way is to take into account the hadron-quark phase
transition which lifts up the pressure in pure quark matter Al05 , while the
transition is expected to occur at much higher densities within a narrow
region of parameters. Instead, one may consider the possible correction to the
gravity. Though the gravitational force was first discovered in the history,
it is still the most poorly characterized, compared to three other fundamental
forces that can be favorably unified within the gauge theory. For the further
grand unification of four forces, the correction to the conventional gravity
seems necessary. The light U-boson, which is proposed beyond the standard
model, can play the role in deviating from the inverse square law of the
gravity due to the Yukawa-type coupling, see Refs. Wen09 ; Ad03 ; Kr09 ; Re09
and references therein. This light U-boson was used as the interaction
propagator of the MeV dark matter and was used to account for the bright 511
keV $\gamma$-ray from the galactic bulge Bo04 ; Boe04 ; Bor06 ; Zhu07 ; Fa07 ;
Je03 . As a consequence of its weak coupling to baryons, the stable neutron
star can be obtained in the presence of the super-soft symmetry energy Wen09 .
In addition, it is noted that through the reanalysis of the FOPI/GSI data with
a different dynamical model another group extracted a contrary density
dependent trend of the symmetry energy at high densities Fen10 . The solution
of the controversy is still in progress.
In pursuit of the covariance in addressing neutron stars bound by the strong
gravity, the relativistic models are favorable to obtain the EOS, though the
fraction, arisen from the relativistic effect of fast particles in the compact
core of neutron stars, is just moderate. Apart from the non-relativistic
models to obtain the EOS of neutron stars in Ref. Wen09 , we will adopt the
relativistic mean-field (RMF) models in this work. The RMF theory which is
based on the Dirac equations for nucleons with the potentials given by the
meson exchanges achieved great success in the past few decades Wal74 ; Bog77 ;
Chin77 ; Ser86 ; Rei89 ; Ring96 ; Ser97 ; Ben03 ; Meng06 ; Ji07 . The original
Lagrangian of the RMF model was first proposed by Walecka more than 30 years
ago Wal74 . The Walecka model and its improved versions were characteristic of
the cancellation between the big attractive scalar field and the big repulsive
vector field. To soften the EOS obtained with the simple Walecka model, the
proper medium effects were accounted with the inclusion of the nonlinear self-
interactions of the $\sigma$ meson proposed by Boguta et. al. Bog77 . A few
successful nonlinear RMF models, such as NL1 Re86 , NL2 Lee86 , NL-SH Sha93 ,
NL3 La97 , and etc., had been obtained by fitting saturation properties and
ground-state properties of a few spherical nuclei. Later on, an extension to
include the self-interaction of $\omega$ meson was implemented to obtain RMF
potentials which were required to be consistent with the Dirac-Brueckner self-
energies Su94 . In this direction, besides the early model TM1 Su94 , there
were recent versions PK1 Lo04 and FSUGold Pie05 .
Although various RMF models reproduce successfully the saturation properties
of nuclear matter and structural properties of finite nuclei, the
corresponding EOS’s may behave quite differently at high densities especially
in isospin-asymmetric nuclear matter. It was reported in the literature Kr09 ;
Wen09 that the light U-boson can significantly modify the EOS in isospin-
asymmetric matter. However, the further systematic work to analyze the effect
of the light U-boson on various nuclear EOS’s is still absent. In this work,
we will investigate in detail the effect of light U-boson on the EOS and
properties of neutron stars with various RMF models. In particular, we will
address the difference of the effects induced by the U-boson in various RMF
models.
The paper is organized as follows. In Sec. II, we present briefly the
formalism based on the Lagrangian of the relativistic mean-field models. In
Sec. III, numerical results and discussions are presented. At last, a summary
is given in Sec. IV.
## II Formalism
In the RMF approach, the nucleon-nucleon interaction is usually described via
the exchange of three mesons: the isoscalar meson $\sigma$, which provides the
medium-range attraction between the nucleons, the isoscalar-vector meson
$\omega$, which offers the short-range repulsion, and the isovector-vector
meson $b_{0}$, which accounts for the isospin dependence of the nuclear force.
The relativistic Lagrangian can be written as:
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle{\overline{\psi}}[i\gamma_{\mu}\partial^{\mu}-M+g_{\sigma}\sigma-
g_{\omega}\gamma_{\mu}\omega^{\mu}-g_{\rho}\gamma_{\mu}\tau_{3}b_{0}^{\mu}]\psi$
(1)
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}b_{0\mu}b_{0}^{\mu}$
$\displaystyle+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})+U_{\rm eff}(\sigma,\omega,b_{0})+{\cal L}_{u},$
where $\psi,\sigma,\omega$,$b_{0}$ are the fields of the nucleon, scalar,
vector, and neutral isovector-vector mesons, with their masses
$M,m_{\sigma},m_{\omega}$, and $m_{\rho}$, respectively.
$g_{i}(i=\sigma,\omega,\rho)$ are the corresponding meson-nucleon couplings.
$F_{\mu\nu}$ and $B_{\mu\nu}$ are the strength tensors of $\omega$ and $\rho$
mesons respectively,
$F_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu},\hbox{
}B_{\mu\nu}=\partial_{\mu}b_{0\nu}-\partial_{\nu}b_{0\mu}.$ (2)
The self-interacting terms of $\sigma$, $\omega$ mesons and the isoscalar-
isovector coupling are given generally as
$\displaystyle U_{\rm eff}(\sigma,\omega^{\mu},b_{0}^{\mu})$ $\displaystyle=$
$\displaystyle-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}+\frac{1}{4}c_{3}(\omega_{\mu}\omega^{\mu})^{2}$
(3)
$\displaystyle+4\Lambda_{V}g^{2}_{\rho}g_{\omega}^{2}\omega_{\mu}\omega^{\mu}b_{0\nu}b_{0}^{\nu}.$
Here, the isoscalar-isovector coupling term is introduced to modify the
density dependence of the symmetry energy Hor01 . In addition, we include in
Lagrangian ${\cal L}_{u}$ for the U-boson that is beyond the standard model. A
very light U-boson can be utilized to interpret the deviation from the
Newton’s gravitational potential which is usually characterized in the form
Wen09 ; Kr09 :
$V(r)=-\frac{G_{\infty}m_{1}m_{2}}{r}(1+\alpha e^{-r/\lambda})$ (4)
where $G_{\infty}$ is the universal gravitational constant,
$\alpha=-g^{2}_{u}/4\pi G_{\infty}M_{B}^{2}$ is a dimensionless strength
parameter with $g_{u}$ and $M_{B}$ being the boson-nucleon coupling constant
and baryon mass, respectively, and $\lambda=1/m_{u}$ is the length scale with
$m_{u}$ being the boson mass. According to the conventional view, the Yukawa-
type correction to the Newtonian gravity resides at the matter part rather
than the geometric part. Thus, following the form of the vector meson, ${\cal
L}_{u}$ is written as:
$\displaystyle{\cal L}_{u}$ $\displaystyle=$
$\displaystyle-{\overline{\psi}}g_{u}\gamma_{\mu}u^{\mu}\psi-\frac{1}{4}U_{\mu\nu}U^{\mu\nu}+\frac{1}{2}m_{u}^{2}u_{\mu}u^{\mu},$
(5)
with $u$ the field of U-boson. $U_{\mu\nu}$ is the strength tensor of U-boson,
$U_{\mu\nu}=\partial_{\mu}u_{\nu}-\partial_{\nu}u_{\mu}.$ (6)
With the standard Euler-Lagrange formala, we can deduce from the Lagrangian
the equations of motion for the nucleon and mesons. They are given as follows:
$[i\gamma_{\mu}\partial^{\mu}-M+g_{\sigma}\sigma-
g_{\omega}\gamma_{\mu}\omega^{\mu}-g_{u}\gamma_{\mu}u^{\mu}-g_{\rho}\gamma_{\mu}\tau_{3}b_{0}^{\mu}]\psi=0$
(7)
$\displaystyle(\partial_{t}^{2}-\bigtriangledown^{2}+m_{\sigma}^{2})\sigma$
$\displaystyle=$ $\displaystyle g_{\sigma}{\overline{\psi}}\psi-
g_{2}\sigma^{2}-g_{3}\sigma^{3},$ (8)
$\displaystyle(\partial_{t}^{2}-\bigtriangledown^{2}+m_{\omega}^{2})\omega_{\mu}$
$\displaystyle=$ $\displaystyle g_{\omega}{\overline{\psi}}\gamma_{\mu}\psi-
c_{3}\omega_{\mu}^{3}$ (9)
$\displaystyle-8\Lambda_{V}g_{\rho}^{2}g_{\omega}^{2}b_{0\nu}b_{0}^{\nu}\omega_{\mu},$
$\displaystyle(\partial_{t}^{2}-\bigtriangledown^{2}+m_{\rho}^{2})b_{0\mu}$
$\displaystyle=$ $\displaystyle
g_{\rho}{\overline{\psi}}\gamma_{\mu}\tau_{3}\psi$ (10)
$\displaystyle-8\Lambda_{V}g_{\rho}^{2}g_{\omega}^{2}\omega_{\nu}\omega^{\nu}b_{0\mu},$
$\displaystyle(\partial_{t}^{2}-\bigtriangledown^{2}+m_{u}^{2})u_{\mu}$
$\displaystyle=$ $\displaystyle g_{u}{\overline{\psi}}\gamma_{\mu}\psi.$ (11)
In the mean-field approximation, all derivative terms drop out and the
expectation values of space-like components of vector fields vanish (only zero
components survive) due to translational invariance and rotational symmetry of
the nuclear matter. In addition, only the third component of isovector fields
survives because of the charge conservation. In the mean-field approximation,
after the Dirac field of nucleons is quantized Ser86 , the fields of mesons
and U-boson, which are replaced by their classical expectation values, obey
following equations:
$\displaystyle m_{\sigma}^{2}\sigma$ $\displaystyle=$ $\displaystyle
g_{\sigma}\rho_{s}-g_{2}\sigma^{2}-g_{3}\sigma^{3},$ (12) $\displaystyle
m_{\omega}^{2}\omega_{0}$ $\displaystyle=$ $\displaystyle
g_{\omega}\rho_{B}-c_{3}\omega_{0}^{3}-8\Lambda_{V}g_{\rho}^{2}g_{\omega}^{2}b_{0}^{2}\omega_{0},$
(13) $\displaystyle m_{\rho}^{2}b_{0}$ $\displaystyle=$ $\displaystyle
g_{\rho}\rho_{3}-8\Lambda_{V}g_{\rho}^{2}g_{\omega}^{2}\omega_{0}^{2}b_{0},$
(14) $\displaystyle m_{u}^{2}u_{0}$ $\displaystyle=$ $\displaystyle
g_{u}\rho_{B},$ (15)
where $\rho_{s}$ and $\rho_{B}$ are the scalar and baryon densities,
respectively, and $\rho_{3}$ is the difference between the proton and neutron
densities, namely, $\rho_{3}=\rho_{p}-\rho_{n}$. The set of coupled equations
can be solved self-consistently using the iteration method. With these mean-
field quantities, the resulting energy density $\varepsilon$ and pressure $P$
are written as:
$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\sum_{i=p,n}\frac{2}{(2\pi)^{3}}\int^{k_{F_{i}}}d^{3}\\!kE^{*}_{i}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}\frac{g_{u}^{2}}{m_{u}^{2}}\rho_{B}^{2}$
(16)
$\displaystyle+\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\rho}^{2}b_{0}^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}$
$\displaystyle+\frac{3}{4}c_{3}\omega_{0}^{4}+12\Lambda_{V}g^{2}_{\rho}g_{\omega}^{2}\omega_{0}^{2}b_{0}^{2},$
$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{1}{3}\sum_{i=p,n}\frac{2}{(2\pi)^{3}}\int^{k_{F_{i}}}d^{3}\\!k\frac{{\bf
k}^{2}}{E^{*}_{i}}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}\frac{g_{u}^{2}}{m_{u}^{2}}\rho_{B}^{2}$
(17)
$\displaystyle-\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\rho}^{2}b_{0}^{2}-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}$
$\displaystyle+\frac{1}{4}c_{3}\omega_{0}^{4}+4\Lambda_{V}g^{2}_{\rho}g_{\omega}^{2}\omega_{0}^{2}b_{0}^{2},$
with $E^{*}_{i}=\sqrt{{\bf k}^{2}+(M^{*}_{i})^{2}}$.
Given above is the formalism for nuclear matter without considering the
$\beta$ equilibrium. For asymmetric nuclear matter at $\beta$ equilibrium, the
chemical equilibrium and charge neutrality conditions need to be additionally
considered, which are written as:
$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\mu_{p}+\mu_{e},$ (18)
$\displaystyle\rho_{e}$ $\displaystyle=$ $\displaystyle\rho_{p},$ (19)
$\displaystyle\rho_{B}$ $\displaystyle=$ $\displaystyle\rho_{n}+\rho_{p},$
(20)
where $\mu_{n},\mu_{p},\mu_{e}$ are the chemical potential of neutron, proton
and electron, respectively, and $\rho_{e}$ is the number density of electrons.
In neutron star matter, the EOS is obtained by adding in Eqs.(16) and (17) the
contribution of the free electron gas.
The neutron star properties are obtained from solving the Tolman-Oppenheimer-
Volkoff (TOV) equation Op39 ; Tol39 :
$\displaystyle\frac{dP(r)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{[P(r)+\varepsilon(r)][M(r)+4\pi r^{3}P(r)]}{r(r-2M(r))},$
(21) $\displaystyle M(r)$ $\displaystyle=$ $\displaystyle
4\pi\int^{r}_{0}d\\!\tilde{r}\tilde{r}^{2}\varepsilon(\tilde{r}),$ (22)
where $r$ is the radial coordinate from the center of the star, $P(r)$ and
$\varepsilon(r)$ are the pressure and energy density at position $r$,
respectively, and $M(r)$ is the mass contained in the sphere of the radius
$r$. Note that here we use units for which the gravitation constant is
$G_{\infty}=c=1$. The radius $R$ and mass $M(R)$ of a neutron star are
obtained from the condition $p(R)=0$. Because the neutron star matter,
consisting of neutrons, protons, and electrons (npe) at $\beta$ equilibrium in
this work, undergoes a phase transition from the homogeneous matter to the
inhomogeneous matter at the low density region, the RMF EOS obtained from the
homogeneous matter does not apply to the low density region. For a thorough
description of neutron stars, we thus adopt the empirical low-density EOS in
the literature Ba71 ; Ii97 .
## III Results and discussions
Among a number of nonlinear RMF parametrizations, we select several typical
best-fit parameter sets, for instance NL1 Re86 , NL-SH Sha93 , NL3 La97 , TM1
Su94 and FSUGold Pie05 , to investigate the effects of the U-boson on the EOS
of isospin-asymmetric nuclear matter and properties of neutron stars. The
nonlinear RMF models usually include the nonlinear self-interactions of the
$\sigma$ meson to simulate appropriate medium dependence of the strong
interaction. This is typical in RMF parameter sets NL1, NL-SH and NL3. In
addition to the nonlinear $\sigma$ meson self-interactions, in TM1 and FSUGold
the nonlinear self-interaction of the $\omega$ meson is also included.
Parameters and saturation properties of these parameter sets are listed in
Table 1.
Table 1: Parameters and saturation properties for various parameter sets. Here, the NL3$\Lambda_{V}$ is the same as the original parameter set NL3 but with the readjusted $g_{\rho}$ after the $\Lambda_{V}$ is included to modify the density dependence of the symmetry energy, and the TM1$\Lambda_{V}$ to the TM1 is the same as the NL3$\Lambda_{V}$ to the NL3. Meson masses, incompressibility and symmetry energy are in units of MeV, and the density is in unit of $fm^{-3}$. | $g_{\sigma}$ | $g_{\omega}$ | $g_{\rho}$ | $m_{\sigma}$ | $m_{\omega}$ | $m_{\rho}$ | $g_{2}$ | $g_{3}$ | $c_{3}$ | $\Lambda_{V}$ | $\rho_{0}$ | $\kappa$ | $M^{*}/M$ | $E_{sym}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
NL1 | 10.138 | 13.285 | 4.976 | 492.250 | 795.359 | 763 | 12.172 | -36.265 | - | - | 0.153 | 211.3 | 0.57 | 43.7
NL-SH | 10.444 | 12.945 | 4.383 | 526.059 | 783.000 | 763 | 6.910 | -15.834 | - | - | 0.146 | 355.4 | 0.60 | 36.1
NL3 | 10.217 | 12.868 | 4.474 | 508.194 | 782.501 | 763 | 10.431 | -28.890 | - | - | 0.148 | 271.8 | 0.60 | 37.4
TM1 | 10.029 | 12.614 | 4.632 | 511.198 | 783.000 | 770 | 7.233 | 0.618 | 71.31 | - | 0.145 | 281.2 | 0.63 | 36.9
FSUGold | 10.592 | 14.302 | 5.884 | 491.500 | 782.500 | 763 | 4.277 | 49.934 | 418.39 | 0.03 | 0.148 | 230.0 | 0.61 | 32.5
NL3$\Lambda_{V}$ | 10.217 | 12.868 | 5.664 | 508.194 | 782.501 | 763 | 10.431 | -28.890 | - | 0.03 | 0.148 | 271.8 | 0.60 | 31.8
TM1$\Lambda_{V}$ | 10.029 | 12.614 | 5.720 | 511.198 | 783.000 | 770 | 7.233 | 0.618 | 71.31 | 0.03 | 0.145 | 281.2 | 0.63 | 32.1
Figure 1: Energy density $\varepsilon$ (upper panel) and pressure P (lower
panel) as a function of density with various RMF parameter sets, NL3, NL1, NL-
SH, TM1, and FSUGold in npe matter at $\beta$ equilibrium.
In Fig. 1, the energy density and pressure of npe matter at $\beta$
equilibrium are shown as a function of nucleon density for various models
without the inclusion of the U-boson. It is seen that the EOS with parameter
sets TM1 and FSUGold is clearly softer than that with the NL1, NL-SH and NL3
with the increase of the density. The softening stems from the inclusion of
the nonlinear self-interaction of the $\omega$ meson that lowers the repulsion
provided by the $\omega$ meson at high densities, while the excess softening
with the FSUGold as compared to that with the TM1 can be attributed dominately
to the larger parameter $c_{3}$ in FSUGold.
Figure 2: The correlation between the pressure and the energy density in npe
matter at $\beta$ equilibrium with various RMF models.
Shown in Fig. 2 is the correlation between the pressure and the energy density
given in Fig. 1. This correlation is usually regarded as the EOS that is used
as the input of the Tolman-Oppenheimer-Volkoff (TOV) equation Op39 ; Tol39
for the evaluation of the neutron star properties. Once again, we see the
large deviations in the EOS with different RMF models especially at high
densities. In the following, it is thus interesting to see how the U-boson
affects the EOS produced by various RMF models that differs largely at high
densities.
Figure 3: Equation of state of neutron star matter with three RMF models, NL3,
NL1 and NL-SH with the inclusion of the U-boson. The numbers in the legend are
the values of $(g_{u}/m_{u})^{2}$ in units of $GeV^{-2}$. Figure 4: The same
as in Fig. 3 but for the RMF models TM1 and FSUGold.
In the RMF approximation, the contribution of the U-boson in a linear form is
just decided by the ratio of the coupling constant to its mass, i.e.,
$g_{u}/m_{u}$, as seen in Eqs.(16) and (17). In Figs. 3 and 4, the EOS’s with
various models are depicted for a set of ratios $(g_{u}/m_{u})^{2}$. It is
shown in Figs. 3 and 4 that the inclusion of the U-boson stiffens the EOS.
This is physically obvious since the vector form of the U-boson provides an
excess repulsion in addition to the vector mesons, whereas an interestingly
large difference appears for different types of models. As shown in Figs. 3
and 4, the EOS’s with the TM1 and FSUGold acquires a much more apparent
stiffening than that with the NL1, NL-SH and NL3 by including the U-boson.
This phenomenon can be understood by the inherent feature of these models. In
models NL1, NL-SH and NL3, the repulsion is quadratic in the density because
the nonlinear self-interaction of the $\omega$ meson is not considered. With
the increase of the density, the repulsion provided by the $\omega$ meson
dominates the attraction provided by the $\sigma$ meson. The cancellation
between the repulsion and attraction in the pressure (see Eq.(17) is not
prominent at high densities so that the U-boson plays a similar role in the
energy density and pressure. Thus, these EOS’s are just moderately modified by
the U-boson, as shown in Fig. 3. For models TM1 and FSUGold that feature a
clearly softer EOS at high densities, the cancellation between the repulsion
and attraction becomes significant and thus sharpens the importance of the
U-boson in the pressure. Comparing to the addition of the big repulsion and
attraction in the energy density, the U-boson just plays a marginal role in
modifying the energy density. Thus, the U-boson can modify appreciably the
correlation between the pressure and energy density in the high-density region
in favorably softened models, for instance, the TM1 and FSUGold, as shown in
Fig. 4. Because in TM1 and FSUGold the nonlinear term of the $\omega$ meson
plays a decisive role in softening the EOS, the larger the parameter $c_{3}$,
the more apparent the modification, as shown comparatively in the upper and
lower panels of Fig. 4.
Figure 5: (Color online) The same as in Fig. 3 but to exhibit the difference
between the cases with and without the modification to the symmetry energy.
Left panels represent the results with the NL3 and NL3$\Lambda_{V}$, and right
panels are the results with the TM1 and TM1$\Lambda_{V}$. Different density
dependencies of the symmetry energy are drawn in the insets of upper panels,
while given in the insets of lower panels are the EOS of two cases in the
absence of the U-boson.
In addition, it is interesting to examine whether the significant difference
in the U-boson-induced modification to the EOS can be created by softening the
symmetry energy. The symmetry energy is softened by including the isoscalar-
isovector coupling term in RMF models (see Eq.(3)). In Fig. 5, we depict the
EOS without (upper panels) and with (lower panels) the softening of the
symmetry energy in NL3 and TM1. However, no visible difference in two cases
with the NL3 is observed, and with the TM1 the difference is not significant.
This observation seems to show a contrast with that in Ref. Wen09 where the
fluffy EOS due to the super-soft symmetry energy can be lifted up by the
U-boson to support a normal neutron star. In deed, the magnitude of the
modification to the EOS caused by the U-boson relies on the softness of the
EOS. As long as the EOS is modified significantly by softening the symmetry
energy, the stiffening role of the U-boson in the EOS can be considerably
enhanced accordingly. Given that the stiff EOS with the NL3 is little modified
by softening the symmetry energy, as shown in the inset of the left lower
panel in Fig. 5, the softening of the symmetry energy can scarcely affect the
role of the U-boson. For models with a softer EOS, the situation can turn out
to be different when the EOS is modified appreciably by softening the symmetry
energy. Indeed, the vital role of the U-boson in the EOS of the non-
relativistic MDI model with a super-soft symmetry energy Wen09 is a typical
case that the role of the U-boson can be largely amplified due to the
softening of the symmetry energy. In RMF models, for instance, the TM1 whose
EOS is softer than that with the NL3, the softening of the symmetry energy can
also result in some visible difference in the EOS and thereby the role of the
U-boson, as shown in right panels of Fig. 5.
Figure 6: The mass-radius relation of neutron stars with various models. The
U-boson is included with various ratio parameters of $(g_{u}/m_{u})^{2}$.
Next, we turn to the consequences in hydrostatic neutron stars with the EOS
modified by the U-boson. Using Eqs.(21) and (22), the mass and radius of
hydrostatic neutron stars can be obtained with the given EOS. In Fig. 6, the
mass-radius (M-R) relation of neutron stars is depicted with different ratio
parameter $(g_{u}/m_{u})^{2}$ for the U-boson in various models. With the
inclusion of the U-boson, we can see that both the maximum mass and radius of
neutron stars increase significantly. It is clearly seen that the star maximum
mass with the soft EOS is modified more significantly by the U-boson. This is
consistent with the corresponding modification to the high-density EOS caused
by the U-boson, as shown in Figs. 3 and 4. The consistency is established on
the fact that the maximum mass of neutron stars is dominated by the high-
density behavior of the EOS. In the past, a few neutron stars with large
masses around $2M_{\odot}$ had been observed Ni05 ; Ni08 ; Oz06 . Though it
can have improvements in experimental aspects, the observation of neutron
stars with large masses is not so scarce. Recently, the mass of the LMXB
4U1608-52 is measured to be 1.74$M_{\odot}$ Gu10 , and most recently a
$2M_{\odot}$ neutron star J1614-2230 was measured through the Shapiro delay
De10 . Note that the model FSUGold which is well consistent with the nuclear
laboratory constraints just produces a maximum mass about 1.7$M_{\odot}$ for
the neutron star without hyperons, whereas the hyperonization can further
reduce the maximum mass to a value below $1.4M_{\odot}$. In this case, the
role of the U-boson is constructive in increasing the maximum mass of neutron
stars, either as the EOS is softened by the creation of new degrees of
freedom, or the EOS is too soft to obtain a large maximum mass.
On the other hand, the radius of neutron stars is primarily determined by the
EOS in the lower density region of $1\rho_{0}$ to $2\rho_{0}$, see Refs.lat00
; Li08 and references therein. Because the symmetry energy in this density
region offers the most important ingredient of the pressure in pure neutron
matter, the density dependence of the symmetry energy plays a crucial role in
determining the radius of neutron stars. While in the present case the
pressure in the lower density region is increased appreciably by the U-boson,
it is not surprising that the sensitive variation of the neutron star radius
is obtained accordingly. This is similar to the non-relativistic case in Ref.
Wen09 . In fact, the radius of neutron stars relies sensitively on the
stiffness of the EOS. Thus, the stiffening of the EOS caused by the U-boson
gives rise to a significant increase of the radius. Concretely, we can see
from Fig. 6 that the larger rise of the radius comes up with the more apparent
stiffening role of the U-boson in softer models. It is known that the radius
of neutron stars extracted from the observation can have a wide range due to
the uncertainties of the distance measurement and theoretical models used for
the spectrum analyses lat00 ; Ha01 ; Lib06 ; Zh07 . A more precise extraction
of the neutron star radius, probably through the coincident measurements, thus
becomes very significant, because it can test the non-Newtonian gravity due to
its promising sensitivity to the star radius.
Figure 7: Mass-radius relations for various models with the
$(g_{u}/m_{u})^{2}=0GeV^{-2}$ (left panel) and the
$(g_{u}/m_{u})^{2}=100GeV^{-2}$ (right panel).
To stress the role of the U-boson in the maximum mass and radius of neutron
stars, we depict in Fig. 7 the M-R relation for various models with and
without the U-boson. Here, for the case with the inclusion of the U-boson, the
calculation is performed with $(g_{u}/m_{u})^{2}=100GeV^{-2}$. It is seen
clearly that the large difference in maximum masses with various types of
models can be reduced largely by the U-boson with suitable parameter
$(g_{u}/m_{u})^{2}$. We can see once again that the reduction of the
difference is mainly attributed to the role of the U-boson in the models
featuring much softer EOS’s. Interestingly, we see that the uncertainty of the
radius for a canonical neutron star (with the mass $1.4M_{\odot}$) can also be
reduced by the U-boson.
In view of interesting and significant roles of the U-boson, we may say that
the task to look for the U-boson and further confirm the non-Newtonian gravity
is also confronted. The recent experimental constraints on the relationship
between parameters $\alpha$ ($g_{u}$) and $\lambda$ ($m_{u}$) can be found in
Ref. Kr09 . To recover the stability of neutron stars using the EOS
constrained by the FOPI/GSI data Xiao09 , the ratio $(g_{u}/m_{u})^{2}\sim
100GeV^{-2}$ was found to be needed Wen09 . In this work, the effect of the
U-boson is investigated within the parameter region $(g_{u}/m_{u})^{2}=0\sim
100GeV^{-2}$. To avoid the visible effect beyond low energy constraints in
finite nuclei, with these values of the ratio parameter we may estimate that
the mass of the U-boson should be of order below $1MeV$ with the coupling
strength being almost or at least three orders less than the fine-structure
constant, while these estimated orders can be compatible with parameter
regions allowed by a few experimental constraints, see Ref. Kr09 . We expect
that more precision experiments will be performed to better determine or
exclude the parameter regions for the non-Newtonian gravity.
At last, it is interesting to discuss the relevance between the parameters of
the non-Newtonian gravity touched upon in this work and the solution to the
dark matter problem. In order to explain the flatness of the rotational curve
of galactic spirals, one needs to assume the non-luminous dark matter being
the additional gravitational source. Alternatively, the Newtonian gravity that
was well tested in the solar system may be assumed to fail at the large
distance scales of galaxies, and hence the Newtonian gravity should be
modified to be the non-Newtonian one Man06 . The Yukawa-type modification to
the Newtonian gravity due to the boson exchange may possibly be considered as
a candidate to solve the dark matter problem. In this work, the vector
coupling of the U-boson that is restrained by the U(1) symmetry produces a
repulsion other than the anticipated attraction. We may thus suppose to solve
the dark matter problem through the introduction of light scalar bosons.
However, since the flatness of the rotational curve requires a supplemental
force roughly linear inversely in the distance from the center of the galaxy,
even if the light scalar boson is assumed to provide the needed attraction in
one region, the exponential suppression factor of the Yukawa-type potential
(see Eq.(4)) actually inhibits the reproduction of the rotational curve in
other regions. In deed, in addition to the introduction of the light scalar
boson, more considerations are necessary to solve the dark matter problem Mb04
. On the other hand, we may explore the constraints from the effect of the
U-boson on the dark matter. However, the coupling of the U-boson with the dark
matter candidates should be assumed to be much stronger than that with the
normal particles to explain the $511keV$ $\gamma$-ray observation while
simultaneously compatible with the low-energy constraints Bo04 ; Boe04 ; Bor06
; Zhu07 . To sum up, we are presently not able to restrain the parameters of
the non-Newtonian gravity originated from the U-boson exchange in this work
directly by using the effect of the U-boson on the dark matter and/or the
solution to the dark matter problem with the modified Newtonian dynamics.
Nonetheless, this deserves further exploration. For instance, the further
first-principle understanding of the underlying origin of the difference in
the U-boson couplings to normal and dark matter particles may open possibility
to extract constraints on the parameters of the non-Newtonian gravity.
## IV Summary
We have studied in this work the effects of the U-boson in RMF models on the
equation of state and subsequently the consequence in neutron stars. All RMF
models are chosen to have similarly nice reproduction of saturation properties
and ground-state properties of finite nuclei, whereas they can give rise to a
significantly large difference in EOS’s at high densities and mass-radius
relations of neutron stars. Interestingly, we find that the U-boson in models
with much softer EOS plays a much more significant role in increasing the
maximum mass of neutron stars. The distinction can be attributed analytically
to the different modification caused by the U-boson in soft and stiff models
to the pressure. Thus, the inclusion of the U-boson may allow the existence of
the non-nucleonic degrees of freedom in the interior of large mass neutron
stars initiated with the favorably soft EOS of normal nuclear matter. In
addition, it is worth notifying that the radius of canonical neutron stars in
all models can be sensitively modified by the U-boson due to its stiffening
role in the EOS. Meanwhile, the difference in the mass-radius relations
predicted by various models can favorably be reduced by increasing the
coupling strength between the U-boson and baryons. At last, constraints on the
parameters of the non-Newtonian gravity are discussed. Presently, we have not
found the direct relevance between the parameters of the non-Newtonian gravity
originated from the U-boson exchange and its effect on the dark matter
concerning the dark matter problem. Together with the future coincident
measurements and more precise extraction of the mass and radius of neutron
stars, the sensitive role of the U-boson in the M-R relation may be helpfully
used to test the physics beyond the standard model and consequently the
existence of the non-Newtonian gravity in the dense neutron star.
## Acknowledgement
Authors thank Professors De-Hua Wen, Lie-Wen Chen and Bao-An Li for useful
discussions. The work was supported in part by the SRTP Grant of the
Educational Ministry of China, the National Natural Science Foundation of
China under Grant No. 10975033, the China Jiangsu Provincial Natural Science
Foundation under Grant No.BK2009261, and the China Major State Basic Research
Development Program under Contract No. 2007CB815004.
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|
arxiv-papers
| 2011-03-12T00:45:58 |
2024-09-04T02:49:17.598930
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dong-Rui Zhang, Ping-Liang Yin, Wei Wang, Qi-Chao Wang, Wei-Zhou Jiang",
"submitter": "Dong-rui Zhang",
"url": "https://arxiv.org/abs/1103.2403"
}
|
1103.2436
|
# Finite temperature calculations for the bulk properties of strange star
using a many-body approach
G.H. Bordbar 1,2111Corresponding author 222E-Mail: bordbar@physics.susc.ac.ir,
A. Poostforush 1 and A. Zamani 1 1Department of Physics, Shiraz University,
Shiraz 71454, Iran333Permanent address,
and
2Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
We have considered a hot strange star matter, just after the collapse of a
supernova, as a composition of strange, up and down quarks to calculate the
bulk properties of this system at finite temperature with the density
dependent bag constant. To parameterize the density dependent bag constant, we
use our results for the lowest order constrained variational (LOCV)
calculations of asymmetric nuclear matter. Our calculations for the structure
properties of the strange star at different temperatures indicate that its
maximum mass decreases by increasing the temperature. We have also compared
our results with those of a fixed value of the bag constant. It can be seen
that the density dependent bag constant leads to higher values of the maximum
mass and radius for the strange star.
Keywords: Strange star, equation of state, structure, density dependent bag
constant
## I Introduction
Strange stars are those which are built mainly from self bound quark matter.
The surface density of strange star is equal to the density of strange quark
matter at zero pressure ($\sim 10^{15}\ g/cm^{3}$), which is fourteen orders
of magnitude greater than the surface density of a normal neutron star. The
central density of these stars is about five times greater than the surface
density haensel ; glendening ; weber . The existence of strange stars which
are made of strange quark matter was first proposed by Itoh a even before the
full developments of QCD. Later Bodmer b discussed the fate of an
astronomical object collapsing to such a state of matter. In 1970s, after the
formulation of QCD, the perturbative calculations of the equation of state of
the strange quark matter was developed, but the region of validity of these
calculations was restricted to very high densities collins . The existence of
strange stars was also discussed by Witten c . He conjectured that a first
order QCD phase transition in the early universe could concentrate most of the
quark excess in dense quark nuggets. He suggested that the true state of
matter was strange quark matter. Based on theoretical works of Witten on
cosmic separation of phases, the transition temperature is approximately $100\
MeV$, an acceptable QCD temperature c . Witten proposal was that the strange
quark matter composed of light quarks is more stable than nuclei, therefore
strange quark matter can be considered as the ground state of matter. The
strange quark matter would be the bulk quark matter phase consisting of almost
equal numbers of up, down and strange quarks plus a small number of electrons
to ensure the charge neutrality. A typical electron fraction is less than
$10^{-3}$ and it decreases from the surface to the center of strange star
haensel ; glendening ; weber . Strange quark matter would have a lower charge
to baryon ratio compared to the nuclear matter and can show itself in the form
of strange stars c ; d ; e ; f .
Just after the collapse of a supernova, a hot strange star may be formed. A
strange star may be also formed from a neutron star and is denser than the
neutron star. If sufficient additional matter is added to a strange star, it
will collapse into a black hole. Neutron stars with masses of
$1.5-1.8M_{\odot}$ with rapid spins are theoretically the best candidates for
conversion to the strange stars. An extrapolation based on this indicates that
up to two quark-novae occur in the observable universe each day. Besides,
recent Chandra observations indicate that objects RX J185635-3754 and 3C58 may
be bare strange stars prakash .
In this article, we consider a hot strange star born just after the collapse
of a supernova. Here we ignore the effects of the presence of electrons, and
consider a strange star purely made up of the quark matter consisting of the
up, down and strange quarks. The energy of quark matter is calculated at
finite temperature, and then its equation of state is derived. Finally using
the equation of state of quark matter, the structure of strange star at
different temperatures is computed by integrating the Tolman-Oppenheimer-
Volkoff (TOV) equations.
## II Calculation of Quark Matter Equation of State
### II.1 Density Dependent Bag Constant
Different models have been used for deriving the equation of state of quark
matter. Therefore there is a great variety of the equations of state for this
system. The model which we use is the MIT bag model which was developed to
take into account the non perturbative effects of quark confinement by
introducing the bag constant. In this model, the energy per volume for the
quark matter is equal to the kinetic energy of the free quarks plus a bag
constant (${\cal B}$) chodos . The bag constant ${\cal B}$ can be interpreted
as the difference between the energy densities of the noninteracting quarks
and the interacting ones. Dynamically it acts as a pressure that keeps the
quark gas in constant density and potential. This constant is shown to have
different values which are $55$ and $90\ \frac{MeV}{fm^{3}}$ in the initial
MIT bag model. Since the density of strange quark matter increases from
surface to the core of the strange star, it is more appropriate to use a
density dependent bag constant rather than a fixed bag constant.
According to the analysis of the experimental data obtained at CERN, the
quark-hadron transition takes place at about seven times the normal nuclear
matter energy density ($156\ MeVfm^{-3}$) aa ; g . Recently, a density
dependent form has been also considered for ${\cal B}$ adami ; jin ; blasch ;
burgio . The density dependence of ${\cal B}$ is highly model dependent. In
this article, the density dependence of ${\cal B}$ will be parameterized, and
we make the asymptotic value of ${\cal B}$ approach a finite value ${\cal
B}_{\infty}$ burgio ,
${\cal B}(n)={\cal B}_{\infty}+({\cal B}_{0}-{\cal
B}_{\infty})e^{-\gamma(n/n_{0})^{2}}.$ (1)
The parameter ${\cal B}_{0}={\cal B}(n=0)$ has constant value which is assumed
to be ${\cal B}_{0}=400\ \frac{MeV}{fm^{3}}$ in this work, and $\gamma$ is the
numerical parameter which is usually equal to $n_{0}\approx 0.17fm^{-3}$, the
normal nuclear matter density. ${\cal B}_{\infty}$ depends only on the free
parameter ${\cal B}_{0}$. We know that the value of the bag constant (${\cal
B}$) should be compatible with experimental data. The experimental results at
CERN-SPS confirms a proton fraction $x_{pt}=0.4$ (data is from experiment on
accelerated Pb nuclei) aa ; burgio . Therefore, in order to evaluate ${\cal
B}_{\infty}$, we use the equation of state of the asymmetric nuclear matter.
The calculations regarding this can be found in the next section.
### II.2 Computation of ${\cal B}_{\infty}$ using the asymmetric nuclear
matter calculations
We use the equation of state of the asymmetric nuclear matter to calculate
${\cal B}_{\infty}$. For calculating the equation of state of asymmetric
nuclear matter, we employ the lowest order constrained variational (LOCV)
many-body method based on the cluster expansion of the energy as follows b2 ;
b3 ; b4 ; b5 ; b6 ; b7 ; b8 ; b9 ; b10 .
The asymmetric nuclear matter is defined as a system consisting of $Z$ protons
($pt$) and $N$ neutrons ($nt$) with the total number density $n=n_{pt}+n_{nt}$
and proton fraction $x_{pt}=\frac{n_{pt}}{n}$, where $n_{pt}$ and $n_{nt}$ are
the number densities of protons and neutrons, respectively. For this system,
we consider a trial wave function as follows,
$\psi=F\phi,$ (2)
where $\phi$ is the slater determinant of the single-particle wave functions
and $F$ is the A-body correlation operator ($A=Z+N$) which is taken to be
$F={\cal S}\prod_{i>j}f(ij)$ (3)
and ${\cal S}$ is a symmetrizing operator. For the asymmetric nuclear matter,
the energy per nucleon up to the two-body term in the cluster expansion is
$E([f])=\frac{1}{A}\frac{<\psi\mid
H\mid\psi>}{<\psi\mid\psi>}=E_{1}+E_{2}\cdot$ (4)
The one-body energy, $E_{1}$, is
$E_{1}=\sum_{i=1}^{2}\sum_{k_{i}}\frac{\hbar^{2}{k_{i}^{2}}}{2m},$ (5)
where labels $1$ and $2$ are used for proton and neutron respectively, and
$k_{i}$ is the momentum of particle $i$. The two-body energy, $E_{2}$, is
$E_{2}=\frac{1}{2A}\sum_{ij}<ij\mid{\cal V}(12)\mid ij-ji>,$ (6)
where
${\cal
V}(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12)\cdot$
(7)
In the above equation, $f(12)$ and $V(12)$ are the two-body correlation and
nucleon-nucleon potential, respectively. In our calculations, we use
$UV_{14}+TNI$ nucleon-nucleon potential Lagaris . Now, we minimize the two-
body energy with respect to the variations in the correlation functions
subject to the normalization constraint. From the minimization of the two-body
energy, we obtain a set of differential equations. We can calculate the
correlation functions by numerically solving these differential equations.
Using these correlation functions, the two-body energy is obtained and then we
can compute the energy of asymmetric nuclear matter. The procedure of these
calculations has been fully discussed in reference b3 .
As it was mentioned in the previous section, the experimental results at CERN-
SPS confirms a proton fraction $x_{pt}=0.4$ aa ; burgio , therefore to compute
${\cal B}_{\infty}$, we proceed in the following manner:
* •
Firstly, we use our results of the previous section for the asymmetric nuclear
matter characterized by a proton fraction $x_{pt}=0.4$. By assuming that the
hadron-quark transition takes place at the energy density equal to
$1100MeVfm^{-3}$ aa ; burgio , we find that the baryonic density of the
nuclear matter is $n_{B}=0.98fm^{-3}$ (transition density). At densities lower
than this value the energy density of the quark matter is higher than that of
the nuclear matter. With increasing the baryonic density these two energy
densities become equal at the transition density, and above this value the
nuclear matter energy density remains always higher.
* •
Secondly, we determine $B_{\infty}=8.99\ \frac{MeV}{fm^{3}}$ by putting the
energy density of the quark matter and that of the nuclear matter equal to
each other.
### II.3 Calculations for the energy of quark matter at finite temperature
To calculate the energy of quark matter, we need to know the density of quarks
in terms of the baryonic density. We do this by considering two conditions of
beta equilibrium and charge neutrality. This leads to the following relations
$\mu_{d}=\mu_{u}-\mu_{e},$ (8) $\mu_{s}=\mu_{u}-\mu_{e},$ (9)
$\mu_{s}=\mu_{d},$ (10) $2/3n_{u}-1/3n_{s}-1/3n_{d}-n_{e}=0,$ (11)
where $\mu_{i}$ and $n_{i}$ are the chemical potential and the number density
of particle $i$, respectively. As mentioned, we consider the system as pure
quark matter ($n_{e}=0$ ) d ; n ; o ; p . Thus according to relation (11), we
have
$n_{u}=1/2(n_{s}+n_{d}).$ (12)
The chemical potential, $\mu_{i}$, at any adopted values of the temperature
($T$) and the number density ($n_{i}$) is determined by applying the following
constraint,
$n_{i}=\frac{g}{2\pi^{2}}\int_{0}^{\infty}{f(n_{i},k,T)}{k^{2}dk},$ (13)
where
$f(n_{i},k,T)=\frac{1}{Exp\\{\beta((m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}-\mu_{i})\\}+1}$
(14)
is the Fermi-Dirac distribution function qq . In the above equation,
$\beta=\frac{1}{k_{B}T}$ and $g$ is the degeneracy number of the system.
As it is previously mentioned, we consider the total energy of the quark
matter as the sum of the kinetic energy of the free quarks and the bag
constant (${\cal B}$). Therefore, the total energy per volume of the quark
matter (${\cal E}_{tot}$) can be obtained using the following relation,
${\cal E}_{tot}={\cal E}_{u}+{\cal E}_{d}+{\cal E}_{s}+{\cal B},$ (15)
where ${\cal E}_{i}$ is the kinetic energy per volume of particle $i$,
${\cal
E}_{i}=\frac{g}{2\pi^{2}}\int_{0}^{\infty}{(m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}}{f(n_{i},k,T)}{k^{2}dk}.$
(16)
After calculating the energy, we can determine the other thermodynamic
properties of the system. The entropy of the quark matter (${\cal S}_{tot}$)
can be derived as follows
${\cal S}_{tot}={\cal S}_{u}+{\cal S}_{d}+{\cal S}_{s},$ (17)
where ${\cal S}_{i}$ is the entropy of particle $i$,
$\displaystyle{\cal S}_{i}(n_{i},T)$ $\displaystyle=$
$\displaystyle-\frac{3}{\pi^{2}}k_{B}\int_{0}^{\infty}[f(n_{i},k,T)\ln(f(n_{i},k,T))$
(18) $\displaystyle+(1-f(n_{i},k,T))\ln(1-f(n_{i},k,T))]k^{2}dk.$
The Helmholtz free energy per volume (${\cal F}$) is given by
${\cal F}={\cal E}_{tot}-T{\cal S}_{tot}.$ (19)
The entropy per particle of the quark matter as a function of the baryonic
density for two cases of the constant and density dependent ${\cal B}$ at
different temperatures are plotted in Figs. 1 and 2. For a fixed temperature,
we see that the entropy per particle decreases by increasing the baryonic
density and for all relevant densities, it is seen that the entropy increases
by increasing the temperature.
In Figs. 3 and 4, the free energy per volume of the quark matter versus the
baryonic density for two cases of the constant and density dependent ${\cal
B}$ are presented at different temperatures. We can see that the free energy
of the quark matter has positive values for all densities and temperatures.
For all densities, it is seen that the free energy decreases by increasing the
temperature.
To obtain the structure of the strange star, the equation of state of the
quark matter is needed. For deriving the equation of state, the following
equation is used,
$P(n,T)=\sum_{i}{n_{i}\frac{\partial{\cal F}_{i}}{\partial n_{i}}-{\cal
F}_{i}},$ (20)
where $P$ is the pressure. The pressure of the quark matter versus the
baryonic density for two cases of the constant and density dependent ${\cal
B}$ are plotted in Figs. 5 and 6. It is seen that by increasing both density
and temperature, the pressure increases. These figures show that for each
temperature, the pressure becomes zero at a specific value of the density. We
see that the density corresponding to zero pressure increases by decreasing
the temperature.
## III Structure of Strange Star
Compact objects like white dwarfs, neutron stars and strange stars have
limiting masses (maximum mass) and with a mass more than the limitting value,
the hydrostatic stability of the star is impossible. For obtaining the maximum
mass of the strange star, we use the Tolman-Oppenheimer-Volkoff (TOV)
equations n ,
$\frac{dP}{dr}=-\frac{G[{\cal E}(r)+\frac{P(r)}{c^{2}}][m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}]}{r^{2}[1-\frac{2Gm(r)}{rc^{2}}]},$ (21)
$\frac{dm}{dr}=4\pi r^{2}{\cal E}(r).$ (22)
By using the equation of state found in the previous section, we integrate the
TOV equations to calculate the structure of the strange star n . The results
of this calculation are given in the following figures and tables.
Figs. 7 and 8 show the gravitational mass versus the central energy density at
different values of temperature for two cases of the constant and density
dependent ${\cal B}$. For each value of the temperature, these figures show
that the gravitational mass increases rapidly by increasing the energy density
and finally reaches to a limiting value at higher energy densities. It is seen
that the limiting value of the gravitational mass increases by decreasing
temperature. Comparing Figs. 7 and 8, one concludes that at all temperatures,
for the density dependent bag constant, the rate of increasing mass with
increasing the central density, at lower values of the central densities, is
substantially higher than that of the case for fixed bag constant, especially
at zero temperature. In Figs. 9 and 10, we have plotted the radius of strange
star versus the central energy density for both ${\cal
B}=90\frac{MeV}{fm^{3}}$ and density dependent ${\cal B}$ at different
temperatures. From Figs. 7$-$10, it can be seen that at each central density,
both mass and the corresponding radius increase by decreasing the temperature.
The gravitational mass of strange star is also plotted as a function of the
radius for the constant and density dependent ${\cal B}$ in Figs. 11 and 12.
It is seen that for all temperatures, the gravitational mass of strange star
increases by increasing the radius and it approaches a limiting value (maximum
mass). Figs. 11 and 12 show that by decreasing the temperature, the limiting
values of mass and the corresponding radius both increase.
In Tables 1 and 2, the maximum mass and the corresponding radius and central
energy density of the strange star at different temperatures for two cases of
the constant and density dependent ${\cal B}$ are given. It is shown that by
decreasing the temperature, the maximum mass of strange star increases. This
behavior is also seen for the radius of strange star versus the temperature.
Meanwhile, the central energy density decreases by decreasing the temperature.
By comparing Tables 1 and 2, we can see that for all temperatures, the maximum
mass and the corresponding radius calculated with the constant ${\cal B}$ are
less than those calculated with the density dependent ${\cal B}$.
## IV Summary and Conclusion
We have considered a pure quark matter for the strange star to calculate the
structure properties of this object at finite temperature. For this purpose,
some thermodynamic properties of the quark matter such as the entropy, free
energy and the equation of state have been computed using the constant and
density dependent bag constant (${\cal B}$). It was shown that the free energy
of the quark matter decreases by increasing the temperature while the entropy
of this system increases by increasing the temperature. It was indicated that
by increasing the temperature, the equation of state of the quark matter
becomes stiffer. We have calculated the gravitational mass of the strange star
by numerically integrating the Tolman-Oppenheimer-Volkoff (TOV) equations. Our
results show that the gravitational mass of the strange star increases by
increasing the central energy density. It was shown that this gravitational
mass reaches a limiting value (maximum mass) at higher values of the central
energy density. We have found that the maximum mass of the strange star
decreases by increasing the temperature. It was also shown that the maximum
mass and radius of the strange star in the case of density dependent ${\cal
B}$ are higher than those in the case of constant ${\cal B}$.
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University Research Council.
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Table 1: Maximum mass ($M_{max}$) in solar mass unit ($M_{\odot}$), and the corresponding radius (R) and central energy density (${\cal E}_{c}$) of the strange star at different temperatures (T) for ${\cal B}=90\ \frac{MeV}{fm^{3}}$. $T$ (MeV) | $M_{max}(M_{\odot})$ | R (km) | ${\cal E}_{c}(10^{14}\frac{gr}{cm^{3}})$
---|---|---|---
0 | 1.354 | 7.698 | 38.24
30 | 1.228 | 7.073 | 47.54
70 | 1.101 | 6.416 | 60.60
80 | 1.039 | 6.142 | 63.65
Table 2: As Table 1 but for the density dependent ${\cal B}$. $T$ (MeV) | $M_{max}(M_{\odot})$ | R (km) | ${\cal E}_{c}(10^{14}\frac{gr}{cm^{3}})$
---|---|---|---
0 | 1.676 | 8.761 | 39.11
30 | 1.341 | 7.442 | 48.47
70 | 1.181 | 6.768 | 61.56
80 | 1.122 | 6.567 | 64.21
Figure 1: The entropy per particle of the quark matter versus the baryonic
density at different temperatures for ${\cal B}=90\ \frac{MeV}{fm^{3}}$.
Figure 2: As Figure 1 but for the density dependent ${\cal B}$. Figure 3: The
free energy per volume of the quark matter versus the baryonic density at
different temperatures for ${\cal B}=90\ \frac{MeV}{fm^{3}}$. Figure 4: As
Figure 3 but for the density dependent ${\cal B}$. Figure 5: The pressure of
the quark matter as a function of the baryonic density at different
temperatures for ${\cal B}=90\ \frac{MeV}{fm^{3}}$. Figure 6: As Figure 5 but
for the density dependent ${\cal B}$. Figure 7: The gravitational mass of the
strange star as a function of the central energy density at different
temperatures for ${\cal B}=90\ \frac{MeV}{fm^{3}}$. Figure 8: As Figure 7 but
for the density dependent ${\cal B}$. Figure 9: The radius of the strange
star as a function of the central energy density at different temperatures for
${\cal B}=90\ \frac{MeV}{fm^{3}}$. Figure 10: As Figure 9 but for the density
dependent ${\cal B}$. Figure 11: The gravitational mass of the strange star
as a function of the radius at different temperatures for ${\cal B}=90\
\frac{MeV}{fm^{3}}$. Figure 12: As Figure 11 but for the density dependent
${\cal B}$.
|
arxiv-papers
| 2011-03-12T10:53:00 |
2024-09-04T02:49:17.605385
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar, A. Poostforush and A. Zamani",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1103.2436"
}
|
1103.2490
|
# Enabling Differentiated Services Using Generalized Power Control Model in
Optical Networks
Quanyan Zhu Department of Electrical and Computer Engineering
University of Toronto, Ontario M5S 3L1
Email: qzhu@control.utoronto.ca Lacra Pavel Department of Electrical and
Computer Engineering
University of Toronto, Ontario M5S 3L1
Email: pavel@control.utoronto.ca Quanyan Zhu, Lacra Pavel Quanyan Zhu is with
the Department of Electrical and Computer Engineering, University of Illinois
at Urbana Champaign, IL, 61801, USA email: zhu31@illinois.edu; L. Pavel is
with the Department of Electrical and Computer Engineering, University of
Toronto, Toronto, ON, M5S 3L1 Canada e-mail:pavel@control.utoronto.ca.
###### Abstract
This paper considers a generalized framework to study OSNR optimization-based
end-to-end link level power control problems in optical networks. We combine
favorable features of game-theoretical approach and central cost approach to
allow different service groups within the network. We develop solutions
concepts for both cases of empty and nonempty feasible sets. In addition, we
derive and prove the convergence of a distributed iterative algorithm for
different classes of users. In the end, we use numerical examples to
illustrate the novel framework.
## I Introduction
Reconfigurable optical Wavelength-Division Multiplexing (WDM) communication
networks with arbitrary topologies are currently enabled by technological
advances in optical devices such as optical add/drop MUXes (OADM), optical
cross connects (OXC) and dynamic gain equalizer (DGE). It is important that
channel transmission performance and quality of service (QoS) be optimized and
maintained after reconfiguration. At the physical transmission level, channel
performance and QoS are directly determined by the bit-error rate (BER), which
in turn depends on optical signal-to-noise ratio (OSNR), dispersion and
nonlinear effects, [1]. Thus, OSNR is considered as the dominant performance
parameter in link-level optimization. Conventional off-line OSNR optimization
is done by adjusting channel input power at transmitter (Tx) to equalize the
dominant impairment of noise accumulation in chains of optical amplifiers.
However, for reconfigurable optical networks, where different channels can
travel via different optical paths, it is more desirable to implement on-line
decentralized iterative algorithms to accomplish such adjustment.
Recently, this problem is addressed in many research works [2],[3],[4], and
two optimization-based approaches are prevalently used: the central cost and
the non-cooperative game approach. The goals and models of the two approaches
are inherently different. Central cost approach satisfies the target OSNR with
minimum total power consumption. The model embeds the OSNR requirements in its
constraints and indirectly optimizes a certain design criterion. Such model
yields a relatively simple closed-form solution; however, it doesn’t optimize
OSNR in a direct fashion, and thus, channel performance can be potentially
improved for users who need higher quality of transmission. On the other hand,
the game approach is a naturally distributed model which directly optimizes
OSNR based on a payoff function in a non-cooperative manner. Each user
optimizes her own utility to achieve the best possible OSNR. The solution from
this approach is given by Nash equilibrium. As a result, this solution concept
yields best achievable OSNR levels for each user. Since the game approach
involves a cost function arising from pricing, it gives an over-allocation of
resources. Some users may wish to avoid such cost and only demand a basic
level of transmission. Apparently, these two approaches are for two different
type of users and different transmission purposes.
To make use of the advantages from each approach, we propose a generalized
model that combines their features. Such a generalization allows to
accommodate different types of users and also provides a novel mixed framework
to study OSNR power control problem. We separate users into two different
categories. One type of users are those who are willing to pay a price to
fully optimize their transmission performance. Another type of users are those
who are content with basic transmission quality, or OSNR level, set by the
network. The quality of service (QoS) can be met for the former by a game-
theoretically based optimization approach; and for the later by a mechanism
similar to central cost approach.
The contribution of this paper lies in the capability of service
differentiation of the generalized model. For simplicity, total capacity
constraints are not considered. The paper is organized as follows. In section
2, we review the network OSNR model and the basic concepts about the two
optimization-based approaches. In section 3, we establish a general framework
and propose two solution concepts for two different cases of feasible sets.
Section 4 gives an iterative algorithm to achieve such solutions in the
framework. This is illustrated in section 5 by numerical examples. Section 6
concludes the paper and points out future directions of research.
## II Background
### II-A Review of Optical Network Model
Consider a network with a set of optical links $\mathcal{L}=\\{1,2,..,L\\}$
connecting the optical nodes, where channel add/drop is realized. A set
$\mathcal{N}=\\{1,2,...,N\\}$ of channels are transmitted, corresponding to a
set of multiplexed wavelengths. Illustrated in Figure 1, a link $l$ has
$K_{l}$ cascaded optically amplified spans. Let $N_{l}$ be the set of channels
transmitted over link $l$. For a channel $i\in\mathcal{N}$, we denote by
$\mathcal{R}_{i}$ its optical path, or collection of links, from source (Tx)
to destination (Rx). Let $u_{i}$ be the $i$th channel input optical power (at
Tx), and $\textbf{u}=[u_{1},...,u_{N}]^{T}$ the vector of all channels’ input
powers. Let $s_{i}$ be the $i$th channel output power (at Rx), and $n_{i}$ the
optical noise power in the $i$th channel bandwidth at Rx. The $i$th channel
optical OSNR is defined as $OSNR_{i}=\frac{s_{i}}{n_{i}}$. In [2], some
assumptions are made to simplify the expression for OSNR, typically for
uniformly designed optical links:
1. 1.
(A1) Amplified spontaneous emission (ASE) noise power does not participate in
amplifier gain saturation.
2. 2.
(A2) All the amplifiers in a link have the same spectral shape with the same
total power target and are operated in automatic power control mode.
Under A1 and A2, dispersion and nonlinearity are considered to be limited, and
ASE noise accumulation will be the dominant impairment. The OSNR for the $i$th
channel is given as
$OSNR_{i}=\frac{u_{i}}{n_{0,i}+\sum_{j\in\mathcal{N}}\Gamma_{i,j}u_{j}},i\in\mathcal{N}$
(1)
where $\mathbf{\Gamma}$ is the full $n\times n$ system matrix which
characterizes the coupling between channels. $n_{0,i}$ denotes the $i$th
channel noise power at the transmitter. System matrix $\mathbf{\Gamma}$
encapsulates the basic physics present in optical fiber transmission and
implements an abstraction from a network to an input-output system. This
approach has been used in [5] for the wireless case to model CDMA uplink
communication. Different from the system matrix used in wireless case, the
matrix $\mathbf{\Gamma}$ given in (2) is commonly asymmetric and is more
complicatedly dependent on parameters such as spontaneous emission noise,
wavelength-dependent gain, and the path channels take.
$\Gamma_{i,j}=\sum_{i\in\mathcal{R}_{i}}\sum_{k=1}^{K_{l}}\frac{G_{l,j}^{k}}{G_{l,i}^{k}}\left(\prod_{q=1}^{l-1}\frac{\mathbf{T}_{q,j}}{\mathbf{T}_{q,i}}\right)\frac{ASE_{l,k,i}}{P_{o,l}},\forall
j\in\mathcal{N}_{l}.$ (2)
where $G_{l,k,i}$ is the wavelength dependent gain at $k$th span in $l$th link
for channel $i$; $\mathbf{T}_{l,i}=\prod_{q=1}^{K_{l}}G_{l,k,i}L_{l,k}$ with
$L_{l,k}$ being the wavelength independent loss at $k$th span in $l$th link;
$ASE_{l,k,i}$ is the wavelength dependent spontaneous emission noise;
$P_{0,l}$ is the output power at each span.
Figure 1: A Typical Optical Link in DWDM Optical Networks
### II-B Central Cost Approach
Similar to the SIR optimization problem in the wireless communication networks
[6, 7], OSNR optimization achieves the target OSNR predefined by each channel
user by allowing the minimum transmission power. Let
$\gamma_{i},i\in\mathcal{N}$ be the target OSNR for each channel. By setting
the OSNR requirement as a constraint, we can arrive at the following central
cost optimization problem (CCP):
(CCP) | $\min_{\mathbf{u}}\sum_{i\in\mathcal{N}}u_{i}$
---|---
subject to | $OSNR_{i}\geq\gamma_{i}\texttt{ }\forall i\in\mathcal{N}.$
(3)
Under certain conditions, it has been shown in [2] that the feasible set of
(CCP) is nonempty and the optimal solution is achievable at the boundary of
the feasible set.
The formulated optimization problem can be extended to incorporate more
constraints such as
$u_{i,\min}\leq u_{i}\leq u_{i,\max},$ (4)
where $u_{i,\min}$ is minimum threshold power required for transmission for
channel $i$ and $u_{i,\max}$ is maximum power channel $i$ can attain. In the
central cost approach, power $u_{i}$ are the parameters to be minimized and
the objective function is linearly separable. In addition, the constraints are
linearly coupled. These nice characteristics in central cost approach leads to
a relatively simple optimization problem.
### II-C Non-cooperative Game Approach
Let’s review the basic game-theoretical model for power control in optical
networks without constraints. Consider a game defined by a triplet
$\langle\mathcal{N},(A_{i}),(J_{i})\rangle$. $\mathcal{N}$ is the index set of
players or channels; $A_{i}$ is the strategy set $\\{u_{i}\mid
u_{i}\in[u_{i,\min},u_{i,\max}]\\}$; and, $J_{i}$ is the cost function. It is
chosen in a way that minimizing the cost is related to maximizing OSNR level.
In [3], $J_{i}$ is defined as
$J_{i}(u_{i},u_{-i})=\alpha_{i}u_{i}-\beta_{i}\ln\left(1+a_{i}\frac{u_{i}}{X_{-i}}\right),i\in\mathcal{N}$
(5)
where $\alpha_{i},\beta_{i}$ are channel specific parameters, that quantify
the willingness to pay the price and the desire to maximize its OSNR,
respectively, $a_{i}$ is a channel specific parameter, $X_{-i}$ is defined as
$X_{-i}=\sum_{j\neq i}\Gamma_{i,j}u_{j}+n_{0,i}$. This specific choice of
utility function is non-separable, nonlinear and coupled. However, $J_{i}$ is
strictly convex in $u_{i}$ and takes a specially designed form such that its
first-order derivative is linear with respect to $\mathbf{u}$.
The solution from the game approach is usually characterized by Nash
equilibrium (NE). Provided that $\sum_{j\neq i}\Gamma_{i,j}<a_{i}$, the
resulting NE solution is uniquely determined in a closed form by
$\mathbf{\widetilde{{\Gamma}}u^{*}=\widetilde{b}},$ (6)
where $\widetilde{\Gamma}_{i,j}=a_{i},$ for $j=i$;
$\widetilde{\Gamma}_{i,j}=\Gamma_{i,j},$ for $j\neq i$ and
$\widetilde{b}=\frac{a_{i}\beta_{i}}{\alpha_{i}}-n_{0,i}$.
Similar to the wireless case [5], we are able to construct iterative
algorithms to achieve the Nash equilibrium. A simple deterministic first order
parallel update algorithm is:
$u_{i}(n+1)=\frac{\beta_{i}}{\alpha_{i}}-\frac{1}{a_{i}}\left(\frac{1}{OSNR_{i}(n)}-\Gamma_{i,i}\right)u_{i}(n).$
(7)
As proved in [3], the algorithm (7) converges to Nash equilibrium
$\mathbf{u}^{*}$ provided that $\frac{1}{a_{i}}\sum_{j\neq
i}\Gamma_{i,j}<1,\forall i$.
## III Generalized Model
In this section, we consider a game designed to allow service differentiation
by separating users into two groups: one group seeking a minimum OSNR target
and another group participating in a game setting for OSNR optimization. The
minimum OSNR for target seekers is set by the network to ensure the minimum
quality of service. However, the game players can submit their parameters and
optimize their service accordingly, but they have to pay a price set by the
network for unit power consumption. This concept is illustrated in Figure 2.
Let’s denote set $\mathcal{N}_{1}=\\{1,2,...,N_{1}\\}$ as the set of
competitors, i.e. users who wish to compete for an optimal OSNR. Let set
$\mathcal{N}_{2}=\\{N_{1}+1,\cdots,N_{2}\\}$ be the group of users with target
OSNR given by $\gamma_{i},i\in\mathcal{N}_{2}$. Let
$\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}$, $m=|\mathcal{N}_{1}|=N_{1}$,
$n=|\mathcal{N}_{2}|$, $N=|\mathcal{N}|=m+n$ and
$\mathbf{u}=[u_{1},\cdots,u_{N_{1}},u_{N_{1}+1},\cdots,u_{N_{2}}]^{T}$.
Figure 2: Game players and target seekers in the network
For the game-theoretical players, using the cost function given in (5), we can
form a system of equations given by
$a_{i}u_{i}+X_{-i}=\frac{a_{i}\beta_{i}}{\alpha_{i}},\forall
i\in\mathcal{N}_{1}$
and thus, $\widetilde{\mathbf{\Gamma}}\mathbf{u}=\widetilde{\mathbf{b}},$
where $\widetilde{\mathbf{\Gamma}}\in\mathcal{R}^{m\times N}$ and
$\widetilde{\mathbf{b}}\in\mathcal{R}^{m}$ are defined as in (6). Users with
target OSNR shall have $\mathbf{u}$ satisfy $OSNR_{i}\geq\gamma_{i},\forall
i\in\mathcal{N}_{2},$ or equivalently from (1),
$\frac{u_{i}}{\Gamma_{i,i}u_{i}+\sum_{j\neq
i}\Gamma_{i,j}u_{j}+n_{0,i}}\geq\gamma_{i}$
and thus in a matrix form,
$\widehat{\mathbf{\Gamma}}\mathbf{u}\geq\widehat{\mathbf{b}},$ where
$\widehat{\mathbf{b}}=[\gamma_{1}n_{0,1},\cdots,\gamma_{N}n_{0,N}]^{T}\in\mathcal{R}^{n}$,
$\widehat{\mathbf{\Gamma}}\in\mathcal{R}^{n\times N}$ and is given in (8).
$\widehat{\mathbf{\Gamma}}=\left[\begin{array}[]{ccccc}-\gamma_{N_{1}+1}\Gamma_{N_{1}+1,1}&\cdots&1-\gamma_{N_{1}+1}\Gamma_{N_{1}+1,N_{1}+1}&\cdots&-\gamma_{N_{1}+1}\Gamma_{N_{1}+1,N}\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\
-\gamma_{N-1}\Gamma_{N-1,1}&-\gamma_{N-1}\Gamma_{N-1,2}&\cdots&1-\gamma_{N-1}\Gamma_{N-1,N-1}&-\gamma_{N-1}\Gamma_{N}\\\
-\gamma_{N}\Gamma_{N,1}&-\gamma_{N}\Gamma_{N,2}&\cdots&\cdots&1-\gamma_{N}\Gamma_{N,N}\\\
\end{array}\right].$ (8)
Let
$F_{1}=\\{\mathbf{u}\in\mathcal{R}^{N}\mid\widetilde{\mathbf{\Gamma}}\mathbf{u}=\widetilde{\mathbf{b}}\\}$
and
$F_{2}=\\{\mathbf{u}\in\mathcal{R}^{N}\mid\widehat{\mathbf{\Gamma}}\mathbf{u}\geq\widehat{\mathbf{b}}\\}$.
In summary, we have a problem formulated as in (DS), where we find solutions
that satisfy $F_{1}$ subject to the constraint described by $F_{2}$.
$\begin{array}[]{cc}\textrm{(DS)}&\widetilde{\mathbf{\Gamma}}\mathbf{u}=\widetilde{\mathbf{b}}\\\
\textrm{s.t.}&\widehat{\mathbf{\Gamma}}\mathbf{u}\geq\widehat{\mathbf{b}}\end{array}$
(9)
In the following discussion, we separate (DS) into two cases: (1) $F=F_{1}\cap
F_{2}\neq\emptyset$, (2)$F=F_{1}\cap F_{2}=\emptyset$, which require different
techniques to find appropriate solutions.
### III-A Non-empty Feasible Set
A non-empty $F$ may give rise to multiple points that solve (DS). We may
impose some design criteria, or, objective function to reformulate DS for
finding an appropriate solution that solves DS and meet the design criteria at
the same time.
We can use the following result to ensure the nonempty feasible set $F$.
###### Theorem III.1
If
$\overline{\mathbf{\Gamma}}=\left[\begin{array}[]{c}\widetilde{\mathbf{\Gamma}}\\\
\widehat{\mathbf{\Gamma}}\\\ \end{array}\right]$ is nonsingular, the feasible
set $F=F_{1}\cap F_{2}$ is non-empty.
###### Proof:
Let $\mu\in\mathcal{R}^{n}_{+}$ a nonnegative vector. Equivalently, we can
express $F_{2}$ into
$F_{2}=\\{\mathbf{u}\in\mathcal{R}^{n}\mid\widehat{\mathbf{\Gamma}}\mathbf{u}=\widehat{\mathbf{b}}+\mu,\textrm{~{}for~{}some~{}}\mu\in\mathcal{R}^{n}_{+}\\}$.
The set $F$ is thus equivalently
$F=\\{\mathbf{u}\in\mathcal{R}^{N}\mid\overline{\mathbf{\Gamma}}\mathbf{u}=\mathbf{\phi},\textrm{~{}for~{}some~{}}\mu\in\mathcal{R}^{n}_{+}\\}$,
where
$\overline{\mathbf{\Gamma}}=\left[\begin{array}[]{c}\widetilde{\mathbf{\Gamma}}\\\
\widehat{\mathbf{\Gamma}}\\\ \end{array}\right]$ and
$\mathbf{\phi}=\left[\begin{array}[]{c}\widetilde{\mathbf{b}}\\\
\widehat{\mathbf{b}}+\mu\\\ \end{array}\right]$. If
$\overline{\mathbf{\Gamma}}$ is nonsingular, there exist a unique
$\mathbf{u}\in\mathcal{R}^{N}$ for every nonnegative $\mu$. Therefore $F$ is
non-empty. ∎
Suppose conditions in Theorem III.1 hold and $F$ is nonempty. We consider an
appropriate solution in $F$ that satisfies a certain design criteria. Thus, we
formulate (DSNP111DSNP stands for “Differentiated Service N-person Problem”.)
in which we minimize total power consumption subject to the conditions arising
from the different service requirements.
$\begin{array}[]{cc}\textrm{(DSNP)}&\min\sum_{i}u_{i}\\\
\textrm{s.t.}&\widetilde{\mathbf{\Gamma}}\mathbf{u}=\widetilde{\mathbf{b}},\widehat{\mathbf{\Gamma}}\mathbf{u}\geq\widehat{\mathbf{b}}\end{array}$
(10)
The constraints of (DSNP) can be relaxed and augmented into
$\overline{\mathbf{\Gamma}}\mathbf{u}\geq\overline{\mathbf{b}}.$ (11)
where
$\overline{\mathbf{\Gamma}}=\left[\begin{array}[]{c}\widetilde{\mathbf{\Gamma}}\\\
\widehat{\mathbf{\Gamma}}\\\ \end{array}\right]\in\mathcal{R}^{N\times N}$ and
$\overline{\mathbf{b}}=\left[\begin{array}[]{c}\widetilde{\mathbf{b}}\\\
\widehat{\mathbf{b}}\\\ \end{array}\right]\in\mathcal{R}^{N}.$
According to the fundamental theorem of linear programming [8], if (DSNP) is
realistic, the solution is obtained at the extreme point of the feasible set
$F$. Since $F$ has only one extreme point when $\overline{\mathbf{\Gamma}}$ is
non-singular, the solution is uniquely given by
$\mathbf{u}=\overline{\mathbf{\Gamma}}^{-1}\overline{\mathbf{b}}.$ (12)
To further characterize the solution $\mathbf{u}$, we assume strict diagonal
dominance of matrix $\overline{\mathbf{\Gamma}}$ [9], which leads to non-
singularity of the matrix and uniqueness of the solution.
###### Theorem III.2
Suppose OSNR targets $\gamma_{i},i\in\mathcal{N}_{2}$ are chosen such that
$\gamma_{i}<\frac{1}{\sum_{j\in\mathcal{N}}\Gamma_{i,j}},i\in\mathcal{N}_{2}$.
In addition, parameters $a_{i}$ are chosen as $a_{i}>\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{ij},\forall i\in\mathcal{N}_{1}.$ The matrix
$\overline{\mathbf{\Gamma}}$ is strictly diagonally dominant. And thus, a
unique solution to (DSNP) is given by (12).
###### Proof:
From the assumption that
$\gamma_{i}\sum_{j\in\mathcal{N}}\Gamma_{ij}<1,i\in\mathcal{N}_{2}$, it is
apparent that $\gamma_{i}<\frac{1}{\Gamma_{ii}}$ and
$\left|1-\gamma_{i}\Gamma_{ii}\right|>\gamma_{i}\sum_{j}\Gamma_{ij},\forall
i\in\mathcal{N}_{2}$. In addition, $a_{i}>\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j},\forall i\in\mathcal{N}_{1}$. Therefore, matrix
$\overline{\mathbf{\Gamma}}$ is strictly diagonally dominant. Using Gershgorin
theorem in [9], we conclude that there exists a unique solution to (DSNP). ∎
The assumption of strict diagonal dominance in Theorem III.2 is reasonable
because typical values of $\Gamma_{ij}$ are found to be on the order of
$10^{-3}$ and desirable levels of OSNR are 20-30dB.
###### Remark III.1
(DSNP) can be seen as a generalized approach that combines central cost
approach in [2] and non-cooperative game approach in [3]. When
$N_{1}=\emptyset,N_{2}\neq\emptyset$, (DSNP) reduces to the central cost
approach. Similarly, when $N_{1}\neq\emptyset,N_{2}=\emptyset$, (DSNP) reduces
to the game-theoretical approach and the given solution is Nash equilibrium
accordingly. This framework allows to study two different types of users at
the same time.
###### Remark III.2
We illustrate a two-person (DSNP), where player 1 chooses to compete and
optimize his utility and player 2 chooses to meet a certain OSNR target
$\gamma_{2}$. We form the 2-by-2 matrix $\overline{\mathbf{\Gamma}}$ and
$\overline{\mathbf{b}}$ as follows.
$\overline{\mathbf{\Gamma}}=\left[\begin{array}[]{cc}a_{1}&\Gamma_{12}\\\
-\Gamma_{21}\gamma_{2}&1-\Gamma_{22}\gamma_{2}\\\
\end{array}\right],\overline{\mathbf{b}}=\left[\begin{array}[]{c}\frac{a_{1}\beta_{1}}{\alpha_{1}}-n_{0,1}\\\
n_{0,2}\gamma_{2}\\\ \end{array}\right]$
The feasible set $F=F_{1}\cap F_{2}$ is shown in Figure 3 by a dotted line.
The relaxed (DSNP) has its relaxed feasible depicted in the shaded region. The
solution is given by
$\mathbf{u}^{*}=\overline{\mathbf{\Gamma}}^{-1}\overline{\mathbf{b}},$ which
is illustrated by the dark point in Figure 3. $\mathbf{u}^{*}$ is nonnegative
componentwise if network price $\alpha_{1}$ is set such that
$s_{2}>\frac{n_{0,2}}{1-\Gamma_{22}}$.
Figure 3: The feasible set of two-person (DSNP).
$s_{1}=\frac{\tilde{b}_{1}}{a_{1}}$; $s_{2}=\frac{\tilde{b}_{1}}{\Gamma_{12}}$
Based on Theorem III.2, we can further investigate how parameters chosen by
game players and target seekers influence the outcome of the allocation. The
result is summarized in Theorem III.3.
###### Theorem III.3
Let $\kappa$ be the condition number of $\overline{\mathbf{\Gamma}}$,
$T_{i}=a_{i}+\sum_{j\neq i,j\in\mathcal{N}}\Gamma_{ij},\forall
i\in\mathcal{N}_{1}$ and $S_{k}=2-2\gamma_{k}\Gamma_{kk},\forall
k\in\mathcal{N}_{2}$. Suppose $\overline{\mathbf{\Gamma}}$ is strictly
diagonally dominant by satisfying conditions in Theorem III.2. In addition,
$T_{i}>S_{k}$ and $\tilde{b}_{i}>\hat{b}_{k},\forall
i\in\mathcal{N}_{1},\forall k\in\mathcal{N}_{2}.$ The maximum allocated power
allocated to users are bound as follows.
$\frac{\max_{i\in\mathcal{N}_{2}}\gamma_{i}n_{0,i}}{\max_{i\in\mathcal{N}_{1}}2a_{i}}\leq\|\mathbf{u}\|_{\infty}\leq\kappa\max_{i\in\mathcal{N}_{1}}\frac{\beta_{i}}{\alpha_{i}}$
###### Proof:
Let $R_{i}$ denote the i-th row absolute sum of matrix
$\overline{\mathbf{\Gamma}}$, i.e.,
$R_{i}=\sum_{j\in\mathcal{N}}\left|\overline{\Gamma}_{ij}\right|.$ (13)
Using conditions from Theorem III.2, we arrive at
$R_{i}=\left\\{\begin{array}[]{ll}1+\gamma_{i}\sum_{j\in\mathcal{N}}\Gamma_{ij}-2\gamma_{i}\Gamma_{ii}<2-2\gamma_{i}\Gamma_{ii},&{i\in\mathcal{N}_{2};}\\\
a_{i}+\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{ij}<2a_{i}.,&{i\in\mathcal{N}_{1}.}\end{array}\right.$
(14)
With the assumption that $a_{i}+\sum_{j\neq
i}\Gamma_{ij}>2-2\gamma_{k}\Gamma_{kk},\forall i\in\mathcal{N}_{1},\forall
k\in\mathcal{N}_{2},$ we obtain
$\|\overline{\mathbf{\Gamma}}\|_{\infty}=\max_{i\in\mathcal{N}}R_{i}=\max_{i\in\mathcal{N}}a_{i}+\sum_{j\neq
i}\Gamma_{ij}.$ Using (14) and the fact that $\Gamma_{ij}\geq 0$, we obtain an
upper and lower bound on $\|\overline{\mathbf{\Gamma}}\|_{\infty}$, i.e.,
$\max_{i\in\mathcal{N}_{1}}a_{i}\leq\|\overline{\mathbf{\Gamma}}\|_{\infty}\leq\max_{i\in\mathcal{N}_{1}}2a_{i}.$
(15)
In addition, from $\tilde{b}_{i}>\hat{b}_{k},\forall
i\in\mathcal{N}_{1},\forall k\in\mathcal{N}_{2},$ we obtain an upper bound and
lower bound for $\|\overline{\mathbf{b}}\|_{\infty}$, given by
$\max_{i\in\mathcal{N}_{2}}\gamma_{i}n_{0,i}\leq\|\overline{\mathbf{b}}\|_{\infty}=\max_{i\in\mathcal{N}}\overline{b}_{i}\leq\max_{i\in\mathcal{N}_{1}}\tilde{b}_{i}=\max_{i\in\mathcal{N}_{1}}\frac{a_{i}\beta_{i}}{\alpha_{i}}$
(16)
Since $\overline{\mathbf{\Gamma}}$ is strictly diagonally dominant, using
matrix norm sub-multiplicativity, we obtain from (12)
$\frac{\|\overline{\mathbf{b}}\|_{\infty}}{\|\overline{\mathbf{\Gamma}}\|_{\infty}}\leq\|\mathbf{u}\|_{\infty}\leq\frac{\kappa\|\overline{\mathbf{b}}\|_{\infty}}{\|\overline{\mathbf{\Gamma}}\|_{\infty}},$
(17)
where $\kappa$ is the condition number of $\overline{\mathbf{\Gamma}}$ given
by
$\kappa=\|\overline{\mathbf{\Gamma}}\|_{\infty}\|\overline{\mathbf{\Gamma}}^{-1}\|_{\infty}\geq
1.$
Using (15), (16) and (17), we obtain
$\displaystyle\frac{\max_{i\in\mathcal{N}_{2}}\gamma_{i}n_{0,i}}{\max_{i\in\mathcal{N}_{1}}2a_{i}}\leq\|\mathbf{u}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\frac{\kappa\max_{i\in\mathcal{N}_{1}}a_{i}\beta_{i}/\alpha_{i}}{\max_{i\in\mathcal{N}_{1}}a_{i}}$
(18) $\displaystyle\leq$
$\displaystyle\frac{\kappa\max_{i\in\mathcal{N}_{1}}a_{i}\max_{i\in\mathcal{N}_{1}}\beta_{i}/\alpha_{i}}{\max_{i\in\mathcal{N}_{1}}a_{i}}$
$\displaystyle\leq$
$\displaystyle\kappa\max_{i\in\mathcal{N}_{1}}\frac{\beta_{i}}{\alpha_{i}}.$
∎
It is easy to observe that the upper bound is dependent on the parameters of
the game players and the lower bound is dependent on the OSNR levels of target
seeker and parameter $a_{i}$ of the game players. In essence, game players
control the outcome of the model and the choice of OSNR target can only affect
the lower bound. Such relation describes a fair scenario in which game
players, who pay for their power at $\alpha_{i}$, have their choices of
parameters $a_{i},\beta_{i}$ to influence the network allocation.
###### Remark III.3
Since
$\|\mathbf{u}\|_{\infty}\leq\|\mathbf{u}\|_{2}\leq\sqrt{N}\|\mathbf{u}\|_{\infty}$,
we can translate the result obtained in (18) directly into Euclidean norm,
i.e.,
$B_{\infty}^{L}\leq\|\mathbf{u}\|_{2}\leq\sqrt{N}B^{U}_{\infty}$ (19)
where
$B_{\infty}^{U}=\kappa\max_{i\in\mathcal{N}_{1}}\frac{\beta_{i}}{\alpha_{i}}$
and
$B_{\infty}^{L}=\frac{\max_{i\in\mathcal{N}_{2}}\gamma_{i}n_{0,i}}{\max_{i\in\mathcal{N}_{1}}2a_{i}}$.
By (19), we can see that the network can encourage uniform channel power
distribution by letting $B_{\infty}^{U}$ close to $\sqrt{N}B_{\infty}^{L}$ and
provide incentive for differentiated services by letting them far apart. It
can be implemented by the network by adjusting OSNR level $\gamma_{i}$ and
pricing $\alpha_{i}$. Decreasing $\alpha_{i}$ encourages more users to be game
players, giving rise to more competitions or service differentiation as a
result of higher upper bound. On the other hand, increasing $\gamma_{i}$
raises the lower bound and encourages more users being target-seekers.
### III-B Empty Feasible Set
In this section, we consider the second case where feasible set $F$ is empty.
Instead of finding an appropriate feasible solution, we find the closest
points between set $F_{1}$ and $F_{2}$. We use a quadratic program (DS2) to
minimize the error norm subject to the constraint described by $F_{2}$.
$\begin{array}[]{cc}\textrm{(DS2)}&\min_{\mathbf{u}}\|\widetilde{\mathbf{\Gamma}}\mathbf{u}-\widetilde{\mathbf{b}}\|_{2}\\\
\textrm{s.t.}&\widehat{\mathbf{\Gamma}}\mathbf{u}\geq\widehat{\mathbf{b}}\end{array}$
(20)
We can turn the constrained problem (20) into an unconstrained problem by
studying its corresponding dual problem. Since
$\|\widetilde{\mathbf{\Gamma}}\mathbf{u}-\widetilde{\mathbf{b}}\|_{2}=\mathbf{u}^{T}\widetilde{\mathbf{\Gamma}}^{T}\widetilde{\mathbf{\Gamma}}\mathbf{u}-2(\widetilde{\mathbf{b}}^{T}\widetilde{\mathbf{\Gamma}})\mathbf{u}+\widetilde{\mathbf{b}}^{T}\widetilde{\mathbf{b}}$,
we denote
$\mathbf{H}=\frac{1}{2}\widetilde{\mathbf{\Gamma}}^{T}\widetilde{\mathbf{\Gamma}},\mathbf{d}=-2(\widetilde{\mathbf{\Gamma}}^{T}\widetilde{\mathbf{b}})$,
$\mathbf{D}=-\widehat{\mathbf{\Gamma}}(\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{H}^{T}\widehat{\mathbf{\Gamma}}^{T}$,
$\mathbf{c}=\widehat{\mathbf{b}}+\widehat{\mathbf{\Gamma}}(\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{H}^{T}\mathbf{d}$;
and form a Lagrangian from the original problem (DS2).
$\displaystyle D(\mu)$ $\displaystyle=$
$\displaystyle\min_{\mathbf{u}}\mathcal{L}(\mathbf{u},\mu)$ $\displaystyle=$
$\displaystyle\min_{\mathbf{u}}\left(\frac{1}{2}\mathbf{u}^{T}\mathbf{H}\mathbf{u}+\mathbf{d}^{T}\mathbf{u}+\widetilde{\mathbf{b}}^{T}\widetilde{\mathbf{b}}+\mu^{T}(-\widehat{\mathbf{\Gamma}}\mathbf{u}+\widetilde{\mathbf{b}})\right)$
Since the objective function is convex, the necessary and sufficient condition
for a minimum is that the gradient must vanish,i.e.,
$\mathbf{H}\mathbf{u}+\mathbf{d}-\hat{\mathbf{\Gamma}}^{T}\mathbf{\mu}=0.$
(22)
For $n<N$, $\widetilde{\mathbf{\Gamma}}$ is not full rank. Therefore,
$\mathbf{H}$ is singular and there exist multiple solutions to (22). Using
pseudoinverse [9], we can find a solution to (22) given by
$\mathbf{u}=-(\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{H}^{T}\left(\mathbf{d}-\hat{\mathbf{\Gamma}}^{T}\mu\right).$
Thus, after replacing into (III-B), we obtain $\mu$ as a solution to the dual
problem (DDS2).
$\textrm{(DDS2)}\max_{\mu\geq
0}\frac{1}{2}\mu^{T}\mathbf{D}\mu+\mu^{T}\mathbf{c}-\frac{1}{2}\mathbf{d}^{T}(\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{H}^{T}\mathbf{d}+\mathbf{b}^{T}\mathbf{b}$
(23)
The problem (LDS2) and dual problem (DDS2) can be solved using unconstrained
optimization algorithms in [10], [8].
## IV Iterative Algorithm
In this section, we develop algorithm for the case of nonempty $F$ set. Let
$u_{i}(n)$ denote the power at channel $i$ at step $n$. An iterative algorithm
is given as follows.
$\left\\{\begin{array}[]{ll}u_{i}(n+1)=\frac{\beta_{i}}{\alpha_{i}}-\frac{1}{a_{i}}\left(\frac{1}{OSNR_{i}(n)}-\Gamma_{i,i}\right)u_{i}(n),&\forall
i\in\mathcal{N}_{1};\\\
u_{i}(n+1)=\frac{\gamma_{i}}{1-\gamma_{i}\Gamma_{i,i}}\left(\frac{1}{OSNR_{i}(n)}-\Gamma_{i,i}\right)u_{i}(n),&\forall
i\in\mathcal{N}_{2}.\end{array}\right.$ (24)
###### Theorem IV.1
Algorithm (24) converges provided that $a_{i}>\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}$ and $\gamma_{i}$ is chosen such that
$\gamma_{i}<\frac{1}{\sum_{j\in\mathcal{N}}\Gamma_{i,j}}$.
###### Proof:
We use a similar approach from [3] to show the convergence of (24). Let’s
define $e_{i}(n)=u_{i}(n)-u_{i}^{*}$, where $u_{i}^{*}$ is given in (12).
Since $\overline{\mathbf{\Gamma}}\mathbf{u}^{*}=\overline{\mathbf{b}}$,
$\widetilde{\Gamma}_{i,i}u_{i}^{*}+\sum_{j\neq
i}\widetilde{\Gamma}_{i,j}u_{j}^{*}=\tilde{b}_{i}$, for $i\in\mathcal{N}_{1}$;
and, $\widehat{\Gamma}_{i,i}u_{i}^{*}+\sum_{j\neq
i}\widehat{\Gamma}_{i,j}u_{j}^{*}=\hat{b}_{i}$, for $i\in\mathcal{N}_{2}$.
Substitute the expression for $u_{i}^{*}$ into $e_{i}(n+1)$, and we obtain
$e_{i}(n+1)=u_{i}(n+1)-u_{i}^{*}=-\frac{1}{{a_{i}}}\left[\sum_{j\neq
i}{\Gamma}_{i,j}(u_{j}(n)-u_{j}^{*})\right]$, for $i\in\mathcal{N}_{1}$; and
$e_{i}(n+1)=u_{i}(n+1)-u_{i}^{*}=\frac{1}{1-{\Gamma}_{i,i}\gamma_{i}}\left[\sum_{j\neq
i}{\Gamma}_{i,j}\gamma_{i}(u_{j}(n)-u_{j}^{*})\right]$, for
$i\in\mathcal{N}_{2}$. Let $\mathbf{e}=[e_{i}(n)],i\in\mathcal{N}$. Therefore,
for $i\in\mathcal{N}_{1}$,
$\displaystyle|e_{i}(n+1)|$ $\displaystyle=$
$\displaystyle\left|\frac{1}{a_{i}}\left[\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}(e_{j}(n))\right]\right|$ (25)
$\displaystyle\leq$ $\displaystyle\frac{1}{a_{i}}\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}\max_{j\in\mathcal{N}}|e_{j}(n)|$ (26)
$\displaystyle\leq$ $\displaystyle\frac{1}{a_{i}}\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}\|\mathbf{e}(n)\|_{\infty}.$ (27)
and similarly, for $i\in\mathcal{N}_{2}$,
$\displaystyle|e_{i}(n+1)|$ $\displaystyle=$
$\displaystyle\left|\frac{1}{1-\Gamma_{i,i}\gamma_{i}}\left[\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}\gamma_{i}(e_{j}(n))\right]\right|$ (28)
$\displaystyle\leq$
$\displaystyle\frac{\gamma_{i}}{|1-\Gamma_{i,i}\gamma_{i}|}\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}\max_{j\in\mathcal{N}}|e_{j}(n)|.$
$\displaystyle\leq$
$\displaystyle\frac{\gamma_{i}}{|1-\Gamma_{i,i}\gamma_{i}|}\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}\|\mathbf{e}(n)\|_{\infty}.$ (29)
Since we assumed that $a_{i}>\sum_{j\neq i,j\in\mathcal{N}}\Gamma_{i,j}$ and
$\gamma_{i}$ is chosen such that
$\gamma_{i}<\frac{1}{\sum_{j\in\mathcal{N}}\Gamma_{i,j}}\leq\frac{1}{\Gamma_{i,i}}$,
we can conclude that $\|\mathbf{e}(n)\|\rightarrow 0$ from the contraction
mapping theorem. As a result, we have $u_{i}(n)\rightarrow u_{i}^{*}$ as
$n\rightarrow\infty$, for $i\in\mathcal{N}$. ∎
###### Remark IV.1
From the proof, we note that the rate of convergence of is determined by
$\sigma=\max\left\\{\max_{i\in\mathcal{N}_{1}}\frac{\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}}{a_{i}},\max_{i\in\mathcal{N}_{2}}\frac{\sum_{j\neq
i,j\in\mathcal{N}}\Gamma_{i,j}\gamma_{i}}{1-\Gamma_{i,i}\gamma_{i}}\right\\}.$
In addition, it is easy to observe that the OSNR target-seeking users are
algorithmically equivalent to competition seeking users by letting
$\beta_{i}/\alpha_{i}=0$ and $a_{i}=\Gamma_{i,i}-\frac{1}{\gamma_{i}}$,
$i\in\mathcal{N}_{2}$. This is because no notion of pricing is used for the
OSNR target seekers and they just have a utility target to meet or
equivalently optimize by letting $a_{i}=\Gamma_{i,i}-\frac{1}{\gamma_{i}}$.
## V Numerical Examples
In this section, we illustrate the concept by a MATLAB simulation. We consider
an end-to-end link described in Figure 1 with 5 amplified spans. We assume
channels are transmitted at wavelengths distributed centered around 1555nm
with channel separation of 1nm. Suppose input noise power is 0.5 percent of
the input signal power. The gain profile for each amplifier is identically
assumed to be parabolic as in Figure 4, which is normalized with respect to
$G_{\max}=30.0$dB. Suppose 20dB is the target OSNR level for users who just
want to meet a satisfactory level of transmission. We first show the case of 3
users, in which 2 users need better quality of service and one user is simply
interested in meeting 20dB as a target. From Figure 5, we can observe that
users who need better services reach an OSNR of 26.33dB and 29.20dB,
respectively. With an appropriate choice of initial conditions, the algorithm
quickly converges in 1-2 steps. In Figure 6, we similarly show the case of 30
users, in which 20 are game players and 10 are target seekers.
Figure 4: Optical Amplifier Spectral Profile Figure 5: OSNR simulation with 3
users in time steps Figure 6: OSNR simulation with 30 users in time steps
## VI Conclusion
In this paper, we examined a generalized power control model in optical
networks, which combines features of central cost approach and game-
theoretical approach. It enables two major service types in the network. One
is game player, who pays for his power consumption and the other is target
seeker, who is satisfied with a minimum service level set by the network. We
discussed two different solutions concepts for nonempty and empty feasible set
respectively and specifically designed an iterative algorithm that converges
to a unique solution for the case of nonempty feasible set. The convergence of
the algorithm was proved and illustrated by numerical examples of a WDM end-
to-end optical link.
In this work, we didn’t include capacity constraints for the sake of
simplicity. We hope this work will lead to future investigations of more
complicated cases where network constraints and nonlinear effects are
considered. In addition, we expect this framework to be used to solve similar
problems in other types of networks, for example, wireless networks.
## References
* [1] G. Agrawal, _Lightwave Technology_. Wiley-Interscience, 2005.
* [2] L. Pavel, “OSNR optimization in optical networks: Modeling and distributed algorithms via a central cost approach,” _IEEE Journal on Selected Areas in Communications_ , vol. 24, no. 4, pp. 54–65, April 2006.
* [3] ——, “A noncooperative game approach to OSNR optimization in optical networks,” _IEEE Transactions on Automatic Control_ , vol. 51, no. 5, pp. 848–852, May 2006.
* [4] Y. Pan and L. Pavel, “OSNR optimization in optical networks: Extension for capacity constraints,” _Proceedings of 2005 American Control Conference_ , pp. 2379–2385, June 2005.
* [5] R. Srikant, E. Altman, T. Alpcan, and T. Basar, “CDMA uplink power control as noncooperative game,” _Wireless Networks_ , vol. 8, p. 659 690, 2002\.
* [6] C. Saraydar, N. Mandayam, and D. Goodman, “Efficient power control via pricing in wireless data networks,” _IEEE Transactions on Communications_ , vol. 50, no. 2, pp. 291–414, February 2002.
* [7] C. Saraydar and D. Goodman, “Pricing and power control in a multicell wireless data network,” _IEEE Journal of Selected Areas of Communications_ , vol. 19, no. 10.
* [8] D. Bertsekas, _Nonlinear Programming_. Athena Scientific, 2003.
* [9] R. Horn and C. Johnson, _Matrix Analysis_. Cambridge University Press, 1990.
* [10] M. Bazaraa, H. Sherali, and C. Shetty, _Nonlinear Programming: Theory and Algorithms_ , 2nd ed. Wiley, 1993.
|
arxiv-papers
| 2011-03-13T03:10:36 |
2024-09-04T02:49:17.611050
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quanyan Zhu and Lacra Pavel",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1103.2490"
}
|
1103.2491
|
# Heterogeneous Learning in Zero-Sum Stochastic Games with Incomplete
Information
Quanyan Zhu†, Hamidou Tembine‡ and Tamer Başar† This work was supported in
part by a grant from AFOSR.†Q. Zhu and T. Başar are with Dept. ECE and CSL,
University of Illinois, 1308 West Main, Urbana, IL, 61801, USA. {zhu31,
basar1}@illinois.edu‡H. Tembine is with Supélec, 3 rue Joliot-Curie 91192 Gif-
sur-Yvette cedex, France tembine@ieee.org
###### Abstract
Learning algorithms are essential for the applications of game theory in a
networking environment. In dynamic and decentralized settings where the
traffic, topology and channel states may vary over time and the communication
between agents is impractical, it is important to formulate and study games of
incomplete information and fully distributed learning algorithms which for
each agent requires a minimal amount of information regarding the remaining
agents. In this paper, we address this major challenge and introduce
heterogeneous learning schemes in which each agent adopts a distinct learning
pattern in the context of games with incomplete information. We use stochastic
approximation techniques to show that the heterogeneous learning schemes can
be studied in terms of their deterministic ordinary differential equation
(ODE) counterparts. Depending on the learning rates of the players, these ODEs
could be different from the standard replicator dynamics, (myopic) best
response (BR) dynamics, logit dynamics, and fictitious play dynamics. We apply
the results to a class of security games in which the attacker and the
defender adopt different learning schemes due to differences in their
rationality levels and the information they acquire.
## I Introduction
Distributed iterative schemes play an important role in the computation of
equilibria and the estimation of payoffs under incomplete information [2].
This paper studies a two-person zero-sum stochastic game with an arbitrary
number of states and a finite number of actions for each player. When each
player has a complete knowledge of its payoff function and has past access to
past actions of the others, then there is an arsenal of tools such as
fictitious play algorithms, best response dynamics, and gradient-based
algorithms, that can be used to arrive at the equilibrium of the game.
However, it is well known that these algorithms may fail to converge even
under the perfect observation of actions and payoffs [5, 11, 10, 3]. A new
learning challenge hence arises when a player does not know its own payoff
function and/or has no information about the past actions of the other
players. In this case, the player needs to interact with the environment to
find out its expected payoff and its optimal strategy.
In practical applications, we are often in search of distributed learning
algorithms that require a minimal amount of information and a minimal amount
of resources. It is then natural to ask whether there exists a learning scheme
that demands less information and less memory within a dynamically evolving
environment, and leads to an efficient, stable and fair outcome. In this
paper, we address this challenge by proposing a class of heterogeneous
learning algorithms in a scenario where the players do not know their own
payoff functions. At each time $t$, each player chooses an action and receives
a numerical value for its payoff or perceived payoff as an outcome of the
instantaneous game. In contrast to fictitious play and best response dynamics
which require the knowledge of the history of actions played by the other
players, our learning algorithm relaxes this assumption. Indeed, it is often
implausible and impractical in applications to assume the capability of
observations of the actions of the other players. Furthermore, we assume that
the state space of the game and its transition law between the states are
unknown to the players. In addition, the players also do not have the
knowledge of the action spaces of the others. The question we will address is
how much the players can expect to learn under such circumstances?
We propose different coupled (or combined) and fully distributed learning
schemes that enable learning optimal strategies and concurrently estimating
the optimal payoffs. In contrast to the standard reinforcement learning
algorithms which focus only on either strategy or payoff reinforcement for the
equilibrium learning, the algorithm that couples the payoff-reinforcement
learning together with strategy-reinforcement learning enables an immediate
prediction and updates the strategies by updated estimations based on recent
experiences. Our learning algorithms also offer the degrees of freedom to
model different levels of rationality and learning rates of the players. The
ordinary differential equations (ODEs) associated with the stochastic learning
algorithms differ from the standard replicator dynamics, best response
dynamics and fictitious play dynamics. Particular connections to logit
dynamics and imitative logit dynamics are also established. Using basic
stochastic approximation techniques from [6, 9, 3, 10] and under suitable
assumptions on the learning rates, we show their convergence to a new class of
game dynamics and asymptotic properties of different learning algorithms
within a class of zero-sum stochastic games.
The paper is structured as follows. In next section, we present the zero-sum
stochastic game model and provide an overview of the basic properties of
reinforcement learning algorithms. Section III presents our main results on
heterogeneous learning algorithms. In Section IV, we apply the learning
algorithms to study security games and provide numerical results. Section V
concludes the paper and discusses future work.
## II Game Model and Learning Algorithms
In this section, we formulate a two-person zero-sum stochastic game model
$\Xi=\langle\mathcal{S},\mathcal{A}_{1},\mathcal{A}_{2},\\{U(s,.)\\}_{s\in\mathcal{S}}\rangle$
where $\mathcal{A}_{1},\mathcal{A}_{2}$ are the finite sets of actions
available to players P1 and P2, respectively, and $\mathcal{S}$ is the set of
possible states. We assume that the state space $\mathcal{S}$ and the
probability distribution on the states are both unknown to the players. A
state $s\in\mathcal{S}$ is an independent and identically distributed random
variable defined on the set $\mathcal{S}$. We assume the action spaces are the
same in each state. The zero-sum game is characterized by a single utility
function
$U:\mathcal{S}\times\mathcal{A}_{1}\times\mathcal{A}_{2}\rightarrow\mathbb{R}$.
P1 collects a payoff $U_{1}(s,a_{1},a_{2})=U(s,a_{1},a_{2})$ when he chooses
$a_{1}\in\mathcal{A}_{1}$ and P2 uses $a_{2}\in\mathcal{A}_{2}$ at state
$s\in\mathcal{S}$, and for the same choices P2 collects a payoff of
$U_{2}(s,a_{1},a_{2})=c-U(s,a_{1},a_{2});$ equivalently, $U(s,a_{1},a_{2})-c$
is cost to P2, where $c$ is a constant. In terms of the single utility
function $U$, P1 is the maximizer and P2 is the minimizer, and both players
are interested in the performance at steady state using mixed strategies, as
to be made clear shortly. The preceding game model can be viewed as a special
class of stochastic games in which the state transitions are independent of
the player actions as well as the current state. Note that what we have here
is a constant-sum game, where the constant is $c$. In the analysis of its
equilibrium, we can let $c=0$ without any loss of generality, and hence view
it as a zero-sum game.
We have slotted time, $t\in\\{0,1,\ldots\\}$, when players pick their mixed
strategies as functions of what has transpired in the past, to the extent the
information available to them allows. Toward this end, we let $f_{t}(a_{1})$
and $g_{t}(a_{2})$ denote the probabilities of P$1$ choosing
$a_{1}\in\mathcal{A}_{1}$ and P2 choosing $a_{2}\in\mathcal{A}_{2}$,
respectively, at time $t$, and let
$\mathbf{f}_{t}=[f_{t}(a_{1})]_{a_{1}\in\mathcal{A}_{1}}$ and
$\mathbf{g}_{t}=[g_{t}(a_{2})]_{a_{2}\in\mathcal{A}_{2}}$ be the mixed
strategies of P1 and P2 respectively (at time $t$), where more precisely
$\displaystyle\mathbf{f}_{t}\in\mathcal{F}:=\left\\{\mathbf{f}:\
f(a_{1})\in[0,1],\sum_{a_{1}\in\mathcal{A}_{1}}f(a_{1})=1\right\\};$ (1)
$\displaystyle\mathbf{g}_{t}\in\mathcal{G}:=\left\\{\mathbf{g}:\
g(a_{2})\in[0,1],\sum_{a_{2}\in\mathcal{A}_{2}}g(a_{2})=1\right\\}.$ (2)
In particular, we define $e_{a_{1}},e_{a_{2}},$ with
$a_{1}\in\mathcal{A}_{1},a_{2}\in\mathcal{A}_{2},$ as unit vectors of sizes
$|\mathcal{A}_{1}|$ and $|\mathcal{A}_{2}|$ , respectively, whose entry that
corresponds to $a_{1}$ or $a_{2}$ is 1 while others are zeros. We assume that
the mixed strategies of the players are independent of the current state $s.$
For any given pair of mixed strategies,
$(\mathbf{f}\in\mathcal{F},\mathbf{g}\in\mathcal{G}),$ and for a fixed $s\in
S$, we define the expected utility (as expected payoff to P1 and expected cost
to P2) as
$\mathbb{U}(s,\mathbf{f},\mathbf{g}):=\mathbb{E}_{\mathbf{f},\mathbf{g}}U(s,a_{1},a_{2}),$
where $\mathbb{E}_{\mathbf{f},\mathbf{g}}$ denotes expectation of $U$ over the
action sets of the players under the given mixed strategies. A further
expectation of this quantity over the states $s$, denoted $\mathbb{E}_{s}$,
yields the performance index of the expected game. We now define the
equilibrium concept of interest for this game, that is the saddle-point
equilibrium:
###### Definition II-A (Saddle Point)
A strategy pair $(\mathbf{f}^{*},\mathbf{g}^{*})$ constitutes a saddle point
for the expected game if and only if $\forall\mathbf{f}\in\mathcal{F}$ and
$\mathbf{g}\in\mathcal{G}$,
$\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f},\mathbf{g}^{*})\leq\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f}^{*},\mathbf{g}^{*})\leq\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f}^{*},\mathbf{g}).$
(3)
This now being a finite zero-sum game (or constant sum game, if $c\neq 0$),
the existence of a saddle point is guaranteed by the minimax theorem.
We now consider this game played over the discrete-time horizon, with the
players generating mixed strategies, say $(\mathbf{f}_{t},\mathbf{g}_{t})$ at
every time point $t$. These strategies will be generated (recursively updated)
according to some rule, which uses the information available to the players.
As indicated before, the players do not know the functional form of $U$, that
is they do not know the entries of the underlying matrix, but at each time $t$
they observe the value $U(s,a_{1,t},a_{2,t})$, where the actions are realized
under $(\mathbf{f}_{t},\mathbf{g}_{t})$, and they recall their own past
actions. With this information, P1 and P2 generate, respectively,
$\mathbf{f}_{t+1}$ and $\mathbf{g}_{t+1}$. The precise way of doing this is
determined by the algorithm picked, and there will be several such algorithms
as will be discussed shortly. For each one, our goal is to show that the
sequences thus generated converge to the pair of mixed saddle-point
strategies, that is $\lim_{t\to\infty}\mathbf{f}_{t}=\mathbf{f}^{*},\
\lim_{t\to\infty}\mathbf{g}_{t}=\mathbf{g}^{*},$ where the limit will be given
a precise meaning later.
### A. Learning Schemes
To achieve the saddle-point solution, we suggest the following reinforcement
learning mechanism for homogeneous learners. We use the abbreviation “RL” for
“reinforcement learning” and “C” for “combined”, suggesting that the algorithm
involves learning the expected utility as well as the strategies. We consider
combined fully distributed, payoff and strategy reinforcement learning
(CODIPAS-RL) in the form:
$\
\left\\{\begin{array}[]{ccc}\mathbf{f}_{t+1}&=&\mathbf{f}_{t}+\Pi_{11}(\lambda_{1,t},a_{1,t},U_{1,t},\hat{\mathbf{u}}_{1,t},\mathbf{f}_{t})\\\
\hat{\mathbf{u}}_{1,t+1}&=&\hat{\mathbf{u}}_{1,t}+\Pi_{12}(\mu_{1,t},a_{1,t},U_{1,t},\mathbf{f}_{t},\hat{\mathbf{u}}_{1,t})\\\
\mathbf{g}_{t+1}&=&\mathbf{g}_{t}+\Pi_{21}(\lambda_{2,t},a_{2,t},U_{2,t},\hat{\mathbf{u}}_{2,t},\mathbf{g}_{t})\\\
\hat{\mathbf{u}}_{2,t+1}&=&\hat{\mathbf{u}}_{2,t}+\Pi_{22}(\mu_{2,t},a_{2,t},U_{2,t},\mathbf{g}_{t},\hat{\mathbf{u}}_{2,t})\\\
&&t\geq 0,a_{i,t}\in\mathcal{A}_{i},i\in\\{1,2\\},\end{array}\right.$
where $\Pi_{i1},\Pi_{i2},i\in\\{1,2\\},$ are properly chosen functions. The
parameters $\lambda_{i,t},\mu_{i,t}$ are learning rates indicating players’
capabilities of information retrieval and update. The vectors
$\mathbf{f}_{t}\in\mathcal{F},\mathbf{g}_{t}\in\mathcal{G}$ are mixed
strategies of the players at time $t$. $\hat{\mathbf{u}}_{i,t},i\in\\{1,2\\},$
are estimated average payoffs updated at each iteration $t$, and
$U_{i,t},i\in\\{1,2\\},$ are the perceived payoffs received by players at time
$t$.
We identify below five different special cases of this general class of
learning algorithms, each one important in its own right.
#### II-A1 CRL0
The first COmbined fully DIstributed PAyoff and Strategy Reinforcement
Learning (CODIPAS-RL) algorithm is CRL0 given in (4) below, which captures the
procedure in [5] for both payoffs and strategies. At every time step $t$, P1
and P2 each chooses an action according to their estimations and their mixed
strategy vectors $\mathbf{f}_{t}$ and $\mathbf{g}_{t}$, respectively. Based on
the joint action, each player perceives his instantaneous payoff $U_{i,t}$,
$i\in\\{1,2\\}$, and updates his strategy vectors. The strategy and utility
updates are not coupled and do not involve optimal choices of the players. The
players make updates by taking a weighted average of the current observed
payoff and the quantities from the previous iteration. The indicator function
${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{i,t}\\}}$ is a unit vector of appropriate dimension
with one of its components corresponding to the action chosen at time $t$,
$a_{i,t}$, being $1$ and the others being zeros. The step size parameters
$\lambda_{i,t}$ need to be small enough such that $\lambda_{i,t}U_{i,t}<1$ for
all $t$.
$\left\\{\begin{array}[]{lll}\mathbf{f}_{t+1}&=&\mathbf{f}_{t}+\lambda_{1,t}{U}_{1,t}\cdot\left({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1,t}=a_{1}\\}}-\mathbf{f}_{t}\right)\\\
\hat{\mathbf{u}}_{1,t+1}&=&\hat{\mathbf{u}}_{1,t}+\mu_{1,t}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1,t}=a_{1}\\}}\left(U_{1,t}-\hat{\mathbf{u}}_{1,t}\right),\
a_{1}\in\mathcal{A}_{1}\\\
\mathbf{g}_{t+1}&=&\mathbf{g}_{t}+\lambda_{2,t}{U}_{2,t}\cdot\left({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{2,t}=a_{2}\\}}-\mathbf{g}_{t}\right)\\\
\hat{\mathbf{u}}_{2,t+1}&=&\hat{\mathbf{u}}_{2,t}+\mu_{2,t}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{2,t}=a_{2}\\}}\left(U_{2,t}-\hat{\mathbf{u}}_{2,t}\right),\
a_{2}\in\mathcal{A}_{2}\\\ \end{array}\right.$ (4)
#### II-A2 CRL1
Algorithm CRL1 given in (5) below is another combined algorithm that learns
the average utility and the mixed strategies concurrently. This is a
Boltzmann-Gibbs based CODIPAS-RL. In a similar fashion as in CRL0, P1 and P2
select their actions based on their current strategy distributions. However,
the updates on the strategies and the average payoff follow reinforcement
learning and $\lambda_{i,t}$ and $\mu_{i,t}$ are the learning rates for the
payoffs and the strategies respectively, satisfying Assumption II-A6 and
$\frac{\lambda_{i,t}}{\mu_{i,t}}\rightarrow 0,i\in\\{1,2\\}$.
$\left\\{\begin{array}[]{lll}\mathbf{f}_{t+1}&=&(1-\lambda_{1,t})\mathbf{f}_{t}+\lambda_{1,t}\tilde{\beta}_{1,\epsilon}(\hat{\mathbf{u}}_{1,t})\\\
\hat{\mathbf{u}}_{1,t+1}&=&\hat{\mathbf{u}}_{1,t}+\frac{\mu_{1,t}}{f_{t}(a_{1})}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1,t}=a_{1}\\}}\left(U_{1,t}-\hat{\mathbf{u}}_{1,t}\right),\
a_{1}\in\mathcal{A}_{1}\\\
\mathbf{g}_{t+1}&=&(1-\lambda_{2,t})\mathbf{g}_{t}+\lambda_{2,t}\tilde{\beta}_{2,\epsilon}(\hat{\mathbf{u}}_{2,t})\\\
\hat{\mathbf{u}}_{2,t+1}&=&\hat{\mathbf{u}}_{2,t}+\frac{\mu_{2,t}}{g_{t}(a_{2})}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{2,t}=a_{2}\\}}\left(U_{2,t}-\hat{\mathbf{u}}_{2,t}\right),\
a_{2}\in\mathcal{A}_{2}\end{array}\right.$ (5)
where
$\tilde{\beta}_{i,\epsilon}:\mathbb{R}^{|\mathcal{A}_{i}|}\rightarrow\mathbb{R}^{|\mathcal{A}_{i}|},i\in\\{1,2\\},$
is the Boltzmann-Gibbs strategy or the soft-max function parameterized by
$\epsilon\geq 0$, which takes in the average payoff vector and produces a
vector that assigns more weight to the maximum component. The weight assigned
to a particular action $a_{i}\in\mathcal{A}_{i},i\in\\{1,2\\}$ is given by
$\tilde{\beta}_{i,\epsilon}(\hat{\mathbf{u}}_{i,t})(a_{i})=\frac{e^{\frac{1}{\epsilon}\hat{u}_{i,t}(a_{i})}}{\sum_{a_{i}^{\prime}}e^{\frac{1}{\epsilon}\hat{u}_{i,t}(a^{\prime}_{i})}},a_{i}\in\mathcal{A}_{i},i\in\\{1,2\\}.$
(6)
It is clear that when $\epsilon$ is high, the output of the
$\tilde{\beta}_{i,\epsilon}$ function does not distinguish among the actions
and assign equal weights to them; when $\epsilon$ approaches zero,
$\tilde{\beta}_{i,\epsilon}$ function bears more resemblance with the maximum
function, assigning $1$ to the action yielding the maximum average payoff but
zeros to the other actions [4].
#### II-A3 CRL2
The procedure for the CODIPAS-RL algorithm CRL2 is similar to CRL1 but only
differs in the use of soft-max function. In place of the Boltzmann-Gibbs
strategy, we adopt imitative Boltzmann-Gibbs strategy which is weighted by the
current strategy vector [7], and is given by
$\sigma_{i}:\mathbb{R}^{|\mathcal{A}_{i}|}\times\mathbb{R}^{|\mathcal{A}_{i}|}\rightarrow\mathbb{R}^{|\mathcal{A}_{i}|},i\in\\{1,2\\}$.
The component-wise mapping for P1 is expressed by
$\sigma_{1}(\mathbf{f}_{t},\hat{\mathbf{u}}_{1,t})(a_{1})=\frac{f_{t}(a_{1})e^{\frac{1}{\epsilon}\hat{u}_{1,t}(a_{1})}}{\sum_{a_{1}^{\prime}\in\mathcal{A}_{1}}f_{t}(a^{\prime}_{1})e^{\frac{1}{\epsilon}\hat{u}_{1,t}(a^{\prime}_{1})}}.$
(7)
Likewise, for P2, we have
$\sigma_{2}(\mathbf{g}_{t},\hat{\mathbf{u}}_{2,t})(a_{2})=\frac{g_{t}(a_{2})e^{\frac{1}{\epsilon}\hat{u}_{2,t}(a_{2})}}{\sum_{a_{2}^{\prime}\in\mathcal{A}_{2}}g_{t}(a^{\prime}_{2})e^{\frac{1}{\epsilon}\hat{u}_{2,t}(a^{\prime}_{2})}}.$
(8)
Collecting all this, the CRL2 algorithm is then as given below:
$\left\\{\begin{array}[]{lll}\mathbf{f}_{t+1}&=&(1-\lambda_{1,t})\mathbf{f}_{t}+\lambda_{1,t}\sigma_{1}(\mathbf{f}_{t},\hat{\mathbf{u}}_{1,t})\\\
\hat{\mathbf{u}}_{1,t+1}&=&\hat{\mathbf{u}}_{1,t}+\frac{\mu_{1,t}}{f_{t}(a_{1})}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1,t}=a_{1}\\}}\left(U_{1,t}-\hat{\mathbf{u}}_{1,t}\right)\\\
\mathbf{g}_{t+1}&=&(1-\lambda_{2,t})\mathbf{g}_{t}+\lambda_{2,t}\sigma_{2}(\mathbf{g}_{t},\hat{\mathbf{u}}_{2,t})\\\
\hat{\mathbf{u}}_{2,t+1}&=&\hat{\mathbf{u}}_{2,t}+\frac{\mu_{2,t}}{g_{t}(a_{2})}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{2,t}=a_{2}\\}}\left(U_{2,t}-\hat{\mathbf{u}}_{2,t}\right)\end{array}\right.$
(9)
#### II-A4 RL2
The learning algorithm (10) updates strategies simultaneously [5, 1].
$\left\\{\begin{array}[]{lll}\mathbf{f}_{t+1}&=&\mathbf{f}_{t}+\lambda_{1,t}U_{1,t}\cdot\left({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1,t}=a_{1}\\}}-\mathbf{f}_{t}\right)\\\
\mathbf{g}_{t+1}&=&\mathbf{g}_{t}+\lambda_{2,t}U_{2,t}\cdot\left({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{2,t}=a_{2}\\}}-\mathbf{g}_{t}\right)\end{array}\right.$
(10)
#### II-A5 RL3
In RL3, we normalize RL2 by some constant $n$ and $C$. This algorithm has
appeared in [1] and is summarized below in (11):
$\left\\{\begin{array}[]{lll}\mathbf{f}_{t+1}&=&\frac{C(n+1)}{nC+U_{1,t}}\left[\mathbf{f}_{t}+U_{1,t}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1,t}=a_{1}\\}}\right]\\\
\mathbf{g}_{t+1}&=&\frac{C(n+1)}{nC+U_{2,t}}\left[\mathbf{g}_{t}+U_{2,t}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{2,t}=a_{2}\\}}\right]\end{array}\right.$ (11)
The following assumption on learning rates is adopted for all the above listed
learning schemes.
###### Assumption II-A6
The learning rates $\lambda_{i,t},\mu_{i,t}$, $i\in\\{1,2\\}$, satisfy the
following conditions:
$\lambda_{i,t}\geq 0,\ \sum_{t\geq 1}\lambda_{i,t}=+\infty,\ \sum_{t\geq
1}\lambda_{i,t}^{2}<+\infty,i\in\\{1,2\\}$ (12) $\mu_{i,t}\geq 0,\ \sum_{t\geq
1}\mu_{i,t}=+\infty,\ \sum_{t\geq 1}\mu_{i,t}^{2}<+\infty,i\in\\{1,2\\}$ (13)
The learning rate which perhaps has the simplest form that satisfies the
conditions of Assumption II-A6 is the harmonic sequence, i.e., $\textrm{(R1)}\
\mu_{i,t}=\frac{1}{t+1}.$ To study learning on different time scales, we need
to consider other learning rates. Typical learning rates are $\textrm{(R2)}\
\mu_{i,t}=\frac{1}{(t+1)\log(t+1)},$ $\textrm{(R3)}\
\mu_{i,t}=\frac{1}{\sqrt{t+1}\log^{2}(t+1)},$ $\textrm{(R4)}\
\mu_{i,t}=\frac{1}{(t+c^{\prime})^{\rho_{i}}},\ \frac{1}{2}<\rho_{i}\leq 1,\
c^{\prime}>0.$ It is clear that the learning rate (R1) is faster than (R2) and
(R3). In addition, by scaling $\rho_{i}$ in (R4), we can obtain learning rates
on different time scales.
### II-B Basic properties
#### II-B1 Properties of RL2, RL3 and CRL0
The algorithm RL2 has been studied by Borgers and Sarin in [5]. The algorithm
RL3 is a normalized version of RL2. This version has been studied by Arthur in
[1]. These authors have shown that RL2 goes to a pseudo-trajectory of the
replicator dynamics when the learning rate $\lambda_{i,t}$ goes to zero.
Similarly the reinforcement learning RL3 goes to a trajectory of an adjusted
version of the replicator equation.
The learning algorithm CRL0 is obtained by combining these strategy
reinforcement learnings with a payoff reinforcement learning (Q-learning). The
Q-learning is known to be convergent to the expected payoffs if all the
actions are sufficiently used and the learning parameters satisfy the standard
conditions. The combination of these two approaches gives a new learning
algorithm called combined fully distributed payoff and strategy reinforcement
learning (CODIPAS-RL). With this new algorithm, the players will be able to
learn both expected payoffs and the associated optimal strategies i.e., if
$(\mathbf{f}_{t},\hat{u}_{1,t},\mathbf{g}_{t},\hat{u}_{2,t})\longrightarrow(\mathbf{f}^{*},\hat{u}_{1}^{*},\mathbf{g}^{*},\hat{u}_{2}^{*})$,
then $(\mathbf{f}^{*},\mathbf{g}^{*})$ is a saddle point of the expected game
and
$\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f}^{*},\mathbf{g}^{*})=\hat{u}_{1}^{*}=c-\hat{u}^{*}_{2}.$
Moreover, the strategies are generated by the replicator equation:
$\displaystyle\dot{f}_{t}(a_{1})$ $\displaystyle=$
$\displaystyle{f}_{t}(a_{1})[{u}_{1}(e_{a_{1}},\mathbf{g}_{t})-\sum_{a^{\prime}_{1}\in\mathcal{A}_{1}}{u}_{2}(e_{a^{\prime}_{1}},\mathbf{g}_{t})f_{t}(a^{\prime}_{1})]$
$\displaystyle\dot{g}_{t}(a_{2})$ $\displaystyle=$
$\displaystyle{g}_{t}(a_{2})[{u}_{2}(\mathbf{f}_{t},e_{a_{2}})-\sum_{a^{\prime}_{2}\in\mathcal{A}_{2}}{u}_{2}(\mathbf{f}_{t},e_{a^{\prime}_{2}})g_{t}(a^{\prime}_{2})]$
where
$u_{1}(\mathbf{f}^{*},\mathbf{g}^{*})=\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f}^{*},\mathbf{g}^{*})$
and $u_{2}(.)=c-u_{1}(.).$
A major inconvenience with CODIPAS-RL, CRL0, RL2 and RL3 is that the rest
points (equilibrium states) of the corresponding ODEs are not necessarily
equilibria of the expected game. For example, all the faces of the simplex are
forward invariant (when started on one face, the trajectory of the replicator
dynamics remains on that face). As well known, the game may not have an
equilibrium on that face. Therefore, the outcome of the replicator dynamics
may not be an equilibrium. To resolve this problem, one can fix the starting
point at the relative interior of the simplex (for example, the uniform
distribution can be chosen as initial point). Then, we have the following
conclusions.
1. (S1)
If started in the interior, the dominated strategies will be eliminated.
2. (S2)
If started in the interior, and if the trajectory goes to the boundary, then
the outcome is an equilibrium.
3. (S3)
If there is a cyclic orbit of the dynamics, the limit cycle contains an
equilibrium in its interior.
4. (S4)
The expected payoff is learned if CODIPAS-RL CRL0 is used: $f(a_{1})>0$
implies that
$\hat{u}_{1,t}(a_{1})\longrightarrow\mathbb{E}_{s}\mathbb{U}(s,e_{a_{1}},\mathbf{g}),$
and similarly for P2, $g(a_{2})>0$ implies that
$\hat{u}_{2,t}(a_{2})\longrightarrow
c-\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f},e_{a_{2}}).$
Another way of eliminating the non-equilibrium rest points is to perturb the
game. The strategy can be perturbed using a small deviation from
$(\mathbf{f},\mathbf{g}),$ i.e., an action $a_{1}$ will be chosen with
probability $(1-\epsilon)f(a_{1})+\frac{\epsilon}{|\mathcal{A}_{1}|}.$
2) Properties of CRL1 and CRL2: Numerically, the approximation of CRL0, RL2
and RL3 can lead to the boundary of the simplex. To solve this problem, we
propose a modified version of CODIPAS-RL based on Boltzmann-Gibbs
distribution. These are the coupled reinforcement learning CRL1 and CRL2.
Since the Boltzmann-Gibbs distribution never vanishes, the new algorithm
CODIPAS-RL CRL1 based on Boltzmann-Gibbs is well defined for any initial
condition and preserves the property that every rest point is a Boltzmann-
Gibbs equilibrium, also called logit equilibrium, i.e., the fixed point of the
mapping
$\tilde{\beta}_{1,\epsilon}(\mathbb{E}_{s}\hat{u}_{1}({s},.,\mathbf{g}))=\mathbf{f},\tilde{\beta}_{2,\epsilon}(\mathbb{E}_{s}\hat{u}_{2}({s},\mathbf{f},.))=\mathbf{g}$
which is an $\epsilon-$saddle-point equilibrium. Thus, by choosing $\epsilon$
arbitrarily small, an approximate solution is obtained. The main advantage of
this Boltzmann-Gibbs distribution is that it is a smooth mapping (a
regularized version of the best-response correspondence).
## III Main results
In this section, we obtain ODE approximations of the learning algorithms in
Section II and show the convergence of different heterogeneous learning
algorithms to saddle-point solutions.
### III-A Convergence to ODE: the combined learning algorithms
We first examine the case where the players learn via different schemes but on
the same time scale or by the same learning rate, i.e., the factor
$\lambda_{i,t}=\lambda_{t},i\in\\{1,2\\},$ independent of the players. We use
${\beta}_{1,\epsilon}(\mathbf{g}_{t}):\Delta(\mathcal{A}_{2})\rightarrow\Delta(\mathcal{A}_{1})$
and
${\beta}_{2,\epsilon}(\mathbf{f}_{t}):\Delta(\mathcal{A}_{1})\rightarrow\Delta(\mathcal{A}_{2})$
to denote P1 and P2’s Boltzmann-Gibbs responses to the other player’s mixed
strategies and
${\beta}_{1,\epsilon}(\mathbf{g}_{t})(a_{1}):=\tilde{\beta}_{1,\epsilon}(u_{1}(e_{a_{1}},\mathbf{g}_{t}))$;
${\beta}_{2,\epsilon}(\mathbf{f}_{t})(a_{2}):=\tilde{\beta}_{2,\epsilon}(u_{2}(\mathbf{f}_{t},e_{a_{2}})),a_{1}\in\mathcal{A}_{1},a_{2}\in\mathcal{A}_{2}.$
###### Theorem III-A1
The combined learning algorithm with different learners using CRL1, RL2, RL3
converges to the joint system of ODEs. In particular, if P1 uses CRL1 and P2
adopts RL2, then the ODE is given by
$\left\\{\begin{array}[]{cll}\frac{d}{dt}\hat{u}_{1,t}(a_{1})&=&{u}_{1}(e_{a_{1}},\mathbf{g}_{t})-\hat{u}_{1,t}(a_{1}),\
a_{1}\in\mathcal{A}_{1},\\\
\dot{\mathbf{f}}_{t}&=&{\beta}_{1,\epsilon}(\mathbf{g}_{t})-\mathbf{f}_{t},\\\
\dot{g}_{t}(a_{2})&=&{g}_{t}(a_{2})[u_{2}(\mathbf{f}_{t},e_{a_{2}})\\\
&&-\sum_{a^{\prime}_{2}\in\mathcal{A}_{2}}u_{2}(\mathbf{f}_{t},e_{a^{\prime}_{2}})g_{t}(a^{\prime}_{2})],a_{2}\in\mathcal{A}_{2}.\end{array}\right.$
(14)
Moreover, if P2 adopts RL3 in lieu of RL2, then one has the adjusted
replicator dynamics instead of the standard replicator equation.
We now have the following corollary corresponding to different learning rates
for the two players.
###### Corollary III-A2
In the heterogeneous learning where players choose to adopt one learning
scheme among CRL1, RL2, RL3 and with different learning rates, we have the
following results.
(C1) If P1 uses CRL1 and P2 learns through RL2 with a rate $k_{2}$ faster than
P1’s rate, then the ODE is given by
$\left\\{\begin{array}[]{ccl}\frac{d}{dt}\hat{u}_{1,t}(a_{1})&=&{u}_{1}(e_{a_{1}},\mathbf{g}_{t})-\hat{u}_{1,t}(a_{1}),\
a_{1}\in\mathcal{A}_{1}\\\
\dot{\mathbf{f}}_{t}&=&{\beta}_{1,\epsilon}(\mathbf{g}_{t})-\mathbf{f}_{t}\\\
\dot{g}_{t}(a_{2})&=&k_{2}{g}_{t}(a_{2})[u_{2}(e_{a_{2}},\mathbf{f}_{t}),\\\
&&-\sum_{a^{\prime}_{2}\in\mathcal{A}_{2}}u_{2}(e_{a^{\prime}_{2}},\mathbf{f}_{t})g_{t}(a^{\prime}_{2})],a_{2}\in\mathcal{A}_{2}.\end{array}\right.$
Moreover, if P2 adopts RL3 in lieu of RL2, then one has the $k_{2}-$adjusted
replicator dynamics instead of the standard replicator equation.
(C2) If P1 uses CRL1 with a rate of learning $k_{1}$ faster than P2 who learns
with RL2, then the ODE is given by
$\left\\{\begin{array}[]{cll}\frac{d}{dt}\hat{u}_{1,t}(a_{1})&=&{u}_{1}(e_{a_{1}},\mathbf{g}_{t})-\hat{u}_{1,t}(a_{1}),\
a_{1}\in\mathcal{A}_{1},\\\
\dot{\mathbf{f}}_{t}&=&k_{1}\left[{\beta}_{1,\epsilon}(\mathbf{g}_{t})-\mathbf{f}_{t}\right],\\\
\dot{g}_{t}(a_{2})&=&{g}_{t}(a_{2})[u_{2}(e_{a_{2}},\mathbf{f}_{t})\\\
&&-\sum_{a^{\prime}_{2}\in\mathcal{A}_{2}}u_{2}(e_{a^{\prime}_{2}},\mathbf{f}_{t})g_{t}(a^{\prime}_{2})],a_{2}\in\mathcal{A}_{2}\end{array}\right.$
###### Lemma III-A3
(Explicit Solutions of Smooth BR Equation): Given P2’s trajectory
$\\{\mathbf{g}_{t^{\prime}}\\}_{t^{\prime}}$ and an initial condition
$\mathbf{f}_{0},$ the smooth best response equation
$\dot{\mathbf{f}}_{t}={\beta}_{1,\epsilon}(\mathbf{g}_{t})-\mathbf{f}_{t}$
(15)
in (14) has a unique solution given by the vectorial function
$\xi_{1}(\mathbf{g}_{t})(a_{1})={f}_{0}(a_{1})e^{-t}+e^{-t}\int_{0}^{t}z_{1,t^{\prime}}(a_{1})\
e^{t^{\prime}}dt^{\prime},\ a_{1}\in\mathcal{A}_{1},$ (16)
where $z_{1,t^{\prime}}={\beta}_{1,\epsilon}(\mathbf{g}_{t^{\prime}}).$ In
particular, if P2 is a slow learner i.e., $\mathbf{g}_{t}=\mathbf{g},$
constant in time, then the smooth best response equation of P1 converges to
$\xi_{1}(\mathbf{g})(a_{1})=(1-e^{-t}){\beta}_{1,\epsilon}(\mathbf{g})(a_{1})+e^{-t}f_{0}(a_{1}),\
a_{1}\in\mathcal{A}_{1},$ (17)
which goes to ${\beta}_{1,\epsilon}(\mathbf{g})$ when
$t\longrightarrow+\infty.$
###### Lemma III-A4
(Explicit Solutions of Replicator Equation): Given P2’s trajectory
$\\{\mathbf{g}_{t^{\prime}}\\}_{t^{\prime}}$ and an interior initial condition
$\mathbf{f}_{0},$ the replicator equation in (14) has a unique solution given
by the vectorial function
$\xi_{1}(\mathbf{g}_{t})(a_{1})=\frac{e^{\int_{0}^{t}u_{1}(e_{a_{1}},\mathbf{g}_{t^{\prime}})\
dt^{\prime}}}{\sum_{a^{\prime}_{1}\in\mathcal{A}_{1}}e^{\int_{0}^{t}u_{1}(e_{a^{\prime}_{1}},\mathbf{g}_{t^{\prime}})\
dt^{\prime}}},\ a_{1}\in\mathcal{A}_{1}$, with a normalization factor $f_{0}.$
In particular, if P2 is a slow learner, i.e. $\mathbf{g}_{t}=\mathbf{g}$,
constant in time, then the replicator equation of P1 converges to
$\xi_{1}(\mathbf{g})(a_{1})=\frac{e^{tu_{1}(e_{a_{1}},\mathbf{g})}}{\sum_{a^{\prime}_{1}\in\mathcal{A}_{1}}e^{tu_{1}(e_{a^{\prime}_{1}},\mathbf{g})}},\
a_{1}\in\mathcal{A}_{1}.$
Note that these solutions are in the interior of the simplex for $t$ finite,
but the trajectory can be arbitrarily close to the boundary when $t$ goes to
infinity. In particular, if we assume that the other player is a slow learner,
i.e., $\frac{\lambda_{2,t}}{\lambda_{1,t}}\to 0,$ then,
$\xi_{1}(\mathbf{g})(a_{1})(t)\rightarrow\frac{{f}_{0}(a_{1})}{\sum_{a^{\prime}_{1}\in
BR_{1}(\mathbf{g})}\ {f}_{0}(a^{\prime}_{1})}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{a_{1}\in BR_{1}(\mathbf{g})\\}},$
when $\epsilon\to 0.$ The set $BR_{1}(\mathbf{g})$ denotes the set of pure
maximizers of $\mathbf{f}$ that maximize
$\mathbb{E}_{s}\mathbb{U}(s,\mathbf{f},\mathbf{g}).$
###### Proposition III-A5
Given any time-varying mixed strategies $\\{\mathbf{g}_{t}\\}_{t},$ the
explicit solution to the replicator equation is
$\xi_{1}(\mathbf{g}_{t})(a_{1})=\tilde{\beta}_{1,\frac{1}{t}}(V)(a_{1})$,
where $V$ is the payoff vector defined by
$V(a_{1}):=u_{1}(e_{a_{1}},\bar{\mathbf{g}}_{t})$, where
$\bar{\mathbf{g}}_{t}=\frac{1}{t}\int_{0}^{t}\mathbf{g}_{t^{\prime}}\
dt^{\prime}.$ In particular, if the time-average sequence
$\bar{\mathbf{g}}_{t}$ converges to $\bar{\mathbf{g}}_{*},$ then the explicit
solution $\xi_{1}(\mathbf{g}_{t})$ converges to a smooth best response to
$\bar{\mathbf{g}}_{*}.$
###### Theorem III-A6 (Two Different Learners)
Consider two learners: one learns faster than the other.
(T1) Assume that P1 is a slow learner of RL2 or RL3 and P2 is a fast learner
of CRL1, i.e., $\frac{\lambda_{1,t}}{\lambda_{2,t}}\longrightarrow 0$ as
$t\rightarrow\infty$ . Then almost surely,
$\|\mathbf{g}_{t}-\xi_{2}(\mathbf{f})\|\longrightarrow 0$ as $t$ goes to
infinity, where $\xi_{2}(\mathbf{f})={\beta}_{2,\epsilon}(\mathbf{f}),$ and
$\dot{f}_{t}(a_{1})={f}_{t}(a_{1})[u_{1}(e_{a_{1}},{\beta}_{2,\epsilon}(\mathbf{f}_{t}))-\sum_{a_{1}^{\prime}\in\mathcal{A}_{1}}{f}_{t}(a^{\prime}_{1})u_{1}(e_{a_{1}^{\prime}},{\beta}_{2,\epsilon}(\mathbf{f}_{t}))]$
(18)
generates the asymptotic pseudo-trajectory of $\\{\mathbf{f}_{t}\\}_{t\geq
0}.$
(T2) Assume that P2 is slow learner of RL2 or RL3 and P1 is a fast learner of
CRL1, i.e., $\frac{\lambda_{2,t}}{\lambda_{1,t}}\longrightarrow 0$ as
$t\rightarrow\infty$ . Then, almost surely,
$\|\mathbf{f}_{t}-\xi_{1}(\mathbf{g})\|\longrightarrow 0$ as $t$ goes to
infinity, where
$\xi_{1}(\mathbf{g})(a_{1})=\frac{e^{tu_{1}(e_{a_{1}},\mathbf{g})}}{\sum_{a^{\prime}_{1}\in\mathcal{A}_{1}}e^{tu_{1}(e_{a^{\prime}_{1}},\mathbf{g})}},\
a_{1}\in\mathcal{A}_{1}$
and the ODE
$\dot{\mathbf{g}}_{t}={\beta}_{2,\epsilon}(\xi_{1}(\mathbf{g}_{t}))-\mathbf{g}_{t}$
(19)
generates the asymptotic pseudo-trajectory of $\\{\mathbf{g}_{t}\\}_{t\geq
0}.$
Note that this last ODE differs from the replicator dynamics, the best
response dynamics, the logit dynamics and fictitious play, etc.
###### Remark III-A7
Note that from Lemma III-A3,
$\xi_{1}(\mathbf{g})(a_{1})=\beta_{1,\frac{1}{t}}(\mathbf{g})(a_{1}).$ This
means that if the trajectories remain in the interior of the simplex, the time
averages of the replicator dynamics and the smooth best-response dynamics are
asymptotically close (the norm of the difference between the two trajectories
is small when $t$ is sufficiently large). The mixed strategy
$\beta_{1,\frac{1}{t}}$ has full support for any $t>0,$ i.e.,
$\xi_{1}(\mathbf{g})$ remains in the relative interior of the simplex for all
$t$.
The following theorem, whose proof can be found in the full report [12], says
that under CRL1, the dominated strategies will be eliminated in the long-term.
###### Theorem III-A8
Consider algorithm CRL1. If a strategy $a_{1}$ is strictly dominated, then
$f_{t}(a_{1})\longrightarrow 0$ when $t\longrightarrow\infty$ and
$\epsilon\longrightarrow 0.$
### III-B Convergence to saddle points
From (T1) of Theorem III-A6, we see that the case with P1 as the slow learner
leads to ODE in (18) whose solution is given by Lemma III-A4, which is in the
form of the smooth best response to P2. Knowing that $\mathbf{g}_{t}$ also
converges almost surely to the smooth best response to P1, we conclude that
the learning algorithm studied in (T1) converges to an $\epsilon-$saddle
point. Similarly, from (T2) of Theorem III-A6, when P1 acts as a fast learner,
the ODE in (19) has its solution given by Lemma III-A3 and leads to the smooth
best response when $t\rightarrow\infty$. In addition, from (T1) and from
Proposition III-A5, $\mathbf{f}_{t}$ converges to
$\xi_{1}=\beta_{1,\frac{1}{t}}$, which is asymptotically close to the smooth
best-response dynamics. Hence we can conclude that the algorithm studied in
(T2) also converges to an $\epsilon-$saddle point. When $\epsilon$ goes to
zero, the stationary points of these heterogeneous dynamics converge to the
saddle points of the expected game. We can extend the preceding argument to
any combination of replicator dynamics and smooth best response dynamics.
Using Theorem III-A1 and its corollary III-A2, we arrive at the following
result.
###### Theorem III-B1
Consider the case of two different learners in which one learns faster than
the other. Let the initial condition be an interior point of the simplex. The
heterogeneous dynamics: (i) CRL0 with CRL1, (ii) CRL0 with CRL2, (iii) CRL1
with CRL2, (iv) CRL1 with RL2, and (v) CRL1 with RL3 lead almost surely to an
$\epsilon-$ saddle point of the expected game.
Figure 1: The payoffs to the players with both players using CRL1.
Figure 2: The mixed strategies of the players with both players using CRL1.
Figure 3: The payoffs to the players with the attacker using CRL1 and the
defender using RL2.
Figure 4: The mixed strategies of the players with the attacker using CRL1 and
the defender using RL2.
## IV Application and Simulation
In this section, we illustrate the heterogeneous learning algorithms with an
example motivated by computer security. In a network intrusion detection
system, an intruder attempts to scan the host machines and seek their
vulnerabilities while the intrusion detector monitors the suspicious behavior
and raises an alarm when attacks are detected. The attacker and the defender
can dynamically adapt their strategies from learning the history of the
behaviors of each other and their own payoffs. It is common that the learning
pattern of the attacker is different from the one used by the defender since
learning schemes depend on an individual’s preference and rationality as well
as the information observed by each person. Hence, in the context of computer
security, heterogeneity of the learning algorithm is essential because it
offers extra degrees of freedom to model agent’s behavior.
Consider a two-person game with one party being the defender (P1) and the
other party the attacker (P2). The defender has two actions available for each
play, i.e., either to defend (D) or not to defend (ND), while the attacker has
two actions either to attack or not to attack. The deterministic payoff matrix
is given by $\mathbf{M}=\left[\begin{array}[]{cc}5&2\\\
1&3\end{array}\right],$ where the columns correspond to the defender
strategies (D) and (ND) whereas the rows correspond to the attacker strategies
(A) and (NA). The stochastic payoff matrix $\mathbf{U}$ is a function of
random matrix $\mathbf{S}=\left[\begin{array}[]{cc}s_{1}&s_{2}\\\
s_{3}&s_{4}\end{array}\right],$ whose components are uniformly distributed on
$[-1,1]$. It is given by $\mathbf{U}=\mathbf{M}+\mathbf{S}.$
At the equilibrium, the attacker selects its actions according to
$\mathbf{f}^{*}=[0.4,0.6]^{T}$ while the defender chooses its actions using
$\mathbf{g}^{*}=[0.2,0.8]^{T}$. The strategy pair
$(\mathbf{f}^{*},\mathbf{g}^{*})$ forms a saddle point solution to the game
$\mathbb{E}\mathbf{U}=\mathbf{M}$, yielding the game value $2.6$. We show in
Figures 4 and 4 the payoffs and the mixed strategies of the players,
respectively, when both adopt the CRL1 learning algorithm. By setting
$\epsilon=\frac{1}{20}$, we observe that the payoffs of P1 choosing actions N
and NA at $t=8000$ are $2.5890$ and $2.6073$ respectively, which are close to
the game value 2.6. For P2, the payoffs at $t=8000$ are $-2.6578$ and
$-2.5855$ for actions N and ND, respectively. The difference between the
payoff and game value is explained by the soft-max parameter $\epsilon$. When
$\epsilon$ approaches $0$, the average payoffs will approach the game value.
The convergence of CRL1 is slow. In Figures 4 and 4, we observe that the
payoff values and the mixed strategy probabilities converge roughly after
$t=6000.$ In Figures 4 and 4, we show the temporal evolution of the payoffs
and mixed strategies of the attacker and defender using the heterogeneous
learning algorithm in which the attacker follows CRL1 whereas the defender
uses RL2. We initialize the payoffs to be $0$ and the strategy vectors
$\mathbf{f}_{0}^{T}=[1/3,2/3],\mathbf{g}_{0}^{T}=[1/3,2/3]$. We set the
parameter $\epsilon=\frac{1}{20}$ in the soft-max best response function of
the attacker. The convergence of the learning process is shown after $t=80s$.
## V Concluding remarks
We have presented heterogeneous distributed learning algorithms for two-person
zero-sum stochastic games along with their general convergence and non-
convergence properties. Our results subsume many known results regarding
learning optimal strategies with different time scales and with different
learning schemes. Interesting work that we leave for the future is to extend
these results to stochastic games with controlled states and nonzero-sum
stochastic games with incomplete information.
## References
* [1] W. B. Arthur, On designing economic agents that behave like human agents. J. Evolutionary Econ. Vol. 3, 1993, pp. 1-22.
* [2] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999.
* [3] M. Benaïm, “Dynamics of Stochastic Approximations. Le Seminaire de Probabilites”. Lectures Notes in Mathematics, Vol. 1709, pp. 1-68, 1999.
* [4] J. S. Shamma and G. Arslan, “Dynamic Fictitious Play, Dynamic Gradient Play, and Distributed Convergence to Nash Equilibria,” IEEE Trans. Automatic Control, Vol. 50, Issue 3, March 2005, pp. 312-327.
* [5] T. Borgers and R. Sarin, Learning Through Reinforcement and Replicator Dynamics , UCSD, Economics Working Paper Series 93-47, 1993. Appeared in Journal of Economic Theory, Vol. 77, Issue 1, November 1997, pp. 1-14.
* [6] V. S. Borkar, “Stochastic approximation with two time scales”, Systems Control Letters, Vol. 29 , Issue 5, 1997, pp. 291-294.
* [7] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
* [8] J. Hofbauer and L. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamics. Annals of Applied Probability, Vol. 19, Aug. 2009, 1347-1368.
* [9] H. J. Kushner and D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems, Springer, New York, 1978.
* [10] D. S. Leslie and E. J. Collins, Convergent multiple timescales reinforcement learning algorithms in normal form games, The Annals of Applied Probability, Vol. 13, No. 4, 2003, pp. 1231-1251.
* [11] H. P. Young, Strategic Learning and Its Limits, Oxford University Press, 2004.
* [12] Q. Zhu, H. Tembine and T. Başar, “Heterogeneous Learning in Zero-Sum Stochastic Games with Incomplete Information,” University of Illinois, CSL, D& C Lab, Research, Sept. 2010.
|
arxiv-papers
| 2011-03-13T03:18:55 |
2024-09-04T02:49:17.616961
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quanyan Zhu, Hamidou Tembine and Tamer Basar",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1103.2491"
}
|
1103.2493
|
# A Constrained Evolutionary Gaussian Multiple Access Channel Game
Quanyan Zhu, Hamidou Tembine, Tamer Başar
###### Abstract
In this paper, we formulate an evolutionary multiple access channel game with
continuous-variable actions and coupled rate constraints. We characterize Nash
equilibria of the game and show that the pure Nash equilibria are Pareto
optimal and also resilient to deviations by coalitions of any size, i.e., they
are strong equilibria. We use the concepts of price of anarchy and strong
price of anarchy to study the performance of the system. The paper also
addresses how to select one specific equilibrium solution using the concepts
of normalized equilibrium and evolutionary stable strategies. We examine the
long-run behavior of these strategies under several classes of evolutionary
game dynamics such as Brown-von Neumann-Nash dynamics, and replicator
dynamics.111Q. Zhu and T. Başar are with the Department of Electrical and
Computer Engineering and the Coordinated Science Laboratory, University of
Illinois at Urbana-Champaign. Postal Address: 1308 West Main, Urbana, IL,
61801, USA. E-mail:{zhu31,tbasar}@decision.csl.uiuc.edu; H. Tembine is with
LIA/CERI, University of Avignon, France. E-mail: hamidou.tembine@univ-
avignon.fr 222This work was done when the second coauthor was visiting
University of Illinois at Urbana Champaign. This work was partially supported
by an INRIA PhD intership grant.
## 1 Introduction
Recently, there has been much interest in understanding the behavior of
multiple access channels under constraints. Considerable amount of work has
been carried out on the problem of how users can obtain an acceptable
throughput by choosing rates independently. Motivated by the interest in
studying a large population of users playing the game over time, evolutionary
game theory was found to be an appropriate framework for communication
networks. It has been applied to problems such as power control in wireless
networks and mobile interference control [1]. In [5], an additive white
Gaussian noise (AWGN) multiple access channel problem was modeled as a
noncooperative game with pairwise interactions, in which users were modeled as
rational entities whose only interest was to maximize their own communication
rates. The authors obtained the Nash equilibria of the two-user game and
introduced a two-player evolutionary game model with pairwise interactions
based on replicator dynamics. However, the case when interactions are not
pairwise arises frequently in communication networks, such the Code Division
Multiple Access (CDMA) or the Orthogonal Frequency-Division Multiple Access
(OFDMA) in Worldwide Interoperability for Microwave Access (WiMAX) environment
[1].
In this work, we extend the study of [5] to wireless communication systems
with an arbitrary number of users corresponding to each receiver. We formulate
a static non-cooperative game with $m$ users subject to rate capacity
constraints and extend the constrained game to a dynamic evolutionary game
with a large number of users whose strategies evolve over time. Different from
evolutionary games with discrete and finite number of actions, our model is
based on a class of continuous games, known as continuous-trait games.
Evolutionary games with continuum action spaces can be seen in a wide variety
of applications in evolutionary ecology, such as evolution of phenology,
germination, nutrient foraging in plants, and predator-prey foraging [10, 20].
### 1.1 Contributions
The main contributions of this work can be summarized as follows. We show that
the static continuous kernel rate allocation game with coupled rate
constraints has a convex set of pure Nash equilibria, coinciding with the
maximal face of the polyhedral capacity region. All the pure equilibria are
Pareto optimal and are also strong equilibria, resilient to simultaneous
deviation by coalition of any size. We show that the pure Nash equilibria in
the rate allocation problem are 100% efficient in terms of Price of Anarchy
(PoA) and constrained Strong Price of Anarchy (CSPoA). We study the stability
of strong equilibria, normalized equilibria, and evolutionary stable
strategies (ESS) using evolutionary game dynamics such as Brown-von Neumann-
Nash dynamics, generalized Smith dynamics, and replicator dynamics.
### 1.2 Organization of the paper
The rest of the paper is structured as follows. We present in the next section
the evolutionary game model of rate allocation in additive white Gaussian
multiple access wireless networks, and analyze its equilibria and Pareto
optimality. In Section 3, we present strong equilibria and price of anarchy of
the game. In Section 4, we discuss how to select one specific equilibrium such
as normalized equilibrium and evolutionary stable strategies. Section 5
studies the stability of equilibria and evolution of strategies using game
dynamics. Section 7 concludes the paper.
## 2 The Game Model
We consider a communication system consisting of several receivers and several
senders (See Figure 1). At each time, there are many local interactions
(typically, at each receiver there is a local interaction) at the same time.
Each local interaction will correspond to a non-cooperative one-shot game with
common constraints. The opponents do not necessarily stay the same from a
given time slot to another time slot. Users revise their rates in view of
their payoffs and the coupled constraints (for example by using an
evolutionary process, a learning process or a trial-and-error updating
process). The game evolves in time. Users are interested in maximizing a
fitness function based on their own communication rates at each time, and they
are aware of the fact that the other users have the same goal. The coupled
power and rate constraints are also common knowledge. Users have to choose
independently their own coding rates at the beginning of the communication,
where the rates selected by a user may be either deterministic, or chosen from
some distribution. If the rate profile arrived at as a result of these
independent decisions lies in the capacity region, users will communicate at
that operating point. Otherwise, either the receiver is unable to decode any
signal and the observed rates are zero, or only one of the signals can be
decoded. The latter case occurs when all the other users are transmitting at
or below a safe rate. With these assumptions, we can define a constrained non-
cooperative game. The set of allowed strategies for user $j$ is the set of all
probability distributions over $[0,+\infty[,$ and the payoff is a function of
the rates. In addition, the rational action (rates) sets are restricted to lie
in the capacity regions (the payoff is zero if the constraint is violated). In
order to study the interactions between the selfish or partially cooperative
users and their stationary rates in the long run, we propose to model the rate
allocation in Gaussian multiple access channels as an evolutionary game with a
continuous action space and coupled constraints. The development of
evolutionary game theory is a major contribution of biology to competitive
decision making and the evolution of cooperation. The key concepts of
evolutionary game theory are (i) Evolutionary Stable Strategies [12], which is
a refinement of equilibria, and (ii) Evolutionary Game Dynamics such as
replicator dynamics [16], which describes the evolution of strategies or
frequencies of use of strategies in time, [20, 7].
Figure 1: A population: distributed receivers and senders, represented by blue
rectangles and red circles respectively.
The single population evolutionary rate allocation game is described as
follows: there is one population of senders (users) and several receivers. The
number of senders is large. At each time, there are many one-shot games called
local interactions. Each sender of the population chooses from the same set of
strategies ${\mathcal{A}}$ which is a non-empty, convex and compact subset of
$\mathbb{R}.$ Without loss of generality, we can suppose that user $j$ chooses
its rate in the interval $\mathcal{A}=[0,C_{\\{j\\}}]$, where $C_{\\{j\\}}$ is
the rate upper bound for user $j$ (to be made precise shortly), as outside of
the capacity region the payoff (as to be defined later) will be zero. Let
$\Delta({\mathcal{A}})$ be the set of probability distributions over the pure
strategy set $\mathcal{A}.$ The set $\Delta({\mathcal{A}})$ can be interpreted
as the set of mixed strategies. It is also interpreted as the set of
distributions of strategies among the population. Let
$\lambda_{t}\in\Delta({\mathcal{A}}),$ and $E$ be a $\lambda_{t}-$ measurable
subset of $\mathbb{R}^{m}$; then $\lambda_{t}(E)$ represents the fraction of
users choosing a strategy out of $E$, at time $t.$ A distribution
$\lambda_{t}\in\Delta({\mathcal{A}})$ is sometimes called the “state” of the
population. We denote by $\mathbb{B}(\mathcal{A})$ the Borel $\sigma-$algebra
on ${\mathcal{A}}$ and by $d(\lambda,\lambda^{\prime})$ the distance between
two states measured with the respect to the weak topology. Each user’s payoff
depends on opponents’ behavior through the distribution of opponents’ choices
and of their strategies. The payoff of a user $j$ in a local interaction with
$(m-1)$ other users is given as a function $u^{j}:\
\mathbb{R}^{m}\longrightarrow\mathbb{R}.$ The rate profile
$\alpha\in\mathbb{R}^{m}$ must belong to a common capacity region
$\mathcal{C}\subset\mathbb{R}^{m}$ defined by $2^{m}-1$ linear inequalities.
The expected payoff of a sender transmitting with the rate $a$ when the state
of the population is $\mu\in\Delta(\mathcal{A})$ is given by $F(a,\mu).$ The
expected payoff is
$F(\lambda,\mu):=\int_{\alpha\in\mathcal{C}}u(\alpha)\
\lambda(d\alpha^{j})\prod_{i\neq j}\mu(d\alpha^{i}).$
The population state is subjected to the “mixed extension” of capacity
constraints $\mathcal{M}(\mathcal{C}).$ This will be discussed in Section 5
and will be made more precise later.
### 2.1 Local Interactions
A local interaction refers to the problem setting of one receiver and its
uplink additive white Gaussian noise (AWGN) multiple access channel with
several senders (say $m\geq 2$) with coupled constraints (or actions). The
signal at the receiver is given by $Y=\xi+\sum_{j=1}^{m}X_{j}$ where $X_{j}$
is a transmitted signal of user $j$ and $\xi$ is zero mean Gaussian noise with
variance $\sigma_{0}^{2}.$ Each user has an individual power constraint
$\mathbb{E}(X_{j}^{2})\leq P.$ The optimal power allocation scheme is to
transmit at the maximum power available, i.e. $P$, for each user. Hence, we
consider the case in which maximum power is attained. The decisions of the
users then consist of choosing their communication rates, and the receiver’s
role is to decode, if possible. The capacity region is a set of all vectors
$\alpha\in{\mathbb{R}}^{m}_{+}$ such that users $j=1,2,\ldots,m$ can reliably
communicate at rate $\alpha^{j},~{}j=1,\ldots,m.$ The capacity region
$\mathcal{C}$ for this channel is the set
$\displaystyle\mathcal{C}=\left\\{\alpha\in{\mathbb{R}}^{m}_{+}~{}\bigg{|}~{}\sum_{j\in
J}\alpha^{j}\leq\log\left(1+|J|\frac{P}{\sigma^{2}_{0}}\right),\right.$
$\displaystyle\left.\forall\ \emptyset\subsetneqq J\subseteq\Omega\right\\}$
(1)
where $\Omega:=\\{1,2,\ldots,m\\}.$ We refer the reader to [21] for more
details on the capacity region. Notice that there is a tradeoff between high
and low rates: if user $j$ wants to communicate at a higher rate, one of the
other users $k$ may need to lower its rate, otherwise the capacity constraint
is violated.
###### Example 2.2.
(Example of capacity region with three users) In this example, we illustrate
the capacity region with three users. Let $\alpha^{1},\alpha^{2},\alpha^{3}$
be the rates of the users. Based on (2.1), we obtain
$\left\\{\begin{array}[]{l}\alpha^{1}\geq 0,\alpha^{2}\geq 0,\alpha^{3}\geq
0\\\ \alpha^{1}\leq\log(1+\frac{P}{\sigma_{0}^{2}})\\\
\alpha^{2}\leq\log(1+\frac{P}{\sigma_{0}^{2}})\\\
\alpha^{3}\leq\log(1+\frac{P}{\sigma_{0}^{2}})\\\
\alpha^{1}+\alpha^{2}\leq\log(1+2\frac{P}{\sigma_{0}^{2}})\\\
\alpha^{1}+\alpha^{3}\leq\log(1+2\frac{P}{\sigma_{0}^{2}})\\\
\alpha^{2}+\alpha^{3}\leq\log(1+2\frac{P}{\sigma_{0}^{2}})\\\
\alpha^{1}+\alpha^{2}+\alpha^{3}\leq\log(1+3\frac{P}{\sigma_{0}^{2}})\\\
\end{array}\right.\Longleftrightarrow M_{3}\gamma_{3}\leq\zeta_{3},\ $
where in the compact notation,
$\gamma_{3}:=\left(\begin{array}[]{c}\alpha^{1}\\\ \alpha^{2}\\\
\alpha^{3}\end{array}\right)\in\mathbb{R}_{+}^{3},\
\zeta_{3}:=\left(\begin{array}[]{c}C_{\\{1\\}}\\\ C_{\\{2\\}}\\\
C_{\\{3\\}}\\\ C_{\\{1,2\\}}\\\ C_{\\{1,3\\}}\\\ C_{\\{2,3\\}}\\\
C_{\\{1,2,3\\}}\end{array}\right),$ $M_{3}:=\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\\\ 1&1&0\\\ 1&0&1\\\ 0&1&1\\\
1&1&1\end{array}\right)\in\mathbb{Z}^{7\times 3}.$
Note that $M_{3}$ is a totally unimodular matrix. By letting
$P=25,\sigma_{0}^{2}=0.1,$ we show in Figure 2 the capacity region with three
users.
Figure 2: Capacity region with three users.
We denote by
$r_{m}=\log\left(1+\frac{P}{\sigma_{0}^{2}+(m-1)P}\right)$
the rate of a user when the signal of the $m-1$ other users is treated as
noise, and $C_{J}=\log(1+|J|\frac{P}{\sigma_{0}^{2}})$ its capacity. Note that
$r_{m}=C_{\\{m\\}}-C_{\\{m-1\\}}.$ The set $\mathcal{C}$ is clearly a non-
empty and bounded subset of ${\mathbb{R}}^{m}.$ $\mathcal{C}$ is closed and is
defined by $2^{m}-1$ convex inequalities. Thus, $\mathcal{C}$ is convex and
compact. From the inequality
$\log\left(1+\sum_{j\in J}x_{j}\right)\leq\log\left(\prod_{j\in
J}(1+x_{j})\right)=\sum_{j\in J}\log(1+x_{j}),$
for all $\forall x\in{\mathbb{R}}^{|J|}_{+},$ we obtain $C_{J}\leq\sum_{j\in
J}C_{\\{j\\}}.$
### 2.3 Payoff
We define the payoff of user $j$ as
$u^{j}(\alpha^{j},\alpha^{-j})=\left\\{\begin{array}[]{ll}g(\alpha^{j})&\mbox{if}\
(\alpha^{j},\alpha^{-j})\in\mathcal{C}\\\
0&\mbox{otherwise}\end{array}\right.,$
where $\alpha^{j}$ is the rate of the user $j$; the vector
$\alpha^{-j}:=(\alpha^{1},\ldots,\alpha^{j-1},\alpha^{j+1},\ldots,\alpha^{m})$
is a profile of rates of the other users; the function $g:\
{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a positive and strictly increasing
function. Given the strategy profile $\alpha^{-j}$ of the others players,
player $j$ has to maximize $u^{j}(\alpha^{j},\alpha^{-j})$ under its action
constraints
$\mathcal{A}(\alpha^{-j}):=\\{\alpha^{j}\in[0,C_{\\{j\\}}],\
(\alpha^{j},\alpha^{-j})\in\mathcal{C}\\}.$
Using the monotonicity of the function $g$ and the inequalities that define
the capacity region, we obtain the following lemma.
###### Lemma 2.3.1.
Let $\overline{BR}(\alpha^{-j})$ be the best reply to the strategy
$\alpha^{-j}$ is
$\overline{BR}(\alpha^{-j})=\arg\max_{y\in\mathcal{A}(\alpha^{-j})}u^{j}(y,\alpha^{-j}).$
$\overline{BR}$ is a non-empty single-valued correspondence (i.e., a standard
function) which is given by
$\max\left(r_{m},\min_{J}\left\\{\ C_{J}-\sum_{k\in J\ \atop k\neq
j}\alpha^{k},\ J\in\Gamma_{j}\right\\}\right)\,$
where $\Gamma_{j}:=\\{J\in 2^{\Omega},\ J\ni j\\}$.
###### Proposition 2.3.2.
The set of Nash equilibria is
$\\{(\alpha^{j},\alpha^{-j})\ |\ \alpha^{j}\geq
r_{m},\sum_{j}\alpha^{j}=C_{\Omega}\\}.$
All these equilibria are optimal in the Pareto sense.333An allocation of
payoffs is Pareto optimal or Pareto efficient if there is no other feasible
allocation that makes every user at least as well off and at least one user
strictly better off under the capacity constraint.
###### Proof.
Let $\beta\in\mathcal{C}.$ If
$\sum_{j=1}^{m}\beta^{j}<C_{\Omega}=\log(1+m\frac{P}{\sigma_{0}^{2}})\,,$
then at least one of the users can improve its rate (hence its payoff) to
reach one of the faces of the capacity region. We now check the strategy
profile in the face
$\\{(\alpha^{j},\alpha^{-j})\ |\ \alpha^{j}\geq
r_{m},\sum_{j=1}^{m}\alpha^{j}=C_{\Omega}\\}.$
If
$\beta\in\\{(\alpha^{j},\alpha^{-j})\ |\ \alpha^{j}\geq
r_{m},\sum_{j=1}^{m}\alpha^{j}=C_{\Omega}\\},$
then from the Lemma, $\overline{BR}(\beta^{-j})=\\{\beta^{j}\\}.$ Hence,
$\beta$ is a strict equilibrium. Moreover, this strategy $\beta$ is Pareto
optimal because the rate of each user is maximized under the capacity
constraint. These strategies are social welfare if the quantity
$\sum_{j=1}^{m}u^{j}(\alpha^{j},\alpha^{-j})=\sum_{j=1}^{m}g(\alpha^{j})$
is maximized. ∎
Note that the set of pure Nash equilibria is a convex subset of the capacity
region.
## 3 Robust equilibria and efficiency measures
### 3.1 Constrained Strong Equilibria and Coalition Proofness
An action profile in a local interaction between $m$ senders is a constrained
$k-$strong equilibrium if it is feasible and no coalition of size $k$ can
improve the rate transmissions of each of its members with respect to the
capacity constraints. An action is a constrained strong equilibrium [4] if it
is a constrained $k-$strong equilibrium for any size $k.$ A strong equilibrium
is then a policy from which no coalition (of any size) can deviate and improve
the transmission rate of every member of the coalition (group of the
simultaneous moves), while possibly lowering the transmission rate of users
outside the coalition group. This notion of constrained strong equilibria
444Note that the set of constrained strong equilibria is a subset of Nash
equilibria (by taking coalitions of size one) and any constrained strong
equilibrium is Pareto optimal (by taking coalition of full size). is very
attractive because it is resilient against coalitions of users. Most of the
games do not admit any strong equilibrium but in our case we will show that
the multiple access channel game has several strong equilibria.
###### Theorem 3.1.1.
Any rate profile on the maximal face of the capacity region $\mathcal{C}:$
$Face_{\max}(\mathcal{C}):=\\{(\alpha^{j},\alpha^{-j})\in\mathbb{R}^{m}\ |\
\alpha^{j}\geq r_{m},\sum_{j=1}^{m}\alpha^{j}=C_{\Omega}\\},$
is a constrained strong equilibrium.
###### Proof.
We remark that if the rate profile $\alpha$ is not on the maximal face of the
capacity region, then $\alpha$ is not resilient to deviation by a single user.
Hence, $\alpha$ cannot be a constrained strong equilibrium. This says that a
necessary condition for a rate profile to be a strong equilibrium is to be in
the subset $Face_{\max}(\mathcal{C}).$ We now prove that the condition:
$\alpha\in Face_{\max}(\mathcal{C})$ is sufficient. Let $\alpha\in
Face_{\max}(\mathcal{C}).$ Suppose that $k$ users deviate simultaneously from
the rate profile $\alpha.$ Denote by $Dev$ the set of users which deviate
simultaneously (eventually by forming a coalition). The rate constraints of
the deviants are
1. 1.
${\alpha^{\prime}}^{j}\geq 0,\ \forall j\in Dev,$
2. 2.
$\sum_{j\in\bar{J}}{\alpha^{\prime}}^{j}\leq C_{\bar{J}},\
\forall\bar{J}\subseteq Dev,$
3. 3.
$\sum_{j\in J\cap Dev}{\alpha^{\prime}}^{j}\leq C_{J}-\sum_{j\in J,j\notin
Dev}\alpha^{j}$, $\ \forall{J}\subseteq\Omega,\ J\cap Dev\neq\emptyset.$
In particular, for $J=\Omega,$ we have $\sum_{j\in
Dev}{\alpha^{\prime}}^{j}\leq C_{\Omega}-\sum_{j\notin Dev}\alpha^{j}.$ The
total rate of the deviants is bounded by $C_{\Omega}-\sum_{j\notin
Dev}\alpha^{j}$, which is not controlled by the deviants. The deviants move to
$({\alpha^{\prime}}^{j})_{j\in Dev}$ with
$\sum_{j\in Dev}{\alpha^{\prime}}^{j}<C_{\Omega}-\sum_{j\notin
Dev}\alpha^{j}\,.$
Then, there exists $j$ such that $\alpha^{j}>{\alpha^{\prime}}^{j}.$ Since $g$
is non-decreasing, this implies that $g(\alpha^{j})>g({\alpha^{\prime}}^{j}).$
The user $j$ who is a member of the coalition $Dev$ does not improve its
payoff. If the rates of some of the deviants are increased, then the rates of
some other users from coalition must decrease. If
$({\alpha^{\prime}}^{j})_{j\in Dev}$ satisfies
$\sum_{j\in Dev}{\alpha^{\prime}}^{j}=C_{\Omega}-\sum_{j\notin
Dev}\alpha^{j}\,,$
then some users in the coalition $Dev$ have increased their rates compared
with $(\alpha^{j})_{j\in Dev}$ and some others in $Dev$ have decreased their
rates of transmission (because the total rate is the constant
$C_{\Omega}-\sum_{j\notin Dev}\alpha^{j}).$ The users in $Dev$ with a lower
rate ${\alpha^{\prime}}^{j}\leq\alpha^{j}$ do not benefit to be member of the
coalition (Shapley criterion of membership of coalition does not hold) . And
this holds for any $\emptyset\subsetneqq Dev\subseteqq\Omega.$ This completes
the proof. ∎
###### Corollary 3.1.2.
In the constrained rate allocation game, Nash equilibria and strong equilibria
in pure strategies coincide.
### 3.2 Constrained Potential Function for Local Interaction
Introduce the following function:
$V(\alpha)=\upharpoonleft_{\mathcal{C}}(\alpha)\sum_{j=1}^{m}g(\alpha^{j})\,,$
where $\upharpoonleft_{\mathcal{C}}$ is the indicator function of
$\mathcal{C},i.e.,\ $ $\upharpoonleft_{\mathcal{C}}(\alpha)=1$ if
$\alpha\in\mathcal{C}$ and $0$ otherwise. The function $V$ satisfies
$V(\alpha)-V(\beta^{j},\alpha^{-j})=g(\alpha^{j})-g(\beta^{j}),\
\forall\alpha,(\beta^{,}\alpha^{-j})\in\mathcal{C}.$
If $g$ is differentiable, then one has
$\frac{\partial}{\partial\alpha^{j}}V(\alpha)=g^{\prime}(\alpha^{j})=\frac{\partial}{\partial\alpha^{j}}u^{j}$
in the interior of the capacity region $\mathcal{C}$, and $V$ is a constrained
potential function [22] in pure strategies.
###### Corollary 3.2.1.
The local maximizers of $V$ in $\mathcal{C}$ are pure Nash equilibria. Global
maximizers of $V$ in $\mathcal{C}$ are both constrained strong equilibria and
social optima for the local interaction.
### 3.3 Strong Price of Anarchy
Throughout this subsection, we assume that the function $g$ is the identity
function, i.e., $g(x)=id(x):=x.$ One of the approaches used to measure how
much the performance of decentralized systems is affected by the selfish
behavior of its components is the price of anarchy. We present a similar price
for strong equilibria under the coupled rate constraints. This notion of Price
of Anarchy can be seen as an efficiency metric that measures the price of
selfishness or decentralization and has been extensively used in the context
of congestion games or routing games where typically users have to minimize a
cost function. In the context of rate allocation in the multiple access
channel, we define an equivalent measure of price of anarchy for rate
maximization problems. One of the advantages of a strong equilibrium is that
it has the potential to reduce the distance between the optimal solution and
the solution obtained as an outcome of selfish behavior, typically in the case
where the capacity constraint is violated at each time. Since the constrained
rate allocation game has strong equilibria, we can define the strong price of
anarchy, introduced in [2], as the ratio between the payoff of the worst
constrained strong equilibrium and the social optimum value which
$C_{\Omega}$.
###### Theorem 3.3.1.
The strong price of anarchy of the constrained rate allocation game is 1 for
$g(x)=x.$
Note that for $g\neq id,$ the CSPoA can be less than one. However, the
optimistic price of anarchy of the best constrained equilibrium also called
price of stability [3] is one for any function $g$ i.e the efficiency of
”best” equilibria is $100\%.$
## 4 Selection of Pure Equilibria
We have shown in previous sections that our rate allocation game has a
continuum of pure Nash equilibria and strong equilibria. We address now the
problem of selecting one equilibrium which has certain desirable properties:
the normalized pure Nash equilibrium, introduced in [13]. See also [15, 6, 9].
We introduce the Lagrangian that corresponds to the constrained maximization
problem faced by every user when the other rates are at the maximal face of
the polytope $\mathcal{C}$:
$\displaystyle\max_{\alpha}$ $\displaystyle u^{j}(\alpha)$ (2) s.t.
$\displaystyle\alpha^{1}+\ldots+\alpha^{m}=C_{\Omega}$ (3)
and the Lagrangian for user $j$ is given by
$L^{j}(\alpha,\zeta)=u^{j}(\alpha)-\zeta^{j}\left(\sum_{j}\alpha^{j}-C_{\Omega}\right).$
From Karush-Kuhn-Tucker optimality conditions, it follows that there exists
$\zeta\in\mathbb{R}^{m}$ such that
$g^{\prime}(\alpha^{j})=\zeta^{j},\ \sum_{j=1}^{m}\alpha^{j}=C_{\Omega}.$
For a fixed vector $\zeta$ with identical entries, define the normal form game
$\Gamma({\zeta})$ with $m$ users, where actions are taken as rates and the
payoffs given by $L(\alpha,\zeta).$ A normalized equilibrium is an equilibrium
of the game $\Gamma(\zeta^{*})$ where $\zeta^{*}$ is normalized into the form
${\zeta^{*}}^{j}=\frac{c}{\tau^{j}},\ c>0,\tau^{j}>0.$ We now have the
following result due to Goodman [6] which implies Rosen’s condition on
uniqueness for strict concave games.
###### Theorem 4.0.1.
Let $u^{j}$ be a smooth and strictly concave function in $\alpha^{j},$ each
$u^{j}$ be convex in $\alpha^{-j}$, and there exist some $\zeta$ such that the
weighted non-negative sum of the payoffs
$\sum_{j=1}^{m}\zeta^{j}u^{j}(\alpha)$ is concave in $\alpha.$ Then the matrix
$G(\alpha,\zeta)+G^{T}(\alpha,\zeta)$
is negative definite (which implies uniqueness) where $G(\alpha,\zeta)$ is the
Jacobian with respect to $\alpha$ of
$h(\alpha,\zeta):=\left[\zeta^{1}\nabla_{1}u^{1}(\alpha),\zeta^{2}\nabla_{2}u^{2}(\alpha),\ldots,\zeta^{m}\nabla_{m}u^{m}(\alpha)\right]^{T}$
and $G^{T}$ is the transpose of the matrix $G.$
This now leads to the following corollary for our problem.
###### Corollary 4.0.2.
If $g$ is a non-decreasing strictly concave function, then the rate allocation
game has a unique normalized equilibrium which corresponds to an equilibrium
of the normal form game with payoff $L(\alpha,\zeta^{*})$ for some
$\zeta^{*}.$
## 5 Stability and Dynamics
In this section, we study the stability of equilibria and several classes of
evolutionary game dynamics. We show that the evolutionary game has a unique
pure constrained evolutionary stable strategy.
###### Proposition 5.1.
The collection of rates
$\alpha=\left(\frac{C_{\Omega}}{m},\ldots,\frac{C_{\Omega}}{m}\right)\,,$
i.e the distribution of Dirac concentrated on the rate $\frac{C_{\Omega}}{m},$
is the unique symmetric pure Nash equilibrium.
###### Proof.
Since the constrained rate allocation game is symmetric, there exists a
symmetric (pure or mixed) Nash equilibrium. If such an equilibrium exists in
pure strategies, each user transmits with the same rate $r^{*}.$ It follows
from Proposition 2.3.2, and the bound $r_{m}\leq\frac{C_{\Omega}}{m}$ that
$r^{*}$ satisfies $mr^{*}=C_{\Omega}$ and $r^{*}$ is feasible. ∎
Since the set of feasible actions is convex, we can define convex combination
of rates in the set of the feasible rates. For example,
$\epsilon\alpha^{\prime}+(1-\epsilon)\alpha$ is a feasible rate if
$\alpha^{\prime}$ and $\alpha$ are feasible. The symmetric rate profile
$(r,r,\ldots,r)$ is feasible if and only if $0\leq r\leq
r^{*}=\frac{C_{\Omega}}{m}.$ We say that the rate $r$ is a constrained
evolutionary stable strategy (ESS) if it is feasible and for every mutant
strategy $mut\neq\alpha$ there exists $\epsilon_{mut}>0$ such that
$\left\\{\begin{array}[]{cc}r_{\epsilon}:=\epsilon\
mut+(1-\epsilon)r\in\mathcal{C}&\forall\epsilon\in(0,\epsilon_{mut})\\\
u(r,r_{\epsilon},\ldots,r_{\epsilon})>u(mut,r_{\epsilon},\ldots,r_{\epsilon})&\forall\epsilon\in(0,\epsilon_{mut})\end{array}\right.$
###### Theorem 5.1.1.
The pure strategy $r^{*}=\frac{C_{\Omega}}{m}$ is a constrained evolutionary
stable strategy.
###### Proof.
Let $mut\leq r^{*}$ The rate $\epsilon\ mut+(1-\epsilon)r^{*}$ is feasible
implies that $mut\leq r^{*}$ (because $r^{*}$ is the maximum symmetric rate
achievable). Since $mut\neq r^{*},$ $mut$ is strictly lower than $r^{*}.$ By
monotonicity of the function $g,$ one has
$u(r^{*},\epsilon\ mut+(1-\epsilon)r^{*})>u(mut,\epsilon\
mut+(1-\epsilon)r^{*}),\ \forall\epsilon.$
This completes the proof. ∎
### 5.2 Symmetric Mixed Strategies
Define the mixed capacity region $\mathcal{M}(\mathcal{C})$ as the set of
measures profile $(\mu^{1},\mu^{2},\ldots,\mu^{m})$ such that
$\int_{\mathbb{R}_{+}^{|J|}}\left(\sum_{j\in J}\alpha^{j}\right)\prod_{j\in
J}\mu^{j}(d\alpha^{j})\leq C_{J},\ \forall J\subseteq 2^{\Omega}.$
Then the payoff of the action $a\in\mathbb{R}_{+}$ satisfying
$(a,\lambda,\ldots,\lambda)\in\mathcal{M}(\mathcal{C})$ can be defined as
$F(a,\mu)=\int_{[0,\infty[^{m-1}}u(a,b_{2},\ldots,b_{m})\ \nu_{m-1}(db)\,,$
where $\nu_{k}=\bigotimes_{1}^{k}\mu$ is the product measure on
$[0,\infty[^{k}.$ The constraint set becomes the set of probability measures
on $\mathcal{R}_{+}$ such that
$0\leq\mathbb{E}(\mu):=\int_{\mathbb{R}_{+}}\ \alpha^{j}\
\mu(d\alpha^{j})\leq\frac{C_{\Omega}}{m}<C_{\\{1\\}}\,.$
###### Lemma 5.2.1.
$F(a,\mu)=\upharpoonleft_{[0,C_{\Omega}-(m-1)\mathbb{E}(\mu)]}\times
g(a)\times$
$\int_{b\in\mathcal{D}_{a}}\
\nu_{m-1}(db)=\upharpoonleft_{[0,C_{\Omega}-(m-1)\mathbb{E}(\mu)]}\times
g(a)\nu_{m-1}(\mathcal{D}_{a})$
where
$\mathcal{D}_{a}=\\{(b_{2},\ldots,b_{m})\ |\
(a,b_{2},\ldots,b_{m})\in\mathcal{C}\\}\,.$
###### Proof.
If the rate does not satisfy the capacity constraints, then the payoff is $0.$
Hence the rational rate for user $j$ is lower than $C_{\\{j\\}}.$ Fix a rate
$a\in[0,C_{\\{j\\}}].$ Let $D^{a}_{J}:=C_{J}-a\delta_{\\{1\in J\\}}.$ Then, a
necessary condition to have a non-zero payoff is
$(b_{2},\ldots,b_{m})\in\mathcal{D}_{a}\,,$
where
$\mathcal{D}_{a}=\\{(b_{2},\ldots,b_{m})\in\mathbb{R}_{+}^{m-1},\ \sum_{j\in
J,j\neq 1}b_{j}\leq D^{a}_{J},\ J\subseteq 2^{\Omega}\\}.$
Thus,
$\displaystyle F(a,\mu)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}^{m-1}}u(a,b_{2},\ldots,b_{m})\
\nu_{m-1}(db)$ $\displaystyle=$ $\displaystyle\int_{b\in\mathbb{R}_{+}^{m-1},\
(a,b)\in\mathcal{C}}g(a)\ \nu_{m-1}(db)$ $\displaystyle=$
$\displaystyle\upharpoonleft_{[0,C_{\Omega}-(m-1)\mathbb{E}(\mu)]}g(a)$
$\displaystyle\times\int_{b\in\mathcal{D}_{a}}\ \nu_{m-1}(db)$
∎
### 5.3 Constrained Evolutionary Game Dynamics
The class of evolutionary games in large population provides a simple
framework for describing strategic interactions among large numbers of users.
In this subsection we turn to modeling the behavior of the users who play
them. Traditionally, predictions of behavior in game theory are based on some
notion of equilibrium, typically Cournot equilibrium, Bertrand equilibrium,
Nash equilibrium, Stackelberg solution, Wardrop equilibrium or some refinement
thereof. These notions require the assumption of equilibrium knowledge, which
posits that each user correctly anticipates how his opponents will act. The
equilibrium knowledge assumption is too strong and is difficult to justify in
particular in contexts with large numbers of users. As an alternative to the
equilibrium approach, we propose an explicitly dynamic updating choice, a
procedure in which users myopically update their behavior in response to their
current strategic environment. This dynamic procedure does not assume the
automatic coordination of users’ actions and beliefs, and it can derive many
specifications of users’ choice procedures. These procedures are specified
formally by defining a revision of rates called revision protocol [14]. A
revision protocol takes current payoffs and current mean rate and maps to
conditional switch rates which describe how frequently users in some class
playing rate $\alpha$ who are considering switching rates switch to strategy
$\alpha^{\prime}.$ Revision protocols are flexible enough to incorporate a
wide variety of paradigms, including ones based on imitation, adaptation,
learning, optimization, etc.
We use a class of continuous evolutionary dynamics. We refer to [17, 19, 18]
for evolutionary game dynamics with or without time delays. The continuous-
time evolutionary game dynamics on the measure space
$(\mathcal{A},\mathcal{B}(\mathcal{A}),\mu)$ is given by
$\dot{\lambda}_{t}(E)=\int_{a\in E}V(a,\lambda_{t})\mu(da)$ (4)
where
$V(a,\lambda_{t})=K\left[\int_{x\in\mathcal{A}}\beta^{x}_{a}(\lambda_{t})\lambda_{t}(dx)-\int_{x\in\mathcal{A}}\beta^{a}_{x}(\lambda_{t})\lambda_{t}(dx)\right],$
and $\beta^{x}_{a}$ represents the rate of mutation from $x$ to $a,$ and $K$
is a growth parameter. $\beta^{x}_{a}(\lambda_{t})=0$ if $(x,\lambda_{t})$ or
$(a,\lambda_{t})$ is not in the (mixed) capacity region, $E$ is a
$\mu-$measurable subset of $\mathcal{A}.$ At each time $t,$ probability
measure $\lambda_{t}$ satisfies $\frac{d}{dt}\lambda_{t}(\mathcal{A})=0$.
Constrained Brown-von Neumann-Nash dynamics.
The constrained revision protocol is
$\beta^{x}_{a}(\lambda_{t})=\left\\{\begin{array}[]{c}\max(F(a,\lambda_{t})-\int_{x}F(x,\lambda_{t})\
dx,0)\\\ \mbox{if}\ (a,\lambda_{t}),\
(x,\lambda_{t})\in\mathcal{M}(\mathcal{C})\\\ 0\
\mbox{otherwise}\end{array}\right.$
Constrained Replicator Dynamics.
$\beta^{x}_{a}(\lambda_{t})=\left\\{\begin{array}[]{c}\max(F(a,\lambda_{t})-F(x,\lambda_{t}),0)\\\
\mbox{if}\ (a,\lambda_{t}),\ (x,\lambda_{t})\in\mathcal{M}(\mathcal{C})\\\ 0\
\mbox{otherwise}\end{array}\right.$
Constrained $\theta-$Smith Dynamics.
$\beta^{x}_{a}(\lambda_{t})=\left\\{\begin{array}[]{cc}\max(F(a,\lambda_{t})-F(x,\lambda_{t}),0)^{\theta}\\\
\mbox{if}\ (a,\lambda_{t}),\ (x,\lambda_{t})\in\mathcal{M}(\mathcal{C})\\\ 0\
\mbox{otherwise}\end{array}\right.,\ \theta\geq 1$
We now provide a common property that applies to all these dynamics: the set
of Nash equilibria is a subset of rest points (stationary points) of the
evolutionary game dynamics. Here we extend to evolutionary game with a
continuous action space and coupled constraints, and more than two-users
interactions. The counterparts of these results in discrete action space can
be found in [7, 14].
###### Theorem 5.3.1.
Any Nash equilibrium of the game is a rest point of the following evolutionary
game dynamics: constrained Brown-von Neumann-Nash, generalized Smith dynamics,
and replicator dynamics. In particular, the evolutionary stable strategies set
is a subset of the rest points of these constrained evolutionary game
dynamics.
###### Proof.
It is clear for pure equilibria by using the revision protocols $\beta$ of
these dynamics. Let $\lambda$ be an equilibrium. For any rate $a$ in the
support of $\lambda,$ $\beta^{a}_{x}=0$ if $F(x,\lambda)\leq F(a,\lambda).$
Thus, if $\lambda$ is an equilibrium the difference between the microscopic
inflow and outflow is $V(a,\lambda)=0$, given that $a$ is the support of the
measure $\lambda.$ ∎
Let $\lambda$ be a finite Borel measure on $[0,C_{\\{j\\}}]$ with full
support. Suppose $g$ is continuous on $[0,C_{\\{j\\}}].$ Then, $\lambda$ is a
rest point of the BNN dynamics if and only if $\lambda$ is a symmetric Nash
equilibrium. Note that the choice of topology is an important issue when
defining dynamics convergence and stability. The most used in this area is the
topology of the weak convergence to measure closeness of two states of the
system. Different distances (Prohorov metric, metric on bounded and Lipschitz
continuous functions on $\mathcal{A}$) have been proposed. We refer the reader
to [11], and the references therein for more details on evolutionary robust
strategy and stability notions.
## 6 Generalization
In this section, we consider the asymmetric case. Each user has its maximum
power $P_{i}$ and a channel gain $h_{i}.$ In addition, the rate of
transmission is subject to a coupled capacity constraint. The capacity region
$\mathcal{C}$ is described by the set
$\left\\{\alpha\in\mathbb{R}^{m}_{+},\sum_{i\in\Omega}\alpha^{i}\leq
C_{\Omega},\ \forall\ \emptyset\subset\Omega\subseteq\mathcal{N}\right\\},$
(5)
where $\Omega$ is any subset of $\mathcal{N}$ and
$C_{\Omega}=\log\left(1+\sum_{i\in\Omega}\frac{P_{i}h_{i}}{\sigma^{2}_{0}}\right),$
(6)
is the capacity for users in $\Omega$. The capacity region reveals a
competitive nature of the interactions among senders: if a user $i$ wants to
communicate at a higher rate, one of the other users has to lower his rate;
otherwise, the capacity constraint is violated. We let
$r_{i,\Omega}:=\log\left(1+\frac{P_{i}h_{i}}{\sigma_{0}^{2}+\sum_{i^{\prime}\in\Omega,i^{\prime}\neq
i}P_{i^{\prime}}h_{i^{\prime}}}\right)$
denote the bound rate of a user when the signals of the $|\Omega|-1$ other
users are treated as noise.
Due to the noncooperative nature of the rate allocation, we can formulate the
one-shot game
$\Xi=\langle\mathcal{N},(\mathcal{A}^{i})_{i\in\mathcal{N}},(u^{i})_{i\in\mathcal{N}}\rangle\,,$
where the set of users $\mathcal{N}$ is the set of players, $\mathcal{A}^{i}$,
$i\in\mathcal{N}$, is the set of actions, and $u^{i}$, $i\in\mathcal{N}$, are
the payoff functions. We define
$u^{i}:\prod_{i=1}^{m}\mathcal{A}^{i}\rightarrow\mathbb{R}_{+}$ as follows.
$\displaystyle u^{i}(\alpha^{i},\alpha^{-i})$ $\displaystyle=$
$\displaystyle\upharpoonleft_{\mathcal{C}}(\alpha)g^{i}(\alpha^{i},\alpha^{-i})$
(7) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}g^{i}(\alpha^{i})&{\textrm{~{}if~{}}}\
(\alpha^{i},\alpha^{-i})\in\mathcal{C}\\\
0&\mbox{otherwise}\end{array}\right.,$ (10)
where $\upharpoonleft_{\mathcal{C}}$ is the indicator function; $\alpha^{-i}$
is a vector consisting of other players’ rates, i.e.,
$\alpha^{-i}=[\alpha^{1},\ldots,\alpha^{i-1},\alpha^{i+1},\ldots,\alpha^{N}]$
and $u^{i}$ is a positive and strictly increasing function for each fixed
$\alpha^{-i}$. Since the game is subject to coupled constraints, the action
set $\mathcal{A}^{i}$ is coupled and dependent on other players’ actions.
Given the strategy profile $\alpha^{-i}$ of other players, the constrained
action set $\mathcal{A}^{i}$ is given by
$\mathcal{A}^{i}(\alpha^{-i}):=\\{\alpha^{i}\in[0,C_{\\{i\\}}],\
(\alpha^{i},\alpha^{-i})\in\mathcal{C}\\}$ (11)
We then have an asymmetric game. The minimum rate that the user $i$ can
guarantee in the feasible regions is $r_{i,\mathcal{N}}$ which is different
than $r_{j,\mathcal{N}}.$
Each user $i$ maximizes $u^{i}(\alpha^{i},\alpha^{-i})$ over the coupled
constraint set. Owing to the monotonicity of the function $g^{i}$ and the
inequalities that define the capacity region, we obtain the following lemma.
###### Lemma 6.0.1.
Let $\overline{BR}^{i}(\alpha^{-i})$ be the best reply to the strategy
$\alpha^{-i}$, defined by
$\overline{BR}^{i}(\alpha^{-i})=\arg\max_{y\in\mathcal{A}^{i}(\alpha^{-i})}u^{i}(y,\alpha^{-i}).$
$\overline{BR}^{i}$ is a non-empty single-valued correspondence (i.e a
standard function), and is given by
$\max\left(r_{i,\mathcal{N}},\min_{\Omega\in\Gamma_{i}}\left\\{C_{\Omega}-\sum_{k\in\Omega\backslash\\{i\\}}\alpha^{k}\right\\}\right),$
(12)
where $\Gamma_{i}=\\{\Omega\in 2^{\mathcal{N}},i\in\Omega\\}$.
###### Proposition 6.1.
The set of Nash equilibria is
$\\{(\alpha^{i},\alpha^{-i})\ |\ \alpha^{i}\geq
r_{i,\mathcal{N}},\sum_{i\in\mathcal{N}}\alpha^{i}=C_{\mathcal{N}}\\}.$
All these equilibria are optimal in Pareto sense.
###### Proof.
Let $\beta$ be a feasible solution, i.e., $\beta\in\mathcal{C}.$ If
$\sum_{i=1}^{N}\beta^{i}<C_{\mathcal{N}}=\log\left(1+\sum_{i\in\mathcal{N}}\frac{P_{i}h_{i}}{\sigma_{0}^{2}}\right),$
then at least one of the users can improve its rate (hence its payoff) to
reach one of the faces of the capacity region. We now check the strategy
profile on the face
$\left\\{(\alpha^{i},\alpha^{-i})\ \bigg{|}\ \alpha^{i}\geq
r_{i,\mathcal{N}},\sum_{i=1}^{N}\alpha^{i}=C_{\mathcal{N}}\right\\}.$
If
$\beta\in\left\\{(\alpha^{i},\alpha^{-i})\ \bigg{|}\ \alpha_{i}\geq
r_{i,\mathcal{N}},\sum_{i=1}^{N}\alpha^{i}=C_{\Omega}\right\\},$
then from the Lemma 12, $\overline{BR}^{i}(\beta^{-i})=\\{\beta^{i}\\}.$
Hence, $\beta$ is a strict equilibrium. Moreover, this strategy $\beta$ is
Pareto optimal because the rate of each user is maximized under the capacity
constraint. These strategies are social welfare optimal if the total utility
$\sum_{i=1}^{N}u^{i}(\alpha^{i},\alpha^{-i})=\sum_{i=1}^{N}g^{i}(\alpha^{i})$
is maximized subject to constraints. ∎
Note that the set of pure Nash equilibria is a convex subset of the capacity
region. The pure equilibria are global optima555This implies that the price of
anarchy is one. if the function $g$ is the identity function.
## 7 Concluding remarks
In this paper, we have studied an evolutionary Multiple Access Channel game
with a continuum action space and coupled rate constraints. We showed that the
game has a continuum of strong equilibria which are 100% efficient in the rate
optimization problem. We proposed the constrained Brown-von Neumann-Nash
dynamics, Smith dynamics, and the replicator dynamics to study the stability
of equilibria in the long run. An interesting question which we leave for
future work is whether similar equilibria structure exist in the case of
multiple access games with non-convex capacity regions. Another extension
would be to the hybrid model in which users can select among several receivers
and control the total rate, which is currently under study.
## References
* [1] Altman, E., El-Azouzi, R., Hayel, Y., and Tembine, H., “Evolutionary power control games in wireless networks,” NETWORKING 2008 Ad Hoc and Sensor Networks, Wireless Networks, Next Generation Internet, Springer Berlin / Heidelberg, pp. 930-942, 2008.
* [2] Andelman, N., Feldman, M., and Mansour, Y., “Strong price of anarchy,” SODA, 2007.
* [3] Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T. and Roughgarden, T., “The price of stability for network design with fair cost allocation,” in Proc. FOCS, pp. 59-73, 2004.
* [4] Aumann, R., “Acceptable points in general cooperative n-person games”, in Contributions to the Theory of Games, volume 4, 1959.
* [5] Gajic, V. and Rimoldi, B., “Game theoretic considerations for the Gaussian multiple access channel,” in Proc. IEEE ISIT, 2008.
* [6] Goodman, J. C., “A note on existence and uniqueness of equilibrium points for concave N-person games,” Econometrica, 48(1),1980, p. 251.
* [7] Hofbauer, J. and Sigmund, K.., Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
* [8] Hofbauer, J., Oechssler, J., and Riedel, F., “Brown-von Neumann-Nash dynamics: The continuous strategy case,” Games and Econ. Behav., 65(2):406-429, 2008.
* [9] Ponstein, J., “Existence of equilibrium points in non-product spaces,” SIAM J. Appl. Math., 14(1):181-190, 1966.
* [10] McGill, B.J. and Brown, J.S., “Evolutionary game theory and adaptive dynamics of continuous traits,” The Annual Rev. of Ecology, Evolution, and Systematics, 38: 403-435, 2007.
* [11] Shaiju, A. J. and Bernhard, P., “Evolutionarily robust strategies: two nontrivial examples and a theorem,” Proc. of ISDG, 2006.
* [12] Smith, J.M. and Price, G.M., “The logic of animal conflict,” Nature, 246:15-18, 1973.
* [13] Rosen, J. B., “Existence and uniqueness of equilibrium points for concave N-person games,” Econometrica, 33:520-534, 1965.
* [14] Sandholm, W. H., Population Games and Evolutionary Dynamics, MIT Press, 2009 (to appear ).
* [15] Takashi, U., “Correlated equilibrium and concave games,” Int. Journal of Game Theory, 37(1):1-13, 2008.
* [16] Taylor, P.D. and Jonker, L., “Evolutionarily stable strategies and game dynamics,” Math. Bioscience, 40:145-156, 1978.
* [17] Tembine, H. , Altman, E. , El-Azouzi, R. and Hayel, Y., “Evolutionary games with random number of interacting players applied to access control”, Proc. of IEEE/ACM WiOpt, March 2008.
* [18] Tembine H., Altman E. and El-Azouzi R., “Delayed evolutionary game dynamics applied to the medium access control”, In Proc. IEEE MASS, 2007.
* [19] Tembine H., Altman E., El-Azouzi R. and Hayel Y. “Multiple access game in ad-hoc networks”, In Proc. GameComm, 2007.
* [20] Vincent, T.L. and Brown, J.S., Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge Univ. Press, 2005.
* [21] Wei Y. and Cioffi, J.M. “Competitive equilibrium in the Gaussian interference channel,” IEEE Internat. Symp. Information Theory (ISIT), 2000.
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|
arxiv-papers
| 2011-03-13T03:43:32 |
2024-09-04T02:49:17.623935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quanyan Zhu, Hamidou Tembine, Tamer Basar",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1103.2493"
}
|
1103.2496
|
# Evolutionary Games for Multiple Access Control222The material in this paper
was partially presented in [9] and [10]. 333This work was supported in part by
a grant from AFOSR and by a MURI grant.
Quanyan Zhu, Hamidou Tembine, Tamer Başar111Q. Zhu and T. Başar are with
Coordinated Science Laboratory, University of Illinois at Urbana-Champaign,
Urbana, IL, USA. (Email: {zhu31, basar1}@illinois.edu) H. Tembine is with
Department of Telecommunications, École Supérieure d’Electricité (SUPELEC),
France. (Email: tembine@ieee.org)
###### Abstract
In this paper, we formulate an evolutionary multiple access control game with
continuous-variable actions and coupled constraints. We characterize
equilibria of the game and show that the pure equilibria are Pareto optimal
and also resilient to deviations by coalitions of any size, i.e., they are
strong equilibria. We use the concepts of price of anarchy and strong price of
anarchy to study the performance of the system. The paper also addresses how
to select one specific equilibrium solution using the concepts of normalized
equilibrium and evolutionarily stable strategies. We examine the long-run
behavior of these strategies under several classes of evolutionary game
dynamics, such as Brown-von Neumann-Nash dynamics, Smith dynamics and
replicator dynamics. In addition, we examine correlated equilibrium for the
single-receiver model. Correlated strategies are based on signaling structures
before making decisions on rates. We then focus on evolutionary games for
hybrid additive white Gaussian noise multiple access channel with multiple
users and multiple receivers, where each user chooses a rate and splits it
over the receivers. Users have coupled constraints determined by the capacity
regions. Building upon the static game, we formulate a system of hybrid
evolutionary game dynamics using G-function dynamics and Smith dynamics on
rate control and channel selection, respectively. We show that the evolving
game has an equilibrium and illustrate these dynamics with numerical examples.
## 1 Introduction
Recently, there has been much interest in understanding the behavior of
multiple access controls under constraints. Considerable amount of work has
been carried out on the problem of how users can obtain an acceptable
throughput by choosing rates independently. Motivated by the interest in
studying a large population of users playing the game over time, evolutionary
game theory was found to be an appropriate framework for communication
networks. It has been applied to problems such as power control in wireless
networks and mobile interference control [11, 1, 5, 6].
The game-theoretical models considered in the previous studies on user
behaviors in CDMA, [37, 4], are static one-shot non-cooperative games in which
users are assumed to be rational and optimize their payoffs independently.
Evolutionary game theory, on the other hand, studies games that are played
repeatedly, and focuses on the strategies that persist over time, yielding the
best fitness of a user in a non-cooperative environment on a large time scale.
In [19], an additive white Gaussian noise (AWGN) multiple access channel
problem was modeled as a noncooperative game with pairwise interactions, in
which users were modeled as rational entities whose only interest was to
maximize their own communication rates. The authors obtained the Nash
equilibrium of the two-user game and introduced a two-player evolutionary game
model with pairwise interactions based on replicator dynamics. However, the
case when interactions are not pairwise arises frequently in communication
networks, such as the Code Division Multiple Access (CDMA) or the Orthogonal
Frequency-Division Multiple Access (OFDMA) in Worldwide Interoperability for
Microwave Access (WiMAX) environment [11].
In this work, we extend the study of [19] to wireless communication systems
with an arbitrary number of users corresponding to each receiver. We formulate
a static non-cooperative game with $m$ users subject to rate capacity
constraints, and extend the constrained game to a dynamic evolutionary game
with a large number of users whose strategies evolve over time. Different from
evolutionary games with discrete and finite number of actions, our model is
based on a class of continuous games, known as continuous-trait games.
Evolutionary games with continuum action spaces can be encountered in a wide
variety of applications in evolutionary ecology, such as evolution of
phenology, germination, nutrient foraging in plants, and predator-prey
foraging [24, 7].
In addition to the single receiver model, we investigate the case with
multiple users and receivers. We first formulate a static hybrid non-
cooperative game with $N$ users who rationally make decisions on the rates as
well as the channel selection subject to rate capacity constraints of each
receiver. We extend the static game to a dynamic evolutionary game by viewing
rate selections governed by a fitness function parameterized by the channel
selections. Such a concept of a hybrid model has appeared earlier in [36] and
[40], in the context of hybrid power control in CDMA systems. The strategies
of channel selections determine the long-term fitness of the rates chosen by
each user. We formulate such dynamics based on generalized Smith dynamics and
generating fitness function (G-function) dynamics.
### 1.1 Contribution
The main contributions of this work can be summarized as follows. We first
introduce a game-theoretic framework for local interactions between many users
and a single receiver. We show that the static continuous-kernel rate
allocation game with coupled rate constraints has a convex set of pure Nash
equilibria, coinciding with the maximal face of the polyhedral capacity
region. All the pure equilibria are Pareto optimal and are also strong
equilibria, resilient to simultaneous deviation by coalition of any size. We
show that the pure Nash equilibria in the rate allocation problem are 100%
efficient in terms of Price of Anarchy (PoA) and constrained Strong Price of
Anarchy (CSPoA). We study the stability of strong equilibria, normalized
equilibria, and evolutionary stable strategies (ESS) using evolutionary game
dynamics such as Brown-von Neumann-Nash dynamics, generalized Smith dynamics,
and replicator dynamics. We further investigate the correlated equilibrium of
the multiple access game where the receiver can send signals to the users to
mediate the behaviors of the transmitters.
Based on the single-receiver model, we then propose an evolutionary game-
theoretic framework for the hybrid additive white Gaussian noise multiple
access channel. We consider a communication system of multiple users and
multiple receivers, where each user chooses a rate and splits it over the
receivers. Users have coupled constraints determined by the capacity regions.
We characterize Nash equilibrium of the static game and show the existence of
the equilibrium under general conditions. Building upon the static game, we
formulate a system of hybrid evolutionary game dynamics using G-function
dynamics and Smith dynamics on rate control and channel selection,
respectively. We show that the evolving game has an equilibrium and illustrate
these dynamics with numerical examples.
### 1.2 Organization of the paper
The rest of the paper is structured as follows. We present in Section 2.1 the
evolutionary game model of rate allocation in additive white Gaussian multiple
access wireless networks, and analyze its equilibria and Pareto optimality in
Section 2.2. In Section 2.3, we present strong equilibria and price of anarchy
of the game. In Section 2.4, we discuss how to select one specific equilibrium
such as normalized equilibrium and evolutionary stable strategies. Section 2.5
studies the stability of equilibria and evolution of strategies using game
dynamics. Section 2.6 analyzes the correlated equilibrium of the multiple
acess game.
In Section 3.1, we present the hybrid rate control model where users can
choose the rates and the probability of the channels to use. In Section 3.2,
we characterize the Nash equilibrium of the constrained hybrid rate control
game model, pointing out the existence of the Nash equilibrium of the hybrid
model and methods to find it. In Section 3.3, we apply evolutionary dynamics
to both rates and channel selection probabilities. We use simulations to
demonstrate the validity of these proposed dynamics and illustrate the
evolution of the overall evolutionary dynamics of the hybrid model. Section 4
concludes the paper. For reader’s convenience, we summarize the notations in
Table 1 and the acronyms in Table 2.
Table 1: List of Notations Symbol | Meaning
---|---
$\mathcal{N}$ | set of $N$ users
${\Omega}$ | a subset of $N$ users
$\mathcal{J}$ | set of $J$ receivers
$\mathcal{A}_{i}$ | action set of user $i$
$P_{i}$ | maximum power of user $i$
$h_{i}$ | channel gain of user $i$
$\alpha_{i}$ | rate of user $i$
${p}_{ij}$ | probability of user $i$ selecting receiver $j$
$u_{i}$ | payoff of user $i$
$\overline{U}_{i}$ | expected payoff of user $i$
$\mathcal{C}$ | capacity region of a set $\mathcal{N}$ of users in a single receiver case
$\mathcal{C}(j)$ | capacity region of a set $\mathcal{N}$ of users at receiver $j$
$\lambda_{i}$ | distribution over the action set $\mathcal{A}_{i}$
$\mu$ | population state
Table 2: List of Acronyms Abbreviation | Meaning
---|---
AGWN | Additive White Gaussian Noise
MAC | Multiple Access Control
MISO | Multi-Input and Multi-Output
CCE | Constrained Correlated Equilibrium
ESS | Evolutionary Stable Equilibrium
NE | Nash Equilibrium
PoA | Price of Anarchy
SPoA | Strong Price of Anarchy
## 2 AWGN Mutiple Access Model: Single Receiver Case
We consider a communication system consisting of several receivers and several
senders (see Figure 1). At each time, there are several simultaneous local
interactions (typically, at each receiver there is a local interaction). Each
local interaction corresponds to a non-cooperative one-shot game with common
constraints. The opponents do not necessarily stay the same from a given time
slot to the next one. Users revise their rates in view of their payoffs and
the coupled constraints (for example by using an evolutionary process, a
learning process or a trial-and-error updating process). The game evolves over
time. Users are interested in maximizing a fitness function based on their own
communication rates at each time, and they are aware of the fact that the
other users have the same goal. The coupled power and rate constraints are
also common knowledge. Users have to choose independently their own coding
rates at the beginning of the communication, where the rates selected by a
user may be either deterministic, or chosen from some distribution. If the
rate profile arrived at as a result of these independent decisions lies in the
capacity region, users will communicate at that operating point. Otherwise,
either the receiver is unable to decode any signal and the observed rates are
zero, or only one of the signals can be decoded. The latter occurs when all
the other users are transmitting at or below a safe rate. With these
assumptions, we can define a constrained non-cooperative game. The set of
allowed strategies for user $i$ is the set of all probability distributions
over $[0,+\infty[,$ and the payoff is a function of the rates. In addition,
the rational action (rate) sets are restricted to lie in the capacity regions
(the payoff is zero if the constraint is violated). In order to study the
interactions between the selfish or partially cooperative users and their
stationary rates in the long run, we propose to model the problem of rate
allocation in Gaussian multiple access channels as an evolutionary game with a
continuous action space and coupled constraints. The development of
evolutionary game theory is a major contribution of biology to competitive
decision making and the evolution of cooperation. The key concepts of
evolutionary game theory are (i) Evolutionary Stable States [27], which is a
refinement of equilibria, and (ii) Evolutionary Game Dynamics such as
replicator dynamics [32], which describes the evolution of strategies or
frequencies of use of strategies in time, [7, 21].
Figure 1: A population: distributed receivers and senders, represented by blue
rectangles and red circles respectively.
The single population evolutionary rate allocation game is described as
follows: there is one population of senders (users) and several receivers. The
number of senders is large. At each time, there are many one-shot games called
local interactions. Each sender of the population chooses from his set of
strategies ${\mathcal{A}_{i}}$ which is a non-empty, convex and compact subset
of $\mathbb{R}.$ Without loss of generality, we can suppose that user $i$
chooses its rate in the interval $\mathcal{A}_{i}=[0,C_{\\{i\\}}]$, where
$C_{\\{i\\}}$ is the rate upper bound for user $i$ (to be made precise
shortly), as outside of the capacity region the payoff (as to be defined
later) will be zero. Let $\Delta({\mathcal{A}}_{i})$ be the set of probability
distributions over the pure strategy set $\mathcal{A}_{i}.$ The set
$\Delta({\mathcal{A}}_{i})$ can be interpreted as the set of mixed strategies.
It is also interpreted as the set of distributions of strategies among the
population. Let $\lambda_{i}\in\Delta({\mathcal{A}}_{i}),$ and $E$ be a
$\lambda_{i}-$ measurable subset of $\mathbb{R}^{N}$; then $\lambda_{i}(E)$
represents the fraction of users choosing a strategy out of $E$, at time $t.$
A distribution $\lambda_{i}\in\Delta({\mathcal{A}}_{i})$ is sometimes called
the “state” of the population. We denote by $\mathbb{B}(\mathcal{A}_{i})$ the
Borel $\sigma-$algebra on ${\mathcal{A}}_{i}$ and by
$d(\lambda,\lambda^{\prime})$ the distance between two states measured with
the respect to the weak topology. Each user’s payoff depends on opponents’
behavior through the distribution of opponents’ choices and of their
strategies. The payoff of a user $i$ in a local interaction with $(N-1)$ other
users is given as a function $u_{i}:\
\mathbb{R}^{N}\longrightarrow\mathbb{R}.$ The rate profile
$\alpha\in\mathbb{R}^{N}$ must belong to a common capacity region
$\mathcal{C}\subset\mathbb{R}^{N}$ defined by $2^{N}-1$ linear inequalities.
The expected payoff of a sender $i$ transmitting at a rate $a$ when the state
of the population is $\mu\in\Delta(\mathcal{A}_{i})$ is given by
$F_{i}(a,\mu).$ The expected payoff for user $i$ is
$F_{i}(\lambda_{i},\mu):=\int_{\alpha\in\mathcal{C}}u_{i}(\alpha)\
\lambda_{i}(d\alpha_{i})\prod_{j\neq i}\mu(d\alpha_{j}).$
The population state is subjected to the “mixed extension” of capacity
constraints $\mathcal{M}(\mathcal{C}).$ This will be discussed in Section 2.5
and will be made more precise later.
### 2.1 Local Interactions
Local interaction refers to the problem setting of one receiver and its uplink
additive white Gaussian noise (AWGN) multiple access channel with $N$ senders
with coupled constraints (or actions). The signal at the receiver is given by
$Y=\xi+\sum_{i=1}^{N}X_{i}$ where $X_{i}$ is a transmitted signal of user $i$
and $\xi$ is a zero-mean Gaussian noise with variance $\sigma_{0}^{2}.$ Each
user has an individual power constraint $\mathbb{E}(X_{i}^{2})\leq P_{i}$ and
the channel gain $h_{i}$. The optimal power allocation scheme is to transmit
at the maximum power available, i.e. $P_{i}$, for each user. Hence, we
consider the case in which maximum power is attained. The decisions of the
users then consist of choosing their communication rates, and the receiver’s
role is to decode, if possible. The capacity region is the set of all vectors
$\alpha\in{\mathbb{R}}^{N}_{+}$ such that users
$i\in\mathcal{N}:=\\{1,2,\ldots,N\\}$ can reliably communicate at rate
$\alpha_{i},~{}i\in\mathcal{N}.$ The capacity region $\mathcal{C}$ for this
channel is the set
$\displaystyle\mathcal{C}=\left\\{\alpha\in{\mathbb{R}}^{N}_{+}~{}\bigg{|}~{}\sum_{i\in\Omega}\alpha_{i}\leq\log\left(1+|\Omega|\frac{P_{i}h_{i}}{\sigma^{2}_{0}}\right).\forall\
\emptyset\subsetneqq\Omega\subseteq\mathcal{N}\right\\},$
###### Example 1.
(Example of capacity region with three users) In this example, we illustrate
the capacity region with three users. Let $\alpha_{1},\alpha_{2},\alpha_{3}$
be the rates of the users and $P_{i}=P,h_{i}=h,\forall i=1,2,3$. Based on
(2.1), we obtain a set of inequalities
$\left\\{\begin{array}[]{l}\alpha_{1}\geq 0,\alpha_{2}\geq 0,\alpha_{3}\geq
0\\\ \alpha_{i}\leq\log\left(1+\frac{Ph}{\sigma_{0}^{2}}\right),i=1,2,3\\\
\alpha_{i}+\alpha_{j}\leq\log\left(1+2\frac{Ph}{\sigma_{0}^{2}}\right),i\not=j,i,j=1,2,3.\\\
\alpha_{1}+\alpha_{2}+\alpha_{3}\leq\log\left(1+3\frac{Ph}{\sigma_{0}^{2}}\right)\\\
\end{array}\right.,$
or in the compact notation, $M_{3}\gamma_{3}\leq\zeta_{3},\ $ where
$\gamma_{3}:=\left[\begin{array}[]{c}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\end{array}\right]\in\mathbb{R}_{+}^{3},\
\zeta_{3}:=\left[\begin{array}[]{c}C_{\\{1\\}}\\\ C_{\\{2\\}}\\\
C_{\\{3\\}}\\\ C_{\\{1,2\\}}\\\ C_{\\{1,3\\}}\\\ C_{\\{2,3\\}}\\\
C_{\\{1,2,3\\}}\end{array}\right],M_{3}:=\left[\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\\\ 1&1&0\\\ 1&0&1\\\ 0&1&1\\\
1&1&1\end{array}\right]\in\mathbb{Z}^{7\times 3}.$
Note that $M_{3}$ is a totally unimodular matrix. By letting
$Ph=25,\sigma_{0}^{2}=0.1,$ we sketch in Figure 2 the capacity region with
three users.
Figure 2: Capacity region with three users.
The capacity region reveals a competitive nature of the interactions among
senders: if a user $i$ wants to communicate at a higher rate, one of the other
users has to lower his rate; otherwise, the capacity constraint is violated.
We let
$r_{i,\Omega}:=\log\left(1+\frac{P_{i}h_{i}}{\sigma_{0}^{2}+\sum_{i^{\prime}\in\Omega,i^{\prime}\neq
i}P_{i^{\prime}}h_{i^{\prime}}}\right),i\in\mathcal{N},\Omega\subseteq\mathcal{N}$
denote the bound on the rate of a user when the signals of the $|\Omega|-1$
other users are treated as noise.
Due to the noncooperative nature of the rate allocation, we can formulate the
one-shot game
$\Xi=\langle\mathcal{N},(\mathcal{A}_{i})_{i\in\mathcal{N}},(u_{i})_{i\in\mathcal{N}}\rangle\,,$
where the set of users $\mathcal{N}$ is the set of players, $\mathcal{A}_{i}$,
$i\in\mathcal{N}$, is the set of actions, and $u_{i}$, $i\in\mathcal{N}$, are
the payoff functions.
### 2.2 Payoffs
We define $u_{i}:\prod_{i=1}^{N}\mathcal{A}_{i}\rightarrow\mathbb{R}_{+}$ as
follows:
$\displaystyle u_{i}(\alpha_{i},\alpha_{-i})$ $\displaystyle=$
$\displaystyle\upharpoonleft_{\mathcal{C}}(\alpha)g_{i}(\alpha_{i},\alpha_{-i})$
(1) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}g_{i}(\alpha_{i})&{\textrm{~{}if~{}}}\
(\alpha_{i},\alpha_{-i})\in\mathcal{C}\\\
0&\mbox{otherwise}\end{array}\right.,$ (4)
where $\upharpoonleft_{\mathcal{C}}$ is the indicator function; $\alpha_{-i}$
is a vector consisting of other players’ rates, i.e.,
$\alpha_{-i}=[\alpha_{1},\ldots,\alpha_{i-1},\alpha_{i+1},\ldots,\alpha_{N}]$
and $u_{i}$ is a positive and strictly increasing function for each fixed
$\alpha_{-i}$. Since the game is subject to coupled constraints, the action
set $\mathcal{A}_{i}$ is coupled and dependent on other players’ actions.
Given the strategy profile $\alpha_{-i}$ of other players, the constrained
action set $\mathcal{A}_{i}$ is given by
$\mathcal{A}_{i}(\alpha_{-i}):=\\{\alpha_{i}\in[0,C_{\\{i\\}}],\
(\alpha_{i},\alpha_{-i})\in\mathcal{C}\\}$ (5)
We then have an asymmetric game. The minimum rate that the user $i$ can
guarantee in the feasible regions is $r_{i,\mathcal{N}}$ which is different
than $r_{j,\mathcal{N}}.$
Each user $i$ maximizes $u_{i}(\alpha_{i},\alpha_{-i})$ over the coupled
constraint set. Owing to the monotonicity of the function $g_{i}$ and the
inequalities that define the capacity region, we obtain the following lemma.
###### Lemma 1.
Let $\overline{BR}_{i}(\alpha_{-i})$ be the best reply to the strategy
$\alpha_{-i}$, defined by
$\overline{BR}^{i}(\alpha_{-i})=\arg\max_{y\in\mathcal{A}_{i}(\alpha^{-i})}u_{i}(y,\alpha_{-i}).$
$\overline{BR}_{i}$ is a non-empty single-valued correspondence (i.e. a
standard function), and is given by
$\max\left(r_{i,\mathcal{N}},\min_{\Omega\in\Gamma_{i}}\left\\{C_{\Omega}-\sum_{k\in\Omega\backslash\\{i\\}}\alpha_{k}\right\\}\right),$
(6)
where $\Gamma_{i}=\\{\Omega\in 2^{\mathcal{N}},i\in\Omega\\}$.
###### Proposition 1.
The set of Nash equilibria is
$\left\\{(\alpha_{i},\alpha_{-i})\ |\ \alpha^{i}\geq
r_{i,\mathcal{N}},\sum_{i\in\mathcal{N}}\alpha_{i}=C_{\mathcal{N}}\right\\}.$
All these equilibria are optimal in Pareto sense.
###### Proof.
Let $\beta$ be a feasible solution, i.e., $\beta\in\mathcal{C}.$ If
$\sum_{i=1}^{N}\beta_{i}<C_{\mathcal{N}}=\log\left(1+\sum_{i\in\mathcal{N}}\frac{P_{i}h_{i}}{\sigma_{0}^{2}}\right),$
then at least one of the users can improve its rate (hence its payoff) to
reach one of the faces of the capacity region. We now check the strategy
profile on the face
$\left\\{(\alpha_{i},\alpha_{-i})\ \bigg{|}\ \alpha^{i}\geq
r_{i,\mathcal{N}},\sum_{i=1}^{N}\alpha_{i}=C_{\mathcal{N}}\right\\}.$
If $\beta\in\left\\{(\alpha_{i},\alpha_{-i})\ \bigg{|}\ \alpha_{i}\geq
r_{i,\mathcal{N}},\sum_{i=1}^{N}\alpha_{i}=C_{\Omega}\right\\},$ then from the
Lemma 6, $\overline{BR}_{i}(\beta_{-i})=\\{\beta_{i}\\}.$ Hence, $\beta$ is a
strict equilibrium. Moreover, this strategy $\beta$ is Pareto optimal because
the rate of each user is maximized under the capacity constraint. These
strategies are social welfare optimal if the total utility
$\sum_{i=1}^{N}u_{i}(\alpha_{i},\alpha_{-i})=\sum_{i=1}^{N}g_{i}(\alpha_{i})$
is maximized subject to constraints. ∎
Note that the set of pure Nash equilibria is a convex subset of the capacity
region.
### 2.3 Robust Equilibria and Efficiency Measures
#### 2.3.1 Constrained Strong Equilibria and Coalition Proofness
An action profile in a local interaction between $N$ senders is a constrained
$k-$strong equilibrium if it is feasible and no coalition of size $k$ can
improve the rate transmissions of each of its members with respect to the
capacity constraints. An action is a constrained strong equilibrium [18] if it
is a constrained $k-$strong equilibrium for any size $k.$ A strong equilibrium
is then a policy from which no coalition (of any size) can deviate and improve
the transmission rate of every member of the coalition (group of the
simultaneous moves), while possibly lowering the transmission rate of users
outside the coalition group. This notion of constrained strong
equilibria444Note that the set of constrained strong equilibria is a subset of
the set of Nash equilibria (by taking coalitions of size one) and any
constrained strong equilibrium is Pareto optimal (by taking coalition of full
size). is very attractive because it is resilient against coalitions of users.
Most of the games do not admit any strong equilibrium but in our case we will
show that the multiple access channel game has several strong equilibria.
###### Theorem 1.
Any rate profile on the maximal face of the capacity region $\mathcal{C}:$
$Face_{\max}(\mathcal{C}):=\left\\{(\alpha_{i},\alpha_{-i})\in\mathbb{R}^{N}\
|\ \alpha_{i}\geq r_{N},\sum_{i=1}^{N}\alpha_{i}=C_{\mathcal{N}}\right\\},$
is a constrained strong equilibrium.
###### Proof.
We remark that if the rate profile $\alpha$ is not on the maximal face of the
capacity region, then $\alpha$ is not resilient to deviation by a single user.
Hence, $\alpha$ cannot be a constrained strong equilibrium. This says that a
necessary condition for a rate profile to be a strong equilibrium is to be in
the subset $Face_{\max}(\mathcal{C}).$ We now prove that the condition:
$\alpha\in Face_{\max}(\mathcal{C})$ is sufficient. Let $\alpha\in
Face_{\max}(\mathcal{C}).$ Suppose that $k$ users deviate simultaneously from
the rate profile $\alpha.$ Denote by $Dev$ the set of users which deviate
simultaneously (eventually by forming a coalition). The rate constraints of
the deviants are
1. 1.
${\alpha}^{\prime}_{i}\geq 0,\ \forall i\in Dev,$
2. 2.
$\sum_{i\in\bar{\Omega}}{\alpha}^{\prime}_{i}\leq C_{\bar{\Omega}},\
\forall\bar{\Omega}\subseteq Dev,$
3. 3.
$\sum_{i\in\Omega\cap Dev}{\alpha}^{\prime}_{i}\leq
C_{\Omega}-\sum_{i\in\Omega,i\notin Dev}\alpha_{i}$, $\
\forall{\Omega}\subseteq\mathcal{N},\ \Omega\cap Dev\neq\emptyset.$
In particular, for $\Omega=\mathcal{N},$ we have $\sum_{i\in
Dev}{\alpha^{\prime}}_{i}\leq C_{\mathcal{N}}-\sum_{i\notin Dev}\alpha_{i}.$
The total rate of the deviants is bounded by $C_{\mathcal{N}}-\sum_{i\notin
Dev}\alpha_{i}$, which is not controlled by the deviants. The deviants move to
$({\alpha}^{\prime}_{i})_{i\in Dev}$ with
$\sum_{i\in Dev}{\alpha}^{\prime}_{i}<C_{\mathcal{N}}-\sum_{i\notin
Dev}\alpha_{i}\,.$
Then, there exists $i$ such that $\alpha_{i}>{\alpha}^{\prime}_{i}.$ Since
$g_{i}$ is non-decreasing, this implies that
$g_{i}(\alpha_{i})>g_{i}({\alpha}^{\prime}_{i}).$ The user $i$ who is a member
of the coalition $Dev$ does not improve its payoff. If the rates of some of
the deviants are increased, then the rates of some other users from coalition
must decrease. If $({\alpha}^{\prime}_{i})_{i\in Dev}$ satisfies
$\sum_{i\in Dev}{\alpha}^{\prime}_{i}=C_{\mathcal{N}}-\sum_{i\notin
Dev}\alpha_{i}\,,$
then some users in the coalition $Dev$ have increased their rates compared
with $(\alpha_{i})_{i\in Dev}$ and some others in $Dev$ have decreased their
rates of transmission (because the total rate is the constant
$C_{\mathcal{N}}-\sum_{i\notin Dev}\alpha_{i}).$ The users in $Dev$ with a
lower rate ${\alpha}^{\prime}_{i}\leq\alpha_{i}$ do not benefit by being a
member of the coalition (Shapley criterion of membership of coalition does not
hold) . And this holds for any $\emptyset\subsetneqq
Dev\subseteqq\mathcal{N}.$ This completes the proof. ∎
###### Corollary 1.
In the constrained rate allocation game, Nash equilibria and strong equilibria
in pure strategies coincide.
#### 2.3.2 Constrained Potential Function for Local Interaction
Introduce the following function:
$V(\alpha)=\upharpoonleft_{\mathcal{C}}(\alpha)\sum_{i=1}^{N}g_{i}(\alpha_{i})\,,$
where $\upharpoonleft_{\mathcal{C}}$ is the indicator function of
$\mathcal{C},i.e.,\ $ $\upharpoonleft_{\mathcal{C}}(\alpha)=1$ if
$\alpha\in\mathcal{C}$ and $0$ otherwise. The function $V$ satisfies
$V(\alpha)-V(\beta_{i},\alpha_{-i})=g_{i}(\alpha_{i})-g_{i}(\beta_{i}),\
\forall\alpha,(\beta_{i},\alpha_{-i})\in\mathcal{C}.$
If $g_{i}$ is differentiable, then one has
$\frac{\partial}{\partial\alpha_{i}}V(\alpha)=g^{\prime}_{i}(\alpha_{i})=\frac{\partial}{\partial\alpha_{i}}u_{i}$
in the interior of the capacity region $\mathcal{C}$, and $V$ is a constrained
potential function [3] in pure strategies.
###### Corollary 2.
The local maximizers of $V$ in $\mathcal{C}$ are pure Nash equilibria. Global
maximizers of $V$ in $\mathcal{C}$ are both constrained strong equilibria and
social optima for the local interaction.
#### 2.3.3 Strong Price of Anarchy
Throughout this subsection, we assume that the functions $g_{i}$ are the
identity function, i.e., $g_{i}(x)=id(x):=x.$ One metric used to measure how
much the performance of decentralized systems is affected by the selfish
behavior of its components is the price of anarchy. We present a similar price
for strong equilibria under the coupled rate constraints. This notion of Price
of Anarchy can be seen as an efficiency metric that measures the price of
selfishness or decentralization and has been extensively used in the context
of congestion games or routing games where typically users have to minimize a
cost function [41, 42]. In the context of rate allocation in the multiple
access channel, we define an equivalent measure of price of anarchy for rate
maximization problems. One of the advantages of a strong equilibrium is that
it has the potential to reduce the distance between the optimal solution and
the solution obtained as an outcome of selfish behavior, typically in the case
where the capacity constraint is violated at each time. Since the constrained
rate allocation game has strong equilibria, we can define the strong price of
anarchy, introduced in [12], as the ratio between the payoff of the worst
constrained strong equilibrium and the social optimum value which
$C_{\mathcal{N}}$.
###### Theorem 2.
The strong price of anarchy of the constrained rate allocation game is 1 for
$g_{i}(x)=x.$
Note that for $g_{i}\neq id,$ the CSPoA can be less than one. However, the
optimistic price of anarchy of the best constrained equilibrium, also called
price of stability [13], is one for any function $g_{i}$ i.e the efficiency of
“best” equilibria is $100\%.$
### 2.4 Selection of Pure Equilibria
We have shown in the previous sections that our rate allocation game has a
continuum of pure Nash equilibria and strong equilibria. We address now the
problem of selecting one equilibrium which has certain desirable properties:
the normalized pure Nash equilibrium, introduced in [29]; see also [31, 20,
23]. We introduce the problem of constrained maximization faced by each user
when the other rates are at the maximal face of the polytope $\mathcal{C}$:
$\displaystyle\max_{\alpha}$ $\displaystyle u_{i}(\alpha)$ (7) s.t.
$\displaystyle\alpha_{1}+\ldots+\alpha_{N}=C_{\mathcal{N}}$ (8)
for which the corresponding Lagrangian for user $i$ is given by
$L_{i}(\alpha,\zeta)=u_{i}(\alpha)-\zeta_{i}\left(\sum_{i=1}^{N}\alpha_{i}-C_{\mathcal{N}}\right).$
From Karush-Kuhn-Tucker optimality conditions, it follows that there exists
$\zeta\in\mathbb{R}^{N}$ such that
$g_{i}^{\prime}(\alpha_{i})=\zeta_{i},\
\sum_{i=1}^{N}\alpha_{i}=C_{\mathcal{N}}.$
For a fixed vector $\zeta$ with identical entries, define the normal form game
$\Gamma({\zeta})$ with $N$ users, where actions are taken as rates and the
payoffs given by $L(\alpha,\zeta).$ A normalized equilibrium is an equilibrium
of the game $\Gamma(\zeta^{*})$ where $\zeta^{*}$ is normalized into the form
${\zeta^{*}_{i}}=\frac{c}{\tau_{i}},\ c>0,\tau_{i}>0.$ We now have the
following result due to Goodman [20] which implies Rosen’s condition on
uniqueness for strict concave games.
###### Theorem 3.
Let $u_{i}$ be a smooth and strictly concave function in $\alpha_{i},$ each
$u_{i}$ be convex in $\alpha_{-i}$, and there exist some $\zeta$ such that the
weighted non-negative sum of the payoffs
$\sum_{i=1}^{N}\zeta_{i}u_{i}(\alpha)$ is concave in $\alpha.$ Then, the
matrix $G(\alpha,\zeta)+G^{T}(\alpha,\zeta)$ is negative definite (which
implies uniqueness) where $G(\alpha,\zeta)$ is the Jacobian with respect to
$\alpha$ of
$h(\alpha,\zeta):=\left[\zeta_{1}\nabla_{1}u_{1}(\alpha),\zeta_{2}\nabla_{2}u_{2}(\alpha),\ldots,\zeta_{N}\nabla_{N}u_{N}(\alpha)\right]^{T}$
and $G^{T}$ is the transpose of the matrix $G.$
This now leads to the following corollary for our problem.
###### Corollary 3.
If $g_{i}$ are non-decreasing strictly concave functions, then the rate
allocation game has a unique normalized equilibrium which corresponds to an
equilibrium of the normal form game with payoff $L(\alpha,\zeta^{*})$ for some
$\zeta^{*}.$
### 2.5 Stability and Dynamics
In this subsection, we study the stability of equilibria and several classes
of evolutionary game dynamics under a symmetric case, i.e.,
$P_{i}=P,h_{i}=h,g_{i}=g,\mathcal{A}_{i}=\mathcal{A},\ \forall
i\in\mathcal{N}$. We will drop subscript index $i$ where appropriate. We show
that the associated evolutionary game has a unique pure constrained
evolutionary stable strategy.
###### Proposition 2.
The collection of rates
$\alpha=\left(\frac{C_{\mathcal{N}}}{N},\ldots,\frac{C_{\mathcal{N}}}{N}\right)\,,$
i.e. the distribution of Dirac concentrated on the rate
$\frac{C_{\mathcal{N}}}{N},$ is the unique symmetric pure Nash equilibrium.
###### Proof.
Since the constrained rate allocation game is symmetric, there exists a
symmetric (pure or mixed) Nash equilibrium. If such an equilibrium exists in
pure strategies, each user transmits with the same rate $r^{*}.$ It follows
from Proposition 1 of Section 2.2, and the bound
$r_{N}\leq\frac{C_{\mathcal{N}}}{N}$ that $r^{*}$ satisfies
$Nr^{*}=C_{\mathcal{N}}$ and $r^{*}$ is feasible. ∎
Since the set of feasible actions is convex, we can define convex combination
of rates in the set of the feasible rates. For example,
$\epsilon\alpha^{\prime}+(1-\epsilon)\alpha$ is a feasible rate if
$\alpha^{\prime}$ and $\alpha$ are feasible. The symmetric rate profile
$(r,r,\ldots,r)$ is feasible if and only if $0\leq r\leq
r^{*}=\frac{C_{\mathcal{N}}}{N}.$ We say that the rate $r$ is a constrained
evolutionarily stable strategy (ESS) if it is feasible and for every mutant
strategy $mut\neq\alpha$ there exists $\epsilon_{mut}>0$ such that
$\left\\{\begin{array}[]{cc}r_{\epsilon}:=\epsilon\
mut+(1-\epsilon)r\in\mathcal{C}&\forall\epsilon\in(0,\epsilon_{mut})\\\
u(r,r_{\epsilon},\ldots,r_{\epsilon})>u(mut,r_{\epsilon},\ldots,r_{\epsilon})&\forall\epsilon\in(0,\epsilon_{mut})\end{array}\right.$
###### Theorem 1.
The pure strategy $r^{*}=\frac{C_{\mathcal{N}}}{N}$ is a constrained
evolutionary stable strategy.
###### Proof.
Let $mut\leq r^{*}$ The rate $\epsilon\ mut+(1-\epsilon)r^{*}$ is feasible
implies that $mut\leq r^{*}$ (because $r^{*}$ is the maximum symmetric rate
achievable). Since $mut\neq r^{*},$ $mut$ is strictly lower than $r^{*}.$ By
monotonicity of the function $g,$ one has $u(r^{*},\epsilon\
mut+(1-\epsilon)r^{*})>u(mut,\epsilon\ mut+(1-\epsilon)r^{*}),\
\forall\epsilon.$ This completes the proof. ∎
#### 2.5.1 Symmetric Mixed Strategies
Define the mixed capacity region $\mathcal{M}(\mathcal{C})$ as the set of
measures profile $(\mu_{1},\mu_{2},\ldots,\mu_{N})$ such that
$\int_{\mathbb{R}_{+}^{|\Omega|}}\left(\sum_{i\in\Omega}\alpha_{i}\right)\prod_{i\in\Omega}\mu_{i}(d\alpha_{i})\leq
C_{\Omega},\ \forall\Omega\subseteq 2^{\mathcal{N}}.$
Then, the payoff of the action $a\in\mathbb{R}_{+}$ satisfying
$(a,\lambda,\ldots,\lambda)\in\mathcal{M}(\mathcal{C})$ can be defined as
$F(a,\mu)=\int_{[0,\infty[^{N-1}}u(a,b_{2},\ldots,b_{N})\ \nu_{N-1}(db)\,,$
(9)
where $\nu_{k}=\bigotimes_{1}^{k}\mu$ is the product measure on
$[0,\infty[^{k}.$ The constraint set becomes the set of probability measures
on $\mathbb{R}_{+}$ such that
$0\leq\mathbb{E}(\mu):=\int_{\mathbb{R}_{+}}\ \alpha_{i}\
\mu(d\alpha_{i})\leq\frac{C_{\mathcal{N}}}{N}<C_{\\{1\\}}\,.$
###### Lemma 2.
The payoff can be obtained as follows:
$F(a,\mu)=\upharpoonleft_{[0,C_{\mathcal{N}}-(N-1)\mathbb{E}(\mu)]}\times
g(a)\times\int_{b\in\mathcal{D}_{a}}\
\nu_{N-1}(db)=\upharpoonleft_{[0,C_{\mathcal{N}}-(m-1)\mathbb{E}(\mu)]}\times
g(a)\nu_{N-1}(\mathcal{D}_{a}),$
where $\mathcal{D}_{a}=\\{(b_{2},\ldots,b_{N})\ |\
(a,b_{2},\ldots,b_{N})\in\mathcal{C}\\}\,.$
###### Proof.
If the rate does not satisfy the capacity constraints, then the payoff is $0.$
Hence the rational rate for user $i$ is lower than $C_{\\{i\\}}.$ Fix a rate
$a\in[0,C_{\\{i\\}}].$ Let
$D^{a}_{\Omega}:=C_{\Omega}-a\delta_{\\{1\in\Omega\\}}.$ Then, a necessary
condition to have a non-zero payoff is
$(b_{2},\ldots,b_{N})\in\mathcal{D}_{a}\,,$ where
$\mathcal{D}_{a}=\\{(b_{2},\ldots,b_{N})\in\mathbb{R}_{+}^{N-1},\
\sum_{i\in\Omega,i\neq 1}b_{i}\leq D^{a}_{\Omega},\ \Omega\subseteq
2^{\mathcal{N}}\\}.$ Thus, we have
$\displaystyle F(a,\mu)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}^{N-1}}u(a,b_{2},\ldots,b_{N})\
\nu_{N-1}(db)$ $\displaystyle=$ $\displaystyle\int_{b\in\mathbb{R}_{+}^{N-1},\
(a,b)\in\mathcal{C}}g(a)\ \nu_{N-1}(db)$ $\displaystyle=$
$\displaystyle\upharpoonleft_{[0,C_{\mathcal{N}}-(N-1)\mathbb{E}(\mu)]}g(a)\times\int_{b\in\mathcal{D}_{a}}\
\nu_{N-1}(db)$
∎
#### 2.5.2 Constrained Evolutionary Game Dynamics
The class of evolutionary games in large population provides a simple
framework for describing strategic interactions among large numbers of users.
In this subsection we turn to modeling the behavior of the users who play
them. Traditionally, predictions of behavior in game theory are based on some
notion of equilibrium, typically Cournot equilibrium, Bertrand equilibrium,
Nash equilibrium, Stackelberg solution, Wardrop equilibrium or some refinement
thereof. These notions require the assumption of equilibrium knowledge, which
posits that each user correctly anticipates how his opponents will act. The
equilibrium knowledge assumption is too strong and is difficult to justify in
particular in contexts with large numbers of users. As an alternative to the
equilibrium approach, we propose an explicitly dynamic updating choice, a
procedure in which users myopically update their behavior in response to their
current strategic environment. This dynamic procedure does not assume the
automatic coordination of users’ actions and beliefs, and it can derive many
specifications of users’ choice procedures. These procedures are specified
formally by defining a revision of rates called revision protocol [30]. A
revision protocol takes current payoffs and current mean rate and maps to
conditional switch rates which describe how frequently users in some class
playing rate $\alpha$ who are considering switching rates switch to strategy
$\alpha^{\prime}.$ Revision protocols are flexible enough to incorporate a
wide variety of paradigms, including ones based on imitation, adaptation,
learning, optimization, etc.
We use here a class of continuous evolutionary dynamics. We refer to [33, 14,
34] for evolutionary game dynamics with or without time delays. The
continuous-time evolutionary game dynamics on the measure space
$(\mathcal{A},\mathcal{B}(\mathcal{A}),\mu)$ is given by
$\dot{\lambda}_{t}(E)=\int_{a\in E}V(a,\lambda_{t})\mu(da)$ (10)
where
$V(a,\lambda_{t})=K\left[\int_{x\in\mathcal{A}}\beta^{x}_{a}(\lambda_{t})\lambda_{t}(dx)-\int_{x\in\mathcal{A}}\beta^{a}_{x}(\lambda_{t})\lambda_{t}(dx)\right],$
and $\beta^{x}_{a}$ represents the rate of mutation from $x$ to $a,$ and $K$
is a growth parameter. $\beta^{x}_{a}(\lambda_{t})=0$ if $(x,\lambda_{t})$ or
$(a,\lambda_{t})$ is not in the (mixed) capacity region, $E$ is a
$\mu-$measurable subset of $\mathcal{A}.$ At each time $t,$ probability
measure $\lambda_{t}$ satisfies $\frac{d}{dt}\lambda_{t}(\mathcal{A})=0$. We
examine the following classes of evolutionary game dynamics, namely, Brown-von
Neumann-Nash dynamics, Smith dynamics and replicator dynamics, where
$F(a,\lambda_{t})$ is the payoff in (9) as defined in the previous subsection.
1. RD-1:
Constrained Brown-von Neumann-Nash dynamics.
$\beta^{x}_{a}(\lambda_{t})=\left\\{\begin{array}[]{cl}\max(F(a,\lambda_{t})-\int_{x}F(x,\lambda_{t})\
dx,0)&\mbox{if}\ (a,\lambda_{t}),\
(x,\lambda_{t})\in\mathcal{M}(\mathcal{C}),\\\
0&\mbox{otherwise.}\end{array}\right.$
2. RD-2:
Constrained Replicator Dynamics.
$\beta^{x}_{a}(\lambda_{t})=\left\\{\begin{array}[]{cl}\max(F(a,\lambda_{t})-F(x,\lambda_{t}),0)&\mbox{if}\
(a,\lambda_{t}),\ (x,\lambda_{t})\in\mathcal{M}(\mathcal{C})\\\
0&\mbox{otherwise.}\end{array}\right.$
3. RD-3:
Constrained $\theta-$Smith Dynamics.
$\beta^{x}_{a}(\lambda_{t})=\left\\{\begin{array}[]{cl}\max(F(a,\lambda_{t})-F(x,\lambda_{t}),0)^{\theta}&\mbox{if}\
(a,\lambda_{t}),\ (x,\lambda_{t})\in\mathcal{M}(\mathcal{C})\\\
0&\mbox{otherwise.}\end{array}\right.,\ \theta\geq 1$
A common property that applies to all these dynamics is that the set of Nash
equilibria is a subset of rest points (stationary points) of the evolutionary
game dynamics. Here we extend the concepts of these dynamics to evolutionary
games with a continuum action space and coupled constraints, and more than
two-users interactions. The counterparts of these results in discrete action
space can be found in [21, 30].
###### Theorem 2.
Any Nash equilibrium of the game is a rest point of the following evolutionary
game dynamics: constrained Brown-von Neumann-Nash, generalized Smith dynamics,
and replicator dynamics. In particular, the evolutionary stable strategies set
is a subset of the rest points of these constrained evolutionary game
dynamics.
###### Proof.
It is clear for pure equilibria by using the revision protocols $\beta$ of
these dynamics. Let $\lambda$ be an equilibrium. For any rate $a$ in the
support of $\lambda,$ $\beta^{a}_{x}=0$ if $F(x,\lambda)\leq F(a,\lambda).$
Thus, if $\lambda$ is an equilibrium, the difference between the microscopic
inflow and outflow is $V(a,\lambda)=0$, given that $a$ is the support of the
measure $\lambda.$ ∎
Let $\lambda$ be a finite Borel measure on $[0,C_{\\{i\\}}]$ with full
support. Suppose $g$ is continuous on $[0,C_{\\{i\\}}].$ Then, $\lambda$ is a
rest point of the BNN dynamics if and only if $\lambda$ is a symmetric Nash
equilibrium. Note that the choice of topology is an important issue when
defining dynamics convergence and stability of the dynamics. The most used in
this area is the topology of the weak convergence to measure closeness of two
states of the system. Different distances (Prohorov metric, metric on bounded
and Lipschitz continuous functions on $\mathcal{A}$) have been proposed. We
refer the reader to [26], and the references therein for more details on
evolutionary robust strategy and stability notions.
### 2.6 Correlated Equilibrium
In this subsection, we analyze constrained correlated equilibria of multiple
access (MAC) games. Building upon the signaling in the one-shot game, we
formulate a system of evolutionary MAC games with evolutionary evolutionary
game dynamics that describe the evolution of signaling, beliefs, rate control
and channel selection, respectively.
We focus on correlated equilibrium in the single-receiver case. Correlated
strategies are based on signaling structures before making decisions on rates.
Different scenarios (with or without mediator, virtual mediator, cryptographic
multi-stage signaling structure) have been proposed in the literature [15, 39,
16, 17].
Figure 3: Signaling between multiple senders and a receiver
In Figure 3, we illustrate the signaling between multiple transmitters and one
receiver. The receiver can act as a signaling device to mediate the behaviors
of the transmitters. The correlated equilibrium has a strong connection with
cryptography in that the private signal sent to the users can be realized by
the coding and decoding in the network [39].
Let $\mathcal{B}$ be the set of signals
$\beta=[\beta_{i},\beta_{-i}]\in\mathbb{R}^{N}.$ The values $\beta$ from the
set of signals need to be in the feasible set
$\mathcal{C}\subset\mathbb{R}^{N}$. Let $\mu\in\Delta{\mathcal{B}}$ be a
probability measure over the set $\mathcal{B}.$ A constrained correlated
equilibrium (CCE) $\mu^{*}$ need to satisfy the following set of inequalities,
$\displaystyle\int
d\mu^{*}(\beta_{i},\beta_{-i})\left[u_{i}(\alpha_{i},\alpha_{-i},\mid\beta_{i})-u_{i}(\alpha_{i}^{\prime},\alpha_{-i}\mid\beta_{i})\right]\geq
0,\forall i\in\mathcal{N},\alpha^{\prime}_{i}\in\mathcal{A}_{i}(\alpha_{-i}).$
Define a rule of assignment of user $i$ as a map its signals to its action’s
set $\bar{r}_{i}:\ \beta_{i}\ \longmapsto\alpha_{i}.$ A CCE is then
characterized by
$\displaystyle\int
d\mu^{*}(\beta)\left[u_{i}(\alpha_{i},\alpha_{-i}\mid\beta_{i})-u_{i}(r_{i}(\beta_{i}),\alpha_{-i})\right]\geq
0,\forall i\in\mathcal{N},\forall r_{i}\ \mbox{such that}\
\bar{r}_{i}(.)\in\mathcal{A}_{i}(\alpha_{-i}).$ (11)
###### Theorem 3.
The set of constrained pure Nash equilibria of the MISO game is given by
$\mbox{max-face}(\mathcal{C})=\left\\{(\alpha_{1},\ldots,\alpha_{N})\ |\
\alpha_{i}\geq 0,\ \sum_{k\in\mathcal{N}}\alpha_{k}=C_{\mathcal{N}}\right\\}$
We can characterize the CCE using the above results as follows.
###### Lemma 3.
Any mixture of constrained pure Nash equilibria of the MISO game is a
constrained correlated equilibrium.
Note that the set of constrained correlated equilibria is bigger than the set
of constrained Nash equilibria. For example, in a two-user case, the
distribution
$\frac{1}{2}\delta_{(r_{1},C_{\\{1,2\\}}-r_{1})}+\frac{1}{2}\delta_{(C_{\\{1,2\\}}-r_{2},r_{2})}$
is different than the Dirac distribution
$\delta_{(\frac{r_{1}+C_{\\{1,2\\}}-r_{2}}{2},\frac{r_{2}+C_{\\{1,2\\}}-r_{1}}{2})}.$
###### Proof.
Let $\mu$ be a probability distribution over some constrained pure equilibria.
Then, $\mu\in\Delta(\mbox{max-face}(\mathcal{C})).$ For any $\beta$ such that
$\mu(\beta)\neq 0,$ one has
$\left[u_{i}(\alpha_{i},\alpha_{-i}\mid\beta_{i})-u_{i}(\bar{r}_{i}(\beta_{i}),\alpha_{-i})\right]\geq
0$
for any measurable function $\bar{r}_{i}:\
[0,C_{\\{i\\}}]\longrightarrow[0,C_{\\{i\\}}].$ Thus, $\mu$ is a constrained
correlated equilibrium.
∎
###### Corollary 4.
Any convex combination of extreme point of the convex compact set
$\mbox{max-face}(\mathcal{C})=\left\\{\alpha=(\alpha_{1},\ldots,\alpha_{N})\
|\ \alpha_{i}\geq r_{i},\
\sum_{k\in\mathcal{N}}\alpha_{k}=C_{\mathcal{N}}\right\\}$
is a constrained correlated equilibrium. Moreover any probability distribution
over the maximal face of the capacity region $\mbox{max-face}(\mathcal{C})$ is
a correlated constrained equilibrium distribution.
## 3 Hybrid AWGN Multiple Access Control
In this section, we extend the single receiver case to one with multiple
receivers. Multi-input and multi-output (MIMO) channel access game has been
studied in the context of power allocation and control. For instance, the
authors in [6] formulate a two-player zero-sum game where the first player is
the group of transmitters and the second one is the set of MIMO sub-channels.
In [5], the authors formulate an $N$-person non-cooperative power allocation
game and study its equilibrium under two different decoding schemes.
Here, we formulate a hybrid multiple access game where users are allowed to
select their rates and channels under capacity constraints. We first obtain
general results on the existence of the equilibrium and methods to
characterize it. In addition, we investigate long-term behavior of the
strategies and apply evolutionary game dynamics to both rates and channel
selection probabilities. We show that G-function based dynamics is appropriate
for our hybrid model by viewing the channel selection probabilities as
strategies that determine of fitness of rate selection. Using the generalized
Smith dynamics for channel selection, we are able to build an overall hybrid
evolutionary dynamics for the static model. Based on simulations, we confirm
the validity of these proposed dynamics and the correspondence between the
rest point of the dynamics and the Nash equilibrium.
### 3.1 Hybrid Model with Rate Control and Channel Selection
In this subsection, we establish a model for multiple users and multiple
receivers. Each user needs to decide the rate at which to transmit and the
channel to pick. We formulate a game
$\overline{\Xi}=\langle\mathcal{N},(\mathcal{A}_{i})_{i\in\mathcal{N}},(\overline{U}_{i})_{i\in\mathcal{N}}\rangle$,
in which the decision variable is $(\alpha_{i},\mathbf{p}_{i})$, and
$\mathbf{p}_{i}=[p_{ij}]_{j\in\mathcal{J}}$ is a $J$-dimensional vector, where
$p_{ij}$ is the probability of user $i\in\mathcal{N}$ to choose channel
$j\in\mathcal{J}$ and $p_{ij}$ needs to satisfy the probability measure
constraints
$\sum_{j\in\mathcal{J}}p_{ij}=1,p_{ij}\geq 0,\forall i\in\mathcal{N}$ (12)
The game $\overline{\Xi}$ is asymmetric in the sense that the strategy sets of
users are different and the payoffs are not symmetric.
Let
$C_{j,\Omega}:=\log\left(1+\sum_{i\in\Omega}\frac{P_{ij}h_{ij}}{\sigma^{2}_{0}}\right)$
be the capacity for a subset $\Omega\subseteq\mathcal{N}$ of users at receiver
$j\in\mathcal{J}$ and
$r_{ij,\Omega}:=\log\left(1+\frac{P_{ij}h_{ij}}{\sigma_{0}^{2}+\sum_{i^{\prime}\in\Omega,i^{\prime}\neq
i}P_{i^{\prime}j}h_{i^{\prime}j}}\right)$ the bound rate of a user $i$ when
the signals of the $|\Omega|-1$ other users are treated as noise at receiver
$j$. Each receiver $j$ has a capacity region $\mathcal{C}(j)$ given by
$\displaystyle\mathcal{C}(j)=\left\\{(\alpha,\mathbf{p}_{j})\in[0,1]^{N}\times\mathbb{R}_{+}^{N}\
\bigg{|}\ \sum_{i\in\mathcal{N}}\alpha_{i}p_{ij}\leq C_{j,\Omega},\forall\
\emptyset\subset\Omega_{j}\subseteq\mathcal{N},\ j\in\mathcal{J}\right\\},$
(13)
The expected payoff function
$\overline{U}_{i}:\prod_{i=1}^{N}\mathcal{A}_{i}\longrightarrow\mathbb{R}_{+}$
of the game is given by
$\displaystyle\overline{U}_{i}(\alpha_{i},\mathbf{p}_{i},\alpha_{-i},\mathbf{p}_{-i})=\mathbb{E}_{\mathbf{p}_{i}}[u_{ij}(\alpha,\mathbf{P})]=\sum_{j\in\mathcal{J}}p_{ij}u_{ij}(\alpha,\mathbf{P}),$
(14)
where $\alpha=(\alpha_{i},\alpha_{-i})\in\mathbb{R}^{N}_{+}$ and
$\mathbf{P}\in[0,1]^{N\times J}=(\mathbf{p}_{i},\mathbf{p}_{-i})$, with
$\mathbf{p}_{i}\in[0,1]^{J},\mathbf{p}_{-i}\in[0,1]^{(N-1)\times J}$. Assume
that the utility $u_{ij}$ of a user $i$ transmitting to receiver $j$ is only
dependent on the user himself and is described by a positive and strictly
increasing function $g_{i}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, i.e.,
$u_{ij}=g_{i},\forall j\in\mathcal{J},$ when capacity constraints are
satisfied.
With the presence of coupled constraints (13) from each receiver and
probability measure constraint (12), each sender has his individual
optimization problem (IOP) given as follows.
$\displaystyle\max_{\mathbf{p}_{i},\alpha_{i}}$
$\displaystyle\sum_{j\in\mathcal{J}}p_{ij}g_{i}(\alpha_{i}p_{ij})$ s.t.
$\displaystyle\sum_{j\in\mathcal{J}}p_{ij}=1,\forall i\in\mathcal{N}$
$\displaystyle p_{ij}\geq 0,\forall i\in\mathcal{N},j\in\mathcal{J}$
$\displaystyle(\alpha,\mathbf{p}_{j})\in\mathcal{C}(j),\forall
j\in\mathcal{J}$
Denote the feasible set of (IOP) by
$\mathcal{F}=\mathcal{F}_{1}\times\mathcal{F}_{2}$, where
$\mathcal{F}_{1}=\left\\{\alpha\in\mathbb{R}^{N}_{+}\mid(\alpha,\mathbf{P})\in\cap_{j\in\mathcal{J}}\mathcal{C}(j),\mathbf{P}\in\mathcal{F}_{2}\right\\},$
(15) $\mathcal{F}_{2}=\left\\{\mathbf{P}\in\mathbb{R}^{N\times
J}\mid\sum_{j\in\mathcal{J}}p_{ij}=1,p_{ij}\geq 0,\forall
i\in\mathcal{N},j\in\mathcal{J}\right\\}.$ (16)
The action set of each user can thus be described by
$\mathcal{A}_{i}(\alpha_{-i},\mathbf{p}_{-i})=\left\\{(\alpha_{i},\alpha_{-i})\in\mathcal{F}_{1},(\mathbf{p}_{i},\mathbf{p}_{-i})\in\mathcal{F}_{2}\right\\}.$
(17)
#### 3.1.1 An Example
Suppose we have three users and three receivers, that is,
$\mathcal{N}=\\{1,2,3\\}$ and $\mathcal{J}=\\{1,2,3\\}$. The capacity region
at receiver $1$ is given by
$\mathcal{C}(1)=\left\\{\begin{array}[]{c}\alpha_{i}\geq 0,\ i=1,2,3\\\
p_{11}\alpha_{1}\leq\log(1+\frac{P_{1}h_{1}}{\sigma_{0}^{2}})\\\
p_{21}\alpha_{2}\leq\log(1+\frac{P_{2}h_{2}}{\sigma_{0}^{2}})\\\
p_{31}\alpha_{3}\leq\log(1+\frac{P_{3}h_{3}}{\sigma_{0}^{2}})\\\
p_{11}\alpha_{1}+p_{21}\alpha_{2}\leq\log(1+\frac{P_{1}h_{1}+P_{2}h_{2}}{\sigma_{0}^{2}})\\\
p_{11}\alpha_{1}+p_{31}\alpha_{3}\leq\log(1+\frac{P_{1}h_{1}+P_{2}h_{2}}{\sigma_{0}^{2}})\\\
p_{21}\alpha_{2}+p_{31}\alpha_{3}\leq\log(1+\frac{P_{1}h_{1}+P_{2}h_{2}}{\sigma_{0}^{2}})\\\
p_{11}\alpha_{1}+p_{21}\alpha_{2}+p_{31}\alpha_{3}\leq\log(1+\frac{P_{1}h_{1}+P_{2}h_{2}+P_{3}h_{3}}{\sigma_{0}^{2}})\\\
0\leq p_{i1}\leq 1,\ i=1,2,3\\\ \end{array}\right\\}.$
This can be written into
$\displaystyle\mathcal{C}(1)=\left\\{\mathbf{p}_{1}=\left[\begin{array}[]{c}p_{11}\\\
p_{21}\\\
p_{31}\end{array}\right]\in[0,1]^{3},\left[\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\\\ \alpha_{3}\end{array}\right]\in\mathbb{R}_{+}^{3}\
\bigg{|}M_{3}\left[\begin{array}[]{c}p_{11}\alpha_{1}\\\ p_{21}\alpha_{2}\\\
p_{31}\alpha_{3}\end{array}\right]\leq\left[\begin{array}[]{c}C_{1,\\{1\\}}\\\
C_{1,\\{2\\}}\\\ C_{1,\\{3\\}}\\\ C_{1,\\{1,2\\}}\\\ C_{1,\\{1,3\\}}\\\
C_{1,\\{2,3\\}}\\\ C_{1,\\{1,2,3\\}}\end{array}\right]\right\\},$
where
$C_{1,\Omega}=\log\left(1+\sum_{i\in\Omega}\frac{P_{i1}h_{i1}}{\sigma_{0}^{2}}\right)$
and $M_{3}$ is a totally unimodular matrix:
$M_{3}:=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&1&0\\\ 1&0&1\\\
0&1&1\\\ 1&1&1\end{array}\right].$ Capacity regions at receivers 2 and 3 can
be obtained in a similar way.
### 3.2 Characterization of Constrained Nash Equilibria
In this subsection, we characterize the Nash equilibria of the defined game
$\overline{\Xi}$ under the given capacity constraint. We use the following
theorem to prove the existence of Nash equilibrium for the case where the
rates are predetermined; this result is then used to solve the game for the
case when both the rates and the connection probabilities are (joint) decision
variables.
###### Theorem 4.
(Başar & Olsder, [38]) Let
$\mathcal{A}=\mathcal{A}_{1}\times\mathcal{A}_{2}\cdots\times\mathcal{A}_{N}$
be a closed, bounded and convex subset of $\mathbb{R}^{N}$, and for each
$i\in\mathcal{N}$, the payoff functional
$\overline{U}_{i}:\mathcal{A}\rightarrow\mathbb{R}$ be jointly continuous in
$\mathcal{A}$ and concave in $a_{i}$ for every
$a_{j}\in\mathcal{A}_{j},j\in\mathcal{N},j\neq i$. Then, the associated
$N$-person nonzero-sum game admits a Nash equilibrium in pure strategies.
Applying Theorem 4, we have the following results immediately.
###### Proposition 3.
Suppose $\alpha_{i},i\in\mathcal{N},$ are predetermined feasible rates. Let
feasible set $\mathcal{F}$ be closed, bounded and convex. If $g_{i}$ in (IOP)
are continuous on $\mathcal{F}$ and concave in $\mathbf{p}_{i}$ (without the
assumption of being positive and strictly increasing), the expected payoff
functions $\overline{U}_{i}:\mathbb{R}^{N}_{+}\times[0,1]^{N\times
J}\rightarrow\mathbb{R}$ are concave in $\mathbf{p}_{i}$ and continuous on
$\mathcal{F}$, then the static game admits a Nash equilibrium.
The existence result in Proposition 3 only captures the case where the rates
$\alpha_{i}$ are predetermined, and relies on the convexity requirement of the
utility functions $g_{i}$. We can actually obtain a stronger existence result
by observing that the formulated game $\overline{\Xi}$ is a potential game
with a potential function given by
$\Psi(\alpha,\mathbf{P})=\sum_{i}f_{i}(\alpha_{i},\mathbf{p}_{i})=\sum_{i}\sum_{j}p_{ij}g_{i}(\alpha_{i}p_{ij}),$
(19)
where $f_{i}=\sum_{j}p_{ij}g_{i}(\alpha_{i}p_{ij})$, the expected payoff to
user $i$.
Note that the feasible set $\mathcal{F}$ is generally nonempty and bounded. We
can conclude the existence result in Proposition 4 of NE from this
observation.
###### Proposition 4.
The hybrid rate control game $\overline{\Xi}$ admits a Nash equilibrium.
###### Proof.
Let us formulate a centralized optimization problem (COP) as follows.
$\begin{array}[]{ccc}&\max_{\alpha,\mathbf{P}}&\Psi(\alpha,\mathbf{P})\\\
&\textrm{s.t.}&{(\alpha,\mathbf{P})}\in\mathcal{F}\end{array}$
Using the result in [3], we can conclude that if there exists a solution to
(COP), then there exists a Nash equilibrium to the game $\overline{\Xi}$.
Since $\mathcal{F}$ is compact and nonempty, and the objective function is
continuous, there exists a solution to (COP) and thus a Nash equilibrium to
the game. ∎
The problem above is generally not convex and uniqueness of the Nash
equilibrium may not be guaranteed. However, we still can further characterize
the Nash equilibrium through the following propositions.
###### Proposition 5.
Let $\beta_{ij}:=\alpha_{i}p_{ij}$. Without predetermining $\alpha$, suppose
that $(\mathbf{p}_{-i},\alpha_{-i})$ is feasible. A best response strategy at
receiver $j\in\mathcal{J}$ for user $i$ must satisfy
$0\leq p_{ij}\alpha_{i}\leq C_{j,\Omega_{j}}-\sum_{k\neq
i}\alpha_{k}p_{kj},\forall\Omega_{j}$ (20)
where $\Omega_{j}:=\\{\Omega\in 2^{\mathcal{N}}\mid
i^{\prime}\in\Omega,p_{i^{\prime}j}>0\\}$ is the set of users transmitting to
receiver $j.$ Since $g_{i}$ is a non-decreasing function, the best
correspondence at $j$ is
$\beta_{ij}^{*}=\alpha_{i}^{*}p_{ij}^{*}=\max\left(r_{ij,\mathcal{N}},\min_{\Omega_{j}}\left(C_{j,\Omega_{j}}{-\sum_{i^{\prime}\neq
i}\alpha_{i^{\prime}}p_{i^{\prime}j}}\right)\right),$ (21)
where $r_{ij,\mathcal{N}}$ is the bound on the rate of user $i$ when signals
of $|{\mathcal{N}}|-1$ other users are treated as noise.
###### Proof.
The proof is immediate by observing that the rate of user $i$ at receiver $j$
must satisfy (20) due to the coupled constraints. Thus, the maximum rate that
user $i$ can use to transmit to receiver $j$ without violating the constraints
is clearly the minimum of $C_{j,{\Omega}_{j}}-\sum_{i^{\prime}\neq
i}\alpha_{i^{\prime}}p_{i^{\prime}j}$ over all $\Omega_{j}$. Since the payoff
is a non-decreasing function, the best response for $i$ at receiver $j$ is
given by (21).∎
###### Proposition 6.
Let $K^{*}_{i}=\textrm{arg}\max_{j\in\mathcal{J}}g_{ij}(\beta_{ij})$. If
$K^{*}_{i}=\\{k^{*}\\}$ is a singleton, then the best reply for user $i$ is to
choose
$\left\\{\begin{array}[]{cc}p_{ij}=1&\textrm{if~{}}j=k^{*}_{i},\\\
p_{ij}=0&\textrm{otherwise,}\end{array}\right.$
and we can determine $\alpha_{i}$ by
$\alpha_{i}=\frac{\beta_{ik^{*}}}{p_{ik^{*}}}$.
If $|K^{*}_{i}|\geq 2$, then the best response correspondence is
$\left\\{\begin{array}[]{ll}\mathbf{p}_{i}\in\Delta(K^{*}_{i})&\textrm{if~{}}j\in
K^{*}_{i},\\\ 0&\textrm{otherwise.~{}}\end{array}\right.$
We can determine $\alpha_{i}$ from $\beta_{ij}$ by $\alpha_{i}=\sum_{j\in
K^{*}_{i}}\beta_{ij}.$
###### Proof.
Since the expected utility is given in the form of
$U_{i}(\alpha_{i},\mathbf{p}_{i},\alpha_{-i},\mathbf{p}_{-i})=\mathbb{E}_{\mathbf{p}_{i}}[u_{ij}(\alpha_{i}p_{ij})],$
the expected utility under best response is
$U_{i}=\mathbb{E}_{\mathbf{p}_{i}}[u_{ij}(\beta_{ij})].$ If for a singleton
$k^{*}$ such that
$k^{*}=\textrm{arg}\max_{i\in\mathcal{N}}g_{ij}(\beta_{ij}),$ we can assign
all the weight $p_{ik^{*}}=1$ to maximize the expected utility. Since
$\beta_{ik^{*}}=\alpha_{i}p_{ik^{*}}$, then
$\alpha=\beta_{ik^{*}}/p_{ik^{*}}=\beta_{ik^{*}}.$ If the set $K^{*}$ is not a
singleton, without loss of generality, we can pick two indices
$j$,$j^{\prime}\in K^{*}$ such that $\beta_{ij}=\alpha_{i}p_{ij}$ and
$\beta_{ij^{\prime}}=\alpha_{i}p_{ij^{\prime}}$, leading to
$u_{ij}(\beta_{ij})=u_{ij^{\prime}}(\beta_{ij^{\prime}})$. Since the utilities
to transmit using $j$ and $j^{\prime}$ are the same, we can assign arbitrary
(two-point) distribution, $p_{ij}$ and $p_{ij^{\prime}}$ over them, with
$p_{ij}+p_{ij^{\prime}}=1$. Therefore,
$\beta_{ij}+\beta_{ij^{\prime}}=\alpha_{i}(p_{ij}+p_{ij^{\prime}})=\alpha_{i}.$
∎
### 3.3 Multiple Access Evolutionary Games
Interactions among users are dynamic and the users can update their rates and
channel selection with respect to their payoffs and the known coupled
constraints. Such a dynamic process can generally be modeled by either an
evolutionary process, a learning process or a trial-and-error updating
process. In classical game theory, the focus is on strategies that optimize
payoffs to the players while, in evolutionary game theory, the focus is on
strategies that will persist through time. In this subsection, we formulate
evolutionary games dynamics based on the static game discussed in Section 3.1.
We use generalized Smith-dynamics for channel selection and G-function based
dynamics for rates. Combining them, we set up a framework of hybrid dynamics
for the overall system.
The action of each user has two components
$(\alpha_{i},\mathbf{p}_{i})\in\mathbb{R}_{+}\times[0,1]^{J}$. We use
$\mathbf{p}_{i}$ as strategies that determine the fitness of user $i$’s rate
$\alpha_{i}$ to receiver $j$. The rate selection evolves according to the
channel selection strategy $\mathbf{P}$. We may view channel selection as an
inner game that involves a process on a short time scale but the rate
selection is an outer game that represents the dynamical link via fitness on a
longer time scale, [7], [8].
#### 3.3.1 Learning the Weight Placed on Receiver
Let $\alpha$ be a fixed rate on the capacity region. We assume that user $i$
occasionally experiments the weights $p_{ij}$ with alternative receivers,
keeping the new strategy if and only if it leads to a strict increase in
payoff. If the choice of receivers’ weights of some users decreases the payoff
or violates the constraints due to a strategy change by another user, he
starts a random search for a new strategy, eventually settling on one with a
probability that increases monotonically with its realized payoff. For the
above generating function based dynamics, the weight of switching from
receiver $j$ to receiver $j^{\prime}$ is given by
$\eta_{jj^{\prime}}^{i}(\alpha,\mathbf{P})=\max(0,u_{ij^{\prime}}(\alpha,\mathbf{P})-u_{ij}(\alpha,\mathbf{P}))^{\theta},\
\theta\geq 1$
if the payoff obtained at receiver $j^{\prime}$ is greater the payoff obtained
receiver $j$ and the constraints are satisfied; otherwise,
$\eta_{jj^{\prime}}^{i}(p,\alpha)=0.$ The frequencies of uses of each receiver
is then seen as the selection strategy of receivers.
The expected change at each receiver is the difference between the incoming
flow and the outgoing flow. The dynamics is also called generalized Smith
dynamics [2] and is given by
$\displaystyle\dot{p}_{ij}(t)=\sum_{j^{\prime}\in\mathcal{J}}p_{ij^{\prime}}(t)\eta_{j^{\prime}j}^{i}(\alpha,\mathbf{P}(t))-p_{ij}(t)\sum_{j^{\prime}\in\mathcal{J}}\eta_{jj^{\prime}}^{i}(\alpha,\mathbf{P}(t)).$
(22)
Let
$\chi_{ij}(\alpha,\mathbf{P}(t)):=\sum_{j^{\prime}\in\mathcal{J}}p_{ij^{\prime}}(t)\eta_{j^{\prime}j}^{i}(\alpha,\mathbf{P}(t))-p_{ij}(t)\sum_{j^{\prime}\in\mathcal{J}}\eta_{jj^{\prime}}^{i}(\alpha,\mathbf{P}(t)).$
Hence, the dynamics can be rewritten as
$\dot{p}_{ij}=\chi_{ij}(\alpha,\mathbf{P}(t))$. For $\theta=1$ the dynamics is
known as Smith dynamics and has been used for describing the evolution of road
traffic congestion in which the fitness is determined by the strategies chosen
by all drivers. It has also been studied in the context of the resource
selection in hybrid systems and migration constraint problem in wireless
networks in [2].
###### Proposition 7.
Any equilibrium of the game $\overline{\Xi}$ with predetermined rates is a
rest points of the generalized Smith dynamics (22).
###### Proof.
The transition rate between receivers preserves the sign in the sense that,
for every user, incoming flow from the receiver $j^{\prime}$ to $j$ is
positive if and only if the constraints are satisfied and the payoff to $j$
exceeds the payoff to $j^{\prime}.$ Let $\alpha$ be a feasible point. We first
remark that if the right hand side of (22) is non-zero for some splitting
strategy $\mathbf{P},$ then
$\displaystyle d$ $\displaystyle:=$ $\displaystyle\sum_{j\in\mathcal{J}}\
\dot{p}_{ij}u_{ij}(\alpha,\mathbf{P})=\sum_{j\in\mathcal{J}}\
\chi_{ij}u_{ij}(\alpha,\mathbf{P})$ $\displaystyle=$
$\displaystyle\sum_{j,j^{\prime}\in\mathcal{J}}p_{ij^{\prime}}\left(u_{ij}(\alpha,\mathbf{P})-u_{ij^{\prime}}(\alpha,\mathbf{P})\right)\eta^{i}_{j^{\prime}j}$
$\displaystyle=$
$\displaystyle\sum_{j,j^{\prime}\in\mathcal{J}}p_{ij^{\prime}}\max\left[0,\left(u_{ij}(\alpha,\mathbf{P})-u_{ij^{\prime}}(\alpha,\mathbf{P})\right)\right]\eta^{i}_{j^{\prime}j}$
which is strictly positive. Thus, if $(\alpha,\mathbf{P})$ is a Nash
equilibrium then $(\alpha,\mathbf{P})$ satisfy the constraints, and
$p_{ij}=0,$ or $\eta^{i}_{jj^{\prime}}(\alpha,\mathbf{P})=0.$ This implies
that $(\alpha,\mathbf{P})$ satisfies also $\chi(\alpha,\mathbf{P}))=0.$ ∎
The following proposition says that the equilibria are exactly the rest point
of (22).
###### Proposition 8.
Any rest point of the dynamics (22) is a Nash equilibrium of the game
$\overline{\Xi}$.
The proof of Proposition 8 can be obtained by using Theorem III in [2]. Since
the probability to switch from receiver $j$ to $j^{\prime}$ is proportional to
$\eta^{i}_{jj^{\prime}}$, which preserves the sign of payoff difference, we
can use the Theorems III in [2]. It follows that the dynamics generated by
$\eta$ satisfy the Nash stationarity property.
#### 3.3.2 G-function Based Dynamics
We introduce here the generating fitness function (G-function) based dynamics
with projection onto the capacity region. The G-function approach has been
successfully applied to non-linear continuous games by Vincent and Brown [7],
[8]. It is appropriate for our hybrid model because we can regard the channel
selection as the variables in a fitness function. Users choose channel
selection probabilities to aim at increasing their fitness of their rate
choice. In our rate allocation game, to deal with constraints, we use
projection into capacity region in order to preserve the trajectories
feasible. Starting from a point in the polytope $\mathcal{C}$, each user
revises and updates its strategy according to a rate proportional to the
gradient and its payoff subject to the capacity constraints. Let $G_{ij}$ be
the fitness generating function of user $i$ at receiver $j$ defined on
$\mathbb{R}^{N}\times\mathbb{R}^{N\times J}$ satisfying
$G_{ij}(v,\alpha,\mathbf{P}){\bigg{|}_{v=\mathbf{p}_{i}}}=\left(C_{j,\mathcal{N}}-p_{ij}\beta_{ij}(t)-\sum_{i^{\prime}\in\mathcal{N}\backslash\\{i\\}}p_{i^{\prime}j}\beta_{i^{\prime}j}(t)\right),$
if $(\alpha,\mathbf{P})$ satisfies in the hybrid capacity region. Notice that
the term $C_{j,\mathcal{N}}-\sum_{i^{\prime}\neq
i}p_{i^{\prime}j}\beta_{i^{\prime}j}(t)$ is maximum rate of $i$ using channel
$j$ at time $t$. Hence, the G-function based dynamics is given by
$\dot{\beta}_{ij}=-\bar{\mu}\left[p_{ij}\beta_{ij}-C_{j,\mathcal{N}}+\sum_{i^{\prime}\neq
i}p_{i^{\prime}j}\beta_{i^{\prime}j}\right]p_{ij}\beta_{ij}.$ (23)
with initial conditions $\beta_{ij}(0)\leq C_{j,\\{i\\}}$, where
$\beta=[\beta_{ij}]$ is defined in Proposition 5, which is of the same
dimension as $\alpha$, and
$\alpha_{i}(t)=\sum_{j\in\mathcal{J}}\beta_{ij}(t)$; $\bar{\mu}$ is an
appropriate parameter chosen for the rate of convergence.
#### 3.3.3 Hybrid Dynamics
We now combine the two evolutionary game dynamics described in the previous
subsections. Variables $\alpha$ and $\mathbf{P}$ are both evolving in time.
The overall dynamics are given by
$\left\\{\begin{array}[]{lll}\dot{p}_{ij}(t)&=&\sum_{j^{\prime}\in\mathcal{J}}p_{ij^{\prime}}(t)\eta_{j^{\prime}j}^{i}(\alpha(t),\mathbf{P}(t))-p_{ij}(t)\sum_{j^{\prime}\in\mathcal{J}}\eta_{jj^{\prime}}^{i}(\alpha(t),\mathbf{P}(t))\\\
\dot{\beta}_{ij}(t)&=&-\bar{\mu}\left[p_{ij}(t)\beta_{ij}(t)-C_{j,\mathcal{N}}+\sum_{i^{\prime}\neq
i}p_{i^{\prime}j}(t)\beta_{i^{\prime}j}(t)\right]p_{ij}(t)\beta_{ij}(t)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\alpha_{i}(t)&=&\sum_{j\in\mathcal{J}}\beta_{ij}(t)\vskip 6.0pt plus 2.0pt
minus 2.0pt,\ \beta_{ij}(0)\leq C_{j,\\{i\\}},\forall j\in\mathcal{J},\
i\in\mathcal{N}\end{array}\right.$ (24)
All the equilibria of the hybrid evolutionary rate control and channel
selection are rest point of the above hybrid dynamics. The following result
can be obtained directly from Proposition 7 and (23).
###### Proposition 9.
Let $(\beta^{*},\mathbf{P}^{*})$ be an interior rest points of the hybrid
dynamics, i.e., $\beta_{ij}^{*}>0,\ p^{*}_{ij}>0$ and
$\chi(\alpha^{*},\mathbf{P})=0.$ Then for all $j$,
$\sum_{i=1}^{N}p^{*}_{ij}\beta^{*}_{ij}=C_{j,\mathcal{N}};~{}~{}\chi\left(\sum_{j=1}^{N}\beta^{*}_{ij},\mathbf{P}^{*}\right)=0.$
### 3.4 Numerical Examples
In this subsection, we illustrate the evolutionary dynamics in (23) and (24)
by examining a two-user and three-receiver communication system as depicted in
Figure 4. Let $h_{i1}=0.1,h_{i2}=0.2,h_{i3}=0.3$, for $i=\\{1,2\\}$. Each
transmission power $P_{i}$ is set to $1$ mW for all $i=1,2$ and the noise
level is set to $\sigma^{2}=-20$ dBm.
Figure 4: Two users and three receivers
In the first experiment, we assume that the rates of the users are
predetermined to be $\alpha=[10,20]^{T}$, the Smith dynamics in (23) yield in
Figure (6) and (6) the response of $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$. It
can be seen that the dynamics converge very fast within less than half a
second.
In the second experiment, we assume that the probability matrix $\mathbf{P}$
has been optimally found by the users using (22). Figures 8 and 8 show that
the $\beta$ values converge to an equilibrium from which we can find the
optimal value for $\alpha$. Since these dynamics are much slower compared to
Smith dynamics on $\mathbf{P}$, our assumption of knowledge of optimal
$\mathbf{P}$ for a slowly varying $\alpha$ becomes valid.
In the next experiment, we simulate the hybrid dynamics in (24). Let the
probability $p_{ij}$ of user $i$ choosing transmitter $j$ and the transmission
rates be initialized as follows:
$\begin{array}[]{ll}\mathbf{P}(0)=\left[\begin{array}[]{ccc}0.2&0.3&0.5\\\
0.25&0.5&0.25\end{array}\right],&\alpha(0)=\left[\begin{array}[]{c}0.2\\\
0.1\end{array}\right].\end{array}$
We let the parameter $\bar{\mu}=0.9$. Figure 11 shows the evolution of the
weights of user 1 on each of the receivers. The weights converge to be
$p_{1j}=1/3$ for all $j$ within two seconds, leading to an unbiased choice
among receivers. In Figure 11, we show the evolution of the weights of the
second user on each receiver. At the equilibrium,
$\mathbf{p}_{2}=[0.3484,0.4847,0.1669]^{T}$. It appears that user 2 favors the
second transmitter over the other ones. Since the utility $u_{ij}$ is of the
same form, the optimal response set $K_{i}^{*}$ is naturally nonempty and
contains all the receivers. Shown in Proposition 6, the probability of
choosing a receiver at the equilibrium is randomized among the three receivers
and can be determined by the rates $\alpha$ chosen by the users.
The $\beta$-dynamics determines the evolution of $\alpha$ in (24). In Figure
11, we see that the evolutionary dynamics yield $\alpha=[15.87,23.19]^{T}$ at
the equilibrium. It is easy to verify that they satisfy the capacity
constraints outlined in Section 2. It converges within 5 seconds and appears
be much slower than in Figures 11 and 11. Hence, it can be seen that
$\mathbf{P}-$dynamics may be seen as the inner loop dynamics while
$\beta-$dynamics can be seen as an outer loop evolutionary dynamics. They
evolve on two different time scales. In addition, thanks to Proposition 8,
finding the rest points for the above dynamics ensures us finding the
equilibrium.
Figure 5: Transmitter 1: Probabilities v.s. Time For Fixed $\alpha_{1}$=10
Figure 6: Transmitter 2: Probabilities v.s. Time For Fixed $\alpha_{2}$=20
Figure 7: Transmitter 1: $\beta$ Value v.s. Time
Figure 8: Transmitter 1: $\beta$ Value v.s. Time
Figure 9: Probability of Transmitter 1 Choosing Receivers
Figure 10: Probability of Transmitter 2 Choosing Receivers
Figure 11: Rates of Each Transmitter
## 4 Concluding Remarks
In this paper, we have studied an evolutionary multiple access channel game
with a continuum action space and coupled rate constraints. We showed that the
game has a continuum of strong equilibria which are 100% efficient in the rate
optimization problem. We proposed the constrained Brown-von Neumann-Nash
dynamics, Smith dynamics, and the replicator dynamics to study the stability
of equilibria in the long run. In addition, we have introduced a hybrid
multiple access game model and its corresponding evolutionary game-theoretic
framework. We have analyzed the Nash equilibrium for the static game and
suggested a system of evolutionary game dynamics based method to find it. It
is found that the Smith dynamics for channel selections are a lot faster than
the $\beta$-dynamics, and the combined dynamics yield a rest point that
corresponds to the Nash equilibrium. An interesting extension that we leave
for future research is to introduce a dynamic channel characteristics: the
gains $h_{ij}(t)$ are time-dependent random variables. Another interesting
question is to find equilibria structure in the case of multiple access games
with non-convex capacity regions.
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|
arxiv-papers
| 2011-03-13T03:55:38 |
2024-09-04T02:49:17.631449
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quanyan Zhu, Hamidou Tembine, Tamer Basar",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1103.2496"
}
|
1103.2579
|
∎
11institutetext: Tamer Başar 22institutetext: Coordinated Science Laboratory
and the Department of Electrical and Computer Engineering, University of
Illinois at Urbana-Champaign, 1308 West Main Street, Urbana, IL, 61801, USA.
Tel.: +1 217-333-3607
Fax: +1 217-265-0997
22email: basar1@illinois.edu 33institutetext: Quanyan Zhu 44institutetext:
Coordinated Science Laboratory and the Department of Electrical and Computer
Engineering, University of Illinois at Urbana-Champaign, 1308 West Main
Street, Urbana, IL, 61801, USA.
44email: zhu31@illinois.edu
# Prices of Anarchy, Information, and Cooperation in Differential
Games††thanks: Research supported in part by grants from AFOSR and DOE.
Tamer Başar Quanyan Zhu
(Received: date / Accepted: date)
###### Abstract
The price of anarchy (PoA) has been widely used in static games to quantify
the loss of efficiency due to noncooperation. Here, we extend this concept to
a general differential games framework. In addition, we introduce the price of
information (PoI) to characterize comparative game performances under
different information structures, as well as the price of cooperation to
capture the extent of benefit or loss a player accrues as a result of
altruistic behavior. We further characterize PoA and PoI for a class of scalar
linear quadratic differential games under open-loop and closed-loop feedback
information structures. We also obtain some explicit bounds on these indices
in a large population regime.
###### Keywords:
Differential games Nash equilibria efficiency price of anarchy price of
information price of cooperation linear-quadratic games information structures
## 1 Introduction
It is well known that the non-cooperative Nash equilibrium in nonzero-sum
games is generally inefficient DUB86 , which means that it would be possible
for all players to do better in terms of attaining higher utilities or lower
costs (than they would attain under Nash equilibria, even if the equilibrium
is unique) through a cooperative behavior. This is true for static
deterministic games, and naturally also for stochastic games as well as
dynamic and differential games. In these latter of classes of games, one could
bring up additional issues with regard to Nash equilibria beyond efficiency or
lack thereof, such as whether an increase in information to one player (or all
or a subset of the players) would be advantageous to that player (or groups of
players), in terms of attaining higher utilities or lower costs, or whether
acquiring more information would be undesirable for a player. In the special
class of games where all players have the same utility function or cost
function (that is, team problems) and what is sought is the global maximum or
global minimum of these functions, the answer to such a query is clean, which
is that additional information (defined as expansion of sigma fields) can
never hurt. The same is true for the special class of zero-sum games. In
stochastic games, or dynamic and differential games which are not team
problems or zero-sum games, however, the answer is not that clean, and one
could encounter quite surprising and at the outset counter-intuitive results.
Perhaps the first demonstration of this was reported in Basar72 and BasHo74 ,
where two classes of two-player stochastic static games were considered, one a
linear-quadratic-Gaussian (LQG) model and the other one a stochastic Cournot
duopoly model, both of which admit unique Nash equilibria. It was shown that
for the LQG model better information (on some stochastic variables) for only
one player leads to lower average Nash equilibrium costs for both players, but
in the duopoly model only the player whose information is improved benefits
while the other one hurts (in the sense that his average Nash equilibrium cost
increases). Another way of comparison would be in terms of the relative values
of the average Nash equilibrium costs attained by the players, when one player
has informational advantage over the other. It was again shown in Basar72
that, in an otherwise completely symmetric game, the player who has better
information attains higher cost than the other player in the LQG model (the
counter-intuitive result), whereas he attains lower cost in the duopoly model
(the intuitive result). Several manifestations of these conclusions can be
seen also in dynamic and differential games; for example time-consistent open-
loop Nash equilibrium is not necessarily inferior to the strongly time-
consistent closed-loop feedback Nash equilibrium BasOls99 .
Now coming back to inefficiency of Nash equilibrium in a fixed nonzero-sum
game, one question of interest is exploration of the extent of this
inefficiency, that is how far off is a Nash equilibrium from the socially
optimal solution, which is obtained as the maximum of the sum of the utilities
of the players, or some convex combination of the utilities (or minimum in the
case of cost functions). The notion of the price of anarchy (PoA)was
introduced in ROU04 as a quantification of this offset, as a utility ratio
between the worst possible Nash solution (among multiple Nash equilibria) and
the social optimum. In a way, this index serves to quantify the loss of
efficiency due to competition. It has been shown that in routing games and
resource allocation games (see, ROU04 and JMT05 ), PoA is bounded by a
constant, allowing agents to achieve some level of efficiency despite being
suboptimal.
The idea of quantifying the gap between social optimality and game equilibrium
solutions sparked many follow-up work in that same vein. In SS08 , price of
simplicity has been introduced for a pricing game in communication networks as
the ratio between the revenue collected from a flat pricing rule and the
maximum possible revenue. In GJC09 , price of uncertainty has been introduced
to measure the relative payoff of an expert user of a security game under
complete information to the one under incomplete information. In ZHU08b ,
price of leadership has been proposed as a measure of comparison of utilities
in a power control game between Nash equilibria and Stackelberg solutions. In
all of these works, primarily communication networks have been used as a
backdrop application domain, be it routing, resource allocation, power
control, or security. Game-theoretical methods along with Nash equilibrium
have found many applications in communication networks, with some selected
recent references being AB98 ; Basar07 ; MahBas03 ; SAAB02 ; ZHU08e ; JOH03 ;
JMT05 ; ABEJW06
In this paper, we discuss several indices which quantify variations or offsets
in the payoff values or costs attained under Nash equilibria in the context of
differential games (DGs). We first extend the notion of PoA to DGs, which
heretofore has been primarily limited to static continuous kernel games. We
provide a characterization of PoA for a class of scalar linear-quadratic (LQ)
DGs, and quantify the efficiency loss in the long run when the players behave
non-cooperatively under the Nash equilibrium concept. We consider both open-
loop (OL) and closed-loop (CL) information structures (ISs). We show that for
the class of scalar LQ DGs with CL IS using the strongly time-consistent CL
feedback Nash equilibrium, the PoA has some appealing computable upper bounds,
which can further be approximated when the number of players is sufficiently
large (that is, the large population regime), whereas, under the OL IS, it is
possible to obtain an expression for the PoA in closed form.
As mentioned earlier, going from static to dynamic (differential) games brings
in the possibility of various ISs, which add richness to the (Nash
equilibrium) solution of a game. Different ISs (generally) yield different
equilibrium solutions, and hence IS is a crucial factor in the investigation
of PoA in DGs. Motivated by this, we introduce another index, the price of
information (PoI), which is a result of the comparison of the equilibrium
utilities or costs under different ISs. For the class of scalar LQ DGs above,
we show that the PoI between the feedback and open-loop ISs is shown to be
bounded from below by ${\sqrt{2}}/{2}$ and from above by $\sqrt{2}$, again in
the large population regime. Finally, motivated by some recent results
reported in AAE10 on the level of cooperation between players in a routing
game, captured by the degree of willingness of a player to place partial
weight on other players’ utilities in his utility function, we introduce the
price of cooperation (PoC) as a measure of benefit or loss to a player on his
base Nash equilibrium payoff due to cooperation.
The structure of the paper is as follows. In Section 2, we introduce a general
$N$-player DG framework with different ISs, and define in this context the
indices, PoA, PoI, and PoC. In Section 3, we investigate the PoA for a class
of scalar LQ feedback DGs. In Section 4, we study the LQ DGs under open-loop
IS, and in Section 5, we establish bounds on the PoI. We conclude and identify
future directions in Section 6. An earlier version of some of the results in
this paper can be found in the recent conference paper ZhuBas10 .
## 2 General Problem Formulation
In this section we first introduce the general nonzero-sum differential games
framework along with the Nash equilibrium solution, and then introduce the
three indices: prices of anarchy, information, and cooperation.
Let $\mathcal{N}=\\{1,2,\cdots,N\\}$ be the set of players, and
$[0,T\rangle$111The notation “$\rangle$” is introduced to capture two cases:
finite horizon when $T$ is finite (in which case we have $[0,T]$), and
infinite horizon when $T$ is infinite (in which case we have $[0,\infty)$). be
the time interval of interest. At each time instant $t\in[0,T\rangle$, each
player, say Player $i$, chooses an $m_{i}$-dimensional control value (action)
$u_{i}(t)$ from his set of feasible control values
$U_{i}\subset\mathbb{R}^{m_{i}}$, where we also make the standard assumption
that as a function of $t$ the control function $u_{i}(\cdot)$ is piecewise
continuous on $[0,T\rangle$. The state variable $x$ is of dimension $n$, and
takes values in $\mathbb{R}^{n}$; as a function of time, $t$, we assume
$x(\cdot)$ to be piecewise continuously differentiable on $[0,T\rangle$, and
evolving according to the differential equation:
$\dot{x}(t)=f(x(t),u_{1}(t),\cdots,u_{N}(t),t)\,,\;\;x(0)=x_{0}\,,$
where $x_{0}\in\mathbb{R}$ is the initial value of the state and the system
dynamics $f(\cdot):\Omega\rightarrow\mathbb{R}^{n}$ is defined on the set
$\Omega=\\{(x,u_{1},\cdots,u_{N},t)|x\in\mathbb{R}^{n},t\in[0,T\rangle,u_{i}\in
U_{i},i\in\mathcal{N}\\}\,,$
as a jointly piecewise continuous function which is also Lipschitz in $x$, and
also possibly Lipschitz in the $u_{i}$’s, depending on whether the underlying
information structure (IS) is open loop of closed loop feedback.
We will consider two different ISs: Open loop (OL),, where the controls are
just functions of time, $t$ (and also of initial state $x_{0}$, which however
is assumed to be fixed and a known parameter of the game), and closed-loop
state-feedback, where the controls are allowed to be functions of current
value of the state and of time, that is, for Player $i$,
$u_{i}(t)=\gamma_{i}(t;x(t))$. In the latter case,
$\gamma_{i}:[0,T\rangle\times\mathbb{R}^{n}\to U_{i}$ is known as the policy
variable (strategy) of Player $i$, which is a mapping from the set of
information available to the player to his control (action) set.222One can
introduce more general ISs, such as those that involve memory, but here we
will restrict the discussion to only OL and CL state-feedback (SF) structures
so as not to encounter informational non-uniqueness of Nash equilibria
BasOls99 . We require each $\gamma_{i}(t;\cdot)$ to be Lipschitz in $x$, in
addition to being jointly piecewise continuous in its arguments, and denote
the class of all such mappings by $\Gamma_{i}$. We further require that $f$ be
Lipschitz not only in $x$ but also in $\\{u_{1},\ldots,u_{N}\\}$, so that the
differential equation generating the state,
$\dot{x}(t)=f(x(t),\gamma_{1}(t;x(t)),\cdots,\gamma_{N}(t;x(t)),t)\,,\;\;x(0)=x_{0}\,,$
admits a unique piecewise continuously differentiable solution for each
$\gamma_{i}\in\Gamma_{i},\;i\in\mathcal{N}$. Clearly, when a particular
$\gamma_{i}$ does not depend on $x$ (such as the OL IS), then it would be
captured as a special case, and hence to capture this also notationally, we
will write $\gamma_{i}\in\Gamma_{i}$ as
$\gamma_{i}^{\eta}\in\Gamma_{i}^{\eta}$, where $\eta$ stands for the
underlying IS (which for the discussion in this paper is either OL or CL
SF).333Even though in general different players can have different ISs, we
will consider here only the case when the IS in the entire DG is either OL or
CL SF. Otherwise, derivation of Nash equilibrium becomes complicated, and one
has to introduce small noise robustness in order to eliminate informational
non-uniqueness, even in LQ DGs Basar89 , BasOls99 . At the conceptual level,
however, the analysis in this paper, and the indices introduced, equally apply
to the mixed IS case.
Each player $i\in\mathcal{N}$ is a cost-minimizer, with the objective function
for Player $i$, as defined on the state and action spaces, is given by
$L_{i}(u)=\int_{0}^{T}F_{i}(x(t),u_{1}(t),\cdots,u_{N}(t),t)dt+S_{i}(x(T))$
when $T<\infty$, and
$L_{i}(u)=\int_{0}^{\infty}F_{i}(x(t),u_{1}(t),\cdots,u_{N}(t),t)dt$
when $T=\infty$, where $u:=\\{u_{1},\ldots,u_{N}\\}$. In the expressions
above, for each $i\in\mathcal{N}$, the function
$F_{i}:\Omega\rightarrow\mathbb{R}$ is Player $i$’s instantaneous (running)
cost function, and in the first expression
$S_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is the terminal value function.
Substituting $u_{i}(t)=\gamma_{i}(t;x(t))$ in the above, we arrive at the ėm
normal or strategic form of the DG, where now the dependence in $L_{i}$ is on
$\gamma_{i}$’s instead of $u_{i}$’s. Let us denote this new cost function
representation by $J_{i}$, for Player $i$, which we write more explicitly
(showing its argument) as $J_{i}(\gamma^{\eta})$, where
$\gamma^{\eta}:=\\{\gamma_{1}^{\eta},\ldots,\gamma_{N}^{\eta}\\}\in\Gamma^{\eta}:=\Gamma^{\eta}_{1}\times\cdots\times\Gamma^{\eta}_{N}$,
where again this covers also the OL IS as a special case; we will occasionally
drop the superscript $\eta$ when the IS is clear from context.
Let $\gamma_{-i}^{\eta}$ denote the collection of policies of all players
except Player $i$, i.e.,
$\gamma_{-i}^{\eta}=(\gamma_{1}^{\eta},\ldots,\gamma_{i-1}^{\eta},\gamma_{i+1}^{\eta},\ldots,\gamma_{N}^{\eta})\,,$
in a game with IS $\eta$. If $\gamma_{-i}^{\eta}$ is fixed as
${\gamma_{-i}^{\eta*}}$, Player $i$ is faced with the dynamic optimization
(optimal control) problem: 444We use “$\textrm{OC}(i)$” to denote Player $i$’s
individual optimal control problem.
$\displaystyle(\textrm{OC}(i))\;\;$
$\displaystyle\min_{\gamma_{i}\in{{\Gamma_{i}^{\eta}}}}J_{i}(\gamma_{i},\gamma_{-i}^{\eta*}):=\int_{0}^{T}F_{i}(x,\gamma_{i}(\eta),{\gamma_{-i}^{\eta*}}(\eta),t)dt+S_{i}(x(T))$
(1)
$\displaystyle\textrm{s.t.~{}}\;\;\dot{x}(t)=f(x,\gamma_{i}(\eta),{{\gamma^{\eta*}_{-i}}}(\eta),t)\,,\;\;x(0)=x_{0}\,.$
In the case of infinite horizon, the problem remains the same with
$S_{i}\equiv 0$ and $T=\infty$. If we denote the solution to $\textrm{OC}(i)$
by ${\gamma_{i}^{\eta}}^{*}$, and carry out the optimization for each $i$,
then what we have is a Nash equilibrium compatible with the IS that defines
the DG. This is made precise below.
###### Definition 1
[$\eta$-Nash equilibrium] For a DG with IS $\eta$, the policy $N$-tuple
$\\{{\gamma_{i}^{\eta}}^{*},\;i\in\mathcal{N}\\}=:{\gamma^{\eta}}^{*}$ is an
$\eta-$Nash equilibrium if, for each $i\in\mathcal{N}$, $\gamma_{i}^{\eta*}$
solves the optimal control problem (OC$(i)$). Let $\Gamma^{\eta*}$ be the set
of all $\eta-$Nash equilibria, as a subset of $\Gamma^{\eta}$.
Now, for the CL IS case, one has to further refine the Nash equilibrium, in
order to eliminate informational non-uniqueness. Consider a family of DGs,
structured the same way, but defined over the time interval $[s,T\rangle$,
where $s>0$ is the parameter that identifies different elements of the family.
We say that an $\eta$-Nash equilibrium, when $\eta$ is the CL IS is strongly
time consistent if its restriction to $[s,T\rangle$ is also an $\eta$-Nash
equilibrium, and this being true for each $s$ and all $x(s)$. Such Nash
equilibria could also be called sub-game perfect equilibria, by direct analogy
with a similar concept in finite games. We will henceforth consider only
strongly time consistent Nash equilibria when $\eta$ is CL, but will suppress
that refinement in the development below.
Let $J^{\eta*}_{i},i\in\mathcal{N}$, denote the achieved values of the
objective functions of the players under a particular $\eta-$Nash equilibrium
$\gamma^{\eta*}$, and a corresponding total cost achieved (as a convex
combination of the individual costs) be given by
$J_{\mu}^{\eta*}=\sum_{i\in\mathcal{N}}\mu_{i}J_{i}^{\eta*}$, where $\mu_{i}$
is a positive weighting factor on Player $i$’th cost, satisfying the
normalization condition $\sum_{i\in\mathcal{N}}\mu_{i}=1$. We assume, without
any loss of generality, that $J^{\eta*}_{i}>0$ for all $i\in\mathcal{N}$, and
hence a fortiori $J_{\mu}^{\eta*}>0$.
Now as a benchmark, let us consider the case of full coordination, where the
players agree on minimizing a single objective function, which is a convex
combination of the individual cost functions. We may call this also a socially
optimal solution. The corresponding underlying optimization problem is the
optimal control problem: 555The acronym “COC” stands for “Centralized Optimal
Control”.
$\displaystyle(\textrm{COC})\;\;\;$
$\displaystyle\min_{\gamma\in\Gamma}\sum_{i=1}^{N}\mu_{i}\left\\{\int_{0}^{T}F_{i}(x(t),\gamma(\eta),t)dt+S_{i}(x(T))\right\\}$
s.t. $\displaystyle\dot{x}(t)=f(x,\gamma(\eta),t)\,,\;\;x(0)=x_{0}\,,$
where the optimization could also be carried out with respect to control
values, $u$, that is in an open-loop fashion, since the problem is
deterministic and also is not strategic. Hence, the optimal value of this
optimal control problem is independent of the IS, which we denote by
$J_{\mu}^{\circ}$, and the corresponding (open-loop) optimal control by
$u^{\circ}=[u_{1}^{\circ},\ldots,u_{N}^{\circ}]$. Note that we necessarily
have $0<J_{\mu}^{\circ}\leq J_{\mu}^{\eta*}\,,\;$ where $J_{\mu}^{\eta*}$ is
under any Nash equilibrium solution out of $\Gamma^{\eta*}$.
###### Definition 2 (Price of Anarchy)
Consider an $N$-person DG as above and its associated optimal control problem
(COC) with $J_{\mu}^{\circ}>0$. The price of anarchy for the DG is666If the
maximum below does not exist, then it is replaced by supremum in the
definition of PoA.
$\rho_{N,\mu,T}^{\eta}=\max_{\gamma^{\eta*}\in\Gamma^{\eta*}}\,J_{\mu}^{\eta*}/{J_{\mu}^{\circ}}$
(2)
as the worst-case ratio of the total game cost to the optimum social cost.
In addition to its dependence on the cost functions, PoA depends on the number
of players in the game, the IS, the weights on individual players and the time
horizon. Note that the PoA as defined in (2) is lower-bounded by 1.
###### Definition 3 (Price of Information (PoI))
Let $\eta_{1}$ and $\eta_{2}$ be two ISs. Consider two $N$-person DGs which
differ only in terms of their ISs, with game $1$ having IS $\eta_{1}$, and
game $2$ having $\eta_{2}$. Let the values of a particular $\mu$ convex
combination of the objective functions be ${J^{\eta_{1}}_{\mu}}^{*}$ and
${J^{\eta_{2}}_{\mu}}^{*}$, respectively, achieved under the Nash equilibria
${\gamma^{\eta_{1}}}^{*}$ and ${\gamma^{\eta_{2}}}^{*}$. The price of
information between the two ISs (under cost minimization) is given by
$\chi_{\eta_{1}}^{\eta_{2}}(\mu)=\max_{\gamma^{\eta_{2}^{*}}\in\Gamma^{\eta^{*}_{2}}}J_{\mu}^{\eta^{*}_{2}}\,/\max_{\gamma^{\eta_{1}^{*}}\in\Gamma^{\eta_{1}^{*}}}J_{\mu}^{\eta_{1}^{*}}.$
(3)
The PoI compares the worst-case costs under two different ISs for the same
convex combination, and quantifies the relative loss or gain when the DG is
played under a different IS. Clearly, when
$\chi_{\eta_{1}}^{\eta_{2}}(\mu)<1$, the IS $\eta_{2}$ is superior to its
counterpart $\eta_{1}$ . The connection between PoI and PoA can be captured by
$\;\chi_{\eta_{1}}^{\eta_{2}}(\mu)={\rho_{N,\mu,T}^{\eta_{2}}}\,/{\rho_{N,\mu,T}^{\eta_{1}}}\,.$
Before introducing the third index (price of cooperation), let us define
another class of DGs, which is an intermediate case between full cooperation
and full non-cooperation. Consider the case where Player $i$, even though his
cost function is $J_{i}$, adopts an altruistic mode and minimizes instead a
cost function that places some weight on other players’ costs. Let
$\lambda_{i}:=\\{\lambda_{i}^{j},j\in\mathcal{N}\\}$ be a set of nonnegative
parameters adding up to $1$, $\sum_{j\in\mathcal{N}}\lambda^{j}_{i}=1$. Let
$\tilde{J}_{i}(\gamma^{\eta};\lambda_{i})\,,\;i\in\mathcal{N}$ be defined by
$\tilde{J}_{i}(\gamma^{\eta};\lambda_{i}):=\sum_{j\in\mathcal{N}}\lambda_{i}^{j}J_{j}(\gamma^{\eta})\,,\;\;i\in\mathcal{N}$
Consider the $\eta$ IS DG with cost functions $\tilde{J}$’s, and let
$\tilde{\Gamma}^{\eta}$ be the set of all its $\eta$-Nash equilibria. For
$\tilde{\gamma}^{\eta}\in\tilde{\Gamma}^{\eta}$, Player $i$ achieves an actual
cost of $J_{i}(\tilde{\gamma}^{\eta})$, which may be better (lower) or worse
(higher) than $J^{\eta*}_{i}$ defined earlier. Note that if
$\lambda_{i}^{j}=\mu_{i}$ for all $i,j\in\mathcal{N}$, then all players have
the same cost function, and every $\eta$-Nash equilibrium solution of the
altruistic game is a solution to COC, assuming that person by person optimal
solutions of COC are globally optimal. Hence, in this limiting case we have
full cooperation. This now brings us to the third index, which is keyed to
individual players.
###### Definition 4 (Price of Cooperation (PoC))
Consider an N-player DG with a fixed IS $\eta$, and with a fixed set of
cooperation vectors $\lambda:=\\{\lambda_{i},\;i\in\mathcal{N}\\}$. Let
$\tilde{J}_{i},\;i\in\mathcal{N}$, and ${\tilde{\Gamma}}^{\eta}$ be as defined
above, and $\Gamma^{\eta}$ be the set of all Nash equilibria of the original
game. Then, the price of cooperation for Player $i$ under the cooperation
scheme $\lambda$ is given by
$\nu_{i}^{\eta}(\lambda)=\max_{\gamma\in{\tilde{\Gamma}}^{\eta}}J_{i}(\gamma)/\max_{\gamma\in\Gamma^{\eta}}J_{i}(\gamma)\,.$
(4)
As indicated earlier, if $\lambda_{i}=\mu$ for all $i$, where
$\mu=\\{\mu_{i},\;i\in\mathcal{N}\\}$ as in PoA, then every NE of
$\\{\tilde{J}_{i},\;i\in\mathcal{N}\\}$ is a person-by-person optimal solution
of the COC with cost function $J_{\mu}$, which would also be globally optimal
under some appropriate convexity conditions. If $\gamma^{0}$ is one such
solution, minimizing $J_{\mu}$, then the PoC is given by
$\nu_{i}^{\eta}(\mu)=J_{i}(\gamma)/\max_{\gamma\in\Gamma^{\eta}}J_{i}(\gamma)\,,$
which can be viewed as the reciprocal of individualized PoA, where the latter
is a measure of the loss or gain an individual player incurs on his individual
cost when he (along with other players) plays the worst NE strategy as opposed
to the globally minimizing strategy (again along with other players).
## 3 Scalar LQ Feedback Differential Games
The analysis of the price of anarchy is complex for general DGs as there often
exist more than one Nash equilibrium, which show strong dependence on the
underlying IS. For specific game structures, however, its analysis may be
tractable provided that we avoid informational non-uniqueness. One such class
is scalar linear quadratic DGs with state feedback IS, which is what we focus
on in this section. These games also enjoy wide applications in economics and
communication networks; see, DJLS06 , AB98 . We first state our model and
recall some important relevant results on LQ feedback DGs; for details, see
BasOls99 , Eng05 .
### 3.1 Game Model
As a special case of the class of DGs considered in the previous section,
consider the infinite-horizon scalar $N-$person LQ DGs, with quadratic cost
function
$L_{i}(u)=\int_{0}^{\infty}\left(q_{i}x^{2}(t)+r_{i}u_{i}^{2}(t)\right)dt,\;\;\;i\in\mathcal{N},$
(5) $\dot{x}(t)=ax(t)+\sum_{i=1}^{N}b_{i}u_{i}(t),\;\;\;x(0)=x_{0}\,,$ (6)
where $q_{i}>0$, $r_{i}>0$, $x_{0}\not=0$, $b_{i}\not=0$ are all scalar
quantities. Let $b:=[b_{1},\dots,b_{N}]$. We are interested in strongly time-
consistent state-feedback (SF) Nash equilibrium (NE), where further the NE
policies are required to be stationary (that is time invariant). We will refer
to such equilibria in short as Feedback NE. The following theorem provides
their characterization.
###### Theorem 1
[Feedback NE, BasOls99 , Eng05 ] Let $\\{k_{i},\;i\in\mathcal{N}\\}$ solve the
set of coupled algebraic Riccati equations
$2\left(a-\sum_{j=1}^{N}s_{j}k_{j}\right)k_{i}+q_{i}+s_{i}k_{i}^{2}=0,\;i\in\mathcal{N}$
(7)
satisfying the stability condition $a-\sum_{i=1}^{N}s_{i}k_{i}<0\,,$ where
$s_{i}:=b_{i}^{2}/r_{i}$. Then, the $N$-tuple of policies
$\gamma_{i}^{*}(x)=-\frac{b_{i}}{r_{i}}k_{i}x,\;i\in\mathcal{N},$ constitutes
a feedback NE, with the corresponding cost for Player $i$ being
$J^{*}_{i}=k_{i}x^{2}_{0}$. Furthermore, the positively weighed total cost is
$J_{\mu}^{*}=\bar{k}x^{2}_{0}$, where $\bar{k}=\sum_{i=1}^{N}\mu_{i}k_{i}$.
If the set of coupled algebraic Riccati equations do not admit a solution
which is also stabilizing, then the DG does not have a feedback NE. $\diamond$
The main challenge in computing the feedback NE solution for this DG is that
equation (7) is a nonlinear coupled system of equations. The fact that we have
a scalar problem alleviates the difficulty somewhat, since it is possible to
turn it into a linear problem through a change of variables, as outlined in
Eng00a ,Eng00b . Let $\sigma_{i}=s_{i}q_{i}$,
$\sigma_{\max}=\max_{i}\sigma_{i}$, $p_{i}=s_{i}k_{i},i=1,\ldots,N$, and
$\lambda=\sum_{i=1}^{N}p_{i}-a.$ (8)
Multiplying (7) by $s_{i}$, we rewrite it as
$p_{i}^{2}-2\lambda p_{i}+\sigma_{i}=0,\;i=1,\ldots,N.$ (9)
Let $\Omega\subset\mathcal{N}$ be an index set,
$\Omega_{-i}=\Omega\backslash\\{i\\}$, and $n_{\Omega}=|\Omega|$. For every
$\Omega\neq\emptyset$, we have (after some manipulations)
$\prod_{j\in\Omega}p_{j}\lambda=\frac{1}{2n_{\Omega}-1}\left\\{\sum_{i\in\Omega}\sigma_{i}\prod_{j\in\Omega_{-i}}p_{j}-\sum_{i\notin\Omega}\prod_{j\in\Omega}p_{j}p_{i}+a\prod_{j\in\Omega}p_{j}\right\\}.$
(10)
When $\Omega=\emptyset$, we define
$\prod_{j\in\Omega}p_{j}\lambda:=\lambda=\sum_{j=1}^{N}p_{j}-a.$ (11)
Hence, for every $\Omega$, we have an equation in the form of either (10) or
(11). Let $\mathbf{p}=[1,p_{1},p_{2},$
$\ldots,p_{N},p_{1}p_{2},\ldots,p_{1}p_{N},p_{2}p_{3},\ldots,p_{N-1}p_{N},\ldots,\prod_{i=1}^{N}p_{i}]^{T}$.
We can write (10) and (11) into
$\widetilde{\mathbf{M}}\mathbf{p}=\lambda\mathbf{p}.$ (12)
Let
$\mathbf{p}:=[1,k_{1},k_{2},\ldots,k_{N},k_{1}k_{2},\ldots,k_{1}k_{N},k_{2}k_{3},\ldots,$
$k_{N-1}k_{N},\ldots,\prod_{i=1}^{N}k_{i}]^{T}$ and
$\mathbf{D}=\textrm{diag}\\{1$, $s_{1}$, $s_{2}$, $\ldots$, $s_{N}$,
$s_{1}s_{2},$ $\ldots$, $s_{1}s_{N}$,$s_{2}s_{3},$ …,
$s_{N-1}s_{N},\ldots,\prod_{i=1}^{N}s_{i}\\}\,.$ Hence, we can rewrite
$\mathbf{p}=\mathbf{D}\mathbf{k}$ and (12) into
$\mathbf{M}\mathbf{k}=\lambda\mathbf{k},\,\mbox{ where
}\;\mathbf{M}:=\mathbf{D}^{-1}\widetilde{\mathbf{M}}\mathbf{D}\,.$ (13)
Equation (13) is an eigenvalue problem with each index set $\Omega$
corresponding to a row enumerated starting from the empty set. It has maximum
$2^{N}$ distinct eigenvalues and $2^{N}$ eigenvectors. The vector formed by
the second entry to the $N+1$-st entry of the eigenvectors yields the solution
to (7) when the first entry of the vector is normalized to $1$ and they
satisfy the stability condition of Theorem 1. This leads to:
###### Theorem 2
[Feedback NE Computation, Eng05 ] Suppose $\mathbf{M}$ is a nondefective
matrix with distinct eigenvalues. Let $(\lambda,\mathbf{k})$ be an eigenvalue-
eigenvector pair such that $\lambda\in\mathbb{R}_{+}$ and
$\lambda>\sigma_{\max}$. Then, a feedback NE
$\gamma_{i}^{*}(x)=-\frac{b_{i}}{r_{i}}k_{i}\,x,\;i\in\mathcal{N}\,,$ is
yielded by $k^{*}=\mathbf{1}^{T}\mathbf{k}$ provided that the resulting
solution is stabilizing, where $\mathbf{1}=[0,1,\ldots,1,0,\ldots,0]^{T}$ is a
vector whose $2$nd to $N+1$-st entries are 1’s.
###### Theorem 3
[Uniqueness of Feedback NE] Let
$\bar{p}:=\sum_{j\in\mathcal{N}}p_{j},p_{-i}:=\sum_{j\in\mathcal{N},j\neq
i}p_{j}$. There exists a unique feedback NE for the LQ DG described by (5) and
(6) under either one of the following two conditions:
(i) $N$ is sufficiently large such that $p_{-i}>a,\forall i$, or (ii) $a=0$.
Furthermore, the solutions to the coupled algebraic Riccati equations that
characterize the feedback NE are of the following forms under the
corresponding conditions above:
1. (s-i)
$p_{i}=(\bar{p}-a)-\sqrt{(\bar{p}-a)^{2}-\sigma_{i}}~{};$
2. (s-ii)
$p_{i}=\bar{p}-\sqrt{\bar{p}^{2}-\sigma_{i}},\;$ where
$\bar{p}-a=\frac{1}{N-1}\left(\sum_{i=1}^{N}\sqrt{(\bar{p}-a)^{2}-\sigma_{i}}+a\right).$
(14)
Moreover, the stability condition $a-\sum_{i=1}^{N}s_{i}k_{i}<0\,$ is
satisfied, and hence the FB NE is stabilizing.
###### Proof
From (9), we obtain
$p_{i}^{2}+2(p_{-i}-a)p_{i}-\sigma_{i}=0,$ (15)
which admits the solutions:
$p_{i}=(a-p_{-i})\pm\sqrt{(a-p_{-i})^{2}+\sigma_{i}}.$ (16)
Since we need $p_{i}>0$, we retain the one with $``+"$ sign. By rearranging
the positive solution of (16), we arrive at
$(\bar{p}-a)^{2}=(p_{-i}-a)^{2}+\sigma_{i}\,,$ (17)
and, therefore, in terms of $\bar{p}$, we have
$p_{i}=(\bar{p}-a)\pm\sqrt{(\bar{p}-a)^{2}-\sigma_{i}}.$ (18)
Under condition (i), we have $p_{i}-\bar{p}+a<0$, hence we obtain the unique
solution (s-i). Under scenario (ii), (18) reduces to
$p_{i}=\bar{p}\pm\sqrt{\bar{p}^{2}-\sigma_{i}}.$ Since, $p_{i}<\bar{p}$, we
again obtain the unique solution (s-ii).
By summing over (18), we have a fixed point equation (14). Let
$\bar{P}(\bar{p}):=\frac{1}{N-1}\left(\sum_{i=1}^{N}\sqrt{(\bar{p}-a)^{2}-\sigma_{i}}+a\right)-(\bar{p}-a)\,.$
Its derivative is given by
$\frac{d\bar{P}}{d\bar{p}}=-1+\frac{\bar{p}-a}{N-1}\left(\sum_{i=1}^{N}\frac{1}{\sqrt{(a-\bar{p})^{2}-\sigma_{i}}}\right).$
Since $\sigma_{i}\geq 0$ and $\bar{p}-a>0$, it follows that
$\displaystyle\frac{d\bar{P}}{d\bar{p}}$ $\displaystyle\geq$
$\displaystyle-1+\frac{\bar{p}-a}{N-1}\left(\frac{N}{(\bar{p}-a)}\right)$ (19)
$\displaystyle=$ $\displaystyle\frac{1}{N-1}>0,\textrm{~{}for~{}}N\geq 2.$
(20)
This says that $\bar{P}$ is a monotonically increasing function, and hence the
solution to $\bar{P}=0$ is unique. Hence, under (i) or (ii), there exists a
unique feedback NE.
The fact that the solution is stabilizing follows directly from (7), where the
first term has to be negative because the second and third terms are positive.
### 3.2 Team Model
When players form a team to achieve an optimal social objective, a specific
total cost is minimized. Let $\bar{q}_{\mu}=\sum_{i=1}^{N}\mu_{i}q_{i}$,
$\overline{R}_{\mu}=\textrm{diag}\\{\mu_{1}r_{1},\ldots,\mu_{N}r_{N}\\}$, and
consider
$\displaystyle(\textrm{FOC})$
$\displaystyle\;\;\;\;\min_{u(t)}\int_{0}^{\infty}\left(\bar{q}_{\mu}x^{2}(t)+u^{T}(t)\overline{R}_{\mu}u(t)\right)dt$
s.t.
$\displaystyle\;\;\;\dot{x}(t)=ax(t)+\sum_{i=1}^{N}b_{i}u_{i}(t)\,,\;\;x(0)=x_{0}\not=0\,.$
The solution to this optimal control problem is standard, and is given below
for future reference (where we suppress the dependence of $\bar{q}$ and
$\overline{R}$ on $\mu$).
###### Theorem 4
[Centralized Optimization] The optimal control problem (FOC) admits a unique
feedback solution which is further stabilizing. The optimal policies are
$\gamma^{\circ}_{i}(x)=-\frac{b_{i}}{\mu_{i}r_{i}}\hat{k}_{\mu}\,x\,,\quad\hat{k}_{\mu}:=\frac{a+\sqrt{a^{2}+\bar{q}\bar{b}}}{\bar{b}}\,,$
(21)
with $\bar{b}:=\sum_{i=1}^{N}(b_{i}^{2}/\mu_{i}r_{i})$, and minimum cost is
$J^{\circ}_{\mu}=\hat{k}_{\mu}x_{0}^{2}$.
The optimal control can also be expressed in open-loop form, as:
$u^{\circ}_{i}=-\frac{b_{i}}{\mu_{i}r_{i}}\hat{k}_{\mu}\Phi(t,0)x_{0},$
where $\Phi(t,0)$ is the unique solution to
$\dot{\Phi}(t,0)=\left(a-\sum_{i=1}^{N}\frac{b_{i}^{2}}{\mu_{i}r_{i}}\hat{k}_{\mu}\right)\Phi(t,0),~{}~{}\Phi(0,0)=1.$
### 3.3 Price of Anarchy (PoA)
Here, we provide a closed-form expression for the PoA in the feedback LQ DG,
where we make the natural assumption that $x_{0}\not=0$, as otherwise the
costs are all zero.
###### Theorem 5
The PoA of the LQ feedback DG described by (5) and (6) is characterized by the
following:
1. (i)
Given a weight vector $\mu$, the PoA $\rho_{\mu}$ is equal to
$\rho_{\mu}^{FB}=\max_{\mathbf{k}\in\mathcal{K}}\,\,[\,{\boldsymbol{\mu}^{T}\mathbf{k}}\,]\,/{\hat{k}}\,,$
(22)
where $\boldsymbol{\mu}=[0,\mu^{T},0,\ldots,0]^{T}$ and $\mathcal{K}$ is the
set of all eigenvectors of the matrix $\mathbf{M}$.
2. (ii)
Suppose
$\mu_{i}=\bar{\mu}_{i}:={s_{i}}\,/{\sum_{j=1}^{N}s_{j}},i\in\mathcal{N}$.
Then,
$\rho^{FB}_{\bar{\mu}}\leq[\,{\varrho({\mathbf{M}})+a}\,]\,/{\sum_{i=1}^{N}s_{i}\hat{k}}\,,$
where $\varrho(\mathbf{M})$ is the spectral radius of $\mathbf{M}$.
3. (iii)
Let $\mu^{s}_{\max}=\max_{i\in\mathcal{N}}\mu_{i}/s_{i}$. Given a weight
vector $\mu$ that satisfies $\sum_{i=1}^{N}\mu_{i}=1$, the PoA is bounded by
$\rho^{FB}_{\mu}\leq{\mu^{s}_{\max}(\varrho(\mathbf{M})+a)}\,/{\hat{k}}.$ (23)
###### Proof
The proof is a direct application of the results in Theorem 1 and Theorem 4.
PoA is the worst-case ratio of the game cost under feedback NE to the optimum
social cost as defined in (2). Under the feedback IS, an LQ DG has
$\rho^{FB}_{\mu}=\max_{k^{*}}\frac{\sum_{i=1}^{N}\mu_{i}k^{*}_{i}(x_{0})^{2}}{\hat{k}(x^{0})^{2}}=\max_{\mathbf{k}\in\mathcal{K}}\frac{\mu^{T}\mathbf{k}}{\hat{k}}\,.$
This leads to statement (i). The price of anarchy under $\bar{\mu}$ is
$\displaystyle\rho^{FB}_{\bar{\mu}}$ $\displaystyle=$
$\displaystyle\max_{k}\frac{\sum_{i=1}^{N}\bar{\mu}_{i}k_{i}}{\hat{k}}=\max_{k}\frac{s_{i}k_{i}}{\sum_{i=1}^{N}{s_{i}}\hat{k}}$
(24) $\displaystyle=$
$\displaystyle\max_{\lambda}\frac{\lambda+a}{\sum_{i=1}^{N}{s_{i}}\hat{k}}.$
The last equality is due to (8). Hence, by taking the largest eigenvalue, we
obtain (ii). The equality is achieved when $\varrho(\mathbf{M})$ is an
eigenvalue in the eigenvalue-eigenvector pair that yields the equilibrium from
Theorem 2. For an arbitrarily picked $\mu$, (22) yields
$\displaystyle\rho^{FB}_{\bar{\mu}}$ $\displaystyle=$
$\displaystyle\max_{k}\frac{\sum_{i=1}^{N}\frac{\mu_{i}}{s_{i}}s_{i}k_{i}}{\hat{k}}\leq\max_{k}\frac{u^{s}_{\max}\sum_{i=1}^{N}s_{i}k_{i}}{\hat{k}}$
(25) $\displaystyle=$
$\displaystyle\max_{\lambda}\frac{u_{\max}^{s}(\lambda+a)}{\hat{k}}\leq\frac{u_{\max}^{s}(\varrho(\mathbf{M})+a)}{\hat{k}}.$
Using (8) and taking the worst case, we obtain statement (iii). Since
$\max_{i\in\mathcal{N}}\frac{\bar{\mu}_{i}}{s_{i}}=\frac{1}{\sum_{j=1}^{N}{s_{j}}}\,,$
the last inequality is achieved when $\mu=\bar{\mu}$.
The next corollary further characterizes the bound on PoA.
###### Corollary 1
The following follow from Theorem 5:
1. (i)
Given a $\mu$ and $a\neq 0$, PoA is bounded above by
$\rho^{FB}_{{\mu}}\leq\left(1+\frac{1}{2a}(N+\sigma_{\max}-1)\right)s^{\bullet},$
(26)
where $\sigma_{\max}=\max_{i\in\mathcal{N}}\sigma_{i}$, and
$s^{\bullet}:=\sum_{i=1}^{N}\frac{s_{i}}{\min_{j\in\mathcal{N}}s_{j}}\,.$
The upper-bound is independent of $\mu$.
2. (ii)
If $a=0$, PoA is bounded above by
$\rho^{FB}_{{\mu}}\leq\frac{\mu^{s}_{\max}}{\sqrt{\bar{q}}\sqrt{\mu^{s}_{\min}}}\sqrt{N}(N+\sigma_{\max}-1),$
(27)
where $\mu^{s}_{\min}=\min_{i\in\mathcal{N}}\mu_{i}/s_{i}$.
###### Proof
The matrices $\mathbf{M}=[m_{ij}]$ and
$\widetilde{\mathbf{M}}=[\tilde{m}_{ij}],i,j=1,\ldots,2^{N},$ share the same
set of eigenvalues. From Gersgorin theorem, we can obtain
$\varrho(\widetilde{\mathbf{M}})\leq\min\left\\{\max_{i}\sum_{j=1}^{2^{N}}|\tilde{m}_{ij}|,\max_{j}\sum_{i=1}^{2^{N}}|\tilde{m}_{ij}|\right\\}\leq\max_{i}\sum_{j=1}^{2^{N}}|\tilde{m}_{ij}|.$
From (10) and (11), the absolute row sum $RS_{k},k=1,\ldots,2^{N}$, can easily
be evaluated by letting $p_{i}=1$:
$\displaystyle
RS_{k}=[{a+\sum_{i\in\Omega}\sigma_{i}+(N-n_{\Omega})}]\,/\,[{2n_{\Omega}-1}],$
where $k$ is the row index corresponding to the set $\Omega$. When
$\Omega=\emptyset$, we let $RS_{1}=N+a$. From (23),
$\rho^{FB}_{\mu}\leq\,\,[\,{\varrho(\mathbf{M})+a}\,]/({\hat{k}/\mu_{\max}^{s}}).$
The numerator is upper-bounded by (skipping some steps):
$\displaystyle\varrho(\mathbf{M})+a$ $\displaystyle\leq$
$\displaystyle\max\left\\{\max_{1\leq n_{\Omega}\leq
N}\frac{(2a+\sigma_{\max}-1)n_{\Omega}+N}{2n_{\Omega}-1},2a+N-1\right\\}$ (28)
$\displaystyle\leq$
$\displaystyle\max\left\\{{2a+N+\sigma_{\max}-1},2a+N-1\right\\}$
$\displaystyle\leq$ $\displaystyle 2a+N+\sigma_{\max}-1.$
The second inequality holds because the quantity
$\frac{(2a+\sigma_{\max}-1)n_{\Omega}+N}{2n_{\Omega}-1}$
increases with $n_{\Omega}$. The denominator has a lower bound:
$\displaystyle\frac{2a}{\bar{b}\mu^{s}_{\max}}$ $\displaystyle\geq$
$\displaystyle\frac{2a}{\sum_{i=1}^{N}\left(\frac{\max_{i\in\mathcal{N}}\mu_{i}/s_{i}}{\mu_{i}}\right)\frac{b_{i}^{2}}{r_{i}}}$
(29) $\displaystyle\geq$
$\displaystyle\frac{2a}{\sum_{i=1}^{N}\frac{s_{i}}{\min_{i\in\mathcal{N}}s_{i}}}=\frac{2a}{s^{\bullet}}\,.$
The last inequality is due to
$\max_{i}\mu_{i}/s_{i}\leq\max_{i}\mu_{i}\max_{i}\frac{1}{s_{i}}$. Combining
(28) and (29), we have, for $a\not=0$,
$\rho^{FB}_{{\mu}}\leq\left(1+\frac{1}{2a}(N+\sigma_{\max}-1)\right)s^{\bullet}$
When $a=0$,
$\hat{k}=\sqrt{\bar{q}/\bar{b}}=\sqrt{\frac{\bar{q}}{\sum_{i=1}^{N}\frac{s_{i}}{\mu_{i}}}}\geq\frac{\sqrt{\bar{q}\mu^{s}_{\min}}}{\sqrt{N}}\,.$
Using this together with (28), we arrive at the inequality (27).
The upper bound on price of anarchy in the preceding corollary provides a
worst case of efficiency loss.
The next result studies the large population game and its proof relies on the
Taylor series expansion of the square-root term in (18).
###### Theorem 6
Suppose the number of players in the LQ DG is sufficiently large so that
$\mbox{(C-i) }p_{-i}>a,\forall i\in\mathcal{N}\,,\;\mbox{(C-ii) }a\ll
N\,,\;\mbox{(C-iii) }\sigma_{\max}\ll\bar{\sigma}\,,$
where $\bar{\sigma}=\sum_{i=1}^{N}\sigma_{i}$. Then, the following quantities
can be approximated as given:
$\mbox{(i) }p_{i}\sim\frac{\sigma_{i}}{\sqrt{2\bar{\sigma}}}\,,\;\;\mbox{(ii)
}u_{i}\sim-\frac{\sigma_{i}}{b_{i}\sqrt{2\bar{\sigma}}}x\,,\;\;$ $\mbox{(iii)
}J^{*}\sim\frac{\bar{q}}{\sqrt{2\bar{\sigma}}}(x_{0})^{2}\,,\;\;\mbox{(iv)
}J^{*}\sim\frac{\bar{q}}{\sqrt{2\bar{\sigma}}}(x_{0})^{2}\,,\;\;$ $\mbox{(v)
}\rho^{FB}_{\mu}\sim\frac{\bar{q}}{\hat{k}\sqrt{2\bar{\sigma}}}\,,\;\mbox{and
for }a=0,\,\rho^{FB}_{\mu}\sim\sqrt{\frac{\bar{q}\bar{b}}{2\bar{\sigma}}}\,.$
###### Proof
By Taylor series expansion, (18) can be written as
$\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle(\bar{p}-a)\left[1-\sqrt{1-\frac{\sigma_{i}}{(\bar{p}-a)^{2}}}\right]$
(30) $\displaystyle=$
$\displaystyle\frac{\sigma_{i}}{2(\bar{p}-a)}\left[1+O\left(\frac{\sigma_{i}}{(\bar{p}-a)^{2}}\right)\right],$
where $O(\cdot)$ is a function such that $\lim_{x\rightarrow 0}O(x)=0$. In a
similar way, (14) can be rewritten as (skipping some steps):
$\displaystyle\bar{p}-a=$
$\displaystyle\frac{\bar{p}-a}{N-1}\left(\sum_{i=1}^{N}\sqrt{1-\frac{\sigma_{i}}{(\bar{p}-a)^{2}}}+a\right)$
$\displaystyle=$
$\displaystyle\frac{\bar{p}-a}{N-1}\left[\frac{N\bar{\sigma}}{2(\bar{p}-a)^{2}}\left(1+O\left(\frac{\sigma_{\max}}{2(\bar{p}-a)^{2}}\right)\right)+a\right].$
(31)
Hence, we obtain for large $N$
$\displaystyle\bar{p}-a$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\bar{\sigma}}{2}}\left[1+O\left(\frac{\sigma_{\max}}{2(\bar{p}-a)^{2}}\right)\right]$
(32)
Note that $\bar{p}-a>0$ due to the stability condition. Let
$\bar{\sigma}=\sum_{i=1}^{N}\sigma_{i}$, as before. Let a solution of (32) be
$\bar{p}=\sqrt{\bar{\sigma}/2}+a$, i.e.,
$\displaystyle\bar{p}-a=\sqrt{\frac{\bar{\sigma}}{2}}\left[1+O\left(\frac{\sigma_{\max}}{\bar{\sigma}}\right)\right].$
(33)
(33) is consistent provided that $\sigma_{\max}\ll\sigma$ and $a\ll N$. Since,
by Theorem 3, the solution is unique under (C-i), $\bar{p}$ can indeed be
approximated by $\bar{p}\sim a+\sqrt{\bar{\sigma}/2}$, which leads to
$p_{i}\sim\frac{\sigma_{i}}{\sqrt{2\bar{\sigma}}}$ from (30). Hence, (ii)-(v)
follow.
## 4 Open-Loop LQ Differential Games
In this section, we go back to the DGs described by (5) and (6), but with
open-loop information. Each player knows only the value of the initial state
of the system. Since the cost runs from zero to infinity, we are interested in
controls that yield finite costs. Accordingly, we restrict the controls of the
players to belong to the set
$\mathcal{U}^{OL}(x_{0})=\\{u\in\mathcal{L}_{2}[0,\infty)\mid
J_{i}(x_{0},u)<\infty,\;\forall i\in\mathcal{N}\\}\,,$
where $\mathcal{L}_{2}[0,\infty)$ is the space of square-integrable functions
on $[0,\infty)$.
###### Theorem 7
[Open-Loop NE, BasOls99 , Eng05 ] Consider the $N-$person LQ DG in (5) and
(6), and assume that there exists a unique solution $\xi^{\star}$ to the set
of equations
$0=2a\xi_{i}+q_{i}-\xi_{i}\left(\sum_{j=1}^{N}s_{j}\xi_{j}\right),$ (34)
such that $a-\sum_{j=1}^{N}s_{j}\xi_{j}^{\star}<0$, where
$s_{i}:=b_{i}^{2}/r_{i}$. Then, the game admits a unique open-loop Nash
equilibrium for every initial state, given by
$u_{i}^{\star}(t)=-\frac{b_{i}}{r_{i}}\xi_{i}^{\star}\exp\left[\left(a-\sum_{j=1}^{N}s_{j}\xi_{j}^{\star}\right)t\right]x_{0}\,.$
(35)
The optimal cost to player $i$ using $u_{i}^{\star}$ is
$\,J_{i}^{\star}=k_{i}^{\star}x_{0},\;$ where $k_{i}^{\star}$ is the unique
solution to
$2\left(a-\sum_{j=1}^{N}s_{j}\xi_{j}^{\star}\right)k_{i}+q_{i}+s_{i}(\xi_{i}^{\star})^{2}=0.$
(36)
The quantities in Theorem 7 can be made more explicit as we discuss below. By
a slight abuse of notation, let $p_{i}:=s_{i}\xi_{i}$ as in the state-feedback
information case. Multiplying (34) and (36) by $s_{i}$, we obtain
$\;0=2ap_{i}+\sigma_{i}-p_{i}\bar{p}\,,\;$ and
$0=2s_{i}k_{i}(a-\bar{p})+\sigma_{i}+p_{i}^{2},$ where
$\bar{p}=\sum_{i=1}^{N}p_{i}$. Hence we can solve for $p_{i},k_{i}$, and
obtain
$p_{i}={\sigma_{i}}\,/\,({\bar{p}-2a})$ (37)
$k_{i}={\sigma_{i}+p_{i}^{2}}\,/\,({2s_{i}(\bar{p}-a)}).$ (38)
To obtain $\bar{p}$, we sum (37) over $i$ and arrive at the quadratic equation
$\bar{p}=\frac{\bar{\sigma}}{\bar{p}-2a}.$ Thus,
$\bar{p}=\sqrt{a^{2}+\bar{\sigma}}+a\,,$ (39)
where we have retained only the positive solution of the quadratic equation
for obvious reasons. It should be pointed out that since the relevant
$\bar{p}$ is unique, we have a unique open-loop NE. Using (39), we can
determine the expression for $\xi_{i}^{\star}$ (and thus the OL NE strategies
of the players 35), as
$\xi_{i}^{\star}=\frac{q_{i}}{\sqrt{a^{2}+\bar{\sigma}}-a}.$ (40)
Note that these are necessarily stabilizing, that is
$a-\sum_{j=1}^{N}s_{j}\xi_{j}^{\star}<0$, in view of (36). Now using (39) and
(37) in (38), we arrive at the closed-form expression for $k_{i}^{\star}$:
$k_{i}^{\star}=\frac{1}{\sqrt{a^{2}+\bar{\sigma}}}\left(\frac{q_{i}}{2}+\frac{\sigma_{i}q_{i}}{2(\sqrt{a^{2}+\bar{\sigma}}-a)^{2}}\right).$
(41)
When $a=0$, $k_{i}^{\star}$ is reduced to
$k^{\star}_{i}=\frac{1}{\sqrt{\bar{\sigma}}}\left(\frac{q_{i}}{2}+\frac{\sigma_{i}q_{i}}{2\bar{\sigma}}\right).$
(42)
Given weighting $\mu$, the open-loop NE yields a total cost of
$J_{\mu}^{\star}=\sum_{i=1}^{N}\mu_{i}J_{i}^{\star}=\sum_{i=1}^{N}\mu_{i}k_{i}^{\star}(x_{0})^{2}=:k_{\mu}^{\star}(x_{0})^{2}\,.$
Since the open-loop NE solution is unique, the PoA under open loop IS can thus
be easily found to be:
$\rho_{\mu}^{OL}={k_{\mu}^{\star}}\,/\,{\hat{k}_{\mu}}\,.$ (43)
We now capture all this in the corollary below.
###### Corollary 2
The OL LQ DG of Theorem 7 admits a unique OL NE given by (35) and (40), which
is also stabilizing. Furthermore, the OL PoA is given by (43).
## 5 Price of Information (PoI)
In the previous sections, we have introduced PoA as a measure of efficiency in
going from cooperative to noncooperative framework, and obtained expressions
for it for FB and OL LQ DGs . Here, we study the price of information (PoI) as
a measure of efficiency with respect to the ISs for again the LQ DG. Following
Definition 3, PoI between open-loop and feedback ISs is defined by
$\chi^{OL}_{FB}={\max_{k^{\star}}J^{OL\star}}\,/\,{\max_{k^{*}}J^{FB*}}\,,$
(44)
which can also be expressed in terms of the PoAs under the two ISs:
$\chi_{FB}^{OL}={\rho_{\mu}^{OL}}\,/\,{\rho_{\mu}^{FB}}\,.$
Using Theorem 5, we can obtain a bound on PoI:
$\chi^{OL}_{FB}\geq\frac{k^{\star}}{\mu_{\max}^{s}(\varrho(\mathbf{M})+a)}\,.$
The following theorem further characterizes the PoI in a special case.
###### Theorem 8
Suppose $a=0$, and the number of players is large so that $N$ satisfies (C-i),
(C-ii), and (C-iii). Then, the PoI is bounded from above and below by two
constants:
${\sqrt{2}}/{2}\leq\chi_{FB}^{OL}\leq\sqrt{2}.$ (45)
###### Proof
Under conditions (C-i), (C-ii), and (C-iii), we have a unique feedback NE that
can be approximated as in statement (iv) of Theorem 6. Hence, from (39) we
obtain
$\displaystyle\chi_{FB}^{OL}$ $\displaystyle=$
$\displaystyle\frac{J^{OL\star}}{J^{FB*}}=\frac{\sqrt{2}}{2}\left(1+\frac{\sum_{i=1}^{N}\mu_{i}q_{i}\sigma_{i}}{\bar{q}\bar{\sigma}}\right)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2}\left(1+\frac{\sum_{i=1}^{N}\mu_{i}q_{i}\sigma_{i}}{\sum_{i=1}^{N}\mu_{i}q_{i}\sum_{i=1}^{N}\sigma_{i}}\right)\leq\sqrt{2}\,,$
where the last inequality is obtained by noting that
$\sum_{i=1}^{N}\mu_{i}q_{i}\sigma_{i}\geq\sum_{i=1}^{N}\mu_{i}q_{i}\sum_{i=1}^{N}\sigma_{i}\,.$
The lower bound can be achieved by noting that $\sigma_{i},q_{i},\mu_{i}$ are
all nonnegative.
Theorem 8 is useful in the design of games via access control or pricing
mechanisms. Let $\bar{\chi}\in(\frac{\sqrt{2}}{2},\sqrt{2}]$ be some target
PoI to achieve so that $\chi_{FB}^{OL}\leq\bar{\chi}$. For example, when
$\bar{\chi}=1$, it means the game needs to be designed so that the open-loop
NE yields no larger cost than the feedback NE. Hence, a necessary condition to
meet such a design criterion is:
$\frac{\sum_{i\in\mathcal{N}}\mu_{i}q_{i}\sigma_{i}}{\bar{q}\bar{\sigma}}\leq\sqrt{2}\chi_{FB}^{OL}-1.$
(46)
An access control is to admit a set $\mathcal{N}$ of players so that (46) is
satisfied when all the system and player parameters are given. When set
$\mathcal{N}$ is fixed and not adjustable, we may use “pricing” mechanisms to
control the parameters $r_{i}$ or $q_{i}$, which reflect the unit “price” of
penalty on the control effort and the state, respectively. In the following
corollary, we capture the special case of homogeneous players.
###### Corollary 3
Suppose the LQ DG satisfies the conditions in Theorem 8. In addition, let the
players be symmetric so that $\sigma_{i}=\sigma,p_{i}=p,\forall
i\in\mathcal{N}$. When $N\geq 3$, the open-loop IS yields better total optimal
cost; otherwise the FB information does better. In addition, as
$N\rightarrow\infty$,
$\lim_{N\rightarrow\infty}\chi_{FB}^{OL}=\frac{\sqrt{2}}{2}$ at the rate of
$O\left(\frac{1}{N}\right)$.
###### Proof
The proof directly follows from Theorem 8. The price of information under the
additional assumptions becomes
$\chi_{FB}^{OL}=\frac{1}{\sqrt{2}}\left(1+\frac{1}{N}\right)$. It is
independent of the parameters of the players and approaches
$\frac{\sqrt{2}}{2}$ as $N\rightarrow\infty$. By letting $\chi_{FB}^{OL}\leq
1$, we obtain $\;N\geq\,{1}\,/\,({\sqrt{2}-1})\,.$ Hence, since $N$ is an
integer, the open-loop NE does better than the feedback NE when there are $3$
or more players.
Theorem 8 and Corollary 3 have implications in the design of games via access
control when open loop is the preferred mode of play.
## 6 Applications and Illustrations
In this section, we apply the results obtained heretofore to two classes of
application scenarios in flow control.
### 6.1 Multiuser Rate-Based Flow Control
We adopt here the communication systems model described in AB98 , where the
players are the users or sources, and the action (control) variables are the
flows into the network. If a link receives more total flow than what it can
accommodate (measured by its capacity), then packets queue up. Having long
queues is not desirable, because it leads to delays in transmission. We call
such links which are congested bottleneck links, and formulate the game around
one such link. Let $q_{l}(t)$ denote the queue length at such a bottleneck
link and let $s(t)$ denote the total effective service rate available at that
link. Assume that each user is assigned a fixed proportion of the available
bandwidth; more specifically, the traffic of source $i,i=1,2,\ldots,N$, has an
allotted bandwidth of $w_{i}s(t)$, where $w_{i}$’s are positive parameters
which add up to $1$. We assume that the users have perfect measurement of
$s(t)$, but occasionally exceed or fall short of the bandwidth allotted to
them due to fluctuations. Hence, if $d_{i}(t)$ denotes the rate of source $i$
at time $t$, we can introduce $u_{i}(t):=d_{i}(t)-w_{i}s_{r}(t)$ as the
control (action) variable of the source. Then, queue build-up is governed by
the differential equation
$\dot{q_{l}}(t)=\sum_{i=1}^{N}u_{i}(t)\,,$ (47)
where we assume that queue is relatively tightly controlled so that end effect
constraints (starvation and exceeding an upper limit) do not become active.
The goal is to ensure that the bottleneck queue size stays around some desired
level $\bar{q}_{l}$, and good tracking between input and output rates is
achieved. Toward that end, we consider the shifted variable
$x(t):=q_{l}(t)-\bar{q}_{l}$, which satisfies the following differential
equation which is the shifted version of (47):
$\dot{x}(t)=\sum_{i=1}^{N}u_{i}\,\;\;x(0)=x_{0}\,.$ (48)
We now consider a noncooperative scenario in which each source determines a
linear feedback policy (or an open-loop policy) to minimize its own individual
cost function
$L_{i}(u)=\int_{0}^{\infty}\left(|x(t)|^{2}+|u_{i}(t)|^{2}\right)dt,$ (49)
which is consistent with the overall goal of keeping $x$ and $u_{i}$’s small.
We can also consider a related team problem in which sources minimize
cooperatively a common cost under the same information structure (where as we
know actually the IS does not make a difference in this case):
$L(u)=\int_{0}^{\infty}\left(N|x(t)|^{2}+\sum_{i=1}^{N}|u_{i}(t)|^{2}\right)dt.$
(50)
This is now within the framework of LQ DGs studied earlier, with the
correspondences being $a=0,x_{0}=1,\sigma_{i}=s_{i}=q_{i}=r_{i}=b_{i}=1$ in
(5) and (6). To obtain some numerical results, let us take $x_{0}=1$.
In the case of the 2-person LQ feedback game, the $M$ matrix introduced
earlier becomes
$\mathbf{M}_{2}=\left[\begin{array}[]{cccc}0&1&1&0\\\ 1&0&0&-1\\\ 1&0&0&-1\\\
0&1/3&1/3&0\end{array}\right]$
and if $N=3$, we have
$\mathbf{M}_{3}=\left[\begin{array}[]{cccccccc}0&1&1&1&0&0&0&0\\\
1&0&0&0&-1&-1&0&0\\\ 1&0&0&0&-1&0&-1&0\\\ 1&0&0&0&0&-1&-1&0\\\
0&1/3&1/3&0&0&0&0&-1/3\\\ 0&1/3&0&1/3&0&0&0&-1/3\\\ 0&0&1/3&1/3&0&0&0&-1/3\\\
0&0&0&0&1/5&1/5&1/5&0\end{array}\right].$
The positive eigenvalue of $\mathbf{M}_{2}$ is $\lambda_{2}=1.1547$ and the
corresponding vector is
$\mathbf{p}_{2}=\mathbf{k}_{2}=[1.0000,0.5774,0.5774,0.3333]^{T}$. The sum of
the optimal costs under equal weights is $J^{*}_{2}=0.5774$ while the optimal
common cost is $J^{\circ}_{2}=0.5$, yielding the price of anarchy value
$\rho^{FB}_{\mu,2}=1.1547$. For the case with 3 players, the eigenvector is
found to be
$\mathbf{p}_{3}=\mathbf{k}_{3}=[1.0000,0.4472,0.4472,0.4472,0.2000,0.2000,0.2000,0.0894]^{T}$
corresponding to $\lambda_{3}=1.3416$. Again under equal weights, the total NE
cost is $J^{*}_{3}=0.4472$ and the minimum social cost is
$J^{\circ}_{3}=0.3333$. Hence, the price of anarchy is given by
$\rho^{{FB}}_{\mu,3}=1.3416$. When the number of players becomes large,
$\rho^{FB}_{\mu}\sim\sqrt{\frac{N}{2}}$ from Theorem 6.
In the case of open-loop flow control, we obtain
$k^{\star}_{i}=\frac{1}{\sqrt{N}}\left(\frac{1}{2}+\frac{1}{2N}\right)$ and
total NE cost as $J^{\star}_{N}=k^{\star}$. In the 2-user game,
$J^{\star}_{2}=0.5303$ yielding the price of information
$\chi_{FB}^{OL}=0.9184$. The open-loop NE thus yields $8.16\%$ less cost in
comparison to the closed-loop FB one. In a 3-user game,
$J^{\star}_{3}=0.3849$, leading to a price of information value of
$\chi_{FB}^{OL}=0.8607$, which yields a $13.93\%$ more cost for the FB IS
case. We also note that as the number of players increases, the open-loop IS
yields a cost approaching $0$, i.e.,
$\lim_{N\rightarrow\infty}J_{N}^{\star}=0$, while in the feedback case, even
though it still converges to $0$, the rate is slower:
$J^{*}\sim\frac{1}{\sqrt{2N}}\rightarrow 0$. We observe that $\chi_{FB}^{OL}$
goes to $\frac{\sqrt{2}}{2}$ at a rate of $\frac{1}{N}$ as $N$ gets large,
i.e.,
$\lim_{N\rightarrow\infty}\chi_{FB}^{OL}=\frac{\sqrt{2}}{2}+\frac{1}{2N}\rightarrow\frac{\sqrt{2}}{2}.$
It is also noted that open-loop NEs always yield less equilibrium costs even
though they require less information.
Due to the symmetry of players in the flow control problem, we can obtain
exact closed-form solutions to the equilibrium costs using (18) and (14)
without approximation. It is not hard to show that under equal weights,
$J^{*}_{\mbox{OL}}=k_{i}=\frac{1}{\sqrt{2N-1}}\,,\;\;J^{\star}_{\mbox{FB}}=\frac{1}{\sqrt{N}}\left(\frac{1}{2}+\frac{1}{2N}\right)\,\;\;\mbox{and}\;\;J^{\circ}=\frac{1}{N}\,.$
In Figure 1, we show the price of information under open-loop and feedback
information structures, and in Figure 2, we show the corresponding prices of
anarchy. By exact calculation, we find when $N=4$, the open-loop NE cost to be
$J^{\star}_{4}=\frac{3}{8}=0.3125$, which catches up with and becomes better
than the feedback NE cost: $J^{*}_{4}=\frac{1}{\sqrt{7}}=0.378$.This is
consistent with our earlier observation based on large population
approximation.
We observe in Figure 1 that the NE costs are the same at $N=1$ (as they should
be), and as $N$ increases, both open-loop and feedback NE costs decrease. As
$N$ becomes large, both costs approach $0$. This happens because the queue
length is fixed. When the number of players goes to infinity, the contribution
from each user is negligible. Moreover, the state $x(t)$ can be driven to zero
very fast as the amount of total control effort increases with the number of
players. The cost incurred from the transient behavior of $x(t)$ then goes to
zero. In addition, for $N\geq 2$, open-loop NE yields better costs. The price
of information $\chi_{FB}^{OL}$ is always below $1$ but maintains its level
above $\frac{\sqrt{2}}{2}$. In Figure 2, the price of anarchy starts at $1$
when $N=1$ and increases as the number of players grows. The cost under the
feedback NE grows faster than the one under open-loop NE.
Figure 1: Price of Information Figure 2: Price of Anarchy
### 6.2 Normalized Flow Control Dynamics
In this section, we investigate a general flow control dynamics, which differs
from (48) by inclusion of a population-dependent normalization factor $f(N)$,
where $f(\cdot)$ is an increasing function of $N$:
$\dot{x}(t)=\frac{1}{f(N)}\sum_{i=1}^{N}u_{i}\,,\;\;\;x(0)=1\,.$ (51)
The introduction of a normalization factor is to adjust the queue length
proportionally when the number of users increases.
###### Proposition 1
The prices of anarchy $\rho_{\mu}^{OL},\rho_{\mu}^{FB}$, and the price of
information $\chi_{FB}^{OL}$ are independent of the normalization factor
$f(N)$, as summarized in Table 1.
Table 1: Various indices for normalized flow control game $J^{*}$ (FB) | $J^{\circ}$ (TP) | $J^{\star}$ (OL) | $\rho_{\mu}^{FB}$ | $\rho_{\mu}^{OL}$ | $\chi_{FB}^{OL}$
---|---|---|---|---|---
$\frac{f(N)}{\sqrt{2N-1}}$ | $\frac{f(N)}{N}$ | $\frac{f(N)}{\sqrt{N}}\left(\frac{1}{2}+\frac{1}{2N}\right)$ | $\frac{N}{\sqrt{2N-1}}$ | $\sqrt{N}\left(\frac{N+1}{2N}\right)$ | $\sqrt{2-\frac{1}{N}}\left(\frac{1}{2}+\frac{1}{N}\right)$
###### Proof
Using (18) and (14), we obtain $p_{i}$ for a given $N$ as follows:
$\displaystyle\bar{p}$ $\displaystyle=$
$\displaystyle\frac{N}{f(N)}\frac{1}{\sqrt{2N-1}},$ $\displaystyle p_{i}$
$\displaystyle=$ $\displaystyle\frac{1}{f(N)\sqrt{2N-1}},$ $\displaystyle
k_{i}$ $\displaystyle=$
$\displaystyle\frac{p_{i}}{s_{i}}=\frac{f(N)}{\sqrt{2N-1}},$ $\displaystyle
J^{*}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\frac{1}{N}k_{i}x_{0}^{2}=k_{i}.$
The team problem yields an optimal cost of
$J^{\circ}=\sqrt{\frac{\bar{q}}{\bar{b}}}=\frac{f(N)}{N}.$ (52)
Hence, the price of anarchy $\rho_{\mu}^{FB}$ under the state-feedback
information structure is independent of $f(N)$, and is given by
$\rho_{\mu}^{FB}=\frac{N}{\sqrt{2N-1}}$ (53)
The open-loop price of anarchy is also independent of the factor $f(N)$. Since
$J^{\star}=\frac{f(N)}{\sqrt{N}}\left(\frac{1}{2}+\frac{1}{N}\right)$, it is
given by
$\rho_{\mu}^{OL}=\sqrt{N}\left(\frac{N+1}{2N}\right).$ (54)
The price of information is also independent of $f(N)$, and given by
$\chi_{FB}^{OL}=\sqrt{2-\frac{1}{N}}\left(\frac{1}{2}+\frac{1}{N}\right).$
(55)
As a case study, we let $f(N)=\frac{1}{N}$. Then, $b_{i}=\frac{1}{N}$,
$s_{i}=\sigma_{i}=\frac{1}{N^{2}}$, for all $i\in\mathcal{N}$. When the
population is large, we have $J^{*}\sim\sqrt{\frac{N}{2}}$ and
$J^{\star}=\sqrt{N}\left(\frac{1}{2}+\frac{1}{N}\right)$. The price of anarchy
remains $\rho\sim\sqrt{\frac{N}{2}}$. The price of information remains
$\chi_{FB}^{OL}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2N}\rightarrow\frac{\sqrt{2}}{2}$
as $N\rightarrow\infty$. It can be shown that $\chi_{FB}^{OL}$ does not change
with the factor $f(N)$. In Figures 3 and 4, we show the prices based on the
exact closed form solution obtained in the same fashion as in the previous
section based on (18) and (14). We observe that the open-loop NE always
outperforms the feedback equilibrium. It should be pointed out that (i) in
Figure 3, the open-loop and feedback costs increase with the number of users.
This is due to the introduction of normalization factor into the system
dynamics. We allocate the queue length as an increasing function of the number
of users; (ii) Figures 4 and 2 are identical due to the above proposition.
If we set $f(N)=\sqrt{N}$, we have the open-loop and feedback optimal costs
approach $\frac{1}{2}$ and $\frac{\sqrt{2}}{2}$ respectively, as
$N\rightarrow\infty$. Figure 5 demonstrates that result.
Figure 3: Price of Information in the Normalized System, $f(N)={N}$ Figure 4:
Price of Anarchy in the Normalized System, $f(N)={N}$ Figure 5: Price of
Information in the Normalized System, $f(N)=\sqrt{N}$
A summary of the results with $f(N)=1$ and $f(N)=\frac{1}{N}$ under large
population approximation is provided in Table 2.
Table 2: Indices under two normalization factors using the large population approximation $f(N)$ | $J^{*}$ (FB) | $J^{\circ}$ (TP) | $J^{\star}$ (OL) | $\rho_{\mu}^{FB}$ | $\rho_{\mu}^{OL}$ | $\chi_{FB}^{OL}$
---|---|---|---|---|---|---
$1$ | $\frac{1}{\sqrt{2N}}$ | $\frac{1}{N}$ | $\frac{1}{\sqrt{N}}\left(\frac{1}{2}+\frac{1}{2N}\right)$ | $\sqrt{\frac{N}{2}}$ | $\sqrt{N}\left(\frac{1}{2}+\frac{1}{2N}\right)$ | $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2N}$
$\frac{1}{N}$ | $\sqrt{\frac{N}{2}}$ | $1$ | $\sqrt{N}\left(\frac{1}{2}+\frac{1}{2N}\right)$ | $\sqrt{\frac{N}{2}}$ | $\sqrt{N}\left(\frac{1}{2}+\frac{1}{2N}\right)$ | $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2N}$
## 7 Conclusion
In this paper, we have introduced the notions of price of anarchy, price of
information, and price of cooperation for nonzero-sum differential games, have
studied the first two extensively for a class of scalar linear-quadratic
differential games, and have obtained bounds and approximations on them, with
computable bounds available in the large population regime. Future promising
work is to extend these results to non-scalar differential games as well as to
obtain their counterparts for the price of cooperation. Also computing these
indices for specific models from communication networks and economics would be
a fruitful area of research.
## References
* (1) T. Alpcan, T. Başar, R. Srikant, and E. Altman, “CDMA uplink power control as noncooperative game,” Wireless Networks, 8:659-690, 2002.
* (2) E. Altman and T. Başar, “Multiuser rate-based flow control,” IEEE Trans. Communications, 46(7):940-949, 1998.
* (3) E. Altman, T. Boulogne, R. El-Azouzi, T. Jimnez, and L. Wynter, “A survey on networking games in telecommunications,” Computers and Operations Research, 33(2):286-311, February 2006.
* (4) A.P. Azad, E. Altman, and R. El-Azouzi, “Routing games: From egoism to altruism,” Proc. 8th International Symp Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt 2010), Workshop WNC3 2010, pp. 442-451, Avignon, France, May 31-June 4, 2010.
* (5) T. Başar, “A contradictory property of the Nash solution for two stochastic nonzero-sum games,” Proc. 10th Allerton Conf on Circuit and System Theory, pp. 819-827, October 1972.
* (6) T. Başar, “Time consistency and robustness of equilibria in noncooperative dynamic games,” in F. Van der Ploeg and A. de Zeeuw, editors, Dynamic Policy Games in Economics, pp. 9–54. North Holland, 1989.
* (7) T. Başar, “Control and game-theoretic tools for communication networks (Overview),” Appl. Comput. Math. , 6(2):104-125, 2007.
* (8) T. Başar and Y.C. Ho, “Informational properties of the Nash solutions of two stochastic nonzero-sum games,” J. Economic Theory, 7(4):370-387, April 1974.
* (9) T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd ed., SIAM Series in Classics in Applied Math., Philadelphia, 1999.
* (10) E. Dockner, S. Jorgensen, N. V. Long, and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2006.
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* (12) J. C. Engwerda, “Feedback Nash equilibria in the scalar infinite horizon LQ-game,” Automatica, 36:135-739, 2000.
* (13) J. C. Engwerda, “The solution set of the N-player scalar feedback Nash algebraic Riccati equations,” IEEE Trans. Automatic Control, 48:847-853, 2000.
* (14) J. C. Engwerda, LQ Dynamic Optimization and Differential Games, Wiley, 2005.
* (15) J. Grossklags, B. Johnson, and N. Christin, “The price of uncertainty in security games,” Proc. Eighth Workshop on the Economics of Information Security (WEIS), 2009.
* (16) R. Johari, S. Mannor, and J. Tsitsiklis, “Efficiency loss in a network resource allocation game: the case of elastic supply,” IEEE Trans. Automatic Control, 50(11):1712-1724, 2005.
* (17) R. Johari and J. Tsitsiklis, “Network resource allocation and a congestion game: The single link case,” Proc. 42nd IEEE Conf. Decision and Control (CDC), pp. 2112-2117, December 2004.
* (18) R.T. Maheswaran and T. Başar, “Nash equilibrium and decentralized negotiation in auctioning divisible resources,” J. Group Decision and Negotiation (GDN) , 13, October 2003.
* (19) T. Roughgarden and E. Tardos, “Bounding the inefficiency of equilibria in nonatomic congestion games,” Games and Economic Behavior, 47:389-403, 2004.
* (20) S. Shakkottai, R. Srikant, A. Ozdaglar, and D. Acemoglu, “The price of simplicity,” IEEE J. Selected Areas in Communication: Game Theory in Communication Systems, 26(7), 2008.
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|
arxiv-papers
| 2011-03-14T03:27:44 |
2024-09-04T02:49:17.644487
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tamer Basar and Quanyan Zhu",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1103.2579"
}
|
1103.2750
|
# Smart Finite State Devices:
A Modeling Framework for Demand Response Technologies
Konstantin Turitsyn, Scott Backhaus, Maxim Ananyev and Michael Chertkov The
work of MC at LANL was carried out under the auspices of the National Nuclear
Security Administration of the U.S. Department of Energy at Los Alamos
National Laboratory under Contract No. DE-AC52-06NA25396.K. Turitsyn is with
MIT, Mechanical Engineering, Cambridge, MA 02139 turitsyn@mit.eduS. Backhaus
is with MPA Division at LANL, Los Alamos, NM 87545 backhaus@lanl.govM. Ananyev
is with New Economic School, Moscow, Russia maksim.ananjev@gmail.comM.
Chertkov is with Theory Division & Center for Nonlinear Studies at LANL, Los
Alamos, NM 87545 and also with New Mexico Consortium, Los Alamos, NM 87544
chertkov@lanl.gov
###### Abstract
We introduce and analyze Markov Decision Process (MDP) machines to model
individual devices which are expected to participate in future demand-response
markets on distribution grids. We differentiate devices into the following
four types: (a) optional loads that can be shed, e.g. light dimming; (b)
deferrable loads that can be delayed, e.g. dishwashers; (c) controllable loads
with inertia, e.g. thermostatically-controlled loads, whose task is to
maintain an auxiliary characteristic (temperature) within pre-defined margins;
and (d) storage devices that can alternate between charging and generating.
Our analysis of the devices seeks to find their optimal price-taking control
strategy under a given stochastic model of the distribution market.
## I Introduction
Automated demand response is often used to manage electrical load during
critical system peaks[1, 2]. During a typical event as the system approaches
peak load, signaling from the utility results in automated customer load
curtailment for a given period of time to avoid overstressing the grid.
Although this type of load control is useful for maintaining system security,
automated demand response must evolve further to meet the coming challenge of
integrating time-intermittent renewables such as wind or photovoltaic
generation. When these resources achieve high penetration and their temporal
fluctuations exceed a level that can be economically mitigated by the
remaining flexible traditional generation (e.g. combustion gas turbines),
automated demand response will play a large role in maintaining the balance
between generation and load. To fill this role, automated demand response must
go beyond today’s peak-shaving capability
To follow intermittent generation, automated demand response must be bi-
directional control, i.e. it should provide for controlled increases and
decreases in load. The response must also be predictable and preferably non-
hysteretic, otherwise the load-generation imbalance may actually be
exacerbated. Predictability would be highly valued by third party companies
that aggregate loads into a pool of demand response resources. Finally,
whatever control methodology is implemented, it must also be stable and not
exhibit temporal oscillations. There are several factors that make achieving
these demand response goals challenging: the different options for demand
response signal, the uncertainty of the aggregate response to that signal, and
the inhomogeneity of the underlying ensemble of loads.
The demand response control signal could take several forms: direct load
control where some number of loads could be disabled via a utility-controlled
switch[3, 4]; end-use parameter control where an ensemble of loads can be
controlled by modifying the set point of the end-use controller, e.g. a
thermostat temperature set point[5, 6, 7]; or indirect control via energy
pricing in either a price taking (open loop) or auction (closed loop)[8]
setting. Today’s automated demand response for peak-shaving is a form of
direct load control which could be adapted and refined for the type of
operation we desire, however, it is difficult to assess the impact of demand
response on the end user because loads are simply disabled and re-enabled with
little concern for the current state of the end use. Direct control is
feasible for a relatively small number of large loads because the
communication overhead is not extreme. Individual direct control of a large
number of small loads would potentially overburden a communication system,
however, “ensemble” control using a single parameter for control has been
proposed, e.g. set point control for thermostatic loads[5, 6, 7] and
connection rate control for electric vehicle charging[9]. However, in these
control models, the underlying loads are assumed to be homogeneous (all of the
same type), which is advantageous because it allows for a quantifiable measure
of the end use impacts and customer discomfort, e.g. increasing all cooling
thermostat set points by 1${}^{o}F$ will generate a decrease in load with a
known end-use impact.
To control a large ensemble of inhomogenous loads with a single demand
response signal requires a quantity that applies to all loads, i.e. energy
pricing [10]. When given access to energy prices, consumers (or automated
controllers acting on their behalf) can make their own local decisions about
whether to consume or not. These local decisions open up new possibilities and
also create problems. The customer is now enabled to automatically modify and
perhaps optimize his consumption of energy to maximize his own welfare, which
is a combination of his total energy costs and the completion of the load’s
end use function. However, without an understanding of how consumers respond
to energy prices, the fidelity of the control allowed by the direct or
ensemble control schemes described above is lost. Retail-level double auction
markets[8] are an effective way of making demand response via pricing a
closed-loop control system, however, a logical outcome of these markets would
be locational prices potentially driven distribution system constraints making
the regulatory implementation troublesome. In contrast, a model where retail
customers are price takers may avoid some regulatory issues, however, price
taking is in essence a form of open loop control which then requires an
understanding of how the aggregate load on the system will respond to price.
Our goal in this initial work is to layout the computational framework for
discovering the end-use response to these price-taking “open-loop” control
systems. We develop state models for several different loads and subject them
to a stochastic price signal that represents how energy prices might behave in
an grid with a large amount of time-intermittent generation. We analyze the
response of these smart loads using a Markov Decision Process (MDP) to
optimize the welfare of the end user. Human owners of the devices have the
ability to program the devices in accordance to their strategies and
preferences, for instance by adjusting their willingness to sacrifice comfort
in exchange for savings on electricity costs. Otherwise, most of the time we
assume that the devices operate automatically in accordance to some optimal
algorithm that was either preprogrammed by their owners, discovered via
adaptive learning[11], or programmed by a third-party aggregator. The
resulting load end-use policies can then be turned around to predict the
effect of a change in prices on electrical load. Our long term strategic
intention is to analyze the aggregated network effect on power flows of many
independent customers and design optimal strategies for both consumers and the
power operator. However, the prime focus of our first publication on the
subject is less ambitious. We focus here on description of different load
models and analyze the optimal behavior of individual consumers.
The material in the manuscript is organized as follows. We formulate our main
assumptions and introduce the general MDP framework in Section II. Models of
four different devices (optional, deferable and control loads and storage
devices) are introduced in Section III. Our enabling simulation example of a
control load (smart thermostat) is presented in Section IV. We summarize our
main results and discuss a path forward in Section V.
## II Setting the Problems
### II-A Basic Assumptions
Future distribution networks are expected to show complex, collective behavior
originating from competitive interaction of individual players of the
following three types:
* •
Market operator, having full or partial control over the signals sent to
devices/customers. The most direct signal is energy price. The operator may
also provide subsidies and incentives or impose penalties, however in this
manuscript, we will mainly focus on direct price control.
* •
Human customers/owners, who are able to reprogram smart-devices or override
their actions.
* •
Smart devices, capable of making decisions about their operations. The devices
are semi-automatic, i.e. pre-programmed to respond to the signal on a short
time scale (measured in seconds-to-minutes) in a specific way, however the
owner of the device may also choose to change the strategy on a longer time-
scale (days or weeks). We model the smart devices as finite state machines
using a Markov Decision Process (MDP) framework. At the beginning of each
interval, a device decides how to change its state based on the current price.
Each change comes with a reward expressing actual transactions between the
provider and the consumer and the level of consumer satisfaction with the
decision. We assume that smart devices are selfish and not collaborative, each
optimizing its own reward.
In this manuscript we restrict our attention to a simple price-taking strategy
of consumer behavior, deferring analysis of more elaborate game-theoretic
interactions between the operator and the individual customers to further
publications.
We model the external states (that include electricity price, weather, and
human behavior) as a stochastic, Markov Chain process, $\\{s^{(e)}(t)\\}$. At
the beginning of the time interval, $t$, the variable describing these factors
is set to $s^{(e)}_{t}$ and changes during the next time step to
$s^{(e)}_{t+1}$ with the transition probability
$T(s^{(e)}_{t+1}|s^{(e)}_{t})$. The transition probabilities are assumed to be
known to the device and statistically stationary, i.e. independent of $t$.
(The later assumption can be easily relaxed to account for natural cycles and
various external factors.) The probability, $p(s^{(e)};t)$, to observe the
external state, $s^{(e)}(t)=s^{(e)}$, at the time $t$, follows the standard
Markov chain equation
$\displaystyle
p(s^{(e)};t+1)=\sum_{s_{t}^{(e)}}T(s^{(e)},s_{t}^{(e)})p(s_{t}^{(e)}|t).$ (1)
We also assume that the Markov chain (1) is ergodic and converges after a
finite transient to the statistically stationary distribution:
$p(s^{(e)};t+1)=p(s^{(e)};t)=p(s^{(e)})$. In the simulation tests that follow
we will restrict ourselves to $s^{(e)}$ drawn from a finite set $S^{(e)}$.
### II-B General Markov Decision Process Framework
Here we adopt the standard (Markov Decision Process) MDP approach [12, 13, 14]
to the problem of interest: description of smart devices responding to the
external (exogenous) Markov process $\\{s^{(e)}(t)\\}$. MDPs provide a
mathematical framework for modeling decision-making in situations where
outcomes are partly random and partly under the control of a decision maker.
Formally, the MDP is a 4-tuple, $(S,A,P(\cdot,\cdot),R(\cdot,\cdot))$, where
* •
$S$ is the finite set of states, in our case a direct product of the machine
states set $S^{(m)}$, and the externality state set $S^{(e)}$,
$S=S^{(m)}\otimes S^{(e)}$.
* •
$A$ is a finite set of actions. $A_{s}$ is the finite set of actions available
from state $s\in S$. Within our framework we model only the decisions made by
the machine, so the set $A$ consists only of actions associated with the
machine, $A=A^{(m)}$.
* •
$P_{a}(s,s^{\prime})=\Pr(s_{t+1}=s^{\prime}\mid s_{t}=s,\,a_{t}=a)$ is the
probability that action $a$ chosen while in state $s=(s^{(m)},s^{(e)})$ at
time $t$ will lead to state $s^{\prime}$ at time $t+1$. The probabilistic
description of the transition allows to account for stochastic nature of the
price fluctuations as well as for the randomness in the dynamics of the smart
devices.
* •
$R_{a}(s,s^{\prime})$ is the reward associated with the transition $s\to
s^{\prime}$ if the action $a$ was chosen. In our models, the reward will
reflect the price paid for electricity consumption associated with the
transition as well as the level of discomfort related to the event.
In the most simple setting analyzed in this work, the behavior of the device
is modeled via the policy function $\pi(s):S\to A$ that determines the action
chosen by the device for a given state: $a_{t}=\pi(s_{t})$. More general
formulations that include randomized decision making process, are not
considered in this paper. Our smart device models seek to operate with the
policy, $\pi(s)$, that maximizes over actions the expectation value of the
total discounted reward,
$\sum_{t=0}^{\infty}\gamma^{t}R_{a_{t}}(s_{t},s_{t+1})$ over the Markov
process, $P_{a}(\cdot,\cdot)$, where $0<\gamma\leq 1$, is the discount rate.
There are numerous algorithms used for optimizing the policies. In our work we
use the algorithms implemented in MDP Matlab toolbox [14].
## III Models of Devices
The specifics of our MDP setting are to be described below for four examples
of loads. Note that these examples are meant to illustrate the power of the
framework and its applicability to ”smart grid” problems. In this first paper,
we do not aim to make the examples realistic. Instead, we focus on the
qualitative features of the loads.The states and actions associated with the
devices are illustrated in the diagrams shown in Figs. 1-4. For simplicity, we
ignore the external part of the state $s^{(e)}$ in these diagrams. Full
diagrams can be produced by taking the Kronecker product of transition graphs
associated with the device and the external factors. In our diagrams, the
states are marked by squares and actions are marked by dashed circles.
Transitions from states to actions and actions to states are marked by dashed
and solid arrows, respectively.
Figure 1: MDP diagram for the model of optional load. See text for
explanations.
### III-A Optional Loads
A smart device described by an “optional load” pattern can operate in two
regimes, at full and limited capacity. An example of such load is a light that
can be automatically dimmed if the electricity price becomes too high (see
Fig. 1). To simplify the mathematical notations, we denote the states of the
machine $s^{(m)}$ by $x$. The machine can be in either of the two states:
$x=0$ and $x=1$, shown as $Idle$ and $Active$ in the diagram (1) respectively.
In the $x=0$ state the machine does not operate (the lights are off). In the
$x=1$ state, the machine is active and the lights are shining at the full
brightness, or are dimmed. Actions of the device are $a_{0}=\mbox{pass}$,
$a_{1}=\mbox{full}$ or $a_{2}=\mbox{shed}$. The $a_{0}$ action represents the
process of waiting for the external signal of switching on the device. If no
external external signal (requesting switching on) appears, the system returns
to the $x=0$ state, otherwise it moves to the $x=1$ state. When the device is
active (in the $x=1$ state), it has two options: operate at full capacity,
corresponding to the action $a_{1}$, or shed the load (dim the lights),
corresponding to action $a_{2}$. Turning the device on or off is an
externality dependent on a human. We assume that the external/human action is
random, with the probability of turning the device ON and turning the device
OFF being $\rho_{ON}$ and $\rho_{OFF}$ respectively. (For simplicity, we
assume that the OFF signal may arrive only by the end of the time interval.)
Assuming additionally that the transition probabilities do not not depend on
the device actions, we arrive to the following expression for the transition
kernel:
$\displaystyle
P_{pass}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{ON}\delta_{x^{\prime},1}+(1-\rho_{ON})\delta_{x^{\prime},0}\right],$
(2) $\displaystyle
P_{full,shed}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{OFF}\delta_{x^{\prime},0}+(1-\rho_{OFF})\delta_{x^{\prime},1}\right],$
(3)
where $\delta_{x_{1},x_{2}}$ is the Kronecker symbol: it is unity if
$x_{1}=x_{2}$ and zero otherwise.
There is no reward associated with either outcome of the $a_{0}=\mbox{pass}$
action, however, the other two actions ($a_{1}$ and $a_{2}$) result in a
reward consisting of two contributions. First is the price paid to the
electricity provider, $E_{full,shed}c$, where $c(t)$ is the cost of
electricity (considered as a component of $s^{(e)}$) and $E_{full,shed}$ is
the amount of energy consumed during the time interval which depends on
whether the lights are fully on or dimmed. ( Here, $E_{full}>E_{shed}>0$ and
both values do not depend on the resulting state of the device). Second, the
reward function accounts for a subjective level of comfort associated with the
$a_{1,2}$ actions: $C_{full,shed}$. The discomfort of the light dimming is
accounted by choosing $C_{full}>C_{shed}$. Summarizing, the cumulative reward
function in this model of the optional load becomes
$\displaystyle R_{pass}(s,s^{\prime})=0,$ (4) $\displaystyle
R_{full}(s,s^{\prime})=C_{full}-E_{full}c,$ (5) $\displaystyle
R_{shed}(s,s^{\prime})=C_{shed}-E_{shed}c.$ (6)
Obviously, our model of optional loads is an oversimplification because there
are a variety of additional effects which may also be important in practice,
however, all these can be readily expressed within the MDP framework. For
example, one may need to limit the wear and tear on the device, thus
encouraging (via a proper reward) minimization of switching. (To account for
this effect would require splitting the $Active$ state in the model explained
above into two states $Active-Full$ and $Active-Shed$.)
### III-B Deferable Loads
Figure 2: MDP diagram for the model of deferable load. See text for
explanations.
Our second example model is a deferable load, i.e. a load whose operation can
be delayed without causing a major consumer discomfort. Practical examples
include dishwashing machines or some maintenance jobs like disk
defragmentation on a computer. A simple model of such a device, shown in Fig.
2, has two states: $x=0$ ($Idle$) when no work is required and $x=1$
($Waiting$) when a job has been requested and the machine is waiting for the
right moment (optimal in terms of the cost) to execute it. As in the previous
model, the only action of the machine in the $Idle$ state is $a_{0}$ ($Pass$),
however, in the $Waiting$ state, there are two possible actions: $a_{1}=Wait$
results in waiting for possible drop of the electricity price and $a_{2}=Work$
results in immediate execution of the job. The transition kernel for the model
is
$\displaystyle
P_{Pass}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{ON}\delta_{x^{\prime},1}+(1-\rho_{ON})\delta_{x^{\prime},0}\right],$
(7) $\displaystyle
P_{Wait}(s,s^{\prime})=T(c^{\prime}|c)\delta_{x^{\prime},1},$ (8)
$\displaystyle P_{Work}(s,s^{\prime})=T(c^{\prime}|c)\delta_{x^{\prime},0},$
(9)
where $\rho_{ON}$ is the probability of an exogeneous job request. In this
model, there is no reward for choosing the $a_{0}=Pass$ action. The reward for
the $a_{2}=Work$ action is equal to minus the price paid for the electricity,
$R_{Work}(s,s^{\prime})=-E*c$, and the reward for the $a_{1}=Wait$ action
represents the level of discomfort associated with the delay,
$R_{Wait}=C_{delay}<0$. As in the model of optional loads, $E$ and $C_{delay}$
are constants parameters.
### III-C Control Loads
Figure 3: MDP diagram for the model of controllable load. See text for
explanations.
A very important class of devices that will likely play a key role in future
demand response technologies are machines tasked to maintain a prescribed
level of physical characteristics of some system. For example, thermostats are
tasked with keeping the temperature in a building within acceptable bounds.
Other examples of the control devices are water heaters, electric ovens,
ventilation systems, CPU coolers etc.
In our enabling, proof-of-principle model of the control load, we consider a
thermostat responsible for temperature control in a residential home. The
state of the device is fully characterized by temperature which can take three
possible values: $x=0,1,2$ corresponding to $Low,Medium,High$ temperatures,
respectively. Each temperature is assumed to be operationally acceptable. For
simplicity, we assume that the thermostat uses an electric heater to modify
the temperature (i.e. the outside temperature is low). The device can choose
between the following three actions. $a_{0}=Cool$ leaves the heater idle for
the forthcoming interval. Since there is some base consumption associated with
the thermostat operation we assume that $E_{Cool}>0$. The next action,
$a_{1}=Keep$, maintains the temperature at the current level and requires some
energy for heater operation: $E_{Keep}>E_{cool}>0$. Finally, $a_{2}=Heat$
corresponds to intensive heating that raises the temperature and requires the
largest amount of energy $E_{Heat}$, and $E_{Heat}>E_{Keep}>E_{Cool}=0$. Our
thermostat state diagram, shown in Fig. (3), assumes that the dynamics of the
thermostat are deterministic, and the resulting state depends only on the
action chosen. The transition probabilities of the thermostat MDP is
$\displaystyle P_{Heat}(s,s^{\prime})=T(c^{\prime}|c)\delta_{x^{\prime},x+1},$
(10) $\displaystyle
P_{Keep}(s,s^{\prime})=T(c^{\prime}|c)\delta_{x^{\prime},x},$ (11)
$\displaystyle P_{Cool}(s,s^{\prime})=T(c^{\prime}|c)\delta_{x^{\prime},x-1}.$
(12)
Assuming that all levels of temperature are equally comfortable, the reward
function depends only on the price and energy consumption associated with the
action,
$\displaystyle R_{Cool,Keep,Heat}(s,s^{\prime})=-cE_{Cool,Keep,Heat}.$ (13)
Our basic model can be generalized to account for different comfort levels of
different states, the possibility for the owner to override an action,
variations of the outside temperature, etc.
### III-D Storage loads
Figure 4: MDP diagram for the model of storage. See text for explanations.
The number of devices with rechargeable batteries is expected to increase
dramatically in the coming years. Currently, these are mostly laptops,
uninterruptable power supplies, etc. In addition, a significant number of
large-scale batteries will be added to the grid most likely via the
anticipated Plug-in Hybrid Electric Vehicles (PHEV) potentially enabled with
Vehicle-to-Grid (V2G) capability. Storage devices, illustrated with the MDP in
Fig. (4), share some similarity with the controlled loads discussed in the
previous Subsection, but they are also different in two aspects. First,
users/owners wants their devices to be charged which leads to a level of
discomfort if the devices are not fully charged. Second, and probably most
significantly, storage devices such as PHEVs are disconnected from the grid
when in use. Having PHEVs in mind, we propose the following model of (mobile)
storage. The system can be in either of the three states, the $x=0=Unplugged$
state (which is similar to the Idle state in the models of Optional and
Deferable loads discussed above), the $x=1=Partially$ state where the storage
is partially charged, and the $x=2=Full$ state where the device is fully
charged.
The four available actions are: $a_{0}=Pass$ when the device is in the
unplugged state, the $a_{1}=Keep$ action possible when the initial state is
$x=1=Partially$ or $x=2=Full$, the $a_{2}=Charge$ action available from the
$x=1=Partially$ state which transitions to the $x=2=Full$ state, and, finally,
the $a_{3}=Discharge$ action, that is an inverse of the $a_{2}$ one, available
from the $x=2=Full$ state resulting in the $x=1=State$. Except for
$a_{0}=Pass$, all these actions can be interrupted by transitioning at the end
of the time interval to the $x=0=Unplugged$ state. As in previous sections, we
assume that the unplugging happens at the end of a time interval. Assuming the
device can be unplugged with the probability $\rho_{OFF}$ and that it can be
reconnected to the grid with the probability $\rho_{ON}$, we arrive at the
following expressions for the transition probability:
$\displaystyle
P_{Pass}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{ON}\delta_{x^{\prime},1}+(1-\rho_{ON})\delta_{x^{\prime},0}\right],$
(14) $\displaystyle
P_{Keep}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{OFF}\delta_{x^{\prime},0}+(1-\rho_{OFF})\delta_{x,x^{\prime}}\right],$
(15) $\displaystyle
P_{Charge}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{OFF}\delta_{x^{\prime},0}+(1-\rho_{OFF})\delta_{x^{\prime},2}\right],$
(16) $\displaystyle
P_{Discharge}(s,s^{\prime})=T(c^{\prime}|c)\left[\rho_{OFF}\delta_{x^{\prime},0}+(1-\rho_{OFF})\delta_{x^{\prime},1}\right].$
(17)
The reward function accounts for the following effects. First, the
$a_{1}=Keep$ action has the cost associated with keeping the battery charged,
$E_{Keep}(x)$, naturally dependent on the state,
$E_{Keep}(2)>E_{Keep}(1)>E_{Keep}(0)=0$. Second, the $a_{2}=Charge$ action
requires $E_{Charge}$ of energy while the $a_{3}=Discharge$ action generates
the $E_{Discharge}<0$ of energy, both nonzero only if the resulting state is
not the $x=0=Unplugged$. Therefore, all the “active” actions,
$Keep,Charge,Discharge$, contribute the reward function in accordance with the
energy price, $c^{\prime}E_{\dots}$. Finally, we also assign an additional
negative reward, $C_{Unplug}<0$, accounting for the discomfort (to the human)
associated with being in the $x=0=Unplugged$ state. The resulting reward
function is
$\displaystyle R_{Pass}(s,s^{\prime})=0,$ (18) $\displaystyle
R_{Keep}(s,s^{\prime})=C_{Unplug}\delta_{x^{\prime},0}\delta_{x,1}-cE_{Keep}(x),$
(19) $\displaystyle R_{Charge}(s,s^{\prime})=-cE_{Charge},$ (20)
$\displaystyle
R_{Discharge}(s,s^{\prime})=C_{Unplug}\delta_{x^{\prime},0}-cE_{Discharge}.$
(21)
## IV Simulations
In order to illustrate the capabilities of the proposed framework, we consider
a simple model of the control load, describing a smart thermostat,
characterized by $N_{T}=10$ levels of the temperature parameter $T$. At every
moment of time the thermostat can choose to raise, lower or keep the same
temperature. The raise and lower options are not available at the highest and
lowest possible temperatures, respectively. The energy consumption associated
with the actions is given by $E_{Keep}=1.0$, $E_{Cool}=0.1$ and
$E_{Heat}=2.1$, respectively, in some normalized energy units. This choice of
energies discourages the system from switching the heater too often: although
the combinations $Heat+Cool$ and $Keep+Keep$ lead to the same temperature
levels, the latter action is preferable as it consumes less energy.
Variations in price are modeled by a Markov chain of $N_{P}=5$ equidistant
levels with the minimum and maximum corresponding to $1.0$ and $2.0$ price
units, respectively. At each time interval, the price either increases with
probability $T(c+1|c)=0.5$ by $1$ level, decreases with probability
$T(c-1|c)=0.3$ by $1$ level, or stays the same. The resulting stationary
probability distribution $p(c)$ is shown in the Figure 5. It is skewed towards
the higher price, mimicking the effect of intermittent renewable generators
that occasionally provide excess power to the grid, thus leading to rapid dips
in the price.
Figure 5: Probability distribution of electricity price in the model example.
The reward function (13) is fully determined by the total cost of energy
consumed by the thermostat within the given time-interval. Our MDP model
imposes upper and lower bounds on the temperature, and we assume that there is
no additional discomfort associated with the variations of temperature between
these bounds, i.e. all of the $N_{T}$ temperature levels are equally
comfortable for the consumer.
Figure 6: Visualization of the policy found as a result of optimization.
This system was analyzed with the Matlab MDP package [14] where we used
different algorithms to verify the stability of the results. The resulting
optimal policy (for the range of parameters tested) is illustrated in 6. As
expected, the thermostat chooses the $Heat$ action when the price is low and
decides to $Cool$ when the price is high; a set of actions that lead to the
skewed probability distribution of temperatures shown in Figure 7. One finds
that the thermostat spends most of the time performing $Keep$ in the low
temperature state waiting for the price to drop.
Figure 7: Probability distribution of temperature levels observed at the
optimal policy.
Perhaps, the most interesting feature of the MDP model is the relation between
consumption and price. We define the expected demand as the average energy
demand for a given price
$\langle
E|c\rangle=\frac{\sum_{x}E_{\pi(x,c)}P_{st}(x,c)}{\sum_{x}P_{st}(x,c)},$ (22)
where $P_{st}(x,c)$ is the stationary joint distribution function of the
temperature and price at the optimal strategy. Dependence of the consumption
on the price for our choice of the parameters is shown in Figure 8, thus
illustrating that variations in price indeed produce demand response. An
interesting feature is that the demand curve is not monotonic. At low
temperatures, the energy consumption shows a slight increase with the price; a
surprising behavior related to saturation of the demand. When the electricity
price decreases gradually from high to low levels, there is a high probability
that the thermostat will reach the highest level of temperature before the
price reaches the lowest level. In this case, the demand will be lower at the
smallest price levels as there will be no unsatisfied demand left in the
system to capitalize on the lowest price. From the economic viewpoint, it is
important to note that this non-monotonicity of the demand curve reflects the
adaptive nature of the MDP algorithm: the smart devices adjust to fluctuations
in price, thus making it more difficult for the electricity providers to
exploit the non-monotonic demand curve for making profit.
Figure 8: Demand of the smart thermostats.
Another interesting result found in our simulations is an increase in average
consumption of the smart (policy optimized) thermostat when compared to its
non-smart counterpart, where the latter is defined as the one ignoring price
fluctuations and sticking to the $Keep$ action. For the set of parameters
chosen in the test case, we observed that the average level of consumption in
the optimal case is $1.03$, i.e. it is $3\%$ higher than in the naive
strategy, an effect associated with the additional penalty (in energy) imposed
on the $Heat$ and $Cool$ actions.
It is also instructive to evaluate savings of the consumer. The average value
of the reward associated with the optimal policy is equal to $-1.6722$, which
should be compared with the reward of $-1.73$ generated by its non-smart
counterpart. Since the reward reflects the customer’s cost of electricity, we
conclude that the customer saves about $3\%$ on the electricity costs
associated with the thermostat. The lower total energy costs for higher energy
consumption was also seen in a related “smart-device” demonstration
project[8].
Note, that the quantitative conclusions drawn and numbers presented above were
meant to illustrate the questions the MDP approach can resolve. The
conclusions and the numbers do not represent any real device as the parameters
used were not justified by actual data.
## V Discussions, Conclusions and Path Forward
To conclude, we have presented a novel modeling framework to analyze future
demand response technologies. The main novel aspect of our approach lies in
the capability of the framework to describe behavior of the smart devices
under varying/fluctuating electricity prices. To achieve this goal, we modeled
the devices as rational agents which seek to maximize a predefined reward
function associated with its actions. In general, the reward function includes
the price paid for the electricity consumption and the level of owner
discomfort associated with the choices made by the device. At the mathematical
level, the system can be described via Markov Decision Processes that have
been extensively studied over the last 50 years. Utilizing the MDP approach,
we showed that a great variety of practical devices can be described within
the same framework by simply changing the set of device states, actions and
reward functions. Specifically, we identified four main device categories and
proposed simple MDP models for each of them. These four categories include
optional loads (like light dimming), deferrable loads (like dishwashing),
control loads (thermostats and ventilation systems), and finally storage loads
(charging of batteries).
To illustrate the approach we experimented with a simple model of a smart
heating thermostat. The MDP-optimized policy of the thermostat followed the
expected pattern: it chooses to not heat or keep the temperature stationary at
high prices and prefers to heat when the price is low. This policy resulted in
$3\%$ of savings in the price paid for electricity, but at the same time led
to the total of $3\%$ increase in the consumption level due to the energy
costs associated with the thermostat actions. The resulting demand curve
showed a noticeable amount of elasticity, thus meeting the main objective of
the demand response technology.
There are many relevant aspects of the model that we did not discuss in the
manuscript. We briefly list some of these and future research challenges and
direction.
* •
_Learning algorithms_. In our model we assumed that smart devices have an
accurate model of stochastic dynamics for external factors (such as price for
electricity), and use this model to find the optimal policy. In reality,
however, this model is not known ab initio and has to be learned from the
observations. Moreover, one can expect that the dynamics of external factors
will be highly non-stationary (i.e. the transition matrix
$T(s^{(e)}_{t+1}|s^{(e)}_{t+1})$ will have an explicit dependence on time).
Therefore, the optimal policy has to be constantly adapted to the varying
dynamics of the external factors. Of a special practical interest is the
generalization of the framework to almost periodic processes, reflecting
natural daily/weekly/yearly cycles in the electricity consumption.
* •
_Price-setting policies_. We did not discuss the price setting policies above,
assuming that the policies are given/pre-defined. However, the electricity
providers might adjust their policies to consumer response. As the electricity
providers pursue their own goals, this setting essentially becomes game-
theoretic and as such it requires more sophisticated approaches for analysis.
Another extension of the model is to introduce auction-based price-setting
schemes, such as in the Olympic Peninsula project [8]. This setting can be
naturally incorporated in the same framework, although the modification may
require simultaneous modeling of multiple (ensemble of) devices.
* •
_Time delays_. Another aspect of the real world not incorporated in our
analysis concerns the separation of the time scales associated with operations
of the device and intervals of the price variations. Multiple time-scale can
be naturally incorporated in the framework by introducing additional states of
the device. These modifications will certainly affect final answer for the
optimal policy, and the resulting demand curve. However, accurate
characterization of the multi-scale behavior will be a challenging task,
requiring analysis of nonlinear response functions and dynamical description
of the underlying non-Markovian processes.
## VI Acknowledgements
We are thankful to the participants of the “Optimization and Control for Smart
Grids” LDRD DR project at Los Alamos and Smart Grid Seminar Series at
CNLS/LANL for multiple fruitful discussions.
## References
* [1] N. Motegi, M. A. Piette, W. D., S. Kiliccote, and P. Xu, “Introduction to commercial building control strategies and techniques for demand response,” LBNL Report Number 59975, Tech. Rep., 2007. [Online]. Available: http://gaia.lbl.gov/btech/papers/59975.pdf
* [2] “U.s. department of energy, “benefits of demand response in electricity markets and recommendations for achieving them”,” U.S. DOE, Tech. Report, 2006\.
* [3] S. S. Oren and S. A. Smith, “Design and management of curtailable electricity service to reduce annual peaks,” _OPERATIONS RESEARCH_ , vol. 40, no. 2, pp. 213–228, 1992. [Online]. Available: http://or.journal.informs.org/cgi/content/abstract/40/2/213
* [4] R. Baldick, S. Kolos, and S. Tompaidis, “Interruptible electricity contracts from an electricity retailer’s point of view: Valuation and optimal interruption,” _OPERATIONS RESEARCH_ , vol. 54, no. 4, pp. 627–642, 2006\. [Online]. Available: http://or.journal.informs.org/cgi/content/abstract/54/4/627
* [5] D. S. Callaway, “Tapping the energy storage potential in electric loads to deliver load following and regulation, with application to wind energy,” _Energy Conversion and Management_ , vol. 50, no. 5, pp. 1389 – 1400, 2009\. [Online]. Available: http://www.sciencedirect.com/science/article/B6V2P-4VS9KPY-1/2/32649b4a9a6779a2cea84379a7c1f9a6
* [6] D. Callaway and I. Hiskens, “Achieving controllability of electric loads,” _Proceedings of the IEEE_ , vol. 99, no. 1, pp. 184 –199, 2011.
* [7] S. Kundu, N. Sinitsyn, S. Backhaus, and I. Hiskens, “Modeling and control of thermostatically controlled loads,” in _17th Power Systems Computation Conference_ , 2011, submitted.
* [8] D. Hammerstrom and et al, “Pacific northwest gridwise testbed demonstration project:part i. olympic peninsuila project,” PNNL-17167, Tech. Rep., 2007. [Online]. Available: http://gridwise.pnl.gov/docs/op˙project˙final˙report˙pnnl17167.pdf
* [9] K. Turitsyn, N. Sinitsyn, S. Backhaus, and M. Chertkov, “Robust broadcast-communication control of electric vehicle charging,” in _Smart Grid Communications (SmartGridComm), 2010 First IEEE International Conference on_ , 2010, pp. 203 –207.
* [10] F. Schweppe, R. Tabors, J. Kirtley, H. Outhred, F. Pickel, and A. Cox, “Homeostatic utility control,” _Power Apparatus and Systems, IEEE Transactions on_ , vol. PAS-99, no. 3, pp. 1151 –1163, May 1980.
* [11] D. O’Neill, M. Levorato, A. Goldsmith, and U. Mitra, “Residential demand response using reinforcement learning,” in _Smart Grid Communications (SmartGridComm), 2010 First IEEE International Conference on_ , 2010, pp. 409 –414.
* [12] R. E. Bellman, “A markovian decision process,” _Jounral of Mathematics and Mechanics_ , vol. 6, 1957.
* [13] M. L. Putterman, _Markov Decision Processes. Discrete Stochastic. Dynamic Programming._ Wiley-Interscience, 2005\.
* [14] “Markov decision process (mdp) toolbox for matlab.” [Online]. Available: http://www.cs.ubc.ca/~murphyk/Software/MDP/mdp.html
|
arxiv-papers
| 2011-03-14T19:35:02 |
2024-09-04T02:49:17.653917
|
{
"license": "Public Domain",
"authors": "Konstantin Turitsyn, Scott Backhaus, Maxim Ananyev and Michael\n Chertkov",
"submitter": "Konstantin Turitsyn",
"url": "https://arxiv.org/abs/1103.2750"
}
|
1103.2845
|
# Langevin process reflected on a partially elastic boundary II
Emmanuel Jacob111 email. emmanuel.jacob@normalesup.org
website. http://www.proba.jussieu.fr/pageperso/jacob
_Laboratoire de Probabilités et Modèles Aléatoires_
_Université Pierre et Marie Curie_
_4 place Jussieu, 75005 Paris, France_
Abstract
A particle subject to a white noise external forcing moves like a Langevin
process. Consider now that the particle is reflected at a boundary which
restores a portion $c$ of the incoming speed at each bounce. For $c$ strictly
smaller than the critical value $c_{crit}=\exp(-\pi/\sqrt{3})$, the bounces of
the reflected process accumulate in a finite time. We show that nonetheless
the particle is not necessarily absorbed after this time. We define a
“resurrected” reflected process as a recurrent extension of the absorbed
process, and study some of its properties. We also prove that this resurrected
reflected process is the unique solution to the stochastic partial
differential equation describing the model. Our approach consists in defining
the process conditioned on never being absorbed, via an $h-$transform, and
then giving the Itō excursion measure of the recurrent extension thanks to a
formula fairly similar to Imhof’s relation.
Key words. Langevin process, second order reflection, recurrent extension,
excursion measure, stochastic partial differential equation, $h$-transform.
A.M.S classification. (MSC2010) 60J50, 60H15
## 1 Introduction
Consider a particle in a one-dimensional space, submitted to a white noise
external forcing. Its velocity is then well-defined and given by a Brownian
motion, while its position is given by a so-called Langevin process. The
Langevin process is non-Markov, therefore its study is often based on that of
the Kolmogorov process, which is Markov. This Kolmogorov process is simply the
two-dimensional process, whose first coordinate is a Langevin process, and
second coordinate its derivative. We refer to Lachal [12] for a detailed
account about it. Further, suppose that the particle is constrained to stay in
$[0,+\infty[$ by a boundary at 0 characterized by an elasticity coefficient
$c\geq 0$. That is, the boundary restores a portion $c$ of the incoming
velocity at each bounce, and the equation of motion that we consider is the
following:
$(SOR)\qquad\left\\{\begin{array}[]{ccl}X_{t}&=&X_{0}+\displaystyle\int_{0}^{t}\dot{X}_{s}\mathrm{d}s\\\
\\\ \dot{X}_{t}&=&\dot{X}_{0}+B_{t}-(1+c)\sum_{0<s\leq
t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},\end{array}\right.$
where $B$ is a standard Brownian motion and $(X_{0},\dot{X}_{0})$ is called
the initial or starting condition. This stochastic partial differential
equation is nice outside the point $(0,0)$. Indeed, if the starting condition
is different from $(0,0)$, there is a simple pathwise construction of the
solution to this equation system, until time $\zeta_{\infty}$, the hitting
time of $(0,0)$ for the process $(X,\dot{X})$. However there is a tough
problem at $(0,0)$. Indeed, there exists an old literature about a
deterministic analogue to theses equations, where the white noise force is
replaced by a deterministic force. See Ballard [1] for a vast review. As early
as in 1960, Bressan [6] pointed out that multiple solutions may occur, even
when the force is $\mathcal{C}^{\infty}$. It appears that the introduction of
a white noise allows to get back a weak uniqueness result. We refer to [4]
(see also [3], [11]) for the particular case $c=0$.
In [10], we have shown for $c>0$ the existence of two different regimes, the
critical elasticity being $c_{crit}:=\exp(-\pi/\sqrt{3})$. It is critical in
the sense that when the starting condition is different from $(0,0)$, then we
have $\zeta_{\infty}=+\infty$ almost surely if $c\geq c_{crit}$, and
$\zeta_{\infty}<+\infty$ almost surely if $c<c_{crit}$. Further, we studied
the super-critical and the critical regimes. In this paper, we study the sub-
critical regime $c<c_{crit}$. The finite time $\zeta_{\infty}$ corresponds to
an accumulation of bounces in a finite time. We write $\mathbb{P}_{x,u}^{c}$
for the law of the reflected Kolmogorov process, with starting condition
$(x,u)\neq(0,0)$, elasticity coefficient $c$, and _killed at time
$\zeta_{\infty}$_. It is the unique strong solution to $(SOR)$ equations, up
to time $\zeta_{\infty}$. We also write $\mathrm{P}_{t}^{c}$ for the
associated semigroup. We will devote ourselves to prove the existence of a
unique recurrent extension to this process that leaves $(0,0)$ continuously.
Moreover, we will prove that this extension gives the unique solution, in the
weak sense, to $(SOR)$ equations.
We point out that this model was encountered by Bect in his thesis ([2],
section III.4.B). He observed the existence of the critical elasticity and
asked several questions on the different regimes. We answer to all of them.
In this work we will be largely inspired by a paper of Rivero [15], in which
he studies the recurrent extensions of a self-similar Markov process with
semigroup $\mathrm{P}_{t}$. Briefly, first, he recalls that recurrent
extensions are equivalent to excursion measures compatible with
$\mathrm{P}_{t}$, thanks to Itō’s program. Then a change of probability allows
him to define the Markov process conditioned on never hitting $0$, where this
conditioning is in the sense of Doob, via an $h-$transform. An inverse
$h-$transform on the Markov process conditioned on never hitting zero _and
starting from 0_ then gives the construction of the excursion measure.
We will not recall it at each step throughout the paper, but a lot of
parallels can be made. However, it is a two-dimensional Markov process that we
consider here. Further, its study will rely on an underlying random walk
$(S_{n})_{n\in\mathbb{N}}$ constructed from the velocities at bouncing times.
In the Preliminaries, we introduce this random walk and use it to estimate the
tail of the variable $\zeta_{\infty}$ under $\mathbb{P}_{0,1}^{c}$. In the
Section 3, we introduce a change of probability, via an $h-$transform, to
define $\widetilde{\mathbb{P}}_{x,u}$, law of a process which can be viewed as
the reflected Kolmogorov process conditioned on never being killed. We then
show in Subsection 3.2 that this law has a weak limit
$\widetilde{\mathbb{P}}_{0^{+}}$ when $(x,u)$ goes to $(0,0)$, using the same
method that was used in [10] to show that for $c>c_{crit}$, the laws
$\mathbb{P}_{0,u}^{c}$ have the weak limit $\mathbb{P}_{0+}^{c}$ when $u$ goes
to zero. All this section can be seen as a long digression to prepare the
construction of the excursion measure in Section 4. This excursion measure is
defined by a formula similar to Imhof’s relation (see [9]), connecting the
excursion measure of Brownian motion and the law of a Bessel(3) process. But
our formula involves the law $\mathbb{P}_{0+}^{c}$ and determines the unique
excursion measure compatible with the semigroup $\mathrm{P}_{t}^{c}$ . We call
_resurrected Kolmogorov process_ the corresponding recurrent extension.
Finally, we prove that this is the (weakly) unique solution to $(SOR)$
equations when the starting condition is $(0,0)$.
## 2 Preliminaries
We largely use the same notations as in [10]. For the sake of simplicity, we
use the same notation (say $P$) for a probability measure and for the
expectation under this measure. We will even authorize ourselves to write
$P(f,A)$ for the quantity $P(f\mathbbm{1}_{A})$, when $f$ is a measurable
functional and $A$ an event. We introduce
$D=(\\{0\\}\times\mathbb{R}_{+}^{*})\cup(\mathbb{R}_{+}^{*}\times\mathbb{R})$
and $D^{0}:=D\cup\\{(0,0)\\}$. Our working space is $\mathcal{C}$, the space
of càdlàg trajectories $(x,\dot{x}):[0,\infty)\to D^{0}$, which satisfy
$x(t)=x(0)+\displaystyle\int_{0}^{t}\dot{x}(s)\mathrm{d}s.$
That space is endowed with the $\sigma-$algebra generated by the coordinate
maps and with the topology induced by the following injection:
$\begin{array}[]{ccc}\mathcal{C}&\to&\mathbb{R}_{+}\times\mathbb{D}\\\
(x,\dot{x})&\mapsto&\big{(}x(0),\dot{x}\big{)},\end{array}$
where $\mathbb{D}$ is the space of càdlàg trajectories on $\mathbb{R}_{+}$,
equipped with Skorohod topology.
We denote by $(X,\dot{X})$ the canonical process and by
$(\mathfrak{F}_{t},t\geq 0)$ its natural filtration, satisfying the usual
conditions of right continuity and completeness. For an initial condition
$(x,u)\in D$, the $(SOR)$ equations
$\left\\{\begin{array}[]{ccl}X_{t}&=&x+\displaystyle\int_{0}^{t}\dot{X}_{s}\mathrm{d}s\\\
\\\ \dot{X}_{t}&=&u+B_{t}-(1+c)\sum_{0<s\leq
t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}\end{array}\right.$
have a unique solution, at least up to the random time
$\zeta_{\infty}:=\inf\\{t>0,X_{t}=0,\dot{X}_{t}=0\\}.$
We call (killed) reflected Kolmogorov process this solution killed at time
$\zeta_{\infty}$, and write $\mathbb{P}_{x,u}^{c}$ for its law. It is Markov.
We also call reflected Langevin process the first coordinate of this process,
which is no longer Markov.
Call $\zeta_{1}$ the first hitting time of zero for the reflected Langevin
process $X$, that is $\zeta_{1}:=\inf\\{t>0,X_{t}=0\\}$. More generally, the
sequence of the successive hitting times of zero $(\zeta_{n})_{n\geq 1}$ is
defined recursively by $\zeta_{n+1}:=\inf\\{t>\zeta_{n},X_{t}=0\\}$. We write
$(V_{n})_{n\geq 1}:=(\dot{X}_{\zeta_{n}})_{n\geq 1}$ for the sequence of the
velocities of the process at these hitting times. That means outgoing
velocities, as we are dealing with right-continuous processes. Finally, when
the starting position is $x=0$, we will simply write $\mathbb{P}^{c}_{u}$ for
$\mathbb{P}^{c}_{0,u}$, and we will also define $\zeta_{0}=0$ and
$V_{0}=\dot{X}_{0}$. We insist on the fact that in each case the starting
condition $(x,u)$ is different from $(0,0)$. Then it is not difficult to see
that $\zeta_{\infty}$ coincides almost surely with $\sup\zeta_{n}$. But we can
say much more.
The sequence
$\left(\dfrac{\zeta_{n+1}-\zeta_{n}}{V_{n}^{2}},\dfrac{V_{n+1}}{V_{n}}\right)_{n\geq
0}$ is i.i.d. and of law independent of $u$, which can be deduced from the
following density:
$\frac{1}{\mathrm{d}s\mathrm{d}v}\mathbb{P}_{1}^{c}\left(({\zeta_{1}},{V_{1}}/c)\in(\mathrm{d}s,\mathrm{d}v)\right)=\frac{3v}{\pi\sqrt{2}s^{2}}\exp(-2\frac{v^{2}-v+1}{s})\int_{0}^{4v/s}e^{-\frac{3\theta}{2}}\frac{\mathrm{d}\theta}{\sqrt{\pi\theta}},$
(2.1)
given by McKean [13]. The second marginal of this density is
$\mathbb{P}_{1}^{c}({V_{1}}/c\in\mathrm{d}v)=\frac{3}{2\pi}\frac{v^{\frac{3}{2}}}{1+v^{3}}\mathrm{d}v.$
(2.2)
In particular, the sequence $S_{n}:=\ln(V_{n})$ is a random walk, with drift
$\mathbb{P}_{1}^{c}(S_{1}-S_{0})=\ln(c)+\frac{\pi}{\sqrt{3}},$
which is zero for the critical value $c_{crit}=\exp(-\pi/\sqrt{3})$. In this
paper we lie in the subcritical case $c<c_{crit}$, when the drift is negative.
A thorough study allows to not only deduce the finiteness of $\zeta_{\infty}$,
but also estimate its tail.
###### Lemma 1.
We have
$\mathbb{P}_{1}^{c}\left(V_{1}^{x}\right)=\frac{c^{x}}{2\cos(\frac{x+1}{3}\pi)}\textrm{
for }x<1/2.$ (2.3)
There exists a unique $k=k(c)$ in $(0,1/4)$ such that
$\mathbb{P}_{1}^{c}\left(V_{1}^{2k}\right)=1$, and
$\mathbb{P}_{1}^{c}(\zeta_{\infty}>t)\underset{t\to\infty}{\sim}C_{1}t^{-k},$
(2.4)
where $C_{1}=C_{1}(c)\in(0,\infty)$ is a constant depending only on $c$, given
by
$C_{1}=\frac{\mathbb{P}_{1}^{c}\left(\zeta_{\infty}^{k}-(\zeta_{\infty}-\zeta_{1})^{k}\right)}{k\mathbb{P}_{1}^{c}(V_{1}^{2k}\ln(V_{1}^{2}))}.$
(2.5)
In other words, $k(c)$ is given implicitly as the unique solution in
$]0,\frac{1}{4}]$ of the equation
$c=\left[2\cos\left(\frac{2k+1}{3}\pi\right)\right]^{\frac{1}{2k}}.$ (2.6)
The upper bound $1/4$ stems from the fact that
$\mathbb{P}_{1}^{c}\left(V_{1}^{2k}\right)$ becomes infinite for $k=1/4$. The
value of $k(c)$ converges to $1/4$ when $c$ goes to 0, and to $0$ when $c$
goes to $c_{crit}$, as illustrated by Figure 1. We may notice that Formula
(2.4) remains true for $c=0$ and $k=1/4$ (and for $c=c_{crit}$ and $k=0$, in a
certain sense).
Figure 1: Graph of the exponent k(c)
###### Proof.
Formula (2.3) is not new. For the convenience of the reader, we still provide
the following calculation. From Formula (2.2), it follows, for $x<1/2$,
$\displaystyle\mathbb{P}_{1}^{c}\left(\left({V_{1}}/c\right)^{x}\right)$
$\displaystyle=$
$\displaystyle\frac{3}{2\pi}\int_{0}^{\infty}\frac{t^{x+3/2}}{1+t^{3}}\mathrm{d}t=\frac{1}{2\pi}\int_{0}^{\infty}\frac{t^{\frac{x}{3}-\frac{1}{6}}}{1+t}\mathrm{d}t.$
Note $\cos(\frac{x+1}{3}\pi)=\sin(\pi y)$, where $y=\frac{x}{3}+\frac{5}{6}$.
Using the variable $y$, which belongs to $(0,1)$, Equation (2.3) becomes
$\int_{0}^{\infty}\frac{t^{y-1}}{1+t}\mathrm{d}t=\frac{\pi}{\sin(\pi y)},$
and follows from:
$\displaystyle\int_{0}^{\infty}\frac{t^{y-1}}{1+t}\mathrm{d}t$
$\displaystyle=$ $\displaystyle\int_{0}^{1}t^{y}(1-t)^{1-y}\mathrm{d}t$
$\displaystyle=$ $\displaystyle\mathrm{B}(y,1-y)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(y)\Gamma(1-y)}{\Gamma(1)}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\sin(\pi y)}.$
where $\mathrm{B}$ and $\Gamma$ are the usual Beta and Gamma function,
respectively.
Now, the function $x\mapsto\mathbb{P}_{1}^{c}\left(V_{1}^{x}\right)$ is
convex, takes value 1 at $x=0$ and becomes infinite at $x=1/2$. Its derivative
at 0 is equal to $\mathbb{P}_{1}^{c}(S_{1}-S_{0})<0.$ We deduce that there is
indeed a unique $k(c)$ in $(0,\frac{1}{4})$ such that
$\mathbb{P}_{1}^{c}\left(V_{1}^{2k}\right)=1.$
Estimate (2.4) will appear as a particular case of an “implicit renewal
theory” result of Goldie [7]. Let us express $\zeta_{\infty}$ as the series:
$\displaystyle\zeta_{\infty}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{\zeta_{n}-\zeta_{n-1}}{V_{n-1}^{2}}V_{n-1}^{2},$
with
$V_{n}^{2}:=V_{1}^{2}\frac{V_{2}^{2}}{V_{1}^{2}}\cdot\cdot\cdot\frac{V_{n}^{2}}{V_{n-1}^{2}}$,
and where
$\left(\dfrac{\zeta_{n}-\zeta_{n-1}}{V_{n-1}^{2}},\dfrac{V_{n}^{2}}{V_{n-1}^{2}}\right)_{n\geq
1}$ is i.i.d. We lie in the setting of Section 4 of Goldie’s paper [7], and
can apply its Theorem (4.1). Indeed, all the following conditions are
satisfied:
$\mathbb{P}_{1}^{c}(V_{1}^{2k})=1,$
$\mathbb{P}_{1}^{c}(V_{1}^{2k}\ln(V_{1}^{2}))<\infty,$
$\mathbb{P}_{1}^{c}(\zeta_{1}^{k})<\infty,$
the last one being a consequence of the inequality $k<1/4$ and of the
following estimate of the queue of the variable $\zeta_{1}$,
$\mathbb{P}_{1}^{c}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{-\frac{1}{4}},$
(2.7)
which was already pointed out in Lemma 1 in [10]. All this is enough to apply
the theorem of Goldie and deduce the requested result, namely
$\mathbb{P}_{1}^{c}(\zeta_{\infty}>t)\underset{t\to\infty}{\sim}C_{1}t^{-k},$
where $C_{1}$ is the constant defined by (2.5), and belongs to $]0,\infty[$. ∎
Next section is devoted to the definition and study of the reflected
Kolmogorov process, _conditioned on never hitting $(0,0)$_. This process will
be of great use for studying the recurrent extensions of the reflected
Kolmogorov process in Section 4.
## 3 The reflected Kolmogorov process conditioned on never hitting $(0,0)$
### 3.1 Definition via an $h-$transform
Recall that under $\mathbb{P}_{1}^{c}$, the sequence $(S_{n})_{n\geq
0}=(\ln(V_{n}))_{n\geq 0}$ is a random walk starting from 0, and write
$\mathbf{P}_{0}$ for its law. The important fact
$\mathbb{P}_{1}^{c}(V_{1}^{2k})=1$ implies $\mathbb{P}_{1}^{c}(V_{n}^{2k})=1$
for any $n>0$, and can be rewritten $\mathbf{P}_{0}(\theta^{S_{n}})=1$, with
$\theta:=\exp(2k)$.
The sequence $\theta^{S_{n}}$ being a martingale, we introduce the change of
probability
$\widetilde{\mathbf{P}}_{0}(S_{n}\in\mathrm{d}t)=\theta^{t}\mathbf{P}_{0}(S_{n}\in\mathrm{d}t).$
Under $\widetilde{\mathbf{P}}_{0}$, $(S_{n})_{n\geq 0}$ becomes a random walk
drifting to $+\infty$. Informally, it can be viewed as being the law of the
random walk $S_{n}$ under $\mathbf{P}_{0}$ conditioned on hitting arbitrary
high levels.
There is a corresponding change of probability for the reflected Kolmogorov
process and its law $\mathbb{P}_{1}^{c}$. We introduce the law
$\widetilde{\mathbb{P}}_{1}$ determined by
$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{n}>T})=\mathbb{P}^{c}_{1}(A\mathbbm{1}_{\zeta_{n}>T}\mathbb{P}_{1}^{c}(V_{n}^{2k}|\mathfrak{F}_{T})),$
for any $n>0$, stopping-time $T$ and $A\in\mathfrak{F}_{T}$. By the strong
Markov property we have
$\mathbb{P}_{1}^{c}(V_{n}^{2k}|\mathfrak{F}_{T})=\mathbb{P}_{X_{T},\dot{X}_{T}}^{c}(V_{1}^{2k})\qquad\text{on
the event }\\{\zeta_{n}>T\\},$
so that there is the identity
$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{n}>T})=\mathbb{P}^{c}_{1}(A\mathbbm{1}_{\zeta_{n}>T}H(X_{T},\dot{X}_{T})),$
where we have written
$H(x,u):=\mathbb{P}_{x,u}^{c}(V_{1}^{2k}).$
Note that $H(0,u)=u^{2k}$. Letting $n$ go to infinity, we get:
$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{\infty}>T})=\mathbb{P}^{c}_{1}(A\mathbbm{1}_{\zeta_{\infty}>T}H(X_{T},\dot{X}_{T})).$
We have $H(0,1)=1$, the function $H$ is harmonic for the semigroup of the
reflected Kolmogorov process, and the process $\widetilde{\mathbb{P}}_{1}$ is
the $h-$transform of $\mathbb{P}_{1}^{c}$, in the sense of Doob.
Under $\widetilde{\mathbb{P}}_{1}$, the law of the sequence $(S_{n})_{n\geq
0}$ is $\widetilde{\mathbf{P}}_{0}$, thus this sequence is diverging to
$+\infty$, and as a consequence the time $\zeta_{\infty}$ is infinite
$\widetilde{\mathbb{P}}_{1}-$almost surely. The term
$\mathbbm{1}_{\zeta_{\infty}>T}$ in
$\widetilde{\mathbb{P}}_{1}(A\mathbbm{1}_{\zeta_{\infty}>T})$ is thus
unnecessary. We may now give a more general definition of this change of
probability, as an $h-$transform, for any starting position $(x,u)$.
###### Definition 1.
The reflected Kolmogorov process _conditioned on never hitting $(0,0)$_ is the
Markov process given by its law $\widetilde{\mathbb{P}}_{x,u}$, for any
starting condition $(x,u)\in D$, which is the unique measure such that for
every stopping-time $T$ we have
$\widetilde{\mathbb{P}}_{x,u}(A)=\frac{1}{H(x,u)}\mathbb{P}_{x,u}^{c}(AH(X_{T},\dot{X}_{T}),T<\zeta_{\infty}),$
(3.1)
for any $A\in\mathfrak{F}_{T}$. We write $\widetilde{\mathrm{P}}_{t}$ its
associated semigroup, and we also write $\widetilde{\mathbb{P}}_{u}$ for
$\widetilde{\mathbb{P}}_{0,u}$.
This denomination is justified by the following proposition.
###### Proposition 1.
For any $(x,u)\in D$ and $t>0$, we have
$\widetilde{\mathbb{P}}_{x,u}(A)=\lim_{s\to\infty}\mathbb{P}^{c}_{x,u}(A|\zeta_{\infty}>s),$
(3.2)
for any $A\in\mathfrak{F}_{t}$.
We stress that in [15], Proposition 2, Rivero defines in a similar way the
self-similar Markov process conditioned on never hitting 0. Incidentally, you
can find in [11] a thorough study of other $h-$transforms regarding the
Kolmogorov process killed at time $\zeta_{1}$.
In order to get Formula (3.2), we first prove the following lemma, which is a
slight improvement of (2.4):
###### Lemma 2.
For any $(x,u)\in D$,
$s^{k}\mathbb{P}^{c}_{x,u}(\zeta_{\infty}>s)\underset{s\to\infty}{\longrightarrow}H(x,u)C_{1}.$
(3.3)
###### Proof.
For $(x,u)=(0,1)$, this is (2.4). For $x=0$, the rescaling invariance property
yields immediately
$s^{k}\mathbb{P}^{c}_{0,u}(\zeta_{\infty}>s)=s^{k}\mathbb{P}^{c}_{0,1}(\zeta_{\infty}>su^{-2})\underset{s\to\infty}{\longrightarrow}u^{2k}C_{1}=H(0,u)C_{1}.$
For $(x,u)\in D$, the Markov property at time $\zeta_{1}$ yields
$\displaystyle s^{k}\mathbb{P}^{c}_{x,u}(\zeta_{\infty}>s)$ $\displaystyle=$
$\displaystyle\mathbb{P}^{c}_{x,u}(s^{k}\mathbb{P}^{c}_{0,V_{1}}(\zeta_{\infty}>s-\zeta_{1}))$
$\displaystyle\underset{s\to\infty}{\longrightarrow}$
$\displaystyle\mathbb{P}^{c}_{x,u}(H(0,V_{1})C_{1})=H(x,u)C_{1},$
where the convergence holds by dominated convergence. The lemma is proved. ∎
Formula (3.2) then results from:
$\displaystyle\mathbb{P}^{c}_{x,u}(A|\zeta_{\infty}>s)$ $\displaystyle=$
$\displaystyle\frac{1}{\mathbb{P}^{c}_{x,u}(\zeta_{\infty}>s)}\mathbb{P}^{c}_{x,u}\left(A\mathbb{P}^{c}_{X_{t},\dot{X}_{t}}(\zeta_{\infty}>s-t),\zeta_{\infty}>t\right)$
$\displaystyle\underset{s\to\infty}{\longrightarrow}$
$\displaystyle\frac{1}{H(x,u)}\mathbb{P}^{c}_{x,u}\left(AH(X_{T},\dot{X}_{T}),\zeta_{\infty}>t\right)$
$\displaystyle=$ $\displaystyle\ \widetilde{\mathbb{P}}_{x,u}(A).$
### 3.2 Starting the conditioned process from $(0,0)$
The study of the reflected Kolmogorov process conditioned on never hitting
$(0,0)$ will happen to be very similar to that of the reflected Kolmogorov
process in the supercritical case $c>c_{crit}$, done in [10]. Observe the
following similarities between the laws $\widetilde{\mathbb{P}}_{u}$, and
$\mathbb{P}^{c}_{u}$ when $c>c_{crit}$: the sequence
$\left(\dfrac{\zeta_{n+1}-\zeta_{n}}{V_{n}^{2}},\dfrac{V_{n+1}}{V_{n}}\right)_{n\geq
0}$ is i.i.d., we know its law explicitly, and the sequence $S_{n}=\ln(V_{n})$
is a random walk with positive drift. It follows that a major part of [10] can
be transcribed _mutatis mutandis_. In particular we will get a convergence
result for the probabilities $\widetilde{\mathbb{P}}_{u}$ when $u$ goes to
zero, similar to Theorem 1 of [10].
Under $\widetilde{\mathbb{P}}_{1}$, the sequence $(S_{n})_{n\geq 0}$ is a
random walk of law $\widetilde{\mathbf{P}}_{0}$. Write $\mu$ for its drift,
that is the expectation of its jump distribution, which is positive and
finite. The associated strictly ascending ladder height process
$(H_{n})_{n\geq 0}$, defined by $H_{k}=S_{n_{k}}$, where $n_{0}=0$ and
$n_{k}=\inf\\{n>n_{k-1},S_{n}>S_{n_{k-1}}\\}$, is a random walk with positive
jumps. Its jump distribution also has positive and finite expectation
$\mu_{H}\geq\mu$. The measure
$m(\mathrm{d}y):=\frac{1}{\mu_{H}}\widetilde{\mathbf{P}}_{0}(H_{1}>y)\mathrm{d}y.$
(3.4)
is the “stationary law of the overshoot”, both for the random walks
$(S_{n})_{n\geq 0}$ and $(H_{n})_{n\geq 0}$. The following proposition holds.
###### Proposition 2.
The family of probability measures $(\widetilde{\mathbb{P}}_{x,u})_{(x,u)\in
D}$ on $\mathcal{C}$ has a weak limit when $(x,u)\to(0,0)$, which we denote by
$\widetilde{\mathbb{P}}_{0^{+}}$. More precisely, write $\tau_{v}$ for the
instant of the first bounce with speed greater than $v$, that is
$\tau_{v}:=\inf\\{t>0,X_{t}=0,\dot{X}_{t}>v\\}.$ Then the law
$\widetilde{\mathbb{P}}_{0^{+}}$ satisfies the following properties:
$\begin{array}[]{ll}(*)&\left\\{\begin{array}[]{l}\displaystyle\lim_{v\to
0^{+}}\tau_{v}=0\quad\text{almost surely}.\\\ \text{For any }u,v>0\text{, and
conditionally on }\dot{X}_{\tau_{v}}=u\text{, the process }\\\
(X_{\tau_{v}+t},\dot{X}_{\tau_{v}+t})_{t\geq 0}\text{ is independent of
}(X_{s},\dot{X}_{s})_{s<\tau_{v}}\text{ and has law
}\widetilde{\mathbb{P}}_{u}.\end{array}\right.\\\ \\\ (**)&\text{For any
}v>0,\text{ the law of }\ln(\dot{X}_{\tau_{v}}/v)\text{ is }m.\end{array}$
In the proof of this proposition we can take $x=0$ and just prove the
convergence result for the laws $\widetilde{\mathbb{P}}_{u}$ when $u\to 0+$.
The general result will follow as an application of the Markov property at
time $\zeta_{1}$.
The complete proof follows mainly the proof of Theorem 1 in [10] and takes
many pages. Here, the reader has three choices. Skip this proof and go
directly to next section about the resurrected process. Or read the following
for an overview of the ideas of the proof, with details given only when
significantly different from that in [10]. Or, read [10] and the following, if
(s)he wants to get the complete proof.
Call $T_{y}(S)$ the hitting time of $(y,\infty)$ for the random walk $S$
starting from $x<y$. Call $\widetilde{\mathbf{P}}_{\mu}$ the law of
$(S_{n})_{n\geq 0}$ obtained by taking $S_{0}$ and $(S_{n}-S_{0})_{n\geq 0}$
independent, with law $m$ and $\widetilde{\mathbf{P}}_{0}$, respectively. That
is, we allow the starting position to be nonconstant and distributed according
to $\mu$. A result of renewal theory states that the law of the overshoot
$(S_{n+T_{y}}-y)_{n\geq 0}$ under $\widetilde{\mathbf{P}}_{x}$, when $x$ goes
to $-\infty$, converges to $\widetilde{\mathbf{P}}_{m}$. Now, for a process
indexed by $I$ an interval of $\mathbb{Z}$, we define a spatial translation
operator by $\Theta^{sp}_{y}((S_{n})_{n\in
I})=(S_{n+T_{y}}-y)_{n\in{I-T_{y}}}$. We get that under
$\widetilde{\mathbf{P}}_{x}$ and when $x$ goes to $-\infty$, the translated
process $\Theta^{sp}_{y}(S)$ converges to a process called the “spatially
stationary random walk", a process indexed by $\mathbb{Z}$ which is spatially
stationary and whose restriction to $\mathbb{N}$ is
$\widetilde{\mathbf{P}}_{m}$ (see [10]). We write $\widetilde{\mathbf{P}}$ for
the law of this spatially stationary random walk.
There exists a link between the law $\widetilde{\mathbf{P}}_{x}$ and the law
$\widetilde{\mathbb{P}}_{e^{x}}$: the first one is the law of the underlying
random walk $(S_{n})_{n\geq 0}=(\ln V_{n})_{n\geq 0}$ for a process
$(X,\dot{X})$ following the second one. Now, in a very brief shortcut, we can
say that the law $\widetilde{\mathbf{P}}$ is linked to a law written
$\widetilde{\mathbb{P}}_{0^{+}}^{*}$. And the convergence results of
$\widetilde{\mathbf{P}}_{x}\circ\Theta^{sp}_{y}$ to $\mathbf{P}$ when
$x\to-\infty$ provide convergence results of $\widetilde{\mathbb{P}}_{u}$ to
$\widetilde{\mathbb{P}}_{0^{+}}^{*}$ when $u\to 0$.
However, this link is different, as the spatially stationary random walk, of
law $\widetilde{\mathbf{P}}$, is a process indexed by $\mathbb{Z}$. The value
$S_{0}$ is thus not equal to the logarithm of the velocity of the process at
time 0, but at time $\tau_{1}$ (recall that
$\tau_{1}=\inf\\{t>0,X_{t}=0,\dot{X}_{t}\geq 1\\}$ is the instant of the first
bounce with speed no less than one). The sequence $(S_{n})_{n\geq 0}$ is then
the sequence of the logarithms of the velocities of the process at the
bouncing times, starting from that bounce. The sequence $(S_{-n})_{n\geq 0}$
is the sequence of the logarithms of the velocities of the process at the
bouncing times happening _before_ that bounce.
The law $\widetilde{\mathbb{P}}_{0^{+}}^{*}$ is the law of a process indexed
by $\mathbb{R}_{+}^{*}$, but we actually construct it “from the random time
$\tau_{1}$”. In order for the definition to be clean, we have to prove that
the random time $\tau_{1}$ is finite a.s. In [10], we used the fact that if
$(\zeta_{1,k})_{k\geq 0}$ is a sequence of i.i.d random variables, with common
law that of $\zeta_{1}$ under $\mathbb{P}^{c}_{1}$, then for any
$\varepsilon>0$ there is almost surely only a finite number of indexes $k$
such that $\ln(\zeta_{1,k})\geq\varepsilon k.$ This was based on Formula 2.7,
which, we recall, states
$\mathbb{P}_{1}^{c}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{-\frac{1}{4}},$
where $c^{\prime}$ is some positive constant. Here the same results holds with
replacing $\mathbb{P}^{c}_{1}$ by $\widetilde{\mathbb{P}}_{1}$ and is a
consequence from the following lemma.
###### Lemma 3.
We have
$\widetilde{\mathbb{P}}_{1}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{k-\frac{1}{4}},$
(3.5)
where $c^{\prime}$ is some positive constant.
###### Proof.
From (3.1) and (2.1), we get that the density of $(\zeta_{1},V_{1}/c)$ under
$\widetilde{\mathbb{P}}_{1}$ is given by
$\displaystyle
f(s,v):=\frac{1}{\mathrm{d}s\mathrm{d}v}\widetilde{\mathbb{P}}_{1}((\zeta_{1},V_{1}/c)\in\mathrm{d}s\mathrm{d}v)$
$\displaystyle=$
$\displaystyle(cv)^{2k}\frac{3v}{\pi\sqrt{2}s^{2}}\exp(-2\frac{v^{2}-v+1}{s})\int_{0}^{\frac{4v}{s}}e^{-\frac{3\theta}{2}}\frac{\mathrm{d}\theta}{\sqrt{\pi\theta}}.$
Thanks to the inequality
$4\sqrt{\frac{v}{s\pi}}e^{-\frac{6v}{s}}\leq\int_{0}^{\frac{4v}{s}}e^{-\frac{3\theta}{2}}\frac{\mathrm{d}\theta}{\sqrt{\pi\theta}}\leq
4\sqrt{\frac{v}{s\pi}},$
we may write
$\displaystyle f(s,v)$ $\displaystyle=$
$\displaystyle(6\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k})s^{-\frac{5}{2}}v^{\frac{3}{2}+2k}e^{-2\frac{v^{2}}{s}+\frac{v}{s}K(s,v)},$
where $(s,v)\mapsto K(s,v)$ is continuous and bounded. The marginal density of
$\zeta_{1}$ is thus given by
$\displaystyle\frac{1}{\mathrm{d}s}\widetilde{\mathbb{P}}_{1}(\zeta_{1}\in\mathrm{d}s)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}_{+}}f(s,v)\mathrm{d}v$
$\displaystyle=$
$\displaystyle(3\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k})s^{-\frac{5}{4}+k}\int_{\mathbb{R}_{+}}w^{\frac{1}{4}+k}e^{-2w+K(s,\sqrt{sw})\sqrt{w/s}}\mathrm{d}w$
$\displaystyle\underset{s\to\infty}{\sim}$
$\displaystyle(3\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k})s^{-\frac{5}{4}+k}\int_{\mathbb{R}_{+}}w^{\frac{1}{4}+k}e^{-2w}\mathrm{d}w,$
where we used successively the change of variables $w=v^{2}/s$ and dominated
convergence theorem. Just integrate this equivalence in the neighborhood of
$+\infty$ to get
$\widetilde{\mathbb{P}}_{1}(\zeta_{1}>t)\underset{t\to\infty}{\sim}c^{\prime}t^{k-\frac{1}{4}},$
with the constant
$c^{\prime}=\frac{3\sqrt{2}.\pi^{-\frac{3}{2}}c^{2k}}{\frac{1}{4}-k}\int_{\mathbb{R}_{+}}w^{\frac{1}{4}+k}e^{-2w}\mathrm{d}w=\frac{3c^{2k}}{\pi^{\frac{3}{2}}2^{\frac{3}{4}+k}}\cdot\frac{1+4k}{1-4k}\
\Gamma\left(\frac{1}{4}+k\right).$
∎
For now, we have introduced $\widetilde{\mathbb{P}}_{0^{+}}^{*}$, law of a
process $(X,\dot{X})$ indexed by $\mathbb{R}_{+}^{*}$. We keep on following
the proof of [10]. First, we get that this law satisfies conditions $(*)$ and
$(**)$, and that for any $v>0$, the joint law of $\tau_{v}$ and
$(X_{\tau_{v}+t},\dot{X}_{\tau_{v}+t})_{t\geq 0}$ under
$\widetilde{\mathbb{P}}_{u}$ converges to that under
$\widetilde{\mathbb{P}}_{0^{+}}^{*}$. Then we establish Proposition 2 by
controlling the behavior of the process just after time 0, through the two
following lemmas:
###### Lemma 4.
Under $\widetilde{\mathbb{P}}_{0^{+}}^{*}$, we have almost surely
$(X_{t},\dot{X}_{t})\underset{t\to 0}{\longrightarrow}(0,0).$
This lemma allows in particular to extend $\widetilde{\mathbb{P}}_{0^{+}}^{*}$
to $\mathbb{R}_{+}$. We call $\widetilde{\mathbb{P}}_{0^{+}}$ this extension.
The second lemma is more technical and controls the behavior of the process on
$[0,\tau_{v}[$ under $\widetilde{\mathbb{P}}_{u}$.
###### Lemma 5.
Write $M_{v}=\sup\\{|\dot{X}_{t}|,t\in[0,\tau_{v}[\\}$. Then,
$\forall\varepsilon>0,\forall\delta>0,\exists v_{0}>0,\exists u_{0}>0,\forall
0<u\leq
u_{0},\quad\widetilde{\mathbb{P}}_{u}(M_{v_{0}}\geq\delta)\leq\varepsilon,$
(3.6)
In [10], we proved these two results by using the stochastic partial
differential equation satisfied by the laws $\mathbb{P}^{c}$. They are of
course not available for the laws $\widetilde{\mathbb{P}}$, and we need a new
proof. We start by showing a rather simple but really useful inequality:
###### Lemma 6.
The following inequality holds for any $(x,u)\in D$,
$\widetilde{\mathbb{P}}_{x,u}\left(V_{1}/c\geq\frac{|u|}{2}\right)\geq
1-\frac{\sqrt{3}}{\pi}.$ (3.7)
For us, the important fact is that the probability is bounded below by a
positive constant, uniformly in $x$ and $u$. The constant $1-\sqrt{3}/\pi$ is
not intended to be the optimal one. Note that this inequality will also be
used again later on in this paper.
###### Proof of Lemma 6.
For $u=0$, there is nothing to prove. By a scaling invariance property we may
suppose $u\in\\{-1,1\\}$, what we do.
The density $f_{x,u}$ of $V_{1}/c$ under $\mathbb{P}_{x,u}^{c}$ is given in
Gor’kov [8]. If you write $p_{t}(x,u;y,v)$ for the transition densities of the
(free) Kolmogorov process, given by
$p_{t}(x,u;y,v)=\frac{\sqrt{3}}{\pi t^{2}}\exp\Big{[}-\frac{6}{t^{3}}(y-x-
tu)^{2}+\frac{6}{t^{2}}(y-x-tu)(v-u)-\frac{2}{t}(v-u)^{2}\Big{]},$
and $\Phi(x,u;y,v)$ for its total occupation time densities, defined by
$\Phi(x,u;y,v):=\int_{0}^{\infty}p_{t}(x,u;y,v)dt,$
then the density $f_{x,u}$ is given by
$\displaystyle f_{x,u}(v)$ $\displaystyle=$ $\displaystyle
v\Big{[}\Phi(x,u;0,-v)-\frac{3}{2\pi}\int_{0}^{\infty}\frac{\mu^{\frac{3}{2}}}{\mu^{3}+1}\Phi(x,u;0,\mu
v)d\mu\Big{]}.\ \ \ \ \ \ \ \ $ (3.8)
Now, knowing the density of $V_{1}$ under $\mathbb{P}_{x,u}^{c}$, we get that
of $V_{1}$ under $\widetilde{\mathbb{P}}_{x,u}$ by multiplying it by the
increasing function $v\mapsto v^{2k}$. This necessarily increases the
probability of being greater than $c/2$. Consequently, it is enough to prove
$\mathbb{P}_{x,u}^{c}(V_{1}/c\geq\frac{1}{2})\geq K^{\prime}$
as soon as $u\in\\{-1,1\\}$. But very rough bounds give
$\displaystyle f_{x,u}(v)$ $\displaystyle\leq$ $\displaystyle v\Phi(x,u;0,-v)$
$\displaystyle\leq$ $\displaystyle v\int_{0}^{\infty}\frac{\sqrt{3}}{\pi
t^{2}}\exp(-\frac{(u+v)^{2}}{2t})\mathrm{d}t.$
For $u\in\\{-1,1\\}$ and $v\in[0,1/2]$ we have $|u+v|\geq 1/2$ and thus
$f_{x,u}(v)\leq\frac{v\sqrt{3}}{\pi}\int_{0}^{\infty}\frac{1}{t^{2}}\exp(-\frac{1}{8t})\mathrm{d}t=\frac{8\sqrt{3}}{\pi}v.$
Consequently,
$\mathbb{P}_{x,u}(V_{1}/c\geq\frac{1}{2})\geq
1-\int_{0}^{1/2}\frac{8\sqrt{3}}{\pi}v\mathrm{d}v=1-\frac{\sqrt{3}}{\pi}>0.$
∎
###### Proof of Lemma 4.
First, observe that conditions $(*)$ and $(**)$ imply that the variables
$\tau_{v}=\inf\\{t>0,X_{t}=0,\dot{X}_{t}>v\\}$ and
$\tau_{v}^{-}:=\sup\\{t<\tau_{v},X_{t}=0\\}$ are almost surely strictly
positive and go to zero when $v$ goes to zero. Then, observe that is is enough
to show the almost sure convergence of $\dot{X}_{t}$ to 0 when $t\to 0$, and
suppose on the contrary that this does not hold.
Then, there would exist a positive $x$ such that
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{x}=0)>0$, where we have written
$T_{x}:=\inf\\{t>0,|\dot{X}_{t}|>x\\}.$ By self-similarity this would be true
for any $x>0$ and in particular we would have
$K:=\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}=0)>0.$ (3.9)
Informally, this, together with (3.7), should induce that $\tau_{c/2}^{-}$
takes the value zero with probability at least $(1-\sqrt{3}/\pi)K$, and give
the desired contradiction. However it is not straightforward, because we
cannot use a Markov property at time $T_{1}$, which can take value 0, while
the process is still not defined at time 0. Consider the stopping time
$T_{1}^{\varepsilon}:=\inf\\{t>\varepsilon,|\dot{X}_{t}|>x\\}$. For any
$\eta>0$, we have
$\liminf_{\varepsilon\to
0}\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}^{\varepsilon}<\eta)\geq\widetilde{\mathbb{P}}_{0^{+}}^{*}(\liminf_{\varepsilon\to
0}\\{T_{1}^{\varepsilon}<\eta\\})\geq\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}<\eta)\geq
K,$
and in particular there is some $\varepsilon_{0}(\eta)$ such that for any
$\varepsilon<\varepsilon_{0}(\eta)$,
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(T_{1}^{\varepsilon}<\eta)\geq\frac{K}{2}.$
(3.10)
Now, write $\theta$ for the translation operator defined by
$\theta_{x}((X_{t})_{t\geq 0})=(X_{x+t})_{t\geq 0}$, so that
$V_{1}\circ\theta_{T_{1}^{\varepsilon}}$ denotes the velocity of the process
at its first bounce after time $T_{1}^{\varepsilon}$. From (3.10) and Lemma 6,
a Markov property gives, for $\varepsilon<\varepsilon_{0}(\eta)$,
$\widetilde{\mathbb{P}}_{0^{+}}^{*}\left(T_{1}^{\varepsilon}<\eta,V_{1}\circ\theta_{T_{1}^{\varepsilon}}\geq\frac{c}{2}\right)\geq
K^{\prime}:=\Bigg{(}1-\frac{\sqrt{3}}{\pi}\Bigg{)}\frac{K}{2}.$
We have _a fortiori_
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(\tau_{c/2}^{-}\leq\eta)\geq K^{\prime}.$
This result true for any $\eta>0$ leads to
$\widetilde{\mathbb{P}}_{0^{+}}^{*}(\tau_{c/2}^{-}=0)\geq K^{\prime}>0$, and
we get a contradiction. This shows $(X_{t},\dot{X}_{t})\underset{t\to
0}{\longrightarrow}(0,0)$ under $\widetilde{\mathbb{P}}_{0^{+}}^{*}$, as
requested. ∎
###### Proof of Lemma 5.
We should prove (3.6). Fix $\varepsilon,\delta>0$. The event
$\\{M_{v}\geq\delta\\}$ coincides with the event $T_{\delta}\leq\tau_{v}$.
From a Markov property at time $T_{\delta}$ and (3.7), we get, for any
$v<c\delta/2$, and any $u$,
$(1-\sqrt{3}/\pi)\widetilde{\mathbb{P}}_{u}(M_{v}\geq\delta)\leq\widetilde{\mathbb{P}}_{u}(\dot{X}_{\tau_{v}}\geq
c\delta/2).$
Choose $v_{0}$ such that
$\widetilde{\mathbb{P}}_{0^{+}}(\dot{X}_{\tau_{v_{0}}}\geq
c\delta/2)\leq\varepsilon$. Then, from the convergence of the law of
$\dot{X}_{\tau_{v_{0}}}$ under $\widetilde{\mathbb{P}}_{u}$ to that under
$\widetilde{\mathbb{P}}_{0^{+}}$, we get, for $u$ small enough,
$\widetilde{\mathbb{P}}_{u}(\dot{X}_{\tau_{v_{0}}}\geq c\delta/2)\leq
2\varepsilon,$
and hence
$\widetilde{\mathbb{P}}_{v}(M_{v_{0}}\geq\delta)\leq\frac{2}{1-\sqrt{3}/\pi}\
\varepsilon.$
∎
In conclusion, all this suffices to show Proposition 2.
## 4 The resurrected process
### 4.1 Itō excursion measure, recurrent extensions,
and $(SOR)$ equations
We finally tackle the problem of interest, that is the recurrent extensions of
the reflected Kolmogorov process. A recurrent extension of the latter is a
Markov process that behaves like the reflected Kolmogorov process until
$\zeta_{\infty}$, the hitting time of $(0,0)$, but that is defined for any
positive times and does not stay at $(0,0)$, in the sense that the Lebesgue
measure of the set of times when the process is at $(0,0)$ is almost surely 0.
More concisely, we will call such a process a resurrected reflected process.
We recall that Itō’s program and results of Blumenthal [5] establish an
equivalence between the law of recurrent extensions of a Markov process and
excursion measures compatible with its semigroup, here $\mathrm{P}_{t}^{c}$
(where as usually in Itō’s excursion theory we identify the measures which are
equal up to a multiplicative constant). The _set of excursions_ $\mathcal{E}$
is defined by
$\mathcal{E}:=\\{(x,\dot{x})\in\mathcal{C}|\zeta_{\infty}>0\text{ and
}x_{t}\mathbbm{1}_{t\geq\zeta_{\infty}}=0\\}.$
An excursion measure $n$ compatible with the semigroup $\mathrm{P}_{t}^{c}$ is
defined by the three following properties:
1. 1.
The measure $n$ is carried by $\mathcal{E}$.
2. 2.
For any $\mathfrak{F}_{\infty}-$measurable function $F$ and any $t>0$, any
$A\in\mathfrak{F}_{t}$,
$n(F\circ\theta_{t},A\cap\\{t<\zeta_{\infty}\\})=n(\mathbb{P}_{X_{t},\dot{X}_{t}}^{c}(F),A\cap\\{t<\zeta_{\infty}\\}).$
3. 3.
$n(1-e^{-\zeta_{\infty}})<\infty.$
We also say that $n$ is a pseudo-excursion measure compatible with the
semigroup $\mathrm{P}_{t}^{c}$ if only the two first properties are satisfied
and not necessarily the third one. We recall that the third property is the
necessary condition in Itō’s program in order for the lengths of the
excursions to be summable, hence in order for Itō’s program to succeed.
Besides, we are here interested in recurrent extensions which leave $(0,0)$
continuously. These extensions correspond to excursion measures $n$ which
satisfy the additional condition $n((X_{0},\dot{X}_{0})\neq(0,0))=0$. Our main
results are the following:
###### Theorem 1.
There exists, up to a multiplicative constant, a unique excursion measure
$\mathbf{n}$ compatible with the semigroup $\mathrm{P}_{t}^{c}$ and such that
$\mathbf{n}((X_{0},\dot{X}_{0})\neq(0,0))=0$. We may choose $\mathbf{n}$ such
that
$\mathbf{n}(\zeta_{\infty}>s)=C_{1}s^{-k},$ (4.1)
where $C_{1}$ is the constant defined by (2.5), and $k=k(c)$ has been
introduced in Lemma 1. The measure $\mathbf{n}$ is then characterized by any
of the two following formulas:
$\displaystyle\mathbf{n}(f(X,\dot{X}),\zeta_{\infty}>T)$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(f(X,\dot{X})H(X_{T},\dot{X}_{T})^{-1}),$
(4.2)
for any $\mathfrak{F}_{t}-$stopping time $T$ and any $f$ positive measurable
functional depending only on $(X_{t},\dot{X}_{t})_{0\leq t\leq T}$.
$\displaystyle\mathbf{n}(f(X,\dot{X}),\zeta_{\infty}>T)$ $\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}H(x,u)^{-1}\mathbb{P}_{x,u}^{c}(f(X,\dot{X}),\zeta_{\infty}>T),$
(4.3)
for any $\mathfrak{F}_{t}-$stopping time $T$ and any $f$ positive _continuous_
functional depending only on $(X_{t},\dot{X}_{t})_{0\leq t\leq T}$.
So Itō’s program constructs a Markov process with associated Itō excursion
measure $\mathbf{n}$ and that spends no time at $(0,0)$, that is a recurrent
extension, that is a resurrected reflected process. We call its law
$\mathbb{P}_{0}^{r}$. The second theorem will be the weak existence and
solution to equations $(SOR)$, the law of any solution being given by
$\mathbb{P}_{0}^{r}$. It is implicit in this theorem and until the end of the
paper that the initial condition is $(0,0)$, though this generalizes easily to
any other initial condition $(x,u)\in D$.
###### Theorem 2.
The law $\mathbb{P}_{0}^{r}$ gives the unique solution, in the weak sense, of
equations $(SOR)$:
$\bullet$ Consider $(X,\dot{X})$ a process of law $\mathbb{P}_{0}^{r}$. Then
the jumps of $\dot{X}$ on any finite interval are summable and the process $W$
defined by
$W_{t}=\dot{X}_{t}+(1+c)\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}$
is a Brownian motion. As a consequence the triplet $(X,\dot{X},W)$ is a
solution to $(SOR)$.
$\bullet$ For any solution $(X,\dot{X},W)$ to $(SOR)$, the law of
$(X,\dot{X})$ is $\mathbb{P}_{0}^{r}$.
Before we tackle the proof these theorems, let us write some comments and
consequences. First, the Itō excursion measure $\mathbf{n}$ is entirely
determined by its entrance law, which is defined by
$\mathbf{n}_{s}(\mathrm{d}x,\mathrm{d}u):=n((X_{s},\dot{X}_{s})\in\mathrm{d}x\otimes\mathrm{d}u,s<\zeta_{\infty})$
for $s>0.$ But Theorem 1 implies that it is characterized by any of the two
following formulas:
$\displaystyle\mathbf{n}_{s}(f)$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(f(X_{s},\dot{X}_{s})H(X_{s},\dot{X}_{s})^{-1}),\quad
s>0,$ (4.4)
for $f:D^{0}\to\mathbb{R}_{+}$ measurable.
$\displaystyle\mathbf{n}_{s}(f)$ $\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}H(x,u)^{-1}\mathbb{P}_{x,u}^{c}(f(X_{s},\dot{X}_{s}),\zeta_{\infty}>s),\quad
s>0,$ (4.5)
for $f:D^{0}\to\mathbb{R}_{+}$ continuous.
Formulas similar to these are found in the case of self-similar Markov
processes studied by Rivero [15]. This ends the parallel between our works.
Rivero underlined that the self-similar Markov process conditioned on never
hitting 0 that he introduced plays the same role as the Bessel process for the
Brownian motion. In our model, this role is played by the reflected Kolmogorov
process conditioned on never hitting $(0,0)$. Here is a short presentation of
this parallel. Write $P_{x}$ for the law of a Brownian motion starting from
position $x$, $\widetilde{P}_{x}$ for the law of the “three-dimensional”
Bessel process starting from $x$. Write $n$ for the Itō excursion measure of
the absolute value of the Brownian motion (that is, the Brownian motion
reflected at 0), and $\zeta$ for the hitting time of 0. Then the inverse
function is excessive (i.e nonnegative and superharmonic) for the Bessel
process and we have the two well-known formulas
$\displaystyle\mathbf{n}(f(X),\zeta>T)$ $\displaystyle=$
$\displaystyle\widetilde{P}_{0}(f(X)/X_{T})$
$\displaystyle\mathbf{n}(f(X),\zeta>T)$ $\displaystyle=$
$\displaystyle\lim_{x\to 0}\frac{1}{x}P_{x}(f(X),\zeta>T),$
for any $\mathfrak{F}_{t}-$stopping time $T$ and any $f$ positive measurable
functional (resp. continuous functional for the second formula) depending only
on $(X_{t})_{0\leq t\leq T}$.
Now, let us give an application of Formula (4.1). Write $l$ for the local time
spent by $X$ at zero, under $\mathbb{P}_{0}^{r}$. Formula (4.1) implies that
the inverse local time $l^{-1}$ is a subordinator with jumping measure $\Pi$
satisfying $\Pi(\zeta_{\infty}>s)\propto s^{-k}.$ That is, it is a stable
subordinator of index $k$. A well-known result of Taylor and Wendel [16] then
gives that the exact Hausdorff function of the closure of its range (the range
is the image of $\mathbb{R}_{+}$ by $l^{-1}$) is given by
$\phi(\varepsilon)=\varepsilon^{k}(\ln\ln 1/\varepsilon)^{1-k}$ almost surely.
The closure of the range of $l^{-1}$ being equal to the zero set
$\mathcal{Z}:=\\{t\geq 0:X_{t}=\dot{X}_{t}=0\\}$, we get the following
corollary:
###### Corollary 1.
The exact Hausdorff function of the set of the passage times to $(0,0)$ of the
resurrected reflected Kolmogorov process is
$\phi(\varepsilon)=\varepsilon^{k}(\ln\ln 1/\varepsilon)^{1-k}$ almost surely.
It is also clear that the set of the bouncing times of the resurrected
reflected Langevin process – the moments when the process is at zero with a
nonzero speed – is countable. Therefore the zero set of the resurrected
reflected Langevin process has the same exact Hausdorff function.
Finally, we should mention that the self-similarity property enjoyed by the
Kolmogorov process easily spreads to all the processes we introduced. If $a$
is a positive constant, denote by $(X^{a},\dot{X}^{a})$ the process
$(a^{3}X_{a^{-2}t},aX_{a^{-2}t})_{t\geq 0}$. Then the law of
$(X^{a},\dot{X}^{a})$ under $\mathbb{P}^{c}_{x,u}$ is simply
$\mathbb{P}^{c}_{a^{3}x,au}$. We have $H(a^{3}x,au)=a^{2k}H(x,u).$ The law of
$(X^{a},\dot{X}^{a})$ under $\widetilde{\mathbb{P}}_{x,u}$, resp.
$\widetilde{\mathbb{P}}_{0^{+}}$, is simply
$\widetilde{\mathbb{P}}_{a^{3}x,au}$, resp. $\widetilde{\mathbb{P}}_{0^{+}}$.
Finally, the measure of $(X^{a},\dot{X}^{a})$ under $\mathbf{n}$ is simply
$a^{2k}\mathbf{n}$.
Last two subsections are devoted to the proof of the two theorems.
### 4.2 The unique recurrent extension compatible with $\mathrm{P}_{t}^{c}$
#### Construction of the excursion measure
The function $1/H$ is excessive for the semigroup $\widetilde{\mathrm{P}}_{t}$
and the corresponding $h-$transform is $\mathrm{P}_{t}^{c}$ (see Definition
1). Write $\mathbf{n}$ for the $h-$tranform of
$\widetilde{\mathbb{P}}_{0^{+}}$ via this excessive function $1/H$. That is,
$\mathbf{n}$ is the unique measure on $\mathcal{C}$ carried by
$\\{\zeta_{\infty}>0\\}$ such that under $\mathbf{n}$ the coordinate process
is Markovian with semigroup $\mathrm{P}_{t}^{c}$, and for any
$\mathfrak{F}_{t}-$stopping time $T$ and any $A_{T}$ in $\mathfrak{F}_{T}$, we
have
$\mathbf{n}(A_{T},T<\zeta_{\infty})=\widetilde{\mathbb{P}}_{0^{+}}(A_{T},H(X_{T},\dot{X}_{T})^{-1}).$
Then, $\mathbf{n}$ is a pseudo-excursion measure compatible with semigroup
$\mathrm{P}_{t}^{c}$, which verifies
$\mathbf{n}((X_{0},\dot{X}_{0})\neq(0,0))=0$ and satisfies Formula (4.2). For
$f$ continuous functional depending only on $(X_{t},\dot{X}_{t})_{t\leq T}$,
we have
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(f(X_{s},\dot{X}_{s})H(X_{s},\dot{X}_{s})^{-1})$
$\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}\widetilde{\mathbb{P}}_{x,u}(f(X_{s},\dot{X}_{s})H(X_{s},\dot{X}_{s})^{-1})$
$\displaystyle=$
$\displaystyle\lim_{(x,u)\to(0,0)}\frac{1}{H(x,u)}\mathbb{P}_{x,u}^{c}(f(X_{s},\dot{X}_{s}),\zeta_{\infty}>s),$
so that the pseudo-excursion measure $\mathbf{n}$ also satisfies Formula
(4.3). In particular, taking $T=s$ and $f=1$, and considering the limit along
the half-line $x=0$, this gives
$\mathbf{n}(\zeta_{\infty}>s)=\lim_{u\to
0}u^{-2k}\mathbb{P}_{0,u}(\zeta_{\infty}>s).$
Using Lemma 2 and the scaling invariance property, we get
$\mathbf{n}(\zeta_{\infty}>s)=C_{1}s^{-k},$
where $C_{1}$ is the constant defined by (2.5). This is exactly Formula (4.1).
This formula gives, in particular,
$\mathbf{n}(1-e^{-\zeta_{\infty}})=C_{1}\Gamma(1-k),$
where $\Gamma$ denotes the usual Gamma function. Hence, $\mathbf{n}$ is an
excursion measure.
Finally, in order to establish Theorem 1 we just should prove that
$\mathbf{n}$ is the only excursion measure compatible with the semigroup
$\mathrm{P}_{t}^{c}$ such that $\mathbf{n}((X_{0},\dot{X}_{0})\neq(0,0))=0$.
That is, we should show the uniqueness of the law of the resurrected reflected
process.
#### Uniqueness of the excursion measure
Let $\mathbf{n}^{\prime}$ be such an excursion measure, compatible with the
semigroup $\mathrm{P}_{t}^{c}$, and satisfying
$\mathbf{n}^{\prime}((X_{0},\dot{X}_{0})\neq(0,0))=0$. We will prove that
$\mathbf{n}$ and $\mathbf{n}^{\prime}$ coincide, up to a multiplicative
constant. Recall that $\zeta_{1}$ is defined as the infimum of
$\\{t>0,X_{t}=0\\}$.
###### Lemma 7.
The measure $\mathbf{n}^{\prime}$ satisfies:
$\mathbf{n}^{\prime}(\zeta_{1}\neq 0)=0$
###### Proof.
This condition will appear to be necessary to have the third property of
excursion measures, that is
$\mathbf{n}^{\prime}(1-e^{-\zeta_{\infty}})<\infty.$ Suppose on the contrary
that $\mathbf{n}^{\prime}(\zeta_{1}\neq 0)>0$ and write
$\tilde{\mathbf{n}}(\cdot)=\mathbf{n}^{\prime}(\cdot\mathbbm{1}_{\zeta_{1}\neq
0})$. The measure $\tilde{\mathbf{n}}$ is an excursion measure compatible with
the semigroup $\mathrm{P}_{t}^{c}$ such that
$\tilde{\mathbf{n}}((X_{0},\dot{X}_{0})\neq(0,0))=0$, satisfying
$\tilde{\mathbf{n}}(\zeta_{1}=0)=0.$ Consider
$\overline{\mathbf{n}}((X_{t},\dot{X}_{t})_{t\geq
0}):=\tilde{\mathbf{n}}((X_{t}\mathbbm{1}_{t<\zeta_{1}},\dot{X}_{t}\mathbbm{1}_{t<\zeta_{1}})_{t\geq
0})$ the excursion measure of the process killed at time $\zeta_{1}$.
The measure $\overline{\mathbf{n}}$ is an excursion measure compatible with
the semigroup $\mathrm{P}_{t}^{0}$, semigroup of the Kolmogorov process killed
at time $\zeta_{1}$ (the first hitting time of $\\{0\\}\times\mathbb{R}$).
Therefore its first marginal must be the excursion measure of the Langevin
process reflected on an inelastic boundary, introduced and studied in [3]. In
particular, under $\overline{\mathbf{n}}$, the absolute value of the incoming
speed at time $\zeta_{1}$, or $|\dot{X}_{\zeta_{1}-}|$, is distributed
proportionally to $v^{-\frac{3}{2}}\mathrm{d}v$ (see [3], Corollary 2, (ii)).
This stays true under $\tilde{\mathbf{n}}$ and implies that
$V_{1}=c|\dot{X}_{\zeta_{1}-}|$ is also distributed proportionally to
$v^{-\frac{3}{2}}\mathrm{d}v$. Now, a Markov property at the stopping time
$\zeta_{1}$ under $\tilde{\mathbf{n}}$ gives
$\tilde{\mathbf{n}}(\zeta_{\infty}-\zeta_{1}>t|V_{1}=v)=\mathbb{P}_{v}^{c}(\zeta_{\infty}>t)=\mathbb{P}_{1}^{c}(\zeta_{\infty}>v^{-2}t)\underset{v^{-2}t\to\infty}{\sim}Cv^{2k}t^{-k}$
As a consequence the function $v\mapsto
v^{-\frac{3}{2}}\tilde{\mathbf{n}}(\zeta_{\infty}-\zeta_{1}>t|V_{1}=v)$ is not
integrable in the neighborhood of 0\. That is
$\tilde{\mathbf{n}}(\zeta_{\infty}-\zeta_{1}>t)=+\infty$, we get a
contradiction. ∎
Recall that we owe to prove that $\mathbf{n}^{\prime}$ and $\mathbf{n}$ are
equal, up to a multiplicative constant. Let us work on the corresponding
entrance laws. Take $s>0$ and $f$ a bounded continuous function. It is
sufficient to prove $\mathbf{n}^{\prime}_{s}(f)=C\mathbf{n}_{s}(f)$, where $C$
is a constant independent of $s$ and $f$.
By reformulating Lemma 7, time $\zeta_{1}$ is zero
$\mathbf{n}^{\prime}$-almost surely, in the sense that the
$\mathbf{n}^{\prime}$-measure of the complementary event is 0. That is,
$\mathbf{n}^{\prime}$-a.s., the first coordinate of the process comes back to
zero just after the initial time, while the second coordinate cannot be zero,
for the simple reason that we are working on an excursion outside from
$(0,0)$. This, together with the fact that the velocity starts from
$\dot{X}_{0}=0$ and is right-continuous, implies that
$\mathbf{n}^{\prime}$-almost surely, the time $\tau_{v}$ (which, we recall, is
the instant of the first bounce with speed greater than $v$) is going to $0$
when $v$ is going to 0.
We deduce, by dominated convergence, from the continuity of $f$, and, again,
from the right-continuity of the paths, that
$\mathbf{n}^{\prime}_{s}(f)=\lim_{u\to
0}\mathbf{n}^{\prime}(f(X_{s+\tau_{v}},\dot{X}_{s+\tau_{v}})\mathbbm{1}_{\tau_{v}<\infty,\zeta_{\infty}>s+\tau_{v}}).$
(4.6)
An application of the Markov property gives
$\displaystyle\mathbf{n}^{\prime}(f(X_{s+\tau_{v}},\dot{X}_{s+\tau_{v}})\mathbbm{1}_{\tau_{v}<\infty,\zeta_{\infty}>s+\tau_{v}})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)\mathbb{P}^{c}_{u}(f(X_{s},\dot{X}_{s})\mathbbm{1}_{\zeta_{\infty}>s})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}g(u),$
where
$g(u)=u^{-2k}\mathbb{P}^{c}_{u}(f(X_{s},\dot{X}_{s})\mathbbm{1}_{\zeta_{\infty}>s})=H(0,u)^{-1}\mathbb{P}^{c}_{u}(f(X_{s},\dot{X}_{s})\mathbbm{1}_{\zeta_{\infty}>s})$
converges to $\mathbf{n}_{s}(f)$ when $u\to 0$, by Formula (4.3). Moreover the
function $u^{2k}g(u)$ is bounded by $\|f\|_{\infty}$, and for any
$\varepsilon>0$ we have
$\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}>\varepsilon)\to 0$ when $v\to 0$.
Informally, all this explains that when $v$ is small, all the mass in the
integral is concentrated in the neighborhood of $0$, where we can replace
$g(u)$ by $\mathbf{n}_{s}(f)$. More precisely, write
$\int_{\mathbb{R}_{+}}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}g(u)=I(v)+J(v),$
where
$\displaystyle I(v)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}\mathbf{n}_{s}(f),$
$\displaystyle J(v)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}(g(u)-\mathbf{n}_{s}(f)\mathbbm{1}_{u\leq
1}).$
By splitting the integral defining $J(v)$, we deduce that $J(v)$ is negligible
compared to $1\vee I(v).$ Recalling that the sum $I(v)+J(v)$ converges to
$\mathbf{n}^{\prime}_{s}(f)$ (Formula (4.6)), we get that $I(v)$ converges to
$\mathbf{n}^{\prime}_{s}(f)$ when $v\to 0$, while $J(v)$ converges to 0.
We thus have
$\mathbf{n}^{\prime}_{s}(f)=C\mathbf{n}_{s}(f),$
where $C$ is independent of $s$ and $f$ and given by
$C=\lim_{v\to
0}\int_{0}^{1}\mathbf{n}^{\prime}(\dot{X}_{\tau_{v}}\in\mathrm{d}u)u^{2k}.$
Uniqueness follows. Theorem 1 is proved.
### 4.3 The weak unique solution to the $(SOR)$ equations
We now prove Theorem 2.
#### Weak solution
We consider, under $\mathbb{P}_{0}^{r}$, the coordinate process $(X,\dot{X})$,
and its natural filtration $(\mathfrak{F}_{t})_{t\geq 0}$. We first prove that
the jumps of $\dot{X}$ are almost-surely summable on any finite interval. As
there are (a.s.) only finitely many jumps of amplitude greater than a given
constant on any finite interval, it is enough to prove that the jumps of
amplitude less than a given constant are (a.s.) summable. Write $L$ for a
local time of the process $(X,\dot{X})$ in $(0,0)$, $L^{-1}$ its inverse, and
$\mathbf{n}$ the associated excursion measure. It is sufficient to prove that
the expectation of the sum of the jumps of amplitude less than $1+1/c$ (jumps
at the bouncing times for which the outgoing velocity is less than one), and
occurring before time $L^{-1}(1)$, is finite. This expectation is equal to
$(1+\frac{1}{c})\int_{0}^{1}\mathbf{n}(N_{[v,1]}(X,\dot{X}))\mathrm{d}v,$
where we write $N_{I}(X,\dot{X})$ for the number of bounces of the process
$(X,\dot{X})$ with outgoing speed included in the interval $I$. For a fixed
$v$, introduce the sequence of stopping times defined by $\tau^{v}_{0}=0$ and
$\tau^{v}_{n+1}=\inf\\{t>\tau^{v}_{n},X_{t}=0,\dot{X}_{t}\in[v,1]\\}$ for
$n\geq 0$. Then $N_{[v,1]}(X,\dot{X})$ is also equal to
$\sup\\{n,\tau^{v}_{n}<\zeta_{\infty}\\}$. Thanks to formula (4.2), for any
$n>0$, we have:
$\displaystyle\mathbf{n}(\zeta_{\infty}>\tau^{v}_{n})$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(H(X_{\tau^{v}_{n}},\dot{X}_{\tau^{v}_{n}})^{-1}\mathbbm{1}_{\tau^{v}_{n}<\infty})$
$\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(\dot{X}_{\tau^{v}_{n}}^{-2k}\mathbbm{1}_{\tau^{v}_{n}<\infty})$
$\displaystyle\leq$ $\displaystyle
v^{-2k}\widetilde{\mathbb{P}}_{0^{+}}(\tau^{v}_{n}<\infty).$
As a consequence, we have
$\displaystyle\mathbf{n}(N_{[v,1]}(X,\dot{X}))$ $\displaystyle\leq$
$\displaystyle
v^{-2k}\widetilde{\mathbb{P}}_{0^{+}}(\sup\\{n,\tau^{v}_{n}<\zeta_{\infty}\\})$
$\displaystyle\leq$ $\displaystyle v^{-2k}\widetilde{\mathbf{P}}(N_{[\ln
v,0]}^{d}(S)),$
where we have written $N_{[\ln v,0]}^{d}(S)$ for the number of instants
$n\in\mathbb{Z}$ such that $S_{n}\in[\ln v,0]$. Recall also that
$\widetilde{\mathbf{P}}$ is the law of the spatially stationary random walk.
It is now a simple verification that $\widetilde{\mathbf{P}}(N_{[\ln
v,0]}^{d}(S))$ is finite and proportional to the length of the interval
$[\ln(v),0]$, that is $-\ln v$. It follows
$\mathbf{n}(N_{[v,1]}(X,\dot{X}))\underset{v\to 0}{=}O(v^{-2k}\ln(1/v))$
and (recall $k<1/4$)
$(1+\frac{1}{c})\int_{0}^{1}\mathbf{n}(N_{[v,1]}(X,\dot{X}))\mathrm{d}v<\infty.$
The jumps are summable.
Now, write
$W_{t}=\dot{X}_{t}+(1+c)\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}.$
We aim to show that the continuous process $W$ is a Brownian motion. For
$\varepsilon>0$, we introduce the sequence of stopping times
$(T_{n}^{\varepsilon})_{n\geq 0}$ defined by $T_{0}^{\varepsilon}=0$ and, for
$n\geq 0$,
$\left\\{\begin{array}[]{rcl}T_{2n+1}^{\varepsilon}&=&\inf\\{t>T_{2n}^{\varepsilon},X_{t}=0,\dot{X}_{t}>\varepsilon\\}\\\
T_{2n+2}^{\varepsilon}&=&\inf\\{t>T_{2n+1}^{\varepsilon},X_{t}=\dot{X}_{t}=0\\}\end{array}\right.$
We also introduce $F^{\varepsilon}=\bigcup_{n\geq
0}[T_{2n}^{\varepsilon},T_{2n+1}^{\varepsilon}]$ and
$H_{t}^{\varepsilon}=\mathbbm{1}_{F^{\varepsilon}}(t)$. For
$0<\varepsilon^{\prime}<\varepsilon$, we have $H^{\varepsilon^{\prime}}\leq
H^{\varepsilon}$, or equivalently, $F^{\varepsilon^{\prime}}\subset
F^{\varepsilon}$. When $\varepsilon$ goes to $0+$, $F^{\varepsilon}$ converges
to the zero set $\mathcal{Z}=\\{t,X_{t}=\dot{X}_{t}=0\\}$, and
$H^{\varepsilon}$ converges pointwisely to $H^{0}=\mathbbm{1}_{\mathcal{Z}}$.
Note that the processes $H^{\varepsilon}$ and $H^{0}$ are
$\mathfrak{F}_{t}-$adapted. Note, also, that Corollary 1 implies in particular
that $\mathcal{Z}$ has zero Lebesgue measure. For ease of notations, we will
sometimes omit the superscript $\varepsilon$.
Conditionally on $\dot{X}_{T_{2n+1}}=u$, the process $(X_{(T_{2n+1}+t)\wedge
T_{2n+2}})_{t\geq 0}$ is independent of $\mathfrak{F}_{T_{2n+1}}$ and has law
$\mathbb{P}_{u}^{c}$. As a consequence the process $(W_{(T_{2n+1}+t)\wedge
T_{2n+2}}-W_{T_{2n+1}})_{t\geq 0}$ is a Brownian motion stopped at time
$T_{2n+2}-T_{2n+1}$. Write
$W_{t}=\int_{0}^{t}H_{s}^{\varepsilon}\mathrm{d}W_{s}+\int_{0}^{t}(1-H_{s}^{\varepsilon})\mathrm{d}W_{s}.$
The process $\int_{0}^{t}(1-H_{s}^{\varepsilon})\mathrm{d}W_{s}$ converges
almost surely to $\int_{0}^{t}(1-H_{s}^{0})\mathrm{d}W_{s}$. But the process
$\int_{0}^{t}(1-H_{s}^{0})\mathrm{d}W_{s}$ is a continuous martingale of
quadratic variation $\int_{0}^{t}(1-H_{s}^{0})\mathrm{d}s=t$ and thus a
Brownian motion. In order to prove that it actually coincides with $W$, we
just need to prove that the term
$D_{t}^{\varepsilon}:=\int_{0}^{t}H_{s}^{\varepsilon}\mathrm{d}W_{s}$ is
almost-surely converging to $0$ when $\varepsilon\to 0$. Without loss of
generality, we just prove it on the event $t\leq L^{-1}(1)$.
This term can be rewritten as
$D_{t}^{\varepsilon}=\left\\{\begin{array}[]{ll}\displaystyle\sum_{k\leq
n}\big{(}W_{T_{2k+1}}-W_{T_{2k}}\big{)}&\text{if }T_{2n+1}\leq t<T_{2n+2},\\\
\\\ \displaystyle
W_{t}-W_{T_{2n}}+\sum_{k<n}\big{(}W_{T_{2k+1}}-W_{T_{2k}}\big{)}&\text{if
}T_{2n}\leq t<T_{2n+1}.\end{array}\right.$
Now, for any $k$, we have
$W_{T_{2k+1}}-W_{T_{2k}}=\dot{X}_{T_{2k+1}}+(1+c)\sum_{T_{2k}<s\leq
T_{2k+1}}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},$
and for any $T_{2n}\leq t<T_{2n+1}$,
$W_{t}-W_{T_{2n}}=\dot{X}_{t}+(1+c)\sum_{T_{2n}<s\leq
t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},$
Hence the term $D_{t}^{\varepsilon}$ involves jumps of amplitude less than
$(1+c)\varepsilon$, whose sum is going to 0 when $\varepsilon$ goes to zero,
plus the fraction $c/(1+c)$ of the jumps occurring at times $T_{2k+1}$, plus
the possible extra term $\dot{X}_{t}$, not corresponding to any jump. We will
prove nonetheless that the jumps occurring at times $T_{2k+1}$, and
$|\dot{X}_{t}|$, are all small when $\varepsilon$ is small enough. It will
follow that $D_{t}^{\varepsilon}$ tends to 0 when $\varepsilon$ goes to 0.
Fix $\eta>0$. Write $A^{\varepsilon}$ for the event
$\sup_{s\leq L^{-1}(1),s\in F^{\varepsilon}}\dot{X}_{s}\geq\eta.$
We will prove that the probability of $A^{\varepsilon}$ is going to 0 when
$\varepsilon$ goes to 0, so that we almost surely don’t lie in
$A^{\varepsilon}$ for $\varepsilon$ small enough, and as a consequence the
jumps occurring at times $T_{2k+1}$ and the possible term $|\dot{X}_{t}|$ will
then all be less than $\eta$, as requested. Write
$\widetilde{T}^{\varepsilon}$ for the infimum of $\\{t:t\in
F^{\varepsilon},|\dot{X}_{t}|\geq\eta\\}$ and $n_{\varepsilon}$ for the
supremum of $\\{n,T_{2n}\leq\widetilde{T}^{\varepsilon}\\}$. The event
$A^{\varepsilon}$ coincides with $\\{\widetilde{T}^{\varepsilon}<L^{-1}(1)\\}$
or $\\{T_{2n_{\varepsilon}+1}<L^{-1}(1)\\}$.
The Markov property at the stopping time $\widetilde{T}^{\varepsilon}$,
together with the inequality (3.7), gives
$\mathbb{P}(\\{\dot{X}_{T_{2n_{\varepsilon}+1}}\geq\eta c/2\\}\cap
A^{\varepsilon})\geq\big{(}1-{\sqrt{3}}/\pi\big{)}\mathbb{P}(A^{\varepsilon}).$
The event $\\{\dot{X}_{T_{2n_{\varepsilon}+1}}\geq\eta c/2\\}\cap
A^{\varepsilon}$ is contained in the event that there is an excursion
occurring before time $L^{-1}(1)$ for which the first bounce with speed
greater than $\varepsilon$ is actually greater than $\eta c/2$. This event has
probability
$\mathbf{n}(T_{1}^{\varepsilon}<\infty,\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2),$
where $T_{1}^{\varepsilon}$ is still defined as the time of the first bounce
with speed greater than $\varepsilon$, here for the excursion. We have:
$\displaystyle\mathbf{n}(\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2,\zeta_{\infty}>T_{1}^{\varepsilon})$ $\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{0^{+}}(H(0,\dot{X}_{T_{1}^{\varepsilon}})^{-1}\mathbbm{1}_{\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2})$ $\displaystyle\leq$ $\displaystyle(\eta
c/2)^{-2k}\widetilde{\mathbb{P}}_{0^{+}}(\dot{X}_{T_{1}^{\varepsilon}}\geq\eta
c/2)$ $\displaystyle\leq$ $\displaystyle(\eta c/2)^{-2k}m\big{(}]\ln({\eta
c}/({2\varepsilon})),\infty[\big{)},$
where we recall that $m$ is the stationary law of the overshoot appearing in
Proposition 2. This probability is thus going to 0 when $\varepsilon$ goes to
0, as well as $\mathbb{P}(A^{\varepsilon})$.
The process $W$ is a Brownian motion, and $(X,\dot{X},W)$ is a solution to
Equations $(SOR)$.
#### Weak uniqueness
Consider $(X,\dot{X},W)$, with law $\mathbb{P}$, be any solution to $(SOR)$,
and its associated filtration $(\mathfrak{F}_{t})_{t\geq 0}$. Then we have
$\dot{X}_{t}=W_{t}-(1+c)\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0},$
with $W$ a Brownian motion.
We start with the observation that the process $\dot{X}$ does not explode and
that the sum just involves positive jumps. Therefore these jumps are summable.
But the process $\sum_{0<s\leq t}\dot{X}_{s-}\mathbbm{1}_{X_{s}=0}$ is
adapted, hence $\dot{X}$ is a semimartingale. As a consequence, it possesses
local times $(L^{a})_{a\in\mathbb{R}}$, and we have an occupation formula (see
for example [14], Theorem 70 Corollary 1, p216):
$\int_{-\infty}^{+\infty}L^{a}_{t}g(a)\mathrm{d}a=\int_{0}^{t}g(\dot{X}_{s-})\mathrm{d}s,$
for any $g$ bounded measurable function. Taking $g=\mathbbm{1}_{\\{0\\}}$
shows that $\dot{X}$ spends no time at zero. It follows that the process
$(X,\dot{X})$ spends no time at $(0,0)$.
Now, exactly as before, introduce, for $\varepsilon>0$, the sequence of
stopping times $T_{n}^{\varepsilon}$, defined by $T_{0}^{\varepsilon}=0$ and
$\left\\{\begin{array}[]{rcl}T_{2n+1}^{\varepsilon}&=&\inf\\{t>T_{2n}^{\varepsilon},X_{t}=0,\dot{X}_{t}>\varepsilon\\}\\\
T_{2n+2}^{\varepsilon}&=&\inf\\{t>T_{2n+1}^{\varepsilon},X_{t}=\dot{X}_{t}=0\\},\end{array}\right.$
as well as $F^{\varepsilon}=\bigcup_{n\geq
0}[T_{2n}^{\varepsilon},T_{2n+1}^{\varepsilon}]$ and
$H^{\varepsilon}=\mathbbm{1}_{F^{\varepsilon}}$. Finally, define the closed
set $F=\lim_{\varepsilon\to 0}F^{\varepsilon}$ and the adapted process
$H^{0}=\mathbbm{1}_{F}.$
###### Lemma 8.
The set $F$ has almost surely zero Lebesgue measure.
This result is not immediate. First, observe that the excursions of the
process may be of two types. Either an excursion bounces on the boundary just
after the initial time, or it doesn’t. We call $\mathcal{E}_{1}$ the set of
excursions of the first type, defined by
$\mathcal{E}_{1}:=\\{(x,\dot{x})\in\mathcal{E}|\zeta_{1}(x,\dot{x}):=\inf\\{t>0,x_{t}=0\\}=0\\},$
and $\mathcal{E}_{2}=\mathcal{E}\backslash\mathcal{E}_{1}$ the set of
excursions of the second type. Unlike before, we do not know _a priori_ that
all the excursions of the process lie in $\mathcal{E}_{1}$. If the process
starts an excursion at time $t$, we write $e^{t}$ for the corresponding
excursion.
A close look at $F$ shows that it contains not only the zero set
$\mathcal{Z}$, but also all the intervals $[t,t+\zeta_{1}(e^{t})]$, where $t$
is the starting time of an excursion $e^{t}\in\mathcal{E}_{2}$. Prove Lemma 8
is equivalent to prove that there is actually no excursion in
$\mathcal{E}_{2}$.
Suppose that this fails. Then the process
$L(t)=\int_{0}^{t}H^{0}_{s}\mathrm{d}s$
is not almost surely constantly equal to zero. We introduce its right-
continuous inverse
$L^{-1}(t):=\inf\\{s>t,L(s)>t\\}.$
There exists a Brownian motion $M$ such that for $t<L(\infty),$
$M_{t}=\int_{0}^{L^{-1}(t)}H^{0}_{s}\mathrm{d}W_{s}.$
Introduce the time-changed process
$(Y_{t},\dot{Y}_{t})=(X_{L^{-1}(t)},\dot{X}_{L^{-1}(t)}),$
stopped at time $L(\infty)$. In order to simplify the redaction, we will often
omit to specify “stopped at time $L(\infty)$”. This time change induces that
the process $(Y,\dot{Y})$ also does not spend any time at zero, and that its
excursions are that of $(X,\dot{X})$ belonging to $\mathcal{E}_{2}$, and
stopped at $\zeta_{1}$ the first return time to $\\{0\\}\times\mathbb{R}$.
###### Lemma 9.
The triplet $(Y_{t},\dot{Y}_{t},M_{t})_{t\leq L(\infty)}$ under $\mathbb{P}$
is a solution of the equations $(SOR)$ with null elasticity coefficient,
stopped at time $L(\infty)$.
###### Proof.
Let $[t,t^{\prime}[$ be the interval corresponding to an excursion of
$(Y,\dot{Y})$. Then the interval $[L^{-1}(t),L^{-1}(t^{\prime}-)]$ is a
maximal interval included in $F$. It follows that the points $L^{-1}(t)$ and
$L^{-1}(t^{\prime})$ belong to $\mathcal{Z}$, and
$Y_{t}=\dot{Y}_{t}=0=Y_{t^{\prime}}=\dot{Y}_{t^{\prime}}$.
Let $s\in[t,t^{\prime}[$. As the process $X$ has no bounce in
$[L^{-1}(t),L^{-1}(s)]$ and $(X,\dot{X},W)$ is a solution to $(SOR)$, we can
write
$\dot{X}_{L^{-1}(s)}=\dot{X}_{L^{-1}(t)}+W_{L^{-1}(s)}-W_{L^{-1}(t)},$
or equivalently
$\dot{Y}_{s}=\dot{Y}_{t}+M_{s}-M_{t}.$
As a consequence, we may write
$\left\\{\begin{array}[]{ccl}Y_{s}&=&Y_{t}+\displaystyle\int_{t}^{s}\dot{Y}_{u}\mathrm{d}u\\\
\dot{Y}_{s}&=&\dot{Y}_{t}+M_{s}-M_{t}-\sum_{t<u\leq
s}\dot{Y}_{u-}\mathbbm{1}_{Y_{u}=0},\end{array}\right.$
where the sum is actually empty. Similarly,
$\left\\{\begin{array}[]{ccl}Y_{t^{\prime}}&=&0=X_{L^{-1}(t^{\prime}-)}=Y_{t^{\prime}-}=Y_{t}+\displaystyle\int_{t}^{t^{\prime}}\dot{Y}_{u}\mathrm{d}u\\\
\dot{Y}_{t^{\prime}}&=&0=\dot{Y}_{t^{\prime}-}-\dot{Y}_{t^{\prime}-}\mathbbm{1}_{Y_{t^{\prime}}=0}=\dot{Y}_{t}+M_{t^{\prime}}-M_{t}-\sum_{t<u\leq
t^{\prime}}\dot{Y}_{u-}\mathbbm{1}_{Y_{u}=0},\end{array}\right.$
where the sum now contains one term.
Adding these equalities on the excursion intervals of $(Y,\dot{Y})$, and
recalling that this process spends no time at $(0,0)$, gives
$\left\\{\begin{array}[]{ccl}Y_{s}&=&\displaystyle\int_{0}^{s}\dot{Y}_{u}\mathrm{d}u\\\
\dot{Y}_{s}&=&M_{s}-\sum_{0<u\leq
s}\dot{Y}_{u-}\mathbbm{1}_{Y_{u}=0},\end{array}\right.$
and $(Y,\dot{Y},M)$ is a solution to $(SOR)$ with null elasticity coefficient
(stopped at time $L(\infty)$). ∎
The article [4], which studied equations $(SOR)$ with null elasticity
coefficient, shows that a solution $(Y,\dot{Y})$ must be a Markov process,
with Itō excursion law $\overline{\mathbf{n}}$. We immediately introduce
another change of time, in a very similar way, but without stopping the
excursions of $\mathcal{E}_{2}$ at time $\zeta_{1}$. Define the random set
$A:=\mathcal{Z}\cup\bigcup_{\\{t|e^{t}\in\mathcal{E}_{2}\\}}[t,t+\zeta_{\infty}(e^{t})],$
and the adapted process $\widetilde{H}=\mathbbm{1}_{A}.$ Define also
$\widetilde{L}(t)=\int_{0}^{t}\widetilde{H}_{s}\mathrm{d}s,$
and $\widetilde{L}^{-1}$ for its right-continuous inverse. Then, there exists
a Brownian motion $\widetilde{M}$ such that
$\widetilde{M}_{t}=\int_{0}^{\widetilde{L}^{-1}(t)}\widetilde{H}_{s}\mathrm{d}W_{s}$
for $t<\widetilde{L}(\infty)$. Finally, the time-changed process
$(\widetilde{Y}_{t},\dot{\widetilde{Y}}_{t})=(X_{\widetilde{L}^{-1}(t)},\dot{X}_{\widetilde{L}^{-1}(t)}),$
stopped at time $\widetilde{L}(\infty)$, spends no time at zero and its
excursions are the excursions of $(X,\dot{X})$ included in $\mathcal{E}_{2}$.
Remark that we have $\widetilde{L}(\infty)\geq L(\infty)$ because $A\supset
F$. We also get the following lemma, similar to Lemma 9, and whose proof we
leave to the reader.
###### Lemma 10.
The triplet
$\Big{(}\widetilde{Y}_{t},\dot{\widetilde{Y}}_{t},\widetilde{M}_{t}\Big{)}_{t\leq\widetilde{L}(\infty)}$
under $\mathbb{P}$ is a solution of the equations $(SOR)$ (with elasticity
coefficient $c$), stopped at time $\widetilde{L}(\infty)$.
The process $(\widetilde{Y},\dot{\widetilde{Y}})$ spends no time at $0$, is a
solution to $(SOR)$, and its excursions, stopped at $\zeta_{1}$, the first
return time to $\\{0\\}\times\mathbb{R}$, are precisely that of $(Y,\dot{Y}).$
This induces that $(\widetilde{Y},\dot{\widetilde{Y}})$ is a Markov process
with Itō excursion measure $\tilde{\mathbf{n}}$ determined by
$\left\\{\begin{array}[]{ccl}\tilde{\mathbf{n}}\left((x_{t\wedge\zeta_{1}})_{t\geq
0}\in\cdot\right)&=&\overline{\mathbf{n}}(x\in\cdot)\\\
\tilde{\mathbf{n}}\left((x_{t+\zeta_{1}})_{t\geq
0}\in\cdot\right|\dot{X}_{\zeta_{1}}=v)&=&\mathbb{P}_{v}^{c}(x\in\cdot)\end{array}\right.$
Now, the result of uniqueness of the excursion measure implies that
$\tilde{\mathbf{n}}$ should be a multiple of $\mathbf{n}$, which is obviously
not the case (for example because $\tilde{\mathbf{n}}(\zeta_{\infty}=0)=0$).
Therefore $\widetilde{L}(\infty)=0=L(\infty)$ a.s. Lemma 8 is proved.
Now, introduce a third time-change,
$(L^{\varepsilon})^{-1}(t):=\inf\\{s>0,L^{\varepsilon}(s)>t\\}$. When
$\varepsilon$ goes to 0, $(L^{\varepsilon})^{-1}$ is going to $L^{-1}=Id$. It
follows that the process
$X^{\varepsilon}:=(X_{(L^{\varepsilon})^{-1}(t)})_{t\geq 0}$ is going
uniformly on compacts to $X$ when $\varepsilon$ goes to 0, almost surely. In
particular the law of $X$ is entirely determined by that of $X^{\varepsilon}$.
The law of $X^{\varepsilon}$ is in turn entirely determined by that of
$(\dot{X}_{T_{2n+1}^{\varepsilon}})_{n\geq 0}.$ We will now determine this
law, which will prove the uniqueness of the law of $X$.
In order to avoid complex notations, we just give the calculation of the law
of $\dot{X}_{T_{1}^{1}}$, which is not fundamentally different from others.
For $\varepsilon>0$ and $n\geq 0$, a Markov property for the process $W$
applied at time $T_{2n+1}^{\varepsilon}$ shows that conditionally on
$\dot{X}_{T_{2n+1}^{\varepsilon}}=u$, the process
$(X_{(T_{2n+1}^{\varepsilon}+t)\wedge T_{2n+2}^{\varepsilon}})_{t\geq 0}$ is
independent from $\mathfrak{F}_{T_{2n+1}^{\varepsilon}}$ and has law
$\mathbb{P}_{u}^{c}$. Write $n_{1}$ for the integer satisfying
$T_{2n_{1}+1}^{\varepsilon}\leq T_{1}^{1}<T_{2n_{1}+2}^{\varepsilon}$.
Conditionally on $\dot{X}_{T_{2n_{1}+1}^{\varepsilon}}=u$, the process
$(X_{(T_{2n_{1}+1}^{\varepsilon}+t)\wedge T_{2n_{1}+2}^{\varepsilon}})_{t\geq
0}$ has the law $\mathbb{P}_{u}^{c}$ conditioned on reaching a speed greater
than one after a bounce.
In other words, the law of $\dot{X}_{T_{1}^{1}}$ under
$\mathbb{P}(\cdot|\dot{X}_{T_{2n_{1}+1}^{\varepsilon}}=u)$ is equal to that of
$\dot{X}_{T_{1}^{1}}$ under $\mathbb{P}_{u}^{c}(\cdot|T_{1}^{1}<\infty)$.
Besides, it should be clear now that $\dot{X}_{T_{2n_{1}+1}^{\varepsilon}}$ is
going to 0 when $\varepsilon$ goes to 0. Recall that $\zeta_{\infty}$, the
hitting time of $(0,0)$, is the lifetime of the excursion (under
$\mathbb{P}_{u}^{c}$ as well as under $\mathbf{n}$). For any $f$ positive
continuous functional, we have:
$\displaystyle\mathbb{P}_{u}^{c}(f(\dot{X}_{T_{1}^{1}})|\
T_{1}^{1}<\zeta_{\infty})$ $\displaystyle=$
$\displaystyle\mathbb{P}_{u}^{c}\big{(}f(\dot{X}_{T_{1}^{1}})\mathbbm{1}_{T_{1}^{1}<\zeta_{\infty}}\big{)}\
/\>\mathbb{P}_{u}^{c}(\mathbbm{1}_{T_{1}^{1}<\zeta_{\infty}})$
$\displaystyle=$
$\displaystyle\widetilde{\mathbb{P}}_{u}\Big{(}f(\dot{X}_{T_{1}^{1}})(H(0,\dot{X}_{T_{1}^{1}}))^{-1}\Big{)}\
/\>\widetilde{\mathbb{P}}_{u}((H(0,\dot{X}_{T_{1}^{1}}))^{-1})$
$\displaystyle\underset{u\to 0}{\longrightarrow}$
$\displaystyle\widetilde{\mathbb{P}}_{0+}\big{(}f(\dot{X}_{T_{1}^{1}})(H(0,\dot{X}_{T_{1}^{1}}))^{-1}\big{)}\
/\>\widetilde{\mathbb{P}}_{0+}((H(0,\dot{X}_{T_{1}^{1}}))^{-1})$
$\displaystyle=$ $\displaystyle\mathbf{n}(f(\dot{X}_{T_{1}^{1}})|\
T_{1}^{1}<\zeta_{\infty}),$
where we used successively (3.1), Proposition 2 and (a generalization of)
(4.2). As a consequence, the law of $\dot{X}_{T_{1}^{1}}$ under $\mathbb{P}$
is entirely determined, and is equal to that of $\dot{X}_{T_{1}^{1}}$ under
$\mathbf{n}(\cdot|\ T_{1}^{1}<\zeta_{\infty})$. Uniqueness of the stochastic
partial differential equation follows.
## References
* [1] P. Ballard. The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Rational Mech. Anal., 154:199–274, 2000.
* [2] J. Bect. Processus de Markov diffusifs par morceaux: outils analytiques et numériques. PhD thesis, Supelec, 2007.
* [3] J. Bertoin. Reflecting a Langevin process at an absorbing boundary. Ann. Probab., 35(6):2021–2037, 2007.
* [4] J. Bertoin. A second order SDE for the Langevin process reflected at a completely inelastic boundary. J. Eur. Math. Soc. (JEMS), 10(3):625–639, 2008.
* [5] R. M. Blumenthal. On construction of Markov processes. Z. Wahrsch. Verw. Gebiete, 63(4):433–444, 1983.
* [6] A. Bressan. Incompatibilità dei teoremi di esistenza e di unicità del moto per un tipo molto comune e regolare di sistemi meccanici. Ann. Scuola Norm. Sup. Pisa Serie III, 14:333–348, 1960.
* [7] C. M. Goldie. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1(1):126–166, 1991.
* [8] J. P. Gor′kov. A formula for the solution of a certain boundary value problem for the stationary equation of Brownian motion. Dokl. Akad. Nauk SSSR, 223(3):525–528, 1975.
* [9] J.-P. Imhof. Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab., 21(3):500–510, 1984.
* [10] E. Jacob. Langevin process reflected on a partially elastic boundary I . http://hal.archives-ouvertes.fr/hal-00472601/en/.
* [11] E. Jacob. Excursions of the integral of the Brownian motion. Ann. Inst. H. Poincaré Probab. Statist., 46(3):869–887, 2010\.
* [12] A. Lachal. Application de la théorie des excursions à l’intégrale du mouvement brownien. In Séminaire de Probabilités XXXVII, volume 1832 of Lecture Notes in Math., pages 109–195. Springer, Berlin, 2003.
* [13] H. P. McKean, Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ., 2:227–235, 1963.
* [14] P. E. Protter. Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2005. Second edition. Version 2.1, Corrected third printing.
* [15] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli, 11(3):471–509, 2005.
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|
arxiv-papers
| 2011-03-15T05:22:37 |
2024-09-04T02:49:17.663298
|
{
"license": "Public Domain",
"authors": "Emmanuel Jacob",
"submitter": "Emmanuel Jacob",
"url": "https://arxiv.org/abs/1103.2845"
}
|
1103.2875
|
# Qubit thermometry for micromechanical resonators
Matteo Brunelli matteo.brunelli@studenti.unimi.it Dipartimento di Fisica,
Università degli Studi di Milano, I-20133 Milano, Italy Stefano Olivares
stefano.olivares@ts.infn.it Dipartimento di Fisica, Università degli Studi di
Trieste, I-34151 Trieste, Italy CNISM, UdR Milano, I-20133 Milano, Italy
Matteo G. A. Paris matteo.paris@fisica.unimi.it Dipartimento di Fisica,
Università degli Studi di Milano, I-20133 Milano, Italy CNISM, UdR Milano,
I-20133 Milano, Italy
###### Abstract
We address estimation of temperature for a micromechanical oscillator lying
arbitrarily close to its quantum ground state. Motivated by recent
experiments, we assume that the oscillator is coupled to a probe qubit via
Jaynes-Cummings interaction and that the estimation of its effective
temperature is achieved via quantum limited measurements on the qubit. We
first consider the ideal unitary evolution in a noiseless environment and then
take into account the noise due to non dissipative decoherence. We exploit
local quantum estimation theory to assess and optimize the precision of
estimation procedures based on the measurement of qubit population, and to
compare their performances with the ultimate limit posed by quantum mechanics.
In particular, we evaluate the Fisher information (FI) for population
measurement, maximize its value over the possible qubit preparations and
interaction times, and compare its behavior with that of the quantum Fisher
information (QFI). We found that the FI for population measurement is equal to
the QFI, i.e., population measurement is optimal, for a suitable initial
preparation of the qubit and a predictable interaction time. The same
configuration also corresponds to the maximum of the QFI itself. Our results
indicate that the achievement of the ultimate bound to precision allowed by
quantum mechanics is in the capabilities of the current technology.
###### pacs:
42.50.-p, 03.65.-w
## I Introduction
The edge between classical and quantum description of a phenomenon is related
to the interactions occurring between the system under investigation and its
environment. As a consequence, if we could, in ideal conditions, avoid
irreversible interactions among them we should observe the emergence of
quantum behavior even in macroscopic systems. As a matter of fact, the
technological developments of the recent years have made it possible to start
inquiring into the quantum limit even in mesoscopic mechanical systems and
experiments have been designed which realize a solid state analogue of cavity
quantum electrodynamics. Many of these experiments focus on detecting the
quantization of vibrational modes in a mechanical oscillator nmr1 ; nmr2 ;
nmr3 ; nmr4 ; nmr5 ; nmr6 ; nmr7 ; nmr8 ; nmr9 ; nmr10 ; nmr11 . Experimental
conditions such that a mechanical object may behave in a quantum fashion are
achieved in the low temperature regime. For example, for a single vibrational
mode of energy $\hbar\omega$ to show quantum features, as the quantization of
lattice vibrations, temperatures $T\ll\frac{\hbar\omega}{k_{B}}$ are required,
which for a micro-sized object oscillating in the microwave band correspond to
few mK.
In this framework it has become increasingly relevant to have a precise
determination of the temperature. However, for a quantum system in equilibrium
with a thermal bath, there is no linear operator that acts as an observable
for temperature. Temperature, thought as a macroscopic manifestation of random
energy exchanges between particles, still retains its meaning but we have lost
any operational definition. This kind of impediment often occurs in physics,
and especially in quantum mechanics, whenever one is interested in quantities
which are not directly accessible, i.e. they do not correspond to observable
quantities. This may either be due to experimental impossibilities, or be a
matter of principle, as it happens for nonlinear functions of the density
operator. In both cases, it turns out that the only way to gain some knowledge
about the quantity of interest is to measure one or more proper observables
somehow related to the parameter we are interested in, and upon suitably
processing the outcomes, to come back and infer its value. Hence, any
conceivable strategy aimed to evaluate the quantity of interest ultimately
reduces to a parameter estimation problem. Relevant examples of this situation
are given by estimation of the quantum phase of a harmonic oscillator Mon06 ;
HDB09 ; asp09 ; PD11 , the amount of entanglement of a bipartite quantum state
EE08 ; EE10 ; EEL11 and the coupling constants of different kinds of
interactions Sar06 ; Hot06 ; Mon07 ; Fuj01 ; Zhe06 ; Boi08 ; ZP07 ; Cam10 ;
Pat06 ; mon10 ; mon11 . Here we focus on the estimation of temperature Man89
and, motivated by recent experimental achievements nmr11 , we specifically
refer to schemes where a micromechanical resonator is coupled to a
superconducting qubit, and then a measurement of the excited state population
is performed on the qubit itself. From the statistics of the population
measurement is then possible to obtain information about the oscillator state,
e.g. infer how close it is to the ground state, and in turn its temperature.
In this context an optimization problem naturally arises, aimed at finding the
most efficient inference procedure leading to minimum fluctuations in the
temperature estimate. In this paper we address this problem in the framework
of local quantum estimation theory (QET) lqe1 ; lqe2 ; lqe3 ; lqe4 ; lqe5 ;
lqe6 . We solve the dynamics of the qubit-resonator coupled system and, in
order to match realistic scenarios, we also take into account an effective
model for non dissipative decoherence. Then, we evaluate the Fisher
information (FI) for the estimation of temperature via population measurement
(hereafter referred to as the FI of the population measurement) and find both
the optimal initial qubit preparation and the smallest temperature value that
can be discriminated. Moreover, we evaluate the Quantum Fisher Information
(QFI) in terms of the symmetric logarithmic derivative in order to calculate
the ultimate bound to precision allowed by quantum mechanics. This enable us
to show that population measurement is indeed optimal for a suitable choice of
the initial preparation of the qubit, and to provide quantum benchmarks for
temperature estimation.
It is worth noting at this point that we are not discussing here temperature
fluctuations in a thermodynamical setting. Although temperature itself may not
fluctuate, as it is suggested by quantum thermodynamical approaches qth , we
expect that fluctuations always appear in the temperature estimates coming
from indirect measurements web07 ; Jan11 . Quantum estimation theory provides
the tools to evaluate lower bounds to the amount of fluctuations for a given
measurement, as well as the ultimate bounds imposed by quantum mechanics.
The paper is structured as follows. In Sec. II we describe the interaction
model: first we briefly review the unitary Jaynes-Cummings dynamics for the
coupled system and describe the measurements performed on the qubit, and then
we take into account the decoherence effects. In Sec. III we show how QET
techniques applies to our system, providing explicit formulas for both the FI
and the QFI. The results are finally shown in detail in Sec. IV both for the
unitary and the noisy dynamics. Sec. V closes the paper with some concluding
remarks.
## II The physical model
As the temperature decreases a mechanical oscillator starts to exhibit its
quantum nature, which mainly manifests itself in quantization of the
vibrational modes. Hence, for our purposes the resonator can be regarded as a
collection of phonons in a thermal equilibrium state. We assume that the
resonator is built as to display an isolated mechanical mode at a given
frequency, so that it can be modeled, rather than a phonon bath with some
spectral distribution, as a single mode phonon field in thermal equilibrium.
### II.1 Unitary dynamics
Let $\mathcal{H}_{R}$ be the infinite dimensional Hilbert space associated
with the single mode phonon field. Upon introducing the creation and
annihilation operators $[a,a^{\dagger}]=1$ one has the number operator
$N=a^{\dagger}a$, and its eigenstates
$\left\\{\left|{n}\right\rangle\right\\}_{n=0}^{\infty}$. The field
Hamiltonian reads:
$H_{\scriptscriptstyle F}=\hbar\,\Omega\,a^{\dagger}a\;,$ (1)
where $\Omega$ denotes the frequency of the vibrational mode. We assume the
resonator in a thermal equilibrium state, i.e. described by the density
operator
$\displaystyle\varrho_{F}$ $\displaystyle=$ $\displaystyle\frac{\exp(-\beta
H_{\scriptscriptstyle F})}{\mathrm{Tr}\left[{\exp(-\beta H_{\scriptscriptstyle
F})}\right]}=\sum_{n=0}^{\infty}p_{n}(\Omega,\beta)\left|{n}\right\rangle\left\langle{n}\right|\;,$
where $\beta=(k_{B}T)^{-1}$ and:
$p_{n}(\Omega,\beta)=e^{-\beta\hbar\Omega
n}\left(1-e^{-\beta\hbar\Omega}\right).$ (2)
The resonator is coupled to a superconducting qubit whose initial preparation
is under control and, after a given interaction time, the excited state
population is detected. The qubit is treated as a normalized vector in a two-
dimensional complex Hilbert space $\mathcal{H}_{Q}$, with
$\\{\left|{e}\right\rangle,\left|{g}\right\rangle\\}$ providing an orthonormal
basis. The qubit is initially prepared in a pure state
$\left|{\psi}\right\rangle=\cos\frac{\vartheta}{2}\left|{e}\right\rangle+e^{i\varphi}\sin\frac{\vartheta}{2}\left|{g}\right\rangle\,,$
(3)
with $\varphi\in\left[0,2\pi\right)$ and $\vartheta\in\left[0,\pi\right]$.
Hence the qubit density operator reduces to the projector
$\varrho_{\scriptscriptstyle
Q}=\left|{\psi}\right\rangle\left\langle{\psi}\right|$. Being a two-level
system, by appropriately choosing the zero energy level and denoting by
$\omega$ its transition frequency, the qubit Hamiltonian can be written as
$H_{q}=\frac{\hbar\omega}{2}\sigma_{z}\;.$
The qubit-resonator interaction is the interaction between a single-mode
bosonic field and a two-level system. In the rotating-wave approximations and
for the near-resonant case, i.e., for small values of the detuning
$\delta=\omega-\Omega$ we have the Jaynes-Cummings (JC) model with Hamiltonian
$\displaystyle\tilde{H}_{{\hbox{\small\sc jc}}}$ $\displaystyle=$
$\displaystyle H_{q}+H_{\scriptscriptstyle F}+H_{int}$ (4) $\displaystyle=$
$\displaystyle\frac{\hbar\omega}{2}\sigma_{z}+\hbar\Omega
a^{\dagger}a+\hbar\lambda\left(\sigma_{+}a+\sigma_{-}{a^{\dagger}}\right)\;.$
The unperturbed Hamiltonian $\tilde{H}_{{\hbox{\small\sc
jc}}}^{(0)}=H_{q}+H_{\scriptscriptstyle F}$ satisfies the eigenvalues
equations
$\tilde{H}_{{\hbox{\small\sc
jc}}}^{(0)}\left|{k,n}\right\rangle=\hbar\left[n\Omega+\frac{1}{2}\,\omega\,(-1)^{k}\right]\left|{k,n}\right\rangle\,,$
with $k=e,g$ and with the correspondences $0\leftrightarrow e$,
$1\leftrightarrow g$. In Eq. (4) $\lambda\in\mathbb{R}$ represents the
coupling strength, $\sigma_{+}a$ and $\sigma_{-}{a^{\dagger}}$ stand
respectively for the operators $\sigma_{+}\otimes a$,
$\sigma_{-}\otimes{a^{\dagger}}$ acting on the tensor product space, where
$\sigma_{\pm}$ are the qubit ladder operators. Upon choosing a suitable
rotating frame one rewrites the Hamiltonian in interaction picture
$H_{{\hbox{\small\sc jc}}}$:
$H_{{\hbox{\small\sc
jc}}}=\frac{\hbar\delta\sigma_{z}}{2}+\hbar\lambda\left(\sigma_{+}a+\sigma_{-}{a^{\dagger}}\right)\;.$
(5)
The interaction only couples, for a given $n$, the states
$\left|{e,n}\right\rangle$ and $\left|{g,n+1}\right\rangle$, and thus it is
possible to study the interaction inside the two-dimensional manifold spanned
by these states leading to a representation – the so called dressed states
basis – where $H_{{\hbox{\small\sc jc}}}$ is diagonal. We further assume the
absence of any initial correlations between the qubit and the oscillator, thus
choosing at time $t=0$ the following factorized density operator
$\varrho(0)=\varrho_{\scriptscriptstyle Q}\otimes\varrho_{{\scriptscriptstyle
F}}\;,$
whose dynamical evolution with respect to the JC Hamiltonian is given by:
$\varrho(t)=U(t)\varrho(0){U}^{\dagger}(t)\;,$
with $U(t)=\exp{\left(-\frac{i}{\hbar}H_{{\hbox{\small\sc jc}}}t\right)}$.
Time evolution entangles the qubit and the resonator sch10 and the
probabilities for the qubit to be found in the ground or excited state are
obtained via the Born rule as
$p(j|\beta)=\mathrm{Tr}_{{\scriptscriptstyle Q}\\!{\scriptscriptstyle
F}}\left[{\varrho(t)\left|{j}\right\rangle\left\langle{j}\right|\otimes{\mathbb{I}}_{\scriptscriptstyle
F}}\right]\qquad j=e,g$ (6)
where $p(j|\beta)$ denotes the conditional probability of obtaining the value
$j$ when the value of the temperature parameter is $\beta$. Upon introducing
the following quantum operation:
$\varrho_{\scriptscriptstyle
Q}\stackrel{{\scriptstyle\mathcal{E}}}{{\longmapsto}}\varrho_{\scriptscriptstyle
P}\equiv\mathrm{Tr}_{{\scriptscriptstyle
F}}\left[{U(t)\,\varrho_{\scriptscriptstyle
Q}\otimes\varrho_{\scriptscriptstyle F}\,{U}^{\dagger}(t)}\right]\;,$ (7)
where $\mathcal{E}:\mathcal{L}(\mathcal{H}_{\scriptscriptstyle
Q})\rightarrow\mathcal{L}(\mathcal{H}_{\scriptscriptstyle Q})$, Eq. (6) can be
equally rewritten at the level of the qubit subsystem alone, namely:
$p(j|\beta)=\mathrm{Tr}_{{\scriptscriptstyle
Q}}\left[{\varrho_{{\scriptscriptstyle
P}}\left|{j}\right\rangle\left\langle{j}\right|}\right]\;.$ (8)
In the following we will refer to $\varrho_{\scriptscriptstyle P}$ as the
probe state: It describes the qubit subsystem at time $t$, obtained as the
partial trace over the phonon field of the overall evolved state of the
coupled system. Since it is a density operator on $\mathcal{H}_{Q}$ it can be
arranged in a 2$\times$2 density matrix. We have
$\displaystyle\varrho_{\scriptscriptstyle P}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}p_{n}(\Omega,\beta)\left(\begin{array}[]{cc}\varrho_{ee}&\varrho_{eg}\\\
\varrho_{ge}&\varrho_{gg}\end{array}\right)\;,$ (11)
where:
$\displaystyle\ \varrho_{ee}$
$\displaystyle=\cos^{2}\frac{\vartheta}{2}\left[\cos^{2}\theta_{n}t+4\,\frac{\delta^{2}}{\theta_{n}^{2}}\sin^{2}\theta_{n}t\right]$
$\displaystyle\hskip
14.22636pt+\sin^{2}\frac{\vartheta}{2}\frac{\lambda^{2}n}{\theta_{n-1}^{2}}\sin^{2}\theta_{n-1}t,$
(12a) $\displaystyle\varrho_{eg}$
$\displaystyle=\frac{1}{2}e^{-i\varphi}\sin\vartheta\left[\cos\theta_{n-1}t+i\frac{2\delta}{\theta_{n-1}}\sin\theta_{n-1}t\right]$
$\displaystyle\hskip
14.22636pt\times\left[\cos\theta_{n}t-i\frac{2\delta}{\theta_{n}}\sin\theta_{n}t\right],$
(12b) $\displaystyle\ \varrho_{ge}$
$\displaystyle=\varrho_{eg}^{\ast}\quad\hbox{and}\quad\
\varrho_{gg}=1-\varrho_{ee},$ (12c)
with:
$\theta_{n}\equiv\theta_{n}(\delta,\lambda)=\frac{1}{2}\sqrt{\delta^{2}+4\lambda^{2}\left(n+1\right)}\,.$
### II.2 Effects of decoherence
A purely Hamiltonian dynamics doesn’t match realistic features. In real-life
scenarios quantum coherence is hard to achieve in mechanical objects, and can
be maintained for relatively small times ($\approx 10^{-9}$s ). Complete Rabi
oscillations between the phonon and the qubit excitation involve only the
first Rabi half periods, then a damping of the probabilities $p(j|\beta)$ to
$\frac{1}{2}$ is observed: the most striking signature of decoherence. Hence
we include in our model the treatment of non dissipative decoherence occurring
between the qubit and the resonator. Following Ref. atomtrap we consider an
effective model provided by adding a power-law term in the thermal
distribution, which leads to probe state matrix elements given by:
$\tilde{\varrho}_{ij}=\sum_{n=0}^{\infty}p_{n}(\Omega,\beta)\left[e^{-\gamma_{n}t}\varrho_{ij}+\frac{1}{2}\left(1-e^{-\gamma_{n}t}\right)\right]$
being $\varrho_{ij}$ the matrix elements of Eq. (12), as evaluated for the
unitary case, $i,j\in\\{e,g\\}$ and
$\gamma_{n}=b(1+n)^{a}\,.$
More explicitly
$\displaystyle\tilde{\varrho}_{ee}$
$\displaystyle=\frac{1}{2}\left[1+\sum_{n=0}^{\infty}p_{n}(\Omega,\beta)e^{-\beta
e^{-\gamma_{n}t}}\left(\varrho_{ee}-\varrho_{gg}\right)\right],$ (13a)
$\displaystyle\tilde{\varrho}_{eg}$
$\displaystyle=\frac{1}{2}\sum_{n=0}^{\infty}p_{n}(\Omega,\beta)e^{-\gamma_{n}t}\varrho_{eg},$
(13b) $\displaystyle\tilde{\varrho}_{ge}$
$\displaystyle=\tilde{\varrho}_{eg}^{\ast}\quad\hbox{and}\quad\tilde{\varrho}_{gg}=1-\tilde{\varrho}_{ee}\,.$
(13c)
One can see that the dynamical evolution now drives the qubit towards the
maximally mixed state, described by the density operator
$\frac{\mathbb{I}}{2}$.
## III Quantum thermometry
In this section we apply the tools of (local) quantum estimation theory (QET)
to the coupled qubit-oscillator system. An estimation problem always consists
in two steps: at first one has to choose a measurement and then, after
collecting a sample of outcomes, one should find an estimator, i.e. a function
to process data and to infer the value of the quantity of interest. In our
case, temperature, expressed as $\beta$, is the unknown parameter which has to
be estimated from the sample of outcomes coming from measurements performed on
the qubit. The results, a string of zeroes and ones for the case of population
measurement, are distributed according to the probabilities
$p(j|\beta)\equiv\varrho_{jj}$ of Eqs. (8) and (12) [or Eq. (13) in presence
of decoherence]. The Cramér-Rao inequality establishes that the variance
Var$(\beta)$ of any unbiased estimator is lower bounded by
$\mbox{Var}(\beta)\geq\frac{1}{MF(\beta)}\ ,$ (14)
where $M$ is the cardinality of the sample, i.e., the number of measurements,
and $F(\beta)$ the so-called Fisher information (FI):
$\displaystyle F(\beta)$ $\displaystyle=$
$\displaystyle\sum_{j=e,g}p(j|\beta)\left[\partial_{\beta}\ln
p(j|\beta)\right]^{2}$ (15) $\displaystyle=$
$\displaystyle\frac{{\left[\partial_{\beta}p(e|\beta)\right]}^{2}}{p(e|\beta)}+\frac{{\left[\partial_{\beta}p(g|\beta)\right]}^{2}}{p(g|\beta)}\;.$
Efficient estimators are those saturating the Cramér-Rao inequality and their
existence depends on the statistical model. However, independently of the
statistical model we have that for sufficiently large samples, i.e., in the
asymptotic regime $M\gg 1$, maximum likelihood estimators are always
efficient.
Quantum mechanically, the probability of obtaining the outcome $j\in\\{e,g\\}$
from a measurement is given according to the Born rule by
$p(j|\beta)=\mathrm{Tr}\left[{\varrho_{\scriptscriptstyle P}\Pi_{j}}\right]$,
where the probe state $\varrho_{\scriptscriptstyle
P}\equiv\varrho_{\scriptscriptstyle P}(\beta)$ parametrized by the unknown
quantity $\beta$ is referred to as the quantum statistical model, and the
collection of operators $\\{\Pi_{j}\\}$, $\Pi_{j}\geq 0$,
$\sum_{j}\Pi_{j}=\mathbb{I}$ is the probability operator-valued measure
describing the measurement taking place on the qubit. In our case the qubit
excited state population is probed and the measurement reduces to a projective
one, $\left|{e}\right\rangle\left\langle{e}\right|$ and
$\left|{g}\right\rangle\left\langle{g}\right|=\mathbb{I}-\left|{e}\right\rangle\left\langle{e}\right|$,
i.e., we are measuring the Pauli operator
$\sigma_{z}=\left|{e}\right\rangle\left\langle{e}\right|-\left|{g}\right\rangle\left\langle{g}\right|$.
Once the observable is fixed, we optimize the estimation procedure by
maximizing the FI over the qubit state parameters, $\vartheta$ and $\varphi$,
as well as over the parameters driving the interaction – i.e., the detuning
$\delta$ and the interaction time $t$. In other words, by employing the
optimal qubit preparation and tuning the interaction parameters one may find a
working regime achieving the maximum precision for that kind of measurement.
On the other hand, one may also maximize the FI over all possible quantum
measurements. Upon defining the symmetric logarithmic derivative (SLD)
$L_{\beta}$ as the selfadjoint operator satisfying the equation
$\frac{L_{\beta}\varrho_{\scriptscriptstyle P}+\varrho_{\scriptscriptstyle
P}L_{\beta}}{2}=\partial_{\beta}\varrho_{\scriptscriptstyle P}\;,$ (16)
it is possible to show that the Fisher information $F(\beta)$ of any quantum
measurement is upper bounded by the following quantity:
$F(\beta)\leq G(\beta)\equiv\mathrm{Tr}\left[{\varrho_{\scriptscriptstyle
P}L_{\beta}^{2}}\right]\;,$ (17)
which is called quantum Fisher information (QFI). QFI does not depend on the
measurement carried on the qubit—indeed being obtained by maximizing over the
possible measurement. It is rather an attribute of the family of states
$\varrho_{\scriptscriptstyle P}(\beta)$ parametrized by the temperature.
Looking back to the Cramér-Rao inequality Eq.(14) one sees that QFI allows one
to write its natural quantum version
$\displaystyle\mbox{Var}(\beta)\geq\frac{1}{MG(\beta)}\,.$ (18)
The above equation represents the Quantum Cramér-Rao bound (QCR), i.e. the
ultimate bound to the precision allowed by quantum mechanics for a given
statistical model $\varrho_{\scriptscriptstyle P}(\beta)$. An optimal
measurement, i.e. a measurement whose FI $F(\beta)=G(\beta)$ equals the QFI
for the parameter $\beta$, is given by the observable corresponding to the
spectral measure of the SLD $L_{\beta}$. On the other hand, other kind of
measurements may achieve optimality for the whole range of values of $\beta$
or for a subset of values. Indeed, we will see in the following that
population measurement is optimal for a suitable choice of the initial qubit
preparation. We remind that for the estimation of a single parameter, as it is
in our case, the QCR may be always attained, and an estimator saturating Ineq.
(18) is called efficient. The existence of an efficient estimator depends on
the statistical model. However, independently of the statistical model, for
sufficiently large samples, i.e., in the asymptotic regime $M\gg 1$, maximum
likelihood and Bayesian estimators are always efficient.
Upon diagonalizing the probe state one achieves the decomposition
$\varrho_{\scriptscriptstyle
P}=\varrho_{+}\left|{\psi_{+}}\right\rangle\left\langle{\psi_{+}}\right|\,+\,\varrho_{-}\left|{\psi_{-}}\right\rangle\left\langle{\psi_{-}}\right|$
and is able to solve the equation for SLD
$\displaystyle L_{\beta}=$
$\displaystyle\,\frac{\left\langle{\psi_{+}}\right|\partial_{\beta}\varrho_{\scriptscriptstyle
P}\left|{\psi_{+}}\right\rangle}{\varrho_{+}}\left|{\psi_{+}}\right\rangle\left\langle{\psi_{+}}\right|$
$\displaystyle+\,\frac{\left\langle{\psi_{-}}\right|\partial_{\beta}\varrho_{\scriptscriptstyle
P}\left|{\psi_{-}}\right\rangle}{\varrho_{-}}\left|{\psi_{-}}\right\rangle\left\langle{\psi_{-}}\right|$
$\displaystyle+\,\frac{2}{\varrho_{+}+\varrho_{-}}\left[\left\langle{\psi_{+}}\right|\partial_{\beta}\varrho_{\scriptscriptstyle
P}\left|{\psi_{-}}\right\rangle\left|{\psi_{+}}\right\rangle\left\langle{\psi_{-}}\right|\right.$
$\displaystyle+\,\left.\left\langle{\psi_{-}}\right|\partial_{\beta}\varrho_{\scriptscriptstyle
P}\left|{\psi_{+}}\right\rangle\left|{\psi_{-}}\right\rangle\left\langle{\psi_{+}}\right|\right],$
(19)
finally obtaining an explicit formula for the QFI
$\displaystyle G(\beta)=$
$\displaystyle\,\frac{\left(\partial_{\beta}\varrho_{+}\right)^{2}}{\varrho_{+}}+\frac{\left(\partial_{\beta}\varrho_{-}\right)^{2}}{\varrho_{-}}$
$\displaystyle+\,2\kappa\,\left[\left|\langle\psi_{-}|\partial_{\beta}\psi_{+}\rangle\right|^{2}+\left|\langle\psi_{+}|\partial_{\beta}\psi_{-}\rangle\right|^{2}\right]$
(20)
where
$|\partial_{\beta}\psi_{\pm}\rangle=\partial_{\beta}\langle
e|\psi_{\pm}\rangle\,|e\rangle+\partial_{\beta}\langle
g|\psi_{\pm}\rangle\,|g\rangle\,,$
and
$\kappa=\frac{\left(\varrho_{+}-\varrho_{-}\right)^{2}}{\varrho_{+}+\varrho_{-}}=(1-2\varrho_{+})^{2}\,.$
Eq. (III) contains a first term which resembles the FI and a second one, truly
quantum in nature, which leads to the QCR and vanishes whenever
$\left|{\psi_{\pm}}\right\rangle$ does not depend on $\beta$.
## IV Dynamics of the Fisher information and optimal working regimes
In this section we report results for the qubit-resonator coupled system with
physical parameters chosen in a range matching the experimental setup of Ref.
nmr11 . More specifically, we present a systematic study of the FI for
population measurement as a function of the state and interaction parameters,
carrying out numerical maximization and finding the optimal working regimes.
We also evaluate the QFI of the family of states $\varrho_{P}(\beta)$ and find
the ultimate bound to precision, i.e. a benchmark in order to assess the
performances of qubit thermometry via population measurement.
Hereafter we work with dimensionless quantities by rescaling times and
frequencies in units of the coupling $\lambda$. We thus substitute time,
detuning and decoherence parameters by their rescaled counterparts
$\displaystyle t\longmapsto\tau\equiv\lambda
t,\quad\delta\longmapsto\gamma\equiv\delta/\lambda,\quad
b\longmapsto\tilde{b}\equiv b/\lambda\,.$
Effective detuning $\gamma$ will range in $|\gamma|\in[0,1.5]$. Also a
dimensionless effective temperature $\tilde{\beta}$ is defined, provided by
the substitution
$\beta\longmapsto\tilde{\beta}\equiv\beta\hbar\Omega\,.$
For convenience, we continue to term $\tilde{\beta}$ and $\tilde{b}$
respectively $\beta$ and $b$.
### IV.1 Resonant Hamiltonian regime
Upon using the expression of the diagonal matrix elements in Eqs. (12) we have
evaluated the FI of Eq. (15). We start the discussion by considering the
resonant case, i.e zero detuning, and analyze the effect of detuning afterward
in this Section. For convenience we adopt the notation $F(\beta)$ for the FI,
but keep in mind the complete dependence $F(\beta;\vartheta,\tau,\gamma)$ on
both the qubit degrees of freedom and the parameters $\gamma$ and $\tau$ which
drive the coupling. Notice that $F(\beta)$ does not depend on the qubit phase
$\varphi$: its building-blocks are in fact the probabilities $p(e|\beta)$ and
$p(g|\beta)$, whereas $\varphi$ only appears in off-diagonal matrix elements.
Varying the parameter $\vartheta$ from $\pi$ to $0$ we span the entire class
of qubit preparation, starting from $\left|{1}\right\rangle$, going trough a
superposition and ending in $\left|{0}\right\rangle$.
Figure 1: (Color online) Upper panel: FI for $\beta=10$ as a function of the
effective time $\tau$, for different $\vartheta$ values: $\vartheta=\pi$
(dashed blue), $\vartheta=0.95\,\pi$ (dot-dashed magenta) and $\vartheta=0$
(solid green). FI takes a pronounced global maximum at
$(\vartheta,\tau)=\left(\pi,\frac{\pi}{2}\right)$ while it is possible to see
a secondary extremely peaked maximum, which occurs for $\tau=\pi$ and
preparing the qubit in $\left|{0}\right\rangle$. Lower panel: log-linear plot
of the FI for $\beta=10$ as a function of $\vartheta$ for,
$\tau=\frac{\pi}{2}$ (dashed blue), $\tau=\frac{\pi}{2}+\varepsilon$ (dot-
dashed magenta), $\tau=\pi$ (solid green), $\tau=\pi+\varepsilon$ (dotted
red), with $\varepsilon=0.01$.
Let us now consider the system at a fixed value of the temperature, e.g. where
the resonator is supposed to be very close to the ground state, say
$\beta=10$. The probabilities $p(j|\beta)=\varrho_{jj}$ evolve periodically in
time according to Eq. (12), as the coupled system undergoes Rabi oscillations.
The corresponding behavior of the FI is shown in the upper panel of Fig. 1.
The FI displays a robust maximum at the optimal time $\tau_{\rm
max}=\frac{\pi}{2}$ for $\vartheta=\pi$, corresponding to prepare the qubit in
its ground state. This maximum is, at the same time, the global and the
smoothest one. In fact, as soon as $\vartheta$ is moved from $\pi$ the FI
suddenly drops to zero, except for a sharp peak centered in $\tau_{\rm max}$,
monotonically decreasing with respect to $\vartheta$, as shown in the lower
panel of Fig. 1. Another maximum of the same order of the global one can be
found at $(\vartheta,\tau)=\left(0,\pi\right)$ but it is extremely peaked,
thus representing a bad (unstable) choice for a possible measurement. Upon
inspecting the temporal evolution of the excited state probability we found
that $p(e|\beta)$ has a minimum at $\tau=\tau_{\rm max}$, a fact which gives
us a physical insight on the FI behavior: since our goal is the estimation of
a vanishing quantity which carries information about thermal disorder, we
expect to find the maximum sensitivity in our predictions where the excitation
is most likely stored – as a phonon – in the resonator, i.e., when
$p(e|\beta)$ is minimum.
Figure 2: (Color online) Log-linear plot of the FI as a function of effective
time $\tau$ for different values of $\beta$. The qubit is prepared in the
ground state $\left|{1}\right\rangle$ ($\vartheta=\pi$). From bottom to top
$\beta=15$ (solid blue), $\beta=10$ (dashed magenta), $\beta=5$ (dot-dashed
green), $\beta=1$ (dotted red). Upon raising the temperature the FI no longer
keeps a scale-free shape: thermal excitations modifies its profile making it
irregular. In particular the global maximum comes earlier in time.
Let us now turn our attention to the dependence of the FI on the temperature
itself. In Fig. 2 we show, on a logarithmic scale, the temporal evolution of
the FI for different values of $\beta$. FI varies over several orders of
magnitude, matching our intuition that the closer we are to the ground state,
the harder is to achieve a given precision in estimation of temperature.
Furthermore, upon lowering the temperature, the temporal evolution of
$p(j|\beta)$ becomes less involved, finally approaching the exactly periodic
one of Rabi oscillations, which in turn freezes the profile of the FI in a
shape independent on the temperature itself.
The qubit preparation $\theta=\pi$ is universally optimal, i.e., it leads to a
maximum of the FI independently of the interaction time. After fixing
$\theta=\pi$ we have numerically maximized $F(\beta)$ with respect to $\tau$.
The solid blue line of the upper panel of Fig. 3 is the the log-plot of
$F_{M}(\beta)=\max_{\tau}F(\beta)\,,$
as a function of $\beta$, from which it is apparent the exponential decrease
of the maximum value achieved by the FI for increasing $\beta$. The Cramér-Rao
inequality immediately relates this fact to an exponential loss of sensitivity
moving towards the quantum ground state of the resonator. An other interesting
feature that emerges from the maximization is a shift in the value of the
optimal interaction time. In the lower panel of Fig. 3 we can recognize the
existence of a steady value for the optimal time $\tau_{\rm
max}=\frac{\pi}{2}$ when approaching the ground state, while for smaller
values of $\beta$ the optimal time comes earlier. In fact, the temporal
evolution of FI (see Fig. 2) not only predicts an exponential increase of the
global maximum when temperatures are raised, but also a shift of its location.
Figure 3: (Color online) Upper panel: log-log plot of the FI maximized over
$\tau$ as a function of $\beta$, with $\theta=\pi$ for different values of
detuning: $\gamma=0$ (solid blue), $\gamma=1$ (dashed magenta), $\gamma=1.5$
(dot-dashed green). Bottom panel: the times $\tau_{\rm max}$ which maximizes
the FI as a function of $\beta$, with $\theta=\pi$ for different values of
$\gamma$ (same values and colors of the upper panel).
### IV.2 Effects of detuning
In this section we take into account the possible existence of a nonzero
detuning $\gamma$ between the oscillator and the qubit frequencies. This has
two main consequences, which are both illustrated in Fig. 3. On the one hand,
the maximum achievable value of the FI slightly decreases and, on the other
hand, the optimal interaction time $\tau_{\rm max}$ at which the maximum takes
place anticipates. Therefore, the best working conditions to achieve the
optimal sensitivity in the estimation of $\beta$ correspond to have the qubit
and the resonator in resonance. It is also worth to notice that $\gamma$ does
not represent a critical parameter, as the initial preparation of the qubit,
since the FI dependence on $\gamma$ is smooth. One can see this in the upper
panel of Fig. 3, where we see that curves corresponding to quite different
values of the detuning are almost superposed.
### IV.3 Quantum Fisher information
In order to assess the performances of the population measurement in the
estimation of temperature we have evaluated the QFI of the family
$\varrho_{\scriptscriptstyle P}(\beta)$. The diagonalization of the probe
state has to be carried out numerically, hence in general analytical
expressions of the QFI are not available. A first fact is that $G(\beta)$
turns out to be independent on the qubit phase $\varphi$, which then does not
represent an extra degree of freedom whereby gain more restrictive bounds to
precision on $\mbox{Var}(\beta)$. Even the optimal qubit preparation for to
the best conceivable measurement involves control of the parameter $\vartheta$
only.
As we have done for the FI, we start to inspect the QFI behavior for a fixed
value of temperature $\beta$ in the resonant case. Also for the QFI the
maximum is achieved by preparing the qubit in the state
$\left|{g}\right\rangle$ and probing it at time $\tau_{\rm max}$. In this case
the behavior of $G(\beta)$ is identical to that of $F(\beta)$, as it is
apparent by comparing Figs. 1 and 4. In other words, for a given value of the
parameter $\beta$ into the range explored, the choice
$(\vartheta,\tau)=(\pi,\tau_{\rm max})$ makes population measurement optimal.
Moreover, the QFI itself reaches its global maximum for that choice. Thus,
provided that an optimal estimator is employed, e.g. maximum likelihood in the
asymptotic regime, this strategy provides optimality in sense that either
inequality (17) is saturated and the right-hand side of QCR is as low as
possible.
This conclusion is confirmed upon a closer inspection of the probe state. When
$\vartheta=\pi$ the off-diagonal terms vanish and $\varrho_{\scriptscriptstyle
P}$ is diagonal, with eigenvalues
$\displaystyle\varrho_{+}$
$\displaystyle=\sum_{n=0}^{\infty}p_{n}(\Omega,\beta)\sin^{2}\left[\sqrt{\gamma^{2}+4n}\,\frac{\tau}{2}\right]\frac{n}{n+\gamma^{2}/4}$
(21a) $\displaystyle\varrho_{-}$ $\displaystyle=1-\varrho_{+}$ (21b)
As a consequence, the QFI reduces to
$G(\beta;\pi,\tau,\gamma)=\frac{\left(\partial_{\beta}\varrho_{+}\right)^{2}}{\varrho_{+}}+\frac{\left(\partial_{\beta}\varrho_{-}\right)^{2}}{\varrho_{-}}\,,$
which coincides with the FI ruling the estimation of $\beta$ via population
measurement.
On the other hand, some striking difference emerges between the performances
of population measurement and that of the optimal one if the qubit is not
prepared in the optimal (ground) state.
Figure 4: (Color online) Upper panel : QFI for $\beta=10$ as a function of
$\tau$, for $\vartheta=\pi$ (dashed blue), $\vartheta=0.95\,\pi$ (dot-dashed
magenta) and $\vartheta=0$ (solid green). QFI behaves like FI for
$\vartheta=\pi$ leading to the same maximum, while for smaller angles it shows
a smoother profile. For angles $0<\vartheta<\pi$ one may find measurements
which improve the precision of temperature estimation. Bottom panel: QFI for
$\beta=10$ as a function of $\vartheta$ for $\tau=\frac{\pi}{2}$ (dashed
blue), $\tau=\frac{\pi}{2}+\varepsilon$ (dot-dashed magenta), $\tau=\pi$
(solid green), $\tau=\pi+\varepsilon$ (dotted red), with $\varepsilon=0.01$.
In the lower panel of Fig. 4 we show $G(\beta)$ as a function of $\tau$ for
different values of $\vartheta$: for $\vartheta<\pi$ the decrease of $G$ is
definitely smoother than that of $F$ and thus, in principle, some measurement
may be found making the initial preparation a less critical parameter.
Moreover inspecting the cut of the QFI along $\tau=\pi$ we note that the
maximum in $\vartheta=0$ becomes more achievable compared to the one of
$F(\beta)$. All these features suggest that for qubit preparations different
from the ground state there will be a sensible difference between the
precision provided by population measurement and the optimal one implementable
on the system. On the other hand, being the overall maximum achievable with
population measurement, our results indicate that the achievement of the
ultimate bound to precision allowed by quantum mechanics is in the
capabilities of the current technology.
### IV.4 Effects of decoherence
In this section we discuss the solution of the reduced qubit dynamics in the
presence of dissipative decoherence, see Eq. (13), and inspect the
corresponding behavior of the FI. For the sake of simplicity we consider zero
detuning. Analogue results are obtained when including the detuning.
The probabilities $p(j|\beta)=\tilde{\varrho}_{jj}$ are damped so that,
waiting for a sufficient long time, whose value depends on $a$ and $b$, we
would find them to be identically $1/2$ or, equally stated, the dynamical
evolution brings the state to the maximally mixed one. The contribution of
decoherence is of the kind exp$\left[-b(1+n)^{a}\tau\right]$ for every $n$,
where $b$ has been rescaled in coupling units $b\longmapsto b/\lambda$. Being
a multiplicative coefficient, as soon as $b$ is different from zero, the
exponential term will participate in killing the sums. Our calculations show a
relevant dependence of the FI on the parameter $b$, namely values $b\approx
10^{-5}$ are sufficient to produce visible effects, while varying $a$ in the
range $(0,1)$ does not deeply influence of FI behavior.
In Fig. 5 we show the temporal evolution of the FI for $\beta=10$, in the
presence of decoherence and for different initial preparations of the qubit.
In the Hamiltonian regime for large $\beta$ the resonator is close to the
ground state, the evolution of $p(j|\beta)$ is periodic and hence, due to Eq.
(15), the same is true for the FI. Upon incorporating decoherence we see that
FI decays at a rate depending on $b$ and thus an irreversible dynamics
emerges, which matches the physical evidence of a limited coherence time. On
the other hand, a clear maximum at $\tau=\pi/2$ still appears, with a slightly
decreased value of $F(\beta)$. In the lower panel of Fig. 5 we show the
maximum value $F_{M}=\max_{\tau}F(\beta)$ for different values of the
decoherence parameter. As it is apparent from the (log-log) plot for high
temperature (smaller $\beta$) the effect of decoherence is negligible, whereas
for increasing $\beta$ the effect is becoming more and more relevant.
Figure 5: (Color online) Upper panel: Fisher information $F(\beta)$ for
$\beta=10$ as a function of $\tau$ in the presence of decoherence and for
different qubit preparations. The decoherence parameters are chosen as to
$a=0.1$ and $b=10^{-5}$. Dashed blue line stands for $\vartheta=\pi$, dot-
dashed magenta for $\vartheta=0.95\,\pi$ while solid green ones for
$\vartheta=0$. Having included decoherence treatment enables us not to
restrict the evolution to the first Rabi half-period. Lower panel: log-log
plot of the Fisher information $F_{M}(\beta)$ maximized over the interaction
time, and in the presence of decoherence, as a function of $\beta$ and for
fixed $\vartheta=\pi$, for $b=0$ (solid green), $b=10^{-5}$ (dashed magenta),
$b=10^{-4}$ (dot-dashed blue).
## V Conclusions
The temperature of a physical object cannot be directly measurable. On the
other hand is can be regarded as a parameter whose value can be indirectly
inferred by measuring some proper observable and then suitably processing the
outcomes, an inference procedure usually referred to as an estimation
procedure. In the case of a micromechanical oscillator with an isolated
vibrational mode, effective schemes have been suggested and realized nmr11
which rely on coupling the resonator to a superconducting qubit and probing
the latter using population measurements. In other words, the qubit is
employed as a quantum thermometer to demonstrate that the resonator has been
cooled to its quantum ground state. In this paper we have analyzed in details
qubit thermometry in these systems, i.e., the estimation of temperature via
quantum limited measurements performed on the qubit. In the framework of
quantum estimation theory we have analyzed precision as a function of both the
qubit initial preparation and the interaction parameters, and we have
evaluated the limits to precision posed by quantum mechanics to qubit
thermometry.
We have computed the FI for population measurement, which is the appropriate
figure of merit to assess the precision of estimation, and have found that its
maximum, and hence the minimum variance in the estimated temperature, is
achieved by preparing the qubit in the ground state, and probing it at an
emergent time $\tau_{\rm max}$, which is predictable. Furthermore, we have
analyzed in details how the maximum depends on the temperature itself, on the
detuning, and on the noise parameter when one takes into account non
dissipative decoherence. In order to evaluate the ultimate bound allowed by
quantum mechanics to the sensitivity of temperature estimation, we have also
computed the quantum Fisher information. We found that QFI is maximized for
the same choice of qubit preparation and measurement time of the FI, and that
for these common values the maxima of FI and QFI coincide. We thus conclude
that population measurement is optimal for temperature estimation.
The range of parameters addressed in our analysis is that of recent
experimental implementations nmr11 . We thus conclude that optimal estimation
of temperature can be done with current technology. Since the FI of population
measurement, and the QFI of the model, both decrease with the decrease of
temperature, the estimation of lower temperature will be intrinsically less
precise. On the other hand, since the are regimes, also in the presence of
decoherence, where the maxima of the FI and the QFI are reasonably smooth as a
function of the qubit preparation and of the interaction time we do not expect
any ”no-go” theorem for temperature estimation. In other words, we expect that
optimal estimation of lower resonator temperatures, perhaps achievable with
further experimental advances, will be still possible with population
measurements. On the other hand, “optimality” will correspond to an inherently
less precise procedure compared to the case of higher temperature.
Our analysis shows the optimality of feasible qubit thermometry in providing
quantum benchmarks for high precision temperature measurement, as well as an
efficient operational quantification of temperature for mechanical modes lying
arbitrary close to their ground state. In other words, achievement of the
ultimate bound to precision allowed by quantum mechanics is in the
capabilities of the current technology. Our results also confirm that QET is a
useful tool for assessing and comparing inference procedures arising in
quantum limited measurements sta10 , even when mesoscopic objects are
involved.
## Acknowledgments
This work has been partially supported the CNR-CNISM agreement.
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|
arxiv-papers
| 2011-03-15T10:14:05 |
2024-09-04T02:49:17.671914
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matteo Brunelli, Stefano Olivares, Matteo G. A. Paris",
"submitter": "Matteo G. A. Paris",
"url": "https://arxiv.org/abs/1103.2875"
}
|
1103.2903
|
11institutetext: DTU Informatics, Technical University of Denmark, Lyngby,
Denmark. 11email: fn@imm.dtu.dk, http://www.imm.dtu.dk/~fn/
# A new ANEW: Evaluation of a word list for sentiment analysis in microblogs
Finn Årup Nielsen
###### Abstract
Sentiment analysis of microblogs such as Twitter has recently gained a fair
amount of attention. One of the simplest sentiment analysis approaches
compares the words of a posting against a labeled word list, where each word
has been scored for valence, — a “sentiment lexicon” or “affective word
lists”. There exist several affective word lists, e.g., ANEW (Affective Norms
for English Words) developed before the advent of microblogging and sentiment
analysis. I wanted to examine how well ANEW and other word lists performs for
the detection of sentiment strength in microblog posts in comparison with a
new word list specifically constructed for microblogs. I used manually labeled
postings from Twitter scored for sentiment. Using a simple word matching I
show that the new word list may perform better than ANEW, though not as good
as the more elaborate approach found in SentiStrength.
## 1 Introduction
Sentiment analysis has become popular in recent years. Web services, such as
socialmention.com, may even score microblog postings on Identi.ca and Twitter
for sentiment in real-time. One approach to sentiment analysis starts with
labeled texts and uses supervised machine learning trained on the labeled text
data to classify the polarity of new texts [1]. Another approach creates a
sentiment lexicon and scores the text based on some function that describes
how the words and phrases of the text matches the lexicon. This approach is,
e.g., at the core of the _SentiStrength_ algorithm [2].
It is unclear how the best way is to build a sentiment lexicon. There exist
several word lists labeled with emotional valence, e.g., ANEW [3], General
Inquirer, OpinionFinder [4], SentiWordNet and WordNet-Affect as well as the
word list included in the SentiStrength software [2]. These word lists differ
by the words they include, e.g., some do not include strong obscene words and
Internet slang acronyms, such as “WTF” and “LOL”. The inclusion of such terms
could be important for reaching good performance when working with short
informal text found in Internet fora and microblogs. Word lists may also
differ in whether the words are scored with sentiment strength or just
positive/negative polarity.
I have begun to construct a new word list with sentiment strength and the
inclusion of Internet slang and obscene words. Although we have used it for
sentiment analysis on Twitter data [5] we have not yet validated it. Data sets
with manually labeled texts can evaluate the performance of the different
sentiment analysis methods. Researchers increasingly use Amazon Mechanical
Turk (AMT) for creating labeled language data, see, e.g., [6]. Here I take
advantage of this approach.
## 2 Construction of word list
My new word list was initially set up in 2009 for tweets downloaded for online
sentiment analysis in relation to the United Nation Climate Conference
(COP15). Since then it has been extended. The version termed AFINN-96
distributed on the
Internet111http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=59819
has 1468 different words, including a few phrases. The newest version has 2477
unique words, including 15 phrases that were not used for this study. As
SentiStrength222http://sentistrength.wlv.ac.uk/ it uses a scoring range from
$-5$ (very negative) to $+5$ (very positive). For ease of labeling I only
scored for valence, leaving out, e.g., subjectivity/objectivity, arousal and
dominance. The words were scored manually by the author.
The word list initiated from a set of obscene words [7, 8] as well as a few
positive words. It was gradually extended by examining Twitter postings
collected for COP15 particularly the postings which scored high on sentiment
using the list as it grew. I included words from the public domain _Original
Balanced Affective Word List_
333http://www.sci.sdsu.edu/CAL/wordlist/origwordlist.html by Greg Siegle.
Later I added Internet slang by browsing the Urban
Dictionary444http://www.urbandictionary.com including acronyms such as WTF,
LOL and ROFL. The most recent additions come from the large word list by
Steven J. DeRose, _The Compass DeRose Guide to Emotion
Words_.555http://www.derose.net/steve/resources/emotionwords/ewords.html The
words of DeRose are categorized but not scored for valence with numerical
values. Together with the DeRose words I browsed Wiktionary and the synonyms
it provided to further enhance the list. In some cases I used Twitter to
determine in which contexts the word appeared. I also used the Microsoft Web
n-gram similarity Web service (“Clustering words based on context
similarity”666http://web-ngram.research.microsoft.com/similarity/) to discover
relevant words. I do not distinguish between word categories so to avoid
ambiguities I excluded words such as patient, firm, mean, power and frank.
Words such as “surprise”—with high arousal but with variable sentiment—were
not included in the word list.
Most of the positive words were labeled with +2 and most of the negative words
with –2, see the histogram in Figure 1. I typically rated strong obscene
words, e.g., as listed in [7], with either –4 or –5. The word list have a bias
towards negative words (1598, corresponding to 65%) compared to positive words
(878). A single phrase was labeled with valence 0. The bias corresponds
closely to the bias found in the OpinionFinder sentiment lexicon (4911 (64%)
negative and 2718 positive words).
Figure 1: Histogram of my valences.
I compared the score of each word with mean valence of ANEW. Figure 2 shows a
scatter plot for this comparison yielding a Spearman’s rank correlation on
0.81 when words are directly matched and including words only in the
intersection of the two word lists. I also tried to match entries in ANEW and
my word list by applying Porter word stemming (on both word lists) and WordNet
lemmatization (on my word list) as implemented in NLTK [9]. The results did
not change significantly.
Figure 2: Correlation between ANEW and my new word list.
When splitting the ANEW at valence 5 and my list at valence 0 I find a few
discrepancies: aggressive, mischief, ennui, hard, silly, alert, mischiefs,
noisy. Word stemming generates a few further discrepancies, e.g.,
alien/alienation, affection/affected, profit/profiteer.
Apart from ANEW I also examined General Inquirer and the OpinionFinder word
lists. As these word lists report polarity I associated words with positive
sentiment with the valence +1 and negative with –1. I furthermore obtained the
sentiment strength from SentiStrength via its Web
service777http://sentistrength.wlv.ac.uk/ and converted its positive and
negative sentiments to one single value by selecting the one with the
numerical largest value and zeroing the sentiment if the positive and negative
sentiment magnitudes were equal.
## 3 Twitter data
For evaluating and comparing the word list with ANEW, General Inquirer,
OpinionFinder and SentiStrength a data set of 1,000 tweets labeled with AMT
was applied. These labeled tweets were collected by Alan Mislove for the
_Twittermood_ /“Pulse of a
Nation”888http://www.ccs.neu.edu/home/amislove/twittermood/ study [10]. Each
tweet was rated ten times to get a more reliable estimate of the human-
perceived mood, and each rating was a sentiment strength with an integer
between 1 (negative) and 9 (positive). The average over the ten values
represented the canonical “ground truth” for this study. The tweets were not
used during the construction of the word list.
To compute a sentiment score of a tweet I identified words and found the
valence for each word by lookup in the sentiment lexicons. The sum of the
valences of the words divided by the number of words represented the combined
sentiment strength for a tweet. I also tried a few other weighting schemes:
The sum of valence without normalization of words, normalizing the sum with
the number of words with non-zero valence, choosing the most extreme valence
among the words and quantisizing the tweet valences to +1, 0 and –1. For ANEW
I also applied a version with match using the NLTK WordNet lemmatizer.
## 4 Results
Figure 3: Scatter plot of sentiment strengths for 1,000 tweets with AMT
sentiment plotted against sentiment found by application or my word list.
My word tokenization identified 15,768 words in total among the 1,000 tweets
with 4,095 unique words. 422 of these 4,095 words hit my 2,477 word sized
list, while the corresponding number for ANEW was 398 of its 1034 words. Of
the 3392 words in General Inquirer I labeled with non-zero sentiment 358 were
found in our Twitter corpus and for OpinionFinder this number was 562 from a
total of 6442.
| My | ANEW | GI | OF | SS
---|---|---|---|---|---
AMT | .564 | .525 | .374 | .458 | .610
My | | .696 | .525 | .675 | .604
ANEW | | | .592 | .624 | .546
GI | | | | .705 | .474
OF | | | | | .512
Table 1: Pearson correlations between sentiment strength detections methods on
1,000 tweets. AMT: Amazon Mechanical Turk, GI: General Inquirer, OF:
OpinionFinder, SS: SentiStrength.
I found my list to have a higher correlation (Pearson correlation: 0.564,
Spearman’s rank correlation: 0.596, see the scatter plot in Figure 3) with the
labeling from the AMT than ANEW had (Pearson: 0.525, Spearman: 0.544). In my
application of the General Inquirer word list it did not perform well having a
considerable lower AMT correlation than my list and ANEW (Pearson: 0.374,
Spearman: 0.422). OpinionFinder with its 90% larger lexicon performed better
than General Inquirer but not as good as my list and ANEW (Pearson: 0.458,
Spearman: 0.491). The SentiStrength analyzer showed superior performance with
a Pearson correlation on 0.610 and Spearman on 0.616, see Table 1.
I saw little effect of the different tweet sentiment scoring approaches: For
ANEW 4 different Pearson correlations were in the range 0.522–0.526. For my
list I observed correlations in the range 0.543–0.581 with the extreme scoring
as the lowest and sum scoring without normalization the highest. With
quantization of the tweet scores to +1, 0 and –1 the correlation only dropped
to 0.548. For the Spearman correlation the sum scoring with normalization for
the number of words appeared as the one with the highest value (0.596).
Figure 4: Evolution of performance as the word list is extended with from 5
words to the full set of words (2477). The upper panel is for the Pearson
correlation while the lower for the Spearman rank correlation. The boxplots
are generated from 50 resamples among the 2477 words.
Figure 4 plots the evolution of the performance of the word list on the
Twitter as the word list is extended from 5 words to the full set of 2477
words.
To examine whether the difference in performance between the application of
ANEW and my list is due to a different lexicon or a different scoring I looked
on the intersection between the two word lists. With a direct match this
intersection consisted of 299 words. Building two new sentiment lexicons with
these 299 words, one with the valences from my list, the other with valences
from ANEW, and applying them on the Twitter data I found that the Pearson
correlations were 0.49 and 0.52 to ANEW’s advantage.
## 5 Discussion
On the simple word list approach for sentiment analysis I found my list
performing slightly ahead of ANEW. However the more elaborate sentiment
analysis in SentiStrength showed the overall best performance with a
correlation to AMT labels on 0.610. This figure is close to the correlations
reported in the evaluation of the SentiStrength algorithm on 1,041 MySpace
comments (0.60 and 0.56) [2].
Even though General Inquirer and OpinionFinder have the largest word lists I
found I could not make them perform as good as SentiStrength, my list and ANEW
for sentiment strength detection in microblog posting. The two former lists
both score words on polarity rather than strength and it could explain the
difference in performance.
Is the difference between my list and ANEW due to better scoring or more
words? The analysis of the intersection between the two word list indicated
that the ANEW scoring is better. The slightly better performance of my list
with the entire lexicon may be due to its inclusion of Internet slang and
obscene words.
Newer methods, e.g., as implemented in SentiStrength, use a range of
techniques: detection of negation, handling of emoticons and spelling
variations [2]. The present application of my list used none of these
approaches and might have benefited. However, the SentiStrength evaluation
showed that valence switching at negation and emoticon detection might not
necessarily increase the performance of sentiment analyzers (Tables 4 and 5 in
[2]).
The evolution of the performance (Figure 4) suggests that the addition of
words to my list might still improve its performance slightly.
Although my list comes slightly ahead of ANEW in Twitter sentiment analysis,
ANEW is still preferable for scientific psycholinguistic studies as the
scoring has been validated across several persons. Also note that ANEW’s
standard deviation was not used in the scoring. It might have improved its
performance.
## Acknowledgment
I am grateful to Alan Mislove and Sune Lehmann for providing the 1,000 tweets
with the Amazon Mechanical Turk labels and to Steven J. DeRose and Greg Siegle
for providing their word lists. Mislove, Lehmann and Daniela Balslev also
provided input to the article. I thank the Danish Strategic Research Councils
for generous support to the ‘Responsible Business in the Blogosphere’ project.
## References
* [1] Pang, B., Lee, L.: Opinion mining and sentiment analysis. Foundations and Trends in Information Retrieval 2(1-2) (2008) 1–135
* [2] Thelwall, M., Buckley, K., Paltoglou, G., Cai, D., Kappas, A.: Sentiment strength detection in short informal text. Journal of the American Society for Information Science and Technology 61(12) (2010) 2544–2558
* [3] Bradley, M.M., Lang, P.J.: Affective norms for English words (ANEW): Instruction manual and affective ratings. Technical Report C-1, The Center for Research in Psychophysiology, University of Florida (1999)
* [4] Wilson, T., Wiebe, J., Hoffmann, P.: Recognizing contextual polarity in phrase-level sentiment analysis. In: Proceedings of the conference on Human Language Technology and Empirical Methods in Natural Language Processing, Stroudsburg, PA, USA, Association for Computational Linguistics (2005)
* [5] Hansen, L.K., Arvidsson, A., Nielsen, F.Å., Colleoni, E., Etter, M.: Good friends, bad news — affect and virality in Twitter. Accepted for The 2011 International Workshop on Social Computing, Network, and Services (SocialComNet 2011) (2011)
* [6] Akkaya, C., Conrad, A., Wiebe, J., Mihalcea, R.: Amazon Mechanical Turk for subjectivity word sense disambiguation. In: Proceedings of the NAACL HLT 2010 Workshop on Creating, Speech and Language Data with Amazon’s Mechanical Turk, Association for Computational Linguistics (2010) 195–203
* [7] Baudhuin, E.S.: Obscene language and evaluative response: an empirical study. Psychological Reports 32 (1973)
* [8] Sapolsky, B.S., Shafer, D.M., Kaye, B.K.: Rating offensive words in three television program contexts. BEA 2008, Research Division (2008)
* [9] Bird, S., Klein, E., Loper, E.: Natural Language Processing with Python. O’Reilly, Sebastopol, California (June 2009)
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|
arxiv-papers
| 2011-03-15T13:39:20 |
2024-09-04T02:49:17.678382
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Finn {\\AA}rup Nielsen",
"submitter": "Finn {\\AA}rup Nielsen",
"url": "https://arxiv.org/abs/1103.2903"
}
|
1103.2929
|
# Measurement of the Muon Charge Asymmetry from W Bosons Produced in $pp$
Collisions at $\sqrt{s}=7{\mathrm{\ Te\kern-1.00006ptV}}$ with the ATLAS
detector
The ATLAS Collaboration
###### Abstract
This letter reports a measurement of the muon charge asymmetry from $W$ bosons
produced in proton-proton collisions at a centre-of-mass energy of
$7~{}{\mathrm{\ Te\kern-1.00006ptV}}$ with the ATLAS experiment at the LHC.
The asymmetry is measured in the $W\rightarrow\mu\nu$ decay mode as a function
of the muon pseudorapidity using a data sample corresponding to a total
integrated luminosity of $31~{}\mbox{pb${}^{-1}$}$. The results are compared
to predictions based on next-to-leading order calculations with various parton
distribution functions. This measurement provides information on the $u$ and
$d$ quark momentum fractions in the proton.
## 1 Introduction
The measurement of the charge asymmetry of leptons originating from the decay
of singly produced $W$ bosons at $pp$, $p\bar{p}$ and $ep$ colliders provides
important information about the proton structure as described by parton
distribution functions (PDFs). The $W$ boson charge asymmetry is mainly
sensitive to valence quark distributions [1] via the dominant production
process $u\bar{d}(\bar{u}d)\rightarrow W^{+(-)}$ and provides complementary
information to that obtained from measurements of inclusive deep inelastic
scattering cross-sections at the HERA electron-proton collider [2, 3, 4, 5].
The HERA data do not strongly constrain the ratio between $u$ and $d$ quarks
in the kinematic regime of low $x$, where $x$ is the proton momentum fraction
carried by the parton [6]. A precise measurement of the $W$ asymmetry at the
Large Hadron Collider (LHC) [7] on the other hand, can contribute
significantly to the understanding of PDFs and quantum chromodynamics (QCD) in
the parton momentum fraction range $10^{-3}\lesssim x\lesssim 10^{-1}$.
In $pp$ collisions the overall production rate of $W^{+}$ bosons is
significantly larger than the corresponding $W^{-}$ rate, since the proton
contains two $u$ and one $d$ valence quarks. The first measurements of the
inclusive $W^{\pm}$ cross-sections at the LHC by the ATLAS [8] and the CMS [9]
Collaborations confirmed the difference predicted by the Standard Model. The
asymmetry in $pp$ collisions is symmetric with respect to the $W$ rapidity,
whereas in $p\bar{p}$ collisions it is antisymmetric; the small sensitivity to
sea quark contributions is strongly suppressed in $p\bar{p}$ compared to $pp$
collisions [10]. Measurements in $p\bar{p}$ collisions have been performed at
the Tevatron by both the CDF [11, 12] and DØ [13, 14] Collaborations, and the
data have been included in global fits of parton distributions [15, 16].
This letter presents a differential measurement of the muon charge asymmetry
from the decay of $W^{\pm}$ bosons in $pp$ collisions at a centre-of-mass
energy of $\sqrt{s}~{}=~{}7{\mathrm{\ Te\kern-1.00006ptV}}$ at the LHC. The
asymmetry varies significantly as a function of the pseudorapidity111The
nominal $pp$ interaction point at the centre of the detector is defined as the
origin of a right-handed coordinate system. The positive $x$-axis is defined
by the direction from the interaction point to the centre of the LHC ring,
with the positive $y$-axis pointing upwards. The azimuthal angle $\phi$ is
measured around the beam axis and the polar angle $\theta$ is the angle from
the $z$-axis. The pseudorapidity is defined as $\eta=-\ln\tan(\theta/2)$.
$\eta_{\mu}$ of the charged decay lepton owing to its strong correlation with
the momentum fraction $x$ of the partons producing the $W$ boson. It is
defined from the cross sections for $W\to\mu\nu$ production
$d\sigma_{\mathrm{W\mu^{\pm}}}/d\eta_{\mu}$ as:
$A_{\mu}=\frac{d\sigma_{\mathrm{W\mu^{+}}}/d\eta_{\mu}-d\sigma_{\mathrm{W\mu^{-}}}/d\eta_{\mu}}{d\sigma_{\mathrm{W\mu^{+}}}/d\eta_{\mu}+d\sigma_{\mathrm{W\mu^{-}}}/d\eta_{\mu}}\,,$
(1)
where the cross sections include the event kinematical cuts used to select
$W\to\mu\nu$ events. No extrapolation to the full phase space is attempted in
order to reduce the dependence on theoretical predictions.
Systematic effects on the $W$-production cross-section measurements are
typically the same for positive and negative muons, mostly canceling in the
asymmetry. The ATLAS detector measures muons with two independent detector
systems. These two independent measurements allow systematic uncertainties to
be controlled. The results presented are based on data collected in 2010 with
an integrated luminosity of $31~{}\mbox{pb${}^{-1}$}$. These results
significantly improve on the previous measurement by the ATLAS Collaboration
[8], which is based on a data set approximately 100 times smaller.
## 2 The ATLAS Detector
The ATLAS detector [17, 18] consists of an inner tracking system (inner
detector, or ID) surrounded by a superconducting solenoid providing a 2T
magnetic field, electromagnetic and hadronic calorimeters and a muon
spectrometer (MS). The ID consists of pixel and silicon microstrip (SCT)
detectors, surrounded by a transition radiation tracker (TRT). The
electromagnetic calorimeter is a lead liquid-argon (LAr) detector in the
barrel and the endcap, and in the forward region copper LAr technology is
used. Hadron calorimetry is based on two different detector technologies, with
scintillator tiles or LAr as the active media, and with either steel, copper,
or tungsten as the absorber material. There is a poorly instrumented
transition region between the barrel and endcap calorimeter,
$1.37<|\eta|<1.52$, where electrons cannot be precisely measured. In view of a
later combination, this motivates the binning in that region for the present
muon analysis. The MS is based on three large superconducting toroids, and a
system of three stations of chambers for trigger and precise tracking
measurements. There is a transition between the barrel and endcap muon
detectors around $|\eta|=1.05$.
## 3 Data and Simulated Event Samples
The data used in this analysis were collected from the end of September to the
end of October 2010. Basic requirements on beam, detector, stable trigger
conditions and data-quality were used in the event selection, resulting in a
total integrated luminosity of $31~{}\mbox{pb${}^{-1}$}$. Events in this
analysis are selected using a single-muon trigger with a requirement on the
momentum transverse to the beam ($p_{\mathrm{T}}$) of at least $13{\mathrm{\
Ge\kern-1.00006ptV}}$. The trigger includes three levels of event selection: a
first level hardware-based selection using hit patterns in the MS and two
higher levels of software-based requirements.
Simulated event samples are used for the background estimation, the acceptance
calculation and for comparison of data with theoretical expectations. The
processes considered are the $W\rightarrow\mu\nu$ signal, and backgrounds from
$W\rightarrow\tau\nu$, $Z\rightarrow\mu\mu$, $Z\rightarrow\tau\tau$,
$t\bar{t}$ and jet production via QCD processes (referred to as “QCD
background” in the text). The signal and background samples (except
$t\bar{t}$) were generated with PYTHIA 6.421 [19] using MRST 2007
$\rm{LO}^{*}$ [20] PDFs. The $t\bar{t}$ sample was generated with POWHEG-HVQ
v1.01 patch 4 [21]; the PDF set was CTEQ 6.6M [22] for the NLO matrix element
calculations, while CTEQ 6L1 was used for the parton showering and underlying
event via the POWHEG interface to PYTHIA. The radiation of photons from
charged leptons was treated using PHOTOS v2.15.4 [23] and TAUOLA v1.0.2 [24]
was used for tau decays. The underlying and pile-up events were simulated
according to the ATLAS MC09 tune [25]. The generated samples were passed
through the GEANT4
[26] simulation of the ATLAS detector [27], reconstructed and analysed with
the same analysis chain as the data. The cross-section predictions for $W$ and
$Z$ were calculated to next-to-next-to-leading-order (NNLO) using FEWZ [28]
with the MSTW 2008 [29] PDFs. The $t\bar{t}$ cross-section was obtained at
next-to-leading-order (plus next-to-next-to-leading-log, NNLL) using POWHEG
[30]. The Monte Carlo (MC) were generated with, on average, two soft inelastic
collisions overlaid on top of the hard-scattering event. Events in the MC
samples were weighted so that the distribution of the number of inelastic
collisions per bunch crossing matched that in data, which has an average of
$2.2$.
## 4 Event Selection
The criteria for the event selection and muon identification follow closely
those used for the $W$ boson inclusive cross-section measurement [8], with an
improved muon quality selection [31]. Events from $pp$ collisions are selected
by requiring a collision vertex with at least three tracks each with
transverse momentum greater than $150{\mathrm{\ Me\kern-1.00006ptV}}$. A beam-
spot constraint has been applied in the collision vertex reconstruction stage
significantly improving the resolution on the collision vertex position in the
transverse plane. To reduce the contribution of cosmic-ray and beam-halo
events, induced by proton losses from the beam, the analysis requires the
collision vertex position along the beam axis to be within $20{\mathrm{\ cm}}$
of the nominal interaction point.
Events with a high transverse momentum muon are selected by imposing stringent
requirements to ensure good discrimination of $W\rightarrow\mu\nu$ events from
background. The muon parameters are first reconstructed separately in the MS
and ID. Subsequently, the tracks from the ID and MS are matched. Their
parameters are then combined, weighted by their respective errors, to form a
combined muon. The $W$ candidate events are required to have at least one
combined muon track with $p_{\mathrm{T}}>20{\mathrm{\ Ge\kern-1.00006ptV}}$
and $p_{\mathrm{T}}$ measured by the MS alone greater than
$p_{\mathrm{T}}^{\mathrm{MS}}>10{\mathrm{\ Ge\kern-1.00006ptV}}$, within the
range $|\eta_{\mu}|<2.4$. The difference between the ID and MS
$p_{\mathrm{T}}$, corrected for the mean energy loss in the material traversed
between the ID and MS, is required to be less than 0.5 times the ID
$p_{\mathrm{T}}$,
$p_{\mathrm{T}}^{\mathrm{MS}}({\mathrm{energy\;loss\;corrected}})-p_{\mathrm{T}}^{ID}<0.5\phantom{0}p_{\mathrm{T}}^{\mathrm{ID}}.$
This requirement increases the robustness against track reconstruction
mismatches, including decays-in-flight of
hadrons. In addition, a minimum number of hits in the ID is required to ensure
high quality tracks [31]. In order to further reduce non-collision
backgrounds, the difference between the $z$ position of the muon track
extrapolated to the beam line and the $z$ coordinate of the collision vertex
is required to be less than $1{\mathrm{\ cm}}$. A track-based isolation for
the muon is defined as $\sum p_{\mathrm{T}}^{\mathrm{ID}}/p_{\mathrm{T}}<0.2$,
where $\sum p_{\mathrm{T}}^{\mathrm{ID}}$ is the scalar sum of transverse
momenta of all other tracks measured in the ID within a cone222$\Delta R$ is
defined as $\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$. $\Delta R<0.4$
around the muon direction excluding the ID track associated with the muon, and
$p_{\mathrm{T}}$ is the transverse momentum of the muon combined track.
The reconstruction of the missing transverse energy
($E_{\mathrm{T}}^{\mathrm{miss}}$) and the transverse mass ($m_{\mathrm{T}}$)
follows the prescription in [8]. The $E_{\mathrm{T}}^{\mathrm{miss}}$ is
determined from the energy deposits of calibrated calorimeter cells in three-
dimensional clusters and is corrected for the momentum of all muons
reconstructed in the event. Jet-quality requirements are applied to remove a
small fraction of events where sporadic calorimeter noise and non-collision
backgrounds can affect the $E_{\mathrm{T}}^{\mathrm{miss}}$ reconstruction
[32]. The transverse mass is defined as
$m_{\mathrm{T}}=\sqrt{2p_{\mathrm{T}}^{\mu}p_{\mathrm{T}}^{\nu}(1-\cos(\phi^{\mu}-\phi^{\nu}))},$
(2)
where the highest $p_{\mathrm{T}}$ muon is used and the $(x,y)$ components of
the neutrino momentum are inferred from the corresponding
$E_{\mathrm{T}}^{\mathrm{miss}}$ components. Events are required to have
$E_{\mathrm{T}}^{\mathrm{miss}}>25{\mathrm{\ Ge\kern-1.00006ptV}}$ and
$m_{\mathrm{T}}>40{\mathrm{\ Ge\kern-1.00006ptV}}$, yielding 129572 $W$
candidates.
## 5 $W^{\pm}$ Signal Yield and Background Estimation
Figure 1: Distribution of the muon pseudorapidity $\eta_{\mu}$ of $W^{+}$ 1 and $W^{-}$ 1 candidates, after final selection. The data are compared to MC simulation, broken down into the signal and various background components. The MC distributions are normalised to the total number of events in data. | $\mu^{+}$ | $\mu^{-}$
---|---|---
| Observed | Exp. Background | $C_{\mathrm{W\mu^{+}}}$ | Observed | Exp. Background | $C_{\mathrm{W\mu^{-}}}$
$0.00<|\eta_{\mu}|<0.21$ | $5052$ | $272\pm 51$ | $0.594\pm 0.005$ | $3726$ | $236\pm 55$ | $0.584\pm 0.004$
$0.21<|\eta_{\mu}|<0.42$ | $6519$ | $385\pm 70$ | $0.779\pm 0.009$ | $4757$ | $334\pm 70$ | $0.759\pm 0.008$
$0.42<|\eta_{\mu}|<0.63$ | $6845$ | $481\pm 88$ | $0.808\pm 0.009$ | $4936$ | $357\pm 70$ | $0.800\pm 0.009$
$0.63<|\eta_{\mu}|<0.84$ | $5963$ | $366\pm 76$ | $0.686\pm 0.008$ | $4212$ | $329\pm 64$ | $0.691\pm 0.008$
$0.84<|\eta_{\mu}|<1.05$ | $5933$ | $395\pm 63$ | $0.672\pm 0.007$ | $4207$ | $358\pm 63$ | $0.681\pm 0.008$
$1.05<|\eta_{\mu}|<1.37$ | $10114$ | $627\pm 93$ | $0.735\pm 0.007$ | $6544$ | $585\pm 101$ | $0.752\pm 0.007$
$1.37<|\eta_{\mu}|<1.52$ | $5726$ | $363\pm 57$ | $0.905\pm 0.009$ | $3601$ | $348\pm 59$ | $0.914\pm 0.009$
$1.52<|\eta_{\mu}|<1.74$ | $8228$ | $542\pm 89$ | $0.905\pm 0.008$ | $5043$ | $518\pm 82$ | $0.925\pm 0.008$
$1.74<|\eta_{\mu}|<1.95$ | $7982$ | $605\pm 114$ | $0.896\pm 0.009$ | $4688$ | $456\pm 80$ | $0.898\pm 0.008$
$1.95<|\eta_{\mu}|<2.18$ | $8392$ | $647\pm 100$ | $0.903\pm 0.009$ | $4971$ | $548\pm 91$ | $0.910\pm 0.009$
$2.18<|\eta_{\mu}|<2.40$ | $7562$ | $534\pm 81$ | $0.881\pm 0.010$ | $4571$ | $492\pm 82$ | $0.896\pm 0.010$
Table 1: Summary of observed number of events, expected background and
correction factor $C_{\mathrm{W\mu^{\pm}}}$ for positive and negative muons in
bins of $|\eta_{\mu}|$. The errors given for the background estimates include
systematic uncertainties, including the uncertainty due to the luminosity,
used in the normalization of the electro-weak and $t\bar{t}$ components. The
$C_{\mathrm{W\mu^{\pm}}}$ factors include trigger and muon reconstruction
scale factors; they include the statistical uncertainty from the MC sample and
the trigger and reconstruction scale factors.
Many components in the $W$ cross-section measurement, such as the luminosity
or detector efficiencies, are in principle the same for positive and negative
muons and therefore mostly cancel in the asymmetry calculation. The main
experimental biases on the asymmetry measurement come from possible
differences in the reconstruction of positive and negative muons. Each effect
(trigger and reconstruction efficiency and momentum scale) is examined to
check that the two charges behave in the same way within the systematic
uncertainties. These studies are performed in absolute pseudorapidity in order
to reduce the uncertainty associated with the limited size of the data samples
used.
As in past $W$ analyses, trigger [31] and muon reconstruction [8, 31]
efficiencies as a function of muon $\eta_{\mu}$ have been measured in data
using a sample of unbiased muons from $Z\rightarrow\mu\mu$ decays, which
provides a source of muons with small background. The trigger efficiency is
determined relative to a reconstructed muon satisfying the selection criteria
of the analysis. The average trigger efficiencies after the full $W$ selection
are $(81\pm 2)$% in the central detector region, $|\eta_{\mu}|<1.05$, and
$(94\pm 1)$% in the forward detector region, $1.05<|\eta_{\mu}|<2.4$, where
the differences are due to the geometrical acceptance of the muon trigger
chambers. In the same muon sample, the muon reconstruction efficiency relative
to an ID track is measured to be $(93\pm 1)$% overall. The efficiency for
reconstructing an ID track is $(99\pm 1)$% [8]. The quoted uncertainties on
these efficiencies are statistical.
Corrections have been applied to the simulated samples to account for
differences in the trigger and reconstruction efficiencies between data and
simulation. These are based on the ratio of the efficiency in data and in
simulation, and are computed as a function of the muon $\eta_{\mu}$ and
charge. The corrections for each charge agree within the statistical
uncertainties, so the charge-averaged result is applied. For the trigger, the
corrections are $0.98$ and $1.03$ in the central and forward MS regions,
respectively. For the reconstruction efficiency, the correction factors are
about $0.99$ per $\eta_{\mu}$ bin except for the central-forward MS transition
region ($|\eta_{\mu}|$ about $1.05$) where the correction factor is $0.94$.
The muon momentum resolution is affected by the amount of material traversed
by the muon, the spatial resolution of the individual track points and the
degree of internal alignment of the ID and MS [33]. This resolution has been
measured as a function of $\eta_{\mu}$ for the main detector regions (in
$\eta_{\mu}$ ranges delimited by $1.05,1.7,2.0$ and $2.4$) from the width of
the di-muon invariant mass distribution in $Z\rightarrow\mu\mu$ decays and
from the comparison of the momentum measurements in the ID and MS in
$Z\rightarrow\mu\mu$ and $W\rightarrow\mu\nu$ decays. The measured resolution
is worse than expected from simulation by $1\;\\!$–$\;\\!5\%$, with the
maximum discrepancy reached in the high-$\eta_{\mu}$ region of the detector.
The discrepancy is due to residual mis-alignments in the ID and MS,
imperfections in the description of the inert material in simulation and an
imperfect mapping of the magnetic field in the MS transition region where the
field is highly non-uniform. Smearing corrections are therefore applied to the
simulation in order to improve the agreement with data.
If the accuracy of the muon momentum measurement is different for positive and
negative muons, this difference can produce a bias in the acceptance of
$\mu^{+}$ with respect to $\mu^{-}$. Differences in the muon $p_{\mathrm{T}}$
measurement between data and simulation have been evaluated comparing the
curvature of muons from $W$ candidates in data and in templates derived from
simulation. A binned likelihood fit for a momentum-scale correction that
yields the best agreement between data and simulation is performed as a
function of $\eta_{\mu}$ separately for positive and negative charges. The
measured biases in the $p_{\mathrm{T}}$ scale between the two charges are
$<1\%$, but they increase to about $3\%$ in the transition and
high-$\eta_{\mu}$ regions due to residual mis-alignments in the ID and MS.
These corrections are applied to the muon momenta in the simulated samples.
Figure 1 shows the pseudorapidity distribution of the selected positive and
negative muons. Data distributions are compared to the MC simulation,
normalised to the total number of events in data. The shape of the simulation
agrees well with the shape of the data after the corrections for the
reconstruction and trigger efficiencies, and the muon-momentum scale and
resolution.
The main backgrounds to $W\rightarrow\mu\nu$ arise from heavy flavour decays
in multijet events and from the electro-weak background from
$W\rightarrow\tau\nu$ where the tau decays to a muon, $Z\rightarrow\mu\mu$
where one muon is not reconstructed and $Z\rightarrow\tau\tau$ where one tau
decays to a muon, as well as semileptonic $t\bar{t}$ decays in the muon
channel. Di-boson and single top backgrounds are found to be negligible. The
$W\rightarrow\tau\nu$ contribution is treated as a background. While this
contribution presents the same asymmetry as the $W\rightarrow\mu\nu$ signal,
it is difficult to include in PDF fits, which assume that the asymmetry is a
function of $\eta_{\ell}$ for $W\rightarrow l\nu$.
The background estimates of the electro-weak and $t\bar{t}$ backgrounds and
the QCD background closely follow the methods used in the $W$ inclusive cross-
section measurement [8]. They are determined separately for positive and
negative muons as a function of $\eta_{\mu}$. The electro-weak and $t\bar{t}$
backgrounds are estimated using MC simulation. The QCD background comes
primarily from $b$ and $c$ quark decays, with a smaller contribution from pion
and kaon decays in flight. This background is estimated using a data-driven
method similar to the one described in [8]. The sample of events fulfilling
the full $W$ selection criteria with the exception of the muon isolation
requirement is compared before and after the isolation requirement. The
isolation efficiency for non-QCD events is measured in data with the
$Z\rightarrow\mu\mu$ sample. The efficiency for QCD events is estimated in a
control sample of low-$p_{\mathrm{T}}$ muons extrapolated to the
high-$p_{\mathrm{T}}$ and high-$E_{\mathrm{T}}^{\mathrm{miss}}$ signal region
using the simulated jet sample. Since the samples before and after isolation
can be defined in terms of a QCD and non-QCD component, the expected number of
QCD events can thus be determined.
Figure 2: Distribution of the transverse momentum of positive and negative
muons after the final selection. The data are compared to MC simulation,
broken down into the signal and various background components. The MC
distributions are normalised to the total number of entries in data.
The expected background amounts to $7\%$ of the selected events; $6\%$ is the
electro-weak and $t\bar{t}$ contribution ($3\%$ $Z\rightarrow\mu\mu$, $2\%$
$W\rightarrow\tau\nu$, and $1\%$ for the sum of $t\bar{t}$ and
$Z\rightarrow\tau\tau$) and the remainder is the QCD background. The cosmic
ray background contamination is estimated to be smaller by a factor of
$10^{5}$ compared to the signal and thus negligible. The $W^{\pm}$ candidate
events and expected background contributions are summarised in Table 1.
Figure 2 shows the transverse momentum distribution for positive and negative
muons after the full event selection. They are compared with the distributions
predicted by the corrected MC simulation normalised to the total number of
events in data. The correction factors, $C_{\mathrm{W\mu^{\pm}}}$,
corresponding to the ratio of reconstructed over generated events in the
simulated $W$ sample, satisfying all kinematic requirements of the event
selection, $p_{\mathrm{T}}^{\mu}>20{\mathrm{\ Ge\kern-1.00006ptV}}$,
$p_{\mathrm{T}}^{\nu}>25{\mathrm{\ Ge\kern-1.00006ptV}}$,
$m_{\mathrm{T}}>40{\mathrm{\ Ge\kern-1.00006ptV}}$, are also listed in Table
1. No correction is made to the full acceptance. The $C_{\mathrm{W\mu^{\pm}}}$
factors include trigger and muon reconstruction scale factors to correct for
observed deviations between data and MC efficiencies. Due to a reduced
geometric acceptance in the trigger, the $C_{\mathrm{W\mu^{\pm}}}$ factors for
the lowest $|\eta_{\mu}|$ bins are significantly smaller than those for the
higher $|\eta_{\mu}|$ regions.
## 6 Systematic Uncertainties
All systematic uncertainties on the asymmetry measurement are determined in
each $|\eta_{\mu}|$ bin accounting for correlations between the charges and
are summarised in Table 2. The dominant sources of systematic uncertainty on
the asymmetry come from the trigger and reconstruction efficiencies. The
determination of these efficiencies is affected by the statistical uncertainty
due to the small available sample of $Z\rightarrow\mu\mu$ events. Systematic
uncertainties on the efficiencies are determined from studies of the impact of
the selection criteria and backgrounds, and no significant charge biases are
found. There is a loss of trigger efficiency in the low pseudorapidity region
due to reduced geometric acceptance, resulting in a larger statistical error.
As a result, the trigger systematic uncertainty on the asymmetry is largest in
the low pseudorapidity bins (6-7% for central $|\eta_{\mu}|$ and 2-3% for
forward $|\eta_{\mu}|$). Similarly, the uncertainties associated with the
reconstruction efficiency are larger in the lowest pseudorapidity bin (about
$7\%$), and in the MS central-forward transition region (about $3\%$), due to
geometrical acceptance effects associated with reduced chamber coverage. In
the remaining regions, the uncertainty is about 1-2%.
| Trigger | Reconstruction | $p_{\mathrm{T}}$ Scale and | QCD | Electro-weak and $t\bar{t}$ | Theoretical
---|---|---|---|---|---|---
| Resolution | Normalisation | Normalisation | Modelling
$0.00<|\eta_{\mu}|<0.21$ | $0.011$ | $0.010$ | $0.003$ | $0.003$ | $<0.001$ | $0.007$
$0.21<|\eta_{\mu}|<0.42$ | $0.010$ | $0.004$ | $0.003$ | $0.003$ | $<0.001$ | $0.005$
$0.42<|\eta_{\mu}|<0.63$ | $0.009$ | $0.004$ | $0.003$ | $0.003$ | $<0.001$ | $0.006$
$0.63<|\eta_{\mu}|<0.84$ | $0.012$ | $0.004$ | $0.003$ | $0.002$ | $0.001$ | $0.007$
$0.84<|\eta_{\mu}|<1.05$ | $0.013$ | $0.006$ | $0.003$ | $0.003$ | $0.001$ | $0.008$
$1.05<|\eta_{\mu}|<1.37$ | $0.006$ | $0.007$ | $0.002$ | $0.002$ | $0.001$ | $0.006$
$1.37<|\eta_{\mu}|<1.52$ | $0.006$ | $0.005$ | $0.002$ | $0.003$ | $0.002$ | $0.005$
$1.52<|\eta_{\mu}|<1.74$ | $0.005$ | $0.004$ | $0.002$ | $0.003$ | $0.002$ | $0.007$
$1.74<|\eta_{\mu}|<1.95$ | $0.006$ | $0.003$ | $0.002$ | $0.002$ | $0.001$ | $0.006$
$1.95<|\eta_{\mu}|<2.18$ | $0.006$ | $0.004$ | $0.002$ | $0.003$ | $0.002$ | $0.009$
$2.18<|\eta_{\mu}|<2.40$ | $0.007$ | $0.005$ | $0.002$ | $0.003$ | $0.002$ | $0.007$
Table 2: Absolute systematic uncertainties on the $W$ charge asymmetry from
different sources as a function of absolute muon pseudorapidity that are
described in the text.
The muon momentum scale and resolution corrections contribute to the
uncertainty primarily due to the limited statistics for the fitting procedures
used to measure the differences between the data and simulation. An additional
source of uncertainty arises from potential biases in the template shapes. The
size of this effect is determined by using different templates created by
shifting the resolution parameters in opposite directions to account for
possible charge biases. Uncertainties associated with the modelling of the
background contributions to the templates, particularly the QCD background,
are also included. The resulting uncertainty on the asymmetry is in the 1-2%
range, with little dependence on $\eta_{\mu}$. The redundant ID and MS
momentum measurements result in a rate of charge mis-identification smaller
than $10^{-4}$ in the $p_{\mathrm{T}}$ range considered, resulting in a
negligible impact on the asymmetry.
The momentum-scale correction procedure is further tested by exploiting the
redundant muon-momentum measurements offered by the ATLAS detector. The full
asymmetry measurement is performed with the ID and MS components of the
combined muon separately, including the scale corrections. Figure 3 compares
the two independently corrected charge-asymmetry distributions, showing good
agreement within the systematic uncertainty associated with the momentum-scale
correction.
Figure 3: $W$ charge asymmetry measured using the ID and MS separately. The MS
measurement is extrapolated to the collision vertex, and corrected for energy-
loss in the calorimeters. The two measurements are independently corrected for
effects of the muon-momentum scale on the muon acceptance. The two
measurements are statistically correlated to a large extent, since they use
the same muons reconstructed by different subdetectors and algorithms. The
error bar reports therefore only the systematic uncertainty associated with
the momentum-scale correction.
The systematic uncertainties on the QCD background arise primarily from the
uncertainty on the isolation efficiency for muons in QCD events due to
possible mis-modellings of the extrapolation of the isolation efficiency to
the large $p_{\mathrm{T}}$ and $E_{\mathrm{T}}^{\mathrm{miss}}$ region in the
QCD simulation (40%). This has been derived from differences in the efficiency
predictions between data and simulation in the low muon $p_{\mathrm{T}}$
control region and in sideband regions where the muon $p_{\mathrm{T}}$ or
$E_{\mathrm{T}}^{\mathrm{miss}}$ cuts are reversed. The electro-weak and
$t\bar{t}$ background and signal contributions are subtracted from data in
these comparisons. Additional uncertainties due to the non-QCD isolation
efficiency and the statistical uncertainty are included in the total
uncertainty on the QCD background estimate. The corresponding systematic
uncertainty on the asymmetry is 1-2%, with little dependence on $\eta_{\mu}$.
For the electro-weak and $t\bar{t}$ backgrounds, the uncertainty in the cross-
sections includes the PDF uncertainties (3%), and the uncertainties estimated
from varying the renormalization and factorization scales: 5% for $W$ and $Z$,
and 6% for $t\bar{t}$ [34, 35, 8]. An additional uncertainty from the
luminosity of $11\%$ is included, since the backgrounds are scaled to the
luminosity measured in data. The combination of all these contributions
results in an uncertainty on the asymmetry of less than $1\%$.
The impact of using an NLO MC rather than Pythia in the
$C_{\mathrm{W\mu^{\pm}}}$ factor calculation has been evaluated and an
additional systematic uncertainty of about $3\%$ is included to account for
the small variations observed. Pythia uses a leading-log calculation for $W$
production and is expected to give a reasonably accurate prediction for the
low $W$ transverse momentum $p_{\mathrm{T}}^{W}$ region whereas MC@NLO [36]
uses higher-order matrix elements and is therefore expected to be more
reliable in the high $p_{\mathrm{T}}^{W}$ region. Therefore the differences in
the scale factors associated with these two MC calculations gives a reasonable
estimate of the associated systematic error.
## 7 Results and Conclusions
The measured particle-level differential charge asymmetry in eleven bins of
muon absolute pseudorapidity is shown in Table 3 and Figure 4. The statistical
and systematic uncertainties per $|\eta_{\mu}|$ bin are included and
contribute comparably to the total uncertainty. Table 3 and Figure 4 also show
particle-level expectations from $W$ predictions at NLO with different PDF
sets: CTEQ 6.6 [16], HERA 1.0 [5] and MSTW 2008 [15]; all predictions are
presented with 90% confidence-level error bands. All MC predictions are
calculated using MC@NLO, with all kinematic selection criteria applied to the
truth particles. The PDF uncertainty bands are obtained by summing in
quadrature the deviations of each of the PDF error sets [37] from the
respective nominal predictions, according to the specifications of the
corresponding PDF collaborations to get 90% C.L. bands. These uncertainties
for all predictions include experimental uncertainties as well as model and
parametrization uncertainties. The HERA 1.0 [5] set also includes the
uncertainty in $\alpha_{s}$, which, however, is not the dominant source of
uncertainty.
While the predictions with different PDF sets differ within their respective
uncertainty bands [38, 39], they follow the same global trend, rising with
$\eta_{\mu}$. The measured asymmetry agrees with this expectation. As
demonstrated graphically in Figure 4, the data are roughly compatible with all
the predictions with different PDF sets, though some are slightly preferred to
others. A $\chi^{2}$-comparison using the measurement uncertainty and the
central value of the PDF predictions yields values per number of degrees of
freedom of $9.16/11$ for the CTEQ 6.6 PDF sets, $35.81/11$ for the HERA 1.0
PDF sets and $27.31/11$ for the MSTW 2008 PDF sets.
In summary, this letter reports a measurement of the $W$ charge asymmetry in
$pp$ collisions at $\sqrt{s}=7{\mathrm{\ Te\kern-1.00006ptV}}$ performed in
the $W\rightarrow\mu\nu$ decay mode using $31~{}\mbox{pb${}^{-1}$}$ of data
recorded with the ATLAS detector at the LHC. Until the start of the LHC, it
has not been kinematically possible to precisely measure the valence quark
distributions and in particular the ratio of $u/d$ quarks below $x\lesssim
0.05$. Whereas none of the predictions with different PDF sets are
inconsistent with these data, the predictions are not fully consistent with
each other since they are all phenomenological extrapolations in $x$. The
input of the data presented here is therefore expected to contribute to the
determination of the next generation of PDF sets, helping reduce PDF
uncertainties, particularly the shapes of the valence quark distributions in
the low-$x$ region.
| Data | MSTW 2008 | CTEQ 6.6 | HERA 1.0
---|---|---|---|---
$0.00<|\eta_{\mu}|<0.21$ | $0.147\pm 0.011\pm 0.017$ | $0.142_{-0.014}^{+0.006}$ | $0.164_{-0.007}^{+0.006}$ | $0.163\pm 0.007$
$0.21<|\eta_{\mu}|<0.42$ | $0.150\pm 0.010\pm 0.012$ | $0.147_{-0.014}^{+0.007}$ | $0.168_{-0.007}^{+0.006}$ | $0.167\pm 0.007$
$0.42<|\eta_{\mu}|<0.63$ | $0.158\pm 0.010\pm 0.012$ | $0.151_{-0.013}^{+0.007}$ | $0.173_{-0.007}^{+0.006}$ | $0.169\pm 0.007$
$0.63<|\eta_{\mu}|<0.84$ | $0.184\pm 0.010\pm 0.015$ | $0.163_{-0.012}^{+0.008}$ | $0.186_{-0.008}^{+0.007}$ | $0.179_{-0.007}^{+0.008}$
$0.84<|\eta_{\mu}|<1.05$ | $0.186\pm 0.011\pm 0.017$ | $0.176_{-0.012}^{+0.009}$ | $0.198_{-0.008}^{+0.007}$ | $0.188\pm 0.008$
$1.05<|\eta_{\mu}|<1.37$ | $0.240\pm 0.008\pm 0.011$ | $0.197\pm 0.010$ | $0.219_{-0.010}^{+0.008}$ | $0.203_{-0.008}^{+0.009}$
$1.37<|\eta_{\mu}|<1.52$ | $0.250\pm 0.011\pm 0.010$ | $0.215_{-0.010}^{+0.011}$ | $0.237_{-0.010}^{+0.009}$ | $0.214\pm 0.009$
$1.52<|\eta_{\mu}|<1.74$ | $0.269\pm 0.009\pm 0.010$ | $0.230_{-0.010}^{+0.012}$ | $0.251_{-0.011}^{+0.009}$ | $0.224\pm 0.009$
$1.74<|\eta_{\mu}|<1.95$ | $0.273\pm 0.009\pm 0.010$ | $0.251_{-0.009}^{+0.013}$ | $0.270_{-0.011}^{+0.010}$ | $0.239_{-0.009}^{+0.010}$
$1.95<|\eta_{\mu}|<2.18$ | $0.276\pm 0.009\pm 0.012$ | $0.266_{-0.010}^{+0.014}$ | $0.284_{-0.011}^{+0.010}$ | $0.251_{-0.010}^{+0.009}$
$2.18<|\eta_{\mu}|<2.40$ | $0.273\pm 0.010\pm 0.012$ | $0.272_{-0.011}^{+0.015}$ | $0.288_{-0.010}^{+0.009}$ | $0.255_{-0.010}^{+0.009}$
Table 3: The muon charge asymmetry from $W$-boson decays in bins of absolute
pseudorapidity. The data measurements are listed with statistical and
systematic uncertainties respectively. Predicted asymmetries of the MSTW 2008,
CTEQ 6.6, and HERA 1.0 PDF sets are shown for comparison. Figure 4: The muon
charge asymmetry from $W$-boson decays in bins of absolute pseudorapidity. The
kinematic requirements applied are $p_{\mathrm{T}}^{\mu}>20{\mathrm{\
Ge\kern-1.00006ptV}}$, $p_{\mathrm{T}}^{\nu}>25{\mathrm{\
Ge\kern-1.00006ptV}}$ and $m_{\mathrm{T}}>40{\mathrm{\ Ge\kern-1.00006ptV}}$.
The data points (shown with error bars including the statistical and
systematic uncertainties) are compared to MC@NLO predictions with different
PDF sets. The PDF uncertainty bands are described in the text and include
experimental uncertainties as well as model and parametrization uncertainties.
## 8 Acknowledgements
We wish to thank CERN for the efficient commissioning and operation of the LHC
during this initial high-energy data-taking period as well as the support
staff from our institutions without whom ATLAS could not be operated
efficiently.
We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC,
Australia; BMWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP,
Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC,
China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic;
DNRF, DNSRC and Lundbeck Foundation, Denmark; ARTEMIS, European Union;
IN2P3-CNRS,
CEA-DSM/IRFU, France; GNAS, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation,
Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel;
INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands;
RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania;
MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR,
Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC
and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva,
Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme
Trust, United Kingdom; DOE and NSF, United States of America.
The crucial computing support from all WLCG partners is acknowledged
gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF
(Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA
(Germany), INFN-CNAF (Italy), NL-T1
(Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the
Tier-2 facilities worldwide.
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R.S.B. King118, J. Kirk129, G.P. Kirsch118, L.E. Kirsch22, A.E. Kiryunin99, D.
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J. Klaiber-Lodewigs42, M. Klein73, U. Klein73, K. Kleinknecht81, M.
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Kocnar113, P. Kodys126, K. Köneke29, A.C. König104, S. Koenig81, L. Köpke81,
F. Koetsveld104, P. Koevesarki20, T. Koffas29, E. Koffeman105, F. Kohn54, Z.
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S. Koperny37, S.V. Kopikov128, K. Korcyl38, K. Kordas154, V. Koreshev128, A.
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H. Kowalski41, T.Z. Kowalski37, W. Kozanecki136, A.S. Kozhin128, V. Kral127,
V.A. Kramarenko97, G. Kramberger74, O. Krasel42, M.W. Krasny78, A.
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A. Leger49, B.C. LeGeyt120, F. Legger98, C. Leggett14, M. Lehmacher20, G.
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Z. Liang118,o, B. Liberti133a, P. Lichard29, M. Lichtnecker98, K. Lie165, W.
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J.P. Martin93, Ph. Martin55, T.A. Martin17, B. Martin dit Latour49, M.
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M.S. Neubauer165, A. Neusiedl81, R.M. Neves108, P. Nevski24, P.R. Newman17,
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D. Romero Maltrana31a, L. Roos78, E. Ros167, S. Rosati132a,132b, M. Rose76,
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Scannicchio163, J. Schaarschmidt115, P. Schacht99, U. Schäfer81, S. Schaepe20,
S. Schaetzel58b, A.C. Schaffer115, D. Schaile98, R.D. Schamberger148, A.G.
Schamov107, V. Scharf58a, V.A. Schegelsky121, D. Scheirich87, M.I. Scherzer14,
C. Schiavi50a,50b, J. Schieck98, M. Schioppa36a,36b, S. Schlenker29, J.L.
Schlereth5, E. Schmidt48, M.P. Schmidt175,∗, K. Schmieden20, C. Schmitt81, M.
Schmitz20, A. Schöning58b, M. Schott29, D. Schouten142, J. Schovancova125, M.
Schram85, C. Schroeder81, N. Schroer58c, S. Schuh29, G. Schuler29, J.
Schultes174, H.-C. Schultz-Coulon58a, H. Schulz15, J.W. Schumacher20, M.
Schumacher48, B.A. Schumm137, Ph. Schune136, C. Schwanenberger82, A.
Schwartzman143, Ph. Schwemling78, R. Schwienhorst88, R. Schwierz43, J.
Schwindling136, W.G. Scott129, J. Searcy114, E. Sedykh121, E. Segura11, S.C.
Seidel103, A. Seiden137, F. Seifert43, J.M. Seixas23a, G. Sekhniaidze102a,
D.M. Seliverstov121, B. Sellden146a, G. Sellers73, M. Seman144b, N. Semprini-
Cesari19a,19b, C. Serfon98, L. Serin115, R. Seuster99, H. Severini111, M.E.
Sevior86, A. Sfyrla29, E. Shabalina54, M. Shamim114, L.Y. Shan32a, J.T.
Shank21, Q.T. Shao86, M. Shapiro14, P.B. Shatalov95, L. Shaver6, C. Shaw53, K.
Shaw164a,164c, D. Sherman175, P. Sherwood77, A. Shibata108, S. Shimizu29, M.
Shimojima100, T. Shin56, A. Shmeleva94, M.J. Shochet30, D. Short118, M.A.
Shupe6, P. Sicho125, A. Sidoti132a,132b, A. Siebel174, F. Siegert48, J.
Siegrist14, Dj. Sijacki12a, O. Silbert171, J. Silva124a,b, Y. Silver153, D.
Silverstein143, S.B. Silverstein146a, V. Simak127, O. Simard136, Lj. Simic12a,
S. Simion115, B. Simmons77, M. Simonyan35, P. Sinervo158, N.B. Sinev114, V.
Sipica141, G. Siragusa81, A.N. Sisakyan65, S.Yu. Sivoklokov97, J.
Sjölin146a,146b, T.B. Sjursen13, L.A. Skinnari14, K. Skovpen107, P. Skubic111,
N. Skvorodnev22, M. Slater17, T. Slavicek127, K. Sliwa161, T.J. Sloan71, J.
Sloper29, V. Smakhtin171, S.Yu. Smirnov96, L.N. Smirnova97, O. Smirnova79,
B.C. Smith57, D. Smith143, K.M. Smith53, M. Smizanska71, K. Smolek127, A.A.
Snesarev94, S.W. Snow82, J. Snow111, J. Snuverink105, S. Snyder24, M.
Soares124a, R. Sobie169,i, J. Sodomka127, A. Soffer153, C.A. Solans167, M.
Solar127, J. Solc127, E. Soldatov96, U. Soldevila167, E. Solfaroli
Camillocci132a,132b, A.A. Solodkov128, O.V. Solovyanov128, J. Sondericker24,
N. Soni2, V. Sopko127, B. Sopko127, M. Sorbi89a,89b, M. Sosebee7, A.
Soukharev107, S. Spagnolo72a,72b, F. Spanò34, R. Spighi19a, G. Spigo29, F.
Spila132a,132b, E. Spiriti134a, R. Spiwoks29, M. Spousta126, T. Spreitzer158,
B. Spurlock7, R.D. St. Denis53, T. Stahl141, J. Stahlman120, R. Stamen58a, E.
Stanecka29, R.W. Stanek5, C. Stanescu134a, S. Stapnes117, E.A. Starchenko128,
J. Stark55, P. Staroba125, P. Starovoitov91, A. Staude98, P. Stavina144a, G.
Stavropoulos14, G. Steele53, P. Steinbach43, P. Steinberg24, I. Stekl127, B.
Stelzer142, H.J. Stelzer41, O. Stelzer-Chilton159a, H. Stenzel52, K.
Stevenson75, G.A. Stewart53, J.A. Stillings20, T. Stockmanns20, M.C.
Stockton29, K. Stoerig48, G. Stoicea25a, S. Stonjek99, P. Strachota126, A.R.
Stradling7, A. Straessner43, J. Strandberg87, S. Strandberg146a,146b, A.
Strandlie117, M. Strang109, E. Strauss143, M. Strauss111, P. Strizenec144b, R.
Ströhmer173, D.M. Strom114, J.A. Strong76,∗, R. Stroynowski39, J. Strube129,
B. Stugu13, I. Stumer24,∗, J. Stupak148, P. Sturm174, D.A. Soh151,o, D. Su143,
HS. Subramania2, A. Succurro11, Y. Sugaya116, T. Sugimoto101, C. Suhr106, K.
Suita67, M. Suk126, V.V. Sulin94, S. Sultansoy3d, T. Sumida29, X. Sun55, J.E.
Sundermann48, K. Suruliz164a,164b, S. Sushkov11, G. Susinno36a,36b, M.R.
Sutton139, Y. Suzuki66, Yu.M. Sviridov128, S. Swedish168, I. Sykora144a, T.
Sykora126, B. Szeless29, J. Sánchez167, D. Ta105, K. Tackmann29, A.
Taffard163, R. Tafirout159a, A. Taga117, N. Taiblum153, Y. Takahashi101, H.
Takai24, R. Takashima69, H. Takeda67, T. Takeshita140, M. Talby83, A.
Talyshev107, M.C. Tamsett24, J. Tanaka155, R. Tanaka115, S. Tanaka131, S.
Tanaka66, Y. Tanaka100, K. Tani67, N. Tannoury83, G.P. Tappern29, S.
Tapprogge81, D. Tardif158, S. Tarem152, F. Tarrade24, G.F. Tartarelli89a, P.
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Tevlin82, J. Thadome174, J. Therhaag20, T. Theveneaux-Pelzer78, M. Thioye175,
S. Thoma48, J.P. Thomas17, E.N. Thompson84, P.D. Thompson17, P.D. Thompson158,
A.S. Thompson53, E. Thomson120, M. Thomson27, R.P. Thun87, T. Tic125, V.O.
Tikhomirov94, Y.A. Tikhonov107, C.J.W.P. Timmermans104, P. Tipton175, F.J.
Tique Aires Viegas29, S. Tisserant83, J. Tobias48, B. Toczek37, T. Todorov4,
S. Todorova-Nova161, B. Toggerson163, J. Tojo66, S. Tokár144a, K. Tokunaga67,
K. Tokushuku66, K. Tollefson88, M. Tomoto101, L. Tompkins14, K. Toms103, G.
Tong32a, A. Tonoyan13, C. Topfel16, N.D. Topilin65, I. Torchiani29, E.
Torrence114, E. Torró Pastor167, J. Toth83,w, F. Touchard83, D.R. Tovey139, D.
Traynor75, T. Trefzger173, J. Treis20, L. Tremblet29, A. Tricoli29, I.M.
Trigger159a, S. Trincaz-Duvoid78, T.N. Trinh78, M.F. Tripiana70, N.
Triplett64, W. Trischuk158, A. Trivedi24,v, B. Trocmé55, C. Troncon89a, M.
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Tsiakiris105, P.V. Tsiareshka90, D. Tsionou4, G. Tsipolitis9, V.
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Tsung20, S. Tsuno66, D. Tsybychev148, A. Tua139, J.M. Tuggle30, M. Turala38,
D. Turecek127, I. Turk Cakir3e, E. Turlay105, R. Turra89a,89b, P.M. Tuts34, A.
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F. Ukegawa160, G. Unal29, D.G. Underwood5, A. Undrus24, G. Unel163, Y. Unno66,
D. Urbaniec34, E. Urkovsky153, P. Urquijo49, P. Urrejola31a, G. Usai7, M.
Uslenghi119a,119b, L. Vacavant83, V. Vacek127, B. Vachon85, S. Vahsen14, C.
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H. van der Graaf105, E. van der Kraaij105, R. Van Der Leeuw105, E. van der
Poel105, D. van der Ster29, B. Van Eijk105, N. van Eldik84, P. van Gemmeren5,
Z. van Kesteren105, I. van Vulpen105, W. Vandelli29, G. Vandoni29, A.
Vaniachine5, P. Vankov41, F. Vannucci78, F. Varela Rodriguez29, R. Vari132a,
E.W. Varnes6, D. Varouchas14, A. Vartapetian7, K.E. Varvell150, V.I.
Vassilakopoulos56, F. Vazeille33, G. Vegni89a,89b, J.J. Veillet115, C.
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Verkerke105, J.C. Vermeulen105, A. Vest43, M.C. Vetterli142,d, I. Vichou165,
T. Vickey145b,y, G.H.A. Viehhauser118, S. Viel168, M. Villa19a,19b, M.
Villaplana Perez167, E. Vilucchi47, M.G. Vincter28, E. Vinek29, V.B.
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Toerne20, V. Vorobel126, A.P. Vorobiev128, V. Vorwerk11, M. Vos167, R. Voss29,
T.T. Voss174, J.H. Vossebeld73, A.S. Vovenko128, N. Vranjes12a, M. Vranjes
Milosavljevic12a, V. Vrba125, M. Vreeswijk105, T. Vu Anh81, R. Vuillermet29,
I. Vukotic115, W. Wagner174, P. Wagner120, H. Wahlen174, J. Wakabayashi101, J.
Walbersloh42, S. Walch87, J. Walder71, R. Walker98, W. Walkowiak141, R.
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M. Warsinsky48, P.M. Watkins17, A.T. Watson17, M.F. Watson17, G. Watts138, S.
Watts82, A.T. Waugh150, B.M. Waugh77, J. Weber42, M. Weber129, M.S. Weber16,
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1 University at Albany, Albany NY, United States of America
2 Department of Physics, University of Alberta, Edmonton AB, Canada
3 (a)Department of Physics, Ankara University, Ankara; (b)Department of
Physics, Dumlupinar University, Kutahya; (c)Department of Physics, Gazi
University, Ankara; (d)Division of Physics, TOBB University of Economics and
Technology, Ankara; (e)Turkish Atomic Energy Authority, Ankara, Turkey
4 LAPP, CNRS/IN2P3 and Université de Savoie, Annecy-le-Vieux, France
5 High Energy Physics Division, Argonne National Laboratory, Argonne IL,
United States of America
6 Department of Physics, University of Arizona, Tucson AZ, United States of
America
7 Department of Physics, The University of Texas at Arlington, Arlington TX,
United States of America
8 Physics Department, University of Athens, Athens, Greece
9 Physics Department, National Technical University of Athens, Zografou,
Greece
10 Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan
11 Institut de Física d’Altes Energies and Universitat Autònoma de Barcelona
and ICREA, Barcelona, Spain
12 (a)Institute of Physics, University of Belgrade, Belgrade; (b)Vinca
Institute of Nuclear Sciences, Belgrade, Serbia
13 Department for Physics and Technology, University of Bergen, Bergen, Norway
14 Physics Division, Lawrence Berkeley National Laboratory and University of
California, Berkeley CA, United States of America
15 Department of Physics, Humboldt University, Berlin, Germany
16 Albert Einstein Center for Fundamental Physics and Laboratory for High
Energy Physics, University of Bern, Bern, Switzerland
17 School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
18 (a)Department of Physics, Bogazici University, Istanbul; (b)Division of
Physics, Dogus University, Istanbul; (c)Department of Physics Engineering,
Gaziantep University, Gaziantep; (d)Department of Physics, Istanbul Technical
University, Istanbul, Turkey
19 (a)INFN Sezione di Bologna; (b)Dipartimento di Fisica, Università di
Bologna, Bologna, Italy
20 Physikalisches Institut, University of Bonn, Bonn, Germany
21 Department of Physics, Boston University, Boston MA, United States of
America
22 Department of Physics, Brandeis University, Waltham MA, United States of
America
23 (a)Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro;
(b)Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil
24 Physics Department, Brookhaven National Laboratory, Upton NY, United States
of America
25 (a)National Institute of Physics and Nuclear Engineering, Bucharest;
(b)University Politehnica Bucharest, Bucharest; (c)West University in
Timisoara, Timisoara, Romania
26 Departamento de Física, Universidad de Buenos Aires, Buenos Aires,
Argentina
27 Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
28 Department of Physics, Carleton University, Ottawa ON, Canada
29 CERN, Geneva, Switzerland
30 Enrico Fermi Institute, University of Chicago, Chicago IL, United States of
America
31 (a)Departamento de Fisica, Pontificia Universidad Católica de Chile,
Santiago; (b)Departamento de Física, Universidad Técnica Federico Santa María,
Valparaíso, Chile
32 (a)Institute of High Energy Physics, Chinese Academy of Sciences, Beijing;
(b)Department of Modern Physics, University of Science and Technology of
China, Anhui; (c)Department of Physics, Nanjing University, Jiangsu; (d)High
Energy Physics Group, Shandong University, Shandong, China
33 Laboratoire de Physique Corpusculaire, Clermont Université and Université
Blaise Pascal and CNRS/IN2P3, Aubiere Cedex, France
34 Nevis Laboratory, Columbia University, Irvington NY, United States of
America
35 Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark
36 (a)INFN Gruppo Collegato di Cosenza; (b)Dipartimento di Fisica, Università
della Calabria, Arcavata di Rende, Italy
37 Faculty of Physics and Applied Computer Science, AGH-University of Science
and Technology, Krakow, Poland
38 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Krakow, Poland
39 Physics Department, Southern Methodist University, Dallas TX, United States
of America
40 Physics Department, University of Texas at Dallas, Richardson TX, United
States of America
41 DESY, Hamburg and Zeuthen, Germany
42 Institut für Experimentelle Physik IV, Technische Universität Dortmund,
Dortmund, Germany
43 Institut für Kern- und Teilchenphysik, Technical University Dresden,
Dresden, Germany
44 Department of Physics, Duke University, Durham NC, United States of America
45 SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh,
United Kingdom
46 Fachhochschule Wiener Neustadt, Wiener Neustadt, Austria
47 INFN Laboratori Nazionali di Frascati, Frascati, Italy
48 Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg
i.Br., Germany
49 Section de Physique, Université de Genève, Geneva, Switzerland
50 (a)INFN Sezione di Genova; (b)Dipartimento di Fisica, Università di Genova,
Genova, Italy
51 Institute of Physics and HEP Institute, Georgian Academy of Sciences and
Tbilisi State University, Tbilisi, Georgia
52 II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen,
Germany
53 SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow,
United Kingdom
54 II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany
55 Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph
Fourier and CNRS/IN2P3 and Institut National Polytechnique de Grenoble,
Grenoble, France
56 Department of Physics, Hampton University, Hampton VA, United States of
America
57 Laboratory for Particle Physics and Cosmology, Harvard University,
Cambridge MA, United States of America
58 (a)Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg,
Heidelberg; (b)Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg; (c)ZITI Institut für technische Informatik, Ruprecht-Karls-
Universität Heidelberg, Mannheim, Germany
59 Faculty of Science, Hiroshima University, Hiroshima, Japan
60 Faculty of Applied Information Science, Hiroshima Institute of Technology,
Hiroshima, Japan
61 Department of Physics, Indiana University, Bloomington IN, United States of
America
62 Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität,
Innsbruck, Austria
63 University of Iowa, Iowa City IA, United States of America
64 Department of Physics and Astronomy, Iowa State University, Ames IA, United
States of America
65 Joint Institute for Nuclear Research, JINR Dubna, Dubna, Russia
66 KEK, High Energy Accelerator Research Organization, Tsukuba, Japan
67 Graduate School of Science, Kobe University, Kobe, Japan
68 Faculty of Science, Kyoto University, Kyoto, Japan
69 Kyoto University of Education, Kyoto, Japan
70 Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET,
La Plata, Argentina
71 Physics Department, Lancaster University, Lancaster, United Kingdom
72 (a)INFN Sezione di Lecce; (b)Dipartimento di Fisica, Università del
Salento, Lecce, Italy
73 Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
74 Department of Physics, Jožef Stefan Institute and University of Ljubljana,
Ljubljana, Slovenia
75 Department of Physics, Queen Mary University of London, London, United
Kingdom
76 Department of Physics, Royal Holloway University of London, Surrey, United
Kingdom
77 Department of Physics and Astronomy, University College London, London,
United Kingdom
78 Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and
Université Paris-Diderot and CNRS/IN2P3, Paris, France
79 Fysiska institutionen, Lunds universitet, Lund, Sweden
80 Departamento de Fisica Teorica C-15, Universidad Autonoma de Madrid,
Madrid, Spain
81 Institut für Physik, Universität Mainz, Mainz, Germany
82 School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
83 CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France
84 Department of Physics, University of Massachusetts, Amherst MA, United
States of America
85 Department of Physics, McGill University, Montreal QC, Canada
86 School of Physics, University of Melbourne, Victoria, Australia
87 Department of Physics, The University of Michigan, Ann Arbor MI, United
States of America
88 Department of Physics and Astronomy, Michigan State University, East
Lansing MI, United States of America
89 (a)INFN Sezione di Milano; (b)Dipartimento di Fisica, Università di Milano,
Milano, Italy
90 B.I. Stepanov Institute of Physics, National Academy of Sciences of
Belarus, Minsk, Republic of Belarus
91 National Scientific and Educational Centre for Particle and High Energy
Physics, Minsk, Republic of Belarus
92 Department of Physics, Massachusetts Institute of Technology, Cambridge MA,
United States of America
93 Group of Particle Physics, University of Montreal, Montreal QC, Canada
94 P.N. Lebedev Institute of Physics, Academy of Sciences, Moscow, Russia
95 Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia
96 Moscow Engineering and Physics Institute (MEPhI), Moscow, Russia
97 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Moscow, Russia
98 Fakultät für Physik, Ludwig-Maximilians-Universität München, München,
Germany
99 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München,
Germany
100 Nagasaki Institute of Applied Science, Nagasaki, Japan
101 Graduate School of Science, Nagoya University, Nagoya, Japan
102 (a)INFN Sezione di Napoli; (b)Dipartimento di Scienze Fisiche, Università
di Napoli, Napoli, Italy
103 Department of Physics and Astronomy, University of New Mexico, Albuquerque
NM, United States of America
104 Institute for Mathematics, Astrophysics and Particle Physics, Radboud
University Nijmegen/Nikhef, Nijmegen, Netherlands
105 Nikhef National Institute for Subatomic Physics and University of
Amsterdam, Amsterdam, Netherlands
106 Department of Physics, Northern Illinois University, DeKalb IL, United
States of America
107 Budker Institute of Nuclear Physics (BINP), Novosibirsk, Russia
108 Department of Physics, New York University, New York NY, United States of
America
109 Ohio State University, Columbus OH, United States of America
110 Faculty of Science, Okayama University, Okayama, Japan
111 Homer L. Dodge Department of Physics and Astronomy, University of
Oklahoma, Norman OK, United States of America
112 Department of Physics, Oklahoma State University, Stillwater OK, United
States of America
113 Palacký University, RCPTM, Olomouc, Czech Republic
114 Center for High Energy Physics, University of Oregon, Eugene OR, United
States of America
115 LAL, Univ. Paris-Sud and CNRS/IN2P3, Orsay, France
116 Graduate School of Science, Osaka University, Osaka, Japan
117 Department of Physics, University of Oslo, Oslo, Norway
118 Department of Physics, Oxford University, Oxford, United Kingdom
119 (a)INFN Sezione di Pavia; (b)Dipartimento di Fisica Nucleare e Teorica,
Università di Pavia, Pavia, Italy
120 Department of Physics, University of Pennsylvania, Philadelphia PA, United
States of America
121 Petersburg Nuclear Physics Institute, Gatchina, Russia
122 (a)INFN Sezione di Pisa; (b)Dipartimento di Fisica E. Fermi, Università di
Pisa, Pisa, Italy
123 Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh
PA, United States of America
124 (a)Laboratorio de Instrumentacao e Fisica Experimental de Particulas -
LIP, Lisboa, Portugal; (b)Departamento de Fisica Teorica y del Cosmos and
CAFPE, Universidad de Granada, Granada, Spain
125 Institute of Physics, Academy of Sciences of the Czech Republic, Praha,
Czech Republic
126 Faculty of Mathematics and Physics, Charles University in Prague, Praha,
Czech Republic
127 Czech Technical University in Prague, Praha, Czech Republic
128 State Research Center Institute for High Energy Physics, Protvino, Russia
129 Particle Physics Department, Rutherford Appleton Laboratory, Didcot,
United Kingdom
130 Physics Department, University of Regina, Regina SK, Canada
131 Ritsumeikan University, Kusatsu, Shiga, Japan
132 (a)INFN Sezione di Roma I; (b)Dipartimento di Fisica, Università La
Sapienza, Roma, Italy
133 (a)INFN Sezione di Roma Tor Vergata; (b)Dipartimento di Fisica, Università
di Roma Tor Vergata, Roma, Italy
134 (a)INFN Sezione di Roma Tre; (b)Dipartimento di Fisica, Università Roma
Tre, Roma, Italy
135 (a)Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des
Hautes Energies - Université Hassan II, Casablanca; (b)Centre National de
l’Energie des Sciences Techniques Nucleaires, Rabat; (c)Université Cadi Ayyad,
Faculté des sciences Semlalia Département de Physique, B.P. 2390 Marrakech
40000; (d)Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda;
(e)Faculté des Sciences, Université Mohammed V, Rabat, Morocco
136 DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l’Univers),
CEA Saclay (Commissariat a l’Energie Atomique), Gif-sur-Yvette, France
137 Santa Cruz Institute for Particle Physics, University of California Santa
Cruz, Santa Cruz CA, United States of America
138 Department of Physics, University of Washington, Seattle WA, United States
of America
139 Department of Physics and Astronomy, University of Sheffield, Sheffield,
United Kingdom
140 Department of Physics, Shinshu University, Nagano, Japan
141 Fachbereich Physik, Universität Siegen, Siegen, Germany
142 Department of Physics, Simon Fraser University, Burnaby BC, Canada
143 SLAC National Accelerator Laboratory, Stanford CA, United States of
America
144 (a)Faculty of Mathematics, Physics & Informatics, Comenius University,
Bratislava; (b)Department of Subnuclear Physics, Institute of Experimental
Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic
145 (a)Department of Physics, University of Johannesburg, Johannesburg;
(b)School of Physics, University of the Witwatersrand, Johannesburg, South
Africa
146 (a)Department of Physics, Stockholm University; (b)The Oskar Klein Centre,
Stockholm, Sweden
147 Physics Department, Royal Institute of Technology, Stockholm, Sweden
148 Department of Physics and Astronomy, Stony Brook University, Stony Brook
NY, United States of America
149 Department of Physics and Astronomy, University of Sussex, Brighton,
United Kingdom
150 School of Physics, University of Sydney, Sydney, Australia
151 Institute of Physics, Academia Sinica, Taipei, Taiwan
152 Department of Physics, Technion: Israel Inst. of Technology, Haifa, Israel
153 Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv
University, Tel Aviv, Israel
154 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki,
Greece
155 International Center for Elementary Particle Physics and Department of
Physics, The University of Tokyo, Tokyo, Japan
156 Graduate School of Science and Technology, Tokyo Metropolitan University,
Tokyo, Japan
157 Department of Physics, Tokyo Institute of Technology, Tokyo, Japan
158 Department of Physics, University of Toronto, Toronto ON, Canada
159 (a)TRIUMF, Vancouver BC; (b)Department of Physics and Astronomy, York
University, Toronto ON, Canada
160 Institute of Pure and Applied Sciences, University of Tsukuba, Ibaraki,
Japan
161 Science and Technology Center, Tufts University, Medford MA, United States
of America
162 Centro de Investigaciones, Universidad Antonio Narino, Bogota, Colombia
163 Department of Physics and Astronomy, University of California Irvine,
Irvine CA, United States of America
164 (a)INFN Gruppo Collegato di Udine; (b)ICTP, Trieste; (c)Dipartimento di
Fisica, Università di Udine, Udine, Italy
165 Department of Physics, University of Illinois, Urbana IL, United States of
America
166 Department of Physics and Astronomy, University of Uppsala, Uppsala,
Sweden
167 Instituto de Física Corpuscular (IFIC) and Departamento de Física Atómica,
Molecular y Nuclear and Departamento de Ingenierá Electrónica and Instituto de
Microelectrónica de Barcelona (IMB-CNM), University of Valencia and CSIC,
Valencia, Spain
168 Department of Physics, University of British Columbia, Vancouver BC,
Canada
169 Department of Physics and Astronomy, University of Victoria, Victoria BC,
Canada
170 Waseda University, Tokyo, Japan
171 Department of Particle Physics, The Weizmann Institute of Science,
Rehovot, Israel
172 Department of Physics, University of Wisconsin, Madison WI, United States
of America
173 Fakultät für Physik und Astronomie, Julius-Maximilians-Universität,
Würzburg, Germany
174 Fachbereich C Physik, Bergische Universität Wuppertal, Wuppertal, Germany
175 Department of Physics, Yale University, New Haven CT, United States of
America
176 Yerevan Physics Institute, Yerevan, Armenia
177 Domaine scientifique de la Doua, Centre de Calcul CNRS/IN2P3, Villeurbanne
Cedex, France
a Also at Laboratorio de Instrumentacao e Fisica Experimental de Particulas -
LIP, Lisboa, Portugal
b Also at Faculdade de Ciencias and CFNUL, Universidade de Lisboa, Lisboa,
Portugal
c Also at CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France
d Also at TRIUMF, Vancouver BC, Canada
e Also at Department of Physics, California State University, Fresno CA,
United States of America
f Also at Faculty of Physics and Applied Computer Science, AGH-University of
Science and Technology, Krakow, Poland
g Also at Department of Physics, University of Coimbra, Coimbra, Portugal
h Also at Università di Napoli Parthenope, Napoli, Italy
i Also at Institute of Particle Physics (IPP), Canada
j Also at Louisiana Tech University, Ruston LA, United States of America
k Also at Group of Particle Physics, University of Montreal, Montreal QC,
Canada
l Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku,
Azerbaijan
m Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg,
Germany
n Also at Manhattan College, New York NY, United States of America
o Also at School of Physics and Engineering, Sun Yat-sen University, Guanzhou,
China
p Also at Academia Sinica Grid Computing, Institute of Physics, Academia
Sinica, Taipei, Taiwan
q Also at High Energy Physics Group, Shandong University, Shandong, China
r Also at California Institute of Technology, Pasadena CA, United States of
America
s Also at Particle Physics Department, Rutherford Appleton Laboratory, Didcot,
United Kingdom
t Also at Section de Physique, Université de Genève, Geneva, Switzerland
u Also at Departamento de Fisica, Universidade de Minho, Braga, Portugal
v Also at Department of Physics and Astronomy, University of South Carolina,
Columbia SC, United States of America
w Also at KFKI Research Institute for Particle and Nuclear Physics, Budapest,
Hungary
x Also at Institute of Physics, Jagiellonian University, Krakow, Poland
y Also at Department of Physics, Oxford University, Oxford, United Kingdom
z has been working on Muon MDT T0 calibration work as author service from
2010/02
aa Also at DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de
l’Univers), CEA Saclay (Commissariat a l’Energie Atomique), Gif-sur-Yvette,
France
ab Also at Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and
Université Paris-Diderot and CNRS/IN2P3, Paris, France
ac Also at Department of Physics, Nanjing University, Jiangsu, China
∗ Deceased
|
arxiv-papers
| 2011-03-15T15:02:37 |
2024-09-04T02:49:17.685591
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "The ATLAS Collaboration",
"submitter": "The ATLAS Collaboration",
"url": "https://arxiv.org/abs/1103.2929"
}
|
1103.3179
|
# Sharp estimates for the global attractor of scalar reaction-diffusion
equations with a Wentzell boundary condition
Ciprian G. Gal
Department of Mathematics
University of Missouri,
Columbia, MO 65211, USA
ciprian@math.missouri.edu
###### Abstract
In this paper, we derive optimal upper and lower bounds on the dimension of
the attractor $\mathcal{A}_{W}$ for scalar reaction-diffusion equations with a
Wentzell (dynamic) boundary condition. We are also interested in obtaining
explicit bounds on the constants involved in our asymptotic estimates, and to
compare these bounds to previously known estimates for the dimension of the
global attractor $\mathcal{A}_{K},$ $K\in\left\\{D,N,P\right\\}$, of reaction-
diffusion equations subject to Dirichlet, Neumann and periodic boundary
conditions. The explicit estimates we obtain show that the dimension of the
global attractor $\mathcal{A}_{W}$ is of different order than the dimension of
$\mathcal{A}_{K},$ for each $K\in\left\\{D,N,P\right\\},$ in all space
dimensions that are greater or equal than three.
11footnotetext: 2000 Mathematics Subject Classification: 35K57, 35K55, 35B40,
35B41, 37L30, 35Q80, 35Q86
## 1 Introduction
It is well-known that the long-time behaviour of solutions of partial
differential equations arising in mathematical physics can, in many cases, be
described in terms of global attractors of the associated semigroups (see [3,
6, 32, 46] and references therein). For a large class of equations of
mathematical physics, including parabolic partial differential equations
modelling reaction, diffusion and drift, hyperbolic type equations, and so on,
the corresponding attractor has finite Hausdorff and fractal dimensions. Thus,
the dynamics on the attractor happens to be finite-dimensional, even though
the system is governed by a set of partial differential equations. As the
dimension of the attractor is indicative of the number of degrees of freedom
needed to simulate a given dynamical system, it is then crucial to obtain more
realistic estimates for its dimension in terms of observable physical
quantities.
Aside from some applied motivation, much of the mathematical interest nowadays
is centered on the dynamics of boundary value problems with _static_ boundary
conditions of Dirichlet and Neumann-Robin type, or even periodic boundary
conditions. The influence of these dissipative boundary conditions on a given
model has only been recently investigated in connection with a class of
reaction-diffusion systems. In [33], a first contribution is made to the
understanding of this problem with a Robin boundary condition. In particular,
it is shown, for a fixed nonlinearity, how the flow defined by the reaction-
diffusion system depends on the interaction between diffusion $\nu$ and
another parameter $\theta$ involved in the boundary condition (cf. also [34]).
A classification of points in $\left(\nu,\theta\right)$-space, as structurally
stable, or bifurcation points, for a one-dimensional scalar reaction-diffusion
equation with a cubic nonlinearity is discussed in detail in [33]. Other
studies on the influence of boundary conditions upon the solution structures
of partial differential equations have also been done by other scientists.
These studies have analyzed the detailed effect of boundary conditions on the
structure of global attractors (see, e.g., [8, 30, 36, 44]). If the
equilibrium is nonhyperbolic and a bifurcation occurs, the structure of
attractors may vary with respect to boundary conditions. This has been
observed in the analysis of pattern formation in a 1D reaction-diffusion
system [8], in lattice systems [43], in the study of steady state bifurcations
[30, 36], and finally in [44], on mode-jumping of the von Karman equations.
Although the global attractors of these systems will depend, for a given
nonlinearity, on the choice of the boundary conditions, their finite dimension
does generally _not_. This result can be easily formulated for a scalar
reaction-diffusion equation, as follows. Consider the parabolic partial
differential equation
$\partial_{t}u=\nu\Delta u-f\left(u\right)+\lambda u+g,\text{
}\left(x,t\right)\in\Omega\times\left(0,+\infty\right),$ (1.1)
where $u=u\left(x,t\right)\in\mathbb{R}$, $\Omega\subset\mathbb{R}^{n}$,
$n\geq 1$, is a bounded domain with sufficiently smooth boundary $\Gamma,$
$g=g\left(x\right)$, and $\nu$, $\lambda$ are positive constants. The function
$f:\mathbb{R\rightarrow R}$ is assumed to be $C^{1,1}$, that is, continuous
and with a Lipschitz continuous first derivative, which satisfies, among other
natural growth conditions (see, e.g., [6, Chapter II]),
$f^{{}^{\prime}}\left(y\right)\geq-c_{f},\text{ for all }y\in\mathbb{R}\text{,
for some }c_{f}>0.$
We may ask that $u$ satisfy either a Dirichlet ($K=D$) boundary condition or a
Neumann ($K=N$) boundary condition, and even a periodic ($K=P$) boundary
condition. It is well-known that equation (1.1), supplemented with an
appropriate initial condition, generates a semigroup $\left\\{S_{t}\right\\}$
acting on a suitable phase space $H$. This semigroup possesses the global
attractor $\mathcal{A}_{K}$, which may depend on the choice of the boundary
conditions, and $\mathcal{A}_{K}$ has finite fractal dimension for each
$K\in\left\\{D,N,P\right\\}$. In particular, the Haussdorf and fractal
dimensions of $\mathcal{A}_{K},$ for any $K\in\left\\{D,N,P\right\\}$, satisfy
the following upper and lower bounds:
$c_{0}\left(\frac{\lambda}{\nu}\right)^{n/2}\left|\Omega\right|\leq\dim_{H}\mathcal{A}_{K}\leq\dim_{F}\mathcal{A}_{K}\leq
c_{1}\left(1+\frac{c_{f}+\lambda}{\nu}\left|\Omega\right|^{2/n}\right)^{n/2},$
(1.2)
for some positive constants $c_{0},c_{1}$ that depend only on $n,$ $f$ and the
shape of $\Omega$ (see, e.g., [3, Chapter III]; cf. also [6], [46, Chapter
VI]). Here, $\left|\Omega\right|$ stands for the Lebesgue measure of $\Omega$.
For a fixed domain $\Omega$, we observe that these estimates are sharp with
respect to $\nu\rightarrow 0^{+}$ (for each fixed $\lambda>0$), or
sufficiently large $\lambda$ (for each fixed $\nu>0$). Hence, these bounds for
the dimension of $\mathcal{A}_{K}$ are of the same order for each
$K\in\left\\{D,N,P\right\\}$. These remarkable estimates also depend linearly
on the ”volume” of the spatial domain $\Omega$, which is consistent with
physical intuition. This property of the dimension of the attractor has not
been proved for all equations, such as, the Kuramoto-Sivashinski equation.
Our main goal in this paper is to investigate the dependance of the dimension
of the global attractor for equation (1.1) subject to a completely new class
of boundary conditions, which are sometimes dubbed as _Wentzell_ boundary
conditions, and which have some applications in probability theory,
specifically, Markov processes. But what are they really? To put them into a
context, let $L$ be an elliptic differential operator of the second order
(e.g., $L=\nu\Delta$) with coefficients that are well-defined over
$\overline{\Omega}$. It is known that there is a one-to-one correspondence
between $\left(C_{0}\right)$-semigroups and Markov processes in
$\overline{\Omega}$ which are homogeneous in time and satisfy the condition of
Feller [9] (that is, the range of the resolvent operator coincides with a
prescribed set). Thus, to each such Markov process there is a corresponding
semigroup of operators
$T_{t}v\left(x\right)=\int\limits_{\overline{\Omega}}v\left(y\right)P\left(t,x,dy\right),$
where the Markov transition function $P\left(t,x,B\right)$ satisfies
$P\left(t,x,B\right)\geq 0,$ for $t\geq 0,$ $x\in\overline{\Omega}$ and any
Borel set $B\subseteq\overline{\Omega}$. As a function of $B$,
$P\left(t,x,\cdot\right)$ is a probability measure. What are the most general
boundary conditions which restrict the given operator $L$ (more correctly, its
closure) to the infinitesimal operator of a semigroup of positive contraction
operators acting on $C\left(\overline{\Omega}\right)$? Wentzell [47] gave a
partial answer to this question in higher space dimensions by finding a
sufficiently large class of boundary conditions which involve differential
operators on the boundary that are of the same order as the operator acting in
$\Omega$. He discovered the following form of boundary conditions:
$Lu+\nu b\partial_{\mathbf{n}}^{L}u=0\text{, on
}\Gamma\times\left(0,+\infty\right),$ (1.3)
where $\mathbf{n}$ denotes the outward normal at $\Gamma$, $b$ is a positive
constant and $\partial_{\mathbf{n}}^{L}u$ is the outward co-normal derivative
of $u$ with respect to $L$. We refer also to the pioneering work of [10], for
generation theorems for $L$ with Wentzell boundary conditions in one space
dimension. Until the work of [11], the study of the operator $L$ with Wentzell
boundary conditions was usually confined to generation properties of this
operator in the space $C\left(\overline{\Omega}\right)$. In 2002, the authors
in [11] have found a way to introduce the Wentzell boundary condition (1.3) in
an $L^{p}$-context, which led to the discovery of the natural space for these
type of problems (see Section 2). The reader is referred to [4, 27] for an
extensive survey of these results and some history.
For the homogeneous linear heat equation (1.1) (that is, $f=g=\lambda=0$), the
Wentzell boundary condition (1.3) is equivalent to a purely differential
equation of the form
$\partial_{t}u+\nu
b\partial_{\mathbf{n}}u=0,\quad\text{on}\;\Gamma\times\left(0,\infty\right).$
(1.4)
Thus, the main attraction here is that there is a dynamic element introduced
into the boundary condition. The heat equation, supplemented by either (1.3)
or (1.4), corresponds to the situation where there is a heat source (if $b>0$)
or sink (if $b<0$) acting on the boundary $\Gamma$. Mathematically speaking,
this kind of conditions (1.4) arises due to the presence of additional
boundary terms in the free energy, which must also account for the action of a
source on $\Gamma$ (see [22]). We refer the reader to [26] (cf. also [16]),
for an extensive derivation and physical interpretation of (1.4) for (1.1).
For the nonlinear parabolic equation (1.1), the boundary condition (1.3) can
be formally be transformed into a condition of the form
$\partial_{t}u+\nu b\partial_{\mathbf{n}}u+f\left(u\right)-\lambda u=g,\text{
on }\Gamma\times\left(0,\infty\right).$ (1.5)
Generally, one may replace $f-\lambda$ in (1.5) by another arbitrary function
$h$, satisfying suitable assumptions. With more sophisticated arguments, using
techniques from semigroup theory, and a variation of parameter formula, it is
possible to prove that the regularity of the solution for (1.1),(1.5)
increases as $f$, $\Omega$ and $g$ become more regular (see Section 2). In
particular, for $g=0$, if $\Omega$ is a bounded $\mathcal{C}^{\infty}$ domain
and $f$ is a $\mathcal{C}^{\infty}$ function, regularity theory implies that
the solution $u\left(t\right)$ to (1.1),(1.5) belongs
$H^{k}\left(\Omega\right),$ for all $k\geq 0$ and all positive times. At least
in this case, the boundary condition (1.3), for equation (1.1), is equivalent
to the boundary condition (1.5). However, in general, this may not be so if
the solution, for the semilinear problem (1.1) and the Wentzell condition
(1.5), is not smooth enough. Since we wish to treat the most general case, by
imposing the least regularity assumptions on $f,$ $g$ and $\Omega,$ we will
devote our attention only to the study of (1.1), subject to linear boundary
conditions of the form (1.4). Our results below can be immediately extended to
other classes of nonlinear Wentzell boundary conditions (see, e.g., [22] and
references therein). Boundary conditions of the form (1.5) arise for many
known equations of mathematical physics. They are motivated by heat control
problems formulated in the book of Duvaut and Lions [7], problems in phase-
transition phenomena [5, 15, 17, 20, 21, 24, 25, 37, 38] (and their
references), special flows in hydrodynamics [12, 22, 42, 39], Stefan problems
[1, 35, 41], models in climatology [40], and many others. The reader is
referred to [18] for a more complete list of references involving the
application of such boundary conditions to real-world phenomena.
By keeping our treatment of the boundary condition simple, we wish to prove
that the problem (1.1), (1.4) generates a dynamical system on a suitable
phase-space, possessing a finite dimensional global attractor
$\mathcal{A}_{W}.$ Then, we establish that the Haussdorf and fractal
dimensions of $\mathcal{A}_{W}$ satisfy the following upper and lower bounds:
$c_{1}\left(\frac{\lambda}{C_{W}\left(\Omega,\Gamma\right)\nu}\right)^{n-1}\leq\dim_{H}\mathcal{A}_{W}\leq\dim_{F}\mathcal{A}_{W}\leq
c_{2}\left(1+\frac{c_{f}+\lambda}{C_{W}\left(\Omega,\Gamma\right)\nu}\right)^{n-1},$
(1.6)
for $n\geq 2,$ and
$c_{3}\left(\frac{\lambda}{C_{D}\left(\Omega\right)\nu}\right)^{1/2}\leq\dim_{H}\mathcal{A}_{W}\leq\dim_{F}\mathcal{A}_{W}\leq
c_{4}\left(1+\frac{c_{f}+\lambda}{C_{D}\left(\Omega\right)\nu}\right)^{1/2},$
(1.7)
in one space dimension. The positive constants $c_{i},$ $i=1,...,4,$ depend
only on $n,$ $f$ and the shape of $\Omega,$ while explicit estimates and
formulas for $C_{W}\left(\Omega,\Gamma\right)$ and $C_{D}\left(\Omega\right),$
respectively, are provided in the Appendix. We note again that, for a fixed
domain $\Omega$, these estimates are sharp with respect to $\nu\rightarrow
0^{+}$ (for each fixed $\lambda>0$), and for sufficiently large $\lambda$ (if
$\nu>0$ is fixed). We remark that the bounds we obtain in (1.6)-(1.7) are
quite simple and their explicit dependance on the physical parameters is
transparent. Moreover, a careful analysis of the constants involved in (1.6)
yields the following more explicit two-sided estimate,
$c_{1}^{{}^{\prime}}\left(\frac{\lambda}{\nu
b}\right)^{n-1}\left|\Gamma\right|\leq\dim_{H}\mathcal{A}_{W}\leq\dim_{F}\mathcal{A}_{W}\leq
c_{2}^{{}^{\prime}}\left(1+\frac{c_{f}+\lambda}{\nu
b}\left|\Gamma\right|^{1/\left(n-1\right)}\right)^{n-1},$ (1.8)
in all space dimensions $n\geq 3$. It is worth pointing out that the bounds in
(1.8) are proportional to the ”surface area” $\left|\Gamma\right|$ of
$\Gamma,$ and _not_ the ”volume” $\left|\Omega\right|$ of $\Omega.$ This is
remarkable; most nonlinear equations arising in mathematical physics,
involving the Laplacian on bounded domains, have the dimension of the
attractor of the order of $\left|\Omega\right|^{\alpha},$ for some $\alpha>0$
and for sufficiently large domains. This property may have profound
implications in the prediction of weather and climate. The reader is referred
to Section 4 where this interesting physical observation is further discussed
for the balance equations governing the large-scale oceanic motion.
Our paper is organized as follows. In Section 2, we obtain upper bounds (cf.
Theorem 2.7) for the fractal dimension of the global attractor for equation
(1.1) with dynamic boundary conditions of the form (1.4). In Section 3, we
employ the same technique of [3] to derive a lower bound for the dimension of
the unstable manifold of a constant stationary solution $u^{\ast}$ of (1.1),
(1.4). As a consequence, we find a lower bound on the dimension of
$\mathcal{A}_{W}$ (see Theorem 3.1). Finally, in the Appendix, we recall some
useful results on the so-called Wentzell Laplacian, and prove an auxiliary
inequality, namely, we derive some kind of Sobolev-Lieb-Thirring inequality
that is required to prove the upper bound in (1.6).
## 2 Upper bounds on the dimension
We use the standard notation and facts from the dynamic theory of parabolic
equations (see, for instance, [4], [11], [17], [22]). We denote by
$\left\|\cdot\right\|_{p}$ and $\left\|\cdot\right\|_{p,\Gamma},$ the norms on
$L^{p}\left(\Omega\right)$ and $L^{p}\left(\Gamma\right),$ respectively. In
the case $p=2$, $\langle\cdot,\cdot\rangle_{2}$ stands for the usual scalar
product. The norms on $H^{r}\left(\Omega\right)$ and
$H^{r}\left(\Gamma\right)$ are indicated by
$\left\|\cdot\right\|_{H^{r}\left(\Omega\right)}$ and
$\left\|\cdot\right\|_{H^{r}\left(\Gamma\right)}$, respectively, for any
$r>0$.
The natural phase-space for problem (1.1), (1.4) is
$\mathbb{X}^{p}:=L^{p}(\Omega)\oplus L^{p}(\Gamma)=\\{F=\binom{f}{g}:f\in
L^{p}(\Omega),\;g\in L^{p}(\Gamma)\\},$
for all $p\in\left[1,\infty\right]$, endowed with the norm
$\left\|F\right\|_{\mathbb{X}^{p}}^{p}=\int_{\Omega}\left|f\left(x\right)\right|^{p}dx+\int_{\Gamma}\left|g(x)\right|^{p}\frac{dS}{b},\text{
}b>0,$ (2.1)
if $p\in[1,\infty),$ and
$\|F\|_{\mathbb{X}^{\infty}}:=\|f\|_{L^{\infty}(\Omega)}+\|g\|_{L^{\infty}(\Gamma)}.$
In the definition of $\mathbb{X}^{p}$, $dx$ denotes the Lebesgue measure on
$\Omega,$ and $dS$ denotes the natural surface measure on $\Gamma$. Moreover,
we have [11],
$\mathbb{X}^{p}=L^{p}\left(\overline{\Omega},d\mu\right),\text{
}p\in\left[1,\infty\right],$
where the measure $d\mu=dx_{\mid\Omega}\oplus\frac{dS}{b}_{\mid\Gamma},$ on
$\overline{\Omega},$ is defined for any measurable set
$B\subset\overline{\Omega}$ by
$\mu(B)=|B\cap\Omega|+\left|B\cap\Gamma\right|$. The Dirichlet trace map
$\mathit{Tr}_{D}:C^{\infty}\left(\overline{\Omega}\right)\rightarrow
C^{\infty}\left(\Gamma\right),$ defined by
$\mathit{Tr}_{D}\left(u\right)=u_{\mid\Gamma}$ extends to a linear continuous
operator $\mathit{Tr}_{D}:H^{r}\left(\Omega\right)\rightarrow
H^{r-1/2}\left(\Gamma\right),$ for all $r>1/2$, which is onto for $1/2<r<3/2.$
This map also possesses a bounded right inverse
$\mathit{Tr}_{D}^{-1}:H^{r-1/2}\left(\Gamma\right)\rightarrow
H^{r}\left(\Omega\right)$ such that
$\mathit{Tr}_{D}\left(\mathit{Tr}_{D}^{-1}\psi\right)=\psi,$ for any $\psi\in
H^{r-1/2}\left(\Gamma\right)$. Identifying each function $v\in
C\left(\overline{\Omega}\right)$ with the vector
$V=\binom{v}{\mathit{Tr}_{D}\left(v\right)}\in
C\left(\overline{\Omega}\right)\times C\left(\Gamma\right)$, it follows that
$C\left(\overline{\Omega}\right)$ is a dense subspace of $\mathbb{X}^{p},$ for
every $p\in[1,\infty),$ and a closed subspace of $\mathbb{X}^{\infty}.$
Finally, we can also introduce the subspaces of
$H^{r}\left(\Omega\right)\times H^{r-1/2}\left(\Gamma\right),$
$\mathbb{V}_{r}:=\left\\{\binom{u}{\psi}\in H^{r}\left(\Omega\right)\times
H^{r-1/2}\left(\Gamma\right):\mathit{Tr}_{D}\left(u\right)=\psi\right\\},$
for every $r>1/2,$ and note that we have the following dense and compact
embeddings $\mathbb{V}_{r_{1}}\subset\mathbb{V}_{r_{2}},$ for any
$r_{1}>r_{2}>1/2$. The linear subspace $\mathbb{V}_{r}$ is densely and
compactly embedded into $\mathbb{X}^{2},$ for any $r>1/2$. We emphasize that
$\mathbb{V}_{r}$ is not a product space and that, due to the boundedness of
the trace operator $\mathit{Tr}_{D},$ $\mathbb{V}_{r}$ is topologically
isomorphic to $H^{r}\left(\Omega\right)$ in the obvious way.
We begin by stating all the hypotheses on $f$ and $g$ that we need. We assume
that $g\in L^{2}\left(\Omega\right)$ and the following conditions for $f\in
C^{1}\left(\mathbb{R}\text{,}\mathbb{R}\right)$ hold:
$f^{{}^{\prime}}\left(y\right)>-c_{f},\text{ for all }y\in\mathbb{R}\text{,}$
(2.2) $\eta_{1}\left|y\right|^{p}-C_{f}\leq
f\left(y\right)y\leq\eta_{2}\left|y\right|^{p}+C_{f},$ (2.3)
for some $\eta_{1}$, $\eta_{2}>0,$ $C_{f}\geq 0$ and $p>2$.
We have the following rigorous notion of weak solution to (1.1), (1.4), with
initial condition $u\left(0\right)=u_{0},$ as in [22].
###### Definition 2.1
The pair $U\left(t\right)=\binom{u\left(t\right)}{\psi\left(t\right)}$ is said
to be a weak solution if $\psi\left(t\right)=\mathit{Tr}_{D}\left(u\right)$
for almost all $t\in\left(0,T\right),$ for any $T>0$, and $U$ fulfills
$\displaystyle U$ $\displaystyle\in
C\left(\left[0,T\right];\mathbb{X}^{2}\right)\cap
L^{\infty}\left(0,T;\mathbb{V}_{1}\right)\cap
L^{p}\left(\Omega\times\left(0,T\right)\right),$ (2.4) $\displaystyle u$
$\displaystyle\in H_{loc}^{1}(0,\infty;L^{2}\left(\Omega\right)),\text{
}\psi\in H_{loc}^{1}(0,\infty;L^{2}\left(\Gamma\right)),$
$\displaystyle\partial_{t}U$ $\displaystyle\in
L^{2}\left(0,T;\mathbb{V}_{1}^{\ast}\right),$
such that the identity
$\left\langle\partial_{t}U,\Xi\right\rangle_{\mathbb{X}^{2}}+\nu\left\langle\nabla
u,\nabla\sigma\right\rangle_{2}+\left\langle f\left(u\right)-\lambda
u,\sigma\right\rangle_{2}=\left\langle g,\sigma\right\rangle_{2},$
holds almost everywhere in $\left(0,T\right)$, for all
$\Xi=\binom{\sigma}{\varpi}\in\mathbb{V}_{1}$. Moreover, we have, in the space
$\mathbb{X}^{2}$,
$U\left(0\right)=\binom{u_{0}}{v_{0}}=:U_{0},$ (2.5)
where $u\left(0\right)=u_{0}$ almost everywhere in $\Omega$, and
$v\left(0\right)=v_{0}$ almost everywhere in $\Gamma$. Note that in this
setting, $v_{0}$ need not be the trace of $u_{0}$ at the boundary. Thus, in
this context equation (1.4) is interpreted as an additional parabolic
equation, acting now on the boundary $\Gamma$.
The following result is a direct consequence of results contained in [22,
Section 2]. The proof is based on the application of a Galerkin approximation
scheme which is not standard due to the nature of the boundary conditions
(see, also, [5]).
###### Theorem 2.2
Let the assumptions of (2.2), (2.3) be satisfied. For any given initial data
$U_{0}\in\mathbb{X}^{2},$ the problem (1.1), (1.4), (2.5) has a unique weak
solution which depends continuously on the initial data in a Lipschitz way.
The following estimate holds:
$\displaystyle\left\|U\left(t\right)\right\|_{\mathbb{X}^{2}}^{2}+\int\limits_{t}^{t+1}\left(\left\|U\left(s\right)\right\|_{\mathbb{V}_{1}}^{2}+\left\|u\left(s\right)\right\|_{L^{p}\left(\Omega\right)}^{p}\right)ds$
(2.6) $\displaystyle\leq
c\left(\left\|U\left(0\right)\right\|_{\mathbb{X}^{2}}^{2}\right)e^{-\rho
t}+c\left(1+\left\|g\right\|_{L^{2}\left(\Omega\right)}^{2}\right),$
for all $t\geq 0$, where the positive constants $c$, $\rho$ are independent of
time and initial data.
As a consequence, problem (1.1), (1.4), (2.5) defines a (nonlinear) continuous
semigroup $\mathcal{S}_{t}$ acting on the phase-space $\mathbb{X}^{2}$,
$\mathcal{S}_{t}:\mathbb{X}^{2}\rightarrow\mathbb{X}^{2},\text{ }t\geq 0,$
given by
$\mathcal{S}_{t}U_{0}=U\left(t\right).$
###### Theorem 2.3
Let $f$ satisfy assumptions (2.2), (2.3), let $g\in
L^{\infty}\left(\Omega\right)$ and $\Gamma\in\mathcal{C}^{2}$. Then,
$\left\\{\mathcal{S}_{t}\right\\}$ possesses the connected global attractor
$\mathcal{A}_{W},$ which is a bounded subset of
$\mathbb{V}_{2}\cap\mathbb{X}^{\infty}$. As a consequence, the global
attractor contains only strong solutions.
Proof. The existence of an absorbing set in $\mathbb{V}_{1}\cap
L^{p}\left(\Omega\right)$ and, hence, the existence of the global attractor
$\mathcal{A}_{W}\subset\mathbb{V}_{1}$ follows from [22, Theorem 2.8 and
Corollary 3.11]. We will now show that the attractor is bounded in
$\mathbb{X}^{\infty}$, and also in $\mathbb{V}_{2}$. All the calculations
below are formal. However, they can be rigorously justified by means of the
approximation procedures devised in [22] and [16] (cf. [5] also). From now on,
$c$ will denote a positive constant that is independent of time and initial
data, which only depends on the other structural parameters of the problem,
that is, $\left|\Omega\right|,$ $\left|\Gamma\right|,$ $\eta_{i},$ $\nu,$ $b,$
$\left\|g\right\|_{\infty}$ and $n$. Such a constant may vary even from line
to line.
Step 1. We will first establish the existence of a bounded absorbing set in
$\mathbb{X}^{\infty}$. First note that by (2.6), there is a constant
$C_{0}>0,$ independent of time and initial data, such that for any bounded
subset $B$ of $\mathbb{X}^{2}$, $\exists$
$\tau=\tau\left(\left\|B\right\|_{\mathbb{X}^{2}}\right)>0$ with
$\sup_{t\geq\tau}\left\|U\left(t\right)\right\|_{\mathbb{X}^{2}}\leq C_{0}.$
(2.7)
We shall now perform an Alikakos-Moser iteration argument. We multiply (1.1)
by $\left|u\right|^{r_{k}-2}u,$ $r_{k}:=2^{k},$ $k\geq 1,$ and integrate over
$\Omega$. We obtain
$\displaystyle\frac{1}{r_{k}}\frac{d}{dt}\left\|u\right\|_{r_{k}}^{r_{k}}+\left\langle
f\left(u\right),\left|u\right|^{r_{k}-2}u\right\rangle_{2}+\nu\int_{\Omega}\nabla
u\cdot\nabla\left(\left|u\right|^{r_{k}-2}u\right)dx$ $\displaystyle=$
$\displaystyle\nu\int_{\Gamma}\partial_{\mathbf{n}}u\left|\psi\right|^{r_{k}-2}\psi
dS+\left\langle\lambda u+g,\left|u\right|^{r_{k}-2}u\right\rangle_{2}.$
Similarly, we multiply (1.4) by $\left|\psi\right|^{r_{k}-2}\psi/b$ and
integrate over $\Gamma$. We have
$\frac{1}{br_{k}}\frac{d}{dt}\left\|\psi\right\|_{r_{k},\Gamma}^{r_{k}}+\nu\int_{\Gamma}\partial_{\mathbf{n}}u\left|\psi\right|^{r_{k}-2}\psi
dS=0.$ (2.9)
Adding the equalities (2), (2.9), we deduce
$\displaystyle\frac{1}{r_{k}}\frac{d}{dt}\left(\left\|U\right\|_{\mathbb{X}^{r_{k}}}^{r_{k}}\right)+\left\langle
f\left(u\right),\left|u\right|^{r_{k}-2}u\right\rangle_{2}+\nu\int_{\Omega}\nabla
u\cdot\nabla\left(\left|u\right|^{r_{k}-2}u\right)dx$ (2.10)
$\displaystyle=\left\langle\lambda
u+g,\left|u\right|^{r_{k}-2}u\right\rangle_{2}.$
A simple manipulation of the third integral in (2.10), and employing
assumption (2.3) on $f$, we readily get the estimate:
$\displaystyle\frac{d}{dt}\left(\left\|U\right\|_{\mathbb{X}^{r_{k}}}^{r_{k}}\right)+\eta_{1}r_{k}\left\|u\right\|_{r_{k}+p-2}^{r_{k}+p-2}+\nu\left(2^{k}-1\right)2^{2-k}\left\|\nabla\left|u\right|^{2^{k-1}}\right\|_{2}^{2}$
(2.11) $\displaystyle\leq r_{k}\left\langle\lambda
u+g+C_{f},\left|u\right|^{r_{k}-2}u\right\rangle_{2}.$
Next, using the fact that
$\left|y\right|^{r_{k}-2}\leq\left|y\right|^{r_{k}}+1,$ for all $k\geq 1$ and
$y\in\mathbb{R}$, we estimate the last term on the right-hand side of (2.11),
$\left\langle\lambda u+g+C_{f},\left|u\right|^{r_{k}-2}u\right\rangle_{2}\leq
c\left(\left\|u\right\|_{r_{k}}^{r_{k}}+1\right),$ (2.12)
for some positive constant $c$ that depends on $\lambda$ and the
$L^{\infty}$-norm of $g$, but is independent of $k$. On the other hand, it
follows from Gagliardo-Nirenberg inequality, and Young’s inequality for
$\varepsilon\in\left(0,1\right),$ that
$\left\|v\right\|_{2}\leq
c\left\|v\right\|_{H^{1}\left(\Omega\right)}^{n/\left(n+2\right)}\left\|v\right\|_{1}^{1-n/\left(n+2\right)}\leq\varepsilon\left\|v\right\|_{H^{1}\left(\Omega\right)}+c\varepsilon^{-n/2}\left\|v\right\|_{1},$
(2.13)
which implies
$\left\|\nabla
v\right\|_{2}^{2}\geq\frac{1-\varepsilon}{\varepsilon}\left\|v\right\|_{2}^{2}-c\varepsilon^{-n/2-1}\left\|v\right\|_{1}^{2}.$
Note that the estimate (2.13) remains valid if one replaces the
$L^{2}\left(\Omega\right)$ and $L^{1}\left(\Omega\right)$-norms by the
$L^{2}\left(\Gamma\right)$ and $L^{1}\left(\Gamma\right)$-norms, respectively,
and $n$ by $n-1$, respectively. Setting $v=\left|u\right|^{r_{k-1}}$ in the
above inequality, noting that $\left(2^{k}-1\right)2^{2-k}\geq 2,$ for each
$k,$ and the fact that $Tr_{D}$ maps $H^{1}\left(\Omega\right)$ boundedly into
$L^{2}\left(\Gamma\right)$, we can estimate the gradient term in (2.11) in
terms of
$c\frac{1-\varepsilon}{\varepsilon}\left(\left\|u\right\|_{r_{k}}^{r_{k}}+\left\|\psi\right\|_{r_{k},\Gamma}^{r_{k}}\right)-c\varepsilon^{-n/2-1}\left(\left\|\left|u\right|^{r_{k-1}}\right\|_{1}^{2}+\left\|\left|\psi\right|^{r_{k-1}}\right\|_{1,\Gamma}^{2}\right).$
(see, e.g., [35, Chapter 5]). This estimate together with (2.11), (2.12) yield
$\displaystyle\frac{d}{dt}\left(\left\|U\right\|_{\mathbb{X}^{r_{k}}}^{r_{k}}\right)+c\left(\nu\frac{1-\varepsilon}{\varepsilon}-r_{k}\right)\left(\left\|u\right\|_{r_{k}}^{r_{k}}+\left\|\psi\right\|_{r_{k},\Gamma}^{r_{k}}\right)$
(2.14) $\displaystyle\leq
c\varepsilon^{-n/2-1}\left(\left\|\left|u\right|^{r_{k-1}}\right\|_{1}^{2}+\left\|\left|\psi\right|^{r_{k-1}}\right\|_{1,\Gamma}^{2}\right)+cr_{k},$
for all $k\geq 1,$ where $c>0$ is independent of $k$.
We shall now make use of an iterative argument to deduce the existence of a
bounded absorbing set in $\mathbb{X}^{r_{k}},$ for all $k\geq 1$. Thus, noting
that $r_{k}\leq\left(r_{k}\right)^{n/2+1}$, then choosing
$\varepsilon=\delta/r_{k}$ with small $\delta=\delta\left(\nu\right)>0$ such
that
$\left(\nu\frac{1-\varepsilon}{\varepsilon}-r_{k}\right)\geq r_{k},$
and setting
$\mathcal{Y}_{k}\left(t\right):=\int_{\Omega}\left|u\left(t,\cdot\right)\right|^{r_{k}}dx+\int_{\Gamma}\left|\psi\left(t,\cdot\right)\right|^{r_{k}}\frac{dS}{b}=\left\|U\right\|_{\mathbb{X}^{r_{k}}}^{r_{k}},$
(2.15)
from (2.14) we derive the following estimate:
$\partial_{t}\mathcal{Y}_{k}\left(t\right)+cr_{k}\mathcal{Y}_{k}\left(t\right)\leq
c\left(r_{k}\right)^{n/2+1}\left(\mathcal{Y}_{k-1}\left(t\right)+1\right)^{2}.$
(2.16)
Let us now take two positive constants $t,$ $\mu$ such that $t-\mu/r_{k}>0,$
for all $k\geq 1$. Their precise values will be chosen later. We claim that
$\mathcal{Y}_{k}\left(t\right)\leq
M_{k}\left(t,\mu\right):=c\left(r_{k}\right)^{n/2+1}(\sup_{s\geq
t-\mu/r_{k}}\mathcal{Y}_{k-1}\left(s\right)+1)^{2}$ (2.17)
holds for $\mathcal{Y}_{k},$ defined by (2.15) and $k\geq 1$. To this end, let
$\zeta\left(s\right)$ be a positive function
$\zeta:\mathbb{R}_{+}\rightarrow\left[0,1\right]$ such that
$\zeta\left(s\right)=0$ for $s\in\left[0,t-\mu/r_{k}\right],$
$\zeta\left(s\right)=1$ if $s\in\left[t,+\infty\right)$ and
$\left|d\zeta/ds\right|\leq Cr_{k}$, if $s\in\left(t-\mu/r_{k},t\right)$. We
define $Z_{k}\left(s\right)=\zeta\left(s\right)\mathcal{Y}_{k}\left(s\right)$
and notice that
$\frac{d}{ds}Z_{k}\left(s\right)\leq
cr_{k}Z_{k}\left(s\right)+\zeta\left(s\right)\frac{d}{ds}\mathcal{Y}_{k}\left(s\right).$
Combining this estimate with (2.16) and noticing that
$Z_{k}\leq\mathcal{Y}_{k}$, we deduce the following estimate for $Z_{k}$:
$\frac{d}{ds}Z_{k}\left(s\right)+cr_{k}Z_{k}\left(s\right)\leq
M_{k}\left(t,\mu\right),\text{ for all }s\in\left[t-\mu/r_{k},+\infty\right).$
(2.18)
Integrating (2.18) with respect to $s$ from $t-\mu/r_{k}$ to $t$ and taking
into account the fact that $Z_{k}\left(t-\mu/r_{k}\right)=0,$ we obtain that
$\mathcal{Y}_{k}\left(t\right)=Z_{k}\left(t\right)\leq
M_{k}\left(t,\mu\right)\left(1-e^{-C\mu}\right)$, which proves the claim
(2.17).
Let now $\tau^{{}^{\prime}}>\tau>0$ be given with $\tau$ as in (2.7), and
define $\mu=2(\tau^{{}^{\prime}}-\tau),$ $t_{0}=\tau^{{}^{\prime}}$ and
$t_{k}=t_{k-1}-\mu/r_{k},$ $k\geq 1$. Using (2.17), we have
$\sup_{t\geq t_{k-1}}\mathcal{Y}_{k}\left(t\right)\leq
c\left(r_{k}\right)^{n/2+1}(\sup_{s\geq
t_{k}}\mathcal{Y}_{k-1}\left(s\right)+1)^{2},\text{ }k\geq 1.$ (2.19)
Note that from (2.7), we have $\left(\sup_{s\geq
t_{1}=\tau}\mathcal{Y}_{1}\left(s\right)+1\right)\leq C_{0}+1=:\overline{C}$.
Thus, we can iterate in (2.19) with respect to $k\geq 1$ and obtain that
$\displaystyle\sup_{t\geq t_{k-1}}\mathcal{Y}_{k}\left(t\right)$
$\displaystyle\leq\left[c\left(r_{k}\right)^{n/2+1}\right]\left[c\left(r_{k-1}\right)^{n/2+1}\right]^{2}\cdot...\cdot\left[c\left(r_{1}\right)^{n/2+1}\right]^{2^{k}}(\overline{C})^{r_{k}}$
$\displaystyle\leq c^{A_{k}}2^{B_{k}n/2+1}\left(\overline{C}\right)^{r_{k}},$
where
$A_{k}:=1+2+2^{2}+...+2^{k}\leq 2^{k}\sum_{i=1}^{\infty}\frac{1}{2^{i}}$
(2.20)
and
$B_{k}:=k+2\left(k-1\right)+2^{2}\left(k-2\right)+...+2^{k}\leq
2^{k}\sum_{i=1}^{\infty}\frac{i}{2^{i}}.$ (2.21)
Therefore,
$\sup_{t\geq t_{0}}\mathcal{Y}_{k}\left(t\right)\leq\sup_{t\geq
t_{k-1}}\mathcal{Y}_{k}\left(t\right)\leq
c^{A_{k}}2^{B_{k}\left(n/2+1\right)}\overline{C}^{r_{k}}.$ (2.22)
Since the series in (2.20) and (2.21) are convergent, we can take the
$r_{k}$-root on both sides of (2.22) and let $k\rightarrow+\infty$. We deduce
$\sup_{t\geq
t_{0}=\tau^{{}^{\prime}}}\left\|U\left(t\right)\right\|_{\mathbb{X}^{\infty}}\leq\lim_{k\rightarrow+\infty}\sup_{t\geq
t_{0}}\left(\mathcal{Y}_{k}\left(t\right)\right)^{1/r_{k}}\leq C_{1},$ (2.23)
for some positive constant $C_{1}$ independent of $t,$ $k$, $U$ and initial
data.
Step 2. We claim that there is a positive constant $C_{2},$ independent of
time and initial data, and there exists $\tau^{{}^{\prime\prime}}>0$ such that
$\left\|U\left(t\right)\right\|_{\mathbb{V}_{2}}\leq C_{2},\qquad\text{for all
}\,t\geq\tau^{{}^{\prime\prime}}.$ (2.24)
Before we prove (2.24), let us recall the following estimate (see [22,
Theorems 3.5, 3.10]):
$\displaystyle\sup_{t\geq\tau_{0}}\left(\left\|U\left(t\right)\right\|_{\mathbb{V}_{1}}^{2}+\left\|\partial_{t}u\left(t\right)\right\|_{2}^{2}+\frac{1}{b}\left\|\partial_{t}\psi\left(t\right)\right\|_{2,\Gamma}^{2}\right)$
(2.25)
$\displaystyle+\sup_{t\geq\tau_{0}}\int_{t}^{t+1}\left\|\partial_{t}u\left(s\right)\right\|_{H^{1}\left(\Omega\right)}^{2}ds$
$\displaystyle\leq C_{3},$
for some positive constant $C_{3}$ that is independent of time and the initial
data. In order to deduce (2.24) from (2.25) and (2.23), we need to
differentiate (1.1) and (1.4) with respect to time. This yields
$\partial_{t}^{2}u=\nu\Delta\partial_{t}u-f^{{}^{\prime}}\left(u\right)\partial_{t}u+\lambda\partial_{t}u,\text{
}\left(\partial_{t}^{2}\psi+\nu
b\partial_{{\mathbf{n}}}\left(\partial_{t}u\right)\right)_{\mid\Gamma}=0.$
(2.26)
Then, we multiply the first equation of (2.26) by $\partial_{t}^{2}u(t)$ and
integrate over $\Omega,$ using the boundary condition of (2.26). After
standard transformations, we obtain
$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\left\|\nabla\partial_{t}u\left(t\right)\right\|_{2}^{2}\right)+\left\|\partial_{t}^{2}u\left(t\right)\right\|_{2}^{2}+\frac{1}{b}\left\|\partial_{t}^{2}\psi\left(t\right)\right\|_{2,\Gamma}^{2}$
$\displaystyle=-\left\langle\left(f^{{}^{\prime}}\left(u\left(t\right)\right)-\lambda\right)\partial_{t}u\left(t\right),\partial_{t}^{2}u\left(t\right)\right\rangle_{2}.$
Using Hölder and Young inequalities, we have
$\displaystyle\frac{d}{dt}\left(\left\|\nabla\partial_{t}u\left(t\right)\right\|_{2}^{2}\right)+\left\|\partial_{t}^{2}u\left(t\right)\right\|_{2}^{2}+\frac{2}{b}\left\|\partial_{t}^{2}\psi\left(t\right)\right\|_{2,\Gamma}^{2}$
$\displaystyle\leq
c\left(\left\|f^{{}^{\prime}}\left(u\left(t\right)\right)\partial_{t}u\left(t\right)\right\|_{2}^{2}+\left\|\partial_{t}u\left(t\right)\right\|_{2}^{2}\right)$
$\displaystyle\leq
Q\left(\left\|u\left(t\right)\right\|_{\infty}\right)\left\|\partial_{t}u\left(t\right)\right\|_{2}^{2},$
for some positive nondecreasing function $Q$ that depends only on $f$ and $c$.
This estimate yields, owing to (2.23), (2.25),
$\frac{d}{dt}\left\|\nabla\partial_{t}u\left(t\right)\right\|_{2}^{2}\leq c.$
Then, we can apply the so-called uniform Gronwall’s lemma (see, e.g., [46,
Chapter III, Lemma 1.1]) to find a time $\tau_{1}\geq 1$, depending on
$\tau_{0}$ and $\tau,$ such that
$\left\|\nabla\partial_{t}u\left(t\right)\right\|_{2}^{2}\leq
c,\qquad\text{for all }\,t\geq\tau_{1}.$ (2.27)
Therefore, (2.27) and (2.25) allow us to deduce from (1.1) and (1.4), via
standard elliptic regularity, the following estimate
$\left\|u\left(t\right)\right\|_{H^{2}\left(\Omega\right)}^{2}\leq
c,\qquad\forall\,t\geq\tau_{1}.$ (2.28)
Summing up, we conclude by observing that (2.24) follows from (2.28) and the
boundedness of the trace map $Tr_{D}:H^{2}\left(\Omega\right)\rightarrow
H^{3/2}\left(\Gamma\right)$. This completes the proof of the theorem.
###### Remark 2.4
The proof of Theorem 2.3 shows how to get an absorbing set in
$\mathbb{V}_{2}$. Because of this, we can also prove the existence of a global
attractor for the dynamical system
$\left(\left\\{\mathcal{S}_{t}\right\\}_{t\geq 0},\mathbb{V}_{1}\right).$
###### Theorem 2.5
If $\Omega$ is a bounded $\mathcal{C}^{\infty}$-domain, and $f,g$ are
$\mathcal{C}^{\infty}$ functions, then the global attractor $\mathcal{A}_{W}$
is a bounded subset of $\mathbb{V}_{k},$ for every $k\geq 1$. In particular,
if $U\in\mathcal{A}_{W}$ then
$u\in\mathcal{C}^{\infty}\left(\overline{\Omega}\right).$
The proof of this result is standard and follows by successive time
differentiation of the equations in (2.26) and an induction argument. We omit
the details.
To prove the finite dimensionality of the global attractor $\mathcal{A}_{W}$,
we can proceed in two different ways. One way is to establish the existence of
a more refined object called exponential attractor $\mathcal{E}_{W}$, whose
existence proof is often based on proper forms of the so-called
squeezing/smoothing property for the differences of solutions. This can be
done by assuming smoother nonlinearities, i.e., $f\in
C^{2}\left(\mathbb{R}\right)$ (see, e.g., [16, 17]). This has been carried out
in [16], and references therein, for a system of reaction-diffusion equations
with dynamic boundary conditions of the form (1.4), without relating the
attractor dimension to the physical parameters of the problem. However, since
we wish to find explicit estimates of fractal or/and Hausdorff dimension of
$\mathcal{A}_{W}$, we shall employ the classical machinery for proving the
finite dimensionality of the global attractor $\mathcal{A}_{W}.$ This is based
on the so-called volume contraction arguments and requires the associated
solution semigroup $\mathcal{S}_{t}$ to be (uniformly quasi-) differentiable
with respect to the initial data, at least on the attractor (see, e.g., [3]).
We give without proof the following result, which follows as a consequence of
the boundedness of $\mathcal{A}_{W}$ into
$\mathbb{V}_{2}\cap\mathbb{X}^{\infty}.$
###### Proposition 2.6
Provided that $f\in C^{2}\left(\mathbb{R}\right)$ satisfies the conditions
(2.2) and (2.3), the flow $\mathcal{S}_{t}$ generated by the reaction-
diffusion equation (1.1) and dynamic boundary condition (1.4) is uniformly
differentiable on $\mathcal{A}_{W},$ with differential
$\mathbf{L}\left(t,U\left(t\right)\right):\Theta=\binom{\xi_{1}}{\xi_{2}}\in\mathbb{X}^{2}\mapsto
V=\binom{v}{\varphi}\in\mathbb{X}^{2},$ (2.29)
where $V$ is the unique solution to
$\displaystyle\partial_{t}v$ $\displaystyle=\nu\Delta
v-f^{{}^{\prime}}\left(u\left(t\right)\right)v+\lambda v,\text{
}\left(\partial_{t}\varphi+\nu b\partial_{\mathbf{n}}v\right)_{\mid\Gamma}=0,$
(2.30) $\displaystyle V\left(0\right)$ $\displaystyle=\Theta.$
Furthermore, $\mathbf{L}\left(t,U\left(t\right)\right)$ is compact for all
$t>0.$
The main result of this section is
###### Theorem 2.7
Let the assumptions of Proposition 2.6 be satisfied. The fractal dimension of
$\mathcal{A}_{W}$ admits the estimate
$\dim_{F}\mathcal{A}_{W}\leq
c_{0}\left(1+\frac{c_{f}+\lambda}{C_{W}\left(\Omega,\Gamma\right)\nu}\right)^{n-1},\text{
for }n\geq 2$ (2.31)
and
$\dim_{F}\mathcal{A}_{W}\leq
c_{0}\left(1+\frac{c_{f}+\lambda}{C_{D}\left(\Omega\right)\nu}\right)^{1/2},\text{
for }n=1,$ (2.32)
where $c_{0}$ depends on the shape of $\Omega$ only. The positive constants
$C_{W},C_{D}$ depend only on $n,$ $\Omega,$ $\Gamma$, $b$ and are given in the
Appendix.
Proof. In order to deduce (2.31)-(2.32), it is sufficient (see, e.g., [6,
Chapter III, Definition 4.1]) to estimate the $j$-trace of the operator
$\mathbf{L}\left(t,U\left(t\right)\right)=\left(\begin{array}[]{cc}\nu\Delta-f^{{}^{\prime}}\left(u\left(t\right)\right)+\lambda
I&0\\\ -b\nu\partial_{\mathbf{n}}&0\end{array}\right).$
We have
$\displaystyle
Trace\left(\mathbf{L}\left(t,U\left(t\right)\right)Q_{m}\right)$
$\displaystyle=\sum_{j=1}^{m}\left\langle\mathbf{L}\left(t,U\left(t\right)\right)\varphi_{j},\varphi_{j}\right\rangle_{\mathbb{X}^{2}}$
$\displaystyle=\sum_{i=1}^{m}\left\langle\nu\Delta\varphi_{j},\varphi_{j}\right\rangle_{2}-\sum_{i=1}^{m}\left\langle\nu\partial_{\mathbf{n}}\varphi_{j},\varphi_{j}\right\rangle_{2,\Gamma}$
$\displaystyle-\sum_{i=1}^{m}\left\langle
f^{{}^{\prime}}\left(u\left(t\right)\right)\varphi_{j},\varphi_{j}\right\rangle_{2}+\sum_{i=1}^{m}\lambda\left\langle\varphi_{j},\varphi_{j}\right\rangle_{2},$
where the set of vector-valued functions
$\varphi_{j}\in\mathbb{X}^{2}\cap\mathbb{V}_{1}$ is an orthonormal basis in
$Q_{m}\mathbb{X}^{2}$. Since the family $\varphi_{j}$ is orthonormal in
$Q_{m}\mathbb{X}^{2},$ using assumption (2.2) on $f$ (i.e.,
$f^{{}^{\prime}}\left(y\right)\geq-c_{f},$ for all $y\in\mathbb{R}$), we find
$Trace\left(\mathbf{L}\left(t,U\right)Q_{m}\right)\leq-\nu\sum_{i=1}^{m}\left\|\nabla\varphi_{j}\right\|_{2}^{2}+\left(c_{f}+\lambda\right)m.$
Let $n\geq 2$. From (5.12) (see Appendix, Proposition 5.5), we obtain
$\displaystyle Trace\left(\mathbf{L}\left(t,U\right)Q_{m}\right)$
$\displaystyle\leq-\nu
c_{1}C_{W}\left(\Omega,\Gamma\right)m^{\frac{1}{n-1}+1}+\left(c_{1}\nu
C_{W}\left(\Omega,\Gamma\right)+c_{f}+\lambda\right)m$
$\displaystyle=:\rho\left(m\right).$
The function $\rho\left(y\right)$ is concave. The root of the equation
$\rho\left(d\right)=0$ is
$d^{\ast}=\left(1+\frac{c_{f}+\lambda}{\nu
c_{1}C_{W}\left(\Omega,\Gamma\right)}\right)^{n-1}.$
Thus, we can apply [6, Corollary 4.2 and Remark 4.1] to deduce that
$\dim_{F}\mathcal{A}_{W}\leq d^{\ast},$ from which (2.31) follows. The case
$n=1$ is similar.
###### Remark 2.8
Concerning the reaction-diffusion equation (1.1), we can also handle dynamic
boundary conditions that involve surface diffusion:
$\partial_{t}u-\alpha\Delta_{\Gamma}u+b\nu\partial_{{\mathbf{n}}}\phi=0,\text{
on }\Gamma,$ (2.33)
where $\alpha>0$ and $\Delta_{\Gamma}$ is the Laplace-Beltrami operator on
$\Gamma$. Our method of establishing upper bounds, comparable to the bounds
(2.31)-(2.32), for the dimension of the global attractor can be also extended
to this case as well. The details will appear elsewhere.
## 3 Lower bounds on the dimension
Lower bounds on the dimension of the global attractor are usually based on the
observation that the unstable manifold of any equilibrium of the system is
always contained in the global attractor (see, e.g., [3]). Thus, a lower bound
on the dimension of the attractor $\mathcal{A}_{W}$ can be found by analyzing
the dimension of an unstable manifold associated with a constant equilibrium
$Z$ for (1.1), (1.4). We begin by assuming that $g$ is constant, for the sake
of simplicity. Steady-state solutions of (1.1), (1.4) satisfy
$L_{0}\left(u\right):=\nu\Delta u-f\left(u\right)+\lambda u-g=0,\text{
}\left(\partial_{\mathbf{n}}u\right)_{\mid\Gamma}=0.$
We seek a solution of this system
$U=\binom{u}{Tr_{D}\left(u\right)}\in\mathbb{X}^{2}$ which coincides with a
constant vector $Z=\mathbf{c}=\binom{c}{c},$ $c$ is a constant. Such a
stationary solution satisfies the equation
$\overline{L}_{0}\left(z\right):=-f\left(z\right)+\lambda z-g=0.$ Since
$f\left(y\right)y\geq\eta_{1}\left|y\right|^{p}-C_{f},\text{ for }p>2,$
we have
$\overline{L}_{0}\left(z\right)z\leq-\widetilde{\eta}_{1}\left|z\right|^{p}+\widetilde{C}_{f},$
for some positive constants $\widetilde{\eta}_{1},\widetilde{C}_{f}$. Thus,
$\overline{L}_{0}\left(z\right)z<0$ on the interval $I_{R}=\left(-R,R\right),$
if $R$ is large enough. It follows that $\overline{L}_{0}\left(z\right)=0$ has
at least one solution $Z=Z\left(\lambda\right)$ (see, e.g., [6, Chapter III]).
By the implicit function theorem, this solution is of order $1/\lambda$ for
sufficiently large $\lambda.$
Now fix this solution. In order to find a lower bound on the dimension of the
global attractor $\mathcal{A}_{W},$ it suffices to establish a lower bound for
$\dim E_{+}\left(Z\right),$ where $E_{+}\left(Z\right)$ is an invariant
subspace of $\mathbf{L}\left(Z\right),$ which corresponds to
$\mathbf{L}\left(Z\right)W=\binom{\nu\Delta
w-f^{{}^{\prime}}\left(z\right)w+\lambda w}{-b\nu\partial_{\mathbf{n}}w}$
with
$\sigma\left(\mathbf{L}\left(Z\right)\right)\subset\left\\{\zeta:\zeta>0\right\\}$.
We note that
$\left(\mathbf{L}\left(Z\right),D\left(\mathbf{L}\left(Z\right)\right)\right)$
is self-adjoint on $X^{2}$ with spectrum contained in
$\left(-\infty,c_{f}+\lambda\right].$
The main result of this section is the following.
###### Theorem 3.1
Let $f\in C^{2}\left(\mathbb{R}\right)$ satisfy assumptions (2.2)-(2.3). There
exist a positive constant $c_{0}$, depending on $f,$ $g$ and the shape of
$\Omega,$ independent of $\lambda,$ $\nu,$ $b$, $\left|\Omega\right|,$
$\left|\Gamma\right|,$ such that
$\dim_{F}\mathcal{A}_{W}\geq\dim_{H}\mathcal{A}_{W}\geq\dim
E_{+}\left(Z\right)\geq
c_{0}\left(\frac{\lambda}{C_{W}\left(\Omega,\Gamma\right)\nu}\right)^{n-1},$
for $n\geq 2$. In one space dimension, the same estimate is valid with $C_{W}$
replaced by $C_{D}$ and $n-1,$ replaced by $1/2$, respectively.
Proof. For a fixed constant solution $Z=\mathbf{c}$ of
$\overline{L}_{0}\left(z\right)=0$ and sufficiently large $\lambda\geq 1,$ we
have $\chi\left(\lambda\right):=-f^{{}^{\prime}}\left(z\right)+\lambda>0$.
Let $\left\\{\varphi_{j}\left(x\right)\right\\}_{ji\in\mathbb{N}_{0}}$ be an
orthonormal basis in $\mathbb{X}^{2}$ consisting of eigenfunctions of the
Wentzell Laplacian $\Delta_{W}$ (see Appendix, Theorem 5.3),
$\Delta_{W}\varphi_{j}=\Lambda_{j}\varphi_{j},\text{
}j\in\mathbb{N}_{0},\text{ }\varphi_{j}\in D\left(\Delta_{W}\right)\cap
C\left(\overline{\Omega}\right)$ (3.1)
such that
$0=\Lambda_{0}<\Lambda_{1}\leq\Lambda_{2}\leq...\leq\Lambda_{,j}\leq\Lambda_{j+1}\leq....$
We shall seek for eigenvectors
$W_{j}=\binom{w_{j}}{Tr_{D}\left(w_{j}\right)}\in\mathbb{X}^{2}$, of the form
$w_{j}\left(x\right)=\varphi_{j}\left(x\right)p_{j},$ $p_{j}\in\mathbb{R}$,
satisfying equation
$\mathbf{L}\left(Z\right)W_{j}=\zeta_{j}W_{j},\text{ }W_{j}\in
D\left(\mathbf{L}\left(Z\right)\right):=D\left(\Delta_{W}\right).$ (3.2)
Note that for $W_{j}\in
D\left(\mathbf{L}\left(Z\right)\right)\subset\mathbb{V}_{1},$ the trace of
$w_{j}$ makes sense as an element of $H^{1/2}\left(\Gamma\right)$.
Substituting such $w_{j}$ into (3.2), taking into account (3.1) and the fact
that
$\mathbf{L}\left(Z\right)W_{j}=-\nu\Delta_{W}W_{j}+\Pi_{\lambda}W_{j},\text{
}\Pi_{\lambda}W_{j}:=\binom{\chi\left(\lambda\right)w_{j}}{0},$
we obtain the equation
$\left(-\nu\Lambda_{j}I+\Pi_{\lambda}\right)p_{j}=\zeta_{j}p_{j},\text{
}\Pi_{\lambda}=\left(\begin{array}[]{cc}\chi\left(\lambda\right)&0\\\
0&0\end{array}\right).$ (3.3)
A nonzero $p_{j}$ exists if $\zeta=\zeta_{j}$ is a root of the equation
$\det\left(-\nu\Lambda_{j}I+\Pi_{\lambda}-\zeta I\right)=0,\text{ }\zeta>0.$
(3.4)
When $\nu=0,$ this equation has at least one root $\zeta>0$ since
$\chi=\chi\left(\lambda\right)>0$ (in fact, $\zeta=\chi\left(\lambda\right)$).
Therefore, there exists $\delta>0$ such that when $\nu\Lambda_{j}<\delta,$ the
equation (3.4) has a root $\zeta_{j}=\zeta_{j}\left(\nu\right)$ with
$\zeta_{j}>0$. Therefore, to any such root $\zeta_{j}$, we can assign a
nontrivial $p_{j},$ which is a solution of (3.3), and thus an eigenvector
$W_{j}=\binom{w_{j}}{Tr_{D}w_{j}},$ $w_{j}=\varphi_{j}p_{j}$. Let us now
compute how many $j$’s satisfy the inequality $\nu\Lambda_{j}<\delta$. The
asymptotic behavior of $\Lambda_{j}$ is $\Lambda_{j}\sim
C_{W}\left(\Omega,\Gamma\right)j^{1/\left(n-1\right)}$ as $j\rightarrow\infty$
(see, Appendix, Theorem 5.4). The last inequality certainly holds when
$1\leq j\leq
c_{1}\delta^{n-1}\left(C_{W}\nu\right)^{1-n}=c_{2}\left(\frac{1}{C_{W}\nu}\right)^{n-1},\text{
for }n\geq 2$
and
$1\leq j\leq
c_{1}\delta^{1/2}\left(C_{D}\nu\right)^{-1/2}=c_{2}\left(\frac{1}{C_{D}\nu}\right)^{1/2},\text{
for }n=1.$
The positive constants $c_{1},$ $c_{2}$ depend on $\lambda.$ In order to get
more explicit estimates for $c_{1},$ $c_{2}$, it is left to remark that
equation (3.4) may be rewritten in the form
$\det\left(-\nu\Lambda_{j}\lambda^{-1}I+\lambda^{-1}\Pi_{\lambda}-\zeta_{1}I\right)=0$
with $\zeta_{1}=\lambda^{-1}\zeta,$ and to observe that a solution of this
equation clearly exists if $\nu\Lambda_{j}\lambda^{-1}\leq\delta,$ for
sufficiently large $\lambda$ and small $\delta$. Employing the asymptotic
formula for $\Lambda_{j}$ once again, we find
$1\leq j\leq
c_{1}^{{}^{\prime}}\delta^{n-1}\lambda^{n-1}\left(C_{W}\nu\right)^{1-n}=c_{2}^{{}^{\prime}}\left(\frac{\lambda}{C_{W}\nu}\right)^{n-1},\text{
for }n\geq 2$
and
$1\leq j\leq
c_{1}^{{}^{\prime}}\delta^{1/2}\lambda^{1/2}\left(C_{D}\nu\right)^{-1/2}=c_{2}^{{}^{\prime}}\left(\frac{\lambda}{C_{D}\nu}\right)^{1/2},\text{
for }n=1.$
It follows that
$\dim E_{+}\left(Z\left(\lambda\right)\right)\geq
c_{2}^{{}^{\prime}}\left(\frac{\lambda}{C_{W}\nu}\right)^{n-1},\text{ for
}n\geq 2$
and
$\dim E_{+}\left(Z\left(\lambda\right)\right)\geq
c_{2}^{{}^{\prime}}\lambda^{1/2}\left(C_{D}\nu\right)^{-1/2},$
in one space dimension. The proof is complete.
## 4 Concluding remarks
In the textbook literature on theoretical geophysics, it was traditional to
use a Robin boundary condition with a nonlinear heat equation to describe
temperature variations at the upper surface of the ocean [28, 29]. But it was
recognized that this was not always the physically correct boundary condition
[40]. Among its applicability to a wide range of phenomena, including phase-
transitions in fluids, and so on [16, 42], the reaction-diffusion system
(1.1)-(1.4) has important applications in climatology and is essentially used
to determine large and rapid temperature changes in the ocean’s surface as a
response to changes into deep water formations [40]. In this paper, we provide
explicit bounds for the dimension of the attractor for this system and study
the effect of the dynamic term $b^{-1}\partial_{t}u$, representing change in
thermal energy in an infinitesimal layer near the surface. Unlike the previous
results, the dimension of the attractor is proportional to the surface area
$\left|\Gamma\right|,$ for large domains $\Omega$ and fixed parameters $\nu,$
$\lambda$ and $b$. Moreover, all the constants involved in our estimates are
given in an explicit form. We also observe that in the case without
$b^{-1}\partial_{t}u$ in (1.4), i.e., $b=+\infty,$ the dimension of the
attractor is much larger (and proportional to the volume $\left|\Omega\right|$
of $\Omega$) than the dimension of the global attractor for the same system
when $0<b\neq+\infty$. Thus, we observe that the addition of the dynamic term
$b^{-1}\partial_{t}u$, $b>0$ drastically changes the situation. This is a
remarkable fact that can have a profound effect onto the long-term dynamics of
other systems that are subject to dynamic boundary conditions of this form. We
will investigate these effects for other systems, such as the Bénard problem
for nonlinear heat conduction, in forthcoming papers. Finally, we note that it
is also possible to extend the results of this paper to the case when the
boundary $\Gamma$ consists of two disjoint open subsets $\Gamma_{1}$ and
$\Gamma_{2}$, each $\overline{\Gamma}_{i}\backprime\Gamma_{i}$ is a $S$-null
subset of $\Gamma$ and $\Gamma=\overline{\Gamma}_{1}\cup\overline{\Gamma}_{2}$
with $\Gamma_{1}\subsetneqq\Gamma$, such that $u$ satisfies a Dirichlet
boundary condition on $\Gamma_{1}$ and a dynamic boundary condition on
$\Gamma_{2}$. We will come back to this issue in a forthcoming article.
## 5 Appendix
In this section, we shall recall several important results concerning a
certain realization of $L=\nu\Delta$ with the Wentzell boundary condition
(1.3). We have the following.
###### Theorem 5.1
Let $\Omega$ be a bounded open set of $\mathbb{R}^{n}$ with Lipschitz boundary
$\Gamma$. Assume that $b>0$ and $0\leq q\in L^{\infty}\left(\Omega\right)$.
Define the operator $\Delta_{W}$ on $\mathbb{X}^{2},$ by
$\Delta_{W}\binom{u_{1}}{u_{2}}:=\binom{-\Delta
u_{1}+q\left(x\right)u_{1}}{b\partial_{\mathbf{n}}u_{1}},$ (5.1)
with
$D\left(\Delta_{W}\right):=\left\\{U=\binom{u_{1}}{u_{2}}\in\mathbb{V}_{1}:-\Delta
u_{1}\in L^{2}\left(\Omega\right),\text{ }\partial_{\mathbf{n}}u_{1}\in
L^{2}\left(\Gamma,\frac{dS}{b}\right)\right\\}.$ (5.2)
Then, $\left(\Delta_{W},D\left(\Delta_{W}\right)\right)$ is self-adjoint on
$\mathbb{X}^{2}.$ Moreover, the resolvent operator
$\left(I+\Delta_{W}\right)^{-1}\in\mathcal{L}\left(\mathbb{X}^{2}\right)$ is
compact.
We refer the reader to [4, 18, 19] for an extensive survey of recent results
concerning the ”Wentzell” Laplacian $\Delta_{W}$.
The eigenvalue problem associated with the operator $\Delta_{W}$ is given by
$\Delta_{W}\varphi=\Lambda\varphi;$ this leads to the following spectral
problem for the perturbed Laplacian
$-\Delta\varphi+q\left(x\right)\varphi=\Lambda\varphi\text{ in }\Omega,$ (5.3)
with a boundary condition that depends on the eigenvalue $\Lambda$ explicitly:
$b\partial_{\mathbf{n}}\varphi=\Lambda\varphi\text{ on }\Gamma.$ (5.4)
Such a function $\varphi$ will be called an eigenfunction associated with
$\Lambda$ and the set of all eigenvalues $\Lambda$ of (5.3)-(5.4) will be
denoted by $\Lambda_{W}.$ Let $\varphi_{j}$ and $\Lambda_{W,j}$, $j\in J$,
denote all the eigenfunctions and eigenvalues of (5.3)-(5.4). We have the
following (see, e.g., [2, 45]).
###### Theorem 5.2
Let $q\geq 0$ with $\int\limits_{\Omega}q\left(x\right)dx>0$. Then, there
exists a sequence of numbers
$0<\Lambda_{W,1}\leq\Lambda_{W,2}\leq...\leq\Lambda_{W,j}\leq\Lambda_{W,j+1}\leq...,$
(5.5)
converging to $+\infty$, with the following properties:
(a) The spectrum of $\Delta_{W}$ is given by
$\sigma\left(\Delta_{W}\right)=\left\\{\Lambda_{W,j}\right\\}_{j\in\mathbb{N}},$
and each number $\Lambda_{W,j},$ $j\in\mathbb{N},$ is an eigenvalue for
$\Delta_{W}$ of finite multiplicity.
(b) There exists a countable family of orthonormal eigenfunctions for
$\Delta_{W}$ which spans $\mathbb{X}^{2}$. More precisely, there exists a
collection of functions $\left\\{\varphi_{j}\right\\}_{j\in\mathbb{N}}$ with
the following properties:
$\displaystyle\varphi_{j}$ $\displaystyle\in D\left(\Delta_{W}\right)\text{
and }\Delta_{W}\varphi_{j}=\Lambda_{W,j}\varphi_{j},\text{ }j\in\mathbb{N},$
(5.6)
$\displaystyle\left\langle\varphi_{j},\varphi_{k}\right\rangle_{\mathbb{X}^{2}}$
$\displaystyle=\delta_{jk}\text{, }j,k\in\mathbb{N}\text{,}$
$\displaystyle\mathbb{X}^{2}$
$\displaystyle=\oplus\overline{lin.span\left\\{\varphi_{j}\right\\}_{j\in\mathbb{N}}}\text{
(orthogonal direct sum).}$
(c) If $\Gamma$ is Lipschitz, then every eigenfunction
$\varphi_{j}\in\mathbb{V}_{1}$, and in fact $\varphi_{j}\in
C(\overline{\Omega})\cap C^{\infty}(\Omega)$, for every $j$. If $\Gamma$ is of
class $C^{2}$, then every eigenfunction $\varphi_{j}\in\mathbb{V}_{1}\cap
C^{2}\left(\overline{\Omega}\right),$ for every $j.$
(d) The following min-max principle holds:
$\Lambda_{W,j}=\min_{\begin{subarray}{c}Y_{j}\subset\mathbb{V}_{1},\\\ \dim
Y_{j}=j\end{subarray}}\max_{0\neq\varphi\in
Y_{j}}R_{W}\left(\varphi,\varphi\right),\text{ }j\in\mathbb{N}\text{,}$ (5.7)
where the Rayleigh quotient $R_{W}$, for the perturbed Wentzell operators, is
given by
$R_{W}\left(\varphi,\varphi\right):=\frac{\left\|\nabla\varphi\right\|_{2}^{2}+\left\langle
q\left(x\right)\varphi,\varphi\right\rangle_{2}}{\left\|\varphi\right\|_{\mathbb{X}^{2}}^{2}},\text{
}0\neq\varphi\in\mathbb{V}_{1}.$ (5.8)
Concerning the case $q\equiv 0$, we have the following.
###### Theorem 5.3
Let $q\equiv 0.$ Then, there exists a sequence of numbers
$0=\Lambda_{W,0}<\Lambda_{W,1}\leq\Lambda_{W,2}\leq...\leq\Lambda_{W,j}\leq\Lambda_{W,j+1}\leq...,$
converging to $+\infty$, with the following properties:
(a) The spectrum of $\Delta_{W}$ is given by
$\sigma\left(\Delta_{W}\right)=\left\\{\Lambda_{W,j}\right\\}_{j\in\mathbb{N\cup}\left\\{0\right\\}},$
and each number $\Lambda_{W,j},$
$j\in\mathbb{N}_{0}=\mathbb{N\cup}\left\\{0\right\\},$ is an eigenvalue for
$\Delta_{W}$ of finite multiplicity. The eigenvalue $\Lambda_{W,0}$ is simple
and its associated eigenfunction is of constant sign.
(b) There exists a countable family of orthonormal eigenfunctions for
$\Delta_{W}$ which spans $\mathbb{X}^{2}$. More precisely, the same conclusion
(b) of Theorem 5.2 holds in this case as well. Finally, both conclusions (c)
and (d) in Theorem 5.2 hold in the case $q\equiv 0$ as well.
The asymptotic behavior of the eigenvalues $\Lambda_{W,j},$ as
$j\rightarrow\infty,$ was established in [13, 14]. We refer the reader to [18]
for more details about the Wentzell Laplacian and other generalizations. Let
$J=\mathbb{N}_{0}$ or $\mathbb{N}$, according to whether $q=0$ or $q>0$
respectively. Set
$C_{D}\left(\Omega\right):=\frac{\left(2\pi\right)^{2}}{\left(v_{n}\left|\Omega\right|\right)^{2/n}}\text{
and
}C_{S}\left(\Gamma\right)=\frac{2\pi}{\left(v_{n-1}\left|\Gamma\right|\right)^{1/\left(n-1\right)}}.$
Here $v_{n}$ denotes the volume of the unit ball in $\mathbb{R}^{n}$, and we
recall that $\left|\Omega\right|$ stands for the $n$-dimensional Euclidean
volume of $\Omega$, while $\left|\Gamma\right|$ stands for the usual
$\left(n-1\right)$-dimensional Lebesgue surface measure on $\Gamma$.
We summarize these results in the following.
###### Theorem 5.4
The eigenvalue sequence $\left\\{\Lambda_{W,j}\right\\}_{j\in J}$ of the
(un)perturbed Wentzell Laplacian $\Delta_{W}$ satisfies:
(i) For $n\geq 2$, we have
$\Lambda_{W,j}=C_{W}\left(\Omega,\Gamma\right)j^{1/\left(n-1\right)}+o\left(j^{1/\left(n-1\right)}\right),\text{
as }j\rightarrow+\infty,$ (5.9)
for some
$C_{W}\left(\Omega,\Gamma\right)\in\left\\{\begin{array}[]{cc}bC_{S}\left(\Gamma\right)\left[2^{-1/\left(n-1\right)},1\right],&\text{for
}n\geq 3,\\\
\left[\frac{C_{D}\left(\Omega\right)C_{S}\left(\Gamma\right)}{2\left(b^{-1}C_{D}\left(\Omega\right)+C_{S}\left(\Gamma\right)\right)},\min\left\\{C_{D}\left(\Omega\right),bC_{S}\left(\Gamma\right)\right\\}\right],&\text{for
}n=2.\end{array}\right.$ (5.10)
(ii) For $n=1,$ we have
$\Lambda_{W,j}=C_{D}\left(\Omega\right)j^{2}+o\left(j^{2}\right),\text{ as
}j\rightarrow+\infty.$ (5.11)
The following version of the Lieb–Thirring inequality is essential.
###### Proposition 5.5
Let $\omega_{j},$ $1\leq j\leq m,$ be a finite family of $\mathbb{V}_{1},$
which is orthonormal in $\mathbb{X}^{2}$. We have
$\sum_{i=1}^{m}\left\|\nabla\omega_{j}\right\|_{2}^{2}\geq
c_{1}C_{W}\left(\Omega,\Gamma\right)\left(m^{\frac{1}{n-1}+1}-m\right).$
(5.12)
The constant $c_{1}>0$ depends only on $n$ and the shape of $\Omega,$ and is
independent of the size of $\Omega,$ $\Gamma,$ of $m,$ and of the
$\omega_{j}$’s.
Proof. Let $B_{W}:=\Delta_{W}+C_{W}\left(\Omega,\Gamma\right)I$ and let
$D\left(B_{W}\right)=D\left(\Delta_{W}\right)$. By Theorems 5.1, 5.2, $B_{W}$
is a linear positive unbounded self-adjoint operator on $\mathbb{X}^{2},$ such
that $B_{W}^{-1}$ is compact. Thus, we can apply the abstract result of [46,
Chapter VI, Lemma 2.1] to deduce that
$\displaystyle\sum_{i=1}^{m}\left(\left\|\nabla\omega_{j}\right\|_{2}^{2}+C_{W}\left\|\omega_{j}\right\|_{\mathbb{X}^{2}}^{2}\right)$
$\displaystyle=\sum_{i=1}^{m}\left\langle
B_{W}\omega_{j},\omega_{j}\right\rangle_{\mathbb{X}^{2}}$ (5.13)
$\displaystyle\geq\Lambda_{W,1}\left(B_{W}\right)+\Lambda_{W,2}\left(B_{W}\right)+...+\Lambda_{W,m}\left(B_{W}\right)$
$\displaystyle\geq
C_{W}\left(1^{1/\left(n-1\right)}+2^{1/\left(n-1\right)}+...+m^{1/\left(n-1\right)}\right)$
$\displaystyle\geq c_{0}C_{W}m^{\frac{1}{n-1}+1},$
since, by (5.9)-(5.11), $\Lambda_{W,j}\left(B_{W}\right)\geq
C_{W}\left(\Omega,\Gamma\right)j^{1/\left(n-1\right)},$ for all $j,$ and some
positive constant $c_{0}$ (indeed, we have
$\Lambda_{W,j}\left(B_{W}\right)=\Lambda_{W,j}\left(\Delta_{W}\right)+C_{W}$).
Thus, the proof of (5.12) follows immediately from (5.13).
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|
arxiv-papers
| 2011-03-16T14:24:05 |
2024-09-04T02:49:17.701310
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ciprian G. Gal",
"submitter": "Ciprian Gal",
"url": "https://arxiv.org/abs/1103.3179"
}
|
1103.3211
|
# Effects of Neutral Hydrogen on Cosmic Ray Precursors in Supernova Remnant
Shock Waves
John C. Raymond,11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden St., Cambridge, MA 02138, USA; jraymond@cfa.harvard.edu J.
Vink,22affiliation: Astronomical Institute, Utrecht University, P.O. Box
80000, 3508TA Utrecht, The Netherlands E.A. Helder,22affiliation: Astronomical
Institute, Utrecht University, P.O. Box 80000, 3508TA Utrecht, The Netherlands
& A. de Laat22affiliation: Astronomical Institute, Utrecht University, P.O.
Box 80000, 3508TA Utrecht, The Netherlands
###### Abstract
Many fast supernova remnant shocks show spectra dominated by Balmer lines. The
H$\alpha$ profiles have a narrow component explained by direct excitations and
a thermally Doppler broadened component due to atoms that undergo charge
exchange in the post-shock region. However, the standard model does not take
into account the cosmic-ray shock precursor, which compresses and accelerates
plasma ahead of the shock. In strong precursors with sufficiently high
densities, the processes of charge exchange, excitation and ionization will
affect the widths of both narrow and broad line components. Moreover, the
difference in velocity between the neutrals and the precursor plasma gives
rise to frictional heating due to charge exchange and ionization in the
precursor. In extreme cases, all neutrals can be ionized by the precursor.
In this paper we compute the ion and electron heating for a wide range of
shock parameters, along with the velocity distribution of the neutrals that
reach the shock. Our calculations predict very large narrow component widths
for some shocks with efficient acceleration, along with changes in the broad-
to-narrow intensity ratio used as a diagnostic for the electron-ion
temperature ratio. Balmer lines may therefore provide a unique diagnostic of
precursor properties. We show that heating by neutrals in the precursor can
account for the observed H$\alpha$ narrow component widths, and that the
acceleration efficiency is modest in most Balmer line shocks observed thus
far.
shock waves — acceleration of particles — ISM: supernova remnants
## 1 Introduction
Cosmic rays are widely believed to originate in supernova remnant (SNR) shock
waves, because the cosmic-ray energy spectrum agrees with model predictions,
because power-law distributions of energetic electrons are seen in SNRs, and
because the power required to maintain the cosmic ray population could be
supplied by about 10% of kinetic energy of Galactic supernovae. The standard
theory for the process is Diffusive Shock Acceleration (DSA), which is a first
order Fermi process requiring that particles scatter between a gasdynamic
subshock and plasma turbulence in a shock precursor. Evidence for non-linear
DSA comes from curved synchrotron spectra (Reynolds & Ellison, 1992; Vink et
al., 2006; Allen et al., 2008), evidence for high compression factors (Warren
et al., 2005; Cassam-Chenaï et al., 2008) and evidence for lower than expected
downstream temperatures (Hughes et al., 2000; Helder et al., 2009). However,
all this evidence is based on observation of downstream properties. The
effects of precursor physics on the H$\alpha$ emission described here offer a
direct probe of the properties of the precursor.
A crucial parameter for these models is the diffusion coefficient $\kappa$, as
it determines the precursor scale length, which is typically $\kappa$ divided
by the shock speed $V_{S}$. Gas is compressed in the precursor and accelerated
to a fraction of the shock speed, and this compression is related to $V_{S}$,
to the efficiency of particle acceleration and to the escape of energetic
particles from the region (Bykov, 2005; Vink et al., 2010). Neutrals can
impede the acceleration process by damping the turbulence needed to scatter
particles back to the shock. However, Drury et al. (1996) found that the
acceleration efficiency can be high as long as the density and neutral
fraction are not too large, though the maximum particle energy is reduced.
One set of diagnostics for the physics of collisionless shocks is based on the
emission from particles in the narrow ionization zone just behind a
nonradiative shock (Raymond, 1991; Heng, 2010). In particular, H$\alpha$
photons from a nonradiative shock in partly neutral gas originate very close
to the shock, and Coulomb collisions do not have time to erase such signatures
as unequal electron and ion temperatures or non-Maxwellian velocity
distributions (Laming et al., 1996; Ghavamian et al., 2001; Raymond et al.,
2008, 2010). In the optical these shocks are seen as pure Balmer line
filaments whose profiles show a narrow component characteristic of the pre-
shock kinetic temperature and a broad component closely related to the post-
shock proton temperature (Chevalier & Raymond, 1978; Heng, 2010; van Adelsberg
et al., 2008). The intensity ratio of the broad and narrow components is
determined by the electron to ion temperature ratio at the shock (Ghavamian et
al., 2001; van Adelsberg et al., 2008; Helder et al., 2010).
The Balmer line profiles also contain signatures of shock precursors. In
general, the narrow component line widths are 40 to 50 $\rm km~{}s^{-1}$,
indicating temperatures around 40,000 K. If that were the ambient ISM
temperature, there would be no neutrals to create the Balmer line filament, so
the width is interpreted as an indication of heating in a narrow precursor too
thin to completely ionize the hydrogen (Smith et al., 1994; Hester et al.,
1994; Lee et al., 2007; Sollerman et al., 2003). Faint emission ahead of the
sharp filament is interpreted as emission from the compressed and heated
precursor gas (Hester et al., 1994; Lee et al., 2007, 2010).
This paper considers the role of neutrals in heating the precursor plasma and
computes the properties of precursor H$\alpha$ emission. While cosmic ray
pressure in the precursor can compress, heat and accelerate ions and electrons
by means of plasma turbulence and magnetic fields, the neutrals only interact
with the precursor by means of collisions with protons and electrons. If the
density is very high, neutrals and protons are tightly coupled by charge
transfer. In that case, the neutrals are compressed along with the protons and
adiabatically heated. They also share in any other heating of the protons,
such as dissipation of Alfvén waves generated by cosmic-ray streaming. On the
other hand, if the density is very low, neutrals pass through the precursor
and the shock without interacting at all, preserving their pre-shock velocity
distribution.
The intermediate case is more complex. A shock that efficiently accelerates
cosmic rays is strongly modified, and gas reaches a significant fraction of
the shock speed in the precursor (Vladimirov et al., 2008; Wagner et al.,
2009). If neutrals and ions are fairly well coupled, they can be described as
fluids whose relative speed gives a frictional heating similar to that in
C-shocks (Draine & McKee, 1993). If a neutral encounters this high speed
compressed plasma without having been brought gradually up to speed by many
previous charge transfers, it can be ionized and become a pickup ion (Raymond
et al., 2008; Ohira & Takahara, 2010) like those observed in the solar wind
(Moebius et al., 1985). It can then have an energy on the order of 1 keV,
which it can share with the other protons. Electron heating is more uncertain,
but it can occur by means of Lower Hybrid waves (Cairns & Zank, 2002). If the
electrons are heated they can excite and ionize H atoms, changing the
H$\alpha$ profile and the broad-to-narrow line ratio used as an electron
temperature diagnostic (Ghavamian et al., 2001).
In this paper we compute the proton, neutral and electron temperatures in the
precursors for a variety of parameters, along with the ionization and
excitation of H atoms. We consider the effects of these processes on Balmer
line diagnostics currently in use. Ohira & Takahara (2010) considered the
effects of neutrals on the velocity structure of the precursor, the
compression ratio and the acceleration process. They found that the pickup
ions can reduce the compression by the subshock and enhance proton injection
into the acceleration process. Morlino et al. (2010) self-consistently
computed the particle acceleration and heating due to neutrals, but within the
fluid approximation for both neutrals and ions. In this paper we emphasize the
effects on the H$\alpha$ line profile.
## 2 Model Calculations
We parameterize the precursor structure in a relatively simple manner. We
assume that the precursor accelerates and compresses the interstellar gas over
a length scale $\kappa$/$V_{S}$, where $\kappa$ is the diffusion coefficient
for cosmic rays near the cutoff. Effective cosmic-ray acceleration requires
$\kappa$ on the order of $10^{24}~{}\rm cm^{2}~{}s^{-1}$, and estimates based
on the scales of H$\alpha$ precursors are 2 to 4$\times 10^{24}~{}\rm
cm^{2}~{}s^{-1}$ (Lee et al., 2007, 2010). We do not consider the second order
effects of momentum and energy deposition by the neutrals on the precursor
length scale.
We assume an exponential form, so that the compression is given by
$\chi~{}=~{}1+(\chi_{1}-1)~{}~{}e^{(xV_{S}/\kappa)}$ (1)
where x is negative ahead of the shock and $\chi_{1}$ is the compression ratio
just upstream of the subshock. It is related to the fractional pressure of
cosmic rays behind the shock, w=$P_{CR}/(P_{G}+P_{CR})$, by equation 9 of Vink
et al. (2010). We simplify this equation with the assumption that for w$<$0.8
the compression in the gas subshock equals 4, so that
$\chi_{1}~{}=~{}(1-w/4)/(1-w)$ (2)
Mass conservation implies velocities $V=V_{S}/\chi$ in the frame of the shock.
To compute the proton and electron temperatures we include adiabatic
compression, Coulomb energy transfer between protons and electrons, energy
losses due to ionization and excitation of Hydrogen, and heating terms.
We assume that any neutral that interacts with the plasma at position $x_{i}$
joins the proton flow at that position. If the interaction was charge
transfer, a new neutral is formed with the bulk speed and thermal speed of the
protons at $x_{i}$. Thus the neutrals arriving at $x_{i}$ are those that last
went through charge transfer at all upstream positions $x_{j}$, and they have
the speeds, $v_{j}$, of the plasma at $x_{j}$. Each ionization of a neutral
from $x_{j}$ at $x_{i}$ deposits energy $0.5m_{p}(v_{j}-v_{i})^{2}$. We assume
that the energy is quickly thermalized among the protons, unlike Ohira &
Takahara (2010), who assumed a pickup ion velocity distribution. The
thermalization time scale is very uncertain, because full kinetic calculations
have not been carried out. However, in the highly turbulent precursor there
are many wave modes besides Alfvén waves that can thermalize the protons, in
particular those associated with bump-on-tail, mirror and firehose-like
instabilities (Winske et al., 1985; Gary, 1978; Sagdeev et al., 1986). We also
ignore heating due to Alfvén wave damping or shocks excited by the cosmic-ray
pressure gradient in order to isolate the effect of the neutrals. Therefore,
we compute a lower limit to the heating.
For electrons, we follow Cairns & Zank (2002), who found that ionization of
fast neutrals forms a ring beam, in which all the particles gyrate around the
magnetic field with the same speed but different phases. The ring beam is
unstable, and provided that the beam velocity (in this case the relative
velocity of bins i and j) is less than 5 times the Alfvén speed, it transfers
a significant amount of heat to electrons via Lower Hybrid waves. We follow
Cairns & Zank (2002) in taking this fraction to be 10%. Again, to isolate the
effects of neutrals we ignore any heating of electrons by Lower Hybrid waves
generated by cosmic-ray streaming (Ghavamian et al., 2007; Rakowski et al.,
2008).
Charge transfer rates are taken from (Schultz et al., 2008) using the
quadrature sum of the thermal speed and the ion-neutral relative speed.
Ionization and excitation rates are computed from cross sections from Janev &
Smith (1993) by integrating over the electron velocity distribution including
the relative electron-neutral flow speed.
## 3 Results
Figure 1 shows a set of models for a shock speed of 2000 $\rm km~{}s^{-1}$
with $\kappa=2.0\times 10^{24}~{}\rm cm^{2}~{}s^{-1}$, a pre-shock density of
0.2 $\rm cm^{-3}$ and a neutral fraction of 0.2. The four models have ratios
of cosmic-ray partial pressure to total pressure behind the subshock, w, of
0.1, 0.3, 0.5 and 0.7. The compression ratios just ahead of the subshock are
1.0833, 1.3214, 1.75 and 2.75. In the high $V_{S}$, high efficiency models the
neutrals are not compressed to this level, because of collisional ionization
and because some pass through without charge transfer. The protons and
electrons are strongly heated in the more efficient models, but the electrons
are much cooler than the protons. The drop in heating just before the subshock
in the 70% efficient model results from the reduced number of neutrals.
Figure 2 shows the velocity distributions of the neutrals perpendicular to the
shock just before the subshock. Note that the w=50% model shows a narrow
component due to neutrals that last experience charge transfer far upstream,
along with a broader component of particles that undergo charge transfer close
to the subshock.
Figure 3 shows a grid of models of the neutral velocity distribution at the
shock for a range of shock speeds and cosmic-ray partial pressures. The panels
show the FWHM measured directly from the computed velocity distribution and
the kurtosis, which would be 3.0 for a Gaussian distribution. Kurtosis is a
problematic statistical moment for real data because it is sensitive to noise
far from the line center and the choice of background level. However, for the
theoretical profiles computed here it highlights cases in which some neutrals
undergo charge transfer close to the subshock and others do not. We also show
the fraction of incident neutrals that survive up to the subshock and the
average number of excitations to the n=3 level per incident H atom. Not all of
these excitations will result in H$\alpha$ photons because some Ly$\beta$
photons escape, but this is a convenient comparison to the 0.2 to 0.25
H$\alpha$ photons per H atom produced in the post-shock region.
## 4 Discussion
The interaction of neutral hydrogen with the ionized plasma in cosmic-ray
precursors described above offers an important tool to measure the properties
of cosmic-ray precursors. The outcome of DSA is very much influenced by
physical processes in the precursor, which are not well determined. For
example, non-adiabatic heating and magnetic field amplification due the
presence of cosmic rays tend to decrease the overall compression factor from
$\chi_{12}>>20$ (e.g. Berezhko & Ellison, 1999; Blasi et al., 2005) to
$7\lesssim\chi_{12}\lesssim 15$ (Vladimirov et al., 2008; Caprioli et al.,
2008). In addition, if the Alfvén waves in the precursor have some drift
velocity this will affect the cosmic-ray pressure profile (Zirakashvili &
Ptuskin, 2008), which limits the escape of energy from the shock region. A
lower energy escape automatically implies a lower downstream cosmic-ray
pressure (Vink et al., 2010).
As shown here, neutrals will influence the physics of the precursor. Morlino
et al. (2010) treated the neutrals as a fluid coupled to the ions by charge
transfer for a unified treatment of the heating and dynamics of the precursor,
but the fluid approximation is only appropriate if neutrals and ions are
coupled fairly well. They obtained a FWHM of 46 $\rm km~{}s^{-1}$ for the
H$\alpha$ line in a 2000 $\rm km~{}s^{-1}$ shock with modest efficiency, a
pre-shock density of 1 $\rm cm^{-3}$ and 50% neutral fraction. For similar
parameters we find a non-Maxwellian profile with smaller FWHM and broader
wings.
Neutrals can also damp plasma waves, which limits the efficiency of cosmic-ray
acceleration. This damping is caused by the central processes described above:
charge exchange and ionization. Drury et al. (1996) found that the maximum
particle energy, and therefore the maximum acceleration efficiency, is
considerably higher than suggested by Draine & McKee (1993). In addition, the
heating due to neutrals penetrating the precursor is a form of non-adiabatic
heating. Energy dissipated in the precursor limits the amount of free energy
available for shock acceleration. If the neutrals ionized in the precursor
behave as pickup ions rather than thermalizing with the protons, the injection
efficiency and particle spectrum will be affected (Ohira & Takahara, 2010). In
any case, the heating of electrons in the precursor is poorly know, and that
will stongly affect the ionization and excitation of H atoms, which in turn
will affect the intensity ratio of the broad and narrow components as well as
the narrow component line width.
The physics of neutral-ion coupling means these processes are not only
sensitive to cosmic-ray pressure and the structure of the precursor, but also
to the pre-shock density and neutral fraction. For example, the protons and
neutrals in the Cygnus Loop nonradiative shocks (Salvesen et al., 2009) are
tightly coupled, and they behave nearly adiabatically, while the pre-shock
density in SN1006 is so low (Acero et al., 2007) that neutrals pass straight
through it. This may explain the narrow line width seen by Sollerman et al.
(2003) in SN 1006 in comparison to broader narrow lines observed for other
young SNRs. The number of charge transfer events for an average neutral in the
precursor can be estimated roughly as
$N_{\rm CT}=nL\sigma\chi_{1}/V_{S},$ (3)
where L is the precursor length scale, and the charge transfer cross section,
$\sigma$, declines slowly with velocity below about 2000 km s-1, then very
rapidly.
Comparing our results to observations, it is obvious that the observed narrow-
line H$\alpha$ widths are in general smaller than predicted by our
calculations for efficient shock acceleration (i.e., $w>0.5$). This may
indicate that none of the shocks investigated so far accelerate particles
efficiently. However, more work is needed before such a conclusion can be
drawn, as the line width depends also on pre-shock density and shock velocity.
For very high shock velocities combined with low densities the neutrals hardly
interact in the precursor, leading to narrow line widths. Such may be the case
for the northeastern region of RCW 86, for which Helder et al. (2009) reported
a high cosmic-ray acceleration efficiency ($w\gtrsim 0.5$, see also Vink et
al., 2010). For this region the pre-shock density may be as low as $n\lesssim
0.1$ (Vink et al., 2006), which, combined with the high velocity
($V_{s}\gtrsim 3000$ km/s), gives few interactions in the precursor and widths
$\lesssim 100$ km/s. Note that this is smaller than could be measured given
the moderate spectral resolution of the measurement.
Perhaps the most striking result from these calculations is that for efficient
shocks near 1000 $\rm km~{}s^{-1}$ a substantial number of H$\alpha$ photons
will be produced in the precursor. These will usually be included in the
narrow component, potentially affecting the electron temperature estimate
based on the broad-to-narrow intensity ratio (Ghavamian et al., 2001). Narrow
component emission from the precursor could explain the broad-to-narrow
intensity ratios that cannot be fit by models of post-shock emission (van
Adelsberg et al., 2008; Rakowski et al., 2009). In extreme cases, emission
from the precursor might also contribute to the broad component, possibly
accounting for the non-Maxwellian profile seen in Tycho’s SNR (Raymond et al.,
2010) and generally leading to an underestimate of the shock speed. Both of
these conclusions depend on the diffusion coefficient and the electron
heating, however.
Other H$\alpha$ line measurements show that the narrow lines are broader than
one might expect for temperatures of typical HII regions, but smaller than 50
km/s (Sollerman et al., 2003). (Not all of the narrow line emission comes from
the precursor, but the narrow line emission downstream is determined by the
velocity distribution in the precursor.) Another effect of charge exchange in
the precursor is that neutrals enter the downstream shock region with a
velocity offset with respect to the local interstellar medium, as seen in
Tycho’s supernova remnant (Lee et al., 2007). For shocks observed face on this
should produce a narrow line offset, which for the combined front and back
side of the remnant should lead to two narrow lines. The spectra of several
LMC remnants (Smith et al., 1994) do not show such an effect. For one of the
remnants in this set, SNR 0509-67.5, the cosmic-ray acceleration efficiency
was estimated to be $w\approx 0.2$ (Helder et al., 2010).
We note that several improvements should be made to the calculations presented
here. Additional heating due to wave dissipation can heat the protons,
resulting in larger narrow component line widths, or it can heat electrons,
increasing the H$\alpha$ narrow component intensity and reducing the number of
neutrals that reach the shock, especially if the electron velocity
distribution is non-Maxwellian (Laming & Lepri, 2007). In addition, ionization
and excitation by proton and helium ion impact are important at high relative
velocities (Laming et al., 1996), and at high shock speeds the velocity
distributions of particles are anisotropic (Heng & McCray, 2007; Heng et al.,
2007; van Adelsberg et al., 2008). Amplification of the magnetic field may
also be important, and radiative transfer calculations in the Ly$\beta$ line
must be done to compute the H$\alpha$ emission. We plan to address these
issues in future work.
This work was carried out while JCR was visiting the Astronomical Institute
Utrecht as Minnaert Professor. It was supported by NASA grant GO-11184.01-A-R
to the Smithsonian Astrophysical Observatory. JV and EH are supported by the
VIDI grant awarded to JV by the Netherlands Science Foundation (NWO).
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Figure 1: Plots of a) proton temperature, b) electron temperature, c) neutral
density and d) heating rate for for models having $\kappa=2.0\times
10^{24}~{}\rm cm^{2}~{}s^{-1}$, pre-shock density = 0.2 $\rm cm^{-3}$ and
neutral fraction 0.2. The models assume a post-shock cosmic-ray pressure of
10%, 30%, 50% and 70% of the total pressure, with the 10% curves at the bottom
and the 70% curves at the top in all four plots. Figure 2: Velocity
distributions perpendicular to the flow direction at the gasdynamic subshock
for the four models shown in Figure 1. Note the narrow central component and
broader wings in the model with 50% cosmic-ray pressure and the large width
predicted by the 70% cosmic-ray pressure model. Figure 3: Model grid for
$\kappa=2.0\times 10^{24}$, a pre-shock density of 0.2 $\rm cm^{-3}$ and
neutral fraction of 0.2. Models are shown for $P_{CR}/(P_{CR}+P_{G})$ = 0.1 to
0.8 at the shock front. The panels show the FWHM and kurtosis of neutrals that
reach the subshock, the fraction of neutrals that reach the subshock and the
number of excitations to n=3 in the precursor per incident neutral hydrogen
atom. As in Figure 1, the 80% models are the extreme cases.
|
arxiv-papers
| 2011-03-16T15:59:29 |
2024-09-04T02:49:17.709382
|
{
"license": "Public Domain",
"authors": "John C. Raymond, J. Vink, E.A. Helder and A. de Laat",
"submitter": "John C. Raymond",
"url": "https://arxiv.org/abs/1103.3211"
}
|
1103.3338
|
#
Possible potentials responsible for stable circular relativistic orbits
Prashant Kumar†, Kaushik Bhattacharya‡
Department of Physics, Indian Institute of Technology, Kanpur
Kanpur 208016, India email: †kprash@iitk.ac.in, ‡kaushikb@iitk.ac.in
###### Abstract
Bertrand’s theorem in classical mechanics of the central force fields attracts
us because of its predictive power. It categorically proves that there can
only be two types of forces which can produce stable, circular orbits. In the
present article an attempt has been made to generalize Bertrand’s theorem to
the central force problem of relativistic systems. The stability criterion for
potentials which can produce stable, circular orbits in the relativistic
central force problem has been deduced and a general solution of it is
presented in the article. It is seen that the inverse square law passes the
relativistic test but the kind of force required for simple harmonic motion
does not. Special relativistic effects do not allow stable, circular orbits in
presence of a force which is proportional to the negative of the displacement
of the particle from the potential center.
## 1 Introduction
The central force problem in non-relativistic classical mechanics is one of
the most useful topics in physics. Closely linked with the central force
problem is the Keplerian orbit theory which is a cornerstone for understanding
planetary motions in the solar system or motion of electrons near the nucleus.
In classical mechanics there is an important theorem called the Bertrand’s
theorem which proposes that there can only be two types of central potentials,
the Coulomb type and the simple harmonic type, which can produce stable,
circular orbits for particles moving around the potential source. A good
presentation of the Bertrand’s theorem can be found in Ref. [1]. The present
article tries to generalize the results of Bertrand’s theorem when the
orbiting particle can have relativistic velocities.
In this article we first set up the relativistic orbit equation for a particle
in a central potential presumed to be dependent on the radial coordinate only.
The relativistic central force orbits were previously studied in Refs. [2, 3,
4, 5]. A brief description of the central force problem in a relativistic
setting in a Coulomb potential was presented in the book on classical theory
of fields by Landau and Lifshitz [6]. Before one starts the main analysis
about the stability of orbits of relativistic particles in a central force
potential it is better to specify the assumptions one makes in arriving at
definite results. In the present article we use the same assumptions and the
approximations as utilized by Boyer in Ref. [2] and Landau in Ref. [6]. In the
specific references cited above, none of them present a Lorentz covariant
treatment of the relativistic central force problem. The main reason being
that all of them assumes a central potential $V(r)$ where $r=|{\bf r}|$ is the
distance between the source and the orbiting particle. The form of the
potential only depends on the position coordinates of the orbiting particle.
The form of $V(r)$ is not Lorentz covariant. In such cases the results of the
whole analysis is valid in a particular frame where the origin of the
coordinate system coincides with the potential center.
The references cited above assumes the particle which produces the potential
$V(r)$ to be static in the specific coordinate system utilized by the
observer. If the source of the potential does not have any velocity then the
retarded nature of the interactions, owing to the finite velocity of light,
does not complicate the calculation of the orbit of the relativistic particle.
A specific example will make the point clear. In classical electrodynamics if
the source of the Coulomb potential $V(r)$ moves with a velocity ${\bf v}_{s}$
and the orbiting particle has a velocity ${\bf v}$ then the potential $V(r)$
gets a relativistic correction. The magnitude of the lowest order relativistic
correction to the Coulomb potential was calculated by Darwin in 1920 and it
looks like
$\frac{V(r)}{2c^{2}}\left[{\bf v}_{s}\cdot{\bf v}+\frac{({\bf v}_{s}\cdot{\bf
r})({\bf v}\cdot{\bf r})}{r^{2}}\right]\,.$
For a better understanding of the Darwin correction one can look at Ref. [7].
In our case ${\bf v}_{s}=0$ and consequently there will be no relativistic
modification of $V(r)$. More over we do not consider any general relativistic
effects due to $V(r)$ into account. We briefly comment on the general
relativistic generalization of the central force problem in section 4. In the
present article the background space-time is assumed to be flat.
In the article it will be shown that the stability condition of the perturbed
orbits around a stable circular orbit gives rise to a non-linear differential
equation for the central potential. The Newtonian or the Coulomb potential
satisfies the resulting differential equation with some restrictions on the
possible value of the angular momentum of the orbiting particle. Except the
Newtonian potential solution we present a more general solution of the
differential equation for the potential which can give rise to stable,
circular orbit for relativistic particles. This solution gives rise to a force
which is not common in physics except its Newtonian inverse square law limit.
The equation of the orbit of a relativistic particle in such a non-trivial
force shows that the orbit will precess and the precession angle can be
calculated.
Unlike the non-relativistic case, in the relativistic case there exist no
radial effective potential minimizing which we can obtain the radius of a
circular orbit. In the relativistic case a first order perturbation from a
circular orbit is enough to determine the stability criterion of the orbit. In
the non-relativistic case one uses higher order perturbations from a circular
orbit to specify the form of the potential. In the relativistic case the
general solution of the form of the potential from first order perturbation
from circular orbit is such that all higher order corrections becomes
irrelevant. As a consequence of this fact the general form of the potential
which can produce stable, circular orbits for relativistic particles contains
more parameters than the corresponding expressions of non-relativistic
potentials.
The material in the article is presented in the following manner. The second
section sets the conventions and derives the orbit equation of a relativistic
particle in a central orbit. Section 3 generalizes the Bertrand’s theorem for
the relativistic case. In this section the stability condition for the
circular orbits will be interpreted as a non-linear differential equation for
the potential. The solutions of the non-linear stability equation will also be
derived in section 3. In section 4 the connection of the present work with
some related works which were existing in the literature are discussed. This
section gives a wider view for the readers who really want to understand the
stability of orbits in special relativity and general relativity. The last
section 5 summarizes the important points presented in the article.
## 2 The orbit equation
In this section we derive the orbit equation of the relativistic particle in
presence of a potential $V(r)$ which is purely a function of the radial
coordinate. The Lagrangian of a relativistic particle of mass $m$ in presence
of a continuous radial potential $V(r)$ is
$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle-
mc^{2}\sqrt{1-{v^{2}}/{c^{2}}}-V(r)\,,$ (1)
where the velocity of the particle ${\bf v}$ in plane polar coordinates is
given as
${\bf v}=\dot{r}\hat{e}_{r}+r\dot{\theta}\hat{e}_{\theta}\,,$
where $\hat{e}_{r}$ , $\hat{e}_{\theta}$ are the mutually orthogonal unit
vectors along the radial and the angular directions. The form of the
Lagrangian in Eq. (1) immediately shows that the angular momentum
$\displaystyle
L=\frac{\partial{\mathcal{L}}}{\partial\dot{\theta}}=mr^{2}\gamma\dot{\theta}\,,$
(2)
is a constant, where
$\gamma=\frac{1}{\sqrt{1-{v^{2}}/{c^{2}}}}\,.$
The total energy $E$ of the particle in presence of the potential $V(r)$ is
$\displaystyle E=mc^{2}\gamma+V(r)\,.$ (3)
Although from the definition of $\gamma$ it looks like that it is a function
of $r$, $\dot{r}$ and $\dot{\theta}$ but it can be shown that in a central
force field $\gamma$ is only a function of the radial coordinate $r$. The
reason for such behavior of $\gamma$ can be understood from the following
reason. As energy and angular momentum are constant functions of $r$,
$\dot{r}$ and $\dot{\theta}$ we can use the conservation conditions of $E$ and
$L$ to re-express $\dot{r}$ and $\dot{\theta}$ as functions of $r$, $E$ and
$L$. As $E$ and $L$ are constants so in a central force field $\dot{r}$ and
$\dot{\theta}$ are functions of $r$ alone. Consequently $\gamma$ is only a
function of $r$. In special relativity the energy of the particle in a central
potential can also be written as
$\displaystyle E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}+V(r)\,,$ (4)
where $p=|{\bf p}|$,
$\displaystyle{\bf p}=m\gamma{\bf v}$ $\displaystyle=$ $\displaystyle
m\gamma(\dot{r}\hat{e}_{r}+r\dot{\theta}\hat{e}_{\theta})$ (5)
$\displaystyle=$ $\displaystyle
p_{r}\hat{e}_{r}+p_{\theta}\hat{e}_{\theta}\,,$
and
$p_{r}=m\gamma\dot{r}\,,\,\,\,\,\,\,p_{\theta}=m\gamma
r\dot{\theta}=\frac{L}{r}\,.$
As because $({p_{r}}/{p_{\theta}})=({\dot{r}}/{r\dot{\theta}})$, we have
$p_{r}=\frac{L}{r^{2}}\frac{dr}{d\theta}\,.$
With the above information on the various momentum components we can now
rewrite Eq. (4) as
$\displaystyle(E-V)^{2}=\left(\frac{L}{r^{2}}\frac{dr}{d\theta}\right)^{2}c^{2}+\frac{L^{2}c^{2}}{r^{2}}+m^{2}c^{4}\,.$
(6)
Instead of $r$ we use the variable $u=\frac{1}{r}$ in terms of which Eq. (6)
becomes
$(E-V)^{2}=L^{2}c^{2}\left(\frac{du}{d\theta}\right)^{2}+u^{2}L^{2}c^{2}+m^{2}c^{4}\,.$
If we differentiate the last equation with respect to $\theta$ and then divide
the resulting equation by $du/d\theta$ we obtain the desired equation of the
orbit of a particle of mass $m$ possessing momentum ${\bf p}$ moving in the
presence of a general central potential $V(r)$ as
$\displaystyle\frac{d^{2}u}{d\theta^{2}}+u=\frac{(V-E)}{L^{2}c^{2}}\frac{dV}{du}\,.$
(7)
Using Eq. (3) we can rewrite the above equation in the form
$\displaystyle\frac{d^{2}u}{d\theta^{2}}+u=-\frac{m\gamma}{L^{2}}\frac{dV}{du}\,.$
(8)
Writing $L=\gamma\ell$, where $\ell=mr^{2}\dot{\theta}$ is the non-
relativistic angular momentum, the above equation in the non-relativistic
limit ($\gamma\to 1$) transforms exactly to the form we get in a conventional
non-relativistic treatment of the problem as given in Ref. [1].
## 3 Circular, Stable closed orbits
Lets define
$\displaystyle J(u)\equiv\frac{(V-E)}{L^{2}c^{2}}\frac{dV}{du}\,.$ (9)
Suppose Eq. (7) admits a circular orbit of radius $r_{0}=1/u_{0}$. For small
perturbations around this circular orbit we can Taylor expand $J(u)$ around
$u_{0}$. Keeping up to first order terms in the perturbation of $u$ we get
$\displaystyle J(u)=J(u_{0})+(u-u_{0})\left(\frac{dJ}{du}\right)_{u_{0}}\,.$
(10)
Noting that $J(u_{0})=u_{0}$ for the circular orbit, we can now write Eq. (7)
as
$\displaystyle\frac{d^{2}u}{d\theta^{2}}+(u-u_{0})=(u-u_{0})\left(\frac{dJ}{du}\right)_{u_{0}}\,.$
If we define $x\equiv u-u_{0}$ then the above equation can be written as
$\displaystyle\frac{d^{2}x}{d\theta^{2}}+\zeta^{2}x=0\,,$ (11)
where $\zeta^{2}$ is defined as
$\displaystyle\zeta^{2}\equiv 1-\left(\frac{dJ}{du}\right)_{u_{0}}\,.$ (12)
From Eq. (11) it is clear that if the orbit of the relativistic particle in a
general central potential has to be stable then $\zeta^{2}>0$ and if the orbit
has to be closed then $\zeta$ must be a rational number.
### 3.1 A differential equation for the potential $V(r)$ producing stable and
closed circular orbits
The rational number $\zeta$ as predicted, in Eq. (12), from the stability
criterion of closed circular orbits in the central force problem is an
interesting input in the theory. The interesting property about this rational
number is that it is a constant and so it does not depend on the details of
the orbit which one tries to perturb. The reason for the constancy of $\zeta$
is the following. For any circular orbit with radius $r_{0}$ a specific
$\zeta$ specifies the number of undulations of the perturbed orbit. If $\zeta$
is a rational number then the number of undulations of the perturbed orbit
will be such that they form a closed geometrical structure. Now suppose one
takes another circular orbit of radius $r_{0}+\delta r$ where $\delta r\ll
r_{0}$. If $\zeta$ has a different value on this orbit then the number of
undulations due to a perturbation will be different. In the limit $\delta r\to
0$ in a continuous manner the two unperturbed circular orbits tends to each
other but the number of undulations on the circular orbits will not match as
$\zeta$ is not a continuous variable but can only have discrete rational
values. Consequently the number of cycles of the perturbations will change
discontinuously with radius and the perturbed orbits cannot be closed at this
discontinuity. As we are only interested in stable, closed orbits we can
conclude that $\zeta$ must be a constant and not change discretely with $r$.
The discussion on the constancy of $\zeta$ as given above closely follows the
analysis given in Ref. [1] where the author gives a nice discussion on the
role of $\zeta$ in the case of non-relativistic orbits.
As $\zeta$ is a constant and must not depend upon the choice of $u_{0}$ or $x$
one can interpret Eq. (12) as an independent differential equation by itself,
$\displaystyle 1-\left(\frac{dJ}{du}\right)=\zeta^{2}\,.$ (13)
whose solutions would give us information about the general form of the
central potential $V(r)$. Using Eq. (9) we can write the last equation as
$\displaystyle(V-E)\frac{d^{2}V}{du^{2}}+\left(\frac{dV}{du}\right)^{2}=L^{2}c^{2}(1-\zeta^{2})\,,$
(14)
which is a non-linear second order differential equation. The right hand side
of the above equation is a constant which can be written as
$\displaystyle d=L^{2}c^{2}(1-\zeta^{2})\,.$ (15)
Eq. (14) admits multiple solutions for $V$. The constant $E$ is the total
energy of the particle.
It is interesting to note that the differential equation for the potential
stemming from the stability of closed, circular orbits in the relativistic
case does not have a non-relativistic analogue. Although the orbit equation
Eq. (8) has a proper non-relativistic limit the same cannot be said about Eq.
(14). The reason for such behavior can be seen clearly if we rewrite Eq. (14)
in a slightly different way. From the expression of the energy of the particle
in the central force field $\gamma$ can always be written as $(E-V)/mc^{2}$.
As the total energy is a constant in the present case we must have
$d\gamma/du=-(1/mc^{2})dV/du$. Consequently Eq. (14) can also be written as
$\displaystyle\gamma\frac{d^{2}\gamma}{du^{2}}+\left(\frac{d\gamma}{du}\right)^{2}=\frac{L^{2}(1-\zeta^{2})}{m^{2}c^{2}}\,,$
(16)
which gives a differential equation of $\gamma$. The equation above obviously
does not have a well defined non-relativistic limit. The relativistic
stability condition produces an ill-defined non-relativistic limit due to the
fact that in the relativistic case $J(u)$ as given in Eq. (9) depends upon the
velocity of the orbiting particle111The $V-E$ in $J(u)$ is proportional to
$\gamma$ which depended upon the velocity of the particle.. In the non-
relativistic $J(u)$ was purely a function of the radial coordinate of the
orbiting particle. A perturbation from the circular orbit in the relativistic
case consists of two kinds of perturbations. One is related to the change in
position of the particle from its previous orbit and the other is the change
in velocity from the velocity it had previously on the circular orbit. In the
non-relativistic case only a radial perturbation from the circular orbit fixes
the shape of the stability condition. As because the stability condition of
the orbit depends upon velocity of the relativistic particle and the
corresponding non-relativistic stability condition does not depend upon the
velocity of the particle, the non-relativistic limit of Eq. (14) or Eq. (16)
is not well defined.
In the case of non-relativistic motion we know that the inverse square law
potential and the simple harmonic potential has the capability to produce
stable, closed circular orbits. In the present case to get the forms of the
potentials which can produce stable, closed orbits we have to solve Eq. (14).
As it is a non-trivial equation we will first try to see whether the the
potentials which produced stable, closed orbits in the non-relativistic regime
still satisfy Eq. (14). Let us try to see whether any power law solution of
the form
$\displaystyle V(u)=-\alpha u^{\tau}\,,$ (17)
where $\alpha>0$ satisfies Eq. (14). In the above equation $\alpha$ and $\tau$
are constants. If we substitute the above form of the potential in Eq. (14) we
get
$u^{2(\tau-1)}\left\\{\alpha^{2}\tau(\tau-1)+\alpha^{2}\tau^{2}\right\\}-u^{\tau-2}\left\\{E\alpha\tau(\tau-1)\right\\}=d\,.$
This directly shows that the above relation can be valid for any $u$ only if
$\tau=1$, when $\alpha^{2}=d$ or
$\displaystyle L=\frac{\alpha}{c\sqrt{1-\zeta^{2}}}\,,$ (18)
on using Eq. (15). In this case we see that choosing $\tau=1$ in Eq. (17) we
get the Coulomb or Newtonian potential. The last equation shows that for
stable, circular orbits the particles angular momentum must satisfy some
condition. Eq. (18) implies that $\zeta^{2}<1$, and as $\zeta^{2}>0$ for a
stable orbit, we have
$\displaystyle 0<\zeta^{2}<1\,.$ (19)
The above equation gives
$\displaystyle L>\frac{\alpha}{c}\,,$ (20)
giving a lower bound on the angular momentum of the orbiting particle. This
lower bound of the orbiting particle was previously obtained in a different
way by T. H. Boyer in Ref. [2]. It must be noted here that except $\tau=1$ no
other values of $\tau$ are allowed in the potential which can produce stable
circular orbits of relativistic particles . In non-relativistic mechanics we
do also have the harmonic-oscillator potential corresponding to $\tau=-2$ and
$\alpha<0$ in Eq. (17), but interestingly relativistic effects forbid this
value of $\tau$.
### 3.2 The general solution of the differential equation for the potential
and the nature of orbits
We can find out the general form of the force which can produce stable,
circular relativistic orbits. Noticing that the left hand side of Eq. (14) can
also be written as
$\frac{d^{2}}{du^{2}}\left[\frac{(V-E)^{2}}{2}\right]\,,$
it can be easily shown that
$\displaystyle V(r)-E=-\sqrt{d\left(b+\frac{1}{r}\right)^{2}+a}\,,$ (21)
satisfies Eq. (14) where $d$ is as given in Eq. (15) and $b$ and $a$ are two
other dimensional, integration constants. For an attractive force $b>0$ and
$d>0$ but $a$ can have any sign. If we assume that as $r\to\infty$, $V(r)\to
0$ then we get a relation between the constants $d$, $b$ and $a$ as
$\displaystyle E=\sqrt{db^{2}+a}\,.$ (22)
From Eq. (21) we get the force acting on the particle,
${\bf F}=-\nabla V(r)\,,$
as
$\displaystyle{\bf
F}=-\frac{d\left(b+\frac{1}{r}\right)}{r^{2}\sqrt{d\left(b+\frac{1}{r}\right)^{2}+a}}\,\,\hat{r}\,,$
(23)
From the form of the force and Eq. (22) we immediately see that if $a=0$ we
have $b=E/\sqrt{d}$ and we get back the Newtonian or the Coulombic potential.
From the form of the potential as written in Eq. (17) we can furthermore
identify $\alpha=\sqrt{d}$ and consequently when $a=0$ we have $b=E/\alpha$.
If $a\neq 0$ then the form of the force is non-trivial. The form of the force
as given in Eq. (23) cannot be reduced to the harmonic oscillator force in any
limits of the constants. This shows that special relativistic effects do not
allow stable circular orbits in the presence of a force which is proportional
to the negative of the displacement vector. Although the force expression in
Eq. (23) is mathematically interesting but in physics we do not encounter such
a force, except the $a=0$ limit.
From the expression of $V-E$ as given in Eq. (21) we get
$\displaystyle J(u)$ $\displaystyle\equiv$
$\displaystyle\frac{(V-E)}{L^{2}c^{2}}\frac{dV}{du}=\frac{d}{L^{2}c^{2}}(u+b)\,.$
(24)
yielding
$\displaystyle\frac{d^{2}u}{d\theta^{2}}+u=\frac{d}{L^{2}c^{2}}(u+b)\,,$ (25)
which gives the orbit equation of the relativistic particle which is acted on
by a force given by Eq. (23). As in general $d=L^{2}c^{2}(1-\zeta^{2})$ for a
stable closed orbit where $\zeta$ must be a rational number, we get
$\displaystyle\frac{1}{r}=\frac{1}{R}\cos(\zeta\theta)+\frac{b(1-\zeta^{2})}{\zeta^{2}}\,,$
(26)
where
$\displaystyle
R=Lc\zeta\left[\frac{b^{2}L^{2}c^{2}(1-\zeta^{2})}{\zeta^{2}}+a-m^{2}c^{4}\right]^{-1/2}\,.$
(27)
The equation of the orbit in Eq. (26) shows that in the most general case we
will have precession of the orbits dictated by the condition
$(2\pi+\delta\theta)\zeta=2\pi$, which predicts that the orbit precesses by an
angle
$\displaystyle\delta\theta=\frac{2\pi(1-\zeta)}{\zeta}\,,$ (28)
per orbit.
### 3.3 The case of large perturbations
Till now we have utilized first order perturbation from a circular orbit as
described in Eq. (10) in the beginning of this section. To include higher
order perturbations from a circular orbit we require more terms in the Taylor
series expansion of $J(u)$ in Eq. (10). The second order effects will come
from terms proportional to $(d^{2}J/du^{2})_{u_{0}}$. If the general form of
the potential $V(r)$ satisfies Eq. (21) then it is immediately clear from Eq.
(24) that all derivatives of $J(u)$, except the first, vanishes. Consequently
in the relativistic case it is impossible to restrict the constants $\zeta$,
$a$ and $b$ by higher order perturbation terms to the circular orbit. For
higher order perturbations from circular orbits the form the potential as
given in Eq. (21) remains the same.
## 4 Connection of the present work with some related works
One of the findings of the present article is related to the absence of
stable, circular orbits for relativistic particles in presence of a harmonic
oscillator potential. The trajectories of relativistic particles in a three
dimensional harmonic oscillator potential has been studied previously by L.
Homorodean in Ref. [8]. The method followed in the referred work is completely
equivalent to the one followed in the present work. It is interesting to note
that in Homorodean’s analysis the general shape of the orbit in the
relativistic case is not an ellipse, or a circle, but a rosette shaped curve.
In presence of the oscillator potential the angular momentum of the orbiting
particle with a specific energy has an upper bound. The trajectory of the
relativistic particle can only be a circle when it has the highest angular
momentum for a fixed energy. In Ref. [8] the author does not give any
information about the stability of the orbits. In the non-relativistic limit
the orbit of the particle can be circular. In the light of the findings in
Ref. [8] of Homorodean the prediction of the absence of a stable, circular
orbit in the oscillator potential is a sensible result.
Bertrand’s theorem in non-relativistic classical mechanics has inspired some
authors to propose a space-time (a metric to be precise) where any bounded
trajectory of a particle is periodic in nature. This kind of a space-time is
named as Bertrand space-time. The works of Perlick, Ballesteros, Enciso,
Herranz and Ragnisco, in Refs. [9, 10], try to generalize the results of the
classical Bertrand’s theorem on a flat 3-space to a curved 3-manifold. In Ref.
[9] the author found that a specific form of a space-time which is
asymptotically flat can support Keplerian orbits. The asymptotically flat
Bertrand space cannot support closed trajectories expected in an oscillator
type of potential. One of the findings of the present article predicts that
even in flat space relativistic effects forbid closed, stable trajectories of
particles in presence of an oscillator potential.
## 5 Conclusion
The outline of the article is based on the well known Bertrand’s theorem on
central potentials and orbits of particles as described in most of the
classical mechanics books. Like the non-relativistic case the relativistic
particle’s orbit around a potential source takes place in a plane where the
angular momentum and presumably the energy of the orbiting particle remains
constant. The main difference between the non-relativistic orbits and
relativistic orbits crops up in the orbit equation itself. Unlike the non-
relativistic case in the relativistic case the orbit equation depends upon the
total energy of the particle. The main aim of the article was to find out
possible forms of central potentials which can produce stable circular orbits
for relativistic particles. The stability condition for the orbits can be
transformed to a non-linear differential equation for the central potential.
It is seen that one of the solutions of the non-linear differential equation
for the central potential is just the normal Coulomb potential. But relativity
affects the properties of the orbits by curtailing the angular momentum values
beyond a certain limit. Except the Coulomb potential solution we find that the
stability equation has another general mathematically interesting solution
which is unlike any potential which we use in conventional physics. In a
specific limit the general solution reproduces the Newtonian or Coulomb form.
In the relativistic version of the central force problem we lack some
restrictions on the potential which can produce stable circular orbits. In the
non-relativistic version minimizing the effective potential one can figure out
the radius of the circular orbits and higher order perturbation corrections to
the stability condition of the orbits could be used for unravelling the exact
nature of the potential. In the relativistic version none of those
restrictions remain and consequently the general solution of the potential
contains some constants whose values cannot be analytically calculated.
An important fact which comes out from the article is about the non-existence
of the harmonic oscillator potential as a solution of the stability equation.
In non-relativistic treatment of the Bertrand’s theorem it is well known that
only two kinds of potentials can produce stable circular orbits, one is of the
Coulomb type and the other is of the harmonic oscillator type. The Coulomb
form of the potential passes the stability test for circular orbits but the
harmonic type does not.
The article was focussed on some mathematical properties of relativistic
particle orbits in a central potential. Before we finally conclude it is
pertinent to say some thing on the practical side of the relativistic central
force problem. Atomic physics always remains a store house of exciting
phenomena and one of the places where one may like to apply the tools of
relativistic central force problems lies inside the atom. This fact was
discussed in Ref. [2]. People have studied about the Schrödinger equation and
the Dirac equation in presence of the Coulomb potential. It can be quite
interesting to study the analogous problems using the potential presented in
this article instead of the Coulomb potential. This attempt can seriously shed
some light on the physics of the atoms.
More over as there exists some work on the general relativistic generalization
of Bertrand’s theorem one may expect that in the simplest of the situations,
where space-time remains flat, the results of the present work can be applied
for the orbits of very fast moving bodies interacting via Newtonian gravity
with a massive source. Acknowledgement: The authors acknowledge the
illuminating comments from H. C. Verma after he read the initial manuscript.
Many of his suggestions were implemented while preparing the final version of
the manuscript.
## References
* [1] H. Goldstein, “Classical Mechanics”, $2^{\rm nd}$ edition, Narosa Publishing House, (1993)
* [2] T. H. Boyer, Am. J. Phys. 72, 992-997, (2004)
* [3] U. Torkelsson, Eur. J. Phys 19, 459-464, (1998)
* [4] Z. Reut, [Q. J] Mech. Appl. Math., 39, 417-423, (1986)
* [5] H. Frommert, Int. J. Theor. Phys., 35, 2631-2643, (1996)
* [6] L. D. Landau, E. M. Lifshitz, “The classical theory of fields”, $4^{\rm th}$ edition, Elsevier, (2008)
* [7] J. D. Jackson, “Classical Electrodynamics”, $2^{\rm nd}$ Edition, Wiley Eastern Limited, (1975).
* [8] L. Homorodean, Europhys. Lett., 66, 8-13, (2004)
* [9] V. Perlick, Class. Quantum Grav., 9, 1009-1021, (1992)
* [10] A. Ballesteros, A. Enciso, F. J. Herranz and O. Ragnisco, Class. Quantum Grav., 9, 165005 (13pp) (2008)
|
arxiv-papers
| 2011-03-17T05:32:40 |
2024-09-04T02:49:17.716046
|
{
"license": "Public Domain",
"authors": "Prashant Kumar, Kaushik Bhattacharya",
"submitter": "Kaushik Bhattacharya",
"url": "https://arxiv.org/abs/1103.3338"
}
|
1103.3359
|
# Dynamics of two topologically entangled chains
F. Ferrari1 J. Paturej1,2, M. Pia̧tek1,3 and T.A. Vilgis2 1 Institute of
Physics, University of Szczecin, Wielkopolska 15, 70451 Szczecin, Poland
2 Max Planck Institute for Polymer Research, 10 Ackermannweg, 55128 Mainz,
Germany
3Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, 141980, Dubna, Russia
###### Abstract
Starting from a given topological invariant, we argue that it is possible to
construct a topological field theory with a finite number of Feynman diagrams
and an amplitude of gauge invariant objects that is a function of that
invariant. This is for example the case of the Gauss linking number and of the
abelian BF models which has been already successfully applied in the
statistical mechanics of polymers. In this work it is shown that a suitable
generalization of the BF model can be applied also to polymer dynamics, where
the polymer trajectories are not static, but change their shape during time.
## I Introduction
There are many situations in which it is necessary to consider topological
relations among one-dimensional objects that are homeomorphic to rings. The
most significant examples are provided by long flexible polymers and
biopolymers, whose trajectories may close themselves and form what in the
polymer scientific literature are called catenanes wasser –ff . The latter are
able to entangle themeselves giving rise to complex links involving two or
more interlocked chains. Additionally, each catenanae may be in the
configuration of a nontrivial knot. Two cases of polymer links are shown in
Fig. 1.
Figure 1: Entangled polymers rings $P_{1}$ and $P_{2}$ with linked
trajectories $C_{1}$ and $C_{2}$. In a) polymer $P_{2}$ is in a nontrivial
knot configuration, while in b) both trajectories are unknots.
Besides polymers, other examples in which topological relations among a system
of one-dimensional objects become relevant can be found in condensed matter
physics (paths around defects in melted crystals) chaikin ; pieranski or in
particle physics (loops in quantum gravity and the so-called hopfions) ash ;
smolin ; hopfions . In order to specify the topological states of a given
system of this kind one uses knots or link invariants. In the following, we
will be interested in the topological relations of a system of a linked rings
without taking into account the fact that these rings could be also in a
nontrivial knot configuration as for example in Fig. 1 a). For this reason, we
will discuss here only link invariants.
It is well known that the correlation functions of the observables of a
topological field theory are topological invariants. Moreover, the
coefficients of the perturbative expansion of those correlation functions are
topological invariants too. In practice, this means that to a finite set of
Feynman diagrams it is possible to associate a given topological invariant.
Our purpose is to solve the inverse problem. This means that, starting from a
given topological invariant, we would like to obtain a topological field
theory with a finite set of Feynman diagrams and a correlation function which
is a function of that invariant. This is the program of topological
engineering that has been stated in Ref. ffbookch . In the last few decades
topological theories with the above characteristics have been extensively
applied in the statistical mechanics of polymers, see for instance edwards
–ferrariTFT and ffbookch ; leal . The most popular approach used in order to
distinguish the different topological configurations of the one-dimensional
objects is based on the Gauss linking number (GLN). The corresponding
topological field theory is an abelian BF model discussed in Ref. blau . The
goal of this work is to extend this approach based on the GLN to the case of
polymer dynamics, in which the shape of the linked trajectories is not static,
but changes in time.
## II The topological engineering program
The program of topological engineering in the case of links may be summarized
as follows:
Let $\cal{T}(\ell)$ be a link invariant, which describes the topological
properties of a $N$–component link $\ell$. It is required that:
* a)
the invariant $\cal{T}(\ell)$ is explicitly written as a functional of
trajectories $C_{1},\ldots,C_{N}$ of knots composing the link.
Given a link invariant of this kind, find a topological field theory with
observables ${\cal{O}}_{1},\ldots,{\cal{O}}_{n}$ such that $\cal{T}(\ell)$, or
equivalently a function $F[\cal{T}(\ell)]$ of it, can be expressed as the
correlator of these observables
$F({\cal{T}})=\int\\!{\cal{D}}\\{\phi\\}e^{-S(\\{\phi\\})}{\cal{O}}_{1}(\\{\phi\\}),\ldots,{\cal{O}}_{n}(\\{\phi\\})$
(1)
where $S(\\{\phi\\})$ is the action of a system and $\\{\phi\\}$ is a set of
fields that can be scalars, vectors or higher order tensors.
The topological field theory and its observables should satisfy the following
conditions:
* b)
Each observable ${\cal{O}}_{i}$, $i=1,\ldots,n$, must depend on the trajectory
of only one knot
* c)
No further regularization should be necessary in order to compute the
correlator $\langle{\cal O}_{1},\ldots,{\cal O}_{n}\rangle$, apart from the
usual regularization schemes required by the possible presence of ultraviolet
divergences.
An example of topological engineering is based on the GLN and the abelian BF
field theory. The GLN is given by:
$\chi(C_{1},C_{2})=\frac{1}{4\pi}\epsilon_{\mu\nu\rho}\oint_{C_{1}}\\!dx_{1}^{\mu}(s_{1})\oint_{C_{2}}\\!dx_{2}^{\nu}(s_{2})\frac{(x_{1}(s_{1})-x_{2}(s_{2}))^{\rho}}{|x_{1}(s_{1})-x_{2}(s_{2})|^{3}}$
(2)
where $x_{1}(s_{1})^{\mu}$ and $x_{2}(s_{2})^{\nu}$ are spatial curves in
three dimensions that represent respectively the closed trajectories $C_{1}$
and $C_{2}$ of two polymers $P_{1}$ and $P_{2}$. The Greek indexes
$\mu,\nu,\rho=1,2,3$ denote the spatial components. Here $s_{1}$ and $s_{2}$
represent the arc-lengths on the curves $C_{1}$ and $C_{2}$. $s_{1}$ and
$s_{2}$ are defined in a such a way that $0\leq s_{1}\leq L$ and $0\leq
s_{2}\leq L$. To find a field theory which is associated to the invariant
$\chi(C_{1},C_{2})$, we rewrite (2) as follows
$\chi(C_{1},C_{2})=\int\\!d^{3}x\int\\!d^{3}y\xi_{1}^{\mu}(x)G_{\mu\nu}(x-y)\xi_{2}^{\nu}(y)$
(3)
where
$\xi_{1}^{\mu}(x)=\oint_{C_{1}}\\!dx_{1}^{\mu}\delta(x-x_{1})\qquad\xi_{2}^{\nu}(x)=\kappa\oint_{C_{2}}\\!dx_{2}^{\nu}\delta(x-x_{2})$
(4)
are called the bond vectors densities and
$G_{\mu\nu}(x-y)=\frac{1}{2\pi\kappa}\epsilon_{\mu\nu\rho}\frac{(x-y)^{\rho}}{|x-y|^{3}}$
(5)
Let us note that $G_{\mu\nu}(x-y)$ coincides with the propagator of the
abelian BF model discussed in Ref. blau . To make the connection with the BF
model even more explicit, we have introduced a new parameter $\kappa$, which
will play later the role of the coupling constant of that model. Clearly, the
addition of this parameter is irrelevant. As a matter of fact, the right hand
side of Eq. (3) does not depend on $\kappa$. Now the quantity
$e^{i\chi(C_{1},C_{2})}=e^{i\int\\!d^{3}x\int\\!d^{3}y\xi_{1}^{\mu}(x)G_{\mu\nu}(x-y)\xi_{2}^{\nu}(y)}$
(6)
can be regarded as the generating functional of a Gaussian field theory with
propagator $G_{\mu\nu}(x-y)$ for the very special choice of currents (4). It
is easy to recognize that the underlaying field theory is an Abelian BF model
with action
$S_{\mbox{\tiny
BF}}=i\kappa\epsilon^{\mu\nu\rho}\int\\!d^{3}xA_{\mu}\partial_{\nu}B_{\rho}$
(7)
It is possible to show that the abelian version of the BF model is actually
equivalent to two Abelian C-S field theories. If we quantize the above
topological field theory using the Lorentz gauge fixing, in which both fields
$A_{\mu}$ and $B_{\mu}$ are completely transverse, we obtain the following
relation
$e^{i\chi(C_{1},C_{2})}=\int\\!{\cal D}A_{\mu}{\cal
D}B_{\mu}e^{-S_{\mbox{\tiny
BF}}}e^{i\int\\!d^{3}x\xi_{1}^{\mu}A_{\mu}}e^{i\kappa\int\\!d^{3}x\xi_{2}^{\mu}B_{\mu}}\delta(\partial^{\mu}A_{\mu})\delta(\partial^{\mu}B_{\mu})$
(8)
The above equation is the analog of Eq. (1) in the present case. There are
just two observables ${\cal O}_{1}$ and ${\cal O}_{2}$, namely the two Abelian
Wilson loops given below:
${\cal O}_{1}=e^{i\int\\!d^{3}x\xi_{1}^{\mu}A_{\mu}}\qquad{\cal
O}_{2}=e^{i\kappa\int\\!d^{3}x\xi_{2}^{\mu}B_{\mu}}$ (9)
## III The case of dynamics
In this Section we would like to extend the program of topological engineering
to the case of two trajectories whose configurations are changing during time.
This problem is very important to study the dynamics of two entangled
polymers. Once again, we choose the Gauss linking invariant in order to impose
topological conditions on two closed trajectories $C_{1}$ and $C_{2}$. The
only difference from the previous static example is that now the curves
$x_{1}$ and $x_{2}$ depend on time, i.e. $x_{1}=x_{1}(t,s_{1})$ and
$x_{2}=x_{2}(t,s_{2})$. The GLN can still be defined, but will be a time
dependent quantities:
$\chi(t,C_{1},C_{2})=\frac{1}{4\pi}\epsilon_{\mu\nu\rho}\oint_{C_{1}}\\!dx_{1}^{\mu}(t,s_{1})\oint_{C_{2}}\\!dx_{2}^{\nu}(t,s_{2})\frac{(x_{1}(t,s_{1})-x_{2}(t,s_{2}))^{\rho}}{|x_{1}(t,s_{1})-x_{2}(t,s_{2})|^{3}}$
(10)
Of course, if the trajectories would be impenetrable, then $\chi$ would be a
constant, since it is not possible to change the topological configuration of
a system of knots if their trajectories are not allowed to cross themselves.
However, in the absence of excluded volume interactions models of polymer
physics are phantom, i.e. crossings are allowed. For this reason, we will
require that only the time average of the GLN is fixed. As a consequence, we
will consider a time averaged version of the GLN on the time interval
$[0,t_{f}]$:
$\langle\chi(t,C_{1},C_{2})\rangle=\int_{0}^{t_{f}}\frac{dt}{t_{f}}\chi(t,C_{1},C_{2})$
(11)
Next, we generalize Eq. (6) to the case of dynamics. To this purpose, we
introduce the following field theory
$S=\frac{1}{t_{f}}\epsilon_{\mu\nu\rho}\int\\!d\eta
d^{3}xA^{\mu}(\eta,x)\partial_{x}^{\nu}B^{\rho}(\eta,x)$ (12)
The above action differs from that of Eq. (7) by the addition of the fourth
dimension represented by variable $\eta$, with $-\infty<\eta<+\infty$. Note
that $S$ is not invariant under diffeomorphism on the whole dimensional space
spanned by the coordinates $x^{1},x^{2},x^{3}$ and $\eta$, but only on its
three dimensional spatial section. As a consequence, strictly speaking $S$
does not describe a topological field theory. The propagator corresponding to
the action (12) in the Lorentz gauge is given by
$G_{\mu\nu}(\eta,\eta^{\prime};x,x^{\prime})=\frac{t_{f}}{2\pi}\epsilon_{\mu\nu\rho}\frac{(x-x^{\prime})^{\rho}}{|x-x^{\prime}|^{3}}\delta(\eta-\eta^{\prime})$
(13)
The analog of Eq. (6) is
$e^{-i\lambda\chi(C_{1},C_{2})}=\int\\!{\cal{D}}A_{\mu}{\cal{D}}B_{\nu}e^{-iS}e^{-i\int\\!d\eta
d^{3}x(J_{1}^{\mu}(\eta,x)A_{\mu}(\eta,x)+J_{2}^{\mu}(\eta,x)B_{\mu}(\eta,x))}$
(14)
where
$J_{1}^{\mu}(\eta,x)=\frac{1}{2t_{f}}\int_{0}^{t_{f}}\\!\frac{dt}{t_{f}}\delta(\eta-t)\int_{0}^{L_{1}}\\!ds_{1}\frac{\partial}{\partial
s_{1}}x_{1}^{\mu}(t_{1},s_{1})\delta^{(3)}(x-x_{1}(t,s_{1}))$ (15)
and
$J_{2}^{\mu}(\eta,x)=\lambda\int_{0}^{t_{f}}\\!\frac{dt}{t_{f}}\delta(\eta-t)\int_{0}^{L_{2}}\\!ds_{2}\frac{\partial}{\partial
s_{2}}x_{2}^{\mu}(t_{1},s_{2})\delta^{(3)}(x-x_{2}(t,s_{2}))$ (16)
The right hand side of Eq. (14) can be seen as the amplitude of the two
observables
${\cal O}_{1}=e^{-i\int\\!d\eta
d^{3}xJ_{1}^{\mu}(\eta,x)A_{\mu}(\eta,x)}\qquad{\cal O}_{2}=e^{-i\int\\!d\eta
d^{3}xJ_{2}^{\mu}(\eta,x)B_{\mu}(\eta,x)}$ (17)
To prove Eq. (14) it is sufficient to perform the Gaussian integration in the
fields $A^{\mu}$ and $B^{\mu}$. The result of that operation is
$e^{-i\int\\!d\eta
d^{3}x(J_{1}^{\mu}(\eta,x)A_{\mu}(\eta,x)+J_{2}^{\mu}(\eta,x)B_{\mu}(\eta,x))}=e^{-i\int\\!d\eta
d^{3}x\int\\!d\eta^{\prime}d^{3}x^{\prime}J_{1}^{\mu}(\eta,x)G_{\mu\nu}(\eta,\eta^{\prime};x,x^{\prime})J_{2}^{\nu}(\eta^{\prime},x^{\prime})}$
(18)
Using the explicit expression of the propagator
$G_{\mu\nu}(\eta,\eta^{\prime};x,x^{\prime})$ given in Eq. (13) it is possible
to verify Eq. (14) after eliminating the spurious variables
$\eta,\eta^{\prime}$ and $x,x^{\prime}$:
$\displaystyle e^{-i\int\\!d\eta
d^{3}x\int\\!d\eta^{\prime}d^{3}x^{\prime}J_{1}^{\mu}(\eta,x)G_{\mu\nu}(\eta,\eta^{\prime};x,x^{\prime})J_{2}^{\nu}(\eta^{\prime},x^{\prime})}=$
(19)
$\displaystyle\\!\\!\\!\\!\\!\\!\exp{\left[-\frac{i\lambda}{4\pi}\int_{0}^{t_{f}}\\!\frac{dt}{t_{f}}\int_{0}^{L_{1}}\\!ds_{1}\int_{0}^{L_{2}}\\!ds_{2}\epsilon_{\mu\nu\rho}\frac{\partial}{\partial
s_{1}}x_{1}^{\mu}(t,s_{1})\frac{\partial}{\partial
s_{2}}x_{2}^{\nu}(t,s_{2})\frac{(x_{1}(t,s_{1})-x_{2}(t,s_{2}))^{\rho}}{|(x_{1}(t,s_{1})-x_{2}(t,s_{2})|^{3}}\right]}$
The right hand side of above equation coincides with
$e^{-i\lambda\chi(C_{1},C_{2})}$. This completes our proof.
## IV Concluding remarks
In this work the program of topological engineering has been extended to the
case of the dynamics of two polymer chains. In particular, the Gauss linking
invariant has been considered. It has been shown that a time average version
of this topological invariant can be reproduced from an amplitude of a field
theory in the form of Eq. (1). This amplitude is given in Eq. (14). Due to the
fact that the conformations of the chains change during time, the underlying
field theory is four dimensional and it is topological only with respect to
diffeomorphisms of the spatial section of four dimensional space.
## V Acknowledgments
One of us – M. Pia̧tek – would like to thank the University of Szczecin and
the Faculty of Mathematics and Physics of that University for the kind
hospitality.
## References
* (1) E. Wasserman, Jour. Am. Chem. Soc. 82, 4433 (1960).
* (2) S.A. Wasserman and N.R. Cozzarelli, Science 232, 951 (1986).
* (3) N.C. Seeman et al., New J. Chem. 17, 739 (1993).
* (4) D.E. Adams, E.M. Shekhtman, E.L. Zechiedrich, M.B. Schmid and N.R. Cozzarelli, Cell 71, 277 (1992).
* (5) S.D. Levene, C. Donahue, T.C. Boles and N.R. Cozzarelli Biophys. J. 69 1036, (1995).
* (6) Molecular Catenanes, Rotaxanes and Knots: A Journey Through the World of Molecular Topology, J.-P. Sauvage and C. Dietrich-Buchecker (Eds.) (Wiley, 1999).
* (7) G. Ten Brinke and G. Haziioannou, Macromolecules 20, 480 (1987).
* (8) A. Vologodskii and V.V. Rybenkov, Phys. Chem. Chem. Phys. 11, 10543 (2009).
* (9) S.F. Edwards, Proc. Phys. Soc. 91, 513 (1967); J. Phys. A (Proc. Phys. Soc.) 1, 15 (1968).
* (10) M.G. Brereton and S. Shah, J. Phys. A:Math. Gen. 13, 2751 (1980); J. Phys. A:Math. Gen. 15, 985 (1982).
* (11) F. Tanaka, Prog. Theor. Phys. 68, 164 (1982); Prog. Theor. Phys. 68, 148 (1982).
* (12) M.G. Brereton and T.A. Vilgis, J. Phys. A: Math. Gen. 28, 1149 (1995).
* (13) M.G. Brereton, J. Phys. A: Math Gen. 34, 5131 (2001).
* (14) T.A. Vilgis and H.L. Frisch, Polymer Bulletin 21, 655 (1989).
* (15) M. Otto and T.A. Vilgis, Phys. Rev. Lett. 80, 881 (1998).
* (16) M. Otto, J. Phys. A: Math. Gen. 34, 2539 (2001); J. Phys. A: Math. Gen. 37, 2881 (2004).
* (17) F. Ferrari, Ann. Phys. (Leipzig) 11 4, 255 (2002); F. Ferrari and I. Lazzizzera Nucl. Phys. B 559 (3), 673 (1999).
* (18) F. Ferrari, H. Kleinert, and I. Lazzizzera, Int. Jour. Mod. Phys. B 14 (32), 3881 (2000); Phys. Lett. A 276, 31 (2000).
* (19) Principles of Condensed Matter Physics, P.M. Chaikin and T.C. Lubensky, (Cambridge, 1995).
* (20) P. Pierański, Eur. Phys. Lett. 81, 66001 (2008).
* (21) A. Ashtekar, Phys. Rev. Lett. 57 18, 2244 (1986).
* (22) C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B331, 80 (1990).
* (23) L. D. Faddeev and A. J. Niemi, Knots and particles, Nature 387 (58) (1997), [arXiv:hep-th/9610193].
* (24) F. Ferrari, Topological field theories with non-semisimple gauge group of symmetry and engineering of topological invariants, in Trends in Field Theory Research, O. Kovras (Ed.) (Nova Science Publisher 2004).
* (25) L. Leal and J. Pineda, Mod. Phys. Lett. A 23, 205 (2008).
* (26) M. Blau and G. Thompson, Ann. Phys. (NY) 205, 130 (1991).
|
arxiv-papers
| 2011-03-17T09:12:21 |
2024-09-04T02:49:17.721637
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Ferrari, J. Paturej, M. Piatek and T.A. Vilgis",
"submitter": "Jaros{\\l}aw Paturej",
"url": "https://arxiv.org/abs/1103.3359"
}
|
1103.3393
|
# Oscillators and relaxation phenomena in Pleistocene climate theory
Michel Crucifix Georges Lemaître Centre for Earth and Climate Research
Earth and Life Institute
Université catholique de Louvain
###### Abstract
palaeoclimates, dynamical systems, limit cycle, ice ages, Dansgaard-Oeschger
events Ice sheets appeared in the northern hemisphere around 3 million years
ago and glacial-interglacial cycles have paced Earth’s climate since then.
Superimposed on these long glacial cycles comes an intricate pattern of
millennial and sub-millennial variability, including Dansgaard-Oeschger and
Heinrich events.
There are numerous theories about theses oscillations. Here, we review a
number of them in order to draw a parallel between climatic concepts and
dynamical system concepts, including, in particular, the relaxation
oscillator, excitability, slow-fast dynamics and homoclinic orbits.
Namely, almost all theories of ice ages reviewed here feature a phenomenon of
synchronisation between internal climate dynamics and the astronomical
forcing. However, these theories differ in their bifurcation structure and
this has an effect on the way the ice age phenomenon could grow 3 million
years ago. All theories on rapid events reviewed here rely on the concept of a
limit cycle, which may be excited by changes in the ocean surface freshwater
balance. The article also reviews basic effects of stochastic fluctuations on
these models, including the phenomenon of phase dispersion, shortening of the
limit cycle and stochastic resonance. It concludes with a more personal
statement about the potential for inference with simple stochastic dynamical
systems in palaeoclimate science.
## 1 Introduction
The Pliocene and the Pleistocene cover approximately the past five million
years. The climatic fluctuations that characterized this period may be
reconstructed from numerous natural archives, including marine, continental
and ice core records. These archives show a complex climate history. Ice
sheets appeared in the northern hemisphere around 3 to 3.5 million years ago
[1, 2]. The volume of these ice sheets fluctuated with the variations of the
seasonal and spatial distributions of incoming solar radiation (insolation),
which are induced by changes in the geometry of the Earth’s orbit and the
angle (obliquity) between Earth’s equator and the ecliptic [3, 4, 5]. This is
called the astronomical forcing 111The astronomical forcing will generally be
taken into account here in the form of a normalised measure of insolation
during the month or on the day of summer solstice at a northerly latitude,
typically 60 or 65∘ N. This is a fairly complex, aperiodic signal, with
dominant harmonics corresponding to the phenomena of precession (23716, 22428
and 18976 years); and obliquity (41000 years) [5]. Glacial cycles had an
average duration of about 40,000 years [6] until about 800,000 years ago. The
dominant period of glacial cycles increased around 800,000 years ago and this
is referred to as the Middle Pleistocene Transition. Data and models about the
Middle Pleistocene Transition are reviewed in ref. [7]. Time-series analyses
based on band-pass filtering provide further evidence of the non-linear nature
of the climate response to the astronomical forcing, from about 1.4 Myr ago
[8]. The latest four glacial cycles, in particular, are distinguished by a
pronounced saw-tooth time-structure: ice accumulates over the continents
during about 80,000 years and then melts in about 10,000 years (Figure 1).
Figure 1: Climatic fluctuations over the late Pleistocene. Ice ages are
reconstructed using the oxygen isotopic ratio of the calcite shells of benthic
foraminifera [9]. Within the last ice ages, large temperature fluctuations
were recorded in Greenland, here depicted by changes in the oxygen isotopic
ratio of ice [10]
Superimposed on these long glacial cycles comes a complex pattern of
millennial and sub-millennial variability [11]. For example, the Greenland
record features at least 20 events of abrupt rise and slower decline in oxygen
isotopic ratio (a proxy for temperature) [12, 13] and methane [14] during the
latest glacial epoch. These events are known as Dansgaard-Oeschger events.
They were found to occur from at least the last glacial inception [15] and the
Antarctic ice core record provide evidence that they are characteristic of
Pleistocene glacial climates [16]. Some of these events follow pulses of
iceberg discharges into the North Atlantic Ocean, called ‘Heinrich events’
[17, 18, 19]. Heinrich events and Dansgaard-Oeschger events have left climatic
footprints all over the globe [20], including in Antarctica [16]. The current
interglacial period is referred to as the Holocene. It is also characterised
by millennial and centennial variability, mainly observed in the North Atantic
[21, 22, 23], but of a much weaker amplitude than during the preceding glacial
period.
The present paper reviews attempts to explain these fluctuations with concepts
that originate in dynamical system theory. These are the concepts of limit
cycle, synchronisation and excitability. The central message of the paper is
that current theories of ice ages and rapid events may often be interpreted in
terms of generic deterministic models, which are also used in other areas of
Science like biology and ecology. However, stochastic parameterisations are an
essential part of any complex system model, and their effects on climatic
oscillations have to be taken into account.
Dynamical system theory entered palaeoclimate science with idealised models
representing the response of ice sheets to the astronomical forcing. These
models were directly derived from the physics of the ice-sheet-atmosphere
system [24, 25, 26, 27]. Ghil and Childress [28], in particular, insisted on
the interest of analysing such models in terms of bifurcation theory. For
modelling the complex carbon cycle response authors sometimes adopted a more
heuristic approach by considering simple models and confronting the results to
palaeoclimate evidence [29].
Nowadays climate research is largely oriented towards large climate simulators
(typically: general circulation models), which are developed to include as
many climate processes as possible. However, thinking in terms of dynamical
system theory remains insightful. Indeed, the behaviour of a complex system at
a certain spatio-temporal scale is in practice often dominated by a few
leading modes, of which the dynamics may be captured fairly convincingly with
a low-order dynamical system. Climate scientists are increasingly using this
property. For example, they formulate simpler models to explain the seemingly
complex behaviours observed in ocean-atmosphere simulators. Examples have been
provided in the recent years focusing on interannual [30], centennial [31] and
millennial [32, 33] variability. In parallel, so-called hysteresis
experiments, which aim at identifying the number of stable states in
individual components of the climate system such as the ocean circulation [34]
or ice sheets [35] contribute to a dynamical-system founded understanding of
the climate system. This approach may also help us to predict and communicate
about the proximity of bifurcations, which may result in catastrophic climatic
changes. Timmermann and Jin [36] termed predictability of the third kind our
ability to anticipate bifurcation phenomena, by reference to the
predictabilities of the first and second kind originally introduced by Lorenz
[37].
The article is structured as follows. Section 2 reviews some of the basic
concepts of oscillator theory. This is no substitute for proper textbook
reading, but the reader will find essential notions and definitions needed to
understand the remainder. Section 3 reviews how these concepts enter theories
of ice ages and rapid events. Section 4 discusses effects of stochastic
fluctuations and, finally, section 5 is a more personal statement about the
potential for inference with simple stochastic dynamical systems in
palaeoclimate science.
Figure 2: Sketch of several forms of relaxation oscillations. _(A.)_ The
vector space is structured by a slow manifold with several stable branches.
All points of the state space are attracted towards the stable branches of the
slow manifold (in full line) along the fast direction, which is here the
horizontal. The slow evolution consists in an upward or downward course along
the slow manifold depending on whether the system lives on the right- or left-
hand side of the null-cline of the slow-variable. The relaxation oscillation
consists in alternate jumps between the two branches of the slow manifold.
_(B.)_ Trajectories are rapidly attracted towards a region of the phase-space
influenced by a saddle point. In the scenario dispayed here, there exists a
combination of parameters for which the limit cycle crosses the saddle point.
In that case, the period of the orbit, which includes the saddle-node, is
infinitely long. It is a ‘homoclinic orbit’, hence this particular bifurcation
is named a homoclinic bifurcation. The homoclinic orbit only exists at the
bifurcation point, but it influences the orbit when the parameter is close to
the bifurcation. This is the reason why one refers to the ‘ghost’ of the
homoclinic orbit. There is another scenario for which different saddle-nodes
are connected to each other. The orbit and the associated bifurcation are then
said to be ’heteroclinic’. _(C.)_ The relaxation oscillation is organised
around a fixed point, with complex eigenvalues with a positive real part. The
bifurcation giving rise to this orbit is a Hopf bifurcation. _(D.)_ One
example of excitability, here depicted for a slow-fast system. The system
resides in a stable space, but a fluctuation may cause an ejection out of the
unstable (dashed) branch of the slow manifold. The system then loops all the
way through the slow manifold before coming back at rest.
## 2 Vocabulary and elementary notions.
The reader will find an accessible introduction to dynamical system theory and
concepts in ref. [38]. More formal background on oscillator theory, albeit a
bit dated, is available in ref. [39]. Bifurcation and oscillator theory is
explicitly connected to climate theory in ref. [28] (see, in particular,
chapter 12) and ref. [40], chapter 7. Background on synchronisation and an
introduction to the phenomenon of excitability is available in ref. [41].
Finally, the Scholarpaedia peer-reviewed web-site is an increasingly rich and
authoritative source of information on dynamical systems. Only the notions
essential for the present article are summarised here.
Oscillator:
The oscillator is a dynamical system that has a globally attracting limit
cycle. In more simple terms, it oscillates even in absence of an external
drive. Here, we are interested in oscillators to describe climate phenomena,
which involve dissipation of energy. The minimal model for a dissipative
oscillator includes two ordinary differential equations, of which at least one
is non-linear.
Relaxation oscillator:
The relaxation oscillator is a particular kind of oscillator featuring an
interplay between relaxation dynamics (generally fast) and a destabilisation
process (generally slow). The relaxation is the process by which the system is
attracted to a region of the phase space. This evokes the relaxation of a
spring. In a relaxation oscillator the system continues to evolve slowly after
the relaxation phase. During this slow evolution phase the system stability
diminishes gradually until the system is ejected out of its relaxation state,
either towards another relaxation state, or to the same relaxation state via a
dissipative loop. In this review we will encounter three kinds of relaxation
oscillators (Figure 2): relaxation founded on slow-fast dynamics (involing a
slow manifold); relaxations structured by a homoclinic orbit (involving only
one relaxation state), and relaxations structured around a focus. More details
are given in the caption of Figure 2.
Excitability:
An excitable system has a globally attracting fixed point (it does not
oscillate spontaneously). However, an external perturbation may have the
effect of exciting it. During this excitation, the system is being ejected far
from its fixed point and then returns to it.
Link between relaxation dynamics and excitability:
In practice it is often found that a relaxation oscillator may be transformed
into an excitable system by a mere change in parameter, and vice-versa. The
reason is the following. A relaxation oscillation is often structured globally
in the phase space, for example by a slow manifold (Figure 2A) or by one or
several saddle points Figure 2B). Suppose now that the oscillation displayed
by such a system ceases because a parameter has been changed. The system is
then no longer an oscillator, but the ‘backbone’ of the oscillation dynamics
are still latent in the phase space because the elements that structured the
limit cycle (the slow manifold or the saddle points) have not disappeared.
Consequently, the system may be run on a trajectory close to the defunct limit
cycle if it is being pushed by some external force (the excitation) into the
region of the phase space previously occupied by this limit cycle. This point
is illustrated on the basis of slow-fast dynamics on Figure 2D, but similar
excitation dynamics generally occur near any kind of ‘explosive bifurcation’,
that is, bifurcations that give rise rapidly to a fully developed limit cycle.
This includes homoclinic, heteroclinic, and certain Hopf bifurcations (two
examples follow and are illustrated on Figure 6).
## 3 Oscillators, relaxation and excitability in palaeoclimates
### 3.1 Models of ice ages
#### 3.1.1 The Saltzman et al. models.
Saltzman established a theory in which ice ages are interpreted as a limit
cycle synchronised on the astronomical forcing. Saltzman and his collaborators
wrote a series of articles on the subject, starting with the introduction of
the limit cycle idea [42] and synchronisation hypothesis [43], the
interpretation of the Middle Pleistocene Transition as a bifurcation [44], and
the more complete models in the mid-1990s [e.g. ref. 45]. The full theory is
developed in a book [40]. Here, we concentrate on two intermediate models [46,
47]. They are called SM90 and SM91, by reference to the authors (Saltzman and
Maasch) and the year of publication. The variables $I$, $\mu$ and $\theta$ are
the continental ice mass, CO2 concentration and deep-ocean temperature,
respectively. The reader is referred to the original publications for the
meaning and value of the different parameters. They are not crucial here; it
suffices to know that they are all positive.
$\left\\{\begin{split}\frac{\mathrm{d}I}{\mathrm{d}t}&=\alpha_{1}-(c\alpha_{2})\mu-\alpha_{3}I-k_{\theta}\alpha_{2}\theta-
k_{R}\alpha_{2}F_{I}(t)\\\
\frac{\mathrm{d}\mu}{\mathrm{d}t}&=\beta_{1}-(\beta_{2}-\beta_{3}\theta+\beta_{4}\theta^{2})\mu-(\beta_{5}-\beta_{6}\theta)\theta+F_{\mu}(t)\\\
\frac{\mathrm{d}\theta}{\mathrm{d}t}&=\gamma_{1}-\gamma_{2}I-\gamma_{3}\theta\end{split}\right.$
(SM90)
and
$\left\\{\begin{split}\frac{\mathrm{d}I}{\mathrm{d}t}&=\alpha_{1}-(c\alpha_{2})\mu-\alpha_{3}I-k_{\theta}\alpha_{2}\theta-
k_{R}\alpha_{2}F_{I}(t)\\\
\frac{\mathrm{d}\mu}{\mathrm{d}t}&=\beta_{1}-(\beta_{2}-\beta_{3}\mu+\beta_{4}\mu^{2})\mu-\beta_{5}\theta+F_{\mu}(t)\\\
\frac{\mathrm{d}\theta}{\mathrm{d}t}&=\gamma_{1}-\gamma_{2}I-\gamma_{3}\theta\end{split}\right.$
(SM91)
In both models the first equation describes the ice mass response to changes
in CO2 ($\mu$) and the astronomical forcing ($F_{I}(t)$) Saltzman adopts the
so-called Milankovitch view 222In fact, this view is introduced by Murphy [48]
but it is developed mathematically in the ‘Canon of Insolation’ [4] authored
by Milankovitch that an increase in insolation causes a decrease in ice mass.
Increases in CO2 or in ocean temperature have the same effects.
The other two equations describe the dynamics of CO2 and the response of deep-
ocean temperature to changes in ice volume. It is further assumed that the
mean state of climate varied slowly throughout the Pliocene-Pleistocene, in
particular in response to a ‘tectonically-driven’ decline in the average
concentration in CO2, consistently with an earlier proposal [49]. This
tectonically-driven decline is here modelled as a slow decrease in the forcing
term $F_{\mu}(t)$ throughout the Pleistocene.
Figure 3: Bifurcation diagrams of the Saltzman and Maasch models (SM90) and
(SM91) as a function of $F_{\mu}$, here treated as a constant control
parameter. Note the difference in scales on both axes. Black lines are fixed
points. Continental ice mass $(I)$ is shown as a function of tectonic forcing
$F_{\mu}$. The red lines indicate limit cycles, shown as the minimum and
maximum values of $I$ along the limit cycle. Unstable fixed points or limit
cycles are denoted by dashed lines. Calculations and figures were made using
the pseudo arc-length continuation software package AUTO [50].
Figure 4: Periodic and fixed point solutions for SM90 (with
$F_{\mu}=1.2$ka/ppm) and SM91 (with $F_{\mu}=0.5$ka/ppm), near the sub-
critical Hopf bifurcation points. The dynamics of SM90 slow down near the
unstable fixed point, while the limit cycle of SM91 is much more decoupled
from the position of the fixed point. Figure 5: Two histories of ice volume
generated with the same models (SM90) and (SM91), using the astronomical
forcing and a decline scenario for the tectonic forcing of CO2. The scenarios
were chosen here to evidence the rise and decline of 100,000-year ice ages and
are not the same as those used by Saltzman. Observe the explosive character of
the appearance and decline of 100,000-year cycles in SM91, with early and late
excitations of the limit cycle by the astronomical forcing, and the smoother
character of the evolution on SM90. Time is running from right to left.
Consider the bifurcation diagrams of the SM90 and SM91 models with respect to
$F_{\mu}$, assuming no astronomical forcing ($F_{I}=0$) (Figure 4). The
systems are then said to be free or autonomous. Depending on $F_{\mu}$, both
models show regimes with a stable fixed point, and regimes for which the fixed
point is unstable so that the system orbits along a limit cycle.
These considerations led Saltzman to interpret the Middle Pleistocene
Transition as a bifurcation between a ‘quasi-linear’ response regime to the
astronomical forcing (in the fixed-point regime) to a regime of non-linear
synchronisation (resonance) on the astronomical forcing. He concluded that ice
ages would occur today even in absence of astronomical forcing. The main
effect of the astronomical forcing is to control the timing of glaciations.
Saltzman’s theory is seductive because it explains in a consistent framework
several aspects of the Pleistocene climate history, including the change from
linear to non-linear regime [8], the presence of 100,000 year periodicity in
climate records [51], the lack of a 400,000-year spectral peak in the ice-
volume record (such a peak appears in the simple piece-wise linear model
devised by Imbrie and Imbrie [52], due to rectification of the precession
signal), the synchronisation of deglaciations on the astronomical forcing [53,
54], and the occurrence of large climatic transitions even when eccentricity,
which modulates the effect of precession on insolation, is at its lowest.
The difficulty for accepting Saltzman’s models as a definitive theory lies in
the physical interpretation of the CO2 equation. This equation encapsulates
all the interesting dynamics of the system and it is thus crucial to the
theory. Some semi-empirical justification for the CO2 equation is given in
ref. [44] but the form of this equation has undergone some somewhat ad hoc
adjustments in SM90. The form present in SM91 is again different, with
important effects on the bifurcation structure, while the authors did not
justify this latter change based on physical or biegeochemical considerations.
To better appreciate the stuctural differences between the two models let us
return to the bifurcation diagrams. Consider SM90. As the forcing is decreased
the fixed point gives rise to a locally unstable limit cycle. The system must
therefore find a stable limit cycle further away from the fixed point, but in
this case not much further. This stable limit cycle is under the influence of
the unstable fixed point, and in particular the system slows down when it
passes near it (Figure 4). This is scenario ‘C’ depicted on Figure 2. The
limit cycle evolves as the tectonic forcing is further decreased, until it
shrinks smoothly around a perpetually glaciated state.
In SM91, the bifurcation induced by the decrease in tectonic forcing is much
more explosive. The system lands on a stable limit cycle that turns out to be
little affected by the position of the unstable point. The cycle dynamics do
not show clear phases of acceleration and the system cannot be regarded as a
relaxation system. The limit cycle disappears abruptly as the tectonic forcing
is further decreased, through a phenomenon called a saddle-bifurcation of
cycles. The consequences of the difference between the bifurcation structures
of SM90 and SM91 may be further appreciated in the transient experiments shown
on Figure 5.
#### 3.1.2 Paillard’s (1998) ice age model (P98).
Paillard has been advocating the concept of relaxation for understanding
palaeoclimate dynamics, both ice ages and the more abrupt events, since the
publication of a seminal paper [55] in 1994. We return to this article later
on, and concentrate on another article published in 1998 [56], in which
Paillard introduces a conceptual model of ice ages. Ice volume dynamics
respond to an ordinary differential equation:
$\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{x_{R}(y)-x}{\tau_{R}(y)}-\frac{F(t)}{\tau_{f}}$
(P98)
In this equation, the ice volume $x$ is linearly relaxed to $x_{R}$ with
characteristic relaxation time $\tau_{R}$. This relaxation process is further
perturbed by the astronomical forcing $F(t)$ with a characteristic time
$\tau_{f}$. Such a system is said to be hybrid [57] because the relaxation
equation involves a discrete state variable, here denoted $y$. Its state may
be ‘deep glacial’ ($G$), ‘mild glacial’ ($g$) or ‘interglacial’ ($i$). The
numerical values of $x_{R}$ and $\tau_{R}$ depend on this climate state.
Climate states $y$ follow a sequence $i\rightarrow g\rightarrow G$ according
to a set of conditions formulated on the level of glaciation $x$ and
insolation. Namely, the transition $g\rightarrow G$ is triggered when the
forcing $F(t)$ exceeds a certain threshold. Occurrence of $G$ drives climate
quickly into an interglacial state $i$ because $x_{R}(G)$ and $\tau_{R}(G)$
are specified in the model to be low.
Paillard is not very specific about the physical meaning of the discrete
variable, but it accommodates the paradigm that the Atlantic ocean circulation
has gone through three different states during the latest glacial period:
intermediate circulation, shut-down of the circulation, and modern, deep-
sinking circulation. The system (P98) features the concept of slow-fast
relaxation dynamics. However, this is not an oscillator because the shift from
$g$ to $G$ is determined by the course of the external forcing. The Middle
Pleistocene Transition is induced in (P98) in a fashion similar to Saltzman,
and on the basis of similar physical assumptions (tectonically-driven decline
in CO2). The drift in climatic conditions induced by tectonics is accounted
for by a term added to the astronomical forcing. In a later review, Paillard
[58] further emphasises empirical evidence for the relevance of the relaxation
concept in the phenomenon of deglaciation.
#### 3.1.3 The Gildor Tziperman model.
Gildor and Tziperman [59] take a moderate step towards higher model complexity
by considering a slightly more explicit representation of atmosphere, ocean,
sea-ice and land-ice dynamics. Namely, the ocean is divided into 8 boxes, and
the atmosphere into 4. Sea-ice fraction responds to standard energy balance
equations. More crucially, land-ice growth is influenced by a somewhat
controversial feedback between sea-ice and precipitation. The feedback is
controversial because it is assumed that cold climate results in a reduction
in ice volume: sea-ice growth causes a reduction in precipitation in ice-
covered areas and, by this mechanism, almost suppresses accumulation of snow
on ice sheets. The latter then no longer compensates for ice ablation and ice
volume shrinks.
A free oscillation arises from the fact that the ice volume thresholds for
switching sea-ice cover ‘on’ and ‘off’ differ. In other words, sea-ice
displays a hysteresis response to variations in ice volume. This is exactly
the principle of the slow-fast relaxation oscillator depicted on Figure 2A :
The curve of equilibrium of sea-ice with respect to ice volume is the slow
manifold, and ice volume integrates the state of sea-ice in time. In turn,
this oscillation can be synchronised on the astronomical forcing.
The Gildor-Tziperman model is coupled to a biogeochemical cycle in a companion
paper [60], but the essential dynamics of the glacial oscillation are
unchanged. Tziperman et al. [61] further comment on the model and its property
of synchronisation on the astronomical forcing, and find that its behaviour is
essentially reducible to a hybrid dynamical system.
#### 3.1.4 The Paillard-Parrenin model.
Paillard and Parrenin [62] propose yet another relaxation model in 2004
(PP04). The prognostic variables are ice volume $I$, the area of the Antarctic
continental ice sheet $A$ and the atmospheric concentration in CO2 ($\mu$)
($a\ldots j$ are parameters):
$\left\\{\begin{split}\frac{\mathrm{d}I}{\mathrm{d}t}&=\frac{1}{\tau_{I}}(-a\mu-
bF(t)+c-I)\\\ \frac{\mathrm{d}A}{\mathrm{d}t}&=\frac{1}{\tau_{A}}(I-A)\\\
\frac{\mathrm{d}\mu}{\mathrm{d}t}&=\frac{1}{\tau_{\mu}}(dF(t)-eI+fH(-D)+g-\mu)\\\
D&=hI-iA+j\end{split}\right.$ (PP04)
As in the other ice age models, ice volume is a slow variable driven by the
astronomical forcing. It is here coupled to a variable with a similar time
scale ($\tau_{A}\sim\tau_{I}$) and a faster one ($\tau_{\mu}=\tau_{I}/3$). The
term $H(-D)$, where $H$ is the Heaviside function, represents the ventilation
of the Southern ocean. CO2 is released into the atmosphere when the Southern
ocean is ventilated ($D<0$), which drives deglaciation. Ice then grows slowly,
until a Southern ocean ventilation flush sends the system back to interglacial
conditions. Ocean ventilation is thus the fast process in this model and it is
the only non-linear process accounted for. Though, contrary to the Gildor-
Tziperman model, it does not present a hysteresis behaviour. Consequently, the
glacial cycles featured by this model cannot be interpreted in terms of shifts
between the branches of a slow manifold.
To better understand the dynamics of glacial cycles in this model we consider
the bifurcation diagram along typical solutions in the phase space for the
free (i.e. unforced) system (Figure 6). The parameter $g$ is taken in this
example as the control parameter, in order to preserve Saltzman’s idea that
ice age cycles appear as the consequence of a slow perturbation of the carbon
cycle. As in SM90, PP04 exhibits a limit cycle arising from a sub-critical
Hopf bifurcation. The dynamics along the limit cycle close to the bifurcation
point are strongly influenced by the presence of the unstable focus. This is
the configuration ‘C’ shown in Figure 2C. Depending on $g$, the focus is
either on the low-ice-volume side of the limit cycle (i.e.: the system spends
most of its time with high CO2) or on the high-volume side of the limit cycle
(i.e.: the system spends most of its time in low CO2). Parrenin and Paillard
estimate that we are currently in the second configuration.
Figure 6: Bifurcation diagrams and phase-space trajectories of the free
Paillard-Parrenin model (PP04) and the van der Pol (VDP) model. Both systems
display explosive bifurcation scenarios, but the details are different. PP04
exhibits a sub-critical Hopf bifurcation, while the limit cycle of VDP is
explicitly framed by a slow manifold. Phase space trajectories are drawn near
the bifurcation points, that is : $g=0.4$ in PP04 and $\beta=0.9$ in VDP. The
Heaviside function in PP04 is approximated as
$H(x)=\mathrm{atan}(500x)/\pi+0.5$ for analysis with AUTO.
#### 3.1.5 A minimal model of ice ages.
It has been claimed [61, 63] that any model that has some form of 100,000 year
internal periodicity could be used to reproduce the course of ice volume over
the last 800,000 years. Taking the argument at face value, Crucifix [64] used
one of the simplest possible slow-fast oscillators: the van der Pol
oscillator, with minimal modifications to account for the astronomical forcing
and the asymmetry between the phase of ice build-up and melt during the late
Pleistocene ($\alpha$, $\beta$, $\gamma$, and $\tau$ are parameters; $F(t)$ is
the astronomical forcing):
$\left\\{\begin{split}\frac{\mathrm{d}x}{\mathrm{d}t}&=(-y+\beta+\gamma
F(t))/\tau\\\
\frac{\mathrm{d}y}{\mathrm{d}t}&=-\alpha(y^{3}/3-y-x)/\tau\end{split}\right.$
(VDP)
The system dynamics are determined by the structure of the slow manifold
$x=y^{3}/3-y$. The parameter $\beta$ controls the position of the fixed-point
on the slow manifold and, consequently, the ratio of times spent by the system
in the two branches (‘glacial’ and ‘interglacial’) of the slow manifold. The
ice age curve can be captured with some tuning (Figure 7), although it is fair
to add that a small change in parameters may shift the timing of one or
several ice age cycles. This minimal model was used to challenge intuitive
arguments about the predictability of ice ages [64].
Figure 7: Astronomical forcing, $x$ and $y$ trajectories obtained using system
(VDP) with $\alpha=30$, $\beta=0.75$, $\gamma=0.4$ and $\tau=36\,\mathrm{ka}$
(1 ka = 1,000 years). Blue dots are an authoritative natural archive thought
to mainly represent fluctuations in ice volume and deep ocean temperature, and
compiled in ref. [9].
### 3.2 Models for millennial climate variability
#### 3.2.1 Dansgaard-Oeschger events as relaxation oscillations.
Welander [65] introduced the concept of relaxation oscillations in the context
of ocean dynamics. He described a heat-salt oscillator involving exchanges of
heat and salt within a single oceanic column, coupled to a phenomenon of
surface temperature relaxation. The destabilisation process needed for the
relaxation oscillation to appear is here related to diffusion between the deep
ocean and the mixed layer. The system dynamics are further controlled by the
mean freshwater flux at the top of the ocean column. It determines the
transitions from a regime characterised by perpetual convection in the oceanic
column, to a regime with intermittent convection (oscillation), and finally to
a regime with no convection [66]. The bifurcations between the different
regimes bear the character of global bifurcations, with the oscillation period
approaching infinity near the bifurcation points (in particular, the second
bifurcation bears the character of a homoclinic bifurcation). The heat-salt
oscillator belongs thus to the class ‘B’ on Figure 2.
Welander [67] and Winton and Sarachik [68] later introduce the concept of
another kind of relaxation oscillator in the ocean. It involves the meridional
structure of the ocean thermohaline circulation, and the key non-linear
process is the meridional advection of heat and salt. The oscillations
featured by this model are termed ‘deep-decoupling’ oscillations [68]. Given
that the slow process now relates to heat accumulating in the global ocean,
the characteristic time of deep-decoupling oscillations is of the order of
1,000 years. The net flux of freshwater delivered to the North Atlantic acts
as a bifurcation parameter controlling the transition between non-oscillating
and oscillating regimes in the Winton-Sarachik model [69].
Millennial oscillations have since been observed across a hierarchy of ocean
models, including 3-D ocean models with prescribed freshwater flux and
restoring conditions to surface temperature [70], and 3-D models coupled to a
simple atmosphere [71, 72]. Sakai and Peltier [73] proposed that millennial
deep-decoupling oscillations could explain Dansgaard-Oeschger events. Colin de
Verdiere et al. [33, 74, 75] complement this early proposal with a fairly
complete theory based on ocean circulation model experiments. The oscillations
described by Colin de Verdiere et al. involve the processes of turbulent
vertical mixing (neglected in Winton and Sarachik [68]), advection, and
convection, which unify the salt oscillator with the deep-decoupling
oscillation model. Incidentally, Colin de Verdiere [74] dismisses the non-
linearity of the equation of state as the cause of the oscillations.
There is, across the model hierarchy, consistency about the fact that the
transition between the oscillating circulation regime and the so-called
diffusive, haline regime (without deep convection) is associated with a
homoclinic bifurcation [33, 76]. The nature of the bifurcation between the
convective regime and the oscillation is more model-dependent. Timmermann et
al. [76], based on experiments with the 8-ocean-box Gildor-Tziperman model,
find a Hopf bifurcation; salt-conserving experiments with a 2-D ocean model
show a transition towards a finite-period cycle, but of increasing period as
the bifurcation is approached; experiments with a more idealised model,
formulated as a 2-equation dynamical system, reveal the signature of an
infinite-period bifurcation [33] 333 This particular case was not illustrated
on Figure 2. There is no saddle point along or near the orbit, but there is a
combination of parameters for which a fixed point appears on the limit cycle.
Beyond this particular parameter value, this fixed point splits into a saddle
and a node. This particular parameter value correpsonds to an ‘infinite-
period’ bifurcation. In practice, as long as the limit cycle exists, the
trajectory slows down near the point where the saddle-node will appear. Some
authors then refer to the influence of the ‘ghost’ of the saddle point [38]. .
The latter implies that Dansgaard-Oeschger events, at the time when they
appear soon after the glacial inception process, should be very long but of a
similar amplitude as the Dansgaard-Oeschger events coming later in the glacial
cycle. This feature is consistent with the Greenland ice core record (Figure
1). More specifically, the first Dansgaard-Oeschger cycles that appeared at
the beginning of the glacial era were characterised by a long ‘plateau’ phase
(also called: interstadial) during which the thermohaline circulation was
certainly very active [15]. In the Colin de Verdiere et al. theory, the
plateau phase is the phase of the trajectory influenced by the ‘ghost of the
saddle point’ [74].
#### 3.2.2 Dansgaard-Oeschger cycles as the manifestation of an excitable
system.
Given the explosive nature of the bifurcations involved in ocean dynamics it
is no surprise to find excitability properties in ocean models. Weaver and
Hughes [70] discuss this effect in salt-conserving experiments with an
idealised-geometry, ocean model. The ocean-atmosphere model of intermediate
complexity CLIMBER (CLIMate BiosphERe mode) was shown to exhibit excitability
properties when boundary conditions are set to be typical of the latest
glacial era [77]. The ocean circulation has then one stable state, with
moderate Atlantic overturning, and a ‘quasi-stable state’ with more intense
overturning. The conceptual sketch of the excitation cycles shown by
Ganopolski and Rahmstorf [78] on their Figure 1 can be interpreted in terms of
slow-fast dynamics, in which the different states of the ocean circulation
constitute the different branches of a slow manifold. The intense overturning
state, which is the ‘plateau’ phase of the Dansgaard-Oeschger event, may thus
be viewed as the repelling branch of the slow manifold in the excitable regime
(Figure 2D). The excitable Dansgaard-Oeschger hypothesis was used as a
possible basis to explain how a weak forcing, exogeneous to the system, could
explain the observed 1500-yr periodicity of Dansgaard-Oeschger cycles (on this
periodicity: see ref. [79, 80] but see the other view in ref. [81]). Two such
theories were developed on the basis of experiments with CLIMBER. One suggests
that Dansgaard-Oeschger events are excited by stochastic fluctuations,
modulated by a weak, hypothetical solar periodic forcing [82] (more on the
effects of stochastic fluctuations in section 4). The alternative theory
suggests that the excitation is induced by the interference between two solar
forcings with periods close to $1470/7$ $(=210)$ and $1470/17$ $(\approx 87)$
years [83], possibly combined with noise [84].
#### 3.2.3 Heinrich cycles as a relaxation oscillation.
MacAyeal [85] proposed an ice-binge/purge theory to explain Heinrich events.
The theory rests on experiments with a 1-spatial direction model of ice flow
dynamics. Suppose, as a starting point, that ice volume grows in response to
net accumulation of snow. The growth continues until the accumulated effect of
geothermal heat flux causes basal sliding. A volume of ice is then released
into the ocean (this is the ‘purge’), causing the release of icebergs
characteristic of Heinrich events. Ice volume thus decreases, until ice
accumulation wins over so that ice volume can grow again. The ice-binge/purge
model is thus a relaxation oscillator combining a slow integrating process
(ice mass accumulation) with a fast lateral discharge process.
#### 3.2.4 Coupling between Heinrich and Dansgaard-Oeschger events.
To what extent Heinrich events may interfere with Dansgaard-Oeschger dynamics?
Paillard [55, 86] investigated this question by coupling the MacAyeal ice
model—but reduced to ordinary differential equations by Galerkin
truncation—with a 3-box ocean model. The coupling simply assumes that ice
released into the ocean causes a net freshening of the surface of the North
Atlantic that alters the deep-ocean circulation. Paillard realised that this
coupling could lead to fairly non-intuitive and complex effects, such as the
succession of Dansgaard-Oeschger events of decreasing amplitude between
Heinrich events. This succession is known in the litterature on palaeoclimate
records as _Bond cycles_ [18]. Paillard also found that the oscillations are
aperiodic in this model under certain parameter configurations.
The issue is further explored in [69], based on the Winton-Sarachik ocean
model, and in [76], based on the slightly more sophisticated Gildor-Tziperman
ocean model [59]. The objective was to study the response of deep-decoupling
ocean oscillations to prescribed Heinrich cycles. Schulz et al. [69] noted
that deep-decoupling oscillations could be synchronised on the Heinrich
cycles. Timmermann et al. [76] then proposed, on the basis of numerical
experiments with a fairly idealised model, that Heinrich events excite
Dansgaard-Oeschger cycles because the variation in ice volume caused by a
Heinrich event modifies slowly the amount of net freshwater released in the
ocean. In turn, they suggested, Dansgaard-Oeschger may have a control on ice
volume growth. This yields a two-way coupling between Dansgaard-Oeschger and
Heinrich events.
Experiments with more comprehensive models of the ocean-ice-sheet-atmosphere
system [87, 88] generally support the idea that the different water and heat
fluxes involved in the different phases of ice build-up and iceberg release
are quantitatively sufficient to support a coupling between ice sheets and
ocean circulation during the latest glacial era. However, it was also noted
that “ three-dimensional thermomechanical ice-sheet models are unable to
satisfactorily reproduce the binge-purge mechanism without an ad hoc basal
parameterisation.” [89].
To address this difficulty a theory in which Dansgaard-Oeschger events trigger
Heinrich events was recently proposed [89]. The ice shelve plays a key role,
in blocking the ice stream flow from the ice sheet to the oceans. Heinrich
events occur when this ice shelve is broken, for example under the influence
of ocean sub-surface warming associated with a Dansgaard-Oeschger event. The
resulting model is a system displaying a slow ice-build-up – Heinrich release
cycle excited by fluctuations in ocean sub-surface temperature.
#### 3.2.5 Holocene oscillations and relationship with Dansgaard-Oeschger
events.
The much smaller ocean oscillations that characterised the Holocene period may
also be a relaxation phenomenon. Schulz et al. [31] observe oscillations in
the atmosphere-ocean model of intermediate complexity ECBILT-CLIO. These
oscillations are related to the convective activity in the Labrador Seas.
Schulz et al. [90] considered the existence of such an oscillator in an
earlier reference and speculated on the possible interactions between the
centennial oscillations, millennial oscillations, and Heinrich cycles. They
considered a model in which each of these three kinds of oscillations is
modelled as a Morris-Lecar relaxation oscillator. Their working hypothesis is
that glacial conditions induce a coupling between these oscillators. They then
observed that a very stable 1500-yr oscillation appears, which they
interpreted as a model equivalent of Dansgaard-Oeschger events.
## 4 Stochastic effects
The myriad of chaotic motions that characterise the dynamics of the ocean and
the atmosphere may be taken into account in the form of parameterisations
involving stochastic time-processes. The method was introduced in climatology
in the 1970s [91] and the theoretical justifications, which allow one to model
chaotic motions as a (linear) stochastic process, are reviewed in ref. [92,
93]. In a statistical inferential framework, the stochastic parameterisations
may also be viewed as a way to account for the distance necessarily existing
between the concepts and dynamics featured by the model, and the complex
system being observed.
The effects of stochastic processes on relaxation oscillators and excitable
systems are generally well documented in the literature because this is a
topic of general interest [94]. Here we review some of them in the specific
context of palaeoclimate dynamics.
### 4.1 Stochastic effects on ice age dynamics
#### 4.1.1 Phase dispersion.
One of the basic effects of noise on oscillators is the phenomenon of phase
dispersion: A weak stochastic forcing on an oscillator causes a fading out of
the memory of the exact initial conditions, even though the gross structure of
the oscillation visualised in the phase space is conserved. The phenomenon is
well known and it is an immediate consequence of the neutral stability of the
phase of a free oscillator with respect to fluctuations. It was early
suggested that this phenomenon of phase dispersion may concern ice ages [95],
but it is more commonly believed that ice ages are phase-locked on the
astronomical forcing. This phase-locking should act against dispersion and
permit a very long predictability horizon of ice ages. Though, a phenomenon of
phase dispersion may happen in oscillators that are locked on a periodic
forcing. A stochastic fluctuation may momentarily cause a burst of
desynchronisation, called phase slip, during which the system is unhooked from
its corresponding deterministic trajectory and attracted to another
trajectory, which leads or lags the original one by one forcing period (ref.
[41], sect. 3.1.3). The difference between phase diffusion in a free system
and in a periodic-forcing-driven oscillator is that the diffusion effect has,
in the latter, a quantum nature. More formally, it is said that the stochastic
forcing disperses the system states around the different attractors that are
compatible with the forcing. In a work in preparation we suggest that the
astronomically-forced climate system may satisfy the conditions for a similar
phenomenon of phase dispersion to occur (B. De Saedeleer, M. Crucifix and S.
Wieczorek, unpublished data, 2011). Given that the astronomical forcing is
aperiodic the description of the phenomenon requires a suitable theoretical
framework, which relies on the notion of a ‘local _pullback_ attractor’. The
equivalent of a phase slip is, in the aperiodic forcing context, a stochastic
shift from one of the deterministic pullback attractors to another one. The
phenomenon is illustrated based on experiments with the VDP model on Figure 8.
Figure 8: The phenomenon of trajectory shift illustrated with the model (VDP)
with astronomical forcing, with parameters $\tau=36\,ka$, $\beta=0.75$,
$\gamma=0.4$, $\alpha=30$. Black: the deterministic system. The generated
history reproduces reasonably well the fluctuations of ice volume of the last
800,000 years. Red: same system but with an additive Wiener process added to
the fast variable $y$, with variance $b=0.2(\mathrm{ka})^{-1/2}$. One possible
realisation of the stochastic system is shown. Observe the solution shift
around 450,000 years ago. Depending on the realisation the shift may occur at
different places.
#### 4.1.2 Reduction of period.
Additive fluctuations generally reduce the period of relaxation oscillators.
In an oscillator presenting a homoclinic orbit such as the Duffing oscillator,
additive fluctuations reduce the time spent near the unstable focus [96]. This
implies that even at the corresponding bifurcation point in the deterministic
system, the return time of oscillations in the stochastically-perturbed system
remains finite. In a slow-fast oscillator such as the van der Pol oscillator,
additive fluctuations generally result in early escapes of the branch of the
slow-manifold on which the system lies (Figure 9). The period of the
oscillator is thus affected by a correction that increases approximately
linearly with the noise variance in the slow-fast van der Pol oscillator [97]
444This result is established assuming fluctuations added to the slow
variable. A much more general theory, suitable for different slow manifolds
and additive fluctuations to slow and fast variables is now available [98]..
This property was used at least once in Pleistocene theory, in the silicate
weathering hypothesis advanced by Toggweiler [99]. Additive fluctuations
reduce the limit-cycle period from 800,000 years to about 100,000 years. The
reasons for the period reduction being so dramatic are left for another
article.
Figure 9: Two possible effects of additive noisy fluctuations in a slow-fast
system. _(A.)_ If the slow-fast system is in the excitable regime, noisy
fluctuations can cause excitations. Excitations will be sporadic if the
amplitude of the oscillations is weak (as shown here), or regular, as in a
limit cycle, in the coherent resonance regime. _(B.)_ If the slow-fast system
is oscillating, additive fluctuations may cause early or delayed escapes from
the slow branches, compared to the deterministic system. The net effet is a
shortening of the cycle duration.
### 4.2 Stochastic effects on Dansgaard-Oeschger dynamics
#### 4.2.1 Stochastic excitation and resonance.
Noise may naturally act as an excitation agent in an excitable system (Figure
9). The topic is extensively reviewed in ref. [94]. Excitation loops are
sporadic if the noise amplitude is weak, in which case the _recurrence time_
of the events is set by the noise amplitude, while the _amplitude of the
events_ is set by the structure of the deterministic vector field. The
frequency of excitation loops increases with the noise amplitude, until the
system behaviour is qualitatively similar to a limit cycle regime. This is the
coherent resonance regime. For yet higher noise amplitudes, the limit cycle
structure is destroyed.
The concept of stochastic excitation has been considered several times in
Dansgaard-Oeschger theories. The idea is introduced based on experiments with
an ocean general circulation model with idealised geometry and forcing [70].
The effect of stochastic fluctuations is not only to excite oscillations in a
system normally at rest, but also to reduce the period of these oscillations
when the system is in oscillatory regime.
A phenomenon of non-autonomous stochastic resonance may occur if the noise is
superimposed to a weak external drive. For this to happen the autonomous
system needs to be stable but excitable. The external forcing must be too weak
to cause excitation by itself. The role of the noise is to provide the
additional power to induce excitation. The timing of the excitation is then
related to the phase of the external forcing. The mechanism was proposed
several times [76, 82] to explain the 1500-yr recurrence time of Dansgaard-
Oeschger events. The idea remains questioned, either on the ground that the
1500-yr recurrence time observed in palaeoclimate records is coincidental [81]
or on the ground that the 1500-yr external forcing is unidentified [83, 33]. A
more subtle case of stochastic resosonance involves the combination of noise
with two solar cycles of 210 and 87 years, which yields the concept of ‘ghost
resonance’ [100] for which some support, albeit not conclusive, is found in
the observations [80].
#### 4.2.2 Decreased sensitivity to noise in resonant oscillators.
Coupled oscillators may exhibit, collectively, a resonance period that is more
robust to external fluctuations than the uncoupled oscillators. Schulz et al.
[90] used this property to explain the stability of the Dansgaard-Oeschger
recurrence period of 1500 years in presence of random fluctuations, without
having to invoke an external forcing.
#### 4.2.3 Pseudo-oscillations in two-well systems.
Finally, a behaviour reminiscent of oscillations may occur in a system that is
neither oscillating nor excitable, but which presents several stable states.
Noise then simply induces jumps between these different states. The simplest
mathematical model is the Langevin equation and this is on this basis that
Schulz et al. [31] interpret the Holocene oscillations observed in the ECBILT-
CLIO climate model.
## 5 Concluding discussion : Can dynamical systems be used for inference?
The review has shown that relaxation oscillations are a popular and powerful
model to explain oscillations observed in the Pleistocene record. The concept
of relaxation implies some form of slow-fast separation, in the sense that at
least one component of the system spends most of its time in ‘quasi-
equilibrium’ states (this may be a ‘slow manifold branch’ or a region
influenced by a saddle-point, depending on the system structure), with
acceleration phases.
Some of these models were constructed following a fairly careful procedure of
truncation of a system of partial differential equations, which describes some
of the fluid dynamics of the climate system. Others were proposed on a more
conceptual basis, the idea being precisely to test a hypothesis based on
palaeoclimate observations. The latter approach is sometimes criticised, on
the ground that box-models, for example, cannot reasonably be taken as an
adequate representation of the complex dynamics of the oceans [101].
This leads us to the last question of this review: can dynamical systems be
used for inference on palaeoclimates? Inference implies that something is
being learned by confronting a model to observations. This inference process
may take the form of a calibration procedure (update our knowledge on
parameters on the basis of observations) or a model selection procedure (which
model, among different alternatives, explains the observations best).
The position taken here is that there is not such a thing as an ‘attractor’ of
the climate system that is to be ‘discovered’. The hope is that some of its
modes of behaviour are sufficiently de-coupled from the rest of the
variability to justify the fact simple dynamical systems may capture the
fundamental dynamical properties of these modes, and we want to learn about
these modes from palaeoclimate observations.
The programme is challenging. Indeed, it was underlined that different
physical assumptions may lead to dynamical systems with dynamical properties
that are similar enough to produce a convincing visual fit on palaeoclimate
data [61]. The message is largely echoed in the present review. The modeller’s
challenge is therefore to operate a model selection on more stringent criteria
than just fitting some standard time series. For example, palaeoclimate
observations may yield constraints on the bifurcation structure of the system.
The Middle-Pleistocene Transition is an attractive test case in this respect.
In a statistical inference process, the observations should be a plausible
outcome or realisation of the model. This makes sense only if the model has a
stochastic component, which describes its uncertainties, limitations, and the
noise that emerges from the chaotic motions of the atmosphere and oceans.
Stochastic dynamical systems begin to be used for inference on palaeoclimate
time series. In a method called ‘potential analysis’, the climate system is
modelled as a Langevin equation, that is, the combination of a down-gradient
drift with a Wiener additive process, and inference is made on the number of
wells of the potential function [102]. The method was applied on Pleistocene
climate records, yielding the conclusion that the number of wells increased
from 2 to 3 over the course of the Pleistocene [103].
However, our position so far has been to favour a Bayesian methodology,
because it allows one to encode physical constraints in the form of prior
distributions on model parameters. The Bayesian formalism is also naturally
designed for model calibration, selection, and probabilistic predictions.
The fact is that Bayesian methods for selection and calibration of dynamical
systems on noisy observations are only emerging. In a recent attempt we
considered a particle filter for parameter and state estimation [104]. To be
honest, there is ample room for progress. Whether the process of inference
with simple dynamical systems on palaeoclimate data will lead new insight in
this context still needs to be demonstrated.
## Acknowledgements
This review benefited from stimulating discussions during the Isaac Newton
Programme Mathematical and Statistical Approaches to Climate Modelling and
Prediction held in Cambridge during the summer-autumn 2010. Guillaume Lenoir,
Bernard De Saedeleer and Jonathan Rougier provided helpful comments on a first
version of the article. Thanks are also due to Didier Paillard, Olivier Arzel,
Alain Colin de Verdière, André Paul and Sebastian Wieczorek for e-mail
correspondence during this review. The author is Research Associate with the
Belgian National Fund of Scientific Research. This research is supported by
the FP7-ERC starting grant ITOP ERG-StG-2009-239604. The editor (Jan Sieber)
and two reviewers are acknowledged.
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|
arxiv-papers
| 2011-03-17T12:32:29 |
2024-09-04T02:49:17.727199
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michel Crucifix",
"submitter": "Michel Crucifix",
"url": "https://arxiv.org/abs/1103.3393"
}
|
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